Derivative Formula, Coupling Property and Strong Fellerfor S(P)DEs Driven by Levy Processes
Z. DongJoint work with: X.H. Peng, Y. L. Song, X.C. Zhang
Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences
7th International Conference on Stochastic Analysis and itsApplications, Seoul national university, August 6-11, 2014
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 1 / 42
Outline
1 Introduction: Framework and Known Results
2 Derivative formula and coupling property for SDEs
3 Derivative formula and coupling property for SPDEs
4 Strong Feller Property for SDEs driven by degenerate additive noise
5 Classical Wiener Space
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 2 / 42
Framework
The Bismut formula (also called Bismut-Elworthy-Li formula) is afundamental tool in stochastic analysis. Let, for instance, Xt be adiffusion process on Rn generated by an elliptic differential operator andPtt≥0 be the associated Markov semigroup. The Bismut formula is oftype
∇ξPt f (x) = Ef (X xt )Mx
t , f ∈ Bb(Rn), t > 0,
where Mxt is a random variable independent of f , and ∇ξ is the directional
derivative along ξ.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 3 / 42
Framework
Applications of Derivative Formula:
Strong Feller properties
Heat kernel estimates
Functional inequalities · · · · · ·
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 4 / 42
Framework
Applications of Derivative Formula:
Strong Feller properties
Heat kernel estimates
Functional inequalities · · · · · ·
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 4 / 42
Framework
Applications of Derivative Formula:
Strong Feller properties
Heat kernel estimates
Functional inequalities
· · · · · ·
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 4 / 42
Framework
Applications of Derivative Formula:
Strong Feller properties
Heat kernel estimates
Functional inequalities · · · · · ·
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 4 / 42
Framework
Development:
Diffusion (jump-diffusion) case:♣ Non-degenerate Wiener processes: J.M. Bismut( Large Deviations and theMalliavin Calculus, Birkhauser, Boston, 1984), K.D. Elworthy and X.M.Li(JFA,1994), S. Peszat and J. Zabczyk(Ann.Prob,1995), A. Takeuchi(JTheor.Prob.,2010), Z.Dong and Y.C.Xie(JDE,2011), B. Xie(Poten. Ana., 2012)· · ·♣ Degenerate Wiener processes: A. Guillin and F.Y. Wang(JDE,2012), F.Y. Wang
and X.C. Zhang(JMPA,2013), · · ·
Purely jump case:♣ Non-degenerate jump processes: R.F. Bass and M. Cranston. (Ann. Probab.,1986), J.R. Norris, (Seminaire de Probabilites XXII. Lect. Notes. Math., 1988),Z.Q.Cheng(2010),F.Y. Wang(SPA, 2012), X.C. Zhang(SPA, 2012), E. Priola andJ. Zabczyk.(PTRF, 2011), J. Wang and F.Y. Wang(SPA, 2012)· · ·♣ Degenerate jump processes: Few results
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 5 / 42
Framework
Development:
Diffusion (jump-diffusion) case:♣ Non-degenerate Wiener processes: J.M. Bismut( Large Deviations and theMalliavin Calculus, Birkhauser, Boston, 1984), K.D. Elworthy and X.M.Li(JFA,1994), S. Peszat and J. Zabczyk(Ann.Prob,1995), A. Takeuchi(JTheor.Prob.,2010), Z.Dong and Y.C.Xie(JDE,2011), B. Xie(Poten. Ana., 2012)· · ·♣ Degenerate Wiener processes: A. Guillin and F.Y. Wang(JDE,2012), F.Y. Wang
and X.C. Zhang(JMPA,2013), · · ·
Purely jump case:♣ Non-degenerate jump processes: R.F. Bass and M. Cranston. (Ann. Probab.,1986), J.R. Norris, (Seminaire de Probabilites XXII. Lect. Notes. Math., 1988),Z.Q.Cheng(2010),F.Y. Wang(SPA, 2012), X.C. Zhang(SPA, 2012), E. Priola andJ. Zabczyk.(PTRF, 2011), J. Wang and F.Y. Wang(SPA, 2012)· · ·♣ Degenerate jump processes: Few results
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 5 / 42
Recall two Facts
Coupling Property:(Cranston, Greven.,1995,SPA)A strong Markov process on a Polish space has coupling property if andonly if
limt→∞
‖Pt(x , ·)− Pt(y , ·)‖Var = 0, x , y ∈ Rn
where Pt(x , ·) is the transition probability and ‖ · ‖Var denotes the totalvariation norm.
