HAL Id: inria-00080360https://hal.inria.fr/inria-00080360v2

Submitted on 20 Jun 2006

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

Nonlinear Markov semigroups and interacting Lévy typeprocesses

Vassili Kolokoltsov

To cite this version:Vassili Kolokoltsov. Nonlinear Markov semigroups and interacting Lévy type processes. [ResearchReport] RR-5932, INRIA. 2006. inria-00080360v2

appor t de r ech er ch e

ISS

N02

49-6

399

ISR

NIN

RIA

/RR

--59

32--

FR

+E

NG

Thème NUM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Nonlinear Markov semigroups and interacting Lévytype processes

Vassili N. Kolokoltsov

N° 5932

June 2006

Unité de recherche INRIA RocquencourtDomaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)

Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30

!"$#%&(')#*,+.-%0/,#1/02&3'4

5(68797;:=<>:%?A@)BCD<=CDE8C8<=FG79CIH∗

J%KML*N0OPQSRUTWVYXIZ\[]L*N0O9Z^Y_MN0`9acbedf_MO9Zghacikj\O[RmlonIpMqr_sZ

tlupsp1ivaw[xIO a]O9yzKMO9a]yzKsO^8|u~vvATWv_M^MOuvkTu0psluvO9Z

&9*oMkk VIO*N0bvaciv_MpsZiusp8ivZcb[]bYb[\XpMacO;ZwO9acYbr^Mqrb^sO9lua&ikp8O*azlo[]ikazZik^(NO;lkZc_Ma]O9ZiuslN0O;lkZc_Ma]lkMqrOZcpslvyO

XxMO9Z]ya]b8Om[]KMOAO9kikqr_I[]biv^sZik(pMa]ikslkMbqrb[\XxIbeZw[ca]bM_M[cbrik^sZiu(RmlkacviopMa]iIyO;ZcZcO9Zik^

X J%KsO*braSxI_slkqZcO*N0brka]ik_MpsZiu)p8ivZcb[]bYb[\XpMa]O9ZcO*a]Yb^s3qbr^MO9lkaivp1O9a]lu[cikazZ%ik^[]KMO$ZcpslvyOikN0O;lkZc_Ma]lkMqrO8ik_s^sxIO9xA_M^sy[cbrik^sZ

B(X)ik^

XxIO;Zcy*acbr1O$[cKsOO9kivq_I[]biv^sZiu lGvO*azluvO9ZiovO*a[cKMO[cazloj\O;y¡[]ika]bO;Zik

[]KMO9ZcO RAlua]kiopsaciIyO;ZcZcO9Z ¢ ^[]KMbeZpslkp1O9a£4O)br^f[ca]iYxM_syO4lu^sxZw[c_1xIX[cKMO4kO9^MO*azluqky*qrlvZcZiu1ZcO*N0brka]ik_Mp1Zik ^Miv^I¤¥qbr^MO9lka p1ifZwb[cbrYb[\XApMacO;ZwO9acYbr^M[cazlu^sZwika]N3lo[cbrik^1ZSiv^2N0O;lkZc_MacO;ZS[cKslu[brZ ^sik^I¤¥qbr^MO;lua Rmlua]kioiva¦MO9qqrO*aZcO*N0brka]ik_MpsZ ¨§ ^©OnIpMqrbeyb[Z\[]ac_sy[c_Ma]OªiukO9^MO*azlo[]ikazZ$ikZc_syzK¨ZwO9N0bvaciv_MpsZbrZkbrkO*^br^y9lkZcO£KMO*^

XbrZ[cKMO «h_syqrbrxMO9lu^mZcpslvyO

Rd ¬ ikaNivacO vO*^MO9a]lkqqrXk­Ml0N3lk^Mbivqrx1®%ZwKMio£br^M0KMio£¯[cKMO;ZwOZcO*N0brka]ik_Mp1ZlkacbeZwO a]ikN°[cKsO4vO*^MO9a]lkqYfbr^MO*[cbey O;df_slo[]biv^sZik8Zw[]lu[cbeZ\[]bry9luqYN0O9yzKslk^Mbry9Zlu^sx$O*vikqr_I[cbrik^slkacXsbivqivkX[cK1lo[(xIO;Zcy*acbr1O[]KMOxIXY^sluN0beyqrlG£±iu%qrlkacvO$^Y_MN$1O9a]ZivaRmlua]kioN0iIxIO*qeZiu%b^f[]O*azlky¡[]b^sp1luac[cbeyqrO9Z ² O*qrqp1ifZwO;xI^MO;ZcZ4a]O9Zc_Mq[zZ ika4[cKMO;ZwOO;dv_1lo[cbrik^1Z4lkacOSkbrkO9^[civkO*[cKMO9a £b[cKªlupspMqbey*lu[cbrik^sZh[cibr^f[cO9a]lvy¡[cbr^Mªpslkaw[]bry*qO;Z*³xIXY^sluN0bey3qelG£±ik%qelua]kO3^Y_MN$1O9a]Z(lu^sx´y*O*^f[cazluq)qrbrNb[([]KMO*ivacO9N­[]KMOqelo[w[]O*a8O*br^M0^MO9£µlkqa]O9lvxIX0ika%[]KMOZ\[zlu^sxsluazxy*ivluv_Mqelo[cbrik^M¤=azluvNO9^f[]lo[]biv^0NiIxIO9qrZ ¶ª·f¸D¹]º(» o¼&k givZcb[cbrfb[\X3pMa]O9ZcO*a]Yb^M$N0O9lkZc_Ma]O¤¥oluqr_MO9x3O9kivq_I[]biv^sZ9­fy*ik^sxIb[cbrik^1luqrqX3p8ivZcb[]bvOSivp1O9aw¤lu[civa]Z9­vRmlkacvio(N0iIxIO*qeZ)ik½b^f[]O*azlky¡[]b^spsluac[cbeyqrO9Z9­YxIXY^sluN0beyqrlG£¨ikqrlkacvO^Y_MN(8O*azZ*­f^MivacN3luqs¾s_sy[c_I¤lu[cbrik^sZ9­Iazlo[cOik&yiv^YkO*a]kO9^syOv­Yfbr^MO*[cbeyO9df_slo[]biv^sZ9­fbr^f[cO9a]lvy¡[cbr^M3Zw[]lusqO4j\_MN0pI¤¿xIbÀ8_sZcbrik^sZ9­1Á`*YX3[\XYp1OpsaciIyO;ZcZcO9Z9­Iyifluk_sqrlu[cbrik^I¤>a]lkkN0O*^f[zlo[cbrik^

∗ Â&Ã\ÄoÅzÆÇ=ÈÃwÉÇ%ÊzËÌ*Ç>Å]Ç>ÍÏÎÐÇ>ÍÏÑ\ÎwÒÓÉ;Í ÔÃwÆÐÎ>Í ÇÖÕÊzË×$ÅzÆØÍÏÑ¥ÙkÒÚDÊcÔÃ\É*Ç=ÆÕÚ8ÛÜ(Ý]Þ&ßÓ&à¨ÅzÉGá3âãäâÖÞäÊ*å9æ;ÃwÉ;ÑwÊ¡æ;ÆÐÇwÒç Æ>ÅzÉGÑ\áèésÈÅzÍÏêeëYÔ9Ù¡Ê¡êÏÊ¡ÙGìËÎ>ÈÅzÍÏêeè ÉGÿÇ

s &(. ! &(s-¡ 4# &3'4 !#8/02 + -%0/W $#%&(')#*

Á½O9Z ZcO*N0b¤>vaciv_Mp8O9Zx ikp8`*azlo[cO9_MazZqrbr^M`9lkba]O9ZZc_Ma qO;ZN0O9Zc_MacO;Z x _M^0O9Zcpslky*O N0O9Zc_MazluMqrOXdf_MbDpMa]`9ZcO*a]kO9^f[ qel(p8ivZcb[]bYb[c` xI`;ya]bvO*^f[4qrO9Z4`9kikqr_I[]biv^sZ%xIO9Z4qrikbeZ4xMO9Z4pMa]iYy*O9Z]Zw_1Z4xMORmlkacvioZc_MaX ÁO9ZZwO9N0b¤¥ka]ik_Mp8O9ZxI_slk_InAx ikp8`*azlo[]O*_MazZqrb^s`9lubra]O9ZZw_Maq O;Zwp1lkyO

B(X)xIO;ZSik^sy[cbrik^sZN0O9Zc_MazluMqrO9Z

8ika]^M`9O9ZZc_MaXdf_MbpMa]`9ZcO*a]kO9^f[0qel p8ivZcb[]bYb[c`AxI`;ya]bvO*^f[qrO9Z3`9kikqr_I[]biv^sZ3xIO9ZN0ioXkO*^s^MO9Z3Zc_MaqrO9Z

[]a]luj\O9y[cikbra]O9Z(xMOy*O9ZpMa]iYy*O9Z]Zw_1Z$xIORmlkacvio lu^sZ0y*O[3luac[cbeyqrOk­)^Miv_sZ$b^f[ca]iIxI_MbeZwiv^sZO[0`[]_sxIbrik^sZqel y*qrlvZcZcOªk`9^M`*azluqrOªxIO9ZZwO9Nb¤¥ka]ik_Mp8O9ZxMOª[cazlu^sZwika]N3lo[]biv^sZ0^Mik^¨qrbr^M`9lkba]O9ZZc_MaqO;ZN0O9Zc_Ma]O9Zdf_Mbpsac`;ZwO9acvO*^f[qrlAp1ifZwb[cbrYb[]`k­hdf_MOq ik^p1O9_I[0lkpMp8O*qrO*a0ZwO9Nb¤¥ka]ik_Mp8O9Z$xIORmlkacvioAiv_xIO¦sO*qrqO9a$^sik^I¤qrbr^M`9lkba]O9Z Pik_sZ%xIiv^M^Miv^sZh_s^MO Z\[]ac_sy[c_Ma]OO*nIpMqbeyb[cO xIO;Z4k`*^s`*azlo[cO9_MazZhxIOyO;Z4ZcO*N0b¤>vaciv_Mp8O9Z)qrikazZ]dv_sOXO9Zw[3q O9Zcpslky*OO9_syqrbexIbO9^

Rd ¬ iv_ pMqr_sZ0v`*^M`9a]lkqO9NO9^f[$_M^MOolua]br`[c`G® O*qrqO*¤¥y*b4p8O*a]N0O[xIOkivbay*ikN0N0O*^f[y*O9ZSZwO9Nb¤¥ka]ik_Mp8O9ZpMacioYbrO*^M^sO*^f[xIO9ZS`9df_slu[cbrik^sZSybr^M`[]brdf_MO;Zv`*^M`9a]lkqO;ZxIO$qrlN`;y*lk^Mbrdf_MOZw[]lu[cbeZ\[]brdf_MOAO[xIOAqrl MbrikqrikvbOª`9kivq_I[]biv^M^slkba]O Pik_1Z3N0ik^f[ca]ik^sZdv_sOAyO;Z3`9df_slu[cbrik^sZZcik^f[sbO9^p8ivZc`*O;ZO[ xMik^M^Miv^sZ xIO;Z)lupMpsqbey*lu[cbrik^sZ lk_InZcXIZ\[]L*N0O9Z xIOp1luac[cbey_MqrO9Z O9^0b^f[cO9a]lvy¡[]biv^³&qivbMxIO9Z)kazlu^sxMZ^sikN(sacO;Z4xMXf^1luN0brdf_MOO*[[cKM`9ika]L*N0O xIOqrlqrbN0b[cOyO9^f[cazluqrOk­MyO xIO9ac^sbO9aO9Zw[%^sik_MvO9lu_O*^pslkaw[]bry*_MqbrO*ap8ik_saqO N0iIxIL*qrO Z\[zlu^sxsluazxxIO y*ivluv_Mqelo[cbrik^M¤=azluvNO9^f[]lo[]biv^ » ; ¹ VYXIZw[cL*N0O;Z&xMXf^1luN0brdf_MO;ZZw_Ma xIO;Z&O9Zcpslky*O9Z&xIO N0O;Zw_Ma]O9ZpMa]`9ZcO*a]olu^f[qelp8ivZcb[cbrfb[c`v­oikp8`¤azlo[]O*_MazZhy*ik^sxMb[]biv^M^MO*qrqrO*N0O*^f[ p8ivZcb[cbÐZ*­IN0iIxIL*qrO9Z N3lua]kioYbrO*^sZ)xMOSpslkaw[]bry*_MqrO9Z)O9^br^f[cO*azlky[cbrik^½­fqrikbDxIO9Zva]lk^sxMZ ^MikN$Ma]O9Z4xIXY^sluN0bedv_sOk­Y¾s_sy[c_slu[cbrik^sZlk_sZcZcbrO*^M^MO;Z*­f[zlu_InxMOy*ik^YkO9acvO*^sy*Ok­f`9df_slo[]biv^sZ4ybr^M`¤[]brdf_MO;Z*­YpMa]iIyO;ZcZc_sZ4Z\[zluMqrO9Z4O*^br^v[]O*azlky[cbrik^½­fpMa]iYy*O9Z]Zw_1Z xIO[\XYp8OÁ`*YXk­My*ivluv_Mqelo[cbrik^M¤=azluvNO9^f[]lo[]biv^

$#3!'h#!#"$! %'&(*),+#-/.103254768492;:

§ ^$brN0p1ivaw[zlu^f[)yqelkZ]Z&iu8RAlua]kio ZwO9N0bvaciv_MpsZ&brZvbvO*^$fX[cKMOZciy9luqrqO;x$¦sO*qrqO9a)ZwO9N0bvaciv_MpsZ9­;b O [cKMOZcO*N0brka]ik_Mp1Z ik&Zw[ca]ik^MvqXy*ik^f[cbr^Y_Mik_1Z ivp1O9a]lu[civa]ZTt­t ≥ 0­Iiv^

C∞(X) ¬ []KMOZcpslkyOik&yiv^f[cbr^f_sik_sZ_s^sy¡[]biv^sZ&ik^$lSqriYy9luqrqXy*ikN0pslvy¡[[]ikp8ikqrikvbry9luqYZwpslvyOXolu^MbeZcKMb^slu[&br^=<s^Mb[\XM®&Zw_syzK[cKslu[

0 ≤ u ≤ 1brN0pMqrbO;Z0 ≤ Ttu ≤ 1

ikaluqrqt ≥ 0 ¢ ^2l£bexIO*aZwO9^sZcOk­YXª¦MO*qrqrO*a ZwO9N0bvaciv_MpsZiv^MO_M^sxIO9a]Zw[]lk^sxMZ[]KMO ZwO9N0bvaciv_MpsZ iup8ivZcb[cbrfb[\X3pMa]O9ZcO*a]Yb^M$qbr^MO;lua%ikp8O*azlo[]ikazZ)br^[]KMO ZwpslvyOiu&1iv_M^sxIO;xy*ik^f[cbr^Y_Mik_sZ

_s^sy¡[]biv^sZC(X)

­Yiv_Ma 1lkZcbryacO*O*a]O*^sy*Oika%¦MO9qqrO*aZcO*N0brka]ik_MpsZh1O9b^s?>Ï@A ¢ [%brZ4O9lvZwX[]i$ZcO*O[]Kslo[%b[]KMO O*vikqr_I[cbrik^ik^C∞(X)

kbrkO*^YX[cKsO O9df_slo[]biv^f = Af

£b[cKªl0qrb^sO9lua ¬ p1ifZcZcbrMqX_M^Y1iv_M^sxIO;xs®ivp1O9a]lu[civaAb^C∞(X)

pMa]O9ZcO*a]kO9Zhp1ifZwb[cbrYb[\Xv­v[]KMO*^AbeZ%yiv^sxIb[cbrik^slkqqrX0p1ifZwb[cbrkOv­fb O (Ag)(x) ≥ 0£KsO*^MO9kO*al_s^sy¡[]biv^

g8O*qrik^MfZ[ci[]KMOxIivN0lkb^ ik

A­beZ^Miv^I¤¥^MO*flo[cbrkO0lu^1x Glk^MbeZwKMO;Zlo[

x ¢ XbeZ[]KMO(«h_syqrbrxMO9lu^mZcpslvyO

Rd ival3N3lu^Mbikqex­1[cKMO ik_Ma]a]L*kO[]KMO*ivacO9NB>DC;A&Zw[]lo[]O9Z[cKslu[l3qrb^sO9luaivp1O9a]lu[civa

Abr^C∞(X)

£KMivZcO3xIikN3lubr^2y*ik^f[]lkb^1Z[]KMOZcpslky*OC2

c (X)ikh[\£4i[cbN0O;Z yik^f[]b^Y_Miv_sZwqrX

xMbÀDO*a]O*^f[cbelusqO_M^sy¡[]biv^sZ £b[]Kl(y*ikN0pslvy¡[4Zc_MpMp8ikac[ beZ4yik^1xIb[]biv^sluqrqrX$p8ivZcb[]bvOblu^1x3ik^MqrX0bb[%KslkZ[]KMOÁ½O9YXv¤FEKMbr^f[]yzKMbr^MOSika]N £b[]KGlkacbelusqO yiYOG3y*bO9^f[]Z9­Ib O b

Ag(x) =a(x)g(x) + (b(x),∇)(g(x) +1

2(c(x),∇,∇)g(x)

+

∫

(g(x+ y) − g(x) − χ(y)(y,∇)g(x))ν(x; dy), ¬ C;®

£KsO*a]Oc(x)brZ%l$p1ifZwb[cbrkOxIO<s^Mb[cO

d× d¤¥N3lo[ca]bnD­

b(x)brZ%l

d¤¥kO;y¡[civa9­

χ(y)beZ%ZwivNO8ik_M^sxMO9x^sik^I¤

^sO*vlu[cbrkO_M^sy[cbrik^m£b[cKmlyikN0pslvy¡[Zc_MpMp8ikac[[cKslu[SO;dv_1luqeZik^MO br^ml3^MO9bvKY1ivacKMiYiIxiu[cKMO(ika]bvb^½­lk^sx

ν(x, ·) brZl0Á`*YXN0O9lvZw_Ma]Ok­Yb O lp1ifZwb[cbrkOIH4ika]O*qDN0O9lvZw_sacOZw_1yzK[]Kslo[∫

min(1, |y|2)ν(x; dy) <∞, ν(x)(0) = 0. ¬ k®¦sika£Kslu[4ivqqrio£Z [cKMO N3lkb^acivqO£brqq8O*qrik^s$[]i[cKMOxI_1luqDika]N(_sqrlu[cbrik^ik&¦sO*qrqO9aZwO9Nbrka]ik_spsZ PSluN0O*qrXk­½l¦MO9qqrO*aZcO*N0brka]ik_Mp

Ttiv^

C∞(X)yqrO9lua]qrXvbvO9ZacbeZcO[ciªlxI_slkq&p8ivZcb[cbrfb[\XmpMa]O9ZcO*a]Yb^s

ZcO*N0brka]ik_MpT ∗

t

ik^ []KMOZcpslky*O M(X)iu8ik_M^sxMO9xJH4ika]O*qhNO;lkZc_Ma]O9Z iv^

X[cKMa]ik_skK.[cKMOxI_slkqb[\X

bexIO9^v[]b[\X(Ttf, µ) = (f, T ∗

t µ)­D£KMO9acO[cKMO$pslubracbr^M

(f, µ)beZvbvO*^ªYX[cKMO$b^f[]O*kazlo[]biv^½­sik)yik_sa]ZcO

¢ A beZh[cKMOkO9^MO*azlo[]ika)ik Tt­Y[cKMO9^

µt = T ∗t µy*lk^8OyzKslka]lvy¡[]O*a]bLK9O9x0YX0[cKMOO9df_slu[cbrik^br^[cKMO£ O;lu

ivacNd

dt(g, µt) = (Ag, µt) = (g,A∗µt),

¬ v®£KsO*a]O

A∗ beZ4[cKMOlvxGj\ikbr^f[[ci A ­I[]Kslo[KsikqexMZ4ikalkqq g a]ikN []KMOxIivN0lkb^iu A MO*a]O £ O ZcKsluqrq½xIO9lkq£b[]K^Mik^Mqrbr^MO9lkalu^slkqivvZ iu ¬ f®¡³

d

dt(g, µt) = Ω(µt)g,

¬ f®

ääÉNPORQRSRT

L

£KsO*a]OΩbeZl$^Mik^Mqrbr^MO9lka []a]lk^sZwika]N0lu[cbrik^0a]ikN lxIO9^sZwOxIivN0lkb^iu M(X)

[ci$[cKMO Zcpslky*OSikqrb^MO;lua_s^sy¡[]biv^sluqeZ%ik^

C(X) &Zcp1O;ybeluqbr^f[cO*a]O9Zw[lua]O[cKMO O;dv_1lo[cbrik^1Z%iu[]KMOika]N

d

dt(g, µt) = (A(µt)g, µt) = (g,A∗(µt)µt),

¬ k®£KsO*a]O

A(µ)beZ l(^Mik^sqbr^MO9lkahN3lupspMb^s(a]ikN lxIO9^sZcOxIivN3lubr^iu M(X)

[ci ¬ p8ivZ]ZwbrMqrX0_M^Y1iv_M^sxIO;xs®qrbr^MO9lka&ivp1O9a]lu[civa]Zb^C∞(X)

­klu^1xO*vO*^$NivacO%Zwp8O9y*b<8y*luqrqrX[cKMO%O9df_slu[cbrik^sZ&£b[cKAb^ ¬ v®&xIO*p8O*^sxMb^Mp8ikqrXY^MikN0beluqrqX3iv^

µ­Mb O []KMO O9df_slu[cbrik^sZ%ik[]KMOika]N

d

dt(g, µt) =

K∑

k=1

∫

· · ·∫

(Akg)(x1, . . . , xk)µt(dx1) · · ·µt(dxk), ¬ v®

£KsO*a]O(O9lvyzKAkbrZl ¬ p8ivZ]ZwbrMqrX_s^f8ik_s^sxIO9x1®ivp1O9a]lu[cikaacivNlxIO9^sZcO$Zc_MsZcpslky*O D ik C∞(X)

[ci[]KMOZcpslky*O

Csym(Xk)iu ZcXfN0N0O[]acbeyyiv^v[]b^Y_Miv_sZ%8ik_M^sxMO9x_M^sy¡[]biv^sZik

kGlkacbelusqO;Z4acivN

X _MalubrN beZ[\£4iivqrxMZ9³ICG®pM_Ma]O0lu^slkqXf[cbeypsacivMqO9N ¬ VYO;y¡[cbrik^1Zo¤>v®¤4[cib^f[ca]iIxI_sy*O$lu^2lupMpMa]iu¤psacbelo[]Olk^sluqrikik[]KMO^Miu[]biv^iu&yik^1xIb[]biv^sluqDp8ivZcb[]bYb[\X0ika4[cKsOwkO9^MO*azlo[]ikazZ

Ωlk^sx

Abr^ ¬ f®­ ¬ k®[]Kslo[%beZ4yzK1luazlky¡[]O*a]brZw[cbey%iva4p8ivZcb[cbrfb[\X3pMa]O9ZcO*a]Yb^M(NO;lkZc_Ma]O¤¥Glkq_sO9x3O9kikqr_I[]biv^sZ*­v[ci0vbvOSlu^OnIpMqrbry*b[

vO*^MO9a]lkq)Z\[]ac_1y¡[c_sacOikyik^1xIb[]biv^sluqrqrXAp8ivZcb[]bvOkO9^MO*azlo[]ikazZ ¬ lª^Miv^Mqbr^MO;lua(lk^sluqrikªik4[]KMO iv_Maca]L*vO[]KMO*ivacO9N®%lu^sx[]i3pMacioYbexIOslkZcbey£ O9qqp1ifZwO;xI^MO9Z]Za]O9Zc_Mq[]Z%iva[cKMO$yika]a]O9Zcp1iv^sxIbr^M$O9kivq_I[]biv^O9df_slo¤[]biv^sZ ¬ v®­ ¬ k® &k®lkpMpMqrbry9lo[cbrik^ ¬ VYO;y¡[]biv^sZ;¤o®S¤%[ciªxIO*vO*qrikp.lu^slkqXf[cbey([ciYikqeZikapMa]ioYb^M[cKMO3qelG£ik qrlkacvO ^f_sN(8O*azZlu^1x[cKMO(y*O*^f[cazluqqbrN0b[S[cKMO9ika]O*N3Z%[]ikkO*[cKMO9a£b[cKªpMa]O9y*brZcOazlo[]O9Zik y*ik^YkO9acvO*^sy*Oival£brxIOy*qrlvZcZ4ik&br^f[cO*azlky[cbr^M0Rmlua]kio3pMa]iIyO;ZcZcO9Z4£b[cKpsZcO*_sxIik¤¥xMbÀDO*a]O*^f[cbeluqDvO*^MO9a]lu[cikazZ9­vb O ikabr^f[cO9a]lvy¡[cbr^M0Á½`9fX[\Xfp8O pMa]iIyO9Z]ZcO9Z !#" + .1&86 .1& + 68:#+ ( +32),+#-?& .=2;:P6! .1&#" 4 6 :1.=& 2 )

§ ZwXIZw[cO9N¯ikMNO;lu^<sO9qrxb^f[cO9a]lvy¡[]b^Mpsluac[cbeyqrO9Z ¬ ika&Rmlua]kiopMaciIy*O9Z]ZwO;Z]®D£b[cK$lu^ _M^sxMO*a]qXYbr^MN0iu[]biv^xMO9Z]ya]b8O9xfXkO*^sO*azlo[civa ¬ C;® brZlZcXYZw[cO9N iu N pslkaw[]bry*qO;Z4b^ X N0ioYb^s$lky9yiva]xIbr^M([ci3l(vO*^MO9a]lu[civaikS[\XYp8O ¬ CG®¡­%£KMO*a]O[cKMO2yiYO GybrO*^f[zZ aj­bj­cj­γjZcp1O;ybXfbr^M []KMOªN0iu[cbrik^©ikO9kO*a]X

j¤=[]K

p1luac[cbeyqrO(xIO*p8O*^1xª^Miu[ik^MqrXik^ª[]KMO(p8ivZcb[]biv^xjik [cKMbeZSpsluac[cbeyqrOk­1M_I[ luqeZwi3iv^ª[cKMO$p1ivb^f[NO;lkZc_Ma]O

µ = h(δx1+ · · · + δxN

)ik^

X ¬ N0O9lk^ <sO*qexs®$Zwp8O9y*b<1O9x´YX.[cKMOp8ivZcb[]biv^ iulkqq4iu[]KMO*apsluac[cbeyqrO9Zx1, . . . , xN

ik[cKMOZwXIZw[cO9N ¬ h 1O9b^Mlp8ivZcb[]bvOZ]y*luqrbr^M(pslka]lkN0O[cO9a9­vO h = 1/N®­vb O aj(xj) =

a(xj , µ)­bj(xj) = b(xj , µ)

­cj(xj) = c(xj , µ)

­νj(xj) = ν(xj , µ) (¢ [ beZ£ O9qqY^Mio£^ ¬ lu^sx2y*lk^8O0O9lvZwbrqXmZcO*O9^1®[cKslu[ pslkZ]Zwbr^M[ci[cKsON0O;lu^ <1O*qex2iva RmyEO;lu^ª¤#$SqrlvZwioªqrbN0b[ b^2[cKMbeZ ZcXYZw[cO9N ¬ lZ]y*lkqbr^MqrbN0b[)£b[]K0[cKMO^Y_MN$1O9a)iuDpsluac[cbeyqrO9Z

N[]O*^sxIbr^M[]i b^=<1^Mb[\Xbr^0Zc_syzK3l£4lGX[cKslu[ [cKMOZ]y*lkqO;x

N0O;lkZc_MacO;Zh(δx1

+ · · ·+δxN)[]O*^sx$[]iZcikN0O <s^Mb[cON0O;lkZc_MacO;Z]®qrO9lvxMZ&ika]N3luqrqX []i lNO;lkZc_Ma]O¤¥Glkq_sO9x

xMXf^1luN0bry9Z%ik^XxMO9Z]ya]b8O9xYXO9df_slu[cbrik^ ¬ u®4£b[cK

A(µ)g(x) = a(x, µ)g(x) + (b(x, µ),∇)g(x) +1

2(c(x, µ)∇,∇)g(x)

+

∫

(g(x+ y) − g(x) − χ(y)(y,∇)g(x))ν(x, µ; dy) ¬ u®

âã&ä½âÖÞ

[]Kslo[Zcp8O9ybL<sO;Z3[cKsOmxIO[]O*a]Nbr^MbeZ\[]bryAxIXY^sluN0beyqrlG£ iu qrlkacvO^f_sN(8O*azZikaN0O9lk^ <sO*qex©b^f[]O*azlky¡[]b^sp1luac[cbeyqrO9Z9­kZwO9Ol^MbeyObr^IivacN3luq1xIbrZ]y_1ZcZcbiv^0b^ >DC9 A¥­f£KMO*a]OluqeZwiik[cKMO9ahp8ikpM_Mqelua)[\Xfp8O9Z)iub^f[]O*azlky¡[]biv^vO*ivNO*[ca]X ¬ O d ¤¿xIbrNO9^sZcbiv^sluqqelo[c[cbeyO lu^sx[]KMOKMbO9a]lka]yzKsbry9luqDb^f[]O*azlky¡[]biv^sZ]® lua]O[civ_syzKMO9x_Mp8ik^ § ^Miu[]KMO*aZwp8O9y*brlkq&y*qrlvZcZiu)br^f[cO*azlky[cbrik^sZ9­s^1luN0O*qrX k ¤¥lkacXka]ik_Mpªbr^f[cO*azlky[cbrik^ªik lkacsb[]a]lkacXikazxIO*a

k ¬ p1ifZcZcbrMqX^Miv^I¤>sb^slkacXv­4lk^sx©p8ivZ]ZwbrMqrX ^Miv^¨pMa]O9ZcO*a]fbr^M.[cKsOm[cik[]luqS^Y_MN(8O*aikpsluac[cbeyqrO9Zz®¡­%£%lkZlvxMxIa]O9Z]ZwO;x.b^ > A ¬ xIO9kO9qivpMb^s_Maw[]KMO*a0ZcikN0ObexIO9lvZ a]ikN > A®¡­ £KsO*a]Ob[£4lvZZcKMio£^ [cKslu[$b[cKMOk¤¥lkacX.b^f[]O*azlky¡[]biv^sZikb^1xIbrZw[cbr^Mv_MbrZcKslkMqOpsluac[cbeyqrO9Z([cK1lo[0pMacO;ZwO9acvO0[cKMO^Y_MN(8O*aiupslkaw[]bry*qO;Z$lkacOxMO9Z]ya]b8O9x YX kO9^MO*azlo[]ikazZ ¬ yik^1xIb[]biv^sluqrqrX p1ifZwb[cbrkOivp1O9a]lu[cikazZz® Bk

b^C∞(Xk)

­k = 1, . . . ,K

­lk^sx[cKsO

k¤¿lua]Xbr^v[]O*azlky[cbrik^sZ%[]Kslo[yzKslu^skO[cKsO(^Y_MN(8O*aSiu pslkaw[]bry*qO;Z ¬ a]ikN k

[cim®lkacOvbvO*^YX

ZcXYN0NO*[ca]bry9luqf[cazlu^1Zwb[cbrik^slkqYkO*a]^MO9qrZPk(x1, . . . , xk; dy1 · · · dym)

­u[]KMO4ika]N3luqIN0O9lvZw_Ma]O¤¥oluqr_MO9xqbrN0b[ik[]KMO qrlkacvOS^Y_MN$1O9aiu&psluac[cbeyqrO9Z%beZ%kbrkO9^YX3[]KMO O9df_slu[cbrik^

d

dt(g, µt) =

K∑

k=1

1

k!

∫

[(Bkg+)(x1, . . . , xk)

+∑

m

∫

(g+(y1, . . . , ym) − g+(x1, . . . , xk))Pk(x1, . . . , xk; dy1 · · · dym)]µt(dx1) · · ·µt(dxk),

¬ @v®£KsO*a]O

g+(x1, . . . , xl) = g(x1) + · · · + g(xl)¬ [cKMOA£ O;lu ika]N ¬ @k®08O*br^M b^f[]aciIxI_sy*O9xbr^ > k~A>­ZcO*O3lkqrZciVIO9y¡[]biv^sZlu^sx ikaxIO*[]lubrqeZ]® Piu[]bry*O$[]Kslo[[]KMik_MvKA[]KMO3ika]brkbr^sluq&b^f[]O*azlky¡[]b^sp1luac[cbeyqrOZcXIZ\[]O*N3ZqrO9lkxMb^M[]i ¬ k®­ ¬ k® ¬ N0O9lk^<1O*qex$br^f[cO9a]lvy¡[cbrik^$pMa]O9ZcO*a]fbr^MS[cKMO^Y_MN$1O9a iuDpsluac[cbeyqrO9Zz®&lu^sx([ci

¬ @v® ¬ br^syqr_sxIbr^M k ¤¥lkacX$j\_MN0psZ£b[cKAp8ivZ]ZcbMqrOazluvN0O*^f[]lu[cbrik^ªikayivlkk_Mqelo[]biv^1®lkacO(df_Mb[cO0xIbÀ8O9acO9^f[9­O;df_slo[]biv^ ¬ @k®y*lk^$8O£acb[w[]O*^0br^(ivacN ¬ k® Rmika]O*iokO9a9­;b[)brZ O;lkZcX[ci_M^sxMO*azZ\[zlu^sx([cKslu[)O9df_slu[cbrik^ ¬ @v®xMO9Z]ya]bMbr^M []KMOqrbrNb[ ikZwXIZw[cO9N0Zh£b[]Kbr^v[]O*azlky[cbrik^sZh^Miu[4pMacO;ZwO9acYbr^M[cKsO^Y_MN(8O*a iu½pslkaw[]bry*qO;Z*­fy*lk^8O$lkqrZci£a]b[c[cO9^Abr^ªika]N ¬ v® S¢ ^mÐlky¡[;­Db lkqq m ≥ k

­8[]KMO*^ ¬ xI_MO([ci[cKMO0ZwXYN0N0O[]acXiu []a]lk^sZcb[]biv^vO*a]^MO*qeZz®∫

g+(y1, . . . , ym)Pk(x1, . . . , xk; dy1 · · · dym)

=

∫

m

kg+(y1, . . . , yk)Pk(x1, . . . , xk; dy1 · · · dym)

= m

∫

g+(y1, . . . , yk)Pmk (x1, . . . , xk ; dy1 · · · dyk)

£b[cKPm

k (x1, . . . , xk ; dy1 · · · dyk) =1

k

∫

Xm−k

Pk(x1, . . . , xk; dy1 · · · dym)

lk^sxb^f[ca]iIxI_sy*b^M[]KMO ikp8O*azlo[]ikazZ

Bkf(x1, . . . , xk) = Bkf(x1, . . . , xk)

+∑

m

∫

[m(f(y1, . . . , yk) − kf(x1, . . . , xk)]Pmk (x1, . . . , xk; dy1 · · · dyk)) ¬ ~v®

ääÉNPORQRSRT

L

iv^MO y*lk^a]O*£a]b[]O ¬ @v®%lvZ

d

dt(g, µt) =

K∑

k=1

1

k!

