XichengZhang StochasticHomeomorphismFlowsofSDEswithSingularDriftsandSobolevDiffusionCoefficients

El e c t ro nic

Journ a l of

Pr

ob a b il i t y

Vol. 16 (2011), Paper no. 38, pages 1096–1116.

Journal URL

http://www.math.washington.edu/~ejpecp/

Stochastic Homeomorphism Flows of SDEs with Singular Drifts and Sobolev Diffusion Coefficients

Xicheng Zhang

School of Mathematics and Statistics Wuhan University, Wuhan, Hubei 430072, P.R.China

Email: XichengZhang@gmail.com

Abstract

In this paper we prove the stochastic homeomorphism flow property and the strong Feller prop- erty for stochastic differential equations with sigular time dependent drifts and Sobolev diffusion coefficients. Moreover, the local well posedness under local assumptions are also obtained. In particular, we extend Krylov and Röckner’s results in[10]to the case of non-constant diffusion coefficients.

Key words:Stochastic homoemorphism flow, Strong Feller property, Singular drift, Krylov’s es- timates, Zvonkin’s transformation.

AMS 2010 Subject Classification:Primary .

Submitted to EJP on November 20, 2010, final version accepted May 2, 2011.

1 Introduction and Main Result

Consider the following stochastic differential equation (SDE) inR^d:

dX_t=b_t(Xt)dt+σt(Xt)dW_t, (1.1) where b:R₊×R^d →R^{d and}σ:R₊×R^d →R^d×R^d are two Borel measurable functions, and {W_t}t¾0is ad-dimensional standard Brownian motion defined on some complete filtered probability space(Ω,F,P;(Ft)t¾0). When σis Lipschitz continuous in x uniformly with respect to t and b is bounded measurable, Veretennikov [14] first proved the existence of a unique strong solution for SDE (1.1). Recently, Krylov and Röckner [10] proved the existence and uniqueness of strong solutions for SDE (1.1) withσ≡Id×d and

Z T

0

‚Z

R^d

|b_t(x)|^pdx

Œ^q_p

dt<+∞, ∀T>0, (1.2) provided that

d p+2

q <1. (1.3)

More recently, following [10], Fedrizzi and Flandoli [4] proved the α-Hölder continuity of x 7→

X_t(x) for anyα ∈(0, 1) basing on Girsanov’s theorem and Khasminskii’s estimate. In the case of non-constant and non-degenerate diffusion coefficient, the present author[15]proved the pathwise uniqueness for SDE (1.1) under stronger integrability assumptions on b and σ (see also [6] for Lipschitz σ and unbounded b). Moreover, there are many works recently devoted to the study of stochastic homeomorphism (or diffeomorphism) flow property of SDE (1.1) under various non- Lipschitz assumptions on coefficients (see[3, 16, 5]and references therein).

We first introduce the class of local strong solutions for SDE (1.1). Letτbe any (Ft)-stopping time andξ anyF0-measurableR^d-valued random variable. Let S_b,σ^τ (ξ) be the class of all R^d-valued (Ft)-adapted continuous stochastic processX_t on[0,τ)satisfying

P (

ω: Z T

0

|b_s(Xs(ω))|ds+ Z T

0

|σs(Xs(ω))|²ds<+∞,∀T ∈[0,τ(ω)) )

=1,

and such that

X_t=ξ+ Z t

0

b_s(Xs)ds+ Z t

0

σs(Xs)dW_s, ∀t∈[0,τ), a.s.

We now state our main result as follows:

Theorem 1.1. In addition to (1.2) with p,q∈(1,∞)satisfying (1.3), we also assume that

(H^σ₁) σt(x) is uniformly continuous in x ∈R^d locally uniformly with respect to t ∈R₊, and there exist positive constants K andδsuch that for all(t,x)∈R₊×R^d,

δ|λ|²¶ X

ik

|σ^ik_t (x)λⁱ|²¶K|λ|², ∀λ∈R^d;

(H^σ₂) |∇σt| ∈L_{l oc}^q (R₊^;L^p(R^d))with the same p,q as required on b, where∇denotes the generalized gradient with respect to x.

