Probabilistic approach for semi-linear stochastic fractal equations

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ScienceDirect Stochastic Processes and their Applications xx (xxxx) xxx–xxx www.elsevier.com/locate/spa

Probabilistic approach for semi-linear stochastic fractal equations Yingchao Xie a , Qi Zhang b,∗ , Xicheng Zhang c

Q1

a School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, PR China b School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China c School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, PR China

Received 9 October 2013; received in revised form 2 July 2014; accepted 4 July 2014

1 2

Abstract

3

In this work we provide a stochastic representation for a class of semi-linear stochastic fractal equations, 1, p and prove the existence and uniqueness of Wρ -solutions to stochastic fractal equations by using purely

4 5

1, p probabilistic argument, where ρ is a suitable weighted function, and Wρ is the associated first order

6

weighted Sobolev space. c 2014 Published by Elsevier B.V. ⃝

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Q2

1. Introduction

9

10

Let (Wt )t60 be a d-dimensional standard Brownian motion on negative real axis R− =: (−∞, 0]. Consider the following stochastic differential equation:  s  s X t,s (x) = x + br (X t,r (x)) dr + σr (X t,r (x)) dWr , s ∈ [t, 0], (1.1) t

13

t

where b : R− × Rd → Rd and σ : R− × Rd → Rd ⊗ Rd are two measurable functions with sup (∥∇bt ∥∞ + ∥∇σt ∥∞ ) < +∞.

t∈R−

∗ Corresponding author. Tel.: +86 13818190615.

E-mail addresses: [email protected] (Y. Xie), [email protected] (Q. Zhang), [email protected] (X. Zhang). http://dx.doi.org/10.1016/j.spa.2014.07.007 c 2014 Published by Elsevier B.V. 0304-4149/⃝

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Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx

Under this condition, for each x ∈ Rd and t < 0, there exists a unique solution (X t,s (x))s∈[t,0] to Eq. (1.1). Given two functions ϕ : Rd → Rd and f : R− × Rd → Rd , consider the function  0 u t (x) := EW ϕ(X t,0 (x)) + EW f s (X t,s (x)) ds, (1.2) t

4 5 6

7

8

where the expectation is with respect to the law of Brownian motion W· . Under some regularity conditions on ϕ and f , it is well-known that u t (x) satisfies the following second order partial differential system in the weak sense (cf. [4,3]): ∂t u + Lt u + f = 0,

u 0 = ϕ,

(1.3)

where jk

9

10 11 12 13 14 15

16

Lt u := 12 σtik σt ∂i2j u + bti ∂i u. Here we have used the usual summation convention. Recently, in [11] we have extended representation (1.2) to more general L´evy processes, and applied to solve the quasi-linear partial integro-differential equation. On the other hand, let (Bt )t60 be another independent m-dimensional standard Brownian motion, and g : R− × Rd → Rd ⊗ Rm another measurable function. Consider the following backward Itˆo’s process:  0  0 ← − u t (x) := EW ϕ(X t,0 (x)) + EW f s (X t,s (x)) ds + EW gs (X t,s (x)) d B s , (1.4) t

17 18 19

20

where the stochastic integral is the backward Itˆo’s integral (see [5, p. 112]). It is also well-known that under some conditions, u t (x) solves the following linear stochastic partial differential equation (SPDE) (cf. [9, p. 181, Theorem 1]):  0  0  0 ← − u t (x) = ϕ(x) + Ls u s (x) ds + f s (x) ds + gs (x) d B s . (1.5) t

21

22

t

t

t

Now, consider the following more general coupled stochastic system:  s  s   X (x) = x + b (X (x), u (X (x))) dr + σr (X t,r (x), u r (X t,r (x))) dWr ,  t,s r t,r r t,r   t t    0 ← − (1.6) u t (x) = EW ϕ(X t,0 (x)) + EW gs (X t,s (x), u s (X t,s (x))) d B s  t   0     + EW f s (X t,s (x), u s (X t,s (x)), ∇u r (X t,s (x))) ds. t

23 24

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If this system allows a “smooth” (FtB )-adapted solution u t (x), where FtB := σ {Bs : s ∈ [t, 0]}, then u t (x) will solve the following quasi-linear SPDEs:  0  ik jk 2 i 1 u t (x) = ϕ(x) + 2 (σs σs )(x, u s (x))∂i j + bs (x, u s (x))∂i u s (x) ds t

 26

+ t

27 28

0

f s (x, u s (x), ∇u s (x)) ds +

0



← − gs (x, u s (x)) d B s .

t

In fact, when b and σ do not depend on u, in the framework of backward stochastic differential equations, Pardoux and Peng [7] have already provided a self-contained probabilistic treatment

Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx

3

for the following semi-linear stochastic partial differential equation:  0  0 u t (x) = ϕ(x) + Ls u s (x) ds + f s (x, u s (x), σs ∇u s (x)) ds t



0

2

t

← − gs (x, u s (x), σs ∇u s (x)) d B s .