∫

(Bkg+)(x1, . . . , xk)µt(dx1) · · ·µt(dxk) ¬ C9v®

£b[cK yik^1xIb[]biv^sluqrqrX.p8ivZcb[cbrkOivp1O9a]lu[civa]ZBkbr^C∞(Xk) H4X ZcbrNbrqelua(N3lk^Mbps_Mqrlu[cbrik^sZ ¬ []Kslo[£ OivN0b[¡®4ik^MO y*lk^xMO9luq£b[cK[]KMOy*lvZwO

m < k «hdf_slu[cbrik^ ¬ C;k®4KslvZ4[cKMOivacN ¬ v®4£b[cK

Akg =1

k!Bkg

+. ¬ CC;®

Kslka]lvy¡[]O*a]bLK9b^Mp1ifZwb[cbrYb[\XpsacO;ZwO9acYbr^MO*vikqr_I[cbrik^1Zbr^y9lkZcOhik X 8O*br^M l«h_syqrbrxMO9lu^Zwp1lkyO ¬ iva&lN3lk^Mbivqrx1®£ OZcKslkqq=<1a]Zw[cqrXZcp8O9ybX$[cKMOy*qrlvZcZ)iuDp1ivqXY^MivNbeluqsO9df_slo[]biv^sZ ¬ v®&[]Kslo[ y9lu^08OpMa]O9ZcO*^f[cO;xbr^ ika]N ¬ C9v®(£b[cK©yiv^sxIb[cbrik^slkqqrX´p8ivZcb[]bvOivp1O9a]lu[cikazZ Bk­h[cKMO9^ vbvOl.Z\[]ac_1y¡[c_sacOiulua]Mb[]a]lkacX

O;df_slo[]biv^sZ ¬ v®psacO;ZwO9acYbr^MAp8ivZcb[]bYb[\Xk­%lu^sxlo[3qelkZw[ZwKMio£ ¬ _M^sxMO*aN0brqrx [cO;yzKM^Mbey*lkqlkZ]Zw_MN0pI[]biv^sZz®[]Kslo[l0^Mik^Mqrbr^MO9lkaikp8O*azlo[civaAb^ªp8ivZcb[cbrfb[\XpsacO;ZwO9acYbr^M0O9df_slo[]biv^sZ ¬ u®%KslvZ%[ci8O iuivacN ¬ u® ¢ [£brqrq[]_Mac^Aik_I[;­1br^Apslkaw[]bry*_Mqelua;­M[cK1lo[Sbr^Ay9lkZcO

K = 2[cKMOp8ivZcb[]bYb[\XpMa]O9ZcO*a]Yb^s3O9df_slo[]biv^sZ ¬ v®y*lk^8O3luqr£4lGXIZS£acb[w[]O*^.b^.ivacN ¬ C9v®S£b[cK´yiv^sxIb[cbrik^slkqqrXªp8ivZcb[]bvO$ivp1O9a]lu[cikazZ Bk

¬ £KMbeyzK brZ^siu[[cKMOy9lkZcOS^sO*b[cKMO9aikal0xIbeZcy*acO*[cOX­I^Miva%ika%[]KMO «)_sy*qbexIO9lk^

Xlk^sx

K > 2®

!#" + . 2 . +#-/.103254768492;: 6 & $& +"#:1684 0 & 6 + (*( 2 .1)

§ [[cKMO O*^sxikf[]KMbeZbr^f[ca]iYxM_sy¡[]biv^£4OhlkacO vikbr^M <sazZ\[½[]ibrqqr_sZw[cazlo[cO)[cKMOhxMbÀDO*a]O*^sy*O)8O[\£4O*O*^ p8ivZcb[cbrfb[\XpsacO;ZwO9acYbr^M(O9kikqr_I[]biv^sZ ¬ k®%lk^sxYb^MO*[cbeyO9df_slo[]biv^sZ ¬ C;k® ¬ br^[cKMO[]O*a]N0b^MivqivkX3ik[cKsO ^MOnY[ZwO;y¡[]biv^[]KMbeZ%NO;lu^sZ4[]KMO xIbÀ8O9acO9^syO8O[\£4O*O*^yiv^sxIb[cbrik^slkqqrXp8ivZcb[]bvOlk^sxZ\[]aciv^MkqrXyiv^sxIb[cbrik^slkqqrX3p1ifZwb[cbrkOivp1O9a]lu[civa]Zz®ik^lAZcbN0pMqrOy*lkZcOiul.xIbrZ]ya]O[]OZw[]lu[cOZwpslvyO

X ´§ [qelkZw[0£ OZwK1luqrq <MnZcikN0OslkZcbey^siu[]lu[cbrik^sZ4[]i01O _sZcO9xbr^[]KMO pslkp1O9a£b[]KMik_I[_saw[]KMO*aa]O*N0br^sxIO9a ¢ ^ VYO;y¡[cbrik^ iv_Ma N3lubr^ Zw[ca]_sy[c_Mazluq a]O9Zc_Mq[]Zik^.[]KMO3p1ifZwb[cbrYb[\X2pMa]O9ZcO*a]fbr^MkO9^MO*azlo[]ikazZ A ­ Akbr^ ¬ k®­ ¬ v®lua]O0ikI[zlubr^MO9x ¬ J%KMO9ika]O*N3Z ;¤\ v®Sika[cKsOy*lkZcOiu X 8O*br^Mml«h_syqrbrxMO9lu^´Zcpslky*O ¬ ikaZcbrNbrqelua]qXlN3lu^Mbikqexs® VYikN0OOnMluN0pMqrO9ZlkacOkbrkO9^ § [[cKMO O9^sxiu)VYO9y[cbrik^ª(£4O xIbeZcy*_sZcZZcKMivaw[]qX[]KMOqbÖ[cbr^M iu[cKMO^Miv^Mqbr^MO;lua0O*vikqr_I[cbrik^sZ ¬ f®([ci.qbr^MO9lka0O*kivq_M[cbrik^sZ$ik^[cKMON(_sq[]b¤¥psluac[cbeyqrOZ\[zlo[cOZcpslvyO § Z)lu^0brNp8ikac[]lk^f[psacO9fbrik_1Z&y*ik^f[ca]brM_I[cbrik^$[ci[]KMO%[cKsO*ika]Xik8p8ivZcb[]bYb[\XpMa]O9ZcO*a]Yb^MO9df_slu[cbrik^sZik[\XYp8O ¬ v®hik^MOZwKsik_Mqex3N0O*^f[cbrik^1iYiv >ÏoA>­Y£KMO9acO[cKMO;ZwOO9df_slu[cbrik^sZ £ O9acOlu^slkqX K9O9x0YX3b^f[]aciIxI_sy*b^sl^slo[]_Mazluq&b^=<1^Mb[]O¤¿xIbrNO9^sZcbiv^sluqZwN0iYiu[]KmN3lu^MbikqexAZw[ca]_sy[c_Ma]O(iv^ª[cKMO0Zcpslky*O(iu)N0O;lkZc_MacO;Zlk^sxmb[]Z

[zlu^MvO*^f[Zcpslky*O9Z½lk^sx£KMO*a]O)ik^sO)y*lu^<1^sx l%acO;Zw_sq[;­9£KMbeyzK beZ½O9Z]ZcO*^f[cbeluqrqXSO9df_MbroluqrO*^f[[ciiv_MaJ%KMO9ika]O*N § Z%b^[cKsOy9lkZcOSikqrb^sO9luavO*^MO9a]lu[civa]Z9­kiv^MO[cKMbr^M0brZ4[]i0kbrkOlZw[ca]_sy[c_Ma]OSikivacN3luqDvO*^MO9a]lu[cikazZ9­lk^sx(lk^Miu[]KMO*apMa]iksqO9NbrZ&[]ixMbrZw[cbr^Mk_sbrZcK([]KMivZcO4ius[]KMO9ZcO4ika]N3luqYkO9^MO*azlo[]ikazZ[cK1lo[)lky[c_slkqqrX kO9^MO*azlo[]O

ZcO*N0brka]ik_Mp1Z J%Ksik_MvK(qriu[]Z&iks£ ivacbr^([]KMbeZ&xMba]O9y[cbrik^(beZxIik^sO 8iu[]K(iva&qrbr^MO9lkalk^sx^sik^Mqrb^sO9luay*lvZwO;Z*­N3lk^fX3df_MO;Z\[]biv^sZha]O*N3lkb^ivp1O9^ tO9y*O*^f[lkyzKMbrO*vO*N0O*^f[]Zhb^[cKMOqrb^MO;lua [cKMO9ika]XlkacOS^MbeyO*qrX3pMa]O9ZcO*^f[cO;xbr^ > @ A J%KsO*a]OSO*nYbeZw[]Z%lu^OnY[cO9^sZwbrkOqrb[]O*azlo[c_sacOik^[cKMO£4O*qrqDp1ifZwO;xI^MO9Z]Z ika%O9df_slu[cbrik^ ¬ k®*­ ¬ u®h£b[cK

âã&ä½âÖÞ

xMbÀDO*a]O*^f[cbeluqkO9^MO*azlo[]ikazZA[cK1lo[lkpMp8O9luaSb^[]KMO$lk^sluqrXIZwbeZik b^f[cO9a]lvy¡[]b^MxIbÀD_sZcbiv^sZ ¬ ZwO;yiv^sxikazxIO*aivp1O9a]lu[civa]Z

A®­ZwO9O > f~ A>­ > vA>­ > k A>­ > uA>­ > k~ Ahlu^1xma]OO*a]O*^1yO9ZS[]KMO*a]O*br^½­lk^sxAikh[]KMO3xIO[]O*a]Nbr^MbeZ\[]bry

psaciIyO;ZcZcO9Z ¬ <sazZw[iva]xIO9aSivp1O9a]lu[civa]Z A ®¡­ZwO9O(O >DC9A>­ >DC9~ A ¢ ^f[cO9kazluq&ikp8O*azlo[]ikazZ A ­ Akbr^ ¬ k®*­ ¬ u®Zw[]lk^sx3iva br^f[cO9a]lvy¡[cbr^MpM_Ma]Ohj\_sNppMa]iIyO9Z]ZcO9Zhlk^sx3[]KMO*braZ\[]_sxIX£%lkZ N0iu[]bolo[]O9xfX0[]KMOH ivq[ K*N3lu^s^

N0iIxIO9qikhyikqrqrbrZcbiv^sZlk^sx[]KMOVYN0ivq_syzKsioYZcYbN0iIxIO*q&iuhy*ivlkk_Mqelo[]biv^lk^sxazluvNO9^f[]lo[]biv^½­DZwO9OO > kA>­#>Ïk A>­#> vA>­#> @A>­#> k~A>­#> f Ahlu^sx2acO*O*a]O*^sy*O9Z[cKMO9acO9b^ VYivNO0pslkaw[]bry*_MqeluayikN$Mbr^slo[]biv^sZiu4xMbÖ¤O9acO9^f[cbeluq½lu^1xbr^f[cO9kazluqDkO9^MO*azlo[]ikazZ4lkacON0ivZw[cqrXy*O*^f[cO9acO;xlkaciv_M^sxZwpslu[cbeluqrqX^Mik^f[ca]brfbeluqvO*azZwbrik^1Z ikH4ivq[ K*N3lu^M^lu^sxVYN0ikqr_syzKMioIZwYb1NiIxIO9qrZ9­IZwO9O >ÏkA¥­ >Ï A¥­ > k A>­D> vA½lu^1xa]OO*a]O*^1yO9Zh[cKsO*a]O*br^½­IZwO9OSluqeZci>ÏuA½ikaN0iIxIO*qeZ%£b[cKªZwp8ik^f[zlu^MO9ik_sZ4Mbraw[]KsZ ¢ ^a]O9y*O*^f[hXvO9luazZ iv^MOivsZwO9acvO9Z)l^MO*£¨£%lGkOik½b^f[cO9acO;Z\[h[ci[]KMOSÁ`*YXpMaciIy*O9Z]ZwO;Z ¬ ZcO*OSO 1iYivYZ> A¥­9> AÖ®lu^sx []KMO*braZcpslo[]brlkqqrX^Mik^f[ca]brfbeluqO*nY[cO*^1Zwbrik^sZ¤Á`*YX [\XYp8OmpMa]iYy*O9Z]ZwO;Z*­4[]KMOmqelo[w[]O*a1O9b^M

1lkZcbry9luqrqXj\_sZw[3[cKsORAlua]kio pMa]iYy*O9Z]ZwO;Z£b[]Kp1ZwO9_sxIiu¤¿xIbÀ8O9acO9^f[cbeluq%kO9^MO*azlo[]ikazZ(ikÁ½`9YXv¤FEKMbr^syzKMbr^[\XYp8Ok­8ZwO9O > @ AivaSl3y*ikN0pMa]O*KMO9^sZwbrkOlk^sluqrXf[cbey Z\[]_sxIXiu[]KMO9ZcOpMa]iIyO9Z]ZcO9Z9­ > k~Aival3pMa]ik1luMbrqbeZw[cbeylk^sluqrXIZwbeZ9­o£KMO*a]O4[cKMO;ZwOpMa]iIyO;ZcZcO9Zlua]O%y*luqrqrO9xj\_MN0pI¤¥xMbÀD_sZcbiv^sZ*­flu^sx0lkqrZci>ÏuAMika)ZcikN0Oyiv^M^MO9y[cbrik^sZ8O[\£4O*O9^.[cKMO;ZwOlkpMpMa]ivlvyzKMO9Z Á½`9YX2lu^1x´Á½`*YXA[\Xfp8OpMa]iIyO;ZcZcO9ZO9^oj\ioX2df_Mb[cOxMbÀDO*a]O*^f[(pMa]ikp8O*ac[cbrO9Z¬ O Ðlo[ []lubrqeZ)lu^sx3xMbrZ]yiv^v[]b^Y_Miv_sZ[]a]luj\O9y¡[]ika]bO;Z]®lvZ _sZc_sluq ² bO9^MO*a pMa]iYy*O9Z]Z)lu^1x0acO9qrlu[cO9x0xIbÀD_sZwbrik^1Z*­£KsbryzKN3lukO%Á½`9YXpMa]iYy*O9Z]ZwO;Z&b^sxMbrZcp1O9^sZ]luMqrO4ika)lupspMqbey*lu[cbrik^sZ&br^0N0lk^YXZwb[c_slu[cbrik^sZ£KsO*a]O%xMbÀD_sZcbiv^psaciIyO;ZcZcO9Z[c_sac^ ik_M[[]i1O0_M^I¤¿lkxMO9df_slo[]O ^A[]KMO0iu[]KMO*a Kslk^sx­[]KMOlu^1luqrXYZcbeZikh[cKsO9ZcO3pMaciIy*O9Z]ZwO;ZbeZZw_sacO9qXN0ika]O$y*ikN0pMqrbry9lo[cO;x[cK1lu^m[]Kslo[iu _sZw_1luq xIbÀ8_sZcbrik^sZ9­xI_MOb^2psluac[cbey_Mqelua]qrX[]i[]O9yzKM^sbry9luqrqXN0ivacOb^YkivqvO9x´lk^sluqrXIZwbeZik[cKMOyivaca]O9Zcp8ik^sxIbr^MmZw[ciIyzKslvZ\[]bryxIbÀDO*a]O*^f[]brlkq O;dv_1lo[cbrik^1Z ¬ ZwO9OO > Alk^sx >ÏA&ika[]KMOqrlu[w[cO9az® tiv_MkKsqXZwp8O9lkYb^M1­M[]KMO$a]O9Zc_Mq[]Ziu)ik_saVYO;y¡[cbrik^.3Z\[zlo[]O[cK1lo[Sp8ivZcb[cbrfb[\XpsacO;ZwO9acYbr^M3N0O9lkZc_Ma]Ooluqr_MO9xmO*kivq_M[cbrik^sZ_M^sxIO9aSN0brqrxm[cO9yzKs^Mbry9luqlkZ]Zw_sNpM[cbrik^sZKslGkOvO*^MO9a]lu[civa]Zik^sik^I¤¥qbr^MO;lua$Zcpslu[cbeluqrqX ^Miv^v[]acbrYbrlkq4Á`*YXf¤ EKsb^syzKsb^´[\XYp8O[]KY_sZacO9pMacO;ZwO9^f[cbr^Mm^Miv^Mqbr^MO;lua0lu^slkqivvZikÁ`*YX3[\XYp8O pMaciIy*O9Z]ZwO;Z4lk^sxZwO9Nbrka]ik_spsZ ¢ ^2VIO9y¡[]biv^sZ£4O(Z\[zluac[ik_salk^sluqrXIZwbeZiu)vO*^MO9a]lkq½p8ivZcb[]bYb[\XpsacO;ZwO9acYbr^M3O*vikqr_I[cbrik^sZYXO;Z\[zluI¤qrbeZwKMbr^M ¬ _sZwbr^Ml <snYO;xp1ivb^f[Slkacv_MN0O*^f[[cK1lo[beZ£4O*qrqY^Mio£^b^ª^sik^Mqrb^sO9luaSlu^slkqXIZcbrZ9­MZwO9OO > @ Aivab[]ZlkpMpMqrbry9lo[]biv^[]i3^Mik^sqbr^MO9lkaSVIyzKMa]iIxIbr^MkO9aO9df_slo[]biv^ªlu^sx,>ÏkAikaZw[ciIyzKslvZ\[]bry ¢ []i ZO9df_slu[cbrik^sZxMO9Z]ya]bMbr^M []KMOAO9kikqr_I[]biv^ ¬ u®£b[]K xIbÀ8O9acO9^v[]brlkqvO*^MO9a]lu[cikazZz®$l kO*^sO*azluq£4O*qrqSp8ivZcO9xM^MO9Z]Z3a]O9Zc_Mq[

¬ J%KMO*ivacO9N C9®luqrqio£br^M2[ci.acO;xI_sy*O[]KMOªZ\[]_sxIX ikS^Miv^Mqrb^MO;luapMa]ikMqrO*N3Z ¬ k®¿¤ ¬ o®([]i ZcikN0OacO9k_I¤qelua]b[\X pMa]ikp8O*ac[cbrO9Ziu []KMO2y*ika]acO;Zwp8ik^sxMb^M´qrbr^MO9lkapMa]ikMqrO*N3Z J%KMbeZa]O9Zc_Mq[y9lu^¨8O.lkpMpMqrbO;x©[cilolkacbrO[\X.iuN0iIxIO9qrZ$£b[]KpsZcO*_1xIiu¤¿xIbÀ8O9acO9^v[]brlkqhvO*^MO9a]lu[cikazZ ¬ 1O9Xkik^1x Z\[zlu^sxsluazx xIbÀ8_sZcbrik^sZz®¡­)£KsO*a]O[]KMO$y*ika]acO;Zwp8ik^1xIb^s3acO9k_Mqelua]b[\XacO;Zw_sq[zZivaqrb^MO;luaSkO*^sO*azlo[civa]Zlua]O(lGolubrqrlkMqOv­MivaSbr^sZw[]lu^1yO(br^Ay9lkZcOikxMO9yivN0p1ifZclkMqO vO*^MO9a]lu[civa]Z ¬ ZcO*O > AÖ®­sa]O*v_MqrlkaO*^Miv_MkKªxIO9kO9^MO*azlo[]OxMbÀD_sZcbiv^sZqrbvO b^f[]O*azlky¡[]b^sy*_Ma]fbrqrb^MO;lua a]^sZw[cO9b^I¤¿QKsqO9^f8O9yz$pMa]iYy*O9Z]ZwO;Z ika Z\[]iYyzK1lkZw[cbeykO9iYxMO9Zcbry4¾sio£Zhik^0[]KMOSy*iu[]lk^MkO9^f[)M_M^I¤xMqO;Z[]i2tbrO*N3lu^s^Mbrlk^ N3lu^MbikqexMZ ¬ ZcO*O >LCRA¥­ > k A®¡­)iva$Zw[]lkMqO(j\_MN0p xIbÀD_sZwbrik^1Z .§ Z$[cKMOqrlu[w[]O*a0lua]Oik%Zcp8O9ybeluqhb^f[cO9acO;Z\[lkNiv^MªÁ½`9fXA[\XYp1O3pMa]iIyO;ZcZcO9Z9­xI_MO0[]i[cKMO9ba$lupMpMqrbey*lo[]biv^sZ br^´l£brxIO3olkacbrO[\XikhN0iIxIO*qeZSacivN pMqrlvZwN3lpMKYXYZcbey*Z[ci?<s^slu^1yO9Z ¬ ZwO9O$O > A>­ > @ A¥­#> A>­ >Ï@ A®¡­£ O0ZwK1luqrq&y*ik^sy*O*^f[cazlo[]O_saw[]KMO*aiv^Zw[]lkMqrO[\Xfp8O pMa]iIyO9Z]ZcO9Z%O*^1xIb^sVYO9y[cbrik^ª£b[]KmlZcbrNpsqO brqqr_sZw[cazlo[]biv^ikJ%KsO*ika]O*N Cbr^[cKMOy9lkZcOSik&^sik^I¤¿xIO*vO*^MO9a]lu[cOZwO;yik^1xiva]xIO9apsluac[iul$p1ZwO9_sxIiu¤¿xIbÀ8O9acO9^f[cbeluqkO9^MO*azlo[]ika ¢ ^mVYO;y¡[cbrik^0£4O b^f[ca]iIxI_sy*Oiv_MaslvZwbeyNiIxIO9q½iubr^v[]O*azlky[cbr^M3Z\[zluMqrO4j\_MN0pI¤¥xMbÀD_sZcbiv^sZ*­spMaciovOl£4O*qrq p8ivZcO9xI^MO;ZcZa]O9Zc_Mq[ ¬ acO;xI_sy*b^MOnIp8ivZcb[]biv^Aiva ZwbrN0pMqbeyb[\Xª[]i[]KMOy*lvZwO£KMO9acO$[cKMOZw[]lkMqrO$pslkaw[

ääÉNPORQRSRT

@ L

beZ ^Mik[(loÀDO9y[cO9x.YXAbr^f[cO9a]lvy¡[cbrik^8®Slk^sx lu^slkqX K9O$[]KMOyivacO0ik []KMOkO9^MO*azlo[]ikaiu4[cKsOyika]a]O9Zcp1iv^sxIbr^M¬ qbr^MO;lua¡® ¦sO*qrqO9apMa]iYy*O9Z]Z%b^ C(M+(X)) ¢ ^¯VIO9y¡[]biv^sZ´lu^sx ´[cKsOAkO9^MO*azluqSlu^slkqXf[cbeyª[]iYikqeZ3ikapMa]iofbr^M´[]KMOAqelG£ iuqelua]kOm^Y_MN$1O9a]Zlk^sxy*O*^f[cazluq1qbrN0b[4[cKMO9ika]O*N3Z)a]O9Zcp8O9y¡[]bvO*qrXlkacOxIO*vO*qrikp8O9xiva br^f[cO*azlky[cbr^M(Rmlkacvio(pMa]iIyO;ZcZcO9Z £b[cK

p1ZwO9_sxIiu¤¿xIbÀ8O9acO9^f[cbeluqIkO9^MO*azlo[]ikazZ J%KMO9ZcO%[ciYivqrZ lua]O4brqrq_sZw[cazlo[]O9x0iv^3ik_Ma)1lkZcbry%O*nIlkN0pMqO%¤&b^f[]O*azlky¡[]b^sZw[]lkMqrO j\_sNpM¤¥xIbÀD_sZwbrik^1Z*­Ms_I[[cKMO9XZcO*O9N [ci08O iubr^f[cO9acO;Z\[O9kO*^ika_sZc_slkq½b^f[]O*azlky¡[]b^sxIbÀD_sZwbrik^1Z Q^MqrbvOlªpMa]ikslkMbqrbeZ\[]bry3lkpMpMa]ivlkyzK ¬ y*ikN0pslky[c^MO;ZcZb^VYvika]ikKMiIx.Zwp1lkyO3ik%[cazloj\O9y[civacbrO9Zz®S^Mika]N3luqrqX_1ZwO;xiva&ivI[]lkb^sb^MS[cKsO9ZcO [\XYp8O%iu1a]O9Zc_Mq[]Z ¬ ZcO*O%O >ÏoA¥­P>Ï A¥­1>ÏkA¥­1> @ Aslk^sx$acO*O*a]O*^sy*O9Z[cKMO9acO9b^(ika[]KMOqelG£©iu½qelua]kO^Y_MN$1O9a]Z9­>LC; A>­ >DC A¥­> A>­ >ÏkA¥­ > k A¥­>Ïk ADlk^sx3a]OO*a]O*^1yO9Z [cKMO9acO9b^3ivah[]KMOSy*O*^f[cazluqqrbrNb[ iusbr^f[cO9a]lvy¡[cbr^MxIbÀD_sZcbiv^sZlu^sx >LC@ A¥­1> k A¥­=> kA¥­=> k AIika[cKMOy*O*^f[cazluqfqbrN0b[[]iZcikN0OhN0iIxIO;Z£b[cKj\_sNp1Z]®­viv_Ma)N0O[]KMiIx­v8O*br^M slvZwO;xOnMyqr_sZcbvO*qrXik^0[]KMO%[cKMO9ika]X(iky*ik^f[cazlky[cbrik^ZwO9Nbrka]ik_spsZ*­kXfbrO*qexMZ^sO*£ b^sZcbrkKf[]Zbr^.[cKMOlk^sluqrXf[cbeylvZwp8O9y[]Zik []KMO3pMa]iYy*O9Z]ZwO;Z_M^sxIO9ayik^1ZwbexIO*azlo[]biv^ ¬ ¦MO*qrqrO*apMa]ikp8O*ac[\Xk­Zw[ca]_sy[c_Ma]Oik½[cKMOyivacO;Z)ik[cKMOkO9^MO*azlo[]ikazZ*­kpMacO;ybeZwOSa]lu[cOSiuy*ik^YkO9acvO*^sy*Ok­uO*[]y9® Rªika]O*iovO*a;­ubr^sZ\[]O9lvxikxIO;Zcy*acbrMbr^M([]KMOqbrN0b[]b^M lk_sZcZcbelu^psaciIyO;ZcZ iu¾s_sy¡[]_slo[]biv^sZ%br^[cKsOvO*a]X0qrlkacvOSZcpslky*O

S′(X)­MlkZ

beZ_1Zw_slkq)br^´[cKMOqb[cO9a]lu[c_Ma]Ok­&ik_Ma$N0O[cKsiYx´lkqqrio£Z[]iAxIimb[(br^l^slo[]_MazluqhZcN3luqrqO9a$Zcpslky*O(C1(X))∗[]Kslo[ brZ½j\_sZ\[ wlMb[4qrlkacvO*az­G[]Kslu^[cKMObr^Mb[cbeluqDZwp1lkyO M(X) ¬ £KsbryzKb[]ZcO*q½beZ)_sZc_sluqrqrX$_M^1lupMpMa]ikpsacbelo[]OivaxMO9Z]ya]bMbr^M3[]KMO$¾s_sy¡[]_slo[]biv^sZz® § ^Mik[cKMO9aZcp8O9ybL<1y[cO9yzKs^Mbry9luqO9lu[c_Ma]O$beZ[cKMO0ZwXIZw[cO9N0lu[cbey_sZcO(ik[]KMO$xMO*a]bolo[]bvO9Ziu)[]KMO$Zcikqr_I[cbrik^1Zik [cKMO$fbr^MO*[cbey(O;dv_1lo[cbrik^1Z£b[cKAa]O9Zcp8O9y¡[S[ci[cKMO$b^Mb[cbeluq xMlu[]l ¬ b^y9lkZcOSik[]KMOIH4iu[ K*N3lu^M^O9df_slu[cbrik^[cKMO;ZwO xIO9acbrolo[]bvO9Z%8O*br^M3Z\[]_sxIbrO9xb^ªxIO*[]lkbq½br^,> f A® VYO;y¡[]biv^ .beZxIO9kiu[]O9x[]i´[cKMO2b^f[]O*azlky¡[]biv^ yzKslu^skbr^M [cKMO2^Y_MN(8O*aiu psluac[cbeyqrO9Z9­£KMifZwOmqelG£

ikqelua]kO^Y_MN$1O9abeZxIO9Z]ya]br1O;x©fX O9df_slu[cbrik^ ¬ @f® °§ Z£4Oª^Mik[cO9x©lk1iovO[cKMbeZ3O9df_slu[cbrik^¨y*lk^¨1OlkqrZci£a]b[c[cO9^Abr^mika]N ¬ C;k®£b[cK yiv^sxIb[cbrik^slkqqrXp1ifZwb[cbrkO$ikp8O*azlo[civa]Z Bkb^C(Xk)

­£KMbeyzKAluqrqio£Z[]i2y*ik^sZcbrxMO*a[]KMO*N lvZ(p1luac[cbey_Mqeluay*lkZcO9Z$iu ¬ C;k® Mio£4O*vO*a;­iva([]KMOlupMpMqrbey*lo[]biv^´[ci2b^f[]O*azlky¡[]b^sp1luac[cbeyqrO9Zlk^mluq[cO*a]^slu[cbrkO a]O9xI_sy[cbrik^[ci ¬ C9f®%8O9yivN0O9ZN0ivacO ^slu[c_Mazluq>­s^1luN0O*qrXYXiksZcO*a]Yb^M0[]Kslo[¬ @v®KslkZlkqa]O9lvxIXika]N ¬ C;k®SM_I[ £b[]K Bk

psacO;ZwO9^v[]b^s^Mik[ j\_1Z\[[cKsOivp1O9a]lu[civa]ZSb^Csym(Xk) ¬ lkZ£4O luqr£4lGXIZ%Zc_MpMp8ivZcO9xb^ ¬ C;k®w®­MM_I[[]KMO ikp8O*azlo[]ikazZ acivN Csym(X )

[]iCsym(Xk)

kbrkO*^YXBkf(x1, . . . , xk) = Bf(x1, . . . , xk)

+∞∑

m=1

∫

(f(y1, . . . , ym) − f(x1, . . . , xk))Pk(x1, . . . , xk; dy1 · · · dym) ¬ C;k®

[]Kslo[0lua]Oyiv^sxIb[cbrik^slkqqrX.p1ifZwb[cbrkObr^´[]KMOZcO*^sZcO3[]Kslo[bSlm^Mik^I¤¥^MO9vlo[]bvOf ∈ Csym(X )

olu^MbeZcKMO9Zlu[

x = (x1, . . . , xk)­[]KMO*^

Bkf(x) ≥ 0 J%KMO[]KMO*ivacX.iuSVYO;y¡[]biv^sZªlu^sxªbrZ(pMa]O9ZcO*^f[cO;x b^l£%lGX[cKslu[ NivacOikaqrO9Z]ZZw[cazlubrkKf[cika]£4lka]xIqrXOnY[cO9^sxMZS[ci[cKsO0RAlua]kioNiIxIO9qrZik br^v[]O*azlky[cbrik^sZS^Miu[psacO;ZwO9acYbr^M.[cKMOª^Y_MN$1O9aiupslkaw[]bry*qO;Z0[cKY_sZ3br^sy*q_sxMb^M [cKMOmpsaciIyO;ZcZcO9Z0ik\­ZclGXv­4y*ivluv_Mqelo[cbrik^©lu^1xazluvN0O*^f[]lu[cbrik^ ¢ ^ VIO9y¡[]biv^ 2£ Oªbrqqr_sZw[cazlo[]O[cKMbeZxIO9kO9qivpMN0O*^f[YXikM[]lubr^Mbr^M´^MO9£ a]O9Zc_Mq[]Z3iv^[]KMO3azlo[cOiuyiv^YkO*a]kO9^syO[ci[cKsOqelG£ ik qelua]kO^Y_MN(8O*azZ lu^sxA[cKsOyO*^f[]a]lkq qbrN0b[br^A[]KMOZw[]lu^1xMluazxVINivq_1yzKMioIZwYbN0iIxIO*qiky*ivlkk_Mqelo[]biv^I¤=azluvN0O*^f[]lu[cbrik^ ¬ [cKY_sZvbYbr^M´l´Zwivq_M[cbrik^[ci ghacivMqO9N C;a]ikNB> Aika%[cKsOy*lkZcOiu&1iv_M^sxIO;xbr^f[cO9^sZwb[cbrO9Zz® ¢ ^ § pMp8O*^sxIbn[\£ iZwbrN0pMqO$lu_InIbrqbelua]Xa]O9Zc_Mq[]ZlkacO(pMacO;ZwO9^f[cO9x½­M[cKsOI<sazZ\[Siu)£KMbeyzKªbrZSxIO*viu[]O9x[ci[]KMO(p8ivZ]ZcbMbrqrb[\Xiu)lupspMaciGnIbrN3lo[cbrik^iuhy*ik^f[cbr^Y_Mik_1Z%_M^sy[cbrik^sZSik^ªN0O9lvZw_sacO;ZYXlk^sluqrXf[cbey_M^sy[cbrik^sZ

br^Zc_syzK3l£4lGX([cKslu[)[cKMbeZhlkpMpMa]iGnIbN3lo[]biv^$a]O9Zcp8O9y¡[zZ[cKsOxIO*a]brGlu[cbrik^ ¬ []Kslo[4xIiYO9Z)^siu[ ikqrqrio£a]ikN±[cKMOZw[]lk^sxMlka]xVf[civ^MO¤ ² O9bO9a]Zw[cazlkZ]Z [cKMO9ika]O*N iv^lkpMpMa]iGnIbN3lo[]biv^1®

âã&ä½âÖÞ

~

¢ [4brZ £4ikac[cK3^siu[cbr^M[]Kslo[)[]KMOivI[]lkb^sO9xqbrN0b[zZhikD¾s_sy¡[]_slo[]biv^3pMa]iYy*O9Z]ZwO;Z Zw_spMpMqrXl qelua]kOy*qrlvZcZ ik^1lo[c_sa]lkqO*nIlkN0pMqO;ZSikab^1<s^Mb[cO3xIbrNO9^sZcbiv^sluq RªO9KMqO9aZwO9N0bvaciv_MpsZ9­8£KsivZcO$lu^1luqrXYZcbeZbeZ£ O9qq_M^1xIO*a£%lGX3br^[]KMO pMa]O9ZcO*^f[qb[cO9a]lu[c_Ma]Ok­IZcO*O O > f Alu^sxa]OO*a]O*^1yO9Z []KMO*a]O*br^ikakO9^MO*azluq8[]KMO*ivacXv­ > 1CRAikaZcikN0OpMa]ikp8O*ac[cbrO9Z4iu lu_sZ]Zcbrlk^RªO9KMqrO*aZcO*N0bvaciv_MpsZ%lu^1x>DC;Aika%[cKsOy*ik^M^sO9y¡[]biv^£b[cKMazlu^1yzKMb^spsaciIyO;ZcZcO9Z4£b[cKbrNN0brkazlo[]biv^ !#" 2 ( 68: ) + &$) :12 .=2 68) 2

§ Z l£4lkacN0br^M_Mp £4O3ZwKslkqq xIbeZ]y_sZ]Z ZwKsikac[cqrX[cKMOy9lkZcO$ik X 8O*br^M[]KMO3ZwO*[ik ^slu[c_Mazluq ^Y_MN$1O9a]Zlk^sxª[]KMO0a K Ziu ¬ v®S8O*br^MlKsikN0ikvO*^MO9ik_sZSdv_1lkxIazlo[]bry$p1ivqXY^MivN0brlkq ¢ ^A[cKMbeZ y*lvZwOO9df_slu[cbrik^ ¬ v®[zluvO9Z [cKMOivacN iu[cKMOZcXIZ\[]O*N ikdf_slvxIa]lu[cbeyO;df_slo[]biv^sZ

xj = (Ajx, x), j = 1, 2, . . . , N, ¬ C;v®£KsO*a]O

NbeZ0lm^slu[c_Mazluq4^Y_MN(8O*a0iva

N = ∞ ­h_M^MY^Mio£^ x = (x1, x2, . . .)brZN¤¥kO;y¡[]ika$lk^sx lkqq

Aj lkacOkbrkO9^2Z]dv_1lua]O N × N¤¥N0lu[ca]bry*O9Z (¢ X beZ[]KMOZcO[ iu4^slo[]_Ma]lkq^f_sN(8O*azZlk^sx2ik^MqrXmMbr^slua]Xbr^f[cO9a]lvy¡[cbrik^1ZpMacO;ZwO9acYbr^M[cKMO^f_sN(8O*ahikpsluac[cbeyqrO9Z)lkacOlkqqrio£ O;x­uN0O9lvZw_sacO*¤>oluqr_MO;x$Ybr^MO[]bryO9df_slo[]biv^

¬ @v® beZacO;xI_sy*O9x[ci0[cKsOxIbrZ]ya]O[]O ZwXIZw[cO*N

xj =1

2

N∑

k=1

K∑

l=1

xkxl

∑

n≤m

Pnmkl (δj

n + δjm − δj

k − δjl ), j = 1, 2, . . . , N, ¬ C9f®

£KMO9acOPnm

kl

brZ%lk^lua]Mb[cazlua]Xy*ikqrqO;y¡[cbrik^iu½p8ivZcb[]bvO^Y_MN(8O*azZh[cKslu[4beZ%ZwXYN0N0O[ca]bey£b[cKa]O9Zcp8O9y¡[[]i3[cKsO(OnMyzKslk^MkO ik

nlu^sx

m ¬ xIO<s^Mbr^M3[]KMO(azlo[]O9Zik[cazlu^sZwika]N3lo[cbrik^ik lk^YXpslkbaSiu pslkaw[]bry*qO;Z jik [\XYp8Oklu^sx

l[ci[]KMOpslkba ik4pslkaw[]bry*qO;Z iu4[\XYp1O

mlu^sx

n®¡­lu^1x2£KsO*a]O

δjn

brZ [cKMO Ea]ik^sO9yzkO9aZcXYN(8ikq ¬ O9df_sluqeZCb j = n

lk^sx$ik[cKMO9ac£beZcO;® »h» Ð » P

N # ∑N

j=1 Aj = 0 = I =7 I/

F P L 1 ! 1I P#" "$% F ¬ C;k®& # R P R #% R 9 7 ' P P (

x0/)F +*P 1

x(t) 9L ,% =P-P,%. (

FR ! ' /)F $ /R0"1* ?,P2% 3 4jP R2

Aj F($ Aj 5%6 L

j)7 = ,

j) 8 /(P

(Ajv, v) ≥ 0 /%

(N − 1))F (

v /)F 9 F

":% ¬ C;k®6 ; < R F 6 R ¬ C9v® IP R == Pnm

kl

" ,% 1 R

Ajkl

/)F >P k 6= j

*l 6= j

* P,% P R2

Aj 1 P,% )F R ? j

² O$ivN0b[lZwbrN0pMqrO(pMa]iYiu iu []KMbrZÐlky[ S¢ [£%lkZpMacO;ZwO9^f[cO9x[]ivbvO(l0O*O9qbr^Miu [cKsO$xIbÀ8O9acO9^syO8O[\£4O*O9^[]KMO yqelkZ]Z ikp8ivZcb[]bYb[\XpMa]O9ZcO*a]fbr^M$O*kivq_M[cbrik^sZ4lu^sx[cKsOy*qrlvZcZ iu&Yb^MO*[cbeyO9df_slo[]biv^sZ ¬ C9f® J%KsbrZxIbÀDO*a]O*^sy*O £bqrq½8OZw[c_1xIbO;xZwXIZw[cO*N3lu[cbey*luqrqrXb^[]KMO ^MOnY[ZcO9y¡[]biv^

ääÉNPORQRSRT

C; L

!#" + .=6 .1& + ),- + : ) 4 6! 2 )J6 ) 2 (*& " :1+34 )J%Ksaciv_MkKMiv_I[[]KMOApslkp1O9a[]KMO.qO*[w[cO9a

XxIO*^Mik[cO;ZlqiIy*lkqqrX©yikN0pslvy¡[NO*[ca]bry2Zwp1lkyOv­lu^1x

C(X)¬ acO;Zwp8O9y[cbrkO9qX C∞(X)®xIO9^Miu[]O9Z3[]KMO H%lk^slkyzK Zcpslky*Omiu 8ik_M^1xIO9x y*ik^f[cbr^Y_Mik_1Z3_M^sy[cbrik^sZik^

X¬ acO;Zwp8O9y[cbrkO9qXb[zZ%yqrivZcO9xZc_MsZcpslky*Oyiv^sZcbrZw[cbr^M(ik_M^sy[cbrik^sZ4olu^MbeZcKMb^s(lo[%b^1<s^Mb[\XI®hO9df_MbrpMp8O9x3£b[cK[]KMOª^MivacN ‖f‖C(X) = supx∈X |f(x)| O*^Mik[cbr^M´YX X0 l iv^MO¤¥p1ivb^f[Zcpslky*Omlu^sx YX Xj [cKMOp8io£4O*azZX × · · · × X ¬ j ¤=[]bN0O9Zz®¡­4£4OZcKslkqqxIO9^Miu[]OYX X [cKMO9baxMbrZÐj\ikbr^f[_M^sbiv^ X =

⋃∞j=0 X

j­ £KsbryzK beZluvlkb^ lmqiIy*lkqqrX´yivNp1lky¡[ZwpslvyO J%KMOO*qrO*N0O*^f[]Z$iu X £bqrq41OxIO9Zcbrk^slu[cO9x YX 8ikqexqrO[c[cO9a]Z9­MO x,y %¢ ^sZw[cO;lkxiu X b[brZiuÖ[cO9^ªN0ika]Oyik^YvO*^MbrO*^f[%[ci3£4ika]£b[cKªb[]ZZcXYN0NO*[ca]bLK;lo[]biv^SX £KMbeyzKmbrZ[cKsO$df_Miu[]bO9^f[Zcpslky*O(ikM[]lubr^MO;xfXbexIO*^f[cbXYb^slu^YX[\£ ipslubrazZ x,y [cKslu[xIbÀDO*aik^MqrXYXl0p1O9acN$_I[]lu[cbrik^iu b[zZO*qrO*N0O9^v[zZ ² OZcKslkqqxMO*^Mik[cO YX Csym(X ) = C(SX ) ¬ a]O9Zcp Bsym(X )

®[]KMO H4lk^slkyzK ZwpslvyO;Z$ikSZcXfN0N0O[]acbeyyik^f[]b^Y_Miv_sZ ¬ acO;Zwp8O9y[cbrkO9qX´8ik_M^1xIO9x N0O9lkZc_MazluMqrO;® _M^sy[cbrik^sZiv^ X lk^sx´YX Csym(Xk) ¬ acO;Zwp Bsym(Xk)

® []KMOyika]a]O9Zcp1iv^sxIbr^MmZwp1lkyO;Z(ik_M^sy[cbrik^sZik^[cKMO<1^Mb[]OSp8io£4O*azZ

Xk M(X) ¬ acO;Zwp8O9y[cbrkO*qrX Msym(X )®4xIO*^Mik[cO;Z)[]KMO9H4lk^slkyzKZcpslvyOik;<s^Mb[cOH4ika]O*q

N0O;lkZc_MacO;Z ik^X ¬ acO;Zwp8O9y[cbrkO9qX0ZwXYN0N0O[]acbeyN0O9lvZw_Ma]O9Z)ik^ X ®)£b[]K3[cKsO_Mqrq1olkacbelo[]biv^lkZ []KMO^sika]N H4X$£ O;lu[]ikp8ikqrikvX(br^0[cKMOZwpslvyOikDN0O;lkZc_MacO;Z £4OZwKslkqq8luqr£4lGXIZ_M^1xIO*azZ\[zlu^sx[]KMO ∗ ¤>£4O9lk [civp1ivqivkX J%KsO$_Mpsp1O9aZw_M1Zcy*acbrpI[##ivalkqq&[]KMO9ZcOZcpslvyO9Z ¬ O M+(X)

®£bqrq)xIO*^siu[cO$[cKsO0y*ika]acO;Zwp8ik^1xIb^sy*ik^MO;Z4iu^sik^I¤¥^MO*flo[]bvOO*qrO*N0O9^v[zZ J%KMOqrio£ O9a4Zc_MsZ]ya]brpI[F<s^ ika%[]KMO ZwpslvyO;Z ik_M^1y¡[cbrik^1Z%ik^ X ­O Csym

fin (X )­s£bqrqxIO*^Mik[cOl$Zc_MsZcpslvyO£b[]Kiv^MqX <s^sb[]O^Y_MN$1O9a%iu^Miv^I¤¥Glk^MbeZwKMbr^M0yivNp8ik^sO*^f[]Z

¬ p1ivqXY^MivN0brlkqD_M^sy¡[]biv^sluqeZz® ² OZcKsluqrqM£4ika]N0ivZw[cqrX$£b[cK X = Rd iva X 8O*br^M(l y*qifZwO;xN0lk^Mbikqex ¢ ^0[]KMbeZhy*lvZwOYX Ck(X)£4O(ZcKsluqrqxMO*^Mik[cO []KMOH%lu^1lkyzKªZwp1lkyO(iu 1iv_M^sxIO;x

k[]bN0O9ZSyiv^v[]b^Y_Miv_sZcqXxIbÀDO*a]O*^f[]brlkMqO_M^sy¡[]biv^sZ

£b[cK8ik_M^1xIO9xxIO9acbrolo[cbrkO;Z4O9df_MbrpMp1O;x£b[]K[]KMO ^MivacN

‖f‖Ck(X) =

k∑

l=0

‖f (l)‖C(X).

y*ik_MazZcO C(X) = C0(X)lk^sx

Ckc (X) ¬ acO;Zwp8O9y¡[]bvO*qrX Ck

∞(X)®)xIO9^Miu[]O9Z []KMOSZc_MsZcpslky*Oiu

Ck(X)y*ik^sZcbeZ\[]b^Miu _M^1y¡[cbrik^1ZS£b[cK.ly*ikN0pslvy¡[Zc_MpMp8ikac[ ¬ acO;Zwp8O9y[cbrkO9qXolu^MbeZcKMb^slo[br^=<s^Mb[\X[civkO[]KMO*a£b[cKªlkqqD[]KMO*braxIO9acbrolo[cbrkO;Z4_Mp[ci3lu^sxbr^syqr_sxIbr^M[cKsO ikazxIO*ak® H4X

L1(X)£ OxIO9^Miu[]O[cKMO _sZc_slkq;H4lk^slkyzKZcpslky*Oik&br^f[cO9kazluMqrO_s^sy¡[]biv^sZ*­8lu^sx

W k(X)beZ%[cKMO

VIik8ikqrO*Zcpslky*OiuN0O;lkZc_Ma]lkMqrO_M^1y¡[cbrik^1ZKslGYb^s <s^sb[]O^Mika]N

‖f‖W k(X) =k∑

t=0

‖f (t)‖L1(X),

£KsO*a]O([]KMO3xIO*a]bolu[cbrkO9ZlkacO0xIO <1^MO9x2b^2[cKsO£4O9lkmZwO9^sZcO $§ [ qelkZw[9­Dival?H%lu^1lkyzK2ZcpslvyO B lk^sxAlp8ivZcb[cbrkOh^Y_MN$1O9at0­o£ O4ZwK1luqrqvxIO9^Miu[]O YX

C([0, t0], B)[cKMO H%lu^slvyzKZwpslvyO)ik1yiv^f[cbr^f_sik_sZ½_M^sy¡[]biv^sZ

a]ikN[0, t0]

[]iBO9df_MbrpMp8O9x£b[]K[cKMO ^MivacN

‖ft‖C([0,t0],B) = supt∈[0,t0]

‖ft‖B .