Then for any (Ft)-stopping timeτ(possibly being infinity) and x ∈R^d, there exists a unique strong solution X_t(x)∈ S_b,^τ_σ(x)to SDE (1.1), which means that for any X_t(x),Y_t(x)∈ S_b,^τ_σ(x),

P{ω:X_t(ω,x) =Y_t(ω,x),∀t∈[0,τ(ω))}=1.

Moreover, for almost allωand all t¾0,

x7→X_t(ω,x)is a homeomorphism onR^d, and for any t>0and bounded measurable functionφ, x,y∈R^d,

|Eφ(X_t(x))−Eφ(X_t(y))|¶C_tkφk_∞|x−y|, where C_t>0satisfieslim_t→0C_t= +∞.

Remark 1.2. The uniqueness proven in this theorem means local uniqueness. We want to emphasize that global uniqueness can not imply local uniqueness since local solution can not in general be extended to a global solution.

By localization technique (cf.[15]), as a corollary of Theorem 1.1, we have the following existence and uniqueness of local strong solutions.

Theorem 1.3. Assume that for any n∈N^{and some p}n,q_n∈(1,∞)satisfying (1.3), (i) |b_t|,|∇σt| ∈L^q_{l oc}ⁿ(R₊^;L^pn(Bn)), where Bn:={x ∈R^{d :}|x|¶n};

(ii) σ^ik_t (x) is uniformly continuous in x ∈ B_n uniformly with respect to t ∈[0,n], and there exist positive constantsδn such that for all(t,x)∈[0,n]×B_n,

X

ik

|σ^ik_t (x)λⁱ|²¾δn|λ|², ∀λ∈R^d.

Then for any x ∈ R^d, there exist an (Ft)-stopping time ζ(x) (called explosion time) and a unique strong solution X_t(x)∈ S_b,^ζ(_σ^x)(x)to SDE (1.1) such that on{ω:ζ(ω,x)<+∞},

lim

t↑ζ(x)X_t(x) = +∞, a.s. (1.4) Proof. For each n∈ N^{, let} χn(t,x) ∈ [0, 1] be a nonnegative smooth function in R₊×R^{d with} χn(t,x) =1 for all(t,x)∈[0,n]×B_nandχn(t,x) =0 for all(t,x)∈/[0,n+1]×B_n+1. Let

bⁿ_t(x):=χn(t,x)b_t(x) and

σⁿ_t(x):=χ_n+1(t,x)σt(x) + (1−χn(t,x)) 1+ sup

(t,x)∈[0,n+2]×B_n+2

|σt(x)|

! I_d×d^.

By Theorem 1.1, for each x ∈R^d, there exists a unique strong solution Xⁿ_t(x)∈ S_b^∞n_,σⁿ(x) to SDE (1.1) with coefficients bⁿ andσⁿ. Forn¾k, define

τn,k(x,ω):=inf{t¾0 :|Xⁿ_t(ω,x)|¾k} ∧n.

It is easy to see that

X_tⁿ(x),X_t^k(x)∈ S_b^τk^n,k,σ^{(xk )}(x). By the local uniqueness proven in Theorem 1.1, we have

P{ω:Xⁿ_t(ω,x) =X^k_t(ω,x),∀t∈[0,τn,k(x,ω))}=1, which implies that forn¾k,

τk,k(x)¶τn,k(x)¶τn,n(x), a.s.

Hence, if we letζk(x):=τk,k(x), thenζk(x)is an increasing sequence of (Ft)-stopping times and forn¾k,

P{ω:Xⁿ_t(x,ω) =X^k_t(x,ω), ∀t∈[0,ζk(x,ω))}=1.

Now, for eachk∈N, we can defineX_t(x,ω) =X_t^k(x,ω)fort< ζk(x,ω)andζ(x) =lim_k→∞ζk(x). It is clear thatX_t(x)∈ S_b,^ζ(x_σ⁾(x)and (1.4) holds.