Strong Feller:(Da Prato, Zabczyk, Erg.Inf.Dimen.Sys.,1995) A Markovsemigroup Pt on Bb(Rn) is strong Feller if ∀f ∈ C 2
b (Rn), one has
|Pt f (x)− Pt f (y)| ≤ C‖f ‖∞|x − y |.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 6 / 42
Recall two Facts
Coupling Property:(Cranston, Greven.,1995,SPA)A strong Markov process on a Polish space has coupling property if andonly if
limt→∞
‖Pt(x , ·)− Pt(y , ·)‖Var = 0, x , y ∈ Rn
where Pt(x , ·) is the transition probability and ‖ · ‖Var denotes the totalvariation norm.
Strong Feller:(Da Prato, Zabczyk, Erg.Inf.Dimen.Sys.,1995) A Markovsemigroup Pt on Bb(Rn) is strong Feller if ∀f ∈ C 2
b (Rn), one has
|Pt f (x)− Pt f (y)| ≤ C‖f ‖∞|x − y |.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 6 / 42
Outline
1 Introduction: Framework and Known Results
2 Derivative formula and coupling property for SDEs
3 Derivative formula and coupling property for SPDEs
4 Strong Feller Property for SDEs driven by degenerate additive noise
5 Classical Wiener Space
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 7 / 42
Framework
Consider following semilinear SDEs:dXt = b(Xt)dt + σtdLt ,
X0 = x ,(1)
where b : Rn → Rn and σ : [0,∞)→ Rn ⊗ Rn are measurable. L is aLevy process with characteristic measure ν.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 8 / 42
Hypothesis
(H2.1) There exists a differentiable function ρ : Rn0 → (0,∞) satisfying
ν(dz) ≥ ν1(dz) := ρ(z)dz .
(H2.2) b ∈ C 1(Rn) with ∇b bounded and Lipschitz continuous. Andthere exists a constant β > 0 such that |σ−1
s | ≤ β for any s > 0.
(H2.3) There is a constant K > 0 such that
〈b(x)− b(y), x − y〉 ≤ −K |x − y |2
for any x , y ∈ Rn.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 9 / 42
Hypothesis
(H2.1) There exists a differentiable function ρ : Rn0 → (0,∞) satisfying
ν(dz) ≥ ν1(dz) := ρ(z)dz .
(H2.2) b ∈ C 1(Rn) with ∇b bounded and Lipschitz continuous. Andthere exists a constant β > 0 such that |σ−1
s | ≤ β for any s > 0.
(H2.3) There is a constant K > 0 such that
〈b(x)− b(y), x − y〉 ≤ −K |x − y |2
for any x , y ∈ Rn.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 9 / 42
Hypothesis
(H2.1) There exists a differentiable function ρ : Rn0 → (0,∞) satisfying
ν(dz) ≥ ν1(dz) := ρ(z)dz .
(H2.2) b ∈ C 1(Rn) with ∇b bounded and Lipschitz continuous. Andthere exists a constant β > 0 such that |σ−1
s | ≤ β for any s > 0.
(H2.3) There is a constant K > 0 such that
〈b(x)− b(y), x − y〉 ≤ −K |x − y |2
for any x , y ∈ Rn.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 9 / 42
Framework
Due to (H2.1), L can be decomposed into two independent parts:
Lt = L1t + L2
t ,
where L1 is purely jump process with ν1(dz). The jump measure of L1 isdenoted by N(dz , dt).
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 10 / 42
Framework
If λ := ν1(Rn0) =∞, we aim to investigate the Bismut type formula for
Pt f (x) := Ef (X xt )
and
P1t f (x) := E
f (X x
t )I[Nt≥1]
where t ≥ 0, x ∈ Rn, f ∈ Bb(Rn),Nt := N([0, t]× Rn).
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 11 / 42
Framework
If λ := ν1(Rn0) =∞, we aim to investigate the Bismut type formula for
Pt f (x) := Ef (X xt )
and
P1t f (x) := E
f (X x
t )I[Nt≥1]
where t ≥ 0, x ∈ Rn, f ∈ Bb(Rn),Nt := N([0, t]× Rn).