H4X3lpMa]ikp1luvlu[civa ¬ a]O9Zcp1O;y¡[]bvO*qrX0l(slkyzY£%luazxpMacivpsluflo[]ika¡®&b^l ÐluN0brqX0ikZwO*[]Z Ms­s ≥ 0

­f£ OZcKslkqq1NO;lu^lÐluN0brqX

U(t, s)­t ≥ s ≥ 0

iu[cazlu^1Z\ivacN3lo[]biv^sZMs 7→Mt

¬ a]O9Zcp1O;y¡[]bvO*qrX Mt 7→ Ms®

âã&ä½âÖÞ

C C

Z]lo[]brZwXYb^sS[]KMOyik¤¥y*XIyqrO4bexIO9^v[]b[\XU(t, s)U(s, r) = U(t, r) ¬ a]O9Zcp1O;y¡[cbrkO9qX U(s, r)U(t, s) = U(t, r)

®iva

t ≥ s ≥ rlu^sx©Zc_syzK[]Kslo[

U(t, t)beZ[cKsOmbrxMO*^f[cb[\X ikp8O*azlo[]ikaikalu^YX

t ¨¢ ^©psluac[cbey_sqrlka9­ P FJ' #%2br^µl ÐluN0brqX¨ik[cKMO,H4ika]O*qZw_ssZwO*[]ZXtiuX ¬ ZwKsikac[cqrXk­br^ X ®beZlpsacivpsluflo[civa$ikSqrb^sO9luay*ik^f[cazlky[cbrik^sZ

V (t, s) : M+(Xs) 7→ M+(Xt) J%KsOmlkxGj\ivb^f[3ikp8O*azlo[]ikazZU(t, s) = V ∗(t, s)

lky[slkyzY£%luazxMZ acivNB(Xt)

[ciB(Xs)

¬ lu^sxivacNl0slkyzY£%luazxpsacivpsluflo[civaz®­[]KMOxI_slkqb[\XO;dv_1lo[cbrik^8O*br^M

(U(t, s)ft, µs) = (ft, V (t, s)µs)

iva_M^sy[cbrik^sZft ∈ B(Xt)

lk^sx2NO;lkZc_Ma]O9Zµs ∈ M+(Xs) 3² OZclGX[]Kslo[(Zc_syzK lpMacivpsluflo[]ikabrZ¦sO*qrqO9ab [cKMO ÐlkN0bqrX

U(t, s)beZlZw[ca]ik^MvqXyik^f[]b^Y_Miv_sZ%ÐluN0brqXiu ikp8O*azlo[]ikazZ

C∞(Xt) 7→ C∞(Xs)¬ b U(t, s)xIO9p1O9^sxik^MqrXiv^[cKMO(xMbÀDO*a]O*^sy*O

t− s­M[]KMO9ZcOivp1O9a]lu[cikazZ ika]N l¦sO*qrqO9aZcO*N0bvaciv_Mp1® §RAlua]kioªpMa]iIyO9Z]ZZcp1O;ybL<sO9x YX

U(t, s)­b O YXA[]KMO3[cazlu^1Zwb[cbrik^ pMa]ikslkMbrqb[cbrO9Z P (t, s, x, dy)

a]ikNXs[ciXtxIO<s^MO;xYX

Et,sx ft =

∫

ft(y)P (t, s, x, dy) = (U(t, s)ft)(x),

£brqrq8O[cKsO*^Ay9luqrqO;xl^Mik^MKsikN0ikvO*^MO9ik_sZ¦sO*qrqO9apMa]iIyO9Z]Z ¬ £KMO*a]O£4O_sZcO [cKMOZ\[zlu^sxsluazx^siu[]lu[cbrik^sZExika%[]KMO OnIp8O9y¡[zlo[]biv^ik[]KMO pMa]iYy*O9Z]Z%Z\[zluac[cbr^M3lo[

x®

½#&'h# ©%& # · »hVI_MpMp8ivZcO

DbrZlqrbr^MO9lkaKY_MqrqIiuslxIO*^1ZwO%Zw_ssZwp1lkyO b^

C∞(X)lu^1x [cKsO4ZcO[iu1y*ik^sZw[]lk^v[

_s^sy¡[]biv^sZ § N0lkpMpMbr^M ΩacivNlxIO*^sZcO(Zc_MsZcpslky*O iu M(X)

y*ik^f[]lkb^sb^M <1^Mb[]O(qrb^sO9luaSyikN$Mbr^slo¤[]biv^sZiu bra]lvyN0O9lvZw_Ma]O9Z[ci ¬ p8ivZ]ZcbMqrXª_M^Y8ik_M^sxMO9xs®qbr^MO;lua_M^sy[cbrik^slkqrZSb^ C(X)

£b[cK xIikN3lubr^sZy*ik^f[]lkb^sb^M

DbeZy9luqrqO;x

x = (x1, . . . , xn)¤+ ,%IP R ival3kbrkO9^ <s^Mb[cO$y*ikqrqO;y¡[]biv^ik

O9qO9N0O*^f[]ZxiuX­ub

Ω(∑n

i=1 ωiδxi)gbeZ^Mik^I¤¥^MO9vlo[]bvOhiva lua]Mb[]a]lkacXp8ivZcb[]bvO%^f_sN(8O*azZ

ω1, . . . , ωn£KsO*^MO9kO*ag ∈ D

brZ%^Miv^I¤¥^MO*flo[cbrkOlu^1xolu^sbrZcKMO9Zlu[p1ivb^f[]Zx1, . . . , xn Ω brZy9luqrqO;x ,%P R bb[brZ

x¤+ ,%IP R ikalkqq

x ∈ X ¢ ^pslkaw[]bry*_Mqrlka9­IlkpMpMqrXfbr^M[cKsbrZxMO <s^Mb[cbrik^[]ip8ikqrXY^MikN0beluqivp1O9a]lu[cikazZ Ω ¬ ZwO9O[cKMO a K Z iuO9df_slo¤[]biv^ ¬ v®c®4£ OZ]lGX0[cKslu[lqbr^MO9lkaN3lup

A = (A1, . . . , AK) : D 7→ (C(X), Csym(X2), . . . , Csym(XK)) ¬ C;k®beZ. 2% 1 bivalk^YX y*ikqrqO;y¡[]biv^ iuSxMbÀDO*a]O*^f[$p8ikbr^f[]Z

x1, . . . , xmiuX­ ika0lu^YX

^sik^I¤¥^MO*flo[]bvO_M^sy[cbrik^g ∈ D

Zw_1yzK[cK1lo[g(xj) = 0

ivaluqrqj = 1, . . . ,m

­Mlu^sxikalu^YXyivqqrO9y[cbrik^ikp8ivZcb[]bvO^f_sN(8O*azZ

ωjik^MOKslvZK∑

k=1

m∑

i1=1

· · ·m∑

ik=1

ωi1 · · ·ωikAkg(xi1 , . . . xik

) ≥ 0. ¬ C;v®

Rmika]O*iokO9a9­£ OZwK1luqrq4Z]lGXA[]Kslo[AbeZ ,% ,%?P R b

Akg(x1, . . . , xk)beZ

x =(x1, . . . , xk)

¤¥y*ik^sxMb[]biv^sluqrqX.p8ivZcb[]bvOika0lu^YXklu^1x

x ¦br^slkqqrXk­)lAqbr^MO9lka$ivp1O9a]lu[civa Ak : D 7→

ääÉNPORQRSRT

CG L

Csym(Xk)beZZ]lubex[]i31O 2%< 2%IP R ika9 ,%I1 b[]KMbeZbeZ%[cKMO

y9lkZcOiva%[cKMOÐlkN0bqrXA = (0, . . . , 0, Ak)

iu[\Xfp8O ¬ CGu® / ¢ Ω beZqrbr^MO9lka9­8b O Ω(µ)g = (Ag, µ)

£b[cK2ZcikN0Oqrbr^MO9lkaSivp1O9a]lu[cikaAbr^C(X)

­D[cKMOlk1iovOxIO <1^Mb[]biv^iu½y*ik^sxIb[cbrik^1luqsp8ivZcb[cbrfb[\XbrZhacO;xI_syO;x[ci []KMOSZw[]lk^sxMlka]x3xIO<s^Mb[cbrik^3ikyik^1xIb[]biv^sluqp8ivZcb[cbrfb[\Xik[]KMO qbr^MO;lua%ikp8O*azlo[]ika

A ¬ ZcO*O br^f[ca]iYxM_sy¡[]biv^1® _MaxMO <s^Mb[cbrik^1Zlua]OSN0ik[cbrGlu[cO;xYX3[]KMOikqrqrio£b^M3ZcbN0pMqrOiksZcO*a]olo[]biv^sZ

»h» Ð » " 1 1 ¬ v®I ( #L D 2% 1 P / R= ?1R P F P ' : )F = R P ¬ f®$1 g ∈ D

* 1Ω ,%9P R

"$ 1 3 P BkC∞(Xk)

*k = 1, . . . ,K

* 3 ,% P R 3 P RP 1 = P( (% (-P ¬ C;k® " 1 P 1

g 7→ Bkg+

C∞(X) 1 R/ P ! $" FC∞(Xk)

/ ,% ,% 1 1 g+ / ( ¬ @f® "

' ¬ bÖ®ÁO[ l^Miv^I¤¥^MO*flo[cbrkO g ∈ D8O%Zw_syzK[cKslu[

g(x1) = · · · = g(xm) = 0ivaZcikN0Ohp8ikbr^f[]Z

x1, . . . , xm KsififZwbr^M µ0 = ω1δx1+ · · · + ωmδxm

£ O(ZcO*O[cKslu[(g, µ0) = 0

­8lk^sxªyiv^sZwO;df_MO*^f[cqrX[]KMOªyiv^sxIb[cbrik^ikp8ivZcb[]bYb[\X´pMacO;ZwO9acolu[cbrik^ brN0pMqbrO9Z$[]Kslo[ d

dt (g, µt) |t=0≥ 0­h£KsbryzK NO;lu^sZ$[]Kslo[

Ω(∑n

i=1 ωiδxi)gbrZ^sik^I¤¥^MO*flo[]bvO Vf[]lu[cO*N0O9^v[ ¬ bbÖ®%beZ%ikYYbiv_sZ

/ )J%KY_sZ ik^[cKMOSika]N3luqsqrO*vO*q>­f[cKMOxIbÀ8O9acO9^syO1O*[\£ O9O*^3[]KMOkO*^sO*azluqsp8ivZcb[cbrfb[\X3pMa]O9ZcO*a]f¤br^M$O*kivq_M[cbrik^sZhiu½ivacN ¬ f®)lk^sx3[]KMOSYbr^MO[]bryO9df_slu[cbrik^sZ iu k ¤¿lua]Xb^f[cO9a]lvy¡[]b^M$psluac[cbeyqrO9Z ¬ C;k®)beZ [cKMOxMbÀDO*a]O*^sy*O8O[\£4O*O*^y*ik^sxMb[]biv^sluqrqX$p1ifZwb[cbrkOSlu^sx3Zw[ca]ik^MvqX$yik^1xIb[]biv^sluqrqrX$p8ivZcb[]bvOkO9^MO*azlo[]ikazZ ¦Mika[]KMOy*lvZwOiu xIbeZcy*acO*[cO

X[cKMbeZik1ZwO9acolo[]biv^beZ%bqrqr_sZ\[]a]lu[cO;xYXghacivp1ifZwb[cbrik^ C C

_Ma <1a]Zw[Z\[]ac_sy[c_Mazluqa]O9Zc_Mq[yzKsluazlky[cO*a]bDK*O9Zh[cKsOyik^1xIb[]biv^sluqp8ivZcb[cbrfb[\Xlu[ <MnIO9xp1ivb^f[zZ ·Y» · P

X = Rd = /1 P D P #

F1C2

c (X) I 1 9

x = (x1, . . . , xn))' ,% 1 *1

Ω

(

n∑

i=1

ωiδxi

)

g =

n∑

j=1

[

aj(x)g(xj) + (bj(x),∇)g(xj) +1

2(cj(x)∇,∇)g(xj )

+

∫

(g(xj + y) − g(xj) − χ(y)(y,∇)g(xj))νj(x; dy)

] ¬ C u®

g ∈ C2

c (X)*>P / 4

cj IP R P F / R2* (

νj %? = *χ

FIL ¬ C9®+* aj , bj , cj , νj P ∑ni=1 ωiδxi

' 0Á½O[ _sZZw[]lkaw[£b[]K.lyivN0NO9^f[iv^m[cKsO iv_Ma]acL9kO []KMO*ivacO9N $§ qriYikik^A[cKMO0pMa]iYiuhikKsbrZ[cKMO9ika]O*N ¬ ZwO9O >DC;A¥­ > ~A>­ >Ï@ A®ZcKMio£Z[cK1lo[([]KMOyzKslka]lvy¡[]O*a]bLK;lo[cbrik^.beZ(lky[c_slkqqrXAvbvO*^ ^Mik[(ik^MqrXivayiv^sxIb[cbrik^slkqqrXp8ivZcb[cbrkOªikp8O*azlo[]ikazZ*­ M_I[lkqrZci ikayik^1xIb[]biv^sluqrqrX p1ifZwb[cbrkOªqrbr^MO9lka3_M^sy[cbrik^slkqrZ

ivI[]lkb^sO9x0YX <snYbr^M []KMOlkacv_MN0O*^f[]Z PSluN0O*qrXk­fb[hbrZ4ZcKMio£^[cKslu[ b[cKMOqrbr^MO9lka)_M^sy[cbrik^slkq (Ag)(x) :C2

c 7→ Rd brZ$ 2%7P R 4lu[ x ­Yb O b Ag(x) ≥ 0£KMO9^MO*vO*ahl^Miv^I¤¥^MO*flo[cbrkO

gGlk^MbeZwKMO;Z

lu[x­[cKsO*^

Ag(x®SKslvZivacN ¬ C;®Sbraca]O9Zcp8O9y¡[]bvO*qrXiu)[]KMO0pMacivp1O9aw[]bO;Ziu Ag(y) b^2iu[]KMO*ap8ikbr^v[zZ y

âã&ä½âÖÞ

C;

Pio£ qrO[ _sZyzKMiYivZcO$lpsluac[cb[cbrik^2iu4_M^Mb[\XmlkZlÐlkN0bqrXmiunZcNiYik[cKA^sik^I¤¥^MO*flo[]bvO _M^sy[cbrik^sZ

χi­

i = 1, . . . , n­Zc_syzKA[cKslu[ ∑n

i=1 χi = 1lu^sx2O9lvyzK

χiO9df_slkqrZik^sO$br^´l^MO9bvKY1ivacKMiYiIxmik

xi¬ lu^1xy*ik^sZcO9df_MO9^f[cqrX2olu^sbrZcKMO9Z$br^lm^MO9bvKY1ivacKMiYiIx iulk^YX2ik[cKMO9a$p8ikbr^f[

xl£b[]K

l 6= i® H4X qbr^MO9lkacb[\X

Ω(∑n

i=1 ωiδxi) =

∑ni=1 Ωi

­%£b[]KΩjg = (

∑ni=1 ωiδxi

)(χjg) qrO9lkacqrX O;lkyzK _M^sy¡[]biv^sluq ΩigbeZy*ik^sxIb[cbrik^1luqrqX p1ifZwb[cbrkOAlu[xib^©[]KMOm_sZc_slkqZwO9^sZcOk­b O g(xi) = 0

brN0pMqbrO9ZΩig ≥ 0

ikal^sik^A^sO*vlu[cbrkO(_M^sy¡[]biv^

g ∈ C2c (X) MO*^sy*O3lupMpMqrXYb^Ml <MnIO9xAp1ivb^f[ kO9a]Zcbrik^ iuh[cKMO ik_saca]L*vO[]KMO*ivacO9N [ci0O;lkyzK

Ωi­Iiv^MOikI[zlubr^sZ%acO9pMa]O9ZcO*^f[]lu[cbrik^ ¬ C o®

§ Zhl yivacivqqelua]X£ OZcKslkqqMivI[]lkb^0^Mio£©l yzKslka]lvy¡[cO9acbDK9lu[cbrik^ikD[cKMOZw[ca]ik^M y*ik^sxIb[cbrik^1luqMp8ivZcb[]bYb[\X ·Y» ·

X

D P/ 9 1

" 1 Ak : D 7→ Csym(Xk)

,%< ,%I1R ,%/

Akg(x1, . . . , xk) =

k∑

j=1

[akΠj1(x1, . . . , xk))g(xj)

+ (bkΠj1(x1, . . . , xk)),∇)g(xj) +1

2(ckΠj1(x1, . . . , xk))∇,∇)g(xj)

+

∫

(g(xj + y)g(xi) − χ(y)(y,∇)g(xj))νk(Πj1(x1, . . . , xk); dy)], ¬ C @v®>1 P? 1

Πjl

Rd (1 1 FR JP= j

l* 4

ck P( ,% νj " I 1R 9 P F9/ R2 1( 2%/ %// R=!"1* χ I L ¬ C;® * >P 1 = 1 ak, bk, ck, νk

(% / R R P( 1 P R F 1! #1 R DR

x2, . . . , xk P R 13 4 x1"

" 1 FAk : D 7→ Csym(Xk)

,% 2%9P R I?,%/

Akg(x1, . . . , xk) =k∑

j=1

ak(Πj1(x1, . . . , xk))g(xj) +Bkg+(x1, . . . , xk), ¬ C9~v®

>1ak

1 / ¬ C k®? Bk 7 ,% P R RP P !"

P FICsym

∞ (Xk)

"; 1 P= #% ! 1 = 1 1 / ' A L 8 1 1 = 1

ak, bk, ck, νkF1 PP R,%9

x1, . . . , xk / R= νk 3 P 49 P

1L %(" ' ¬ be®%¦sacivN J%KMO9ika]O*N Cb[ikqrqio£Z [cK1lo[ikalkacsb[]a]lkacX3xIbÀ8O9acO9^f[ x1, . . . , xk

Akg(x1, . . . , xk) =k∑

j=1

[ajk(x1, . . . , xk)g(xj)

+ (bjk(x1, . . . , xk),∇)g(xj) +1

2(cjk(x1, . . . , xk)∇,∇, )g(xj)

+

∫

(g(xj + y) − g(xj) − χ(y)(y,∇)g(xj))νjk(x1, . . . , xk; dy)],

ääÉNPORQRSRT

C9 L

£KsO*a]OO9lvyzKcjkbeZ l p8ivZcb[cbrkOSxMO <s^Mb[cOSN3lo[ca]bnD­fO9lvyzK

νjk

beZ l$Á½`9fXN0O9lvZw_Ma]O RªivacO9iokO*a;­kxI_sO[ci([cKMOZcXYN0NO*[ca]Xiu_M^1y¡[cbrik^1ZAkgika

g ∈ C2c (X)­M[]KMO_M^sy¡[]biv^sZ

ajk, b

jk, c

jk, ν

jk

xMO*p8O*^sxik^x1, . . . , xkZcXYN0NO*[ca]bry9luqrqX3br^[cKMO ZcO*^sZcO[cKslu[9­sZ]lGX

ajΠjl = al ikalkqq j, l lk^sx ajΠlm = aj ika m 6= j­l 6= j

_MO0[]i[]KMbeZZwXYN0N0O[]acX2b[beZp1ifZcZcbrMqO0[]iacO9£acb[cO[cKsOlu8iokO$ika]N(_MqellkZ ¬ C@k® C0£b[]K ak = a1k

­bk = b1k

­ck = c1k

­νk = ν1

k ¢ [ a]O*N3lubr^sZ[ci^Miu[]bry*Oh[]Kslo[)xI_sO []i[cKMOyiv^f[cbr^f_sb[\Xv­G[cKMbeZacO9pMa]O9ZcO*^f[]lu[cbrik^a]O*N3lkb^sZ4oluqrbexO9kO9^iva^Miu[^MO;yO;ZcZ]lua]bqrXxIbÀ8O9acO9^f[%p8ikbr^f[]Zx1, . . . , xk ¬ bbÖ®#H4X$[cKMO iv_Ma]acL9kO4[cKMO9ika]O*N lupMpsqbrO9x[]i Xk ­fl y*ik^sxMb[]biv^sluqrqX$p1ifZwb[cbrkO Bk

br^C∞(Xk)

KslkZivacN

Bkf(x1, . . . , xk) = a(x1, . . . , xk)f(x1, . . . , xk)

+

k∑

j=1

(bj(x1, . . . , xk),∇xj)f(x1, . . . , xk)) +

1

2(c(x1, . . . , xk)∇,∇)f(x1, . . . , xk)

+

∫

(f(x1 + y1, . . . , xk + yk) − f(x1, . . . , xk) −k∑

i=1

(χ(yi)(yi,∇xi)f(x1, . . . , xk))

× ν(x1, . . . , xk; dy1 · · · dyk). ¬ uv®

§ pMpMqrXYb^M[]KMbrZ[ci f = g+ lu^1xy*ikN0pslkacbr^M£b[]K ¬ C o®4XYbrO*qexMZ4[cKsO acO;dv_sba]O9xacO;Zw_sq[ ¬ bbrbe® H4XSiv^MOhlk^sx[cKsO Z]luN0O)pMa]iIyO9xM_MacOv­9iv^MO)pMa]iokO;Z <sazZ\[[cKMO%yiv^v[]b^Y_Mb[\XSikaxMbÀDO*a]O*^f[ x1, . . . , xk­S[cKsO*^¯ika[\£ i iu$[cKMO9N yivb^sy*brxMb^M1­O*[]y ¢ lkqq x1, . . . , xklua]O2xIbÀDO*a]O*^f[9­ik^MO y*lk^ _sZcO.[cKMO

xMO9yivN0p1ifZwb[cbrik^ a]ikN°[]KMO4pMaciYikMiu1J%KMO9ika]O*N Cha]O9xM_sybr^M[]KMO%pMa]ikMqrO*Nµ[cipMa]ioYb^M[cKMOyiv^f[cbr^f_sb[\Xiva&[cKMOivp1O9a]lu[cikaAk(χjg)(x1, . . . , xk) KsififZwbr^M g [ci1Olyiv^sZw[]lu^f[;­oik^MO%pMa]iokO;Z½[]KMOyiv^v[]b^Y_Mb[\Xik½[cKsOy*iYO GybrO*^f[]Zakiva []KMbeZ ivp1O9a]lu[civa J%KMO*^yzKMiYivZcb^s g [ci1O lyiv^sZ\[zlu^f[ br^l$^MO9bvKf8ika]KMiYiIxikhlyO9aw[zlubr^ªp1ivb^f[

xik^sO(pMa]iokO;Z%[cKMOyiv^v[]b^Y_Mb[\Xiu

νk J%KMO*^AyzKsififZwbr^M[ci8O(qrbr^MO9lkaSlua]ik_s^sxxiv^MOkO*[]Z4[cKsOyik^f[]b^Y_Mb[\Xiu

bk­Mlu^sxlu[%qelkZw[[cKsOy*ik^f[cbr^Y_Mb[\Xiu

ckikqrqio£Z J%KY_sZ ivp1O9a]lu[civa]Z

Akb^ ¬ Y® lua]OZ\[]aciv^MkqrX(y*ik^sxIb[cbrik^1luqrqX$p1ifZwb[cbrkOb½lu^sxik^sqX$bD[cKsO*XKslGkO4ika]N

Akg = Bkg+/k!

£b[]KmZwivNOy*ik^sxMb[]biv^sluqrqXp1ifZwb[cbrkOBkbr^C∞(X

® ¬ _Mpª[]iZcikN0O N(_Mq[cbrpMqrbry9lo[]biv^ivp1O9a]lu[civa]Zz®lk^sx([cKY_sZ bD[cKMO9X$yivaca]O9Zcp8ik^sx[ci lYb^MO*[cbey%O9df_slu[cbrik^$ika)ZcikN0ORmlua]kio N0iIxIO*qsiuk¤¥lkacX

br^f[cO9a]lvy¡[cbr^Mpsluac[cbeyqrO9Z ·Y» · P P

X = Rd D

FPC2

c (X) ¬ C;k®

,%P R P,% I ( ! 1I 1 Ak

1 ? R ¬ C@v®$ 1 / % %. ¬ C @k® ck

νk 3( R R2%

P R = P,% R/( 1 19/ R2K∑

k=1

k

m∑

i1=1

· · ·m∑

ik−1=1

ωi1 · · ·ωik−1ck(x, xi1 , . . . , xik−1

) ¬ =C;®

P R = # 1 = K∑

k=1

k

m∑

i1=1

· · ·m∑

ik−1=1

ωi1 · · ·ωik−1νk(x, xi1 , . . . , xik−1

) ¬ kk®

âã&ä½âÖÞ

CG

P R > 7%m* /% L( 17! P R 7P=

ω1, . . . , ωm P

x, x1, . . . , xm νk

(% ¬ k® * ! = !" R P =L(* AkP3 R ¬ C9~f®= Bk

5%¬ uk®9 4 R,% P R c ν ' · ; MO*a]O4£4OZwK1luqrqIpMaciovOh[]Kslo[hlkqq Ak

KslGkO ika]N ¬ C u®&£b[cK n = k­uM_I[)p8ivZ]ZwbrMqrX

£b[cKsik_I[4p1ifZwb[cbrYb[\X0iucklu^sx

νk ¦Miva4lkacMb[cazlua]XI<snYO;x x1, . . . , xm[cKMOS_M^sy[cbrik^slkq1ik

gvbvO*^3YX

K∑

k=1

m∑

i1=1

· · ·m∑

ik=1

ωi1 · · ·ωikAkg(xi1 , . . . , xik

) ¬ uv®

beZx = (x1, . . . , xm)

¤%yiv^sxIb[cbrik^slkqqrXp8ivZcb[]bvOlk^sxyiv^sZwO;df_MO*^f[cqrXKslvZ%ika]N ¬ C o® ¬ £b[]K m br^sZ\[]O9lvxikn®­slkZikqrqio£Z4a]ikN J%KsO*ika]O*N C QSZwbr^M εωj

b^1Z\[]O9lkxiuωj­8xIbrfbexIbr^M3YX

εlk^sx[cKMO9^ªqO*[w[]b^M

ε → 0iv^MO3ikM[]lubr^sZ[]Kslo[ ∑m

i=1A1g(xi)KslvZ[cKsOZclkNO0ivacN § Z m beZlkacMb[cazlua]Xk­D[]KMbeZbrNpsqbrO9Ziv^A[]KMO0ik^MO0Kslk^sxA[cKslu[

A1g(x)KslkZ[]KMO3ZclkNOivacN ivalua]Mb[cazlua]X

x ¬ [cKY_sZ vbYbr^M[]KMO3a]O9df_MbracO;xZw[ca]_sy[c_MazluqDa]O9Zc_Mq[ivaA1®%lk^sxiv^[cKMOiu[]KMO*aK1lu^sx[cK1lo[

K∑

k=2

εk−2m∑

i1=1

· · ·m∑

ik=1

ωi1 · · ·ωikAkg(xi1 , . . . , xik

)

K1lkZ []KMOa]O9df_Mbra]O9xika]N § vlkb^3qrO[c[cbr^M ε→ 0XYbO9qrxsZ)[]KMOSZ]luN0Oa]O*pMa]O9ZcO*^f[zlo[cbrik^ivah[]KMO_M^1y¡[cbrik^1luq

m∑

i1=1

m∑

i2=1

ωi1ωi2A2g(xi1 , xi2).

§ Zlu8iokO[cKsbrZ%brN0pMqbrO9Zik^[cKMO iv^MO Kslu^1x[]Kslo[ω2

1A2g(x1, x1) + 2ω1ω2A2g(x1, x2) + ω22A2g(x2, x2)

K1lkZh[cKMOacO;df_Mba]O9xivacN lk^sx3KMO9^syOluqeZciA2g(x1, x1)

KslkZh[cKMbeZ ika]N ika%lua]Mb[]a]lkacXx1¬ ps_I[ ω2 = 0br^[cKMOªpsacO9fbrik_1Z$O*nIpMacO;ZcZcbrik^1®­ lu^1x KsO*^sy*OmluqeZwi

A2g(x1, x2)KslvZ[cKMOmZ]luN0Oika]N ivalua]Mb[]a]lkacX

x1, x2¬ [cKY_sZvbYbr^M([]KMO a]O9df_MbracO;xZw[ca]_sy¡[]_MazluqDacO;Zw_sq[iva A2

®­slu^1xiv^[]KMO iu[]KMO*aK1lu^sx[cKslu[K∑

k=3

εk−3m∑

i1=1

· · ·m∑

ik=1

ωi1 · · ·ωikAkg(xi1 , . . . , xik

)

K1lkZ4[cKsO acO;dv_sba]O9xika]N ¦sikqrqio£br^M$[cKMbeZpMa]iIyO;xI_Ma]OSbr^sxI_sy[cbrkO9qXXfbrO*qexMZ4[]KMOyqelubrN ikVf[]O*p,C · ¦Ma]ikN []KMO ikI[zlubr^MO9xa]O*pMa]O9ZcO*^f[]lu[cbrik^ika Ak

b[ikqrqio£Z []Kslo[K∑

k=1

m∑

i1=1

· · ·m∑

ik=1

ωi1 · · ·ωikAkg(xi1 , . . . , xik

) =

K∑

k=1

k

K∑

l=1

m∑

i1=1

· · ·m∑

ik−1=1

ωlωi1 · · ·ωik−1

[

ak(xl, xi1 , . . . , xik−1)g(xl) + (bk(xl, xi1 , . . . , xik−1

),∇)g(xl) +1

2(ck(xl, xi1 , . . . , xik−1

)∇,∇)g(xl)

+

∫

(g(xj + y) − g(xj) − χ(y)(y,∇)g(xj))νk(xl, xi1 , . . . , xik−1; dy)

]

.

ääÉNPORQRSRT

C; L

§ Z)[]KMbeZh_s^sy¡[]biv^sluq8KslkZh[ci$1OZw[ca]ik^MvqX0yiv^sxIb[cbrik^slkqqrX0p1ifZwb[cbrkOv­k[]KMOSa]O9df_Mbra]O9x0p8ivZcb[]bYb[\Xpsacivp1O9aw[\Xikclu^1x

νikqrqio£Z4a]ikN J%KMO*ivacO9N C

» » ÐI ¸ P9 K = 2

19 P9' 11R % / P R #%3 ¬ C;k® ' )¢ ^ªy*lvZwO K = 2

[cKsO p1ifZwb[cbrYb[\Xik ¬ 1C9®4acO;lkxMZlvZ [cKMO p8ivZcb[]bYb[\Xik[cKsO N0lu[ca]bn

c1(x) + 2

m∑

i=1

ωic2(x, xi)¬ of®

iva luqrq ^slo[]_Ma]lkqm­½p8ivZcb[cbrkO0^Y_MN(8O*azZ

ωjlk^sx2p1ivb^f[]Z

x, xj­j = 1, . . . ,m MO*^sy*O c1 brZ lkq£%lGXIZp8ivZcb[cbrkO ¬ pM_I[ ωj = 0

ika%luqrqj® JipMa]iokOZw[ca]ik^M$yik^1xIb[]biv^sluq8p8ivZcb[]bYb[\X3ik^MOSKslkZh[cipMa]iokO[]Kslo[

c(x, y)beZ%p8ivZcb[]bvOxMO <s^Mb[cOivaluqrq

x, y H4_I[b[]KMO*a]OOnIbrZw[ x, y Zw_1yzK[]Kslo[ c(x, y) beZ%^Miu[p8ivZcb[]bvOxMO <s^Mb[cOv­I[cKsO*^fXyzKMiYivZcb^Mqelua]kOO9^Mik_skK^f_sN(8O*a%iup1ivb^f[zZx1, . . . , xm

^MO9lkay­Iik^sO£4ik_sqrxkO[

l0N3lu[ca]bniu&ika]N ¬ ov®[cKslu[beZ^Mik[p1ifZwb[cbrkO(xMO <s^Mb[cO ¬ O*vO*^ivaluqrq ωj = 1® J%KMbeZyiv^f[cazlkxIbey¡[]biv^y*ikN0pMqrO[]O9Z4[cKsO pMaciYik J&luYbr^M$br^f[cilky*y*ik_M^f[%OnIpMqrbry*b[]qX0[]KMOZwXYN0N0O[ca]X3iukO*^sO*azlo[civa]Zhb^ ¬ f® ¬ ^Miu[]OS[]Kslo[%br^ ¬ f® ika ¬ @v®[]KMO0_sZcOik ^Miv^sZcXfN0N0O[]acbey$kO*^sO*azlo[civa]ZSika[]KMO*braZwXYN0N0O[]acbDK9lu[cbrik^sZZcp1O;ybL<sO9Z[]KMO3ZclkNOO9df_slu[cbrik^1®

lkqqrio£Z&[ci(kO[hl_sZwO*_MqsO;df_MbolkqO9^v[)a]O*psacO;ZwO9^v[zlo[]biv^(iva ¬ v®¡­ ¬ u® PluN0O9qXv­u[cKsOikqrqio£br^MZw[]lu[cO*N0O9^v[beZ%ikYYbrik_sZ » » ÐI ¸ " A < ,% P R 6P 6+% P 7P

1 R ¬ C @k®+* 9 5 R ¬ k®

d

dt(g, µt) =

K∑

k=1

k

∫

(A1kg)(x, y1, . . . , yk−1)µt(dx)µt(dy1) · · ·µt(dyk−1),

¬ kk®

A1

k : C∞(X) 7→ Csym(Xk) . # 4?

A1kg(x, y1, . . . , yk−1) = ak(x, y1, . . . , yk−1)g(x)

+ (bk(x, y1, . . . , yk−1),∇g(x)) +1

2(ck(x, y1, . . . , yk−1)∇,∇)g(x) + Γk(y1, . . . , yk−1)g(x),

¬ uv®>1 ∇ (* ! = +*#1 9 P = R P( #F 1 R L

x ∈ Rd >1

Γk(y1, . . . , yk−1)g(x) =

∫

(g(x+ z)− g(x) − χ(z)(z,∇)g(x))νk(x, y1, . . . , yk−1; dz).

¬ u®

" Ak9 5% ¬ C C9® 1 ¬ v® 1 R ¬ C9v® "1* 1 A1

k = (1/k!)Bkπ* >1

PI 1 π 7 +%

πg(x1, . . . , xk) = g(x1)

âã&ä½âÖÞ

C

" 1 R ! (- P ¬ vu® ) ¬ o® / R= µt1 1R R$ 1(

R$PI/ R=+*# &P $ Rµt(dx) = ft(x)dx

ft ∈ L1(X)

d

dtft(x) =

K∑

k=1

k

∫ [

1

2(ck(x, y1, . . . , yk−1)∇,∇)ft(x) + ((∇ck − bk)(x, y1, . . . , yk−1),∇ft(x))

+(ak −∇bk +1

2(∇,∇ck))(x, y1, . . . , yk−1)ft(x) + Γ∗

k(y1, . . . , yk−1)ft(x)

] k−1∏

l=1

ft(yl)dyl,

¬ @v®>1

(∇ck(x, y1, . . . , yk−1))j =

d∑

i=1

∂

∂xicij(x, y1, . . . , yk−1),

(∇,∇ck)(x, y1, . . . , yk−1) =

d∑

i,j=1

∂2

∂xi∂xjcij(x, y1, . . . , yk−1),

>P Γ∗

k(y1, . . . , yk−1) 1 P P F 19 =1 ¬ o® ' ¬ uv®

§ Zl0ZwbexIO a]O9Zc_Mq[;­I£ O y9lu^kbrkO^Mio£ []KMOZw[ca]_sy¡[]_MacOikp8ivZcb[]bYb[\XpMa]O9ZcO*a]Yb^M$O*vikqr_I[cbrik^sZ ¬ k® ·Y» · P

X

D 1

A I / 3 1

R/ 1 D

' M(X) F P F 1 ! / R= F? / 1

C∞(X)

1< F D #1 ,L' ¬ k® 6 2% 1R ?

