Derivative Formula, Coupling Property and Strong Feller ...pkim/7ICSAA/Slides/Zhao_Dong.pdf · Z....
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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
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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
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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.
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Framework
Applications of Derivative Formula:
Strong Feller properties
Heat kernel estimates
Functional inequalities · · · · · ·
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Framework
Applications of Derivative Formula:
Strong Feller properties
Heat kernel estimates
Functional inequalities · · · · · ·
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Framework
Applications of Derivative Formula:
Strong Feller properties
Heat kernel estimates
Functional inequalities
· · · · · ·
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Framework
Applications of Derivative Formula:
Strong Feller properties
Heat kernel estimates
Functional inequalities · · · · · ·
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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
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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
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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 |.
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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 |.
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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
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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.
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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.
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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.
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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.
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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).
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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).
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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).
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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.
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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.
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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).
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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 .
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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.
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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..
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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..
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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..
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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..
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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).
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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.
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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.
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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.
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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
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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
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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
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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.
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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 →∞.
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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 →∞.
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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 →∞.
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(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).
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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).
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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).
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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)
.
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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.
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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
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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.
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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.
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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)
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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)
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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.
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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.
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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.
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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).
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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.
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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 ).
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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.
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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
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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,
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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,
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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,
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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,
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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).
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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).
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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).
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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).
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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.
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Thank you !
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