+

1

3

t

In this work we are concerning with the forward SDE (1.1) replaced by the following SDE driven by a symmetric α-stable process:  s  s X t,s (x) = x + br (X t,r (x)) dr + σr dW Sr , s ∈ [t, 0], (1.7) t

α/2

6

t

where (St )t60 is an independent α2 -subordinator with Laplace transform: Ee−λSt = e−t|λ|

4 5

,

λ > 0,

7

8

so that (L t )t60 := (W St )t60 is a symmetric α-stable process. Notice that the generator of SDE (1.7) is given by  1 dz Lsb u(x) := (1.8) (u(x + σs z) + u(x − σs z) − 2u(x)) d+α + bs (x) · ∇u(x). 2 Rd |z| Remark 1.1. If b has bounded and continuous first order partial derivatives with respect to x and locally uniformly in t, then the solution of SDE (1.7) defines aC 1 -stochastic diffeomorphism s flow. In fact, if we let Yt,s (x) := X t,s (x) − Z t,s , where Z t,s := t σr dW Sr , then Yt,s (x) solves the following random ODE:  s Yt,s (x) = x + br (Yt,r (x) + Z t,r ) dr.

9 10

11

12 13 14 15

16

t

Since b has bounded and continuous derivatives, x → Yt,s (x) is differentiable, so does X t,s (x). We shall prove the following result. Theorem 1.2. Let α ∈ (1, 2), p >

α α−1

18

1, p

∨ d and Wρ

be defined in Section 2. Suppose that

(H1) b : R− × Rd → Rd is continuous and has bounded and continuous first order partial derivatives with respect to x and locally uniformly in t, and σ : R− → Rd ⊗ Rd is locally bounded and satisfies sup t∈[T,0]

∥σt−1 ∥

< ∞,

∀T < 0.

19

20 21 22 23

(H2) f satisfies that for all t, x and u, u ′ ∈ Rd , U, U ′ ∈ Rd ⊗ Rd , | f t (x, u, U ) − f t (x, u ′ , U ′ )| 6 C(|u − u ′ | + |U − U ′ |), and for some γ1 ∈

17

p L ρ (Rd ),

24 25 26

| f t (x, u, U )| 6 γ1 (x) + C(|u| + |U |). ′

(H3) g has first order derivatives with respect to x and u, and satisfies |∇x gt (x, u) − ∇x gt (x, u ′ )| + |∇u gt (x, u) − ∇u gt (x, u ′ )| 6 C|u − u ′ |,

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Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx p

1 2

and for some γ2 ∈ L ρ (Rd ), |gt (x, u)| + |∇x gt (x, u)| 6 γ2 (x) + C|u|,

|∇u gt (x, u)| 6 C.

1, p

3 4

5

∞ (R− ; Then for any ϕ ∈ Wρ , there exists a unique (FtB )-adapted process u ∈ L p (Ω ; L loc 1, p Wρ )) satisfying  0 ← − L u t (x) = E ϕ(X t,0 (x)) + E L gs (X t,s (x), u s (X t,s (x))) d B s t 0

 6

+

E L f s (X t,s (x), u s (X t,s (x)), ∇u s (X t,s (x))) ds,

(1.9)

t 7 8 9

10 11

12

where the expectation E L is with respect to the L´evy process L · = W S· , and the stochastic integral is a backward Itˆo’s integral w.r.t. the m-dimensional standard Brownian motion B. As a result, we have: Theorem 1.3. In the situation of Theorem 1.2, u t (x) is the unique weak solution of the following ∞ (R− ; W1, p )): semi-linear SPDE in the class of all (FtB )-adapted processes in L p (Ω ; L loc ρ  0  0  0 ← − u t (x) = ϕ(x) + Lsb u s (x) ds + f s (x, u s (x), ∇u s (x)) ds + gs (x, u s (x)) d B s , t

13

14

i.e., for all ψ ∈

t

C0∞ (Rd ; Rd )

⟨u t , ψ⟩ = ⟨ϕ, ψ⟩ +

 t

 15

+

0

t

and t 6 0,

0

⟨u s , Lsb∗ ψ⟩ ds +

0



⟨ f s (u s , ∇u s ), ψ⟩ ds

t

← − ⟨gs (u s ), ψ⟩ d B s ,

(1.10)

t 16

17

18 19 20 21 22 23 24 25 26 27 28 29 30 31

32

where Lsb∗ is the formally adjoint operator of Lsb and given by  1 dz b∗ Ls φ(x) := (φ(x + σs z) + φ(x − σs z) − 2φ(x)) d+α − div(bs φ)(x). 2 Rd |z| The novelty of this result is that b is not necessarily bounded and we are working in the weighted space. For example, let us take bs (x) = x and σs = I, then L b becomes the fractional α Ornstein–Uhlenbeck operator, which exhibits different behaviour as fractional Laplacian ∆ 2 . When b is bounded, a H¨older theory for stochastic integro-partial differential equations has been established in [6]. Compared with [6], our treatment is purely probabilistic. Notice that the method of backward doubly SDEs used in [7] seems not valid for nonlocal fractional Laplacian. This is due to the absence of a suitable martingale representation theorem. Here, the key point of proving Theorem 1.3 is to show the equivalence between (1.9) and (1.10), i.e., mild solutions and weak solutions. We emphasise that the unboundedness of b causes some technical difficulties for proving such an equivalence. This paper is organised as follows: In Section 2, we give some necessary preliminaries. In Section 3, we prove Theorem 1.2. In Section 4, we prove Theorem 1.3. Before concluding this introduction, we make the following conventions: C denotes an irrelevant positive constant, whose values may change in different places, and A ≼ B means that A 6 C B

for some constant C > 0.

Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx

5

2. Preliminaries

1

In the remainder of this paper, we fix α ∈ (1, 2), γ ∈ [0, α), and set

2

ρ(x) := (1 + |x|)γ . p

3

p

Let L ρ := L ρ (Rd ) be the weighted L p -space with norm:  ∥ f ∥ p;ρ :=

Rd k, p

For k ∈ N, let Wρ

1



p

=

|f| ρ p

1

p

.