D A

PP 1 1 1 / µn → µ

,%1*µ, µn ∈ D

g ∈ D

* P A(µn)g → A(µ)g

C(X)

P A(µ)

1 R ¬ u® ' ¦Ma]ikN J%KMO*ivacO9N Cmlu^sx []KMOAxIO<s^Mb[cbrik^¨iu yiv^sxIb[cbrik^slkqp1ifZwb[cbrYb[\Xb[3ikqrqio£Z0[]Kslo[

A(µ)KslkZSivacN ¬ u®iva µ 1O9b^M <s^sb[]Oqrbr^MO9lkay*ikN$Mb^1lo[cbrik^1ZSik bazlky$N0O9lkZc_Ma]O9Z ¦Miva lua]Mb[]a]lkacX

µ = lim µn ∈ D£b[cK

µn8O*br^M <1^Mb[]O$y*ikN(sb^slu[cbrik^sZiu brazlky N0O9lvZw_sacO;Z*­ A(µn)g

yiv^YkO*a]kO;Z4[cilªy*ik^f[cbr^Y_Mik_1Z_M^sy[cbrik^ ika$lua]Mb[]a]lkacX

g ∈ D J%KMbeZ(xIO<s^MO;Z[cKMOyiYO GybrO*^f[zZ a, b, c, ν br^ ¬ k® lkZy*ik^f[cbr^Y_Mik_1Z_M^sy¡[]biv^sZ iva(lua]Mb[cazlua]Xµ ∈ D YX <sazZ\[psacioYbr^M[cKMOy*ik^f[cbr^Y_Mb[\XAiu a YX.yzKMiYivZcb^M

g = 1­8[cKsO*^m[]KMO0yik^f[]b^Y_Mb[\Xiu

νYXªyzKMiYivZcbr^M

g[ci1O0yiv^sZ\[zlu^f[Sbr^2lk^mikp8O*^.ZwO*[9­lu^1x[cKMO9^A[]KMO

y*ik^f[cbr^Y_Mb[\XiubYXyzKsififZwbr^M

g[ci38O qbr^MO9lka%b^ªlu^ikp8O*^ªZcO[ VYivN0O acO9N0lkacIZ4lu^sxO*nIlkN0pMqO;Z4[ci[cKsO ikI[zlubr^MO9xacO;Zw_sq[zZlua]Obr^ikazxIO*a

/ ¢ [brZy*qO;lua4[cKslu[b&ik^MqrX3[\£4i3yikN0p8ik^MO9^f[]Z9­YZ]lGX Ailu^sx

Aj­sxIi0^Miu[olu^sbrZcKbr^[cKMO

N3lkpMpMbr^M ¬ C;k®­8lk^sx A brZSyik^1xIb[]biv^sluqrqrXp8ivZcb[]bvOk­1[cKMO9^mO9lvyzKª^Mik^M¤>olu^sbrZcKMbr^MyivNp8ik^sO*^f[ Ailu^1x

AjbrZ4yiv^sxIb[cbrik^slkqqrXp1ifZwb[cbrkOlkZ)£4O*qrq ¬ []lkkO εωj

b^sZw[cO;lkx3ikωjbr^[cKsOSxIO<s^Mb[cbrik^lk^sx[cKsO*^p1lkZ]Z [ci

[]KMO(qrbrNb[]Zε → 0

lu^sxε → ∞ ® ¢ ^Ay9lkZcOiu N0ika]O [cK1lu^[\£4i^Mik^M¤>olu^sbrZcKMbr^MyivNp8ik^sO*^f[]Zb^ª[]KMOÐlkN0bqrX

A­D[]KMOlk^sluqrikvik_sZSZ\[zlo[]O*N0O*^f[brZSÐluqeZcO ¬ br^.psluac[cbey_Mqelua;­ ika]ikqrqrlkacX Cy9lu^A^siu[S8O0OnY[cO9^sxIO;x[]i

K > 2® PSluN0O*qrXk­Ib A beZyiv^sxIb[cbrik^slkqqrX3p1ifZwb[cbrkOv­f[cKMO#w1iv_M^sxMlkacX ikp8O*azlo[]ikazZ A1

lu^sxAKlua]O

ääÉNPORQRSRT

C@ L

y*ik^sxMb[]biv^sluqrqX0p8ivZcb[]bvOlvZh£4O*qrq ¬ []KMO ZclkNOlua]k_MN0O9^v[¡®¡­YM_I[4[cKsObr^f[cO9acN0O9xMbrlu[cOikp8O*azlo[civa]Z Ak^MO*O;x

^siu[%[ci38Ok­slvZZwKsio£Z4lkqa]O9lvxIXl0ZwbrN0pMqrO OnMluN0pMqrOiu&[cKMO ivp1O9a]lu[cikaA = (A1, A2, A3)

£b[]KAig(x1, . . . , xi)) = ai(∆g(x1) + · · · + ∆g(xi)), i = 1, 2, 3,

£b[cKa1 = a3 = 1

lu^sx£b[cKa21O9b^MlZwN3lkqq½O9^Mik_skKª^MO9vlo[]bvO^f_sN(8O*a # )y*ik_MazZwOv­Mb[brZO9lvZwX[]i0£acb[cOlk^O*nYpsqbeyb[Zwivq_M[cbrik^[ci0O9df_slu[cbrik^ ¬ f®4br^[cKMbeZy*lvZwO

/ ² OvlGkOik_Ma0a]O9Zc_Mq[zZiva X = Rd ­)M_I[3_sZcb^s2qriIy*luqrbDK9lo[]biv^ lua]k_MN0O9^v[zZ ¬ qbrkObr^ qrb^sO9luay*lvZwOv­)ZwO9O > ~AÖ® [cKMOZ]luN0Oa]O9Zc_Mq[zZ$y9lu^´8OO;lkZcbqrX.OnY[cO9^sxIO9x´[ci2y*qifZwO;x´N3lu^MbikqexMZ A¢ [brZZcO*O9N0b^MvqXp1ifZcZcbrMqO[ciyzKslka]lvy¡[]O*a]bLK9Obr^[cKsO$Z]luN0O£%lGX3[cKMOyivaca]O9Zcp1iv^sxIbr^M01iv_M^sxMlkacXyiv^sxIb[cbrik^sZ¬ kO9^MO*azluqrbLK9b^slkqrZci([]KMO qrb^MO;luay9lkZcOSa]ikNB> ~ A®¡­s[cKMiv_MkK[cKsbrZbeZluqra]O9lkxMX3^Miu[Zci3Z\[]a]lkbvKv[cika]£4lka]x / H4lvZwbey(y*ik^sxMb[]biv^sZiuhp8ivZcb[]bYb[\Xªiu ¬ 1C9®¡­ ¬ kk®Sy*lk^A8O0£acb[w[]O*^Abr^2lk^2lkq[]O*a]^slo[]bvObr^f[cO9kazluqika]N PluN0O9qXv­[]KMOp8ivZcb[]bYb[\X2ik4N3lu[ca]bry*O9Z ¬ =C9® brZ(O9df_MbrGlkqO9^f[ ¬ lu[(qO;lkZw[ ika$1iv_M^sxIO;xy*ik^f[cbr^Y_Mik_1Zh_s^sy¡[]biv^sZ

c® []i$[]KMO p8ivZcb[]bYb[\Xiu[cKMO N3lu[ca]bry*O9Z

K∑

k=1

k

∫

ck(x, y1, . . . , yk−1)µ(dy1) · · ·µ(dyk−1)¬ u~v®

ivahluqrqM^Miv^I¤¥^MO*flo[cbrkO H4ika]O*qIN0O;lkZc_MacO;Zµ(dy) ik^sxIb[cbrik^1Z ¬ 1C;®­ ¬ vk® lvy¡[]_sluqrqX(acO9pMacO;ZwO9^f[)NiIxIbL<sO;xN$_Mq[cb¤¥xIbrN0O*^sZcbrik^slkqN3lo[]acbnY¤>oluqr_MO;x ivaN0O9lvZw_sacO*¤>oluqr_MO;x vO*azZwbrik^1Z0iu [cKsOA_sZc_sluqS^Miu[]biv^¨iu p8ivZcb¤

[]bvOAxIO<s^Mb[cOm_s^sy¡[]biv^sZ ¦Mikab^1Z\[zlu^sy*Ok­br^ y*lkZcO k = K = 3lu^sx

d = 1­ ¬ k~k®3N0O9lk^sZ3[]Kslo[

∫

c3(x, y, z)ω(y)ω(z)dydzbeZhl^Mik^I¤¥^MO9vlo[]bvO%^f_sN(8O*a)iva lk^fX$^Mik^M¤>^MO9vlu[cbrkObr^v[]O*va]lkMqO4_M^1y¡[cbrik^

ω J%KMO_sZc_slkqM^Miu[]biv^$ikDlp8ivZcb[cbrkOxIO <1^Mb[]O_M^sy[cbrik^ c3 ¬ lkZ)lS_M^1y¡[cbrik^3ik1[]KMOqelkZw[ [\£ iolua]brlkMqrO9Zz®£4ik_sqrxacO;df_Mba]OS[cKMOZ]luN0Op1ifZwb[cbrYb[\XikalkacMb[cazlua]X ¬ ^Mik[^MO9y*O9Z]ZclkacbrqrX3p1ifZwb[cbrkOG®hbr^f[cO9kazluMqrO ω / § Z3ZwKMio£Zg)a]ikp8ivZcb[cbrik^ C C[cKsOmZ\[zlo[cO9N0O*^f[3iu ivacivqqelua]X CxMifO;Z0^Miu[3KMivqrx ikaxMbrZ]ya]O[]O

X ¢ ^y9lkZcOikyiv^v[]b^Y_Miv_sZ4Z\[zlo[cOZcpslky*O X iv^MOy*lu^Z\[zluac[ O9O*qrb^M$[cKsOxIbÀDO*a]O*^sy*OS8O[\£4O*O9^Zw[ca]bey¡[cqrX¨yiv^sxIb[cbrik^slkqqrX p8ivZcb[cbrkO2lk^sx y*ik^sxMb[]biv^sluqrqXp8ivZcb[]bvOAikp8O*azlo[]ikazZ3ik^sqX©£b[cKk = 3 §ZcbrNpsqOSOnMluN0pMqrOiklyiv^sxIb[cbrik^slkqqrX0p1ifZwb[cbrkOv­vM_M[ ^Mik[4l(Zw[ca]bey¡[cqrX3yiv^sxIb[cbrik^slkqqrX$p8ivZcb[cbrkOivp1O9a]lu[civa

a]O*psacO;ZwO9^v[zZ [cKMO ivp1O9a]lu[cikaxIO<s^MO;xlvZA3g(x1, x2, x3) = cos(x2 − x3)∆g(x1) + cos(x1 − x3)∆g(x2) + cos(x1 − x2)∆g(x3).

¬ kv®« dv_1lo[cbrik^ ¬ @v®4br^[]KMbeZy*lvZwO[]lkkO;Zh[]KMOika]N

d

dtft(x) = ∆ft(x)

∫

ft(y)ft(z) cos(y − z)dydz. ¬ 1C;®J%KsOw^Miv^Mqbr^MO;luaxIbÀ8_1Zwbrik^yiYO Gy*bO9^v[beZ^Mik[Zw[ca]bey¡[cqrXp8ivZcb[cbrkOKMO*a]O PSluN0O*qrXk­sZcbr^syO

∫

f(y)f(z) cos(y − z)dydz =1

2(|f(1)|2 + |f(−1)|2),

£KsO*a]Of(p)brZh[cKsO¦sik_Ma]bO9a)[]a]lk^sZ\ivacN iu

f­f[cKMbeZ OnIpMa]O9Z]Zwbrik^xIi$^Miu[4KslGkOxMi(8OZw[ca]bry[cqrX$p8ivZcb[cbrkO

iva3luqrq^Mik^I¤¥olu^MbeZcKMb^s2^Miv^I¤¥^MO*flo[cbrkOf Mio£ O9kO9a9­ iv^MOªy*lu^ <s^sx [cKsOO*nIpMqbeyb[Zwivq_M[cbrik^ [ci.[cKMO

lk_syzKYX3pMa]ikMqrO*N ik ¬ 1C;®³

ft =1√

2πωt

∫

exp

− (x− y)2

2ωt

f0(y)dy,

âã&ä½âÖÞ

C;~

£b[cKωt = ln(1 + t(|f0(1)|2 + |f0(−1)|2).J%KsbrZ beZO;lkZcbqrXAivI[]lkb^sO9x.YX2pslvZcZcbr^M[ciª[]KMO¦Miv_Ma]bO9a [cazlu^sZwika]Nik%O;df_slo[]biv^ ¬ 1C9®[cKslu[(KslkZ[cKMOivacN

d

dtft(p) = −1

2p2(|ft(1)|2 + |ft(−1)|2)ft(p),

lk^sx£KsbryzKbrZZcikqrkO;x3YX3iksZcO*a]Yb^M([cKslu[ξt = |ft(1)|2 + |ft(−1)|2 ZcikqrkO;Z)[]KMOO9df_slo[]biv^ ξt = −ξ2tlk^sxyik^1ZwO;dv_sO*^f[cqrXO9df_slkqrZ

ξt = (t+ ξ−10 )−1

/ (J%KsO*a]O(beZl^slo[]_MazluqwxIO;yivNp8ivZ]lusqO y*qrlvZcZSiu ikp8O*azlo[civa]Z9­1ika£KMbryzK ¬ k®Sa]O9xI_1yO9ZZw[cazlubrkKf[cika]£4lka]xIqrX[ci2lªqbr^MO;luapMa]iksqO9N PSluN0O*qrXk­ Zc_MpMp8ivZcO k!Akg = Bkg+ = (Bkg)

+ iva$lkqqk = 1, . . . ,K

£b[cKªZcikN0OBkbr^C∞(X)

[cK1lo[kO9^MO*azlo[]ORmlkacvio0pMaciIy*O9Z]ZwO;Z)[]KMO*bra ¬ b O Bk1 = 0® J%KsO*^ ¬ kv® []lkkO9Z []KMOika]N

d

dt(g, µt) =

K∑

k=1

1

(k − 1)!(Bkg, µt)‖µt‖(k−1),

£KsbryzKbrZl$qrbr^MO9lkaO9df_slo[]biv^xMO*p8O*^sxIbr^M0ik^ ‖µt‖ = ‖µ0‖lvZ%ik^lpsluazluN0O[]O*a ¦sikayiv^syqr_sZcbiv^2ik []KMbeZZwO;y¡[]biv^2qrO[_sZ(xIbrZ]y_1ZcZ[]KMO3qbÖ[cbr^Mmik ¬ v®­ ¬ C9k®[]iqbr^MO9lka O*vikqr_I[cbrik^sZiv^.[cKMON$_Mq[cb¤>pslkaw[]bry*qOZw[]lu[cOZcpslky*O X £KMbeyzK´brZ ikbN0p8ikac[]lu^1yO0ika[cKMOy*ika]acO;Zwp8ik^1xIb^sp1luac[cbeyqrOZcXIZ\[]O*N3Z4xIO9Z]ya]bsb^M([cKMO9ba wpMa]ikpslkvlo[]biv^0iuyzKslkivZ pMacivp1O9aw[\X ¢ ^Ðlky[4O;dv_1lo[cbrik^1Z ¬ uv®*­ ¬ vk®)8O*qrio£lkacO%xIO9acbrkO;x(br^?>AsacivN lZcy9luqrb^MqrbN0b[)iu8[cKMO%N0ikN0O*^f[)N0O9lvZw_sacO;Zik1br^f[cO*azlky[cbr^Mpsluac[cbeyqrO%ZwXIZw[cO*N3Z

br^[cKMOZcpMbracb[iuO > k Aiva> k A ¦sikal?<s^Mb[cO3Zc_MsZcO[I = i1, . . . , ik

iu%lyik_s^v[zluMqrO0ZwO*[J­£ O3xIO9^Miu[]OYX |I | [cKMO0^Y_MN(8O*a ikO9qO9N0O*^f[]Zb^

I­;YX |I | b[]Z&yivN0pMqO9N0O*^f[ J\I ­GYX xI

[]KMO y*ikqrqO;y¡[cbrik^ikf[]KMOholkacbeluMqrO9Zxi1 , . . . , xik

lu^1xYX

dxI[cKsO(N0O9lvZw_Ma]O

dxi1 · · · dxik qO;lua]qXO9lvyzK f ∈ Bsym(X )

beZxIO<s^MO;xªYXb[]ZyivNp8ik^sO*^f[]Zfk ik^ Xk Zwi([cKslu[ iva x = (x1, . . . , xk) ∈ Xk ⊂ X ­YZ]lGXk­fik^sOy9lu^£a]b[]O f(x) = f(x1, . . . , xk) =fk(x1, . . . , xk) ¬ []KMO2_spMp1O9abr^sxIO*n k lu[ f beZivpI[cbrik^slkqSlk^sx¨beZ_sZcO9x¨[ciZw[ca]O9Z]Z3[]KMOA^Y_MN$1O9aikolkacbeluMqrO9Zbr^.lk^2O*nIpMacO;ZcZcbrik^1® VYbrN0bqelua ^Mik[]lu[cbrik^sZlua]O$ika N0O9lvZw_Ma]O9Z (¢ ^ pslkaw[]bry*_Mqelua;­1[]KMO3pslkba]b^s8O[\£4O*O9^

Csym(X )lu^1x M(X )

y*lk^8O £acb[w[]O*^ªlkZ

(f, ρ) =

∫

f(x)ρ(dx) = f0ρ0 +

∞∑

n=1

(x1, . . . , xn)ρ(dx1 · · · dxn),

f ∈ Csym(X ), ρ ∈ M(X ), ¬ vk®Zci3[]Kslo[ ‖ρ‖ = (1, ρ)

ivaρ ∈ M+(X ) Jilu^Alkacsb[]a]lkacX Y (dx) ∈ M(X)

[]KMO*a]O(y*ika]acO;Zwp8ik^sxsZlN0O;lkZc_MacO

Y ⊗ ∈Msym(X )xIO <s^sO9xYXb[zZyivN0p1iv^MO*^f[]Z

(Y ⊗)n(dx1 · · · dxn) =1

n!Y ⊗n(dx1 · · · dxn) =

1

n!Y (dx1) · · ·Y (dxn). ¬ kv®

JiªO9lvyzKg ∈ Csym(X )

[cKMO9acOyika]a]O9Zcp1iv^sxMZlJ ,% $ = (g, Y ⊗)

ik^ M(X) ¬ luqeZwiy9luqrqO;x3ZwivN0O[cbrN0O9Zh[cKMOSkO9^MO*azlo[]b^M_s^sy¡[]biv^sluq1ikag® VI_syzKl _s^sy¡[]biv^sluq8£bqrq81Oy*luqrqrO9x3lp8ikqrXY^Miu¤N0beluqik^ M(X)

­sbg ∈ Csym

fin (X )­8b O b)iv^MqrXl <1^Mb[]O^Y_MN(8O*aSiu[cKsO$yivN0p1iv^MO*^f[]Ziu g xIi^Miu[olk^MbrZcK

ääÉNPORQRSRT

k L

·Y» · " µt ¬ C;k®(*1 νt = (µt)

⊗ ∈ Msym(X ) 1

(- P

d

dtνl

t(dx1 · · · dxl) =l∑

j=1

K∑

k=1

Cll+k−1

∫

xl+1,...,xl+k−1

(Bj,l+1,...,l+k−1k )∗νk+l−1

t (dx1 · · · dxl+k−1),

¬ uf®>1Cl

m

1 RP ; (*B∗

k

P P FBk

(BI

k)∗νt(dx1 · · · dxm) 1 P ,! B∗

k

1I R LR9 4 R I ⊂ 1, . . . ,m

" 1 ' νt ∈ Msym(X )

1( ( 5% ¬ kv®(* P 1 P3 Csym(X)

7 +% 1 (- P

g(x1, . . . , xl) = (LBg)(x1, . . . , xl) =∑

I⊂1,...,l

∑

j 6∈I

(Bj,I|I|+1gI)(x1, . . . , xl),

¬ vk®

>1gI(x1, . . . , xl) = g(xI )

B

j1,...,jk

k

P 1 ! Bk

P R L j1, . . . , jk

R P =L(*(LBg

1)(x1, . . . , xl) = (Bl(g1)+)(x1, . . . , xl).

' 1ZwO9acvO[]Kslo[%[cKsOZ\[]aciv^Mika]N iu ¬ C;k®4beZ

µt(dx) =K∑

k=1

1

(k − 1)!B∗

k(µt ⊗ · · · ⊗ µt)(dxdy1 · · · dyk−1),

£KsbryzKbN0pMqrbrO9Z ¬ uv®4YXZw[cazlubrkKf[cika]£4lka]x0N3lu^sbpM_sqrlu[cbrik^sZ ¦Ma]ikN ¬ uf®4b[ikqrqrio£Zh[]Kslo[

d

dt(g, νt) =

∞∑

l=0

K∑

k=1

∑

I⊂1,...,l+k−1,|I|=k−1

∑

j 6∈I

Bj,Ik gI(x1, . . . , xl+k−1)νt(dx1 · · · dxl+k−1)

=

∞∑

m=0

∑

I⊂1,...,m

∑

j 6∈I

Bj,I|I|+1gI(x1, . . . , xm)νt(dx1 · · · dxm),

£KsbryzKbN0pMqrbrO9Z ¬ vu® ¦sacivN [cKMO xI_slkqb[\X8O[\£4O*O9^ ¬ uv®4lu^1x ¬ vu® ik^sOy*ik^sy*q_sxMO9Z4[cK1lo[£KMO*^sO*kO9a iv^MOKslkZ []KMO£ O9qqp8ivZcO9xM^MO9Z]Ziva[cKMO lu_syzKYXpMa]ikMqrO*N ikhO9df_slu[cbrik^sZ ¬ fu®Slu^sx ¬ C9f®iva ZwivNOxIO*^1ZwO0Zw_ssZwp1lkyO;Zikbr^Mb[cbeluqIyiv^f[cbr^f_sik_sZ_M^sy[cbrik^sZ

gik^ X lu^sx(b^Mb[cbeluqYN0O9lvZw_sacO;Z µ0

ik^X­Gik^MO4KslvZ½[]KMO4xM_sluqrb[\X a]O*qelo[]biv^

(gt, µ⊗0 ) = (g0, µ

⊗t ) ¬ kv®

brN0pMqrXfbr^M´[]KMOAbr^Yolua]brlk^syOªik [cKMO.yika]a]O9Zcp1iv^sxIbr^M´ZwpslvyOmik wlk^sluqrXf[cbey _M^sy[cbrik^slkqrZiu [cKsOm[\XYp1O(g, Y ⊗)

ik^ M(X )_M^sxIO9a[cKMOlky[cbrik^ik[cKsOZcO*N0brka]ik_Mp

TtF (µ) = F (µt)ik^

C(M(X)) ¢ ^N3lk^fX$y*lkZcO9Z&iv^MOy*lu^pMa]iokO [cKslu[ [cKMbeZ)Zcpslky*O%pMacioYbexIO9ZlyivacO4iva[cKMO%kO9^MO*azlo[]ikaiu1[cKMbeZ)ZcO*N0brka]ik_Mp½­£b[cK[]KMO a K Z LB

iu ¬ fk®4Zwp8O9y*bXYbr^M0[cKsOika]N iu&[cKMO vO*^MO9a]lu[civa iv^[]KMbrZyika]O

âã&ä½âÖÞ

1C

´2(!±z#VI_MpMp8ivZcO

X = Rd _MahZw[cazlo[]O*kX$iuZwivqYbr^M ¬ @v® £bqrqs8O[]i qiYik$ika l<MnIO9x0p8ikbr^f[hiu½lN3lupMpsb^Ma]ikNut ∈ C([0, t0], L1(X)

® []i$[]KMOZwivq_M[cbrik^Ft ∈ C([0, t0], L1(X))

ik[]KMO qrb^MO;luaO;dv_1lo[cbrik^

d

dtFt(x) =

K∑

k=1

k

∫ [

1

2(ck(x, y1, . . . yk−1)∇,∇)Ft(x) + ((∇ck − bk)(x, y1, . . . , yk−1),∇Ft(x))

+(∇bk − 1

2(∇,∇ck) + ak)(x, y1, . . . , yk−1)Ft(x) + Γ∗

k(y1, . . . , yk−1)Ft(x)

] k−1∏

t=1

ut(yl)dyl.

¬ u® ·Y» · P P ; ' ¬ o®/ = 1 ' x L

C2(X) ,

= 6 = R J R P4 , P ? R LR R R ! P %/ R=R 1? P Γk

F 16 P " P $ /%t0

/%ut ∈

S0C([0, t0], L+1 (X)) 19R/R C([0, t0], L

+1 (X))

! ? = P&P $> (L1

)FR ‖u0‖ /R0" P P P P F0"Ut,s([u])

*0 ≤ s ≤ t ≤ t0

F 1 /4 %I L ¬ o® L1(X)

( 7( ,%?RP 1 Ut,s([u])

1 2%,' P R #%/ RR ? / R R?

L1(X) F.P R R ? RP,% =/ 1

ak P ¬ o®!" P ,%< PP R,% t, s / /( 1

" Ut,t([u]) PI #% P F

t*

" 1I , %Ut,s([u])Us,r([u]) = Ut,r([u])

1 t ≥ s ≥ r

* " 1 = gt = Ut,s([u])gs

L FC2(X)

19=- P F ¬ o® PI gs = g

>P g ∈ C2

c (X)*

" P9 P W 2(X) R R(<5%

Ut,s([u])

‖Us,r([u])g‖W 2(X) ≤ C2(t0)‖g‖X2(X)¬ @v®

1 C2(t0)

FP P,% ‖u0‖

1 /5 %7 f0 ∈W 2(X)∩C2

c (X)1 4 3=- P

L R ft ∈W 2(X) ∩C2(X)

' ¬ @k® P9P f0 # ¬ @k®3 ? t ≥ 0

7 f0

t = 0

" 1 / f0 7→ ft

4 F 2% PP ' 0" P F (W 2(X))+

1 RR PL1)F R !

f0 % / P /( % L ! ¬ kv®

' ² OlkacO vikbr^M[ciZcKMio£ []Kslo[Sb^ªy*lvZwO u0 ∈ W 2(X)[cKsO*a]O OnIbrZw[]Zl0_s^Mbrdf_MOI<MnIO9xp1ivb^f[

ik[cKsOªN0lkpMpMbr^Mut 7→ Ft = Ut,0([u])u0 ¦&bra]Zw[3iksZcO*a]kO[cKslu[xIbÀDO*a]O*^f[]brlu[cbr^MA[]KMOªZwO9N0bvaciv_MpbexIO9^v[]b[\Xika

Ut,s([u])iv^MOkO*[]Z%[]Kslo[

d

dsUt,s([u])g = Ut,s([u])L(s)g ¬ v~v®

ääÉNPORQRSRT

v L

ivag ∈W 2(X)

­v£KMO*a]OL(s)xIO*^Mik[cO[]KMOvO*^MO9a]lu[civa]Z ¬ xMO*p8O*^sxIbr^M$ik^ u ® lo[)[]bN0O s ¬ []KMOivp1O9a]lu[civa]Ziv^[]KMO a K Z ik&O;df_slo[]biv^ ¬ o®c® ÁO[

u08O/<MnIO9x´lk^sx.qO*[

u1, u2 yivb^sy*brxMO£b[]K u0lo[ [cbrN0O K*O9acimlk^sx 8O*qrik^M[]iª[cKMOZwpMKsO*a]O

S0C([0, t0]), L+1 (X)) _MO[ci ¬ k~v®4ik^MOKslvZ

Ft([u1]) − Ft([u

2]) =

∫ t

0

d

dsUt,s([u

2])Us,0([u1])u0 ds

=

∫ t

0

Ut,s([u2])(L1(s) − L2(s))Us,0([u

1])u0 ds,¬ vv®

£KsO*a]OL1(s), L2(s)

xIO*^siu[cO[cKMOkO9^MO*azlo[]ikazZ lo[[]bN0O9Zsiu[]KMOO;dv_1lo[cbrik^ ¬ u® £b[cK u1

lk^sxu2a]O9Zcp8O9y¡[]bvO*qrX qrO9lua]qrX

‖(L1(s) − L2(s))g‖L1(X) ≤ C1‖u1(s) − u2(s)‖L1(X)‖g‖W 2(X)¬ PC;®

£KsO*a]OC1xIO*p8O*^sxsZik^ ‖u0‖L1(X)

lu^sx [cKsOm1iv_M^sxMZika3[]KMOAxIO9acbrolo[]bvO9Z3ik[]KMOAy*ifOGybrO*^f[]Z3ikO;df_slo[]biv^ ¬ o® ik^sZcO9df_MO9^v[]qX ¬ @k®­ ¬ vf®¡­ ¬ PC9®4brN0pMqX[cKslu[

sup0≤s≤t

‖Fs([u1]) − Fs([u

2])‖L1(X) ≤ tC1C2(t0)‖u0‖W 2(X) sup0≤s≤t

‖u1(s) − u2(s)‖L1(X).¬ fk®

MO*^sy*O[]KMO$N3lkpMpMbr^Mut[]iFtb^A[cKMOZw_ssZwO*[Sik

S0C([0, t0], L1(X))ikI[zlubr^MO9xªYX?<MnIb^s

u0beZSl

y*ik^f[cazlky[cbrik^£KMO*^MO9kO9atC1C2(t0)‖u0‖W 2(X) < 1. ¬ fv®

iv^sZcO9df_MO*^f[]qX2ik^MOivI[]lkb^1Zlª_M^Mbedf_MOZcikqr_I[cbrik^ []i ¬ v~k®iv^ [cKMOb^f[cO9acolkq [0, t1] ikalu^YX t1 ≤ t0Z]lo[]brZwXYb^s ¬ vv®¡­slk^sxN0ivacO9iokO*asup

0≤s≤t1

‖us‖W 2(X) ≤ C2(t0)‖u0‖W 2(X).¬ kf®

¢ t1 < t0ik^MO.y*lk^ acO9p1O;lo[[cKMbeZpMa]iYy*O9xI_sacO.Z\[zluac[cbr^M a]ikN [cKsO2[]bN0O

t1ivI[]lkb^sb^Ml_s^Mbrdf_MO

y*ik^f[cbr^Y_slo[]biv^ik[]KMOZwivq_M[cbrik^[ci[cKMO br^f[cO9acoluq[t1, t2]

­I£KMO9acO(t2 − t1)C1(C2(t0))

2‖u0‖W 2(X) < 1. ¬ Yk®§ vO*X ik1ZwO9acolo[]biv^^Mio£ beZ[cKslu[)lkZ[cKMbeZ^MO9£ Zcikqr_I[cbrik^luvlkb^$ZwivqvO9Z ¬ k®ik^MO%Z\[]bqrqIKslvZ[]KMO%O9Zw[cbrN3lo[cO¬ kv® £b[cK t1 a]O*psqrlvyO9xfX t2 ¬ b O C2

xIi$^siu[%KslGkO[ci1Oa]O*pMqelky*O9x3YXC2

2

KMO*a]O;®­Y£KMbryzKluqrqio£Zh[cilu[w[zlubr^[cKsOh[]bN0O

t0YX_sZcb^s[]KMbrZ&pMa]iIyO;xI_Ma]O

p[cbrNO;Z&£b[]K

p8O*br^MS[cKMO%N0br^MbN3lkqfbr^f[cO9kO*a&OnMy*O*O9xMb^M

t0C1(C2(t0))2‖u0‖W 2(X).J%KsbrZy*ikN0pMqrO[cO;Z4[cKMO psaciYiu&iu&[cKMOJ%KsO*ika]O*N

² O´lua]OAkivb^s´[cibqrqr_sZ\[]a]lu[cO2[cKMO.kO9^MO*azluqa]O9Zc_Mq[ikI[zlubr^MO9x YX lupMpMqrXYb^sb[[]i[]KMO y*lkZcOAikO;df_slo[]biv^sZ4£b[]Kl3Zw[ca]bey¡[cqrX^Miv^I¤¿xIO*vO*^MO9a]lu[cOxIbÀD_sZcbiv^p1luac[Zc_syzK[cK1lo[K∑

k=1

k

∫

ck(x, y1, . . . , yk−1)µ(dy1) · · ·µ(dyk−1) ≥ ‖µ‖m ¬ vv®

ivaluqrqxlu^sx ¬ ^Mik^I¤¥^MO9vlo[]bvO;® µ lk^sxZwivNO <MnIO;xa]O9lkq m

âã&ä½âÖÞ

k

·Y» · P /ak

' ¬ ku® * ¬ uk® * ck (% ¬ fk® 1 / R= νk

¬ o®$1 =1R R R P4F $P / R=

νk(x, y1, . . . , yk−1; dz) = ψk(x, y1, . . . , yk−1; z)dz¬ u®

R/( 1 ψk(x, y1, . . . , yk−1; z) ≤ max

(

C

|z|β1+d,

C

1 + |z|β2+d

) ¬ @v®

k, x, z, yj /9 1

β1 ∈ (0, 2)*β2 > 0

*C > 0

∫

|y|≤1

sup|z|≤1

∣

∣

∣

∣

∂ψ

∂x(x+ z, y)

∣

∣

∣

∣

|y|2 <∞,

∫

|y|≤1

sup|z|≤1

∣

∣

∣

∣

∂2ψ

∂x2(x+ z, y)

∣

∣

∣

∣

|y|2 <∞.

P P b ∈ C3(X)

c ∈ C4(X)

= =P R R P4F P R DR1 P 1 1 P R 10"/' 1 ( ' 1ZwO9acvO7<sazZw[4[]Kslo[[]KMOxI_slkq Γ∗

k

[ci0ivp1O9a]lu[cika ¬ o®4KslvZ4ika]N

Γ∗k(y1, . . . , yk−1)g(x) =

∫

[g(x− z)ψ(x− z, y1, . . . , yk−1; z) − g(x)ψ(x, y1, . . . , yk−1; z)

+ (∇g(x), z)χ(z)ψ(x, y1, . . . , yk−1; z) + g(x)χ(z)(∇xψ(x, y1, . . . , yk−1; z), z)] dz£KsbryzKy*lk^8O £acb[w[]O*^br^[]KMOika]N∫

(g(x− z) − g(x) − (∇g(x), z))χ(z))ψ(x − z; z) dz

+ g(x)

∫

(ψ(x− z; z) − ψ(x; z) + χ(z)(∇xψ(x; z), z)) dz

+ (∇g(x),∫

zχ(z)(ψ(x, z) − ψ(x − z, z)) dz) ¬ v~v®¬ £KMO*a]O£ O ivN0b[c[cO9x[]KMOlua]k_MN0O9^v[zZ y1, . . . , yk−1

iu&[cKsO_M^sy¡[]biv^ψ®­Mb O b[KslvZ4[cKMOZ]luN0Oika]N¬ o® pMq_1Zhl <sazZ\[)iva]xMO*ahxIbÀDO*a]O*^f[cbeluqMivp1O9a]lu[civa £b[]K31iv_M^sxIO;x3yiYO GybrO*^f[zZ ¬ []KMOqelo[w[]O*a 1O9b^MxM_MO[ciiv_MalkZ]Zw_sNpM[cbrik^sZ%iv^

ψ® J%KMO£4O*qrqp1ifZwO;xI^MO9Z]Z4iu&[cKMO lk_syzKYX3pMacivMqrO*N ivaO9df_slo[]biv^ ¬ u® _M^1xIO*a[]KMO%yik^1xIb[]biv^sZ&iuM[]KMO%J%KMO*ivacO9N°lk^sx£b[cK$1iv_M^sxIO;x

ψ£%lkZ&ZwKMio£^I<sa]Zw[ ZwO9O*N0b^skqrXbr^ > f A ¢ ^?> v A>­> kA&b[S£%lkZZcKMio£^[]Kslo[b^ª[]KMbeZSy9lkZcO [cKMO9acO(OnIbrZw[]ZSla]O*O9^_M^sy¡[]biv^

G(t, x, x0; [u])ik)O9df_slo[]biv^

¬ o®­)£KMbeyzK beZ$y*ik^f[cbr^Y_Mik_sZcqrX xMbÀDO*a]O*^f[cbelusqOb^ t, x, x0ika

t > 0lk^sx Z]lo[cbeZ$<sO9Z([cKMO KslkpMN3lu^I¤ESikqrN0ikvika]ioªO9df_slo[]biv^ .¢ ^ > 1CRA4[cKMOZ]luN0Oa]O9Zc_Mq[$£%lkZ$ikI[zlubr^MO9x iva Γk

8O*br^M2l <s^Mb[cOZw_MN ikazlky[cbrik^slkqp1io£4O*azZ%iu&[cKsO(ÁlkpMqrlvyOikp8O*azlo[]ika ¬ Zw[]lkMqrOqrlG£Zz® H ik[cKpMa]iYiuÐZ4a]ikNB> v A&lu^1x > 1C Ay*lk^8OO;lkZcbqrX(vO*^MO9a]lkqbDK*O;x([]ikbrkO%[]KMOO*nYbeZw[cO*^1yOikl xMbÀDO*a]O*^f[cbelusqO a]O*O9^0_M^sy[cbrik^3_M^sxMO*a)[]KMOlk1iovOy*ik^sxMb[]biv^sZ%ik^

ψ Rªika]O*iovO*a;­vlvZ4[cKMOZ]luN0OKMikqexMZ4iva%[cKMOxI_1luqO9df_slu[cbrik^lu^sx[cKsOxI_sluqO9df_slo[]biv^psacO;ZwO9acvO9Z%yiv^sZ\[zlu^f[]Zb[ivqqrio£Z4[cK1lo[[]KMO(Zcikqr_I[cbrik^iu ¬ k®%pMa]O9ZcO*a]kO9Z []KMO L1¤h^MivacN J%KMbrZZwKMio£ZO9kO9acXf[cKsb^M$OnMyO9pI[ ¬ @v®¡­IM_I[%[]KMOSqelo[c[cO9a4O;Z\[]bN3lu[cObrZ%Zw[cazlubrkKf[wivac£%luazx ¢ ^Ðlky[9­Iik^[cKMOik^sOSKslk^sx­[]KMO%pMa]O9ZcO*a]Glu[cbrik^ik8ZcN0ifik[cKM^sO9Z]Z*­obr^$p1luac[cbey_Mqelua ¬ @k®­kbrZ&£4O*qrqIf^sio£^ika^Miv^I¤¿xIO*vO*^MO9a]lu[cO xMbÀD_sZcbiv^O;df_slo[]biv^sZ*­b O ika ψ = 0 ^.[cKsO3iu[cKsO*aKslk^sx­[cKMOZcO*a]brO9Za]O*pMa]O9ZcO*^f[]lu[cbrik^2ika G(t, x, x0; [u])ivI[]lkb^sO9xb^,> vA½iva^Mik^M¤>olu^sbrZcKMbr^M

ψZcKMio£Z [cK1lo[[cKsO b^f[cO9kazluqDikp8O*azlo[]ika

Γy*lk^8O[ca]O9lu[cO9xlkZl

ZcN3luqrqp1O9aw[]_Ma]slo[]biv^[]Kslo[xIiYO;Z%^Miu[luÀ8O;y¡[%[cKMO olkqbexIb[\Xiu ¬ @k®

ääÉNPORQRSRT

u L

$#%'h# #&*½ M!.