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 11 / 42
L1-derivative
For T > 0, let V = V (s, z)s≤T ,z∈Rn0
be a predictable and integrableprocess. Let ε > 0, define
Nε(B × [0, t]) =
∫ t
0
∫Rn
0
IB(z + εV (s, z))N(dz , ds),
where ν(B) <∞.
Definition (Bass,1986, Ann. Prob.)
A functional Ft(N) := F (N(dz , ds)|s≤t) is called to have anL1-derivative in the direction V , if there exists an integrable randomvariable denoted by DVFt(N), such that
limε→0
E∣∣Ft(N
ε)− Ft(N)
ε− DVFt(N)
∣∣ = 0.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 12 / 42
L1-derivative
For T > 0, let V = V (s, z)s≤T ,z∈Rn0
be a predictable and integrableprocess. Let ε > 0, define
Nε(B × [0, t]) =
∫ t
0
∫Rn
0
IB(z + εV (s, z))N(dz , ds),
where ν(B) <∞.
Definition (Bass,1986, Ann. Prob.)
A functional Ft(N) := F (N(dz , ds)|s≤t) is called to have anL1-derivative in the direction V , if there exists an integrable randomvariable denoted by DVFt(N), such that
limε→0
E∣∣Ft(N
ε)− Ft(N)
ε− DVFt(N)
∣∣ = 0.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 12 / 42
Integration by Parts Formula
Denote
V =V : Ω× [0,T ]× Rn
0 → Rn∣∣∣V is predictable with V and DzV bounded,
∃U0 ⊂ Rn0 compact, s.t. SuppV ⊂ [0,T ]× U0.
Proposition 2.1(Norris,1988)
Let (H2.1) with ρ ∈ C 1(Rn0). If a functional Ft(N) has an L1-derivative
DVFt(N) for V ∈ V, then
EDVFt(N) = −EFt(N)Rt,
where
Rt =
∫ t
0
∫Rn
0
div(ρ(z)V (s, z))
ρ(z)N(dz , ds).
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 13 / 42
Derivative formula for P1t
Let Jt be the derivative of X xt w.r.t. the initial value x .
Theorem 2.2(Dong, Song, 2013.)
Let (H2.1)-(H2.2) hold and ρ ∈ C 1(Rn) with∫Rn |∇ρ(z)|dz <∞. For
t > 0, ξ ∈ Rn and f ∈ Cb(Rn), we have
∇ξP1t f (x) = −E
f (X x
t )I[Nt≥1]
Nt
∫ t
0
∫Rn
∇ log ρ(z) · (σ−1s Jsξ)N(dz , ds)
.
Furthermore,
‖∇P1t f ‖∞ ≤
4βet‖∇b‖∞
λ(1− e−λt − e−λtλ0t)‖f ‖∞
∫Rn
|∇ρ(z)|dz .
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 14 / 42
Theorem 2.3(Dong, Song, 2013.)
Let (H2.1)-(H2.3) hold and ρ ∈ C 1(Rn) with∫Rn |∇ρ(z)|dz <∞. Then
for t > K+λK ,
‖Pt(x , ·)− Pt(y , ·)‖Var ≤ 4β
Kλ
∫Rn
|∇ρ(z)|dz |x − y |+ 2e−
KλK+λ
t .
Remark: From Theorem 2.2 or Theorem 2.3, we can not obtain the strongFeller of Pt under the condition λ <∞. The reason is that with a positiveprobability the process does not jump before a fixed time t > 0. But thecoupling property of Pt is investigated.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 15 / 42
Notations
Denote
Li =h : Rn → R|
∫Rn
0
|h(z)|iν1(dz) <∞, i = 1, 2;
C =h : Rn → R
∣∣∣h is differential and has compact support in Rn0
.
For h ∈ C , we define a weighted norm as
‖h‖ρ =∫
Rn0
|∇h(z)|2ν1(dz) 1
2+∫
Rn0
h2(z)|∇ log ρ(z)|2ν1(dz) 1
2.
Let C‖·‖ρ
be the closure of C under ‖ · ‖ρ. Denote
Hρ =h ∈ L1 ∩ L2
∣∣∣h ≥ 0, h ∈ C‖·‖ρ
and ∇h is bounded..
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 16 / 42
Notations
Denote
Li =h : Rn → R|
∫Rn
0
|h(z)|iν1(dz) <∞, i = 1, 2;
C =h : Rn → R
∣∣∣h is differential and has compact support in Rn0
.