The aim of this paper is now to prove Theorem 1.1. We organize it as follows: In Section 2, we prove two new estimates of Krylov’s type, which is the key point for our proof and has some independent interest. In Section 3, we prove Theorem 1.1 in the case of b=0. For the stochastic homeomorphism flow, we adopt Kunita’s simple argument (cf.[11]). For the strong Feller property, we use Bismut-Elworthy-Li’s formula (cf. [2]). In Section 4, we use Zvonkin’s transformation to fully prove Theorem 1.1. In Appendix, we recall some well known facts used in the present paper.

2 Two estimates of Krylov’s type

We first introduce some spaces and notations. Forp,q∈[1,∞)and 0¶S<T <∞, we denote by L^qp(S,T)the space of all real Borel measurable functions on[S,T]×R^d with the norm

kfk_L^q_p(S,T):=





 Z T

S

‚Z

R^d

f(t,x)^pdx

Œ^q

p







1 q

<+∞.

Form∈N^andp¾1, letH_p^mbe the usual Sobolev space overR^d with the norm

kfkH_p^m:=

m

X

k=0

k∇^kfkL^p <+∞,

where ∇ denotes the gradient operator, and k · kL^p is the usual L^p-norm. We also introduce for 0¶S<T<∞,

H^2,q_p (S,T) =L^q(S,T;H²_p),

and the spaceHp^2,q(S,T)consisting of functionu=u(t)defined on[S,T]with values in the space of distributions onR^{d such thatu}∈H^2,qp (S,T)and∂_tu∈L^qp(S,T). For simplicity, we write

L^q_p(T) =L^q_p(0,T), H^2,q_p (T) =H^2,q_p (0,T), H_p^2,q(T) =H_p^2,q(0,T) and

L_tu(x):= ¹₂σ^ik_t (x)σ_t^jk(x)∂i∂ju(x) +bⁱ_t(x)∂iu(x). (2.1) Here and below, we use the convention that the repeated indices in a product will be summed automatically. Moreover, the letterC will denote an unimportant constant, whose dependence on the functions or parameters can be traced from the context.

We first prove the following estimate of Krylov’s type (cf. [8, p.54, Theorem 4]).

Theorem 2.1. Suppose that σ satisfies (H^σ₁) and b is bounded measurable. Fix an (Ft)-stopping timeτand anF0-measurableR^d-valued random variable ξand let X_t∈ S_b,^τ_σ(ξ). Given T0 >0and p,q∈(1,∞)with

d p+2

q <2, (2.2)

there exists a positive constant C=C(K,δ,d,p,q,T₀,kbk_∞)such that for all f ∈L^qp(T0)and0¶S<

T¶T₀,

E





 Z T∧τ

S∧τ

f(s,X_s)ds FS





¶Ckfk_L^q_p(S,T). (2.3)

Proof. Letr=d+1. SinceL^r_r(T₀)∩L^qp(T₀)is dense inL^qp(T₀), it suffices to prove (2.3) for f ∈L^r_r(T0)∩L^q_p(T0).

FixT∈[0,T₀]. By Theorem 5.2 in appendix, there exists a unique solutionu∈ H_r^2,r(T)∩ Hp^2,q(T) for the following backward PDE on[0,T]:

∂tu(t,x) +L_tu(t,x) = f(t,x), u(T,x) =0.

Moreover, for some constantC =C(K,δ,d,p,q,T₀,kbk_∞),

k∂tukL^r_r(S,T)+kuk_H^2,r_{r (S,T)}¶CkfkL^r_r(S,T), ∀S∈[0,T] (2.4) and

k∂tuk_L^q_p(S,T)+kuk_H^2,q

p (S,T)¶Ckfk_L^q_p(S,T), ∀S∈[0,T]. In particular, by (2.2) and[10, Lemma 10.2],

sup

(t,x)∈[S,T]×R^d|u(t,x)|¶Ckfk_L^q_p(S,T). (2.5)

Let ρ be a nonnegative smooth function in R^d+1 with support in {x ∈ R^{d+1 :} |x| ¶ 1} and R

