Stochastic differential equations with coefficients in Sobolev spaces

31 Pages·0.26 MB·English

by  Shizan Fang

Stochastic differential equations with
coefficients in Sobolev spaces
Shizan Fangc∗, Dejun Luoa,b, Anton Thalmaiera
0 aUR Math´ematiques, Universit´e duLuxembourg, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg
1
bKey Laboratory of Random Complex Structuresand Data Science, Academy of Mathematics and
0
2 Systems Science, Chinese Academy of Sciences, Beijing 100190, China
n cI.M.B, BP 47870, Universit´edeBourgogne, Dijon, France
a
J

8
1
Abstract

] We consider Itoˆ SDE dX = m A (X )dwj +A (X )dt on Rd. The diffusion coef-
R t j=1 j t t 0 t
P ficients A1,··· ,Am are supposedPto be in the Sobolev space Wl1o,cp(Rd) with p > d, and to
h. have linear growth; for the drift coefficient A0, we consider two cases: (i) A0 is continuous
whose distributional divergenceδ(A ) w.r.t. the Gaussianmeasure γ exists, (ii) A has the
t 0 d 0
a Sobolev regularity W1,p′ for some p′ > 1. Assume exp λ δ(A ) + m (δ(A )2 +
m loc Rd 0 | 0 | j=1 | j |
A 2) dγ <+ for some λ >0, in the case (i), if the pathwise uniqueness of solutions
[ |∇ j| d ∞ 0 R (cid:2) (cid:0) P
holds, th(cid:1)e(cid:3)n the push-forward (Xt)#γd admits a density with respect to γd. In particular, if
1 thecoefficientsareboundedLipschitzcontinuous,thenX leavestheLebesguemeasureLeb
t d
v quasi-invariant. In the case (ii), we develop a method used by G. Crippa and C. De Lellis
7
for ODE and implemented by X. Zhang for SDE, to establish the existence and uniqueness
0
of stochastic flow of maps.
0
3
. MSC 2000: primary 60H10, 34F05; secondary 60J60, 37C10, 37H10.
1
Key words: Stochastic flows, Sobolev space coefficients, density, density estimate, pathwise
0
0 uniqueness, Gaussian measure, Ornstein-Uhlenbeck semigroup.
1
:
v
1 Introduction
i
X
r Let A ,A ,...,A : Rd Rd be continuous vector fields on Rd. We consider the following Itˆo
a 0 1 m
→
stochastic differential equation on Rd (abbreviated as SDE)
m
j
dX = A (X )dw +A (X )dt, X = x, (1.1)
t j t t 0 t 0
j=1
X
where w = (w1,...,wm) is the standard Brownian motion on Rm. It is a classical fact in the
t t t
theory of SDE (see [16, 17, 21, 30]) that, if the coefficients A are globally Lipschitz continuous,
j
then SDE (1.1) has a unique strong solution which defines a stochastic flow of homeomorphisms
on Rd; however contrary to ordinary differential equations (abbreviated as ODE), the regularity
of the homeomorphisms is only H¨older continuity of order 0 < α < 1. Thus it is not clear
whether the Lebesgue measure Leb on Rd admits a density under the flow X . In the case
d t
where the vector fields A ,j = 0,1,...,m, are in C∞(Rd,Rd), the SDE (1.1) defines a flow of
j b
diffeomorphisms, and Kunita [21] showed that the measures on Rd which have a strictly positive
∗[email protected]
1
smooth density with respect to Leb are quasi-invariant underthe flow. This resultwas recently
d
generalized in [27] to the case where the drift A is allowed to be only log-Lipschitz continuous.
0
Studies on SDE beyond the Lipschitz setting attracted great interest during the last years, see
for instance [10, 11, 13, 19, 20, 23, 24, 29, 34, 35].
In the context of ODE, existence of a flow of quasi-invariant measurable maps associated to
a vector field A belonging to Sobolev spaces appeared first in [6]. In the seminar paper [7],
0
Di Perna and Lions developed transport equations to solve ODE without involving exponential
integrability of A . On the other hand, L. Ambrosio [1] took advantage of using continuity
0
|∇ |
equations which allowed him to construct quasi-invariant flows associated to vector fields A
0
with only BV regularity. In the framework for Gaussian measures, the Di Perna-Lions method
was developed in [4], also in [2, 12] on the Wiener space.
Thesituation forSDE isquitedifferent: even forthevector fieldsA ,A ,...,A in C∞ with
0 1 m
linear growth, if no conditions were imposed on the growth of the derivatives, the SDE (1.1)