5

be the k-order weighted Sobolev space with norm:

k 

∥ f ∥k, p;ρ :=

| f (x)| ρ(x) dx p

4

6

∥∇ j f ∥ p;ρ ,

7

j=0 p

where ∇ j denotes the j-order generalised gradient. If ρ ≡ 1, we shall drop the subscript ρ in L ρ k, p and Wρ . Notice that by the Sobolev embedding theorem, ∥ f ∥∞ ≼ ∥ f ∥1, p 6 ∥ f ∥1, p;ρ ,

p > d.

8 9

(2.1)

10

Suppose now that (H1) is satisfied. For t 6 s 6 0 and x ∈ Rd , let X t,s (x) be the unique solution of SDE (1.7). For any bounded and measurable function f on Rd , define

12

Tt,s f (x) := E f (X t,s (x)).

13

By the Markov property, it is easy to see that Tt,s f (x) = Tt,r Tr,s f (x),

14

t < r < s 6 0.

(2.2)

Moreover, for any f ∈ Cb∞ (Rd ), by (2.2) and Itˆo’s formula, we have for t < s, dTt,s f = −Ltb Tt,s f. dt

dTt,s f = Tt,s Lsb f, ds

(2.3)

s∈[t,0]

,

(2.4)

L 1ρ ,

and for any f ∈    E f (X t,s (x))ρ(x) dx 6 C Rd

Rd

19 20

21

22

f (x)ρ(x) dx.

(2.5)

In particular, for any k ∈ N ∪ {0} and p ∈ [1, ∞], if b has bounded derivatives up to order k, p k + 1, then for any T < 0, there is a constant C = C(T, k, p) such that for any f ∈ Wρ and T 6 t < s 6 0, ∥Tt,s f ∥k, p;ρ 6 C∥ f ∥k, p;ρ .

17

18

Theorem 2.1. Assume (H1). Let X t,s (x) solve SDE (1.7) and Jt,s (x) = ∇ X t,s (x) be the Jacobian matrix. Then we have |t| sup ∥∇bs ∥∞

15

16

Let us first prove the following result.

|Jt,s (x)| 6 e

11

(2.6)

23

24 25 26

27

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Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx

Proof. Notice that Jt,s (x) solves the following linear SDE:  s Jt,s (x) = I + ∇br (X t,r (x))Jt,r (x) dr.

(2.7)

t 3 4

5

Estimate (2.4) follows by Gronwall’s inequality. For fixed t < s 6 0, let Yrt,s (x) solve the following SDE:  r  s ′ Yrt,s (x) = x − bt+s−r ′ (Yrt,s (x)) dr − σr ′ dW Sr ′ . ′ t

6

By the uniqueness of SDE, it is easy to see that Yrt,s (X t,s (x)) = X t,t+s−r (x),

7

8

9

10

11

12

(2.8)

t+s−r

∀r ∈ [t, s].

So, −1 Yst,s (X t,s (x)) = x ⇒ X t,s (x) = Yst,s (x).

(2.9)

On the other hand, by (2.8) we have for any γ ∈ [1, α),  s γ  r   t,s t,s γ γ γ ′  E|Yr (x)| ≼ |x| + E|bt+s−r ′ (Yr ′ (x))| dr + E  σr ′ dW Sr ′  t t+s−r  r γ ′ ≼ |x|γ + 1 + E|Yrt,s ′ (x)| dr , t

13 14

15

where we have used the fact that α-stable-like process has finite moment of γ -order with γ ∈ [0, α) (cf. [10]). By Gronwall’s inequality, we obtain sup E|Yrt,s (x)|γ ≼ |x|γ + 1.

(2.10)

r ∈[t,s] 16

17

Notice that the above estimate still holds for γ ∈ [0, 1) since sup E|Yrt,s (x)|γ 6 sup (E|Yrt,s (x)|)γ ≼ (|x| + 1)γ ≼ |x|γ + 1.

r ∈[t,s] 18

19

r ∈[t,s]

Similar to (2.4), we also have sup |∇Yrt,s (x)| 6 e

|t| sup ∥∇bs ∥∞ s∈[t,0]

.

(2.11)

r ∈[t,s] 20

21

22

Now by the change of variables, we have   −1 −1 E f (X t,s (x))ρ(x) dx = E f (x)ρ(X t,s (x)) det(∇ X t,s (x)) dx d d R R  (2.9) = E f (x)ρ(Yst,s (x)) det(∇Yst,s (x)) dx, Rd

23 24

25 26

27

which then implies (2.5) by (2.10) and (2.11). As for (2.6), it follows by the chain rule and (2.4), (2.5). We omit the details.  Theorem 2.2. Assume (H1). Let X t,s (x) solve SDE (1.1) and Jt,s (x) = ∇ X t,s (x). For any f ∈ Cb1 (Rd ), we have    s 1 f (X t,s (x)) σr−1 Jt,r (x) dW Sr . (2.12) ∇E f (X t,s (x)) = E Ss − St t

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Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx

In particular, for any k ∈ N and p ∈ (1, ∞], if b has bounded derivatives up to order k, then k−1, p and T 6 t for any T < 0, there is a constant C = C(T, k, p) such that for any f ∈ Wρ < s 6 0, ∥∇ k Tt,s f ∥ p;ρ 6

C 1

(s − t) α

∥ f ∥k−1, p;ρ .