R^d+1ρ(t,x)dtdx=1. Setρn(t,x):=n^d+1ρ(nt,nx)and extendu(s)toR^{by setting}u(s,·) =0 for s¾T andu(s,·) =u(0,·)fors¶0. Define

u_n(t,x):=

Z

R^d+1

u(s,y)ρ_n(t−s,x−y)dsdy (2.6) and

f_n(t,x):=∂tu_n(t,x) +L_tu_n(t,x). Then by (2.4) and the property of convolutions, we have

kf_n−fkL^r_r(T)¶k∂_t(u_n−u)k_L^r

r(T)+kbⁱ∂_i(u_n−u)k_L^r

r(T)+Kk∂_i∂_j(u_n−u)k_L^r

r(T)

¶k∂t(un−u)k_L^r_r(T)+kbk_∞k∇(un−u)k_L^r_r(T)+Kku_n−uk_H^2,r_{r (T)}

¶k∂_t(u_n−u)k_L^r_r(T)+Cku_n−uk_H^2,r_{r (T)}→0 asn→ ∞.

So, by the classical Krylov’s estimate (cf. [9, Lemma 5.1]or[6, Lemma 3.1]), we have

n→∞lim E

Z T∧τ

0

|f_n(s,X_s)−f(s,X_s)|ds

!

¶ lim

n→∞kf_n− fkL^r_r(T)=0. (2.7) Now using Itô’s formula foru_n(t,x), we have

u_n(t,X_t) =u_n(0,X₀) + Z t

0

f_n(s,X_s)ds+ Z t

0

∂iu_n(s,X_s)σ^ik_s (Xs)dW_s^k, ∀t< τ. In view of

sup

s,x |∂iu_n(s,x)|¶C_n, by Doob’s optional theorem, we have

E





 Z T∧τ

S∧τ

∂iu_n(s,X_s)σ^ik_s (Xs)dW_s^k FS





=0.

Hence,

E





 Z _T∧τ

S∧τ

f_n(s,X_s)ds FS





=E

–

(un(T∧τ,X_T∧τ)−u_n(S∧τ,X_S∧τ)) FS

™

(2.8)

¶2 sup

(t,x)∈[S,T]×R^d|u_n(t,x)|¶2 sup

(t,x)∈[S,T]×R^d|u(t,x)|^(2.5¶⁾Ckfk_L^q_p(S,T). The proof is thus completed by (2.7) and lettingn→ ∞.

Next, we want to relax the boundedness assumption on b. The price to pay is that a stronger integrability assumption is required.

Theorem 2.2. Suppose thatσsatisfies(H^σ₁)and b∈L^q(R₊^,L^p(R^d))provided with d

p+2

q <1. (2.9)

Fix an (Ft)-stopping timeτand anF0-measurableR^d-valued random variableξand let X_t∈ S_b,^τ_σ(ξ).

Given T₀ > 0, there exists a positive constant C =C(K,δ,d,p,q,T₀,kbk_L^q_p(T₀))such that for all f ∈ L^qp(T₀)and0¶S<T ¶T₀,

E





 Z T∧τ

S∧τ

f(s,X_s)ds FS





¶Ckfk_L^q_p(S,T). (2.10)

Proof. Following the proof of Theorem 2.1, we letr=d+1 and assume that f ∈L^r_r(T0)∩L^q_p(T0).

Below, forN>0, we write

L^N_t u(x):= ¹₂σ^ik_t (x)σ_t^jk(x)∂i∂ju(x) +1_{|b

t(x)|¶N}bⁱ_t(x)∂iu(x).

Fix T ∈ [0,T₀]. By Theorem 5.2, there exists a unique solution u∈ H_r^2,r(T)∩ Hp^2,q(T) for the following backward PDE on[0,T]:

∂_tu(t,x) +L^N_t u(t,x) = f(t,x), u(T,x) =0.

Moreover, for some constantC₁=C₁(K,δ,d,p,q,T₀,N),

k∂tukL^r_r(S,T)+kuk_H^2,r_{r (S,T)}¶C₁kfkL^r_r(S,T), ∀S∈[0,T], (2.11) and for some constantC₂=C₂(K,δ,d,p,q,T₀,kbk_L^q_p(T)),

k∂_tuk_L^q_p(S,T)+kuk_H^2,q

p (S,T)¶C₂kfk_L^q_p(S,T), ∀S∈[0,T]. In particular, by (2.9) and[10, Lemma 10.2],

sup

(t,x)∈[S,T]×R^d|u(t,x)|+ sup

(t,x)∈[S,T]×R^d|∇u(t,x)|¶C₂kfk_L^q_p(S,T). (2.12) ForR>0, define

τR:=inf

¨

t∈[0,τ): Z t

0

|b_s(Xs)|ds¾R

« .