_saslvZwbeyN0iIxIO*q&iu)br^f[cO9a]lvy¡[cbr^MZ\[zluMqrOj\_MN0pI¤¿xIbÀ8_sZcbrik^sZbeZSxIO<s^MO;xmYX X = Rd lu^sx K = 2b^

¬ v® ika ¬ C9k®%y*ik^M^sO9y¡[]O9x[cKsaciv_MkK ¬ C C9®¡­slk^sxfX B1, B2kbrkO*^YX

(B1f)(x) = −σ(x)|∆|α/2f(x), f ∈ C2c (X), ¬ kv®

(B2f)(x, y) = V (x, y)∂f

∂x(x, y) + V (y, x)

∂f

∂y(x, y)

+

∫

(f(x+ x1, y + y1) − f(x, y))ψ(x, y;x1, y1)dx1dy1, f ∈ C2c (X2), ¬ =C;®

£KsO*a]OV, σ, ψ

lua]OkbrkO9^_M^sy[cbrik^sZS£b[]Kψ8O*br^MZcXYN0NO*[ca]bry£b[]KAa]O9Zcp1O;y¡[[]i[cKsO(p8O*a]N(_I[zlo[cbrik^

ik)O*b[cKMO9ax, yikax1, y1

­lu^sx[]KMO$br^sxIO*nªiuhZw[]lusbqrb[\XαbeZSl <snYO;xª^Y_MN(8O*abr^ª[cKMObr^v[]O*a]oluq ¬ Ck­½k®

¬ £ O3xIi^Mik[ y*ik^sZcbexIO*aKMO9acO$[cKMO3br^sxIbeyO;Z α ≤ 1­lkZS[cKsO$[]O9yzKM^sbrdf_MO3lupspMqbrO9x28O*qrio£ £4ik_Mqexma]O9df_MbracO

lNiIxIbL<1y9lo[cbrik^ªbr^ª[]KMbrZy*lvZwOG® J%KMO <sazZ\[ ¬ acO;Zwp8O9y[cbrkO9qX[cKMOZwO;yiv^sxs®%pslkaw[Siu ¬ =C9®Z\[zlu^sxMZika[cKMOp8iu[]O*^f[cbeluq ¬ acO;Zwp8O9y[cbrkO9qX3[]KMO%j\_MN0p[\XYp1OG®4b^f[cO9a]lvy¡[]biv^ / J%KMOy9lkZcO

α = 2yivaca]O9Zcp8ik^sxMZ []i3[cKMO$Z\[zlu^sxMlka]xxIbÀ8_1Zwbrik^sZSlu^sxbeZivNb[w[]O9x­s[cKMiv_MkK

lkqq a]O9Zc_Mq[]Z8O*qrio£±lkacO3Zw[cbrqq)olkqbexmivaα = 2 0¢ ^ psluac[cbey_sqrlka9­br^´y*lkZcO α = 2

­σ = 1

­ψ = 0

[cKsOy*ika]acO;Zwp8ik^1xIb^sRmlua]kioN0iIxIO*qIiksb^f[]O*azlky¡[]b^sSpslkaw[]bry*qO;ZbeZkbrkO9^YX[cKMOZcXIZ\[]O*Niu ¢ [ci Z&O9df_slu[cbrik^sZ

dXi = dWi +∑

j

V (Xi, Xj)dt, i = 1, 2, · · ·

« dv_1lo[cbrik^1Z ¬ kk®%lu^sx ¬ @k® ikavO*^MO9a]lu[civa]Z ¬ kk®­ ¬ =CG® []luvOivacNd

dt(g, µt) = −(σ(x)|∆|α/2g, µt) +

∫

(V (x, y),∇g(x))µt(dx)µt(dy)

+

∫

(g(x+ z) − g(x))ψ(x, y; z)dzµt(dx)µt(dy)¬ kk®

lk^sxd

dtft(x) = −|∆|α/2(σ(x)ft(x)) −

∫

[(V (x, y),∇)ft(x)ft(y) + divxV (x, y)ft(x)ft(y)]dy,

+

∫

(ft(x− z)ψ(x− z, y; z)− ft(x)ψ(x, y; z))ft(y)dydz¬ uv®

a]O9Zcp8O9y¡[]bvO*qrXk­I£KMO9acO

divxV (x, y) =

d∑

i=1

∂Vi

∂xi(x, y), ψ(x, y; z) =

∫

ψ(x, y; z, w)dw,

Vilu^1x

xi8O*br^M1­½iu y*ik_MazZcOk­D[cKsO0y*iYikazxIb^1lo[cO;Zik)[cKsOvO9y[cikazZ

Vlk^sx

x $¢ [ beZp1ifZcZcbsqO$[cikO*[l£4O*qrq p8ivZcO9xI^MO;ZcZik ¬ kk®lkZ lªy*ik^sZcO9df_MO9^syOiuJ%KMO*ivacO9NU C Mio£ O9kO9a9­£4O3lkZ]Zw_MN0O;xAKMO9acO0[]Kslo[

âã&ä½âÖÞ

v

σ(x)brZ^Miu[loÀDO9y¡[]O9x©YX©b^f[]O*azlky¡[]biv^½­%£KMbeyzK £ iv_Mqex©qO;lkx[ci1O*[w[cO9aa]O9Zc_Mq[]Z ¬ lua]Mb[cazlua]Xbr^Mb[cbeluqy*ik^sxMb[]biv^sZ]®­M[]Kslu^ª[]KMivZcOivI[]lkb^MO;x[cKMa]ik_skKmJ%KMO9ika]O*N C § ZSl3pMa]O*qrbN0br^slua]XZw[cO9pª£ O(^MO*O;x[cKMO1lkZcbrypMa]ikp8O*ac[cbrO9Z4iu&[cKsOyika]a]O9Zcp1iv^sxIbr^M(qrb^sO9luapsacivMqO9N kbrkO9^YX3[]KMOikqrqrio£b^M$[cKsO*ika]O*N [cKslu[brZ

ikbr^sxIO9p1O9^sxIO*^f[br^f[cO9acO;Z\[ ·Y» · > P

α ∈ (1, 2)*σ(x)

*At(x) R P( ,% Bt(x)

*φt(x)

" ) R,% L P( ,%; PP 0"= = P '

x

σ, φ*6 )F

At, Bt, φt

FP PP R,% . */ R=,% ? 1!" t ≥ 0

*R/( 1

supt≤t0

max(σ(x), σ−1(x), |∇σ(x)|, |At(x)|, |∇At(x)|, |Bt(x)|, φt(x, z)(1 + |z|β+d)) ≤ C0(t0)

¬ of® /9 P β ∈ (0, 2)

*C0(t0) > 0

?x, z

P " 1 (- P

d

dtGt(x) = −σ(x)|∆|α/2Gt(x) − (At(x),∇Gt(x)) +Bt(x)Gt(x)

+

∫

(Gt(x+ z) −Gt(x))φt(x, z) dz¬ kk®

1 = G(t, s, x, y)

*t0 ≥ t ≥ s ≥ 0

* / ; P P

G(s, s, x, y) = δ(x− y)*#R/( P

G(t, s, x, y) = Sα(t, s, x− y)(1 +O(1) min(1, |x− y|) + O(t1/α))

+O(t)(1 + |x− y|d+min(α,β))−1, ¬ uv®>1

O(1), O(t1/α) P P,%

t0*C0(t0)

*

Sα(t, s, x− y) = (2π)−d

∫

exp

−tσ(y)|p|α + i(p, x,−y −∫ t

s

Ar(y)dr)

dp ¬ u®

F( L:1R % 1 4α 1 = R P4 = "

>1 PID F R ¬ uv® ( >P β ≥ α

" G(t, s, x, y) %8>1/)F #P P /) / 4-P8)

/ (* 7

Bt = 0 $P ∫

G(t, s, x, y)dy = 1

x

t > s

" G(t, s, x, y) 9 P R,% 7 L

t, s, x, y>1

t > s

∣

∣

∣

∣

∂G

∂x(t, s, x, y)

∣

∣

∣

∣

= O((t− s)−1/α)G(t, s, x, y) ¬ @v®

=P R,% 0 ≤ s < t ≤ t0

*x, y ∈ Rd

ääÉNPORQRSRT

k L

" /% Gs ∈ C∞(X)1 =P-P L R " Gt

C∞(X)

F1 /( % L ! /(-P ¬ ku® PP

t → Gt ∈ C∞(X)P

R ¬ kv® t > s R

Gs t = s

"/

Gt ∈ C1(X)

t > s

‖Gt‖C1(X) = O(t− s)−1/α‖Gs‖C(X); *

σ,A,Bt, φt ∈ Ck(X)*k > 0

*P / Gs 7→ Gt

. = ( P Ck(X)

=P R,% t ∈ [0, t0]

' ¢ ^(y*lvZwO Bt(x) = 0

lu^sxAt(x) = A(x)

8O*br^M[]bN0Ohbr^sxIO9p1O9^sxIO9^v[&[cKsbrZacO;Zw_sq[&brZikI[zlubr^MO9xbr^ > vA ¬ [cKMOSOnIbrZw[cO9^syOSiu[]KMO a]O*O9^3_M^sy[cbrik^b^y9lkZcO ψ = 0

1O9b^s(pMa]O*Ybrik_sZcqXikM[]lubr^MO;x3b^> PCRA® ¢ [a]O*N3lubr^sZ[ciiksZcO*a]kO []Kslo[[]KMO$psaciYiuhacivN > v AvO*^MO9a]lkqbDK*O;ZSZw[cazlubrkKf[cika]£4lka]xIqrX3[]i[]KMO0pMacO;ZwO9^f[^sik^I¤¥KMikN0ivkO*^sO*ik_1Z Zcb[]_slo[]biv^ / ¢ ^ > vAiv^MO y*lk^ <s^1xlkqrZci[\£ ik¤¥ZcbexIO9xO9Zw[cbrN0lu[cO;Z%ika%[]KMO acO9O*^_M^sy[cbrik^ ¬ kk®

» » ÐI ¸ 1/ =7 P ! 1 P Gs 7→ Gt

4 "1 =(/ 9/ M(X) 7→ M(X)1 P 1 ∗ ) 1L% R/( P 2 8% 171R #% 1(F P / R=!" 1 ; (-P ¬ vk® t > s

F "1 =( / (C1

∞(X))∗ 7→ (C1∞(X))∗1 < P 1 ∗ ) 1L%

' ¦sikqrqio£Z a]ikN J%KsO*ika]O*N C ¬ bM®%lk^sx[]KMOxI_slkqb[\Xlkacv_MN0O*^f[]Z J%KMOslvZwbeypMa]ikp8O*ac[cbrO9ZSiuh[cKMO0Zwivq_M[cbrik^m[]i[]KMO lk_syzKYXpMacivMqrO*N iuhO;df_slo[]biv^sZ ¬ ku®*­ ¬ kv®Slua]Oy*ikqrqO;y¡[]O9xb^[cKsOikqrqio£br^M$[cKMO9ika]O*N ·Y» · > P

α ∈ (1, 2)*σ(x)

1R = R/( P σ−1 . =(8*

σ ∈ C1(X)*V ∈ C1(X

" / = J! P # R L< = = =P R R P4F 1I( ?+* PP =

ψ8% 1 =

ψ(x, y; z) ≤ C

1 + |z|β+d¬ u~v®

x, z, y

/9 P β ∈ (0, 2)

*C > 0

P P L8 1 " 5 %I/)F f0 ∈ C∞∩L1(X)

P 4 7I=- P7/)F $L R

ft ∈ C∞ ∩ L1(X) ∩ C1(X)! 1 /( % L F ¬ uv® t 7→ ft

/ PP =

R+ 7→ C∞(X)P R ¬ uf® t > 0

" " 17/ f0 7→ ft

F 99 ,% PP ! 0" )P I / R ! L+

1 (X) (t, f0) 7→ ft . PP 9 ! R L

f0

ft

1R( 17( 1L%?' L1(X)

" " P/ f0 7→ ft

4 F F , ! 8 = P P R 1L %(" / R

µ0 7→ µt M+(X)

R/( P ¬ kk®91 3 % g ∈ C2(X) ∩C∞(X)

1I/ R=µt

1 1R #%ft ∈ L+

1 (X) ∩ C∞(X)

t > 0

âã&ä½âÖÞ

" P / µ0 7→ µt ? #P F,% 7 L 1(

µ0 4

kP ( " = δkµt

δµk0

(x1, . . . , xk;µ0) 9 # (1* =P R ,% = (

µ0 % =( . P R,% 7 L x1, . . . , xk

' (* σ(x)

*V (x, y)

*ψ(x, y; z)(1 + |z|β+d)

L C2(X)

& = ! x=P R2%

P( F P ? R LR(* 1 ( = δkµt

δµk0

(x1, . . . , xk;µ0)F1 #

PP R,% 7 Lx1, . . . , xk

PF PL %! 1 7(/ P (C2(X))∗

R 1 =L(*

∥

∥

∥

∥

∇2

(

g,δ2µt

δµ20

)∥

∥

∥

∥

≤ ‖g‖C2(X)C(t, ‖µ0‖) ,/6 P

C(t, ‖µ0‖) # PP 2% 7 D = P

g

" 1 R=3 1I! '" 19/ f0 7→ ft4 F J!

PP PI R F PD%(" / µ0 7→ µt

(C1(X))∗

R/( P ‖µt‖M(X) ≤ C(t)t−1/α‖µ0‖(C1(X))∗ .

/ § Z&beZO;lkZcbqrXZcO*O9^½­G[cKsO lu8iokOhN0lkpMpMbr^M t 7→ µtbeZ^siu[&y*ik^f[cbr^Y_Mik_1Z½br^ M+(X) ¬ _s^MqbrkOb[]Za]O9Zw[ca]bry[cbrik^ª[]i

L1(X)®b^2[cKsO$^MivacN []ikp8ikqrikkXªlvZSl_M^sy[cbrik^.iu

tlu[t = 0

iva lua]Mb[]a]lkacXµ0£KsbryzKªbeZ^siu[lksZcikqr_I[cO9qXyiv^v[]b^Y_Miv_sZ£b[cKma]O9Zcp8O9y¡[S[ciÁO*8O9Zck_MO$N0O9lkZc_Ma]O MO*^sy*O[cKsO(^MO;yO9Z]Zcb[\X[]i0£ ivac3br^[cKMO £4O9lk3[civp1ivqivkX3iu

M(X)­M£KMbeyzK£brqq1O O*nYpsqivb[]O9xb^[cKsO ^MOnY[J%KMO9ika]O*N

' _sa%Zw[cazlo[]O*vX3b^xIO9lkqbr^M£b[]KO9df_slo[]biv^ ¬ uv® beZ4[cKMOZ]luN0OlkZ%br^[]KMOpMa]O*Ybiv_sZ%ZwO;y¡[]biv^lk^sxbrZ)slkZcO9x$ik^[cKsOiksZcO*a]olo[]biv^[cK1lo[hl Zcikqr_I[cbrik^[ci[]KMbeZ O9df_slu[cbrik^0brZ l <snYO;xp1ivb^f[hikDlN3lupMpsb^Mut 7→ Ft

­M£KMO*a]Oikaut ∈ C([0, t0],M+(X))

[cKMO_s^sy¡[]biv^FtZwivqvO9Z [cKMO O;df_slo[]biv^

d

dtFt(x) = −|∆|α/2(σ(x)Ft(x)) − (Vt(x, [u]),∇Ft(x)) − divVt(x, [u])Ft(x)

+

∫

(Ft(x− z)ψt(x− z, z, [u])− Ft(x)ψt(x, z, [u]))) dz¬ kv®

£b[cK[]KMO b^sb[]brlkqxMlo[zlF0 = u0

­I£KMO*a]O

Vt(x, [u]) =

∫

V (x, y)ut(dy), ψt(x, z, [u]) =

∫

ψ(x, y; z)ut(dy).

¢ ^ iva]xIO9a[ciªZwivqvO[cKMbeZO9df_slu[cbrik^._sZwbr^MmJ%KMO9ika]O*N C0£4O0yzK1lu^MvO$[]KMO3_M^MY^Mio£^2_M^sy[cbrik^ Ft[ci

Gt(x) = σ(x)Ft(x)qO;lkxIbr^M[ci[cKsO O9df_slo[]biv^

d

dtGt(x) = −σ(x)|∆|α/2Gt(x) − (Vt(x, [u]),∇Gt(x))

+ [(Vt(x, [u]),∇ ln σ(x)) − divVt(x, [u])]Gt(x)

+

∫

(σ(x)σ−1(x − z)Gt(x− z)ψt(x− z, z, [u])−Gt(x)ψt(x, z, [u]))dzdy.¬ 1C;®

ääÉNPORQRSRT

@ L

§ pMpMqrXYb^M J%KMO*ivacO9N C[ci O;df_slo[]biv^ ¬ 1C9®3XfbrO*qexMZl´Zwivq_I[]biv^ Ft = U(t, s)Fs[]i´[cKsO lu_syzKYXpsacivMqO9N iu&O9df_slu[cbrik^ ¬ kf® b^[cKsOika]N

Ft(x) =

∫

σ−1(x)G(t, s, x, y; [u])σ(y)Fs(dy),

£KsbryzK0beZ _M^Mbedf_MO%ikaFs1O9b^Ma]ikN

C∞(X)lk^sx£KMO*a]O

G(t, s, x, y; [u])beZ[cKMO acO9O*^$_M^sy¡[]biv^3iu

O;df_slo[]biv^ ¬ 1C;® &§ Z)beZ)luqeZciS[]KMOy*lvZwObr^3J%KMO*ivacO9N CO9df_slu[cbrik^ ¬ vv®&beZ)xI_slkqI[cil ZcbN0brqrlka)O9df_slu[cbrik^½­s_I[h£KMbeyzK3_M^MqrbvO ¬ vk®pMa]O9ZcO*a]kO;Z yiv^sZ\[zlu^f[]Z MO*^1yO[]KMOÐluN0brqXiuD[]KMOikp8O*azlo[]ikazZ U(t, s)OnY[cO9^sxMZ

[]i l Zw[ca]ik^MvqXy*ik^f[cbr^Y_Mik_1ZZwO9N0bvaciv_MpsZ0iu beZwivN0O[ca]brO9Z3br^L+

1 (X)lu^sx M+(X)

­£KMO9acOmZw[ca]ik^My*ik^f[cbr^Y_Mb[\Xmb^ y*lvZwO M(X)

beZ_M^sxMO*azZ\[]ifiIx2b^.[]KMO3ZwO9^sZwO0ik []KMO0£ O;luª[]ikp8ikqrikkXªik M(X) J%KMOvO*Xpsacivp1O9aw[\X3ik[]KMbrZZcO*N0brka]ik_Mpikqrqrio£Zha]ikN ¬ @f®¡³ Ft ∈W 1(X)lu^1x

‖Ft‖W 1(X) ≤ C1(t0)(t− s)−1/α‖Fs‖M(X)¬ vk®

£KsO*^MO9kO*at > s

lu^sxFs ∈ M(X)

² O(ZcKsluqrqikqrqrio£ ^Mio£¯[cKMOZ]luN0O qrb^sO iu)lua]k_MN0O9^v[lvZ%b^mJ%KMO9ika]O*N CM_M[£b[]K u08O*br^Mlk^

lkacsb[]a]lkacX0O*qrO*N0O9^v[ikL+

1 (X)iva M+(X) ¢ ^1Z\[]O9lkxiu ¬ PC9®4iv^MO KslkZ

‖(L1(s) − L2(s))g‖L1(X) ≤ C1‖u1(s) − u2(s)‖M(X)‖g‖W1(X).¬ kv®

iv^sZcO9df_MO*^f[]qXv­ ¬ vv®¡­ ¬ vu®­ ¬ vk®4XYbO9qrx

sup0<s≤t

‖Fs([u1]) − Fs([u

2])‖L1(X) ≤ t1−1/αC1C2(t0)‖u0‖M(X) sup0<s≤t

‖u1(s) − u2(s)‖L1(X).

¬ uf®MO*^sy*O[]KMON3lupMpMbr^M

ut[]iFtbr^[]KMO Zw_ssZwO*[4ik

S0C([0, t0],M+(X))ivI[]lkb^sO9xYX <MnIbr^M

u0beZ4l

y*ik^f[cazlky[cbrik^£KMO*^MO9kO9at1−1/αC1C2(t0)‖u0‖M(X) < 1. ¬ fk®

iv^sZcO9df_MO*^f[]qXiv^MO ikM[]lubr^sZl3Zcikqr_I[]biv^[]i ¬ kk®%ik^[]KMO b^f[]O*a]Glkq [0, t1] ikaSlu^YX t1 ≤ t0Z]lo[]brZwXYb^s

¬ vu®­£KsbryzK´beZ _M^Mbedf_MO3iva u0 ∈ C∞(X) ¢ t1 < t0ik^MOy*lu^ a]O*p8O9lu[ [cKMbeZ pMa]iIyO9xM_MacOZw[]lkaw[]b^s

a]ikN [cKMO([cbrN0Ot1ikM[]lubr^Mbr^Ml3_M^sbrdf_MO(y*ik^f[cbr^Y_slo[]biv^iu[cKMO$Zwivq_I[]biv^[ci3[]KMObr^v[]O*a]oluq

[0, t2]£b[cK

t2 − t1Z]lo[cbeZwXfbr^M ¬ vk®­MO[zy ÁO[_sZZwKMio£ [cKsOmacO;df_Mba]O9xyiv^f[cbr^f_sb[\X pMacivp1O9aw[]bO;Z0iu[cKsOmyik^1Z\[]ac_sy[cO;xZcikqr_I[]biv^ VY_spMp1ifZwO

u1(t), u2(t)lua]O[cKMO<MnIO;xp8ikbr^f[]Z%[]i

ut 7→ Ft¬ b O []KMOyiv^sZ\[]ac_1y¡[cO;xZcikqr_I[]biv^sZ [ci3O9df_slu[cbrik^ ¬ kk®c®£b[cK3xIbÀDO*a]O*^f[)b^sb[]brlkqIGlkq_sO9Z

u10, u

20 ∈ M(X)

8iu[cK3Z]lo[]brZwXYb^M ¬ vk® Á½O*[ U i(t, s), i = 1, 2­kxIO9^Miu[]O

[]KMOyivaca]O9Zcp8ik^sxIbr^M$pMacivpsluflo[]ikazZ)[]i ¬ kk® J%KMO9^

u1(t) − u2(t) = U1(t, 0)u10 − U2(t, 0)u2

0 = (U1(t, 0) − U2(t, 0))u10 + U2(t, 0)(u1

0 − u20)

=

∫

U2(t, s)(L1s − L2

s)U1(s, 0)u1

0 ds+ U2(t, 0)(u10 − u2

0).¬ kv®

iv^sZcO9df_MO*^f[]qX

sup0≤s≤t

‖u1(s) − u2(s)‖M(X) ≤ ω sup0≤s≤t

‖u1(s) − u2(s)‖M(X) + ‖u10 − u2

0‖M(X)

âã&ä½âÖÞ

k~

£b[cKªlω ∈ (0, 1) ¬ 1O9b^M3vbvO*^YX3[cKMO q K Z iu ¬ vk®c®­slu^sx[cKY_sZ

(1 − ω) sup0≤s≤t

‖u1(s) − u2(s)‖M(X) ≤ ‖u10 − u2

0‖M(X).

J%KMbeZbrN0pMqrbO;Z [cKsOyiv^v[]b^Y_Mb[\X ¬ O*vO*^´Á½brpsZcKMb[Ky*ik^f[cbr^Y_Mb[\XI®ik%[cKMON3lkpMpMbr^M µ0 7→ µtbr^´[cKMO

^sika]N [civp1ivqivkX38iu[cKbr^L+

1 (X)lk^sx M(X) J%KMOy*ik^f[cbr^Y_Mb[\X ik

f0 7→ ftlkZl

L1(X)¤>oluqr_MO;x_M^1y¡[cbrik^iu

tikqrqrio£Z½a]ikN[]KMOZclkN0O pMa]ikp8O*ac[\X

ik[]KMO qbr^MO;luapMa]ikMqrO*N Vf[zlo[]O*N0O*^f[ ¬ bM®ikqrqio£ZDa]ikN¯[cKsOhp8ivZ]ZwbrMbrqb[\X[]ixIbÀ8O9acO9^f[cbelo[cO)[cKMO O9df_slo[]biv^ika µt£b[cKa]O9Zcp8O9y¡[

[]i0[cKMO(b^Mb[cbeluq&yiv^sxIb[cbrik^ªlua]Mb[cazlua]X3^f_sN(8O*aik[]bN0O;Z*­8lkZ%[]KMOa K Z iu[]KMO O9df_slu[cbrik^ªxIO*p8O*^1xMZiv^µtdf_slkxMa]lu[cbey*luqrqrX bÀ8O9acO9^f[cbelo[cbrik^qrO9lvxMZ[]i3[cKMO[cbrNO(^Mik^I¤¥KMivNivkO9^MO*iv_sZ%O9df_slu[cbrik^iu[\XYp1O ¬ kv®¡­£KsivZcO£4O*qrq¤>p8ivZcO9xI^sO9Z]Zhivqqrio£Z%luflubr^a]ikN J%KsO*ika]O*N C Ji3kO*[ ¬ M®%ik^MOO*nf[]O*^sxsZ%[cKMOlk1iovO <MnIO9xp1ivb^f[lkacv_MN0O*^f[]Z a]ikN M(X)

[ci(C1(X))∗

_sZcb^Mbr^sZw[cO;lkxiu ¬ vk®%lu^sx ¬ uv® []KMO b^sO9df_sluqrb[cbrO9Z

‖(L1(s) − L2(s))g‖(C1(X))∗ ≤ C1‖u1(s) − u2(s)‖(C1(X))∗‖g‖L1(X)lk^sxacO;Zwp8O9y[cbrkO*qrX

sup0<s≤t

‖Fs([u1]) − Fs([u

2])‖(C1(X))∗

≤ t1−1/αC1C2(t0)‖u0‖(C1(X))∗ sup0<s≤t

‖u1(s) − u2(s)‖(C1(X))∗ .

² O$ZcKsluqrqlupMpMqrX[cKMO0lu8iokO a]O9Zc_Mq[]Z[]i[]KMO$Zw[c_sxMXik [cKMOZwO9N0bvaciv_Mpiu)p8ivZcb[]bYb[\XpsacO;ZwO9acYbr^My*ik^f[cazlky[cbrik^sZTtF (µ) = F (µt)

iv^([]KMOZcpslky*O4ik88ik_s^sxIO9x$_M^1y¡[cbrik^1Z ik^ M+M (X)

­k£KMbeyzKbeZ)lZc_MI¤ZcO[br^ M+(X)

yik^1ZwbeZ\[]b^s$iu&N0O9lkZc_Ma]O9Z £b[]K^Mika]N3Z 8ik_s^sxIO9xYXM Á½O*[ C0(M+

M (X))xIO*^Mik[cO

[]KMO0yqrivZc_MacO(b^C(M+(X)) ¬ [cKMO3Zcpslky*O(ik)NO;lkZc_Ma]O9ZSbrZyiv^sZwbexIO9acO;xªb^A[cKMO ∗ ¤¥£4O9lu[civp1ivqivkXM®ik[]KMOZwO*[%ikp8ikqrXf^sikN0brlkqs_s^sy¡[]biv^sluqeZ

Csym∞,fin(X )

yik^1ZwbeZ\[]b^s(iu <s^Mb[cOqbr^MO;luayivN(Mbr^slo[]biv^sZhiu[cKMO_s^sy¡[]biv^sluqeZ%iu&[cKMOivacN

F (µ) =1

m!

∫

gm(x1, . . . , xm)µ(dx1) · · ·µ(dxm) = (gm, µ⊗m) ¬ u®

£b[cKgm ∈ Csym

∞ (Xm) m§ Z$p8ikqrXf^sikN0brlkq _M^sy[cbrik^slkqrZ$lua]O3Y^Mio£^ [cim8OxMO*^sZcObr^ C(M+M (Z))ivayikN0pslvy¡[Zcpslky*O9Z

Z ¬ £KMbeyzKbrZl0xIbracO;y¡[yiv^sZcO9df_MO*^1yOiu[]KMO ² O9bO9a]Zw[cazlkZ]Z [cKMO9ika]O*N®¡­Yb[ikqrqio£Z[]Kslo[ iva(lqriYy9luqrqX.yivNp1lky¡[N0O*[ca]bry3Zcpslky*OX[cKMOZcpslky*O

C0(M+M (X))

y*ik^sZcbrZw[]ZikO*qrO*N0O*^f[]ZFa]ikN

C(M+M (X))

Zc_syzK[cKslu[Fn → F

b^C(M+

M (X))­I£KMO*a]O

Fn(µ) = F (χnµ) ¬ χn8O*br^M0[cKMO

br^sxIbey*lu[civa _s^sy¡[]biv^ªiu[]KMO slkqqik&azlkxIbr_sZn£b[]KZcikN0O <MnIO9xy*O*^f[cO9az® tO;y*lkqq[cKslu[[cKMO(Glkacbelo[]biv^sluq½xMO*a]bolo[]bvO δF䵬 ikaZcKMikac[cqrX δF ®ikl_s^sy¡[]biv^sluq F ∈ C(M(X))beZxIO<s^MO9xlvZ

δF (x, µ) = limh→0,h>0

1

h(F (µ+ hδx) − F (µ)).

J%KsOZwpslvyOC1(M+

M (X))beZ xIO<s^MO9x lvZ[cKMOZcpslvyO9Zik _M^1y¡[cbrik^1luqeZZw_1yzKA[]Kslo[

δFOnIbeZ\[zZikaluqrq

x ∈ Xlu^sx

µ ∈ M+M (X)

lu^1xmbeZlyik^f[]b^Y_Miv_sZ_s^sy¡[]biv^2ik)[\£ iolua]brlkMqO;Z[]KMO*a]O ^MO0ZcO*O9ZSYX

ääÉNPORQRSRT

v L

br^sZcp1O;y¡[]biv^[]Kslo[ivaF ∈ C1(M+

M (X))lk^sxlua]Mb[]a]lkacX

µ, ν ∈ M+M (X)

[cKMOivacN(_sqrl

F (ν) − F (µ) =

∫ 1

0

ds

∫

δF (x, µ+ s(ν − µ))(ν − µ)(dx)

KsikqexMZlu^1x[cKslu[[cKMOZwp1lkyOC0(M+

M (X))yiv^v[zlubr^sZ%[]KMO$Zcpslky*O

C10 (M+(X))

iu_M^1y¡[cbrik^1luqeZ%acivNC1(M+

M (X))Zw_1yzK[]Kslo[

δF (·, µ) ∈ C∞(X)_M^Mbika]N0qX3iva

µ ∈ M+M (X)

·Y» · > P7 R=7 P3' 1 1 ,% P V (x, y)

ψ(x, y; z)(1 + |z|d+β)

L FC∞(X) ∩ C1(X)

? = PI' y=P R ,% R P4

P 9R DR 1 " 19/ (µ0, t) 7→ µt

I P ! / R LR µ0

*µt

1R ) 4 1 1L%

" P9 2% ' P F TtF (µ) = F (µt)

# R C(M+(X))

C(M+

M (X)) 5 %

M* >=4 ? ,% PP

C0(M+M (X))

" P R/R P C1,20 (M+

M (X))! C1

0 (M+M (X))

1R /! > = F

δF (x, µ) # PP 2% 7 D x 1 # ( 7R

L FC∞(X)

= RR2%9 µ ∈ M+

M (X)F ? PR ' 1

LB! 1I

Tt

C0(M+M (X))

"1 P I! 1,% = C2,sym∞,fin(X)

P /! 7 $ P! 1 = ' R ¬ o® g ∈ C∞(X)

# /R 7 L # /( 6R 9 L

C∞(X) I ?6 ! 1 F LB

' ¬ bÖ®0J%KMOªyiv^f[cbr^f_sb[\X ik[]KMON0lkpMpMbr^M t → µt

br^ [cKMO£4O9lu []ikp8ikqrikvX ikqrqio£Z$a]ikNxM_sluqrb[\X Á½O*[_sZZcKMio£¯[cKslu[b un

0 ∈ M+(X)yiv^fvO*a]kO;Zh£4O9luYqrX3[ci

u0 ∈ M+(X)­n → ∞ ­I[]KMO*^

un(t)yiv^fvO*a]kO;Z%[ci

u(t)£4O9lkfqrXikaO9lkyzK

t > 0­D£KMO*a]O

un(t)­n = 1, 2, . . . , u(t)

lua]O[]KMO<MnIO9xp8ikbr^f[]Z ikD[cKMON3lupspMb^s

ut 7→ Ft¬ yik^1ZwbexIO*a]O9x$br^0[cKMOpsaciYiuDiu8[cKsOpMa]O*Ybiv_sZ[]KMO*ivacO9N3®&£b[cK3br^Mb[cbeluqxslo[]l

un0 , u0

acO;Zwp8O9y[cbrkO9qX ÁO[ Un(t, s)lu^sx

U(t, s)xIO*^siu[cO$[cKsO$pMa]ikp1luvlu[civa]Zy*ika]acO;Zwp8ik^sxMb^M[ci

un(t)lu^sx

u(t) Piu[]bry*O[cK1lo[[]KMO$lvxGj\ikbr^f[Sivp1O9a]lu[cika[ci[]KMOikp8O*azlo[civa Lta]ikN[]KMO(a K Z Lt(Ft)ik ¬ kv®4O9df_sluqeZ

L∗t g = −σ(x)|∆|α/2g + (Vt(x, [u]),∇F(t)) +

∫

(g(x− z) − g(x))ψt(x, z, [u]))dz.¬ @v®

O*^siu[cbr^M

γnt (x) = Vt(x, [u

n − u]) =

∫

Vt(x, y)(un(t) − u(t))(dy),

δnt (x, z) = ψ(x, z, [un − u])(1 + |z|d+β)

âã&ä½âÖÞ

PC

¬ £ O_sZwO%[]KMO^siu[]lu[cbrik^sZ)b^f[ca]iIxI_sy*O9x0br^0[cKMOpsaciYiuDiuD[]KMOpMa]O*Ybrik_sZ&[]KMO*ivacO9N® lu^sx0_1Zwbr^M ¬ kk® XfbrO*qexMZ

γnt (x) =

(∫ t

0

(Uns,0)

∗(L∗n,s − L∗

s)U∗t,sV (x, ·) ds, un

0

)

+ (U∗t (t, 0)V (x, ·), un

0 − u0),

δnt (x, z) =

(∫ t

0

(Uns,0)

∗(L∗n,s − L∗

s)U∗t,sψ(x, ·; z)(1 + |z|d+β)ds, un

0

)

+ (U∗t (t, 0)ψ(x, ·; z)(1 + |z|d+β), un

0 − u0).

MO*^sy*O

|γnt (x)| ≤ K

(∫ t

0

(supx

|γns (x)| + sup

x,z|δn

s (x, z)|)ds, un0

)

+ |(U∗t (t, 0)V (x, ·), un

0 − u0)|

ivaZwivN0Oy*ik^sZw[]lk^f[K£b[cKªl0ZwbrN0bqeluaO;Z\[]bN3lu[cOika

δtXYbO9qrxMb^Miva

ξt = supx

|γnt (x)| + sup

x,z|δn

t (x, z)|

[]KMO O9Zw[cbrN3lo[]O

ξt ≤ K

∫ t

0

ξsds+supx

|(U∗t (t, 0)V (x, ·), un

0 −u0)|+sup |(U∗t (t, 0)ψ(x, ·, z)(1+ |z|d+β), un

0 −u0)|

¬ £b[]KZcikN0OSiu[cKsO*ayiv^sZ\[zlu^f[ K ® § Z []KMO ZwO*[]Z4iu½_s^sy¡[]biv^sZ V (x, ·) lk^sx ψ(x, ·; z)(1 + |z|d+β)lua]O

y*ikN0pslvy¡[b^C∞¬ 1iv_M^sxIO;xA£b[cK._M^Mbika]N0qXm8ik_s^sxIO9x.xIO9acbrolo[cbrkO;Z]®­D[cKMO0qelkZw[[\£ i[cO9acN3Zik^2[cKMOa K Z ik[cKMbeZbr^MO9df_slkqb[\X[]O*^sx[]i/K*O9acilvZ n → ∞ _M^sbivacN0qrXiva t ∈ [0, T ]

ivalua]Mb[cazlua]XT > 0

iv^sZcO9df_MO*^f[]qX ¬ xI_sO[ci aciv^f£%luqrq Z qrO*N0N3lv® [cKMOZclkN0OKMivqrxMZika ξt ­ lu^sx´KsO*^sy*Oika γtlk^sx

δt iv^sZcO9df_MO*^f[]qXv­ ¬ kv®4bN0pMqrbO;Z

(g, un(t) − u(t)) = O(1) sups∈[0,t]

ξs

∫ t

0

s−1/αds+ |(U∗t (t, 0)g, un

0 − u0)|

iva lua]Mb[]a]lkacXg ∈ C∞

­£KMbryzK2bN0pMqrbrO9Z[cKsO0y*ik^YkO9acvO*^sy*O ¬ g, un(t) − u(t)) → 0lk^sxmKMO9^syO[]KMO

£4O9lkyiv^fvO*a]kO9^syOun(t) → u(t) ¬ bbÖ® H X ¬ be®[]KMO3ZwO9Nbrka]ik_sp Tt

pMacO;ZwO9acvO9Z[cKMO0Zwp1lkyO;ZC(M+

M (X))lk^sx

C(M+(X)) ÁO[_sZZcKMio£[cKslu[)[cKMOZcpslvyOC0(M+

M (X))beZ)br^YGlkacbelu^f[ _M^sxMO*a []KMOlky[cbrik^0iu8[cKMbeZhZcO*N0brka]ik_Mp JixIi[cKMbeZiv^MO^MO9O9xMZh[ciZwKsio£ []Kslo[ [cKMO_M^sy[cbrik^slkq

F (µt)8O*qrik^MfZ [ci

C0(M+M (X))

iva lk^fXt > 0

lk^sxFiu

ivacN ¬ u® ¦sika4ZwbrN0pMqbeyb[\X0qO*[ _sZ kbrkO[cKMOlua]k_MN0O*^f[zZikah[cKMOy*lvZwO k = 1ik^MqrXk­IxIO9^Miu[]b^M

g1 YX g¬ [cKMO vO*^MO9a]lkq½y*lvZwObeZdf_Mb[]OZwbrN0bqelua¡® ÁO[ χnxIO9^Miu[]Ol3ZwN0iYiu[]KMO9xbr^sxIbey*lo[]ika%_M^1y¡[cbrik^ªik[]KMO slkqq

ik8azlkxMb_sZnbr^X ¬ l yiv^v[]b^Y_Miv_sZ&_M^sy[cbrik^0[]Kslo[)O;dv_1luqeZ ik^MObr^[cKMbeZ)1luqrq=­vGlk^MbeZwKMO;Zik_I[zZwbexIO%[]KMOslkqqikazlkxIbr_sZ

n+ 1lk^sxbrZ%8ik_M^1xIO9xYXiv^MOO*kO9acXY£KMO9acOG® VYbr^syOival0yikN0pslvy¡[ X [cKMO p8ikqrXY^MikN0beluq_s^sy¡[]biv^sluqeZSlkacO(f^sio£^ª[ci1O0xIO*^1ZwO$b^

C(M+M (X)) ¬ £KsbryzK.R brZlk^mO9lvZwXyiv^sZwO;df_MO*^sy*Oiuh[cKMO

² O*brO*azZw[cazlkZ]Z½[]KMO*ivacO9N3®­o[cKsO _s^sy¡[]biv^sluq (g, µt(χnµ0))8O*qrik^MfZ[]i

C0(M+M (X))

ika lu^YXn MO9^syO

ääÉNPORQRSRT

f L

b[acO9N0lkb^1Z[ciZwKsio£¯[]Kslo[(g, µt(χnµ0))

yik^YvO*a]kO9Z4[]i(g, µt(µ0))

ivalkacsb[]a]lkacXt_s^MbivacN0qrXika

lkqqµ0 ∈ MM (X) )§ flubr^_sZcb^s ¬ vv®4ik^sO£a]b[cO9Z

(g, µt(χnµ0)−µt(µ0)) =

(∫ t

0

(Uns,0)

∗(L∗n,s − L∗

s)U∗t,sV (x, ·)ds, un

0

)

+(g, U(t, 0)(χnµ0 −µ0)).