(2.13)

Proof. Formula (2.12) was proven in [12, Theorem 1.1]. Let us prove (2.13). First of all, by (2.7), it is easy to see that for any m = 0, 1, . . . , k and T < 0, sup x∈Rd ,T 6t
|∇ Jt,s (x)| 6 C. m

(2.14)

Below we drop the variable “x”. By the chain rule and H¨older’s inequality, we have  s  p     1 k p m −1 n  |∇ E f (X t,s )| ≼ E |∇ ( f (X t,s ))| ·  σr ∇ Jt,r dW Sr  m+n=k−1 Ss − St t  s  p  m    (2.14)  1 j −1 n  ≼ E |(∇ f )(X t,s )| ·  σr ∇ Jt,r dW Sr  m+n=k−1 Ss − St t j=0    p  p−1  s m     p−1 1 −1 n j p   . ≼ E σ ∇ J dW E|(∇ f )(X )| t,r S t,s r r   m+n=k−1 (Ss − St ) t j=0 By Burkholder’s inequality (cf. [12, (2.11)]), we have    s  p  p   s    p−1  p−1 1 1 −1 n −1 n    E 6E σr ∇ Jt,r dW Sr  σr ∇ Jt,r dW Sr  p E S·  (Ss − St ) t t |Ss − St | p−1   s  p    2( p−1) 1  ≼E |σr−1 ∇ n Jt,r |2 dSr  p E S·  t |Ss − St | p−1   (2.14) 1 6 CE . p |Ss − St | 2( p−1) Here, E S· denotes the conditional expectation with respect to S· On the other hand, by the property of the α/2-stable subordinator (cf. [12]), we also have     p 1 1  6 C(s − t)− α( p−1) . = E E p p 2( p−1) |Ss − St | 2( p−1) Ss−t Thus, combining the above calculations and by (2.5), we obtain  k−1  k−1    1 1 k p j p |∇ E f (X t,s )| ρ ≼ E|(∇ f )(X t,s )| ρ ≼ |∇ j f | p ρ. p p |s − t| α j=0 |s − t| α j=0 The proof is finished.



1 2 3

4

5 6

7

8

9

10

11

12

13

14

15

16 17

18

19

20

21

The following lemma is similar to [2].

22

k, p

Lemma 2.3. Assume p > 2, k ∈ N ∪ {0} and T < 0. Let f ∈ L p (Ω × [T, 0]; Wρ ) be a 0 ← − measurable (FtB )-adapted process. Then t → t Tt,s f s d B s is almost surely continuous in

23

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Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx k, p

Wρ , and for any (FtB )-stopping time τ ,  p       0 0 ← −   p  6 CE Tt,s f s d B s  ∥ f s ∥k, p;ρ ds . E  sup   T ∨τ t∈[T ∨τ,0]  t

1

2

k, p;ρ

Proof. Let

3

1 p−1 <β< . 2 p

4

(2.15)

Noticing that for any t < s,  sin(πβ) s (s − r )β−1 (r − t)−β dr = 1, π t

5

6

by (2.2) and stochastic Fubini’s theorem (cf. [8, p. 207, Theorem 64]), we can write  0   sin(πβ) 0 s ← − ← − Tt,s f s (x) d B s = (s − r )β−1 (r − t)−β Tt,s f s (x) dr d B s π t t t   sin(πβ) 0 0 ← − 1[t,s] (r )(s − r )β−1 (r − t)−β Tt,r Tr,s f s (x) dr d B s = π t t sin(πβ) 0 0 ← − = 1[t,s] (r )(s − r )β−1 (r − t)−β Tt,r Tr,s f s (x) d B s dr π t t  sin(πβ) 0 Tt,r gr (x)(r − t)−β dr, = π t

7

8

9

10

11

where

12

gr (x) :=

13



0

← − (s − r )β−1 Tr,s f s (x) d B s .

r 14

Q3

15

Thus, by Minkowski’s inequality and H¨older’s inequality, we have  p p    0 sin(πβ) 0 ← −   −β ∥Tt,r gr ∥k, p;ρ (r − t) dr Tt,s f s d B s  6   t  π t k, p;ρ   p−1  0

(2.6)

t (2.15)



≼

17

t 18

19

Q4

pβ − p−1

(r − t)

dr

t

p

∥gr ∥k, p;ρ dr.

On the other hand, by Burkholder’s inequality we have p E∥gr ∨τ ∥k, p;ρ

 ≼ E (2.6)

20

0

0

p

∥gr ∥k, p;ρ dr

≼

16

p 2

0

(s − r ∨ τ )2(β−1) ∥Tr ∨τ,s f s ∥2k, p;ρ ds r ∨τ   0

≼ E r ∨τ

p

(s − r ∨ τ )2(β−1) ∥ f s ∥k, p;ρ ds .

(2.16)

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Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx

Hence,

1





p    0 ← −   Tt,s f s d B s  E  sup   t∈[T ∨τ,0]  t



0

 sup

≼E



0

≼E T



2

 p ∥gr ∨τ ∥k, p;ρ

0 0

≼E T



p

∥gr ∥k, p;ρ dr

t∈[T ∨τ,0] t

k, p;ρ



r ∨τ 0

dr

3

 (s − r ∨ τ ) 

2(β−1)

p ∥ f s ∥k, p;ρ

ds dr

4

p

≼E T ∨τ

∥ f s ∥k, p;ρ ds .

0 ← − k, p The continuity of t → t Tt,s f s (x) d B s in Wρ is due to (2.16) and the following easy fact: k, p For p > 1, β ∈ (0, 1 − 1p ) and f ∈ L p ([T, 0]; Wρ ), the mapping  0 p t → Tt,s f s (x)(s − t)−β ds is a continuous function in Wk, ρ .