Letu_n be defined by (2.6). As in the proof of Theorem 2.1 (see (2.8)), by (2.12), we have

E





 Z T∧τR

S∧τR

(∂su_n+L_su_n)(s,X_s)ds FS





¶C₂kfk_L^q_p(S,T). (2.13)

Now if we set

f_n^N(t,x):=∂tu_n(t,x) +L^N_t u_n(t,x), then

E





 Z T∧τR

S∧τR

f_n^N(s,X_s)ds FS





=E





 Z T∧τR

S∧τR

(∂su_n+L_su_n)(s,X_s)ds FS







−E





 Z T∧τR

S∧τR

1_{|b

s(X_s)|>N}b_sⁱ(Xs)∂iu_n(s,X_s)ds FS





. Hence, by (2.12) and (2.13),

E





 Z T∧τR

S∧τR

f_n^N(s,X_s)ds FS





¶Ckfk_L^q_p(S,T)+CE





 Z T∧τR

S∧τR

1_{|bs(Xs)|>N}|b_s(X_s)|ds FS





, (2.14) whereC =C(K,δ,d,p,q,T₀,kbk_L^q_p(T0))is independent ofnandR,N. Observe that for fixed N>0, by (2.11),

n→∞lim kf_n^N− fkL^r_r(T)=0, and for fixedR>0, by the dominated convergence theorem,

Nlim→∞E

Z T∧τR

S∧τR

1_{|b

s(X_s)|>N}|b_s(Xs)|ds

!

=0.

Taking limits for both sides of (2.14) in order: n→ ∞,N→ ∞andR→ ∞, we obtain (2.10).

3 SDE with Sobolev diffusion coefficient and zero drift

In this section we consider the following SDE without drift:

X_t(x) =x+ Z t

0

σs(X_s(x))dW_s. (3.1)

We first prove that:

Theorem 3.1. Under(H^σ₁) and(H^σ₂), the local pathwise uniqueness holds for SDE (3.1). More pre- cisely, for any (Ft)-stopping timeτ(possibly being infinity) and x∈R^{d, let X}t,Y_t ∈ S0,^τσ(x), then

P{ω:X_t(ω) =Y_t(ω),∀t∈[0,τ(ω))}=1.

In particular, there exists a unique strong solution for SDE (3.1).

Proof. SetZ_t:=X_t−Y_t. By Itô’s formula, we have

|Z_t∧τ|²=2 Z _t∧τ

0

〈Z_s,[σs(Xs)−σs(Ys)]dW_s〉+ Z _t∧τ

0

kσs(Xs)−σs(Ys)k²ds.

If we set

M_t:=2 Z t

0

〈Z_s,[σs(Xs)−σs(Ys)]dW_s〉

|Z_s|² and

A_t:= Z t

0

kσs(X_s)−σs(Y_s)k²

|Z_s|² ds, then

|Z_t∧τ|²= Z t∧τ

0

|Z_s|²d(Ms+A_s).

Here and below, we use the convention that ⁰₀ ≡0. Thus, if we can show that t 7→M_t∧τ+A_t∧τ is a continuous semimartingale, then the uniqueness follows. For this, it suffices to prove that for any t¾0,

E|M_t∧τ|²<+∞, EA_t∧τ<+∞. Set

σ_sⁿ(x):=σ_s∗ρ_n(x),

whereρ_nis a mollifier inR^d as used in Theorem 2.1. By Fatou’s lemma, we have E^At∧τ¶lim