For h ∈ C , we define a weighted norm as
‖h‖ρ =∫
Rn0
|∇h(z)|2ν1(dz) 1
2+∫
Rn0
h2(z)|∇ log ρ(z)|2ν1(dz) 1
2.
Let C‖·‖ρ
be the closure of C under ‖ · ‖ρ. Denote
Hρ =h ∈ L1 ∩ L2
∣∣∣h ≥ 0, h ∈ C‖·‖ρ
and ∇h is bounded..
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 16 / 42
Notations
Denote
Li =h : Rn → R|
∫Rn
0
|h(z)|iν1(dz) <∞, i = 1, 2;
C =h : Rn → R
∣∣∣h is differential and has compact support in Rn0
.
For h ∈ C , we define a weighted norm as
‖h‖ρ =∫
Rn0
|∇h(z)|2ν1(dz) 1
2+∫
Rn0
h2(z)|∇ log ρ(z)|2ν1(dz) 1
2.
Let C‖·‖ρ
be the closure of C under ‖ · ‖ρ. Denote
Hρ =h ∈ L1 ∩ L2
∣∣∣h ≥ 0, h ∈ C‖·‖ρ
and ∇h is bounded..
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 16 / 42
Notations
Denote
Li =h : Rn → R|
∫Rn
0
|h(z)|iν1(dz) <∞, i = 1, 2;
C =h : Rn → R
∣∣∣h is differential and has compact support in Rn0
.
For h ∈ C , we define a weighted norm as
‖h‖ρ =∫
Rn0
|∇h(z)|2ν1(dz) 1
2+∫
Rn0
h2(z)|∇ log ρ(z)|2ν1(dz) 1
2.
Let C‖·‖ρ
be the closure of C under ‖ · ‖ρ. Denote
Hρ =h ∈ L1 ∩ L2
∣∣∣h ≥ 0, h ∈ C‖·‖ρ
and ∇h is bounded..
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 16 / 42
Derivative Formula for Pt
Theorem 2.6(Dong, Song, 2013.)
Let (H2.1)-(H2.2) hold and ρ ∈ C 1(Rn0). If θ := lim inf
x→∞ν1([h≥x−1])
log x > 0 for
some h ∈ Hρ, then for t > 8(θ∧1)(1−e−1)
, ξ ∈ Rn and f ∈ Cb(Rn),
∇ξPt f (x) =− E
f (X x
t )[H−1
t
∫ t
0
∫Rn
0
〈∇(ρ(z)h(z))
ρ(z), σ−1
s Jsξ〉N(dz , ds)
+ H−2t
∫ t
0
∫Rn
0
〈∇h(z), σ−1s Jsh(z)ξ〉N(dz , ds)
],
where Ht =∫ t
0
∫Rn
0h(z)N(dz , ds).
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 17 / 42
Derivative Formula for Pt
Continuation of Theorem 2.6
Furthermore,
‖∇Pt f ‖∞ ≤C(
1 +1
t − 8(θ∧1)(1−e−1)
)e‖∇b‖∞tβ‖f ‖∞
×∥∥ |∇(hρ)|
ρ
∥∥L2 + 2‖∇h‖∞
√‖h‖2
L2 + ‖h‖2L1
,
where C is a constant independent of t.
Remark: The condition θ > 0 can ensure Levy measure ν1 is infinite.Indeed, from Theorem 2.6 and classical approximation argument, we canderive strong Feller property of Pt when b is Lipschitz continuous.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 18 / 42
Derivative Formula for Pt
Continuation of Theorem 2.6
Furthermore,
‖∇Pt f ‖∞ ≤C(
1 +1
t − 8(θ∧1)(1−e−1)
)e‖∇b‖∞tβ‖f ‖∞
×∥∥ |∇(hρ)|
ρ
∥∥L2 + 2‖∇h‖∞
√‖h‖2
L2 + ‖h‖2L1
,
where C is a constant independent of t.
