Nonlinear Markov semigroups and interacting Lévy type ...

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

Submitted on 20 Jun 2006

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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

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

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X = Rd = /1 P D P #

c (X) I 1 9

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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

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xl£b[]K

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D P/ 9 1

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,%< ,%I1R ,%/

Akg(x1, . . . , xk) =

[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

Rd (1 1 FR JP= j

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∑

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

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

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[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)],

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

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bk = b1k

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br^C∞(Xk)

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

(χ(yi)(yi,∇xi)f(x1, . . . , xk))

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

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Akg = Bkg+/k!

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

X = Rd D

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∑

· · ·m∑

ik−1=1

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

) ¬ =C;®

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) ¬ kk®

âã&ä½âÖÞ

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x, x1, . . . , xm νk

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) ¬ uv®

beZx = (x1, . . . , xm)

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

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1 R ¬ C @k®+* 9 5 R ¬ k®

dt(g, µt) =

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¬ kk®

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

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

R$PI/ R=+*# &P $ Rµt(dx) = ft(x)dx

ft ∈ L1(X)

dtft(x) =

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∏

ft(yl)dyl,

¬ @v®>1

(∇ck(x, y1, . . . , yk−1))j =

∂xicij(x, y1, . . . , yk−1),

(∇,∇ck)(x, y1, . . . , yk−1) =

∂xi∂xjcij(x, y1, . . . , yk−1),

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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

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/ 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

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)

âã&ä½âÖÞ

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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(*

g,δ2µt

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C(t, ‖µ0‖) # PP 2% 7 D = P

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PP PI R F PD%(" / µ0 7→ µt

(C1(X))∗

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

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tlu[t = 0

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M£KMO*a]Oikaut ∈ C([0, t0],M+(X))

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

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®

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I£KMO*a]O

Vt(x, [u]) =

V (x, y)ut(dy), ψt(x, z, [u]) =

ψ(x, y; z)ut(dy).

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

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

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

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

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£KsO*^MO9kO*at > s

lu^sxFs ∈ M(X)

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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

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‖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

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u10, u

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

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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)

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(1 − ω) sup0≤s≤t

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0‖M(X).

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‖(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

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≤ t1−1/αC1C2(t0)‖u0‖(C1(X))∗ sup0<s≤t

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F (µ) =1

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C(M+M (X))

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C10 (M+(X))

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L FC∞(X) ∩ C1(X)

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

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O*^siu[cbr^M

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âã&ä½âÖÞ

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γnt (x) =

(∫ t

(Uns,0)

∗(L∗n,s − L∗

s)U∗t,sV (x, ·) ds, un

+ (U∗t (t, 0)V (x, ·), un

0 − u0),

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(∫ t

(Uns,0)

∗(L∗n,s − L∗

s)U∗t,sψ(x, ·; z)(1 + |z|d+β)ds, un

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

0 − u0).

MO*^sy*O

|γnt (x)| ≤ K

(∫ t

|γns (x)| + sup

x,z|δn

s (x, z)|)ds, un0

+ |(U∗t (t, 0)V (x, ·), un

0 − u0)|

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x,z|δn

t (x, z)|

[]KMO O9Zw[cbrN3lo[]O

ξt ≤ K

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|(U∗t (t, 0)V (x, ·), un

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k = 1 ¢ B1

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(B1g)(x) =

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

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1F (hδx)O;dv_1luqeZ

∫ ∫ [

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

δµ(x1;hδx), yχ(y)

ν(x1, dy).

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

Lh1F (µ) =

∫ ∫ [

h(F (µ+ h(δz+y − δz)) − F (µ)) −

∇zδF

δµ(z;µ), yχ(y)

µ(dz)ν(z, dy)

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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)

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lvZ l_s^sy¡[]biv^ªiu

x, y_M^Mbika]N0qXika

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Lh1F (µ) =

δµ, µ

∫ ∫ (∫ 1

δµ2(µ+ th(δz+y − δz))dt, (δz+y − δz)

ν(z, dy)µ(dz). ¬ @vk®

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»h» Ð » "

B2g(x, y) = V (x, y)∂g

∂x(x, y) + V (y, x)

∂y(x, y), ¬ @vv®

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

∂x2+ 2γ(x, y)

∂x∂y+ c(y, x)

∂y2,

γ7 %/ R / R

γ(x, y) = γ(y, x)*!P

Bl2 = 0

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l(x1, x2) =

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âã&ä½âÖÞ

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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)

FP m,T, ‖µ‖

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»h» Ð » P Ψt

P L ! D R Zt

*9 LΩt

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

Yt = Ωt(Zt) P ,% '

R/ Ωt(Z)

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

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9 1L ? 1 P

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ξt*t ≥ 0

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X P 1</ ! PI

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R/ 1 D

! C∞(X) ∩ C2(X)

1 1= ! P9 FΛt

/% / 7 1 +% 1 R =L

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∂z, ξt

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»h» Ð » F (Y )

5% ¬ o® " g = gm ∈ C2,sym

∞ (Xm)

µt(B) 1 (- P ¬ C9k® ΛB,h

1 ' 1 L R h−1/2(Zh

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L = LhB

1ΛBk,h

t F 1,% = ' P R

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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;®

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Xm−l

B1gz(y)(dy)Y⊗(m−1)(dz)

h−1+l/2

Xm−l

Bl1gz(y)(

√hY + µt)(dy)Y

⊗(m−l)(dz). ¬ ~vk®

' § pMpMqrXYb^M ¬ u® [ci

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(−1)m−p(g, Y ⊗p ⊗ µ⊗(m−p)t )

XYbrO*qexMZ

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Xm−p

Xp−l

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Y ⊗(p−l)(dw)µ⊗(m−p)t (du),

âã&ä½âÖÞ

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(√hY + µt)

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= h−m/2m∑

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Cm−pm−l−r(−1)m−p

Xm−l−r

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√hY + µt)

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= h−1/2

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» » ÐI ¸

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t F (Y ) +O(√h) ¬ ~kv®

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Xm−1

(Bk(g+w

))(v)(Y ⊗ µ⊗(k−1)t )(dv)Y ⊗(m−1)(dw)

Xm−2

(B2k(g+

w))(v)(µ⊗k

t )(dv)Y ⊗(m−2)(dw) ¬ ~uf®

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" *9O(

, P,% = ! Y

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*#=P R,% =(µt

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» » ÐI ¸ = F' R ¬ u® P ! P F

¬ ~uv®I +%

ΛBt F (Y ) =

δY, Y ⊗ µ

⊗(k−1)t

δY 2, µ⊗k

, ¬ ~vk®9 ,%

t F (Y ) =

, Y ⊗ µ⊗(k−1)t

δY (yi)δY (yj)−

By1,...,yk

δY (zi)δY (yj)

|∀i zi=yi, µ⊗k

. ¬ ~kv®

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∫ ∞

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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)

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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

Nonlinear Markov semigroups and interacting Lévy type ...

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