"↓0

E Z _t∧τ

0

kσs(X_s)−σs(Y_s)k²

|Z_s|² ·1_|Z

s|>"ds

¶3

‚ lim

"↓0

sup

n∈NE Z _t∧τ

0

kσ_sⁿ(Xs)−σ_sⁿ(Ys)k²

|Z_s|² ·1_|Zs|>"ds +lim

"↓0 n→∞lim E

Z t∧τ

0

kσⁿ_s(X_s)−σs(X_s)k²

|Z_s|² ·1_|Z

s|>"ds +lim

"↓0 nlim→∞E

Z _t∧τ

0

kσⁿ_s(Ys)−σs(Ys)k²

|Z_s|² ·1_|Z

s|>"ds

Œ

=: 3(I₁(t) +I₂(t) +I₃(t)). By estimate (2.3), we have

I₂(t)¶lim

"↓0

1

"² lim

n→∞E Z _t∧τ

0

kσ_sⁿ(Xs)−σs(Xs)k²ds

¶lim

"↓0

1

"² lim

n→∞k|σⁿ−σ|²k_L^q/2

p/2(t)=lim

"↓0

1

"² lim

n→∞kσⁿ−σk²_L^q

p(t)=0, and also,

I₃(t) =0.

ForI₁(t), we have

I₁(t)^(5.2)¶ Csup

n∈NE Z t∧τ

0

hM |∇σ_sⁿ|(X_s) +M |∇σ_sⁿ|(Y_s)i2

ds

¶Csup

n∈Nk(M |∇σ_·ⁿ|)²k_L^q/2

p/2(t)=Csup

n∈NkM |∇σⁿ_·|k²_L^q

p(t)

(5.3)

¶ Csup

n∈N

k∇σ_·ⁿk²_L^q

p(t)¶Ck∇σ_·k²_L^q

p(t). Combining the above calculations, we obtain that for allt¾0,

E^A_t∧τ¶Ck∇σ_·k²_L^q

p(t). (3.2)

Similarly, we can prove that E|M_t∧τ|²=4E

Z _t∧τ

0

|[σs(X_s)−σs(Y_s)]^∗Z_s|²

|Z_s|⁴ ds¶Ck∇σ_·k²_L^q

p(t),

where the star denotes the transpose of a matrix. The existence of a unique strong solution now follows from the classical Yamada-Watanabe theorem (cf.[7]).

Below, we prove better regularities of solutions with respect to the initial values.

Lemma 3.2. Under (H^σ₁) and (H^σ₂), let X_t(x) be the unique strong solution of SDE (3.1). For any T>0,γ∈R^{and all x}6= y ∈R^{d, we have}

sup

t∈[0,T]E^€|X_t(x)−X_t(y)|^2γŠ

¶C|x−y|^2γ, where C=C(K,δ,p,q,d,γ,T).

Proof. Forx 6= y and"∈(0,|x−y|), define

τ_":=inf{t¾0 :|X_t(x)−X_t(y)|¶"}. SetZ_t^":=X_t∧τ"(x)−X_t∧τ"(y). For anyγ∈R, by Itô’s formula, we have

|Z_t^"|^2γ=|x−y|^2γ+2γ

Z t∧τ_"

0

|Z_s^"|^2(γ−1)〈Z_s^",[σs(X_s(x))−σs(X_s(y))]dW_s〉 +2γ

Z _t∧τ"

0

|Z_s^"|^2(γ−1)kσs(Xs(x))−σs(Xs(y))k²ds +2γ(γ−1)

Z _t∧τ"

0

|Z_s^"|^2(γ−2)|[σs(X_s(x))−σs(X_s(y))]^∗Z_s^"|²ds

=:|x−y|^2γ+

Z t∧τ"

0

|Z_s^"|^2γ

α(s)dW_s+β(s)ds , where

α(s):= 2γ[σs(Xs(x))−σs(Ys(y))]^∗Z_s^"

|Z_s^"|² and

β(s):=2γkσs(X_s(x))−σs(Y_s(y))k²

|Z_s^"|² +2γ(γ−1)|[σs(Xs(x))−σs(Ys(y))]^∗Z_s^"|²

|Z_s^"|⁴ .