Remark: The condition θ > 0 can ensure Levy measure ν1 is infinite.Indeed, from Theorem 2.6 and classical approximation argument, we canderive strong Feller property of Pt when b is Lipschitz continuous.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 18 / 42
SDEs Driven by α-stable Process
As an example, we have
Example
Let (H2.2) hold. If ρ(z) = Cα|z|n+α with Cα > 0 and 0 < α < 2, then for
t ≥ 1 + 81−e−1 , ξ ∈ Rn and f ∈ Cb(Rn),
‖∇ξPt f ‖∞ ≤ C (n, α)‖f ‖∞|ξ|βe‖∇b‖∞t ,
where C (n, α) denotes a constant only depending on n and α.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 19 / 42
Outline
1 Introduction: Framework and Known Results
2 Derivative formula and coupling property for SDEs
3 Derivative formula and coupling property for SPDEs
4 Strong Feller Property for SDEs driven by degenerate additive noise
5 Classical Wiener Space
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 20 / 42
Framework
Let (H, 〈·, ·〉) be a separable Hilbert space and µ be a Gaussian measureon H with covariance operator Q.
Quasi-invariant Property:
Under the shift z 7→ z + h for any h ∈ImQ12 , µ(·+ h) and µ are mutually
absolutely continuous.
ϕ(z , h) :=µ(dz + h)
µ(dz)= exp〈h, z〉0 −
1
2〈h, h〉0, µ− a.s,
where 〈·, ·〉0 stands for the inner product induced by Q12 and equipped on
ImQ12
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 21 / 42
Framework
Let (H, 〈·, ·〉) be a separable Hilbert space and µ be a Gaussian measureon H with covariance operator Q.
Quasi-invariant Property:
Under the shift z 7→ z + h for any h ∈ImQ12 , µ(·+ h) and µ are mutually
absolutely continuous.
ϕ(z , h) :=µ(dz + h)
µ(dz)= exp〈h, z〉0 −
1
2〈h, h〉0, µ− a.s,
where 〈·, ·〉0 stands for the inner product induced by Q12 and equipped on
ImQ12
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 21 / 42
Framework
Consider SDEs on H:dXt = AXtdt + F (Xt)dt + dLt + dZt ,
X0 = x ,(2)
where A : D(A) ⊂ H→ H is an adjoint, unbounded and linear operatorgenerating a C0-semigroup Stt≥0 on H. L := Ltt≥0 is a Levy processon H with Levy measure ν. Z := Ztt≥0 is another square-integrableLevy process independent of L.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 22 / 42
Hypothesis
(H3.1) ImS(t) ⊂ ImQ holds for any t > 0.
(H3.2) F : H→ H is Frechet differentiable with ∇F bounded andLipschitz continuous.
(H3.3) A is a dissipative operator defined by
A =∑k≥1
(−γk)ek ⊗ ek , (3)
for 0 < γ1 ≤ γ2 ≤ · · · ≤ γk ≤ · · · and γk →∞ as k →∞.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 23 / 42
Hypothesis
(H3.1) ImS(t) ⊂ ImQ holds for any t > 0.
(H3.2) F : H→ H is Frechet differentiable with ∇F bounded andLipschitz continuous.
(H3.3) A is a dissipative operator defined by
A =∑k≥1
(−γk)ek ⊗ ek , (3)
for 0 < γ1 ≤ γ2 ≤ · · · ≤ γk ≤ · · · and γk →∞ as k →∞.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 23 / 42
Hypothesis
(H3.1) ImS(t) ⊂ ImQ holds for any t > 0.
(H3.2) F : H→ H is Frechet differentiable with ∇F bounded andLipschitz continuous.
(H3.3) A is a dissipative operator defined by
A =∑k≥1
(−γk)ek ⊗ ek , (3)
for 0 < γ1 ≤ γ2 ≤ · · · ≤ γk ≤ · · · and γk →∞ as k →∞.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 23 / 42
(H3.4) There exists a differentiable function ρ : H→ (0,∞) with ∇ρbounded and satisfying
λ :=
∫Hρ(z)µ(dz) <∞ and
∫H|z |2ρ(z)µ(dz) <∞,
such that
ν(dz) = ρ(z)µ(dz).
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 24 / 42
Integration by Parts Formula
Set
V =V : Ω× [0,T ]→ ImQ
∣∣∣V is predictable and
∫ T
0
E|Q−1V (s)|ds <∞..
Theorem 3.1(Dong, Song, Xu 2013)
Suppose (H3.4) holds. For V ∈ V and f ∈ C 2b (H),
EDV f (Lt)
= −E
f (Lt)Mt
, t ≤ T , (4)
where Mt =∫ t
0
∫H
(〈z ,Q−1V (s)〉 + 〈∇ log ρ(z),V (s)〉
)N(dz , ds).