¬ k~v®J%KsO2ZcO9yiv^sx[cO9acNKMO9acO2brZ_M^Mbika]N0qX¨ZcN3luqrq=­8O9y9lu_sZcOg ∈ C∞(X)

­Slu^sxU∗

t,0

pMa]O9ZcO*a]kO;Z0[cKsbrZpsacivp1O9aw[\Xª_M^sbivacN0qrXk­lvZ[cKMOy*iYO GybrO*^f[]Zik%lkqq vO*^MO9a]lu[civa]Zlua]O0_M^Mbika]NqrXm8ik_M^1xIO9x xI_MO0[]i[cKMOlvZcZc_MN0pI[]biv^ [cK1lo[ |µ0| ≤ M ¦Miva$[]KMO <sazZ\[[cO9acN []KMbeZ0brZ^Mik[3xMba]O9y[cqrX´ikYYbiv_sZ H _M[0£4Oy*lu^_1ZwO[cKsO$[]acbeyzªa]ikN []KMO3psluac[ ¬ bÖ® PlkN0O*qrXk­y*ik^sZcbrxMO*a <sazZ\[[cKsO$_s^sy¡[]biv^sZ g iu ika]N V (x, ·) lu^sxψ(x, ·; z)(1 + |z|d+β)

lu^sx2pMa]iokO$[cKMO3ZcN3luqrq^MO;ZcZ_sZcbr^M aciv^f£%luqrq ZSqO9N0N0l $² KsO*^.[cKMbeZbeZ xIik^MO[]KMOZcN0lkqqr^MO;ZcZ4iu&[cKsO<sa]Zw[%[cO9acN b^ ¬ v~k®48O9yivN0O9ZyqrO9lka ¢ [a]O*N3lubr^sZ4[]i0ZcKMio£¨[]Kslo[ Tt

beZZw[ca]ik^MvqXy*ik^f[cbr^Y_Mik_1Z iv^p8ikqrXY^MikN0beluqD_M^1y¡[cbrik^1luqeZ%iu&[cKMOika]N¬ o® )§ flubr^qrO[_sZ%a]O9xM_syO[cKsOxIbrZ]y_1ZcZcbiv^[]i[cKMOy9lkZcO k = 1

iv^MqX J%Kf_1Z%£ O^MO9O9x[ci3ZcKMio£ []Kslo[(g, µt)

[cO*^1xMZ[]i(g, µ0)

lvZt → 0

ikalk^YXkbrkO9^g ∈ C∞(X)

_M^sbivacN0qXikaµ0£b[]Kml01iv_M^sxIO;x

^sika]N ¦sika%[cKsbrZb[beZ%O*^Miv_MkK[ci3pMa]iokO[cKslu[

sup‖µ0‖≤M

‖U(t, 0)∗g − g‖C∞(X) → 0,

£KsO*a]OU(t, 0)

brZ [cKMOpsacivpsluflo[civay*ika]acO;Zwp8ik^1xIb^s[]iµ0 H4_I[[cKsbrZivqqrio£Zlkvlubr^2acivN[cKsOikI¤ZcO*a]olo[]biv^ [cKslu[([cKsOyiYOG3y*bO9^f[]Z$iu[cKsOy*ika]acO;Zwp8ik^sxMb^MªvO*^MO9a]lu[civa]Z lkacO_s^MbivacN0qrX 1iv_M^sxIO;x ika

‖µ0‖ ≤M ¬ bbrbe®«hdf_slo[]biv^ ¬ kk® XYbrO*qexMZ

d

dtF (µt = −

∫

σ(x)|∆|α/2δF (x, µt)µt(dx) +

∫ ∫

(V (x, y),∇xδF (x, µt))µt(dx)µt(dy)

+

∫

(δF (x + z, µt) − δF (x, µt))ψ(x, y; z) dzµt(dx)µt(dy)¬ ov®

ivaF ∈ C1,2

0 (M+M (X)) Ji3psaciovOS[]Kslo[ F 8O*qrik^MfZ%[ci0[]KMO(xIivN3lubr^ik LB

­1ik^MO ^MO9O9xMZ[ciZcKMio£[]Kslo[

(F (µt)−F (µ0))/tKslkZlqbrN0b[lkZ

t→ 0_s^MbivacN0qrXikalkqq

µ0acivN M+

M J%KMO acO9pMa]O9ZcO*^f[]lu[cbrik^

F (µt) − F (µ0) = td

dt|s=0 F (µs) +

∫ t

0

(

d

dsF (µs) −

d

ds|s=0 F (µs)

)

ds,

brN0pMqrbO;Z [cK1lo[(b[$beZ(Zc_=GybrO*^f[[]iAZwKsio£ []Kslo[ ddtF (µt) − d

ds |s=0 F (µs)[cO9^sxMZ([ci K9O*a]iªlkZ

t → 0_s^MbivacN0qrX.ikaµ0a]ikN M+

M H4_I[0xM_MO[]i ¬ uk®*­[cKMbeZ$xMbÀDO*a]O*^sy*Oy9lu^ 8O£a]b[c[cO9^ lkZ$lmqrb^sO9luay*ikN$Mb^1lo[cbrik^ ¬ £b[]K¨1iv_M^sxIO;x¨yiYO GybrO*^f[zZ]®0ik[cKMOAxMbÀDO*a]O*^sy*O9Z3ik[cKMOªivacN (φ, µt − µ0)£b[cK

vbvO*^_M^sy[cbrik^sZφa]ikN

C∞lu^sx[]KMOmbr^v[]O*va]lkqrZ ∫

(δF (x, µt) − δF (x, µ0))µ0(dx) J%KsO <sazZ\[xMbÀDO*a]O*^sy*O9Z[]O*^sx2[ci K9O*a]iYXA[cKMO3pslkaw[ ¬ bbÖ®iu [cKMOJ%KsO*ika]O*N J%KMO0br^f[cO*va]lkq brZ ZcN3luqrq)8O9y*lk_sZcOik[]KMO0yik^f[]b^Y_Mb[\XiuδF (x, µ)

b^µ ¬ _s^MbivacN iv^ x lu^sx µ acivN y*ikN0pslvy¡[ZcO[zZ]® H4XJ%KsO*ika]O*N ¬ bM®[cKsO0ZcpslvyO C1,2

0 (M+M (X))

beZbr^folkacbelu^f[S_s^sxIO*a[]KMO3lky[cbrik^Aikh[]KMO3ZwO9Nbrka]ik_spTtlu^1xAKMO9^syO

a]O*psacO;ZwO9^v[zZ%ly*ika]O

âã&ä½âÖÞ

v

¬ bM®¦sikqrqio£ZacivN []KMOmp8ivZ]ZwbrMbrqb[\X[cilkpMpMa]iGnIbN3lo[]O[cKMOm_M^sy¡[]biv^sluqeZacivN C1,20 (M+

M (X))¬ [civkO[]KMO*a%£b[cK[cKsO*braxIO*a]bolu[cbrkO9Zz® YXp1ivqXY^MivN0brlkqrZ9­Y£KMbryzKikqrqrio£Z4acivN g)a]ikp8ivZcb[]biv^ § Ciu&[cKMO§ pMp8O*^sxIbn

* © 0 .%&ÁO[

h > 01O0lp8ivZcb[]bvO$p1luazluN0O[]O*a ² OZcKsluqrqxIO*^Mik[cOYX δx []KMO bra]lvyN0O9lvZw_sacOlo[ x § vO*Xa]ikqrOb^[cKsO[cKMO9ika]Xiu&NO;lkZc_Ma]O¤¥Glkq_sO9xqrbrNb[]ZbrZ%psqrlGXvO9xYX[cKMO br^syqr_sZcbiv^

x = (x1, . . . , xl) 7→ hδx = h(δx1+ · · · + δxl

), ¬ C;®£KsbryzKmxIO<s^MO;ZSlKMikN0O*ivN0ika]pMKMbeZwN 1O*[\£ O9O*^

SX lu^sx[cKsOZcO[ M+δ,h(X)

iuh¤¥Z]y*lkqO;x <s^Mb[cO$Zc_MN3Z

ik䤥N0O9lkZc_Ma]O9Z ÁO[

Xlk^sx

D8O[cKMOªZ]luN0OªlkZb^©J%KMO9ika]O*N Cu­%lu^sx qrO[ B1, . . . , BK

1Oªl yivqqrO9y[cbrik^iky*ik^sxMb[]biv^sluqrqX.p8ivZcb[]bvOikp8O*azlo[]ikazZ br^

C∞(X), . . . , C∞(XK)a]O9Zcp8O9y¡[]bvO*qrXA[]Kslo[kO9^MO*azlo[]O¦MO9qqrO*a

ZcO*N0brka]ik_Mp1Zlu^sx­;br^ psluac[cbey_Mqelua;­;KslGkOa]O*pMa]O9ZcO*^f[]lu[cbrik^ ¬ kk® &§ ZB(t)' K¤ %7

X ­D£KMO*a]O(lk^fXyivqqrO9y[cbrik^ªik k ≤ Kp1luac[cbeyqrO9Zbr^f[cO9a]lvy¡[Slvy*yiva]xMb^M0[]i

Bk­DbrZxIO<s^MO9xª[cKMa]ik_MvK

b[]ZkO*^sO*azlo[civa

LBf(x1, . . . , xl) =

K∑

k=1

Lkf(x1, . . . , xl) =

min(K,n)∑

k=1

∑

I⊂1,...,l:|I|=k

(BIkf)(x1, . . . , xl)

¬ uk®

¬ ZwO9O^Miu[zlo[cbrik^1Z4ika BIk

b^ªJ%KMO9ika]O*N u® ² OZwK1luqrq½Z\[]_sxIX^sio£ l0Zcy9luqrb^M0qrbN0b[iu[cKMO qelua]kO^Y_MN(8O*aikpslkaw[]bry*qO;Z%£KMO*a]O k ¤¥lkacX3pslkaw[ Lkiks[]KMOkO9^MO*azlo[]ika&brZ Z]y*lkqO;xYX [cKsO%Ðlky¡[]ika

hk−1 lk^sx x = (x1, . . . , xl)beZ Zc_MsZw[cb[c_I[]O9xYX(lNO;lkZc_Ma]O

lvy*y*ikazxIb^s[]i ¬ CG® O9^Miu[]b^M F (hδx) = f(x)ika

f ∈ Csym(X )XfbrO*qexMZ[]KMOZ]y*lkqO;xAvO*^MO9a]lu[cika

iv^C(M+

δ,h(X))vbvO*^YX

(LhBF )(hδx) =

K∑

k=1

hk−1 ¬ ov®£b[cK

LhkF (hδx) = (Lkf)(x). ¬ of®

¦sikalqbr^MO;luaikp8O*azlo[]ikaBkb^Csym(Xk)

qO*[Bl

k, l = 1, 2, . . . ,xMO*^Mik[cO[cKMO3qrbr^MO9lkaivp1O9a]lu[civa]Z

Csym(X l) → Csym(Xk)xIO<s^MO;xlvZ

(Blkg(x) =

l∑

q=1

(−1)l−q

∫

Xl−q

Bx

k

(∫

Xq

g(y, z)δ⊗qx

(dy)

)

δ⊗(l−q)x

(dz),

£KsO*a]Oiky*ik_MazZcOx ∈ Xk ­ y ∈ Xq ­ z ∈ X l−q lu^1x Bx

k

N0O9lk^sZ%[]KMOlky[cbrik^iuBkiv^[]KMO olua]brlkMqrO

x ¢ ^pslkaw[]bry*_Mqelua;­(B1

kg(x) = Bk

(∫

g(y)δx(dy)

)

= (Bkg+)(x), ¬ kk®

ääÉNPORQRSRT

k L

(Blkg(x) =

l∑

q=1

(−1)l−q

q!(l − q)!(Bx

1 g(x, . . . , x, z, . . . , z)) |z=x,¬ ov®

£KsO*a]OxbeZ%£a]b[c[cO*^

q[cbrN0O9Zlk^sx

zbeZ%£acb[w[]O*^

l − q[cbrN0O9Zbr^[cKMO qelkZw[ika]N(_Mqel

»h» Ð » Bk

P R +* Bk1 = 0

F (µ)

+% ¬ o® g = gm ∈ C2,sym

∞ (Xm) L

x = (x1, . . . , xn)

n ≥ max(2, k) 9(1 1

hk−1LhkF (hδx) =

m∑

l=1

hl−1

∫

Xm−l

(

Φkh[Bl

kgw](hδx))

(hδx)⊗(m−l)(dw), ¬ u®

>P gw(z) = g(w, z)

Φk

h[Blkgw]

7 5% 1R ! 12 ' H4X ¬ uk®¿¤ ¬ ov®

hk−1LhkF (hδx) =

1

m!hm+k−1

∑

I⊂1,...,n:|I|=k

BIk

∑

j1,...,jm=1

g(xj1 , . . . , xjm),

£KsO*a]OBI

k

N0O9lk^sZh[cKMOlky[cbrik^ikBkik^3[]KMOSolkacbeluMqrO9Z

xI O9^Miu[]b^s$YX q [cKsO^Y_MN(8O*a%ik½b^sxMOnIO9Zbr^ j1, . . . , jm []Kslo[8O*qrik^s$[]i I ik^MOa]O*£a]b[]O9Z4b[lkZ

hm+k−1

m!

∑

I⊂1,...,n:|I|=k

BIk

m∑

q=1

Cqm

∑

j1,...,jq∈I

∑

i1,...im−q 6∈I

g(xj1 , . . . , xjq, xi1 , . . . xim−q

)

=hm+k−1

m!

m∑

q=1

Cqm

∑

I⊂1,...,n:|I|=k

BIk

∫

Xq

∫

Xm−q

g(y, z)δ⊗qxI

(dy)(δx − δxI)⊗(m−q)(dz)

=hm+k−1

m!

m∑

q=1

Cqm

m−q∑

p=0

Cpm−q(−1)p

∑

I⊂1,...,n:|I|=k∫ ∫ (

BIk

∫

g(y, z,w)δ⊗qxI

(dy)

)

δ⊗pxI

(dz)δ⊗(m−q−p)x

(dw),

£KsbryzKYXyzKslu^Mvb^s([]KMOb^sxMOnp[ci[]KMO b^1xIOn

l = q + plu^1x_sZcbr^M

CqmC

pm−q = Cl

mCpl

acO;lkxMZ%lkZ

hk

m!

m∑

l=1

hl−1Clm

l∑

q=1

Cql (−1)l−q

∑

I⊂1,...,n:|I|=k∫ [∫

BIk

(∫

g(y, z,w)δ⊗qxI

(dy)

)

δ⊗(l−q)xI

(dz)

]

(hδx)⊗(m−l)(dw),

£KsbryzKXYbO9qrxMZ ¬ u®4fX ¬ C C*v®

âã&ä½âÖÞ

f

§ Z%ika F iuika]N ¬ u®4ik^sOK1lkZ

δlF

δY l(x1, . . . , xl;Y ) = (gm(x1, . . . , xl, ·), Y ⊗(m−l))

b[ikqrqio£ZacivN ¬ k®[]Kslo[ik^sO$y*lk^AO*nf[]O*^sxm[cKMOikp8O*azlo[]ika Lhk

[cilky¡[ik^ANivacO$kO9^MO*azluqb^=<1^Mb[]O*qrXxMbÀDO*a]O*^f[cbelusqO_M^sy[cbrik^sZ

F (Y )ikN0O9lvZw_Ma]O9Z4YX[]KMOika]N(_sqrl

hk−1LhkF (Y ) =

∞∑

l=1

hl−1Φkh

[

Blk

δlF

δY l(Y )

]

(Y ). ¬ @v®

§ Zl0xIba]O9y[yiv^sZwO;df_MO*^sy*OSik ¬ @v®4ik^MOikM[]lubr^sZlkqrZci[cKMOivqqrio£b^s »h» Ð »

Bk 7 1 R *

Bk1 = 0

F (Y ) = (g, Y ⊗)

g ∈ Csym∞,fin(X ) ∩ C2(X )

f(x) = F (hδx) = (g, (hδx)⊗) =

M∑

m=1

hm

m!

l∑

i1,...,im=1

g(xi1 , . . . , xim) ¬ o~v®

x = (x1, . . . , xl)

1

(LhBF )hδx) = (LBg, (hδx)⊗) +O(h), ¬ @vv®

>P LB 75% ¬ vu® O(h)

FP IM P9/5 / /)F '= 71 '

g" ‖g‖C2(X )

J%KMOlu^1luqrXv[]brylupspMaciflkyzK0[]i$pMa]ioYb^s[cKsOSqelG£ iuqrlkacvO^Y_MN(8O*azZ4brZ%xIO;Zcy*acbr1O;xYX0[cKsOivqqrio£b^s

a]O9Zc_Mq[ ·Y» · P P P ¬ ku® < F C2

c (X )" FR D

T h

t

Csym

∞ (X )* P J(-P ¬ C9v® : 14' 1 R 1 )

,% P Tt

C0(M+

M (X)) 1 P

C2,sym∞,fin(X )

& 9 F P 1& ,%! L T h

t P ,% F P Tt

h → ∞ P =D(* 1 ,%J! 9 R

ZhB(t)

M+δ,h(X)

1( (.+% F ¬ ov®9 R9 2% F 1= F R P ) 5 R (65% 1 (-P 7>1 1 P P P ' ¦Mivqqrio£Z3a]ikN ¬ @vv®lu^sx l£ O9qqY^Mio£^¨kO9^MO*azluqSacO;Zw_sq[ ¬ ZcO*OAO >Ï=CRA iva> Aika

xMbÀDO*a]O*^f[ pMa]ifikÐZ]®Siv^2[]KMOyiv^YkO*a]kO9^syO$iu%yik^f[]a]lvy¡[cbrik^.ZwO9N0bvaciv_MpsZ Rmika]O$pMa]O9y*brZcO*qrXk­½[cKMOlk1iovOa]O9Zc_Mq[Zw[]lu[cO9Z[cKsOy*ik^YkO9acvO*^sy*Oika[]KMOb^Mb[cbeluq y*ik^sxIb[cbrik^1Zik [cKMOYb^MO*[cbeyO;dv_1lo[cbrik^1Z8O*br^MO;df_sluq[]i[cKslu[iu

ZhB(t)­1lk^sxiv_MayqelubrN []KMO*^ikqrqrio£Z4acivN [cKMO £4O9lkyiv^v[]b^Y_Mb[\Xik

Tt

ääÉNPORQRSRT

v L

¢ ^$VIO9y¡[]biv^£4OhxIbexlkqqv[cKMO ^MO9y*O9Z]ZclkacXhj\ik []iSlkpMpMqrX[]KMbrZ[cKsO*ika]O*Nµ[]i[cKMO%ZcXYZw[cO9N¯ikMb^f[]O*azlky¡[]b^sZw[]lkMqrO0j\_MN0p©xMbÀD_sZcbiv^sZZcp1O;ybL<sO9xYX []KMOªkO9^MO*azlo[]ikazZ ¬ uv®¡­ ¬ 1C9® ¢ ^ []KMbeZy*lvZwOivp1O9a]lu[cika ¬ uk®8O9y*ikN0O9Z

Lf(x1, . . . , xl) = −l∑

j=1

σ(xj)|∆xj|α/2f(x1, . . . , xl) + 2

∑

i<j

[

V (xi, xj)∂f

∂xi+ V (xj , xi)

∂f

∂xj

+

∫

(f(xi + x1, xj + x2) − f(xi, xj))ψ(xi, xj ;x1, x2)dx1dx2

]

, ¬ @1C;®

£KsO*a]Ofbr^ []KMOªqelkZw[b^f[cO9kazluqxIO9p1O9^sxMZiky*ik_MazZcOiv^ik[cKMO9alua]k_MN0O9^v[zZ[cKslu[lua]O^Miu[£a]b[c[cO9^

O*nIpMqbeyb[cqrX [ci ZcKMikac[cO9^©[cKMOªivacN$_Mqel qrO9lkacqrX C∞(X l)beZbr^YGlkacbelu^f[3_M^sxMO*a

LivaO9lvyzK

l­lu^sx

[]KMO3a]O9Zw[ca]bry[cbrik^AikL[]iO;lkyzK

C∞(X l)vO*^MO9a]lu[cO;ZSlª¦sO*qrqO9aZwO9Nbrka]ik_spm[cKsO*a]O ¬ YX2J%KMO9ika]O*N C9®£KsO*^MO9kO*a

ψ(x, y; z, w) ≤ C

1 + (|z| + |w|)2d+β¬ @vk®

£b[cK.ZwivN0O(p8ivZcb[]bvOClk^sx

β ¬ Zw[ca]bry[cqrXmZwp8O9lkfbr^M[cKMO0yiv^sxIb[cbrik^miv^m[cKsOxIO9^sZcb[\Xiuh[cKMOb^f[]O*kazluqivp1O9a]lu[civab^ J%KMO*ivacO9N CxMbÀDO*azZa]ikN ¬ @fu®*­&s_I[$[]KMOiv^MqrX.a]O*qrO*olk^v[$pMacivp1O9aw[\X2iu[]KMbrZxIO9^sZwb[\XbeZS[cKslu[b[ xIO <s^sO9Zl8ik_s^sxIO9xAb^f[]O*kazluqikp8O*azlo[]ika;­1£KMbeyzKAbeZyqrO9lkacqrX[cKMO0y*lvZwO$_M^sxIO9a ¬ @vu® § flubr^YXJ%KMO*ivacO9N C[]KMO$y*ika]Oiu[]KMO(vO*^MO9a]lu[cika L iu Tt1O9qiv^MvZ[ci[]KMO$xMikN3lubr^ªiu

L ¬ []lkkO*^ªbr^mb[]Za]O*psacO;ZwO9^v[zlo[]biv^br^ M+δ,h(X)) J%KMbeZ%bN0pMqrbO;Z%[cKsOikqrqio£br^MacO;Zw_Mq[

·Y» · ¬ @vk® 1 =7 P ' 1 1 P ¬ kk®(* ¬ =C;® R5R % F (% ? P PI P P / 9! 1 § Zl <sazZ\[Z\[]O*p2b^2[cKMO0xIbracO;y¡[]biv^miu)[]KMO0yO*^f[]a]lkqqrbN0b[9­½£4O$ZcKslkqqpMa]ikp8ivZcO^Mio£ lN0O[cKsiYxmikaO;Z\[]bN3lu[cbr^M.[cKMOazlo[cOikyiv^YkO*a]kO9^syOb^¨J%KMO9ika]O*N Cu­4bqrqr_sZ\[]a]lu[cbr^M b[3iv^ iv_Ma3N3lubr^ N0iIxIO*qikbr^f[cO9a]lvy¡[cbr^M.Z\[zluMqrOpsaciIyO;ZcZcO9Z Ji2[cKsbrZ0O9^sx­)iv^MO^sO*O9xsZ([]i O9Zw[cbrN0lu[cOZwO9acbrO9Z ¬ @k® J%KMbeZ$beZ^Miu[xMbGy_sq[iva8ik_M^sxMO9xkO9^MO*azlo[]ikazZ

Bk¬ ZcO*O O*nMluN0pMqrO9Zbr^AVYO9y[cbrik^ o®­sM_I[S1O;yikN0O;ZbN0p8ivZ]ZwbrMqrObr^vO*^MO9a]lkqMy*lvZwO Ji(ybrazy_MN$kO*^f[ []KMbrZ xIbLG3y*_Mq[\X$£4OZcKsluqrqMkbrkOSlu^3luq[cO9ac^slu[cbrkOa]O*pMa]O9ZcO*^f[zlo[cbrik^$ikah[cKMOq K Z iu ¬ @k® ¦sika%[]KMOkO*^sO*azlo[civa]Zhiu[cKMO%j\_MN0p[\XYp1Ov­I£ O ZcKsluqrqxIi0b[%ivaZwbrN0pMqbeyb[\Xiv^MqX0iva%[cKMOy9lkZcO

k = 1 ¢ B1

beZvbvO*^YX[cKMO br^f[cO*va]lkqDpsluac[iu ¬ CG®¡­ ¬ u®­Ib O

(B1g)(x) =

∫

(g(x+ y) − g(x) − (∇g(x), y)χ(y))ν(x, dy), ¬ @kf®

[]KMO*^Lh

1F (hδx)O;dv_1luqeZ

∫ ∫ [

1

h(F (hδx + h(δz+y − δz)) − F (hδx)) −

(

∇zδF

δµ(z;µ), yχ(y)

)]

(hδx)(dz)ν(z, dy)

âã&ä½âÖÞ

ivaF (hδx) = f(x)

­M8O9y9lu_sZcO yqrO9lua]qrX∫

(f(x1 + y, x2, . . . , xn) − f(x1, . . . , xn) − (∇x1f(x), yχ(y)))ν(x1, dy)

=

∫

(F (hδx + h(δx1+y − δx1)) − F (hδx) − h

(

∇x1

δF

δµ(x1;hδx), yχ(y)

)

ν(x1, dy).

MO*^sy*O[cKMO ivp1O9a]lu[cikaLh

1Fy*lu^8O OnY[cO9^sxIO9xYX3[]KMOika]N(_Mqel

Lh1F (µ) =

∫ ∫ [

1

h(F (µ+ h(δz+y − δz)) − F (µ)) −

(

∇zδF

δµ(z;µ), yχ(y)

)]

µ(dz)ν(z, dy)

¬ @uf®[]iA[]KMO_M^sy[cbrik^sZFiv^N0O9lvZw_sacO;ZZc_syzK[cKslu[ δF

δµ (x;µ)OnIbeZ\[zZ*­ brZ0xIbÀ8O9acO9^f[cbeluMqrObr^

x­ lk^sx[cKMO

br^f[cO9kazluq8b^ ¬ @kv® brZ4£4O*qrqDxMO <s^MO;x ¢ N0ivacO9iokO*a;­ δ2Fδµ2 (x, y;µ)

OnIbrZw[]Z%lk^sx8O*qrik^MfZ [ciC2(R2d)

lvZ l_s^sy¡[]biv^ªiu

x, y_M^Mbika]N0qXika

µa]ikN lk^YX8ik_M^sxMO9x ¬ br^ª^MivacN®%ZcO[9­s[cKMO9^ªOnIpslk^sxIbr^M[cKMOa K Z ik ¬ @uf®4b^ h lu^1x_1Zwbr^M ¬ @vk®4XYbO9qrxsZ

Lh1F (µ) =

(

B1δF

δµ, µ

)

+ h

∫ ∫ (∫ 1

0

δ2F

δµ2(µ+ th(δz+y − δz))dt, (δz+y − δz)

⊗2

)

ν(z, dy)µ(dz). ¬ @vk®

J%KsbrZ%beZ%[cKsO acO9pMacO;ZwO9^f[]lo[]biv^£4O ^MO*O;x[]i3O9Zw[cbrN0lu[cO[cKsO a]lu[cOiu yiv^fvO*a]kO9^syOSb^J%KMO*ivacO9N ^[]KMO iu[]KMO*aK1lu^sx­8ZwO9acbrO9Z ¬ @f®%1O;yivNO;Z <s^Mb[cO(b^my9lkZcOik pM_Ma]OxIbÀDO*a]O*^f[]brlkq½kO9^MO*azlo[]ikazZ Bk

­lvZ0b^©[]KMbeZy*lvZwO[]KMOmikp8O*azlo[]ikazZ

Blk

olu^sbrZcKikaqelua]kOl ¬ £KMbeyzK¨kbrkO;Z3lu^¨lu^1luqrXv[]bryO*nIpMqrlk^slo[]biv^iva[]KMOÐlvy¡[[]Kslo[b^f[cO9a]lvy¡[]b^MxMbÀD_sZcbiv^sZSlkacOZcbrNpsqO9aikalu^slkqXIZcbrZz® %² O$ZcKsluqrqxMO*N0ik^sZw[cazlo[]O [cKsbrZZw[]lu[cO9NO9^f[%iv^MqrX3ika%[]KMO N0ivZw[bN0p8ikac[]lu^f[y*lvZwOikMbr^slua]Xbr^f[cO*azlky[cbrik^sZ

»h» Ð » "

B2g(x, y) = V (x, y)∂g

∂x(x, y) + V (y, x)

∂g

∂y(x, y), ¬ @vv®

P Bl

2 = 0

l > 1

" B2g(x, y) = c(x, y)

∂2g

∂x2+ 2γ(x, y)

∂2g

∂x∂y+ c(y, x)

∂2g

∂y2,

>1c

γ7 %/ R / R

γ(x, y) = γ(y, x)*!P

Bl2 = 0

l > 2

ääÉNPORQRSRT

@ L

' ¬ be®0QSZwbr^M.[cKsOªkO*^sO*azluqxIO <1^Mb[]biv^©iu Bl2

br^[]KMOªpMa]O9ZcO*^f[Zwb[c_1lo[cbrik^XYbO9qrxsZ0ikagl ∈

Csym(X l)

Bl2g

l(x1, x2) =

l∑

q=1

(−1)l−q

q!(l − q!

q∑

j=0

l−q∑

i=0

CjqC

il−q

×[

V (x1, x2)∂

∂x1+ V (x2, x1)

∂

∂x2

]

g(x1, . . . , x1, x2, . . . , x2, z1, . . . , z1, z2, . . . , z2) |z1=x1,z2=x2,

£KsO*a]Ob^A[cKMO$qrlvZ\[S[cO9acN[cKsO*a]O$lua]Ojlua]k_sNO9^f[]Z

x1­q − j

lua]k_MN0O9^v[zZx2­ilua]k_MN0O9^v[zZ

z1­lu^1x

l − q − ilua]k_MN0O*^f[zZ

z2 ÁO[_sZpMa]iokO []Kslo[[cKMO3y*ifOGybrO*^f[]Zlo[ V (x1, x2)lk^sx

V (x2, x1)8iu[]K

olk^MbrZcK3br^y*lkZcOl > 1 H4X0ZwXYN0N0O[]acXb[4brZhO*^Miv_MkK0[]i(xIO;luqs£b[cK[cKMO <1a]Zw[ yiYOG3y*bO9^f[hik^MqrX MO*^sy*Ok­£4O^MO*O;x[ci0pMa]iokOS[cKslu[

l∑

q=1

Cql (−1)l−q

q∑

j=0

l−q∑

i=0

CjqC

il−qj

∂g

∂x1(x, . . . , x, y, . . . , y) = 0

¬ m = i + jolua]brlkMqO;Z

xlu^1x

l − molua]beluMqrO9Z

y®iva(lu^YX xIbÀDO*a]O*^f[]brlkMqO

g Á½O*[(_sZ(ZcKMio£±[cKslu[[]KMOªyiYO Gy*bO9^v[lu[ ∂g∂x1

(x, . . . , x, y, . . . , y) ¬ m []bN0O;Z x lu^sx l − m[]bN0O9Z

y®$olu^MbeZwKsO9Zikalk^YX

m = 1, . . . , l,b O [cK1lo[

m∑

j=1

j+l−m∑

q=j

Cql C

jqC

m−jl−q j(−1)l−q = 0.

H4XZcKMbÖ[cbr^M[cKMObr^sxIOnqb^O9lvyzKZc_MN­I^sluN0O*qrXxIO9^Miu[]b^M

q +m− jYX

q­Ia]O9xI_sy*O9Z [cKsbrZ%O;dv_1lo[cbrik^

[]il∑

q=m

m∑

j=1

Cq+j−mj Cj

q+j−mCm−jl−q−j+mj(−1)l−q/m−j = 0,

£KsbryzKloÖ[]O*aZcbN0pMqrOluqrkO9Ma]lkbrySN3lu^MbrpM_Mqelo[]biv^sZ%£b[cKMbr^MikN0beluq½yiYOG3y*bO9^f[]Z%a]O*£a]b[]O9ZlkZl∑

q=m

(−1)l−qCql C

mq

m∑

j=1

Cjmj(−1)m−j = 0. ¬ @ u®

H4_I[4[cKsbrZ%KMivqrxsZ*­IlkZ []KMO Zw_MN iovO*ajolu^MbeZwKsO9Z ika%O9lvyzK <MnIO;x

m > 1­Ilu^sxKMO*^1yOS[cKMOq K Z%a]O9xI_sy*O9ZK[]i ∑l

q=1(−1)l−qqCql

­M£KMbeyzKlkvlubr^olk^MbrZcKMO;Z4ikal > 1 ¬ bbÖ®J%KMO psaciYiubeZlu^slkqivkiv_sZ N0b[w[]b^M[cKMO(xMO[]lkbq^Miu[]O ik^MqrX[]Kslo[br^sZw[cO;lkxiu ¬ @ u®4iv^MO _sZcO9ZKsO*a]OS[]KMO bexIO*^f[cb[\X

l∑

q=m

(−1)l−qCql C

mq

m∑

j=1

Cjmj(j − 1)(−1)m−j = 0.

VYbrN0bqelua]qrXiv^MOO;lkZcbqrX3ZwKsio£Z [cK1lo[Bl

1

olu^MbeZwKsO9Zhika%luqrql > 2

ika%lu^YX0ZwO;yiv^sx3ikazxIO9a xIbÀ8O9acO9^f[cbeluqivp1O9a]lu[civa

B1­MZcKMio£br^M[cKMO<1^Mb[]O*^MO;ZcZ%ikZcO*a]brO9Z ¬ @v®4br^y9lkZcOik&br^f[cO9a]lvy¡[cbr^M0xIbÀ8_sZcbrik^sZ

² Oy*lk^pMa]iokO^Mio£ [cKMO N3lubr^acO;Zw_sq[iu[cKMbeZZcO9y[cbrik^

âã&ä½âÖÞ

v~

·Y» · 1 R=3 1! P R= 3 9R3 #% P ψ =

0

gm(x1, . . . , xm) ∈ C2(Xm)

F (µ) = (gm, µ⊗m) R=/

µ = hδx

x =(x1, . . . , xn)

P supt≤T

‖T ht F (µ) − TtF (µ)‖ ≤ hC(m,T, ‖µ‖)‖gm‖C2(Xm)

1 F C

FP m,T, ‖µ‖

' hPiu[cbeyO7<sazZ\[%iulkqq1[cKslu[ ¬ @vu®hbeZ lkpMpMqrbry9luMqrO[]i B1ikika]N ¬ uv®¡­IZwbr^sy*Ob[4beZ4£ O9qq8Y^Mio£^

¬ ZwO9O(O > fA®%[cK1lo[[]KMOazlky[cbrik^slkqÁlupMqelky*brlk^ |∆|α KslvZivacN ¬ @vk®£b[cK ν(x, dy) = |y|−d−αdy POnY[9­D[]KMO3Z\[zlu^sxsluazxªacO9pMa]O9ZcO*^f[]lu[cbrik^ªivaS[]KMO0xIbÀDO*a]O*^1yO0iu)[]KMO3lky[cbrik^mikh[\£4iZcO*N0bvaciv_MpsZSacO;lkxMZlvZ

TtF (µ) − T ht (µ) =

∫ t

0

d

dsT h

t−sPhTsF (µ) ds =

∫ t

0

Uht−s(PhLB − Lh

BPh)TsF (µ) ds, ¬ @ @v®£KsO*a]O

PhxIO*^siu[cO;Z [cKMOpMa]iuj\O;y¡[cbrik^ik^[cKMO_M^1y¡[cbrik^1ZxIO*p8O*^sxMb^M0iv^ bazlkySN0O9lkZc_Ma]O9Z4iu[cKMOika]N

hδx QSZwbr^M ¬ @vk®¡­oJ%KMO*ivacO9N¯ ¬ bM®lu^sxg)a]ikp8ivZcb[cbrik^ ¬ bÖ®XYbrO*qexMZiva ‖(PhLB−LhBPh)TsF (µ)‖[]KMO O9Zw[cbrN3lo[]O

hC(m,T, ‖µ‖)‖gm‖C2(Xm)

­M£KMbeyzKxIba]O9y[cqrXbrNpsqbrO9Z%[]KMO Z\[zlo[]O*N0O*^f[iu[cKMOJ%KsO*iu¤a]O*N '4$# *¡#

² O0ZwKslkqq Zw[]lkaw[£b[]K2O9qO9N0O*^f[]lkacXa]O9Zc_Mq[zZ ¬ g)a]ikp8ivZcb[]biv^2 C0lu^sx2 38O*qrio£®a]ikN []KMO[cKMO9ika]Xik¦sO*qrqO9apMa]iYy*O9Z]ZwO;Z H4Xm[]KMO ¬ [cbrNOxIO*p8O*^sxMO*^f[z® Fik%lª¦sO*qrqO9aslkyzY£%luazxmpMa]ikp1luvlu[civa U(t, s)br^C∞(X)£4O ZwK1luqrqNO;lu^[cKMO ivp1O9a]lu[civa

Λtg = lims→t,s≤t

U(t, s)g − g

t− s= lim

r→t,r≥t

U(r, t)g − g

r − t£b[cK$[cKsOxIikN3lubr^1O9b^M[]KMOZwp1lkyO4iu

g ∈ C∞(X)ika £KsbryzK$[cKMO;ZwO4[\£4iqrbrNb[]ZO*nIbrZw[)lu^sxyivb^sy*brxMO

»h» Ð » P Ψt

P L ! D R Zt

7 / Z

! X

*9 LΩt

,%: P ,% ' 1 / / R ! X ( P PP 1 1 P 0" P 1 R

Yt = Ωt(Zt) P ,% '

R/ Ωt(Z)

! X

J 1/ L R >P 4(" P F +%UY (t, s)f(ys) = Et−s

Ω−1s ys

Ωtf = (Ω−1s )Ψt−sΩtf(ys),

¬ @k~v®>P

Ωtf(y) = f(Ωt(y)) >1

Et−sz = Et,s

z

FR 1 P4F : 1(F 1 #% R P4 (6+% 1 R

Zt F

z

' ¦MivacN$_Mqel ¬ @k~f®4ikqrqio£Z4a]ikNxIO <1^Mb[]biv^ ^MO ^MO9O9xMZiv^MqrX[]i3iksZcO*a]kO[cK1lo[ UY (t, s)beZ

l0Zw[ca]ik^MvqXy*ik^f[cbr^Y_Mik_1Z4ÐluN0bqrXiky*ik^f[cazlky[cbrik^sZ4lkZ%b[brZly*ikN0p8ivZcb[]biv^ik&Zc_syzKÐluN0bqrbrO9Z

ääÉNPORQRSRT

f L

»h» Ð » P9 R=3 1' P R RP 1 X

9 1L ? 1 P

Ωt(z) = (z − ξt)/a*>P

a 1

ξt*t ≥ 0

*9 7 D<=R

X P 1</ ! PI

L!

Ψt FP 1

R/ 1 D

! C∞(X) ∩ C2(X)

1 1= ! P9 FΛt

! Yt

1D

/% / 7 1 +% 1 R =L

Λtf = Ω−1t LΩtf − 1

a

(

∂f

∂z, ξt

)

. ¬ ~kv®

' ¦sikqrqio£Z4brN0NO;xIbelo[cO9qXYXxIbÀ8O9acO9^v[]brlu[cbr^M ¬ @v~v®4_sZcbr^M[cKMO pMa]iIxI_sy[%a]_MqrO ² Oy9lu^ Zw[]luac[^Mio£ [cKsOlu^slkqXIZcbrZiu%[cKMOpMaciIy*O9Z]Z iu%¾s_sy[c_slu[cbrik^sZ$lua]ik_M^1xA[]KMOO9kivq_I[]biv^´xIO¤Z]ya]br1O;xYX3[]KMO Yb^MO*[cbeyO9df_slu[cbrik^sZ

»h» Ð » F (Y )

5% ¬ o® " g = gm ∈ C2,sym

∞ (Xm)

µt =

µt(B) 1 (- P ¬ C9k® ΛB,h

t

1 ' 1 L R h−1/2(Zh

B(t) − µt) >1 ZhB(t)

1( ( +% 1 ¬ u~k®(* ¬ @kk® " 5% ¬ ~kk® a = h1/2

L = LhB

1ΛBk,h

t F 1,% = ' P R

ΛBkht F (Y ) =

m∑

l=2

h−1+l/2

∫

Xm−l

Φkh[Bl

kgz](√hY + µt)Y

⊗(m−l)(dz)

+ h−1/2

∫

Xm−1

(Φkh[B1

kgz](√hY + µt) − (Bkg

+z, µ⊗k

t ))Y ⊗(m−1)(dz). ¬ ~1C;®

P =L(*

ΛB1,ht F (Y ) =

∫

Xm−l

∫

X

B1gz(y)(dy)Y⊗(m−1)(dz)

+

m∑

l=2

h−1+l/2

∫

Xm−l

∫

X

Bl1gz(y)(

√hY + µt)(dy)Y

⊗(m−l)(dz). ¬ ~vk®

' § pMpMqrXYb^M ¬ u® [ci

ΩtF (Y ) = F ((Y − µt)/√h) = h−m/2

m∑

p=0

(−1)m−p(g, Y ⊗p ⊗ µ⊗(m−p)t )

XYbrO*qexMZ

hk−1LhkΩtF (Y ) = h−m/2

m∑

p=1

(−1)m−p

p∑

l=1

hl−1

×∫

Xm−p

∫

Xp−l

(

Φkh[Blgw,u](Y )

)

Y ⊗(p−l)(dw)µ⊗(m−p)t (du),

âã&ä½âÖÞ

C

lk^sxyik^1ZwO;dv_sO*^f[cqrX

hk−1Ω−1t Lh

kΩtF (Y )) = h−m/2m∑

l=1

hl−1m∑

p=l

(−1)m−p

×∫

Xm−p

∫

Xp−l

(

Φh[Blgw,u](√hY + µt)

)

(√hY + µt)

⊗(p−l)(dw)µ⊗(m−p)t (du)

= h−m/2m∑

l=1

hl−1m−l∑

r=0

m∑

p=r+l

Cm−pm−l−r(−1)m−p

×∫

Xr

∫

Xm−l−r

(

Φkh[Blgz,y](

√hY + µt)

)

hr/2Y ⊗r(dz)µ⊗(m−l−r)t (dy),

£KsbryzKXYbO9qrxluqrqM_I[[]KMO qelkZw[4[]O*a]N3Zik^[cKMO a K Z iu ¬ ~PC;®4YX3[]KMO ikYYbiv_sZ4brxMO*^f[cb[\Xm∑

p=n

(−1)m−pCm−pm−n =

1, n = m

0, n < m.