By the Doléans-Dade’s exponential (cf.[13]), we have

|Z_t^"|^2γ=|x−y|^2γexp

¨Z _t∧τ"

0

α(s)dW_t−1 2

Z _t∧τ"

0

|α(s)|²ds+

Z _t∧τ"

0

β(s)ds

« . FixT>0 below. Using (2.3) and as in the proof of (3.2), we have for any 0¶s<t¶T,

E





 Z t

s

|β(r∧τ_")|dr Fs





¶Ck∇σk²_L^q

p(s,t), whereC =C(K,δ,p,q,d,γ,T). Thus, by Lemma 5.3, we get for anyλ >0,

E^exp λ

Z T∧τ_"

0

|β(s)|ds

!

¶E^exp λ Z T

0

|β(s∧τ_")|ds

!

<+∞. Similarly, we have

E^exp λ

Z T∧τ"

0

|α(s)|²ds

!

<+∞, ∀λ >0.

In particular, by Novikov’s criterion, t7→exp

¨ 2

Z _t∧τ"

0

α(s)dW_s−2

Z _t∧τ"

0

|α(s)|²ds

«

=:M_t^"

is a continuous exponential martingale. Hence, by Hölder’s inequality, we have

E|Z_t^"|^2γ¶|x−y|^2γ(E^M"_t)¹²

‚ E^exp

¨Z _t∧τ"

0

|α(s)|²ds+2

Z _t∧τ"

0

β(s)ds

«Œ¹₂

¶C|x− y|^2γ, whereC is independent of"and x,y.

Noting that

lim"↓0τ_"=τ:=inf{t¾0 :X_t(x) =X_t(y)}, by Fatou’s lemma, we obtain

E|X_t∧τ(x)−X_t∧τ(y)|^2γ=lim

"→0

E|Z_t^"|^2γ¶C|x−y|^2γ. Lettingγ=−1 yields that

τ¾t, a.s.

The proof is thus complete.

Sinceσis bounded, the following lemma is standard, and we omit the details.

Lemma 3.3. Under(H^σ₁), let X_t(x)solve SDE (3.1). For any T >0,γ∈Rand all x∈R^d, we have E

‚ sup

t∈[0,T](1+|X_t(x)|²)^γ

Œ

¶C₁(1+|x|²)^γ, where C₁=C₁(K,γ,T), and for anyγ¾1and t,s¾0,

sup

x∈R^d

E|X_t(x)−X_s(x)|^2γ¶C₂|t−s|^γ, where C₂=C₂(K,γ).

Basing on Lemmas 3.2 and 3.3, it is by now standard to prove the following theorem (cf. [11, Theorem 4.5.1]). For the reader’s convenience, we sketch the proof here.

Theorem 3.4. Under(H^σ₁)and(H^σ₂), let X_t(x)∈ S0,^∞σ(x)be the unique strong solution of SDE (3.1), then for almost allωand all t∈R₊^{, x} 7→X_t(ω,x)is a homeomorphism onR^d.

Proof. Forx 6= y∈R^{d, define}

Rt(x,y):=|X_t(x)−X_t(y)|⁻¹.

For anyx,y,x⁰,y⁰∈R^{d withx} 6= y, x⁰6=y⁰ands6=t, it is easy to see that

|Rt(x,y)− Rs(x⁰,y⁰)|¶Rt(x,y)· Rs(x⁰,y⁰)·[|X_t(x)−X_s(x⁰)|+|X_t(y)−X_s(y⁰)|]. By Lemmas 3.2 and 3.3, for anyγ¾1 ands,t∈[0,T], we have

E|Rt(x,y)− Rs(x⁰,y⁰)|^γ¶C|x−y|^−γ|x⁰−y⁰|^−γ(|t−s|^γ/2+|x−x⁰|^γ+|y−y⁰|^γ).

Choosingγ > 4(d+1), by Kolmogorov’s continuity criterion, there exists a continuous version to the mapping(t,x,y)7→ Rt(x,y) on{(t,x,y)∈R₊×R^d×R^{d : x} 6= y}. In particular, this proves that for almost allω, the mappingx 7→X_t(ω,x)is one-to-one for allt ¾0.

As for the onto property, let us define Jt(x) =

¨ (1+|X_t(x|x|⁻²)|)⁻¹, x 6=0,

0, x =0.