5

6 7

8

t



The proof is complete.

9

3. Proof of Theorem 1.2

10

Proof of Theorem 1.2. For the simplicity of notation, we drop the variable “x” below. Set u 0t ≡ Tt,0 ϕ and define the following Picard’s iteration:  0  0 ← − n u n+1 = T ϕ + T (g (u )) d B + Tt,s ( f s (u ns , ∇u ns )) ds. (3.1) t,s s s s t,0 t t

t∈[T ∨τ,0]

t∈[T ∨τ,0]

13

14 15

16

t∈[T,0]

By (3.1) and Minkowski’s inequality, we have    sup

12

t

(Step 1). Let FtB := σ {Bs : s ∈ [t, 0]}. We shall prove the following claim: for any T < 0, there exists a constant C > 0 such that for any n ∈ N and (FtB )-stopping time τ ,       p p p E sup ∥u nt ∥1, p;ρ = E sup ∥u nt∨τ ∥1, p;ρ 6 C ∥ϕ∥1, p;ρ + 1 . (3.2)

E

11

p

∥u n+1 ∥1, p;ρ t

sup

≼ E

t∈[T ∨τ,0]

17

 p

∥Tt,0 ϕ∥1, p;ρ

 p   0  ← −    + E  sup  Tt,s (gs (u ns )) d B s    t t∈[T ∨τ,0] 1, p;ρ  p 

18



0

+E

sup t∈[T ∨τ,0] t

=: I1 + I2 + I3 . For I1 , by (2.6) we have I1 ≼

p ∥ϕ∥1, p;ρ .

∥Tt,s ( f s (u ns , ∇u ns ))∥1, p;ρ ds

19

20

21

22

23

10 1

2

3

Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx

For I2 , by Lemma 2.3 and (H3), we have   0 p I2 ≼ E ∥gs (u ns )∥1, p;ρ ds T ∨τ    0  (H3) n p ≼ E 1 + ∥u s ∥1, p;ρ ds T ∨τ

 0

6

4

T 5

6

 p 1 + E∥u ns∨τ ∥1, p;ρ ds.

For I3 , by (2.13) and H¨older’s inequality, we have p   0 n n ∥∇Tt,s ( f s (u s , ∇u s ))∥ p;ρ ds I31 := E sup t∈[T ∨τ,0] t

p ds ≼ E sup (s − t)1/α t∈[T ∨τ,0] t    p/q   0 0 ds p , ≼ E  sup ∥ f s (u ns , ∇u ns )∥ p;ρ ds q/α t (s − t) t∈[T ∨τ,0] t 

7

8

9

where

1 p

I31

10

0



∥ f s (u ns , ∇u ns )∥ p;ρ

= 1. Since q ∈ (1, α), by (H2), we further have  0  p ≼ 1 + E∥u ns∨τ ∥1, p;ρ ds.

+

1 q

T

11

Similarly, 

 sup

I32 := E

12

0

t∈[T ∨τ,0] t 13

p ∥Tt,s ( f s (u ns , ∇u ns ))∥ p;ρ

Combining the above calculations, we get   E

14

p

sup ∥u n+1 t∨τ ∥1, p;ρ

T

t∈[T,0] 15

h TN

16

19 20 21

22 23 24

 0  p 1 + E∥u ns∨τ ∥1, p;ρ ds. T

p

E∥u ns∨τ ∥1, p;ρ ds.



:= sup E

sup

n6N

t∈[T,0]

p ∥u nt∨τ ∥1, p;ρ

,

then 

18

≼

If we let 

17

0



≼ ∥ϕ∥1, p;ρ + 1 +

ds

h TN +1 6 C

p ∥ϕ∥1, p;ρ

0

 +1+ T

 h sN ds

 6C

p ∥ϕ∥1, p;ρ

0

 +1+ T

 h sN +1 ds ,

where the constant C is independent of N and τ . By Gronwall’s inequality and letting N → ∞, we obtain (3.2). (Step 2). Write Utn,m (x) := u nt (x) − u m t (x), n G n,m (x) := g (x, u (x)) − gt (x, u m t t t t (x)), n,m n n m Ft (x) := f t (x, u t (x), ∇u t (x)) − f t (x, u m t (x), ∇u t (x)).

11

Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx

Then by (3.1), we have  0  ← − Utn+1,m+1 = Tt,s G n,m d B + s s t

1

0 t

Tt,s Fsn,m ds.

2

For R > 0, define a backward (FtB )-stopping time   τ R := sup t < 0 : ∥u nt ∥1, p;ρ > R . By Minkowski’s inequality, we have   E

sup

t∈[T ∨τ R ,0]

p ∥Utn+1,m+1 ∥1, p;ρ

4

5

p



   0  ← −   n,m ≼ E  sup  Tt,s G s d B s   t∈[T ∨τ R ,0]  t

6



1, p;ρ



 sup

+E

3

p

0

 Tt,s F n,m  ds s 1, p;ρ

t∈[T ∨τ R ,0] t

=: J1 + J2 .

(3.3)

By (H3) and (2.1), it is easy to see that ∥G n,m s ∥1, p;ρ

≼

8

∥Usn,m ∥1, p;ρ (∥u ns ∥1, p;ρ

+ 1).