VI_MI[cazlky[cbr^M

h−1/2

(

δF

δY(Y ), µt

)

= h−1/2

∫ ∫

(Bkg+z

(y)Y ⊗(m−1)(dz)µ⊗kt (dy)

XYbrO*qexMZ ¬ ~PC9® § Z%[cKMO[]O*a]N3Z%iuikazxIO*a h−1/2 y9lu^sy*O*qbr^ ¬ ~1CG®4ik^MOikM[]lubr^sZ

» » ÐI ¸

ΛBk,ht F (Y ) = ΛBk

t F (Y ) +O(√h) ¬ ~kv®

ΛBk,ht F (Y ) =

∫

Xm−1

∫

Xk

(Bk(g+w

))(v)(Y ⊗ µ⊗(k−1)t )(dv)Y ⊗(m−1)(dw)

+

∫

Xm−2

∫

Xk

(B2k(g+

w))(v)(µ⊗k

t )(dv)Y ⊗(m−2)(dw) ¬ ~uf®

1 4 R 1 m = 1

" *9O(

√h)

, P,% = ! Y

'

m+k−1 = 4=5% 1 /5 = ! P )FR ' 1 = P

Blkg, l = 1, . . . ,m

*#=P R,% =(µt

J%KMO(ikp8O*azlo[civa]Z ¬ ~uv®lua]Odv_sb[]O _M^sxMlkN0O*^f[]lkq )yiv_Ma]ZcOk­M[cKMO9XlkacO O*nf[]O*^sxMO9xYXqrb^sO9lua]b[\X[cilkacsb[]a]lkacXp8ikqrXY^MikN0beluqk_M^sy¡[]biv^sluqeZF J%KMO)^sOnY[&Zw[]lu[cO*N0O9^v[&kbrkO;ZD[cKMO4luq[cO9ac^slu[cbrkO a]O*pMa]O9ZcO*^f[zlo[cbrik^1Zik[]KMO9ZcO ikp8O*azlo[]ikazZhbr^[cO*a]N3Z%iu&_M^sy[cbrik^slkqxIO*a]brGlu[cbrkO;Z

ääÉNPORQRSRT

Y L

» » ÐI ¸ = F' R ¬ u® P ! P F

ΛBk

t

¬ ~uv®I +%

ΛBt F (Y ) =

(

B1k

δF

δY, Y ⊗ µ

⊗(k−1)t

)

+

(

B2k

δ2F

δY 2, µ⊗k

t

)

, ¬ ~vk®9 ,%

ΛBk

t F (Y ) =

(

Bk

(

δF

δY

)+

, Y ⊗ µ⊗(k−1)t

)

+

1

2Bk

k∑

i,j=1

δ2F

δY (yi)δY (yj)−

By1,...,yk

k

k∑

i,j=1

δ2F

δY (zi)δY (yj)

|∀i zi=yi, µ⊗k

t

. ¬ ~kv®

¦sika]N(_Mqel ¬ ~kk®&y9lu^(8Oyiv^sZwbexIO9acO;xlkZlSslkZcbrZivalk^$lk^sluqrXf[cbey Zw[c_sxMXiuM[]KMO%qbrN0b[]b^sS¾1_sy¡[]_slo[]biv^psaciIyO;ZcZ9­olkZ&b[ ZcKMio£Z[cKslu[[cKMO%kO9^MO*azlo[]ikaiuM[]KMO4¾s_sy¡[]_slo[]biv^$psaciIyO;ZcZh−1/2(Zh

B(t)−µt)yiv^YkO*a]kO;Z

iv^$p8ikqrXY^MikN0beluqf_M^sy[cbrik^slkqrZ&[]i[cKMO%ikp8O*azlo[]ika ¬ ~kf® &² Kslu[iv^MO%^MO*O;xMZKMO9acO%beZ*­uik8y*ik_MazZwOv­ula]brkika]ik_1Zy*ik^YkO9acvO*^sy*Oa]O9Zc_Mq[ika[]KMOyivaca]O9Zcp1iv^sxIbr^MpMacivpsluflo[]ikazZ*­Ilo[qrO9lvZ\[iv^ZcikN0Oy*qrlvZcZ%ik_M^1y¡[cbrik^1luqeZ ² OZcKsluqrqsxIO9N0ik^sZw[cazlo[]O%lk^3lu^slkqXf[cbeylupspMaciflkyzK(ika)ivI[]lkb^Mbr^MZc_syzK3lacO;Zw_sq[)iv^iv_Ma)slvZwbey4N0iYxMO*qsikVIO9y¡[]biv^$ Vf[]bryzYbr^M[ci[cKsO []a]lvxIb[cbrik^½­u£4OZwK1luqrqYy*ik^sZcbexIO*a&[cKsOqbr^MO9lka&_M^sy¡[]biv^sluqeZik^N0O9lvZw_Ma]O9Z9­GM_I[£brqrqkbrkO[]KMOpMa]O9y*brZcO O9Zw[cbrN3lo[]O9Z%iva[cKsOacO9N3lubr^sxIO*a;­M£KMbeyzKªZwO9O*N [ci8O^MO9£°O*kO9^£KMO*^mlkpMpMqrbO;x[]imbr^v[]O*azlky[cbr^MmxIbÀ8_sZcbrik^sZ ¬ ZcO*O >Ïu A¥­£KMO*a]Olkqa]O9lvxIXª[cKsO_M^Mbika]N yik^YvO*a]kO*^1yO0£b[]K´acO;Zwp8O9y[[ci[]KMO^Mika]N iulAqrbr^MO9lka$_s^sy¡[]biv^©lk^sx £b[cKsik_I[iv_Ma √

hO;Z\[]bN3lu[cObrZ0pMa]O9ZcO*^f[]O9xlvZ0l.Zcbrk^MbL<1y*lk^f[

psacivka]O9Z]Z4y*ikN0pslkacO;x[]i3[cKMO(_sZw_1luq½a]O9Zc_Mq[ikI[zlubr^MO9xYXZw[]lk^sxMlka]xpMa]ikslkMbqrbeZ\[]bryN0O[]KMiIxslkZcO9xiv^[]KMOyivN0pslky[c^MO;ZcZ4iu lupspMaciGnIbrN3lo[cbr^M$pMaciIy*O9Z]ZwO;Z ik¾1_sy¡[]_slo[]biv^1® H4_I[ <sazZw[&[]KMa]O*OkO9^MO*azluqYivsZwO9acolu[cbrik^sZik^$[cKMOvO*^MO9a]lu[civa ¬ ~uv®lkacO4br^iva]xMO*a;³ ¬ bÖ®[cKMOpsacivpsluflo[civalkacbeZcb^Ma]ikN ¬ ~kv®)y9lu^38OiuÖ[]O*^3O9lvZwbrqrXy*ik^sZw[ca]_sy¡[]O9x­v1O;y*lk_sZwOv­vlvZ)ik^sOZwO9O9Z a]ikN ¬ ~fu®¡­v[cKsbrZhpMacivpslo¤flo[]ika4pMacO;ZwO9acvO9Z)[cKMO Zcpslky*OSikp8ikqrXf^sikN0brlkqrZ4iklu^YX3kbrkO9^iva]xIO9a ¬ bbÖ®hika]N(_Mqel ¬ ~vu® br^sxIbey*lu[cO9Z4[]Kslo[[]KMOqrbN0b[cbr^MpMa]iIyO9Z]Z9­u£KMO*^sO*kO9a b[ brZ £4O*qrqsxIO<s^MO;x­kbeZ)lu^0br^=<s^Mb[cOSxIbN0O9^sZwbrik^1luq lk_sZcZcbelu^ a]Zw[cO9b^I¤QKMqrO*^Y1O;yz´pMa]iYy*O9Z]Z*­M_M[0£4OZcKslkqq%^Mik[$8Oy*ik^sy*O*a]^MO9x KsO*a]O£b[cK[cKMOdf_MO9Zw[cbrik^ iuO*nYbeZw[cO*^1yOikZc_syzK3l pMa]iIyO;ZcZ iva)£b[cK3b[]Z)psacivp1O9aw[]bO;Z ¬ KMio£4O*vO*a;­kZcO*OI> fADlu^sx0a]OO9acO9^syO;Z[cKsO*a]O*br^0ika []KMOkO9^MO*azluq[]KMO*ivacX.iuZc_syzK psaciIyO;ZcZcO9Z9­ lu^sx > vA>­ >ÏuA iva([]KMOy*ika]acO;Zwp8ik^1xIb^sªb^=<1^Mb[]OxMbN0O*^1Zwbrik^slkq ZcXIZ\[]O*Nik ¢ []i ZZw[ciIyzKslvZ\[]bryO9df_slu[cbrik^sZ0lkacbeZwbr^M.b^ [cKsOy*ik^f[cO*nY[ikSbr^v[]O*azlky[cbr^M.xMbÀD_sZcbiv^sZ]® ¬ brbbÖ®br^y9lkZcO£KsO*^mZcikqr_I[cbrik^sZ[ciYbr^MO[]bry(O9df_slu[cbrik^sZSlua]O acO9k_Mqelua]bLK9b^sbr^ª[]KMO$ZcO*^sZcO []Kslo[[]KMO$Zcikqr_I[]biv^ªNO;lkZc_Ma]ObeZlusZcikqr_I[cO9qXyiv^f[cbr^f_sik_sZ£b[]KmacO;Zwp8O9y[[]iÁ½O91O;Zwv_MON0O9lvZw_sacO ivaSlkqq&p1ifZwb[cbrkO([cbrN0O9Z ¬ lkZb^2ik_Ma1lkZcbry%O*nIlkN0pMqOiub^f[]O*azlky¡[]b^sZ\[zluMqrOpMa]iIyO9Z]ZcO9Zz®¡­o[]KMO^MivacN±iuD[cKsON0O9lvZw_Ma]O

h−1/2(ZhB(t)−µt)

brZik½ikazxIO*a

h−1/2 ika%luqrqs[]bN0O9Z9­Ilu^sxKMO*^sy*O[]KMOÐlkN0bqrX3iuZc_syzKN0O9lvZw_sacO;Zhy*lk^^Mik[4yiv^YkO*a]kO£4O9lkYqX[]i(l N0O9lvZw_sacO ¢ ^3ik[cKMO9ah£4ikazxMZ*­vik^MOSy*lu^3O*nIp1O;y¡[hiv^MqX wN(_syzK3£4O9lkkO*a%y*ik^YkO9acvO*^sy*O ^0[cKMOqrO*vO*qik½pMa]ikpslkvlo[]ikazZ*­vik^MOy*lk^OnIp8O9y¡[yiv^YkO*a]kO9^syOiv^p8ikqrXY^MikN0beluq1_M^sy¡[]biv^sluqeZ £b[]Kiv^MqX3Zc_=GybrO*^f[cqrXZcN0iYiu[cKmy*iYO GybrO*^f[]Z Rmika]O*iokO9a9­1lu^YXN0O;lkZc_MacO(£b[]K2lZcN0ifik[cKAxMO*^sZcb[\Xiv^ Rd y*lk^ª1O0lupMpsaciGnIb¤N3lu[cO9xYX3[cKsOZw_MN iu[]KMO bazlkyN0O9lvZw_Ma]O9Z hδx br^Zw_syzKl£4lGX0[]Kslo[ |hδx − ν|h−1/2 brZ1iv_M^sxIO;xivaO*kO9^.y*ik^YkO9acvO*^f[[]i K*O*a]ib^2[cKMO3^sika]N iuh[]KMOZcpslky*O

(C1(X))∗ MO*^1yO[cKMO3lkpMp1O;luazlu^sy*O$ik[]KMO OnIpMa]O9Z]Zcbiv^ ‖(ZhB(0) − µ0)/

√h‖(C1(X))∗

1O9qio£

âã&ä½âÖÞ

f

² O$ZcKsluqrqy*ik^sZcbexIO*aS^Mio£°ik_MaSslkZcbryNiIxIO9qik VYO9y[cbrik^ª S§ ZSb[Sikqrqio£Z%a]ikN ¬ ~kf®­1_M^1xIO*aS[cKMOlvZcZc_MN0pI[]biv^sZikJ%KMO*ivacO9N ª[cKMOkO9^MO*azlo[]ikaik[cKMOqbrN0b[]b^sApMa]iIyO9Z]Z ΛB1

t + ΛB2

t

xIO <s^sO9Z$[cKMOO9kivq_I[]biv^ik&qrb^sO9lua%_s^sy¡[]biv^sluqeZ

Fg(µ) = (g, µ)YXN0O9lk^sZik[cKsO O9df_slo[]biv^

g(x) = −σ(x)|∆|α/2g(x) +

∫

(V (x, y)∇g(x) + V (y, x)∇g(y))µt(dy),¬ ~ u®

£KsbryzKbr^3[]KMOSbr^YkO9a]ZcO[cbrN0OvbvO9Z []KMOS[cazlu^sZwika]N3lo[cbrik^Gt,s : gt 7→ gs

[cK1lo[ beZ 1iv_M^sxIO;xbr^C∞(X)lk^sxb^

C∞ ∩ C2(X) ¬ luflubr^lvy*yiva]xIbr^M([ciJ%KMO9ika]O*N C lu^1xb[]Z%ivYfbrik_1Z%NiIxIbL<1y9lo[cbrik^8® Ji3pMa]iokOiv_MayO9^f[cazluqqrbN0b[£ O ZcKsluqrq^MO*O;x[cKMOivqqrio£b^slk_InIbqrbrlkacXacO;Zw_Mq[ · 1 R=7 1 ' 1 ?L

g2 ∈ Csym∞ (X2) ∩ C2(X2)

1

supt≤T

E

(

g2,

(

ZhB(t) − µt(µ0)√

h

)⊗2)

≤ C(T, ‖ZhB(0)‖)‖g2‖C2(X)

(

1 +√h

∥

∥

∥

∥

ZhB(0) − µ0)√

h

∥

∥

∥

∥

C1(X))∗+

∥

∥

∥

∥

ZhB(0) − µ0)√

h

∥

∥

∥

∥

2

(C1(X))∗

)

¬ ~@v®

C(T, ‖µ0‖) < 1 FP

h

' ^sOK1lkZ

E

(

g2,

(

ZhB(t) − µt(Z

hB(0))√

h

)⊗2)

= E

(

g2,

(

ZhB(t) − µt(Z

hB(0))√

h

)⊗2)

+

(

g2,

(

µt, ZhB(0) − µt(µ0)√

h

)⊗2)

+ 2E

(

g2,Zh

B(t) − µt(ZhB(0))√

h⊗ µt, Z

hB(0) − µt(µ0)√

h

)

.

J%KsO(O9Zw[cbrN3lo[]O(iuh[cKMO <sa]Zw[[]O*a]N yika]a]O9Zcp1iv^sxMZ[]i[]KMO <1a]Zw[[]O*a]Nbr^ª[]KMO$sa]lvyzkO[Sik^ª[]KMO(a K Z ik¬ ~@k®­M£KMbeyzKikqrqio£Z a]ikN J%KMO9ika]O*N 0lk^sx[]KMOika]N(_sqrl

E

(

g2,

(

ZhB(t) − µt(Z

hB(0))√

h

)⊗2)

=1

2hE(g2, Zh

B(t)⊗2 − µt(ZhB(0))⊗2

+ µt(ZhB(0)) ⊗ (µt(Z

hB(0)) − Zh

B(t)) + (µt(ZhB(0)) − Zh

B(t)) ⊗ µt(ZhB(0))).

J%KsO4a]O9df_Mbra]O9x$O9Zw[cbrN0lu[cO4ika[]KMOZcO9yiv^sxlu^sx([cKMbrazx([]O*a]N3Z lua]O vbvO*^fX[cKMO%[]KMbazx0lu^sxZcO9yiv^sx([cO*a]Na]O9Zcp8O9y¡[]bvO*qrX3b^[]KMO MazlkyzkO*[4iv^[]KMO a K Z iu ¬ ~ @k®¡­M£KsbryzKikqrqrio£Zha]ikN J%KMO*ivacO9N ¬ M® ² Oy*lk^pMa]iokO^Mio£ [cKMO N3lubr^acO;Zw_sq[iu[cKMbeZZcO9y[cbrik^

ääÉNPORQRSRT

v L

·Y» · P =7 P ! 1 ?Lg ∈ C2

∞(X)1

supt≤T

∣

∣

∣

∣

E

(

g,Zh

B(t) − µt(µ0)√h

)

−(

Gt,0g,Zh

B(0) − µ0√h

)∣

∣

∣

∣

≤ C(T, ‖ZhB(0)‖)

√h‖g‖C2(X)

(

1 +

∥

∥

∥

∥

ZhB(0) − µ0√

h

∥

∥

∥

∥

2

(C1(X))∗

)

. ¬ ~k~v®

' )Á½O*[Ufluc

h (t, s)8O%[cKMOslvyzY£4lka]x psacivpsluflo[civa&y*ika]acO;Zwp8ik^1xIb^s[ci[]KMOpMa]iIyO9Z]Z

(ZhB(t)−

µt)/√h J%KsOq K Z iu ¬ ~@v®%y9lu^1O £a]b[w[cO9^lvZ

supt≤T

∣

∣

∣(Ufluch (t, 0)Fg(ξ0) − (Gt,0g, ξ0)

∣

∣

∣

£b[cKξ0 = (Zh

B(0) − µ0)/√h­s£KMbryzKO9df_slkqrZ

supt≤T

∫ t

0

Ufluch (s, 0)(ΛB,h

s − ΛBs )Gt,sg ds(ξ0).

§ Z (ΛB,hs − ΛB

s )Gt,sgbeZ(lªdf_slvxIa]lu[cbey_M^sy[cbrik^slkqhpMa]ikp8ikac[cbrik^slkq[ci √

h­&[cKMOa]O9df_Mbra]O9x.O9Zw[cbrN3lo[cO

ivqqrio£Z acivN ÁO*N0N3l C &3'4 ' # .% #&' *J%KsOSN0O[]KMiIxMZxIO9kO*qrikp8O9xlu8iokOy*lu^1O lkpMpMqrbO;xlkqrZci[]i$[]KMO lu^slkqXIZcbrZ iupMaciIy*O9Z]ZwO;Z yzKslk^Mkbr^M$[cKMO^Y_MN$1O9a(ikpsluac[cbeyqrO9Z9­&lvZ£4Olua]Ovikbr^M[ci.xIO*N0ik^1Z\[]a]lu[cO3^Mio£ ik^ [cKMOOnMluN0pMqrOiuy*ivluv_Mqelo[cbrik^M¤azluvN0O*^f[]lu[cbrik^N0iIxIO9qrZ JikO9^MO*azluqrbDK*OVYO9y[cbrik^A0br^Al0£%lGX[]ib^1yqr_sxIOpMa]iIyO;ZcZcO9ZyzK1lu^Mvb^M0[]KMO^Y_MN$1O9aiu pslkaw[]bry*qO;Z*­lvZcZc_MN0OS[cKslu[

Dlu^1x

Xlua]O[]KMO ZclkNO lvZ br^J%KMO*ivacO9N Clk^sxqrO[ Bk : Csym(X ) 7→ Csym(Xk)

­k = 1, . . . ,K

8O4ikM[cKsO ivacN ¬ CGu®£b[]K Bk1O9b^Myiv^sxIb[cbrik^slkqqrX p8ivZcb[]bvO ivp1O9a]lu[cikazZ½br^

Csym(Xk)¬ [cKslu[SZw[]lu^1xika[]KMOpMa]iIyO9Z]ZcO9Z%pMa]O9ZcO*a]Yb^M[]KMO(^Y_MN$1O9aiu pslkaw[]bry*qO;Zlu^sx[]Kslo[£4O*a]OxIO9^Miu[]O9x Bkbr^mVYO9y[cbrik^sZS$lk^sxk®lk^sxPk8O*br^Ml$ÐluN0bqrXikZcXYN0NO*[ca]bry[]a]lk^sZcb[]biv^vO*a]^MO*qeZ4acivN

Xk [ci X § pMa]iYy*O9Z]Z ZB(t)

iuk¤¿lua]Xbr^f[cO*azlky[cbrik^ªbr^

X ¬ p8ivZ]ZcbMqrXªyzKslu^skbr^M0[cKMO^Y_MN(8O*aSiu)pslkaw[]bry*qO;Z]®y*lu^8OxIO<s^MO9xlkvlubr^[]KMa]ik_MvK[]KMO kO9^MO*azlo[]ika ¬ uk®­I£KMbeyzKªy*lk^8O £acb[w[]O*^br^N0ivacOxIO[zlubrqlkZ

LBf(x) =K∑

k=1

Lkf(x) =K∑

k=1

∑

I⊂1,...,l:|I|=k

[(BIkf)(x) +

∫

(f(y,xI ) − f(x))Pk(x, dy)] ¬ C;kv®

ivax = (x1, . . . , xl)

£b[cKlkacMb[cazlua]Xl J%KMOSZ]y*lkqO;x3pMa]iYy*O9Z]Z)iv^ M+

δ,h(X)beZ lkvlubr^3kbrkO9^3fX ¬ of®¡­

¬ Gv® _sa$y*lkqry*_Mqelo[cbrik^1Zbr^ VYO9y[cbrik^sZ0ªlk^sx m£4O*a]Oy9lua]acbrO9x.ik_M[$br^ Zc_syzK lm£%lGXm[]Kslo[[cKMO9X´lua]OZw[cazlubrkKf[cika]£4lka]xIqrXkO*^sO*azluqrbLK9O9x3[]i[cKMO^MO9£¯N0iIxIO*qik[]KMbrZZwO;y¡[cbrik^ ¦sikabr^sZw[]lu^1yOk­ ¬ o®­ ¬ @k®%Zw[cbrqqKsikqex­I£KMO*a]Oika%[]KMO b^f[]O*kazluqivp1O9a]lu[civaBk : Csym(X ) 7→ Csym(Xk)

iu[cKMOivacN

Bf(x) =

∫

X

(f(y) − f(x))P (x, dy), x ∈ Xk,

âã&ä½âÖÞ

Y

[]KMO ikp8O*azlo[]ikaBl

k : Csym(X l) 7→ Csym(Xk)beZkbrkO9^YX

Blkg(x) =

l∑

q=1

(−1)l−q

∫ ∫ ∫

g(y, z)(δu − δx)⊗q(dy)P (x, du)δ⊗(l−q)x

(dz). ¬ C;1C;®

J%KsbrZ qrO9lvxMZ&[ci[]KMOyivaca]O9Zcp8ik^sxIbr^MkO9^MO*azluqrbLK;lo[]biv^(iuJ%KMO9ika]O*N3Z) C%£b[]K$[]KMOqelG£ik8qelua]kO4^Y_MN(8O*a8O*br^M3luflubr^Zcp8O9ybL<sO;xYX ¬ C9v®%lu^sx[cKsO qbrN0b[]b^M0pMa]iIyO;ZcZ%ik¾s_sy[c_slu[cbrik^sZfXkO*^sO*azlo[civa]Z ¬ ~kv® / _MaxIO;yivNp8ivZcb[cbrik^ iu Bk

acivN ¬ C C;v®Db^f[]i[cKMO Zw_MN°ik Bk¬ [cKMO ^Y_MN(8O*aiuIpsluac[cbeyqrO9ZpsacO;ZwO9acYbr^M3psluac[z®lu^sxª[cKMO$acO9N0lkb^sb^M(j\_MN0pI¤>[\Xfp8O(p1luac[SbeZ^Mik[S_M^sbrdf_MOv­DlvZ[]KMOj\_sNp1ZSpsacO;ZwO9acYbr^M

[]KMO^Y_MN(8O*ahiu1p1luac[cbeyqrO9Z ¬ qrbrkObr^ H4ivq[ K*N3lu^M^3y*ikqrqbeZwbrik^1Z]®&y*lk^$8OpM_I[ b^0O*b[cKsO*a)ik1[]KMO9ZcO%[\£ ipslkaw[zZ ¢ ^´y*lvZwO0ik4pM_sacOj\_MN0p´br^f[cO*azlky[cbrik^sZ Bk = 0

b^ ¬ C9vk®lu^sx.Ybr^MO[]bry0O;dv_1lo[cbrik^ ¬ C;k®[zlukO;Z[cKMOivacNd

dt(g, µt) =

K∑

k=1

∫ ∫

(g+(y) − g+(x1, . . . , xk))Pk(x1, . . . , xk; dy)µ⊗kt (dx). ¬ C;vk®

MO*^sy*O[cKMO vO*^MO9a]lu[cika ¬ ~uv®4ik[]KMOqbrN0b[]b^M3psaciIyO;ZcZ4iva%¾s_sy[c_slu[cbrik^y*lu^8O £acb[w[]O*^lkZ

ΛBk

t F (Y ) =

∫ ∫

(δF (z) − δF (y))Pk(y, dz)(Y ⊗ µ⊗k−1)t )(dy)

+1

2

∫ ∫

(δ2F (z, z) + δ2F (y,y) − 2δ2F (y, z))Pk(y, dz)µ⊗kt (dy) ¬ C;kf®

lk^sx[cKMO xIXY^slkNbey*Z4ik[cKsO b^Yolua]brlk^f[ qrb^MO;lua%_M^1y¡[cbrik^1ZFg(Y ) = (g, Y )

brZ4kbrkO9^YX0[]KMO O9df_slu[cbrik^

g(x) =

K∑

k=1

N∑

n=1

∫

[

n∑

i=1

g(zi) − g(x) −k−1∑

i=1

g(yi)

]

Pk(x, y1, . . . , yk−1; dz1 · · · dzn)µ⊗(k−1)t (dy).

¬ C;uY®§ Zb^.[]KMO3pMa]O*Ybiv_sZZwO;y¡[]biv^½­£4O3ZwKslkqq)xMO*^Mik[cO3YX Gt,s

[cKMOy*ika]acO;Zwp8ik^sxMb^MO*vikqr_I[cbrik^.ikp8O*azlo[]ikazZiv^

gbr^b^YvO*azZwOS[cbrN0O ¬ slvyzf£%luazxpsacivpsluflo[civaz® ² OZcKslkqq yiv^sZcbrxIO9aS^sio£°[cKMO0Z\[zlu^sxsluazxªN0iYxMO*qiuhy*ivluv_Mqelo[cbrik^mlk^sxªazluvN0O*^f[]lu[cbrik^myivN(Mbr^MO;x£b[cK.p8ivZ]ZwbrMqrOy*ikqrqbeZwbrik^AMacO;luoluvOk­D£KMO9acO

X = R+­PkxIiYO9Z^Mik[ Glk^MbeZwK2ik^MqrXika

k8O*br^M[\£4i

lk^sxik^MOv­MZwi[cK1lo[P2(x1, x2; dy) = K(x1, x2)δ(x1 + x2 − y)dy + C(x1, x2, y1)δ(x1 + x2 − y1 − y2)dy1dy2,lk^sxP1(x; dy1dy2) = F (x, y1)δ(y1 + y2 − x) dy1dy2 J%KsOyik^f[]b^Y_Miv_sZ_M^sy[cbrik^sZ K,C, F lkacOy9luqrqO;x[cKMOyivlkk_Mqelo[]biv^½­MyivqqrbeZwbrik^ªlu^1xa]lkkN0O*^f[]lu[cbrik^kO9ac^sO*qeZacO;Zwp8O9y[cbrkO9qX J%KMO$yika]a]O9Zcp1iv^sxIbr^MqelG£ ikqelua]kO^Y_MN(8O*azZ ¬ C;vu® [zlukO;Z4[cKMOivacN

d

dt(g, µt) =

∫ ∞

0

∫ ∞

0

(g(x1 + x2) − g(x1) − g(x2))K(x1, x2)µ⊗2t (dx1dx2)

+

∫ ∞

0

∫ ∞

0

∫ x1+x2

0

dz(g(z) + g(x1 + x2 − z) − g(x1) − g(x2))C(x1, x2, z)µ⊗2t (dx1dx2)

+

∫ ∞

0

∫ x

0

dz(g(z) + g(x− z) − g(x))F (x, z)µt(dx)¬ C;vv®

ääÉNPORQRSRT

f L

¬ y*lkqqrO9xVYN0ikqr_syzKMioIZwYb Z O9df_slu[cbrik^b^ªy*lvZwOSiu&Glk^MbeZwKMbr^M C lk^sx F ® )§ Z]Zc_MN0OSivaZwbrN0pMqbeyb[\X3[]Kslo[lkqqbr^f[cO9^sZwb[cbrO9Zlua]OS8ik_M^1xIO9x ¬ N0ivacOa]O9luqrbeZ\[]brylkZ]Zc_MN0pI[cbrik^sZ4£brqq1OxIbeZ]y_sZ]ZwO;xbr^,> CRA®¡³

supx1,x2,x

max(K(x1, x2),

∫

C(x1, x2, z)dz,

∫

F (x, z)dz) <∞. ¬ C9vv®J%KsO*^ b[$beZ(£4O*qrqhY^Mio£^ []Kslo[ ¬ C;vu®brZ$£ O9qq p1ifZwO;x ¬ ZcO*OO >ÏkA4ika0yifluv_Mqrlu[cbrik^sZ lk^sx'> vA%br^vO*^MO9a]lkqMy*lvZwOG®¡­ub O ikahlk^fX <s^Mb[cO µ0

£b[cKl <1^Mb[]OZcO9yiv^sxN0ikN0O*^f[ ∫x2µ0(dx) <∞ []KMO*a]OOnIbeZ\[zZl_M^Mbedv_sO(8ik_M^sxMO9xmZcikqr_I[cbrik^

µtik ¬ C;vu®£b[]KAl38ik_M^sxMO9xmZcO9y*ik^sxN0ikN0O9^v[lu^sxªpsacO;ZwO9acYbr^M0[cKMON3lvZcZ9­sb O Zw_1yzK[cKslu[ ∫ xµ0(dx) =∫

xµt(dx) ¢ ^ > fAlk^sx,> k~ Ab[SbrZZwKsio£^[]Kslo[S[cKMbeZSO9df_slo[]biv^KsikqexMZ lkqrZcibr^[]KMOZw[ca]ik^s(ZwO9^sZcO£b[cK[]KMOxIO*a]bolu[cbrkO8O*br^M_M^sxIO9a]Zw[ciYiIx3br^[cKsOSZcO*^sZcOik½[cKMO^MivacNik M(X) J%KMbeZlkqqrio£Z%[cixMbÀDO*a]O*^f[cbelo[]O [cKsbrZO9df_slo[]biv^£b[]KmacO;Zwp8O9y[[ci0[]KMO(br^Mb[cbeluqNO;lkZc_Ma]O µ0 « dv_1lo[cbrik^ikalkqqxIO*a]bolu[cbrkO9Z%lkacOikYYbrik_sZcqX3qrbr^MO9lka%lk^sxYXl0ZwbrN0pMqO br^sxI_1y¡[cbrik^iv^MO ZwO9O9Z4[cK1lo[

supx1,...,xl

∥

∥

∥

∥

δlµt

δµl0

(x1, . . . , xl)

∥

∥

∥

∥

≤ Cl(µ0)tll! ¬ C; k®

ivaluqrql£b[]KZcikN0O yiv^sZw[]lu^f[

C(µ0) VYbrN0bqelua [ci$J%KMO*ivacO9N £4OvO[h^Mio£[]KMOivqqrio£b^sacO;Zw_sq[hiv^0[cKMOSa]lu[cOikyiv^fvO*a]kO9^syO4[ci([cKMOqelG£ ikqelua]kO^Y_MN(8O*azZ ¬ C;vu® ·Y» · R=/ ¬ C9vk®? 1 =L )' R ), / / ¬ C9fu® gm ∈ C∞(Xm)

F (µ) = (gm, µ⊗m)

R= µ = hδx

x = (x1, . . . , xn)

1supt≤T

‖T ht F (µ) − TtF (µ)‖ ≤ hC(m,T, ‖µ‖)‖g‖C(Xm)

1 F C

FP m,T, ‖µ‖

' ² O%ZcKslkqqYluflubr^(_sZcO ¬ @@k® TsF (µ)beZ lu^$b^=<1^Mb[]O*qrX(xIbÀ8O9acO9^f[cbeluMqrOh_s^sy¡[]biv^½­o[cKsO48ik_s^sxMZ

iva%[cKMOxMO*a]bolo[]bvO9Z48O*br^M3kbrkO9^YX ¬ C9 o® ¦sacivN ¬ C;1C9®4b[ikqrqio£Z []Kslo[‖Bl

kg‖ ≤ 5lC‖g‖/l!, k = 1, 2,

lk^sxKMO*^1yOZcO*a]bO;Z ¬ @f® [c_Ma]^sZiv_I[[]i31Oy*ik^YkO9acvO*^f[ iva Ft(µ) = F (µt)lk^sx[cKMO$Zw_MN iokO9a

l ≥ 2beZ%O9Zw[cbrN3lo[]O9xYX

1

k!

∞∑

l=2

hl−1

∥

∥

∥

∥

Blk

δlFt

δµt

∥

∥

∥

∥

‖hδx‖k ≤ h

k!

∞∑

l=2

hl−2(C(µ0)t)l‖hδx‖k, k = 1, 2,

Zci[]Kslo[S[cKMbeZbrZyiv^YkO*a]kO9^f[lu^sxAiu)ikazxIO9aO(h)

1iv_M^sxIO;xªikaO9lvyzKt ¬ lk^sxmZc_=GybrO*^f[cqrXmZcN0lkqq h ® MO*^sy*O ¬ PhLB − Lh

BPh)TsF (µ)beZikiva]xIO9a

hbr^ ¬ @ @k®¡­£KsbryzK©brNpsqbrO9Z[cKMOAZw[]lu[cO9NO9^f[iu[cKMOJ%KsO*ika]O*N

² Oy9lu^´^Mio£ ikM[]lubr^ [cKsOyO9^f[cazluqhqrbrNb[$a]O9Zc_Mq[ika([cKMbeZ$NiIxIO9q 1lkZcbry9luqrqX.yikpYXYbr^Mª[cKsOlua]k_M¤N0O9^v[zZ4iu½[]KMOpMacO9Ybiv_sZ VYO;y¡[]biv^½­IO*vO*^£b[cKlkxMxMb[]biv^sluqZcbN0pMqrb<8y*lo[]biv^sZh[cKslu[%£4OxIi(^Mik[^MO*O;xlu^YXZcN0iYiu[cKs^MO9Z]Z(ik[cKsOy*iYO GybrO*^f[]Z$iuSp1ivqXY^MivNbeluq4_M^1y¡[cbrik^1Z ¬ Zwbr^sy*Olkqq Bk

lua]O8ik_s^sxIO9x1® J%KY_sZbr^sZw[cO;lkxiu Á½O*N0N3l0 C£ OkO*[

âã&ä½âÖÞ

· 1 R=7 1 ' 1 L g2 ∈ Csym

∞ (X2) P

supt≤T

E

(

g2,

(

ZhB(t) − µt(µ0)√

h

)⊗2)

≤ C(T, ‖ZhB(0)‖)‖g2‖

(

1 +

∥

∥

∥

∥

ZhB(0) − µ0√

h

∥

∥

∥

∥

2

M(X)

)

C(T, ‖µ0‖)

< 1 FP h

§ ^sx[cKMO9^ªZwbrN0bqeluaN0iIxIbL<1y*lu[cbrik^iu J%KMO*ivacO9N CXYbrO*qexMZ4[cKsOikqrqio£br^M3yO9^v[]a]lkqDqbrN0b[a]O9Zc_Mq[ ·Y» · P =7 P ! 1 ?L

g ∈ C∞(X)1

supt≤T

∣

∣

∣

∣

E

(

g,Zh

B(t) − µt(µ0)√h

)

−(

Gt,0g,Zh

B(0) − µ0√h

)∣

∣

∣

∣

≤ C(T, ‖ZhB(0)‖)

√h‖g‖

(

1 +

∥

∥

∥

∥

ZhB(0) − µ0√

h

∥

∥

∥

∥

2

M(X)

)

. ¬ C;@f®

J%KsbrZacO;Zw_Mq[beZ^MO9£O*vO*^ikayifluk_sqrlu[cbrik^NiIxIO9q ¬ £b[]Kmolu^MbeZcKMb^s C lu^sx F ®lk^sxªO9kO*^ª£b[cKsik_I[[]KMO0O9Zw[cbrN3lo[cOiu4yik^YvO*a]kO*^1yO J%KMO0iv^MqXªpsacO9fbrik_1Za]O9Zc_Mq[iv^m[cKsO0y*O*^f[cazluqqrbN0b[ika yifluk_sqrlu[cbrik^N0iIxIO9q%lkpMp8O9lua]O9xb^ >DC @ A¥­ lk^sx b[3brZ0xIO9kiu[]O9x [ci2[cKMOªy*lvZwOiuSik^MqrX xIbeZcy*acO*[cON0lvZcZ0xIbeZw[ca]bM_M[cbrik^ J%KsO(lua]k_MN0O9^v[zZ%b^ >LC@ A&lkacOdf_Mb[cO(xMbÀDO*a]O*^f[a]ikNik_MazZlk^sx[cKMO9XxMi^Miu[SkbrkOlu^YXO9Zw[cbrN0lu[cO;Z%ikay*ik^YkO9acvO*^sy*O &39# z# s!* o')#*ÁO[

Blu^sx

B∗ 1Ol3acO;luqZcO*p1luazluMqrOIH%lu^slvyzKmZwpslvyOlk^sxªb[]ZxI_sluq£b[]KAxI_slkqb[\XxIO*^Mik[cO;xªYX (·, ·)lk^sxA[cKMO3_M^sb[(1luqrqrZ xIO9^Miu[]O9x.YXB1lu^sx

B∗1 ¢ [ ivqqrio£ZSacivN[cKsOVf[civ^MO¤ ² O9bO9a]Zw[cazlkZ]ZS[cKMO9ika]O*N[]Kslo[ <s^Mb[cOSxMbN0O*^1Zwbrik^slkq ¬ ivahyXYqrb^1xIacbey*lkqe®_M^sy¡[]biv^sZ)ikDika]N Ff (v) = f((g1, v), . . . , (gm, v))

£b[cKg1, . . . , gm ∈ B

lk^sxf ∈ C(Rm)

lkacOAxMO*^sZcO2br^¨[]KMO.ZcpslvyOC(B∗

1)ik ∗ ¤¥£ O;luYqrX yiv^f[cbr^f_sik_sZ8ik_s^sxIO9x2_M^1y¡[cbrik^1Zik^.[cKMO3_s^Mb[1luqrq b^

B∗ 3² O3^MO9O9x.lNivacO0pMa]O9y*brZcO3Z\[zlo[]O*N0O*^f[[cKslu[ [cKMO;ZwOlkpMpMa]iGnIbN3lo[]biv^sZy9lu^ª8OyzKMifZwO9^ªb^.Zw_1yzKml£%lGX[]Kslo[S[cKMO9XªacO;Zwp8O9y[xIbÀ8O9acO9^f[cbelo[cbrik^ J%KMbeZÐlvy¡[brZy*ac_1ybeluq ivaik_Ma$OnIp8ivZcb[]biv^ lk^sx.[cKsOlu_I[]KMika$xIbrx ^Miu[I<1^sx b[b^ []KMOqrb[]O*azlo[]_MacO ² OZcKslkqq ^MO*O;xb[$iv^MqrXAiva[cKsOy*lvZwOiuB = C∞(X)

£b[cKX8O*br^M

Rn ikab[]ZZc_MMN3lu^Mbikqex­)lk^sx £ OacO;xI_syOlu[w[]O*^f[cbrik^ik^MqrX3[]i0[cKMbeZy*lvZwO ² OZwK1luqrq½ZclGX3[]Kslo[l$ÐlkNbrqrX P1, P2, · · ·

iu[]KMO qrb^MO;luay*ik^f[cazlky[cbrik^sZ4br^Biu&[cKsOika]N

Pjv =

Lj∑

l=1

(wlj , v)φ

lj ,

¬ C;k~f®

£KsO*a]Oφl

j

lk^sxwl

j

lua]OmZwivN0O <s^Mb[cOAqrbr^MO9lkabr^sxIO*p8O*^1xIO*^f[ZcO[zZ0a]ikN [cKMO2_M^Mb[slkqqeZB∗

1

lu^1xB1a]O9Zcp8O9y¡[]bvO*qrXk­ika]NUlu^´lupspMaciGnIbrN3lo[cbrkObexIO*^f[cb[\Xk­&b4[]KMOZcO9df_MO9^syO

Pjy*ik^YkO9acvO9ZZw[ca]ik^MvqX[]iª[cKMO

bexIO9^v[]b[\Xikp8O*azlo[civalkZj → ∞

ääÉNPORQRSRT

@ L

JiZcO*O$[cKsO$O*nYbeZw[cO*^1yOiu Zc_syzKAlÐluN0brqXv­qO*[_sZyzKMiYifZwOl <s^sb[]O 1j

¤>^sO[x1, x2, . . . , xLj

b^2[cKMO1luqrq ‖x‖ ≤ j ­lu^1x2qrO[ φl

j

1Olªy*ikqrqO;y¡[cbrik^.iuyiv^f[cbr^f_sik_sZ^Miv^I¤¥^MO*flo[cbrkO_M^1y¡[cbrik^1ZZw_1yzK2[]Kslo[φj(x) =

∑

l φlj(x)1O*qrik^svZ$[ci

[0, 1]O*vO*a]Xf£KsO*a]Ok­)O;dv_1luqeZ$iv^MOika ‖x‖ ≤ j

lu^sxolu^sbrZcKMO9Z$iva‖x‖ ≥ j + 1

lk^sx Zc_syzK [cK1lo[$O;lkyzKφl

j

O9df_slkqrZ(ik^MOb^lA^MO*brkKY8ika]KMiYiIx iuxl

j

lk^sx´olu^sbrZcKMO9Zika‖x− xl

j‖ ≥ 2/j J%KsO*^[cKMO ivp1O9a]lu[cikazZ

Pjf(x) =

Lj∑

l=1

f(xlj)φ

lj(x) =

Lj∑

l=1

(f, δxlj)φl

j(x)

ivacN lu^lupMpMa]iGnIbN3lu[cbrkObrxIO9^f[cb[\Xb^B = C∞(X)

»h» Ð » P /$ ,%P1, P2, · · ·

! # = 1R 8 = 1 B 5% ¬ C;k~k® R 5 / #%

B P

"3 % F ∈ C(B∗1 )

1 ,%Fj = F (P ∗

j ) R F

F=P R2% PI R

1L % ! C(B∗

1 )"1*

" P 4 P R == ε1, ε2, · · ·

2% ! PΠj

C(B∗

1 )

P 1R ! # = 1/! P ,% = 1 ! µ ∈ B∗

! 1 R

Fj,ν(µ) = exp

−εjLj∑

l=1

(φlj , µ− ν)2

, ν ∈ B∗,

R/4 P Πj(F )

P F=P R ,%

B∗1

* " F

k /R PP R,%. 7 L 1 1 P

δkF (µ)(v1, . . . , vk)4

∗ ) 2%: PP = ! k + 1

R LR(* 1 P6 R R?' P

k!