As above, using Lemmas 3.2 and 3.3, one can show that(t,x)7→ Jt(x)admits a continuous version.

Thus,(t,x)7→X_t(ω,x)can be extended to a continuous map from R₊×Rˆ^{d to}R^{ˆd, where} R^ˆd = R^d∪ {∞}is the one-point compactification ofR^{d. Hence, X}t(ω,·):Rˆ^d →Rˆ^d is homotopic to the identity mapping X₀(·) so that it is an onto map by the well known fact in homotopic theory. In particular, for almost allω, x 7→ X_t(ω,x) is a homeomorphism onRˆ^{d for all t} ¾ 0. Clearly, the restriction ofX_t(ω,·)toR^d is still a homeomorphism sinceX_t(ω,∞) =∞.

Now we turn to the proof of the strong Feller property.

Theorem 3.5. Under(H^σ₁)and(H^σ₂), let X_t(x)∈ S_0,^∞_σ(x)be the unique strong solution of SDE (3.1), then for any bounded measurable functionφ, T >0and x,y ∈R^d,

|E(φ(Xt(x)))−E(φ(Xt(y)))|¶ C_T

ptkφk_∞|x−y|, ∀t∈(0,T]. (3.3)

Proof. Defineσⁿ_t(x):=σt∗ρn(x), whereρn is a mollifier inR^{d. By(Hσ}₁), it is easy to see that for all(t,x)∈R₊×R^d,

δ|λ|²¶ X

ik

|[σⁿ_t(x)]^ikλⁱ|²¶K|λ|², ∀λ∈R^{d. (3.4)} Let Xⁿ_t(x) ∈ S_0,σ^∞n(x) be the unique strong solution of SDE (3.1) corresponding to σⁿ. By the monotone class theorem, it suffices to prove (3.3) for any bounded Lipschitz continuous functionφ. First of all, by Bismut-Elworthy-Li’s formula (cf.[2]), for anyh∈R^{d, we have}

∇hEφ(Xⁿ_t(x)) =1 tE

–

φ(Xⁿ_t(x)) Z t

0

[σⁿ_s(X_sⁿ(x))]⁻¹∇hX_sⁿ(x)dW_s

™

, (3.5)

where for a smooth function f, we denote∇hf :=〈∇f,h〉. Noting that

∇hX_tⁿ(x) =h+ Z t

0

∇σⁿ_s(X_sⁿ(x))· ∇hX_sⁿ(x)dW_s, by Itô’s formula, we have

|∇hX_tⁿ(x)|²=|h|²+2 Z t

0

〈∇hX_sⁿ(x),∇σ_sⁿ(X_sⁿ(x))· ∇hX_sⁿ(x)dW_s〉 +

Z t

0

k∇σⁿ_s(X_sⁿ(x))· ∇hX_sⁿ(x)k²ds

=:|h|²+ Z t

0

|∇hX_sⁿ(x)|²

α_hⁿ(s)dW_s+β_hⁿ(s)ds , where

α_hⁿ(s):= (∇hX_sⁿ(x))^∗· ∇σⁿ_s(X_sⁿ(x))· ∇hX_sⁿ(x)

|∇hX_sⁿ(x)|² and

β_hⁿ(s):= k∇σⁿ_s(X_sⁿ(x))· ∇hX_sⁿ(x)k²

|∇hX_sⁿ(x)|² . By the Doléans-Dade’s exponential again, we have

|∇hXⁿ_t(x)|²=|h|²exp

¨Z t

0

αⁿ_h(s)dW_s− 1 2

Z t

0

|αⁿ_h(s)|²ds+ Z t

0

β_hⁿ(s)ds

« . FixT>0. By (2.3), we have for any 0¶s<t¶T,

E





 Z t

s

|β_hⁿ(r)|dr Fs





¶Ck∇σⁿk²_L^q

p(s,t)¶Ck∇σk²_L^q

p(s,t),

where C = C(K,δ,p,q,d,T) is independent of n,x and h. Thus, by Lemma 5.3, we get for any λ >0,

sup

n

sup

h∈R^d

E^exp λ Z T

0

|β_hⁿ(s)|ds

!

<+∞.