Hence, by Lemma 2.3 we have    0 n,m p J1 ≼ E ∥G s ∥1, p;ρ ds 6 C R E T ∨τ R 0

 6 CR T

9

10

0 T ∨τ R

 p

∥Usn,m ∥1, p;ρ ds

p

0

sup

t∈[T ∨τ R ,0] t 0

 ≼ T

(3.4)

0

p

∥Fsn,m ∥ p;ρ ds

≼ T

p

15

0

6 CR T

p

n+1,m+1 sup ∥Ut∨τ ∥1, p;ρ R

p

n,m E∥Us∨τ ∥ ds. R 1, p;ρ

18

19

 6 CR

t∈[T,0]



0

lim E

T n,m→∞

 p

n,m sup ∥Ut∨τ ∥ R 1, p;ρ

ds,

20

t∈[s,0]

which in turn implies that for any R > 0,   lim E

n,m→∞

p

n,m sup ∥Ut∨τ ∥ R 1, p;ρ

t∈[T,0]

16

17



By (3.2) and Fatou’s lemma, we obtain   n,m→∞

p

n,m ∥ ds E∥Fs∨τ R p;ρ

(3.5)

t∈[T,0]

lim E

13

p

n+1,m+1 ∥1, p;ρ sup ∥Ut∨τ R

12

14

n,m E∥Us∨τ ∥ ds. R 1, p;ρ

Combining (3.3)–(3.5), we obtain   E

11

n,m ds, E∥Us∨τ ∥ R 1, p;ρ

where the constant C R is independent of n and m. For J2 , as in treating I3 in Step 1, and by (H2) we have     J2 ≼ E

7

= 0.

21

(3.6)

22

12 1

2

Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx

Now, by (3.2) we have  sup 

E

t∈[T,0] 3



∥Utn,m ∥21, p;ρ  sup

=E

t∈[T,0]

∥Utn,m ∥21, p;ρ 1τ R
 4

6E

t∈[T ∨τ R ,0]

∥Utn,m ∥21, p;ρ

5

6E

sup t∈[T,0]

n,m 2 ∥ ∥Ut∨τ R 1, p;ρ

 6

6E

sup t∈[T,0]

7

8

+

 2

p

sup

+ E

t∈[T,0]





∥Utn,m ∥21, p;ρ 1τ R >T

 

+C

n,m 2 ∥Ut∨τ ∥ R 1, p;ρ

sup t∈[T,0]







+E

 sup



1 E Rp

p ∥Utn,m ∥1, p;ρ

P(τ R > T )( p−2)/ p

( p−2)/ p

 sup t∈[T,0]

p ∥u nt ∥1, p;ρ

C , R p−2

which, together with (3.6) and p > 2, yields that for any T < 0,   lim E

n,m→∞

sup ∥Utn,m ∥21, p;ρ

= 0.

(3.7)

t∈[T,0]

1, p

9

10

11 12

Thus, there exists a continuous Wρ -valued (FtB )-adapted process u t (x) such that   lim E

n→∞

sup ∥u nt − u t ∥21, p;ρ

= 0.

t∈[T,0]

(Step 3). By taking limits for (3.1), it is easy to see that u satisfies Eq. (1.9). Moreover, the uniqueness follows by a similar calculation as in Step 2. 

14

Remark 3.1. If we assume the coefficients b, f, g and initial value ϕ satisfy some smoothness conditions, then using (2.6) and (2.13), we can further obtain the existence of smooth solutions.

15

4. Proof of Theorem 1.3

13

16

17

Let ϱ(x) ∈ C0∞ (Rd ) be a nonnegative smooth function with support in the unit ball and  ϱ(x) dx = 1. Rd

18

19

20

21

22

23

For ε ∈ (0, 1), let us set ϱε (x) := ε −d ϱ(x/ε), 1 (Rd ), define and for f ∈ L loc

f ε (x) := f ∗ ϱε (x) :=

 Rd

f (y)ϱε (x − y) dy.

Let χ ∈ C0∞ (Rd ) be another nonnegative smooth function with χ (x) = 1,

|x| 6 1,

χ (x) = 0,

|x| > 2.

Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx

13

For R > 0 and a function f , let us set χ R (x) := χ (x/R),

1

f R (x) := f (x)χ R (x).

2

We need the following two simple lemmas.

3

Lemma 4.1. For α ∈ (1, 2), there exists a constant C > 0 only depending on α, d such that for any f ∈ Cb2 (Rd ) and R > 1, |Ltb ( f χ R )(x) − (Ltb f )(x)χ R (x)| 6

C(∥ f ∥∞ (∥σ ∥∞ + |bs (x)|) + ∥∇ f ∥∞ ∥σ ∥2∞ ) . R (4.1)

Proof. Notice that by definition (1.8), Lsb ( f χ R )(x) − (Lsb

4 5

6

7

8

f )(x)χ R (x)

9



dz = ( f (x + σs z) − f (x))(χ R (x + σs z) − χ R (x)) d+α |z| Rd  f (x) dz + (χ R (x + σs z) + χ R (x − σs z) − 2χ R (x)) d+α 2 |z| Rd + f (x)bs (x) · ∇χ R (x) =: I1 (x) + I2 (x) + I3 (x). For I1 (x), in view of α ∈ (1, 2), we have  ∥∇χ ∥∞ dz |I1 (x)| 6 ∥∇ f ∥∞ ∥σ ∥2∞ d+α−2 R |z|61 |z|  ∥∇χ∥∞ ∥σ ∥∞ dz + 2∥ f ∥∞ d+α−1 R |z|>1 |z| ∥∇χ∥∞ 6 C(∥ f ∥∞ ∥σ ∥∞ + ∥∇ f ∥∞ ∥σ ∥2∞ ) . R Similarly, we have |I2 (x)| 6

C∥ f ∥∞ ∥∇χ ∥∞ ∥σ ∥∞ , R

10

11

12

13

14

15

16

17

18

and

19

C∥ f ∥∞ ∥∇χ ∥∞ |bs (x)| . R Combining the above calculations, we obtain (4.1). |I3 (x)| 6

20



Lemma 4.2. Assume that b : Rd → Rd is Lipschitz continuous and u ∈ L p (Rd ) for some p > 1. Then lim ∥(b · ∇u) ∗ ϱε − b · ∇(u ∗ ϱε )∥ p = 0,

ε→0

21

22 23

24

where the gradient ∇u is taken in the distributional sense.