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 25 / 42
Integration by Parts Formula
Set
V =V : Ω× [0,T ]→ ImQ
∣∣∣V is predictable and
∫ T
0
E|Q−1V (s)|ds <∞..
Theorem 3.1(Dong, Song, Xu 2013)
Suppose (H3.4) holds. For V ∈ V and f ∈ C 2b (H),
EDV f (Lt)
= −E
f (Lt)Mt
, t ≤ T , (4)
where Mt =∫ t
0
∫H
(〈z ,Q−1V (s)〉 + 〈∇ log ρ(z),V (s)〉
)N(dz , ds).
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 25 / 42
Derivative Formula
Set
C 1b (H) =
G : H→ H
∣∣G and its first order derivatives are continuous and bounded..
Theorem 3.2(Dong, Song, Xu 2013)
Let (H3.1), (H3.2) and (H3.4) hold. If∫ t
0 ‖Q−1S(s)‖ds <∞, then for
f ∈ C 1b (H) and ξ ∈ H,
∇ξP1t f (x) = −E
f (X x
t )I[Nt≥1]
Nt
∫ t
0
∫H
(〈z ,Q−1Jsξ〉+ 〈∇ log ρ(z), Jsξ〉
)N(dz , ds)
.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 26 / 42
Example
Let (H3.3) hold. For 0 < δ < 12 , (−A)δ denotes the fractional power of
−A, defined by
(−A)δ =1
Γ(δ)
∫ ∞0
t−δS(t)dt,
where Γ is the Euler function. It can be proved that S(t)H ⊂ D((−A)δ),for any t > 0, and
‖(−A)δS(t)‖ ≤ Cδt−δ
for a suitable positive constant Cδ. Take Q =((−A)δ
)−1, then we have
S(t)H ⊂ ImQ. Moreover,
limt→∞
∫ t0 ‖Q
−1S(s)‖2ds
t≤ lim
t→∞
C 2δ
∫ t0 s−2δds
t= 0.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 27 / 42
Outline
1 Introduction: Framework and Known Results
2 Derivative formula and coupling property for SDEs
3 Derivative formula and coupling property for SPDEs
4 Strong Feller Property for SDEs driven by degenerate additive noise
5 Classical Wiener Space
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 28 / 42
Brownian motion
W :=w : [0,∞)→ Rm|w is continuous with w0 = 0.
W is endowed with the locally uniform topology and the probabilitymeasure µW so that the coordinate process
Wt(w) := wt = (w1t , · · · ,wm
t )
is an m-dimensional Brownian motion.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 29 / 42
Brownian motion
W :=w : [0,∞)→ Rm|w is continuous with w0 = 0.
W is endowed with the locally uniform topology and the probabilitymeasure µW so that the coordinate process
Wt(w) := wt = (w1t , · · · ,wm
t )
is an m-dimensional Brownian motion.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 29 / 42
Subordinator
S :=` : R+ → Rm
+|` is cadlag with `0 = 0, each component
being increasing and purely jumping.
S is endowed with the Skorohod metric and the probability measureµS so that the coordinate process
St(`) := `t = (`1t , · · · , `mt )
is an m-dimensional Levy process with Laplace transform
EµS(e−z·St ) = exp
∫Rm
+
(e−z·u − 1)νS(du)
. (5)
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 30 / 42
Subordinator
S :=` : R+ → Rm
+|` is cadlag with `0 = 0, each component
being increasing and purely jumping.
S is endowed with the Skorohod metric and the probability measureµS so that the coordinate process
St(`) := `t = (`1t , · · · , `mt )
is an m-dimensional Levy process with Laplace transform
EµS(e−z·St ) = exp
∫Rm
+
(e−z·u − 1)νS(du)
. (5)
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 30 / 42
Subordinated Brownian motion
Consider the following product probability space
(Ω,F ,P) :=(W× S,B(W)×B(S), µW × µS
).
Lift Wt and St to this probability space, then Wt and St are independent,and the subordinated Brownian motion
WSt :=(W 1
S1t, · · · ,Wm
Smt
)is an m-dimensional Levy process.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 31 / 42
Subordinated Brownian motion
Assume
P(ω ∈ Ω : ∃j = 1, · · · ,m and ∃t > 0 such that S jt (ω) = 0) = 0, (6)
which means that St is nondegenerate along each direction.