Πj(F ) 1 R 1 .R !

F= R ,%?

B∗1

' ¬ be®%Piu[]bry*O[cKslu[

P ∗j (µ) =

Lj∑

l=1

(φlj , µ)wl

j .

J%KsOacO;df_Mba]O9x.yiv^YkO*a]kO9^syOiva [cKMO_M^sy¡[]biv^sZik%[cKMO3ivacNFg(µ) = exp(g, µ) ­ g ∈ C∞(X)ivqqrio£Z acivN []KMO xIO <s^sb[]biv^ik[]KMOlupspMaciGnIbrN3lo[cbrkOSbrxMO*^f[cb[\Xk­Iiva

Fg(P∗j (µ)) = exp(Pjg, µ).

¦sika lua]Mb[cazlua]XF ∈ C(B∗

1 )[cKMO3Zw[]lu[cO*N0O9^v[beZSivI[]lkb^sO9xA[cKMa]ik_MvKmb[]Z lupspMaciGnIbrN3lo[cbrik^AfX[]KMO0qbr^I¤

O;luayikN$Mbr^slo[]biv^sZ iuO*nIp1iv^MO*^f[cbeluqh_M^sy[cbrik^sZFg¬ £KMbryzKbrZ$p1ifZcZcbrMqOfX.[]KMOVf[]ik^MO*¤ ² O*brO*azZw[cazlkZ]Z[]KMO*ivacO9N® ¬ bbÖ®&¦Mikalu^YX j [cKMO _M^sy[cbrik^slkq Fj(µ) = F (P ∗

j (µ))yqrO9lua]qrXy9lu^8O4£acb[w[]O*^(br^[]KMO ika]N

Fj(µ) =

fj(y(µ))£b[cK

y(µ) = (φ1j , µ), . . . , (φ

Lj

j , µ) lk^sx fj1O9b^M8ik_M^sxMO9xAy*ik^f[cbr^Y_Mik_1Z_M^1y¡[cbrik^1ZSik

âã&ä½âÖÞ

f~

LjGlkacbelusqO;Z § pMpsaciGnIbrN0lu[cbr^M fj

<sazZ\[$fX2[cKsO3_M^sy[cbrik^sZeε∆fj

lu^sx.[]KMOqelo[c[cO*a(_M^sy[cbrik^´YX.lqrbr^MO9lkayikN$Mbr^slo[]biv^

hjik []KMO lu_1ZcZcbrlk^A_s^sy¡[]biv^sZ iu4[cKsO0[\Xfp8O

exp−εj∑Lj

l=1(yj − ξj)2 ­£ O[]KMO*^ªxIO<s^MO

Πj(F (µ)) = hj(y(µ))­1£KMbryzKO9^oj\ioXIZh[]KMO a]O9df_MbracO;xpMa]ikp8O*ac[\X ¬ bbrbe® ^MOik^MqrX^MO9O9xMZ4[]i3^Miu[]bry*OS[]Kslo[b F beZ k []bN0O;Zyiv^v[]b^Y_Miv_sZcqXxIbÀDO*a]O*^f[cbeluMqrOk­Y[]KMO*^

δkFj(µ)(v1, . . . , vk) = δkF (P ∗j (µ))(P ∗

j v1, . . . , P∗j vk)

lk^sx[cKMO9^[]KMO a]O9Zc_Mq[%ivqqrio£Z4acivN ¬ bÖ® '4( # »h» Ð »< /% =

k*19 7 7/

f 7→ Φk[f ] = Φ[f ]

1I 1 Csym(Xk)

F 1 1,% = / R=R ' I kR/( 1

∑

I⊂1,...,n,|I|=k

f(xI ) = Φk[f ](δx) ¬ C C9f®

,5 %P x = (x1, . . . , xn)

X I/ =P 1 R

Φk[f ](Y ) = (f, Y ⊗k) +

k−1∑

l=1

(−1)l(Φkl [f ], Y ⊗(k−l)) ¬ C CCG®

Φk

l [f ] 31R %7 RR = (I 1

Csym(Xk)F1 )

P = P ! k − l

R DR /P =L(*

Φ2[f ](Y ) =1

2

∫ ∫

f(y1, y2)Y (dy1)Y (dy2) −1

2f(y, y)Y (dy). ¬ CCGk®

/ H4ik[cKZwbexIO9Z%ik ¬ CC;k®4olu^MbeZcKbr^ªy*lkZcO n < k ' 1ZwO9acYbr^M([]Kslo[

∑

I⊂1,...,n,|I|=k

f(x1) = (f, (δx)⊗k) − σ,

£KsO*a]OσxMO*^Mik[cO9Z[cKMO(Zc_MNiokO9a%[cKMO(y*ikN$Mb^1lo[cbrik^1Z£b[]Kª^Miu[SNivacO[]Kslu^

k − 1xIbÀ8O9acO9^v[Sp1ivb^f[]Z9­

iv^MOO9lvZwbrqXikI[zlubr^sZ4[cKslu[∑

I⊂1,...,n,|I|=k

f(x1) = (f, (δx)⊗k) − 1

k!

k−1∑

j=1

jj!∑

J⊂1,...,n,|J|=j

P jf(xJ , z)δ⊗(k−j−l)x

(dz),

£KsO*a]OPjNO;lu^sZ4[]KMOZcXfN0N0O[]acbDK9lu[cbrik^iokO*a4[]KMO<sazZ\[

jolkacbeluMqrO9Z4ik[]KMO kqr_Mbr^M0ikp8O*azlo[civa

Pf(x1, . . . , xn−1) = f(x1, x1, x2, . . . , xn−1).

ääÉNPORQRSRT

k L

¦sacivN[]KMbrZika]N(_Mqel[cKsOacO;df_Mba]O9x Zw[]lu[cO*N0O9^v[(ikqrqio£ZYX Z\[]a]lkbvKv[cika]£4lka]xAb^1xI_sy¡[]biv^ [civkO[]KMO*a£b[cK[]KMO OnIpMqrbry*b[ivacN$_Mqel

Φkl (f) =

k−1∑

il=l

il

il−1∑

il−1=l−1

il−1 · · · i2i2−1∑

i1=1

i1Pi1 · · ·P i1f.

§ Z)£4Olua]O4br^f[cO9acO;Z\[]O9x0br^$[]KMOZ]y*lkqO;x([]a]lk^sZwika]N0lu[cbrik^sZ9­o£4OZwKslkqqsb^f[ca]iIxI_sy*O%[cKMO^1lo[c_sa]lkqsZcy9luqrb^sbr^Φ[f ]xIO<s^Mbr^M

Φkl [f ](Y ) = hkΦk[f ](Y/h). ¬ CC;v®

¢ ^[]KMbrZ%^siu[]lu[cbrik^sZ ¬ CC;k®4XYbO9qrxMZ4[]KMOikqrqrio£b^M3ZcbN0pMqrOM_I[_M^1xMluN0O*^f[zluqivacN$_Mqrl

hk∑

I⊂1,...,n,|I|=k

f(xI) = Φkh[f ](hδx) = (f, (hδx)⊗k) +

k−1∑

l=1

(−h)l(Φkl [f ], (hδx)⊗(k−l)). ¬ C C*Y®

0 &»Yº · ¼ · 9 J%KsOlu_M[cKMiva(beZ [cK1lu^Mf_Mq [ci.tik8O*ac[ciS_MO K9lvxMl H%lo[zluqrqrliva(KMbeZ(KsivZcpMb¤[zluqrb[\X br^´[cKsOQ^MbvO*azZwbexMlvx § _I[]ik^MivN0l2RªO[]acivp1ivqb[]lk^sl ¬ Q § R ­RmOnIbry*i2y*b[\XM®¡­[]KMbeZ$£4ika].1O9b^Mbr^Mb[cbelo[]O9x´xI_Ma]br^M[cKMOlu_I[]KMika ZYbeZwb[[]imQ § R b^ lk_I[c_MN0^kku J%KMOlk_I[cKMivabeZ kazlo[]O_Mq []i2V lk_M8O*ac[Slu^sxªR 8§ fbelu^iva[]KMO*braKMifZwpsb[zluqrb[\Xb^ ¢ PSt ¢\§ ¬ ¦Mazlu^sy*O;®­s[]KMbrZS£4ika]8O*br^MyivN0pMqO*[cO;xxM_Macbr^M[cKMO0lu_I[]KMika ZYbeZwb[S[]i ¢ PSt ¢\§ b^A[cKMO3ZcpMacbr^Mukv J%KMOlk_I[cKsikaS[cKslk^MIZ[]KMOacO*O*a]O*O9Zika[]KMO*bra yiv^sZ\[]ac_1y¡[cbrkOya]b[]brdf_MOiu [cKMO<sazZ\[ xIazloÖ[ iuh[cKMbeZpslup8O*a lu^1xARmlkaw[]b^MO $ O*a]^MO9_MbqrqrO$ivadf_MbeyzpsacikO9Z]Zwbrik^slkq8[\Xfpsb^M0ik[cKsbrZN3lk^f_1Zcy*acbrpI[b^ªÁlo[cO*n o%'4>LCAV s§ qr1O9kO9acbris­ § Mbrq8O*ac[9­ $ Eivqivkivq[zZwio VY_saqO yivN0p1ivaw[]O*N0O*^f[lkZcXYNpM[ciu[]brdf_MOxI_^MioXvlk_lvZcZciIybr`(_M^sOxIbÀD_sZwbrik^.xI`*v`*^M`9ac` ¬ ¦sacO9^syzK1® > § ZwXYN0pI[cik[cbey(8O*K1lGfbrikaSiu)[]KMOkO*a]^MO9q&lvZcZciIybelo[cO;x£b[cKlxIO*vO*^MO9a]lu[cOxIbÀD_sZwbrik^1A S t Rmlu[cK f§ y*lkx VIy*b VYiIy t lk^ ¬ kkvv®¡­C;1C¤ C;u~ > A $ 8§ qexIik_sZ O[]O*a]Nbr^MbeZ\[]bry lk^sxZ\[]iIyzKslkZw[cbeyNiIxIO9qrZikay*ivluqrO9Z]yO9^syO ¬ lukvacO9vlu[cbrik^lk^sxyiu¤lkk_Mqelo[]biv^1®¡³ªl.a]O*YbO9£ iu[cKsOªNO;lu^I¤ <sO9qrx[cKMO9ika]X ika3pMa]ik1luMbrqbeZw[]Z H4O9ac^Miv_Mqrqb ¬ C;~k~v~v®¡­u¤> @ > A $k§ pMpMqrO*slk_MN Á½`9fX(pMaciIy*O9Z]ZwO;Zlk^sx$Zw[ciIyzKslvZ\[]bry4y*lkqry*_Mqr_sZ lkN(Ma]bexIkOVf[c_1xIbO;Z br^ § xIolk^syO;xRAlo[cKsO*N3lo[]bry9Z*­s~v luN$Ma]brxIvO Q^MbrkO9a]Zcb[\X3ghacO;ZcZ9­ luN$MacbexIvOk­Muvu > A $ « H%lua]^sxIikacÀ8¤¥PbO9qrZcO*^ O[±lkq ¬ «hxMZ ® Á½`*YX pMa]iYy*O9Z]ZwO;Z*³ J%KMO*ivacX lu^sx § pMpMqrbry9lo[cbrik^1Z H4bracYKk_sZcO*a;­IkkPC > A t H%lkZ]Z Vf[ciIyzKslvZ\[]brymxMbÀDO*a]O*^f[cbeluqSO9df_slu[cbrik^sZ3£b[cK j\_sNp1Z g)a]ikslk VY_Ma] C ¬ ukkf®­ C*¤ C;~¬ O*qrO9y[ca]ik^Mbey9®

âã&ä½âÖÞ

1C

> A $ g H4O*qelGfYbr^ S_slk^f[c_MN Mazlu^syzKMbr^M pMa]iIyO;ZcZcO9Z lk^sx0^Mik^sqbr^MO9lkahxIXY^slkNbey*ZhiuN(_sq[]brdf_slk^v[]_MNZcXIZ\[]O*N3Z ikYq 8§ y*lvx PSlu_MªVMVMVIt ¬ C;~@ @v®¡­ C9u @o¤ C9fk «)^Mvq J&lu^sZcq b^AVIio g)KYXIZ ikYq ¬ C9~ @@v®­s@PC¤\@v

> A $ H4O*qelGYfbr^½­ $ Eivqivkivq[zZwio ! ^ kO9^MO*azluq%Yb^sO[cbeyO9df_slu[cbrik^ iva3N3lu^YX´psluac[cbeyqrOZwXIZw[cO*N3Z£b[cK b^f[]O*azlky¡[]biv^½­&a]lkkN0O*^f[zlo[cbrik^´lk^sx yifluv_Mqrlu[cbrik^ g)a]iYy t VYiIy Áik^sx § ¬ kkfk®¡­C*¤¿v

> @A $ H4O*^Mbr^MO[luq ¦Mazlky[cbrik^slkqqrXmZ\[zluMqrO0xIbrZw[ca]brM_I[cbrik^1Z ¢ ^½³ $ _ ESika]ikqrO*D­DP P VYYkivaw[zZwiool¬ «hxMZ ®hVf[ciIyzKslvZ\[]bry%RªiIxIO9qrZikDVf[]ac_sy[c_MazluqMJ_Ma]M_MqrO*^sy*Ob^g)qelkZcN3l RªifZcy*io£ Vf[]lu[cOQ^MbvO*azZwb[\Xk­RmivZ]yio£¯kkk ¬ br^ªt_sZ]Zwbelu^8® > ~A ¤¥R H4ik^YXv­8ghK ik_Ma]a]L*kOv­1g g)a]brik_Ma]O[ VIO*N0b¤¥ka]ik_sp1O;ZSxIO0¦MO9qqrO*a Zc_Ma_M^MOolua]b`*[c`8ikazxy*ikN0pslvy¡[cO0O*[psacivMqL9N0O9Zlu_InAqrbrNb[cO;Zb^f[]`*ka]iu¤¿xIbÀ8`9acO9^f[cbrO*qrqOxI_.ZcO9y*ik^sxAikazxIa]O$xIiv^M^slu^f[ qrbO9_lk_pMacbr^sy*bp8O xI_N0lunIbN$_MN s§ ^M^ M¢ ^sZw[ ¦sik_Ma]bO9a9­ a]O*^MivMqrO ¬ C;~k@f®¡­Mvk~o¤\k1C

>DC9A ² H4a]lk_M^½­ E MO9pMp J%KsO $qelkZcioxIXY^sluN0bey*Zlk^sxb[zZ0¾s_sy[c_slu[cbrik^sZb^©[]KMO CuP qbrN0b[ikbr^f[cO9a]lvy¡[cbr^M0pslkaw[]bry*qO;Z ivNN Rmlu[cK g)KYXYZ ¬ C;~ k®¡­ C9PC¤ CC; >DCCAt §( lua]Niv^slM­DÁ _ bÀD_sZwbrkOKYXIxIaciIxIXY^slkN0bry qrbrNb[]ZivaZcXIZ\[]O*N3Ziuhbr^f[cO9a]lvy¡[cbr^MxMbÀD_I¤Zcbrik^sZh£b[]K <s^Mb[cOSa]lk^MkOazlu^1xIikN b^f[]O*azlky¡[]biv^ ivN0N Rmlo[]K g)KYXYZ ½ ¬ C;~k~ u®¡­YufG¤\@u >DC;AghK ik_saca]L*vO VY_Maqel$ika]NObr^v[]O*vacik¤¥xIbÀD`*a]O*^f[cbr`*qrqOxI_k`9^M`*azlo[]O*_Ma4br^=<s^Mb[c`;ZwbrN3luqx _M^ZwO9N0b¤vaciv_Mp8OxMO¦MO*qrqrO*aZc_Ma_M^MO olua]br`[c` VY`*N J%Ks`*ika]bxI_giu[cO9^f[cbrO*q>­;C9~vvG¤¿ks­k« nIp8ivZc` >DC9A $ lG£Zwiv^ acb[cbey*lkq)xIXY^sluN0bey*Zlu^1xm¾s_1y¡[c_1lo[cbrik^1ZikalNO;lu^ <sO*qex2NiIxIO9q)iu%yiYikp8O*azlo[]bvO8O*K1lGfbrika Vf[]lu[ g)KYXIZ ¬ C9~ @kv®­Mu~o¤F@v >DC*A $ lG£Zcik^ RªO9lvZw_sacO*¤ $)lkq_MO;x Rmlua]kio2g)aciIy*O9Z]ZwO;Z MO*^M^sO9df_Mbr^ g Á ¬ O9x ®« yikqrOx « []`xIOpsacivsluMbrqrb[]`9ZxIO(VIlkb^f[c¤¥¦&qrik_Ma S¢ ¤ C;~k~PC VYpMa]b^skO*aÁO9y¡[ Pik[cO;ZRmlo[]K C;uPC ¬ C;~k~kf®¡­ C¤\uv >DC;A $ lG£Zcik^.O[$lkq O*^MO9a]lkqbDK*O;x RªO*KMqrO*aVYO*N0brka]ik_MpsZ lk^sx lo[zluqrXf[cbey/H azlu^1yzKMb^sghaciIyO;ZcZcO9Z£b[cK ¢ N0N0bva]lu[cbrik^ gik[cO*^f[]brlkq § ^slkq ¬ kkuY®¡­ kG¤¿~ >DC9A $ § lG£Zwiv^½­u kaw[]^MO*a Á½ik^s[]bN0O%1O9KslGYbivaik1br^f[cO*azlky[cbr^MSxMbÀD_sZcbiv^sZ o¢ ^³ t lPivaca]brZ¬ «hx ® Vf[ciIyzKslvZ\[]bry luqey_Mqr_sZ&br^ § pspMqbey*lu[cbrik^ g)b[cN3lk^tO;ZwO;luazyzKPik[cO9Z&br^$Rmlu[cKMO9N3lo[cbey*Z VYO9acbrO9Z C;~ ¬ C;~@@f®¡­Mu~u¤¿u >DC A $ § lG£Zcik^­G uac[c^MO9a Rª_Mq[cbrqO9kO9qvqelua]kOhxMO*Ybrlu[cbrik^sZlu^sxb^f[cO9a]lvy¡[]b^MSxIbÀD_sZcbiv^sZ ghacivslu J%KsO*ika]XtO9qrlu[ ¦&brO*qexMZ ½ ¬ C9~v~uf®­Ifku @ >DC @AR O9lvyiv^f_­P ¦Miv_Ma]^MbO9a9­« Jlk^Mac` tlu[cO.iu(y*ik^YkO9acvO*^sy*Oªiu$l Z\[]iYyzK1lkZw[cbeyªpsluac[cbeyqrOAOZcXIZ\[]O*N iva%[cKMOVYN0ivq_syzKsio£ZwYbyifluk_sqrlu[cbrik^O;dv_1lo[cbrik^ RmO[cKsiYxMikq ikN0pM_I[ s§ pMpsq ghacivslu

¬ uvkv®­ C91C*¤ CG@ >DC9~At Á ivMa]_sZwKsb^ $qelkZcio3O9df_slo[]biv^sZ ¦s_M^sy[ 1§ ^sluq M§ pspMq ¬ C9~ o~v®­ CC;

ääÉNPORQRSRT

v L

>ÏuAt ¦sO*a]qrlk^sx­ $ ¦sO*a]^Mbedv_sOk­ ba]ik_In ikN0pslvy¡[c^sO9Z]Z(ik[cKMO¾s_sy[c_slu[cbrik^sZlkZ]ZwiIy*brlu[cO9x £b[cKZcikN0OvO*^MO9a]lkqbDK*O;x^Mik^Mqrbr^MO9lka7H4ikq[K*N3lk^M^ªO9df_slu[cbrik^sZ lk^slkx Rmlo[]K ¬ C9~v~vk®­;CC;~vo¤CGuf

>Ï=CA V P «)[cKMbrO*a;­)J E_Mac[K Rmlkacviomg)a]iIyO9Z]ZcO9Z Kslka]lvy¡[cO9acbDK9lu[cbrik^ lu^sxyik^YvO*a]kO*^1yO vikKs^² brqrO*XVIik^sZ9­ C9~ @k

>ÏkAH ¦MO9ac^slk^sxIO Kk­oV Rm`*qr`9luazx *§ Mbrq8O*ac[cbelu^$lupMpMa]ivlvyzKiva½¾s_1y¡[c_1lo[cbrik^1Z½ik^[]KMO RAyEO9lk^I¤ $qelkZcioN0iIxIO9q VY[ciIyzKslkZw[cbeyg)a]iYy*O9Z]ZwO;Z § pMpsq ¬ C;~k~ u®¡­Ivo¤\u >ÏuAH ¦MO9ac^1lu^sxIO Kk­IV Rª`*qr`9lka]x v§ ZcXYN0pI[cik[cbey8O*KslGYbrikahika4b^f[cO9a]lvy¡[]b^M$xIbÀD_sZcbiv^pMa]iIyO;ZcZcO9Z £b[cKZcpslvyO¤>[cbrN0Oa]lk^sxIikN Mbac[cK H4O*a]^Mik_Mqrqrb ¬ uvkv®­I~1C¤ CC C @>ÏoA uac[c^MO9a ^m[]KMORAyEO9lk^I¤ $qelkZcioqrbN0b[SivaSbr^f[cO*azlky[cbr^MxMbÀD_sZcbiv^sZ Rmlu[cK PlvyzKMa

¬ C9~ @@v®­ C9~ ;¤\o @ >ÏkA S azluKslkN­ V Rª`9q`;luazx VY[ciIyzKslkZw[cbeyp1luac[cbeyqrOlupspMaciGnIbrN3lo[cbrik^1ZikakO*^sO*azluqrbLK9O9x O*O H4ikq[K¤N3lk^M^NiIxIO9qrZlk^sxyiv^fvO*a]kO9^syOSO9Zw[cbrN0lu[cO;Z M§ ^M^ g)a]ik ¬ C9~k~ u®­C C;o¤ C;v >ÏuA« br^M`v­f §(v² O*qrqr^MO*a Q^MbivacN y*ik^YkO9acvO*^sy*O4br^ZcikN0OqrbN0b[ []KMO*ivacO9N ivahN(_sq[]bpMqrOp1luac[cbeyqrOZcXIZ\[]O*N3Z Vf[ciIyzKslvZ\[]bryg)a]iIyO;ZcZcO9Z%lu^1x[]KMO*bra § pspMqbey*lu[cbrik^sZ ¬ C;~k~ u®¡­I ;¤u >Ï A¦ _MbelkZ ik^YkO9acvO*^sy*O pMacivp1O9aw[]bO;Zik lZw[ciIyzKslvZ\[]bry(N0iYxMO*q&ikayifluk_sqrlu[cbrik^I¤>a]lkkN0O*^f[zlo[cbrik^psaciIyO;ZcZcO9Z4£b[cKªxIbÀ8_1Zwbrik^ Vf[ciIyzK s§ ^sluq s§ pMpMq ¬ ukPC;®­Iufo¤¿ @ >Ï@AP flky*ik ghZcO*_1xIiu¤¿xIbÀ8O9acO9^v[]brlkqikp8O*azlo[]ikazZ4lk^sxRmlua]kio3psaciIyO;ZcZcO9Z $ ikq M¢w¢ ³ O*^MO9a]lu[civa]Z%lu^1x[]KMO*brap8iu[]O*^f[cbeluqD[]KMO*ivacX Á½ik^1xIik^½³ ¢ N0p8O*a]brlkq ikqrqO9kOg)a]O9Z]Z*­skkv >Ïu~A vlvyiIx­ § VYKMbracXfluO* Á½brNb[ []KMO*ivacO9N3Z¨ikaµZ\[]iIyzKslkZw[cbey°pMa]iYy*O9Z]ZwO;Z VYO9y*ik^sx O;xIb[cbrik^ a]_M^sxMqO9KMacO9^ xIO*aRmlo[]KMO*N3lo[]brZ]yzKMO9^ ² beZ]ZwO9^sZcyzK1loÖ[cO9^5> ¦s_M^sxMlkNO9^f[]luqSg)a]b^sy*bpsqO;Ziu Rmlu[cKI¤O9N3lo[cbey*lkqVIybrO*^1yO9Z$A¥­1@ @ VYpMa]b^skO*ac¤ $ O*a]qrlks­ H O9acqrbr^½­suvk

> kAR E lky g)acivslusbqrb[\Xlu^sxacO9qrlu[cO;x[]ikpMbey*Z%br^pMKYXYZcbey*luq½Z]ybrO*^1yO I² brqrO*X ¢ ^f[]O*azZcy*bO9^syOv­C;~vk~ > 1CA §( P EiIyzKY_M8O*b g lka]lk1ivqbeypsZcO*_1xIiu¤¿xIbÀ8O9acO9^v[]brlkMqrOO9df_slu[cbrik^½­)Zc_Mp8O*azZwbr^Mv_Mqrlka$br^f[cO9kazluqeZ(lu^1xRAlua]kiopMaciIy*O9Z]ZwO;Z 8¢ K*vO9Zw[cbel § olvx Plk_MD­VYO*a Rmlo[]O*N ¬ C;~@@f®¡­~kv~o¤¿~kk ¬ t_sZ]Zcbrlk^1®¡­RAlo[cK ¤¿QSVMVMt ¢ K9 ¬ C9~@v~v®­Ikku¤¿vu~ ¬ «)^MvqbeZwK8® > vA $ P ESikqrikkivq[zZwio VYXYN0N0O[]acbey4VY[]lusqO4ÁlG£Z&lu^sxVf[]lkMqO*¤¥ÁbvOhv_MN0pI¤ bÀ8_sZcbrik^sZ g)aciIy Áik^I¤xMik^ªRmlo[]K VYiIy ¬ kkvv®¡­ kko¤ u@ > kA $ P ESikqrikvikq[]Zcio VIO*N0bry*qrlvZcZcbey*luq § ^slkqXIZcbrZ ika bÀD_sZcbiv^sZlu^sx Vf[]bryzK1lkZw[cbey¨g)a]iYy*O9Z]ZwO;Z VIpMacbr^MvO*aÁ½O;y¡[]_MacO Piu[cO;Z%b^ªRmlu[cK C uus­Muvk > uA $ P ESikqrikvikq[]Zcio RmO9lkZc_Ma]O¤¥oluqr_MO9x.qrbN0b[]Z$iub^f[]O*azlky¡[]b^smpsluac[cbeyqrOZcXIZ\[]O*N3Z £b[cK k ¤¥^slua]Xbr^f[cO9a]lvy¡[cbrik^1Z ¢¡ g)acivslu J%KMO*ivacXtO*qelo[ ¦&brO*qexMZ ¬ uvkv®­Ikuk¤¥v~u

âã&ä½âÖÞ

k

> vA $ P ESikqrikvikq[]Zcio ^ « nY[]O*^sZcbiv^iuMRªivqqrbL<sO9x9H4ikq[K9N3lu^M^lk^sxVYN0ikqr_syzKMioIZcfbo« dv_1lo[cbrik^1Z[cig lkaw[]bry*qOVYXIZ\[]O*N3Z4£b[]Klk¤¥lkacX ¢ ^f[cO*azlky[cbrik^ t_sZ]Zcbrlk^kiv_Ma]^sluqsikRmlo[]K g)KYXYZ ¬ kkkf®¡­k@u¤¿k~v

> kA $ P ESikqrikvikq[]Zcio RmO9lkZc_Ma]O¤¥oluqr_MO9x.qrbN0b[]Z$iub^f[]O*azlky¡[]b^smpsluac[cbeyqrOZcXIZ\[]O*N3Z £b[cK k ¤¥^slua]Xbr^f[cO9a]lvy¡[cbrik^1Z ¢w¢¡ Vf[]iIyzKslkZw[cbey*Z%lu^1xVY[ciIyzKslkZw[cbey*ZtO*p8ikac[]Z ¬ uvuf®­YfG¤\@ > A $ P ESikqrikkivq[zZwio ^ªRmlua]kio3pMa]iIyO;ZcZcO9Z4£b[cKmxIO;yikN0p8ivZ]luMqrOpsZcO*_sxIik¤¥xMbÀDO*a]O*^f[cbeluqvO*^MO9aw¤lu[civa]Z Vf[ciIyzKslvZ\[]bry9Z%lu^sxªVf[]iYyzK1lkZw[cbey*ZtO9p1ivaw[zZ ¬ kkkf®¡­ C*¤>v > @A $ P ESikqrikkivq[zZwio MXIxIa]iIxIXY^sluN0beyÁ½brNb[hik ifluk_sqrlu[cbrik^I¤¿¦MazluvNO9^f[]lo[]biv^(J Xfp8ORªiIxIO9qrZ ik

k¤>^slkacX ¢ ^f[cO9a]lvy¡[]b^M$g luac[cbeyqrO9Z vik_sac^slkq1ik&Vf[zlo[cbeZw[cbey*luqDg)KYXIZcbry9Z ½­M u ¬ kkkf®¡­C9f=C*¤ C;vu

> k~A $ P ESikqrikkivq[zZwio Ebr^MO[]bry0O;dv_1lo[cbrik^1Zika []KMOpM_sacOj\_MN0p N0iIxIO9qrZ ik k ¤>^slkacXAb^f[cO9a]lvy¡[]b^Mp1luac[cbeyqrOZcXIZ\[]O*N3Z Rmlua]kio3g)a]iIyO;ZcZcO9Z%lu^1xtO9qrlu[cO;x¦&brO*qexMZ ¬ uvkf®¡­I~vo¤ C;@ > vA $ P ESikqrikvikq[]Zcio ^W[cKMO acO9k_Mqelua]b[\X iu Zwivq_M[cbrik^sZ []i[cKMO Zwp1lo[cbeluqrqrX KMivNivkO*¤^sO*ik_1Z H4ikq[K9N3lu^M^ O;dv_1lo[cbrik^ £b[cK p1ivqXY^MivNbeluqrqrXvacio£br^MyivqqrbeZwbrik^ kO9ac^sO*q g)a]O*pMa]br^v[Q^MbrkO*azZcbrxMlvx § _I[civ^MikN3l RªO*[ca]ikp8ikqrb[zlu^sls­mu ufv C ¢¡ kv ukfI­RmOnIbry*i g)_MsqbeZwKsO9x br^§ xM Vf[]_sx ik^f[ Rmlo[]K ¬ kkkf®

> PCA $ P ESikqrikvikq[]Zcio J%KMO$yO*^f[]a]lkqqbrN0b[S[cKMO9ika]O*N iva[cKsO$VYN0ikqr_syzKMio£ZcYb½yivlkk_Mqelo[]biv^N0iIxIO9q ¢ ^pMa]O*p1luazlo[cbrik^

> fA §( EivqiIxIvis­E VIlu8O*qO*qex­ ²f² lkk^MO9a f§ Zw[ciIyzKslvZ\[]bryN0O*[cKMiIx0iva4ZcikqrYb^MVYN0ikqr_syzKMio£Zcfb Zy*ivlkk_Mqelo[]biv^O9df_slu[cbrik^ RAlo[cK ikN0pM_I[ VYbrN(_Mqelo[]biv^ ¬ C;~k~v~v®¡­M ;¤o~ > vAJ ESikN3lo[zZw_ ^0[cKMOSN0lkaw[]b^svluqrOpMa]ikMqrO*N±ika kO9^MO*azlo[]ikazZikZ\[zluMqrOpMa]iIyO;ZcZcO9Z £b[cK3p8O*ac[c_saw¤1lo[cbrik^1Z Z]luol0 RAlo[cK ¬ C9~ @uf®­ CC9u¤ C;v > kAR ÁlkyzKMio£bey Kv­ghK Álu_sacO9^syik[9­ (² aK9ivZcO* ! ^ [cKMO ikac[w¤FM_MqeZw[w¤\VIlGYaciv^Miomyifluk_sqrlu[cbrik^O;df_slo[]biv^lu^sxb[]ZacO9qrlu[cbrik^[ci[cKMO4VYN0ikqr_syzKMio£ZcfbkO9df_slo[]biv^ V ¢\§ R Rmlu[cK G§ ^1luq ¬ kkvv®¡­

C;k~v~o¤ C*f1C > fAg Á½O;Zcy*iu[;­fR tiYO;yzf^sO*a gO*ac[c_Ma]slu[cbrik^sZhiu O*^sO*azluqrbLK9O9x3RªO9KMqO9a%VIO*N0bvaciv_MpsZ lu^sx § pMpMqrbry9lo¤[]biv^sZ4[ci3Vf[]iYyzK1lkZw[cbey MO9lu[4«hdf_slu[cbrik^£b[]KÁ`*YXPivbrZcO lu^sxVYbr^Mk_Mqelua acbÖ[ g&ik[cO9^v[]brlkq § ^slkq

¬ kkkf®¡­IPC ;¤¿uv > vA¦ Á½O9XfYazlK VIy*lkqbr^M [cKsO*ika]X lk^sx©OnMlvy¡[cqrX ZcikqrkO;xNiIxIO9qrZb^©[]KMOmYbr^MO[]bry9Ziu braca]O*vO*azZwbrMqrOlkkvacO9vlo[]biv^ g)KYXYZcbey*ZtO9p1ivaw[zZ ½­so¤¥ ¬ ukvv®­Y~fG¤\=CG > A $ g RmlkZcqrio g&O9aw[]_Ma]slo[]biv^J%KsO*ika]Xlk^sx § ZcXYNpM[ciu[]bry9luqfRªO[]KMiIxMZ RªivZ]yio£ Vf[]lu[cOhQ^MbrkO*azZcb[\XghacO;ZcZ9­ C9~vv ¬ br^t_sZcZcbelu^1® ¦Ma]O*^syzKJa]lk^sZwq _s^MiIx­sg lua]beZ*­ C9~ u > @A $ g RmlkZcqrio ivN0pMqO*nµRmlkacvio Kslubr^sZmlk^sx°¦M_M^1y¡[cbrik^1luq$¦sO*XY^MN3lu^ ¢ ^f[cO9kazluq RªivZ]yio£ ­PSlu_Mols­;C9~ o ¬ b^ªt_1ZcZcbrlk^1®

ääÉNPORQRSRT

u L

> v~AM g RmyEO;lu^ § yqelkZ]Ziu%Rmlua]kiopMa]iIyO9Z]ZcO9ZlkZ]ZwiIybelo[]O9xA£b[]K.^sik^Mqrb^sO9luapslka]lk1ivqbey$O9df_slo¤[]biv^sZ g)a]iYy Plu[ s§ y9lkx VIy*b ¬ C9~kvv®­C;~k ;¤ C9~1C C >ÏuA >V Rª`*qr`9lka]x ik^YvO*a]kO*^1yOmik([cKsO2¾1_sy¡[]_slo[]biv^sZikab^f[]O*azlky¡[]b^s xIbÀ8_1Zwbrik^sZ£b[cK j\_MN0psZlvZcZciIybelo[]O9x £b[]K H4ivq[ K*N3lu^M^©O;dv_1lo[cbrik^1Z Vf[]iIyzKslkZw[cbey*ZVf[]iIyzKslkZw[cbey*Z3tO9p M¹ ¬ C;~k~ @v®¡­

C;~vo¤¿vk >Ï=CA R olk^mPO*O9acO9^ iv^f[cbr^f_sb[\Xlu^sxAtO*pMa]O9ZcO*^f[zlo[cbrik^ik lk_sZ]Zwbelu^mRªO9KMqrO*aVIO*N0bvaciv_MpsZ giu¤[]O*^f[cbeluq § ^slkq ¬ kkkf®¡­ C9~v~o¤\=C C >ÏkA Pika]acbeZ qr_sZw[cO*a ifluv_Mqrlu[cbrik^ ivNN RAlo[cK g)KYXYZ ¬ kkvv®¡­fv ;¤¥vf >ÏuAt Á VIyzKMbrqrqbr^M acio£%[]Klk^sx M qrxMO*ayiv^sxIb[cbrik^sZika[]KMO(Z]luN0pMqrO pslo[]KsZik¦sO*qrqO9apMa]iYy*O9Z]ZwO;Z ghacivslu J%KsO*ika]XtO9qrlu[ ¦&brO*qexMZ ¬ C9~k~ @v®­Ikvo¤¥PCC >ÏoA $ VY[ca]ifiIyz RmlkacvioghaciIyO;ZcZcO9Z4a]ikN E I¢ [ ZgO*azZcp1O;y¡[cbrkO M§ ^M^1luqeZiuRmlo[]K Vf[]_sxIbrO9Z9CGkI­ghacbr^syO*[civ^Q^MbrkO*azZcb[\X3g)a]O9Z]Z9­1uvk >ÏkA §( ¤\V V K*^sb[]N0lk^ Pik^sqbr^MO9lkaStO*¾sO9y[cbr^M bÀ8_sZcbrik^mg)a]iIyO;ZcZSlu^sx[]KMO$ghacivpsluflo[cbrik^iu K1luivZlk^sx¦&q_sy[c_slu[cbrik^ § Z]ZciYy*brlu[cO;x kiv_Ma]^ ¦M_M^sy[ 8§ ^slkq ¬ C;~@uY®¡­MPCC¤¿kv >ÏuA §( ¤\V V K9^Mb[]N3lu^ JikpMbey*Z(b^ g)acivpsluflo[]biv^.ik KslkivZ ¢ ^½³3«hy*ikqrOx « []`xMOg)a]ikslkMbrqb[c`;Z$xIOVMlubr^v[c¤¥¦&qrik_Ma S¢ ¤ C9~ @k~ VYpsacbr^MkO9aÁ½O9y[c_Ma]O Pik[cO9ZRmlo[]K C*fu ¬ C9~v~1CG®¡­ C9 ;¤\kk >Ï AM J&lu^1luolM­fR Mb[zZw_1xMl O*^f[cazluq1qbrN0b[ [cKMO9ika]O*N3Zika4lZcbN0pMqrOSxIbÀD_sZwbrik^3N0iIxIO9q1ikDbr^f[cO9a]lvy¡[w¤br^M3pslkaw[]bry*qO;Z Mba]ivZcKMbrN0l0Rmlu[cK ¬ C;~@1CG®¡­IC;G¤¥fk>Ï@A $ $ QyzK1lubrfbr^½­$ R ikqriu[]lkacO9 Kslk^syO0lu^1x.Vf[zluMbrqrb[\XD³ Vf[]lkMqO brZw[ca]brM_I[cbrik^1Zlk^sxA[cKMO9ba§ pMpMqrbry9lo[]biv^sZ $VYg­;C;~k~k~

>Ïu~AE QSyzKMbXfluN3l VIy9luqrb^M$qbrN0b[ikb^f[]O*azlky¡[]b^s$xIbÀ8_1Zwbrik^sZ4£b[cKlua]Mb[]a]lkacXbr^Mb[]brlkqxIbrZw[ca]brM_I[cbrik^1Z ghaciv J%KMO9ika]XtO*qelo[ ¦&bO9qrxMZ ¬ C9~k~kf®­Y~ ;¤ CC;

âã&ä½âÖÞ

Unité de recherche INRIA RocquencourtDomaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)

Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)

ISSN 0249-6399