25

Proof. We write

26

(b · ∇u) ∗ ϱε − b · ∇(u ∗ ϱε ) =

 Rd

u(y)(b(·) − b(y)) · ∇ϱε (· − y) dy − (udivb) ∗ ϱε .

27

14

Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx

By the property of convolutions, we have

1

lim ∥(udivb) ∗ ϱε − udivb∥ p = 0,

2

ε→0

and for Lebesgue-almost all x ∈ Rd ,      (b(x) − b(y)) · ∇ϱε (x − y) dy − divb(x) = 0. lim  d ε→0

3

4

R

Hence, by the dominated convergence theorem,       lim u (b(·) − b(y)) · ∇ϱε (· − y) dy − udivb  = 0.

5

6

ε→0

Rd

p

On the other hand, observe that     |gε (x)| :=  (u(x) − u(y))(b(x) − b(y)) · ∇ϱε (x − y) dy  Rd   6 ∥∇b∥∞ |u(x) − u(y)| · |ε −d ∇ϱ|((x − y)/ε) dy d  R |u(x) − u(x − y)| · |ε −d ∇ϱ|(y/ε) dy . = ∥∇b∥∞

7

8

9

10

|y|6ε 11

12

13

Q5

By Minkowski’s inequality, we obtain   ∥gε ∥ p 6 ∥∇b∥∞ ∥u(·) − u(· − y)∥ p · |ε −d ∇ϱ|(y/ε) dy |y|6ε   |∇ϱ|(y) dy , 6 ∥∇b∥∞ sup ∥u(·) − u(· − y)∥ p |y|61

|y|6ε 14 15

which converges to zero as ε → 0.



Now, we can prove the following equivalence between mild solutions and weak solutions. p

16 17

18

Proposition 4.3. Given p > 2, let ϕ ∈ L p and u t , f t , gt ∈ L loc (R− ; L p (Ω ; L p )) be three (FtB )-adapted processes. Then u t (x) satisfies  0  0 ← − u t = Tt,0 ϕ + Tt,s f s ds + Tt,s gs d B s , (4.2) t

t

C0∞ (Rd ),

19

if and only if for any ψ ∈

20

⟨u t , ψ⟩ = ⟨ϕ, ψ⟩ +

 t

21

22

23

0

⟨u s , Lsb∗ ψ⟩ ds +

25

0

⟨ f s , ψ⟩ ds +

t

where Lsb∗ is the formally adjoint operator of Lsb . Proof. “⇒” For ε ∈ (0, 1), define  0  ε ε ε Tt,s f s ds + u t := Tt,0 ϕ + t

24



0 t

← − Tt,s gsε d B s ,

where ϕ ε := ϕ ∗ ϱε ,

f ε := f ∗ ϱε ,

g ε := g ∗ ϱε .

0

 t

← − ⟨gs , ψ⟩ d B s ,

(4.3)

Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx

15

By the property of convolutions, (2.6) and Lemma 2.3, it is easy to see that for each t < 0,

1

lim E∥u εt − u t ∥2p = 0.

2

ε→0

By Fubini’s theorem, we have for any ψ ∈ C0∞ (Rd ),  0  0  ′ ′ ⟨u εt ′ , Ltb∗ = ⟨Tt ′ ,0 ϕ ε , Ltb∗ ′ ψ⟩ dt ′ ψ⟩ dt + t

t

0 0

 +

t′

t

 = t

0

0

s

+ t

t

 t

 t

0 t

t

5

⟨Ltb′ Tt ′ ,s f sε , ψ⟩ dt ′ ds

6

0

7

(⟨Tt,s f sε , ψ⟩ − ⟨ f sε , ψ⟩) ds

8



0

t

⟨ f sε , ψ⟩ ds

9

0

 − t

← − ⟨gsε , ψ⟩ d B s .

By taking limits ε → 0, we obtain (4.3). “⇐” Choosing ψ = ϱε (x − ·) in (4.3), we then have  0  0  0 ← − u εt = ϕ ε + Lsb u εs ds + h εs ds + gsε d B s , t

4

← − (⟨Tt,s gsε , ψ⟩ − ⟨gsε , ψ⟩) d B s

⟨u εt , ψ⟩ − ⟨ϕ ε , ψ⟩ −

=

′ ⟨Tt ′ ,s f sε , Ltb∗ ′ ψ⟩ ds dt

← − ⟨Ltb′ Tt ′ ,s gsε , ψ⟩ dt ′ d B s

= ⟨Tt,0 ϕ ε , ψ⟩ − ⟨ϕ ε , ψ⟩ + 

0 s

+

(2.3)

+

t′

t

0

← − ′ ⟨Tt ′ ,s gsε , Ltb∗ ′ ψ⟩ d B s dt

⟨Ltb′ Tt ′ ,0 ϕ ε , ψ⟩ dt ′



3

0

t

10

11 12

(4.4)

13

t

where h εs

14

:=

f sε

+ (Lsb u s ) ∗ ϱε

− Lsb u εs

=

f sε

+ (bs · ∇u s ) ∗ ϱε − bs · ∇u εs .