Remark
α-stable subordinator meets this assumption.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 32 / 42
The SDE driven by subordinated Brownian motion
Consider the following SDE:
dXt = b(Xt)dt + σdWSt , X0 = x , (7)
where b : Rd → Rd is a smooth function, σ is a d ×m matrix.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 33 / 42
Hormander’s type condition
Hormander’s type condition at point x ∈ Rd :
∃n = n(x) ∈ N, s.t.
Rank[σ,B1(x)σ,B2(x)σ, · · · ,Bn(x)σ] = d , (8)
where B1(x) := (∇b)ij(x) = (∂jbi (x))ij , and for n ≥ 2,
Bn(x) := (b · ∇)Bn−1(x)− (∇b · Bn−1)(x).
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 34 / 42
Strong Feller property
Theorem 4.1(Dong, Peng, Song, Zhang 2013)
Assume that
the solution to (7) globally exists
(b, σ) satisfy Hormander’s type condition (8) at each point x ∈ Rd .
Then for any t > 0, the law of Xt(x) is continuous w.r.t. variable x in thetotal variation distance. In particular, the semigroup (Pt)t>0 has thestrong Feller property, i.e., for any t > 0 and f ∈ Bb(Rd),
x 7→ Ef (Xt(x)) is continuous.
Remark
This result has been much extended to multiplicative noise by Xi ChengZhang 2013.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 35 / 42
Example
Stochastic oscillators:dzi (t) = ui (t)dt, i = 1, · · · , d ,dui (t) = −∂ziH(z(t), u(t))dt, i = 2, · · · , d − 1,
dui (t) = −[∂ziH(z(t), u(t)) + γiui (t)]dt +√TidW
iS it, i = 1, d ,
where d ≥ 3, γ1, γd ∈ R, T1,Td > 0, and
H(z , u) :=d∑
i=1
(1
2|ui |2 + V (zi )
)+
d−1∑i=1
U(zi+1 − zi ).
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 36 / 42
Example
Proposition 4.2
Assume that V ,U ∈ C∞(R) are nonnegative and lim|z|→∞ V (z) =∞. If
U is strictly convex, then for any f ∈ Bb(Rd × Rd), the map
(z0, u0) 7→ Ez0,u0f (zt , ut)
is continuous.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 37 / 42
Outline
1 Introduction: Framework and Known Results
2 Derivative formula and coupling property for SDEs
3 Derivative formula and coupling property for SPDEs
4 Strong Feller Property for SDEs driven by degenerate additive noise
5 Classical Wiener Space
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 38 / 42
Classical Wiener Space
Let W be the space of all continuous functions from R+ := [0,∞) toRm vanishing at starting point 0 and the Wiener measure µW so thatthe coordinate process
Wt(ω) = ωt
is a standard m-dimensional Brownian motion.
Let H ⊂W be the Cameron-Martin space. The inner product in H isdenoted by
〈h1, h2〉H :=m∑i=1
∫ ∞0
hi1(s)hi2(s)ds.
Let D be the Malliavin derivative operator.
Let Dk,p be the associated Wiener-Sobolev space with the norm
‖F‖k,p := ‖F‖p + ‖DF‖p + · · ·+ ‖DkF‖p,
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 39 / 42
Classical Wiener Space
Let W be the space of all continuous functions from R+ := [0,∞) toRm vanishing at starting point 0 and the Wiener measure µW so thatthe coordinate process
Wt(ω) = ωt
is a standard m-dimensional Brownian motion.
Let H ⊂W be the Cameron-Martin space. The inner product in H isdenoted by
〈h1, h2〉H :=m∑i=1
∫ ∞0
hi1(s)hi2(s)ds.
Let D be the Malliavin derivative operator.
Let Dk,p be the associated Wiener-Sobolev space with the norm
‖F‖k,p := ‖F‖p + ‖DF‖p + · · ·+ ‖DkF‖p,
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 39 / 42
Classical Wiener Space
Let W be the space of all continuous functions from R+ := [0,∞) toRm vanishing at starting point 0 and the Wiener measure µW so thatthe coordinate process
Wt(ω) = ωt
is a standard m-dimensional Brownian motion.
Let H ⊂W be the Cameron-Martin space. The inner product in H isdenoted by
〈h1, h2〉H :=m∑i=1
∫ ∞0
hi1(s)hi2(s)ds.
Let D be the Malliavin derivative operator.
Let Dk,p be the associated Wiener-Sobolev space with the norm
‖F‖k,p := ‖F‖p + ‖DF‖p + · · ·+ ‖DkF‖p,
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 39 / 42
Classical Wiener Space
Let W be the space of all continuous functions from R+ := [0,∞) toRm vanishing at starting point 0 and the Wiener measure µW so thatthe coordinate process
Wt(ω) = ωt
is a standard m-dimensional Brownian motion.
Let H ⊂W be the Cameron-Martin space. The inner product in H isdenoted by
〈h1, h2〉H :=m∑i=1
∫ ∞0
hi1(s)hi2(s)ds.
Let D be the Malliavin derivative operator.
Let Dk,p be the associated Wiener-Sobolev space with the norm
‖F‖k,p := ‖F‖p + ‖DF‖p + · · ·+ ‖DkF‖p,
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 39 / 42
Strong continuity
Theorem 5.3(Dong, Peng, Song, Zhang 2013)
Let (Xλ)λ∈Λ be a family of Rd -valued Wiener functionals over W.Suppose that for some p > 1:(1) Xλ ∈ D2,p for each λ ∈ Λ, and λ 7→ ‖Xλ‖2,p is locally bounded.
(2) λ 7→ Xλ is continuous in probability, i.e., for any ε > 0 and λ0 ∈ Λ,
limλ→λ0
P(|Xλ − Xλ0 | ≥ ε) = 0.
(3) For each λ ∈ Λ, the Malliavin covariance matrix ΣXλ of Xλ is invertible
almost surely.Then the law of Xλ in Rd admits a density ρλ(x) so that λ 7→ ρλ iscontinuous in L1(Rd).
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 40 / 42
Strong continuity
Theorem 5.3(Dong, Peng, Song, Zhang 2013)
Let (Xλ)λ∈Λ be a family of Rd -valued Wiener functionals over W.Suppose that for some p > 1:(1) Xλ ∈ D2,p for each λ ∈ Λ, and λ 7→ ‖Xλ‖2,p is locally bounded.(2) λ 7→ Xλ is continuous in probability, i.e., for any ε > 0 and λ0 ∈ Λ,
limλ→λ0
P(|Xλ − Xλ0 | ≥ ε) = 0.
(3) For each λ ∈ Λ, the Malliavin covariance matrix ΣXλ of Xλ is invertible
almost surely.Then the law of Xλ in Rd admits a density ρλ(x) so that λ 7→ ρλ iscontinuous in L1(Rd).
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 40 / 42
Strong continuity
Theorem 5.3(Dong, Peng, Song, Zhang 2013)
Let (Xλ)λ∈Λ be a family of Rd -valued Wiener functionals over W.Suppose that for some p > 1:(1) Xλ ∈ D2,p for each λ ∈ Λ, and λ 7→ ‖Xλ‖2,p is locally bounded.(2) λ 7→ Xλ is continuous in probability, i.e., for any ε > 0 and λ0 ∈ Λ,
limλ→λ0
P(|Xλ − Xλ0 | ≥ ε) = 0.
(3) For each λ ∈ Λ, the Malliavin covariance matrix ΣXλ of Xλ is invertible
almost surely.
Then the law of Xλ in Rd admits a density ρλ(x) so that λ 7→ ρλ iscontinuous in L1(Rd).
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 40 / 42
Strong continuity
Theorem 5.3(Dong, Peng, Song, Zhang 2013)
Let (Xλ)λ∈Λ be a family of Rd -valued Wiener functionals over W.Suppose that for some p > 1:(1) Xλ ∈ D2,p for each λ ∈ Λ, and λ 7→ ‖Xλ‖2,p is locally bounded.(2) λ 7→ Xλ is continuous in probability, i.e., for any ε > 0 and λ0 ∈ Λ,
limλ→λ0
P(|Xλ − Xλ0 | ≥ ε) = 0.
(3) For each λ ∈ Λ, the Malliavin covariance matrix ΣXλ of Xλ is invertible
almost surely.Then the law of Xλ in Rd admits a density ρλ(x) so that λ 7→ ρλ iscontinuous in L1(Rd).
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 40 / 42
Remark
Bogachev (AMS, 2010) have already showed such a result in the first orderSobolev space W 1,p(Rd ,Rd) provided p ≥ d . Theorem 5.3 does notdepend on the dimension of space.
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 41 / 42
Thank you !
Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 42 / 42