(4.5)

Notice that h εs (x) is linear growth with respect to x. Since (2.3) holds only for bounded smooth functions, we cannot directly obtain (4.2) for u εt by using Duhamel’s formula, and we need to further cutoff it. Multiplying both sides of (4.4) by χ R , we obtain  0  0  0 ← − ε,R ε,R ε,R b ε,R gsε,R d B s , (4.6) cs ds + ut = ϕ + Ls u s ds + t

t

15

Q6

16 17 18

19

t

where

20

b ε b ε,R csε,R := h ε,R s + (Ls u s )χ R − Ls u s .

(4.7)

Now, if we define

21

22

wtε,R := Tt,0 ϕ ε,R +

 t

0

Tt,s csε,R ds +

 t

0

← − Tt,s gsε,R d B s ,

then using the same argument as in the first part, we find that  0  0  0 ← − ε,R ε,R b ε,R ε,R wt = ϕ + Ls ws ds + cs ds + gsε,R d B s , t

t

t

23

24

25

16 1

2

Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx

which together with (4.6) yields that  0 wtε,R − u ε,R = Lsb (wsε,R − u ε,R t s ) ds. t

3

4

By the maximal principal for nonlocal parabolic equations (cf. [13, Lemma 3.1]), we have  0  0 ← − ε,R ε,R ε,R ε,R u t = wt = Tt,0 ϕ + Tt,s cs ds + Tt,s gsε,R d B s . t

5

lim lim E∥u ε,R − u t ∥2p = 0, t

6

7

t

We need to take limits R → ∞ and ε → 0. By definitions, it is easy to prove that ε→0 R→∞

lim lim ∥Tt,0 ϕ ε,R − Tt,0 ϕ∥ p = 0,

ε→0 R→∞

and  2  0 ← −   ε,R Tt,s (gs − gs ) d B s  = 0, lim lim E   ε→0 R→∞  t

8

p

9

and  2  0     ε,R lim E  lim Tt,s ( f s − f s ) ds  = 0.  ε→0  R→∞ t

10

(4.8)

p

11

12

13

14

15 16

17

18

19

20

21

Moreover, notice that by (4.5) and (4.7), csε,R = f sε,R + γsε,R + (Lsb u εs )χ R − Lsb u ε,R s , where γsε := (bs · ∇u s ) ∗ ϱε − bs · ∇u εs . By Lemma 4.1 and the dominated convergence theorem, we have for each ε ∈ (0, 1) and x ∈ Rd , t < 0,    0    ε,R ε b ε b ε,R Tt,s (γs − γs + (Ls u s )χ R − Ls u s )(x) ds  = 0. (4.9) lim   R→∞  t On the other hand, by Lemma 4.2, we have for each s < 0, lim ∥γsε ∥ p = 0.

ε→0

Thus, by the dominated convergence theorem again, we have  2  0    lim E  Tt,s γsε ds  = 0.  ε→0  t p

22

23

Combining (4.8)–(4.10) we obtain  2  0     lim E  lim Tt,s (csε,R − f s ) ds  = 0.  ε→0  R→∞ t p



24

The proof is complete.

25

Proof of Theorem 1.3. It is a direct conclusion of Theorem 1.2 and Proposition 4.3.

(4.10)

Y. Xie et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx

17

Uncited references

1

Q7

[1]. Acknowledgements We gratefully thank the referee for his/her very useful suggestions. Yingchao Xie is supported by NSFs of China (No. 11271169) and Project Funded by the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions. Qi Zhang is supported by NSFs of China (No. 11101090). Xicheng Zhang is supported by NSFs of China (Nos. 11271294, 11325105). References [1] D. Applebaum, L´evy Processes and Stochastic Calculus, in: Cambridge Studies in Advance Mathematics, vol. 93, Cambridge University Press, 2004. [2] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1991. [3] M. Freidlin, Functional Intergration and Partial Differential Equations, in: Annals of Math. Studies, Princeton Univ. Press, Princeton, 1985. [4] A. Friedman, Stochastic Differential Equations and Applications. Vol. 1, Academic Press, New York, 1975. [5] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990. [6] R. Mikulevicius, H. Pragarauskas, On H¨older solutions of the integro-differential Zakai equation, Stochastic Process. Appl. 119 (2009) 3319–3355. [7] E. Pardoux, S. Peng, Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Related Fields 98 (1994) 209–227. [8] P. Protter, Stochastic Integration and Differential Equations, Springer-Verlag, Berlin, 2004. [9] B.L. Rozovskii, Stochastic Evolution Systems: Linear Theory and Applications to Non-linear Filtering, Kluwer Academic Publishers, Boston/London, 1990. [10] K. Sato, L´evy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999. [11] X. Zhang, Stochastic functional differential equations driven by L´evy processes and quasi-linear partial integrodifferential equations, Ann. Appl. Probab. 22 (2012) 2505–2538. [12] X. Zhang, Derivative formula and gradient estimate for SDEs driven by α-stable processes, Stochastic Process. Appl. 123 (2013) 1213–1228. [13] X. Zhang, L p -maximal regularity of nonlocal parabolic equations and applications, Ann. Inst. H. Poincar´e Anal. Non-Lin´erare 30 (2013) 573–614.

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