could not define a flow of diffeomorphisms (see [25, 26]). More precisely, let τ be the life time
x
of the solution to (1.1) starting from x. The SDE (1.1) is said to be complete if for each x Rd,
∈
P(τ = + ) =1; it is said to be strongly complete if P(τ = + , x Rd) = 1. The goal in [26]
x x
∞ ∞ ∈
is to construct examples for which the coefficients are smooth, but the SDE (1.1) is not strongly
complete (see [11, 25] for positive examples). Now consider
Σ = (w,x) Ω Rd; τ (w) = + .
x
{ ∈ × ∞}
Suppose that the SDE (1.1) is complete, then for any probability measure µ on Rd,
1 (w,x)dP(w) dµ(x) = 1.
Σ
ZRd(cid:18)ZΩ (cid:19)
By Fubini’s theorem, 1 (w,x)dµ(x) dP(w) = 1. It follows that there exists a full
Ω Rd Σ
measure subset Ω Ω such that for all w Ω , τ (w) = + holds for µ-almost every x Rd.
0 0 x
⊂ R (cid:0)R ∈(cid:1) ∞ ∈
Now under the existence of a complete unique strong solution to SDE (1.1), we have a flow of
measurable maps x X (w,x).
t
→
Recently, inspiredbyapreviousworkduetoAmbrosio,LecumberryandManiglia[3],Crippa
andDeLellis[5]obtainedsomenewtypeofestimates ofperturbationforODEwhosecoefficients
have Sobolev regularity. More precisely, the absence of Lipschitz condition was filled by the
following inequality: for f W1,1(Rd),
∈ loc
f(x) f(y) C x y M f (x)+M f (y)
d R R
| − | ≤ | − | |∇ | |∇ |
holds for x,y Nc and x y R, where N i(cid:0)s a negligible set of Rd and(cid:1)M g is the maximal
R
∈ | − | ≤
function defined by
1
M g(x) = sup g(y) dy,
R
Leb (B(x,r)) | |
0<r≤R d ZB(x,r)
here B(x,r) = y Rd; y x r ; the classical moment estimate was replaced by estimating
{ ∈ | − | ≤ }
the quantity
X (x) X˜ (x)
t t
log | − | +1 dx,
σ
ZB(0,r) (cid:18) (cid:19)
where σ > 0 is a small parameter. This method has recently been successfully implemented to
SDE by X. Zhang in [36].
Theaiminthispaperistwo-fold: firstweshallstudyabsolutecontinuity ofthepush-forward
measure (X ) Leb with respect to Leb , once the SDE (1.1) has a unique strong solution;
t # d d
2
secondly we shall construct strong solutions (for almost all initial values) using the approach
mentioned above for SDE with coefficients in Sobolev space. The key point is to obtain a priori
Lp estimate for the density. To this end, we shall work with the standard Gaussian measure γ ;
d
this will be done in Section 2. The main result in Section 3 is the following
Theorem 1.1. Let A ,A ,...,A be continuous vector fields on Rd of linear growth. Assume
0 1 m
that the diffusion coefficients A ,...,A are in the Sobolev space Dq(γ ) and that δ(A )
1 m ∩q>1 1 d 0
exists; furthermore there exists a constant λ > 0 such that
0
m
exp λ δ(A ) + δ(A )2 + A 2 dγ < + . (1.2)
0 0 j j d
Rd | | | | |∇ | ∞
Z (cid:20) (cid:18) j=1 (cid:19)(cid:21)
X(cid:0) (cid:1)
Suppose that pathwise uniqueness holds for SDE (1.1). Then (X ) γ is absolutely continuous
t # d
with respect to γ and the density is in the space L1logL1.
d
A consequence of this theorem concerns the following classical situation.
Theorem 1.2. Let A ,A ,...,A be globally Lipschitz continuous. Suppose that there exists a
0 1 m
constant C > 0 such that
m
x,A (x) 2 C(1+ x 2) for all x Rd. (1.3)
j
h i ≤ | | ∈
j=1
X
Then the stochastic flow of homeomorphisms X generated by SDE (1.1) leaves the Lebesgue
t
measure Leb quasi-invariant.
d
Remark that the condition (1.3) not only includes the case of bounded Lipschitz diffusion
coefficients, but also, maybe more significant, indicates the role of dispersion: the vector fields
A , ,A should not go radically into infinity. The purpose of Section 4 is to find conditions
1 m
···
that guarantee strict positivity of the density, in the case where the existence of the inverse flow
is not known, see Theorem 4.4.
The main result in Section 5 is
Theorem 1.3. Assume that the diffusion coefficients A , A belong to the Sobolev space
1 m
Dq(γ ) and the drift A Dq(γ ) for some q > 1. Ass·u·m· e (1.2) and that the coefficients
∩q>1 1 d 0 ∈ 1 d
A ,A , ,A are of linear growth, then there is a unique stochastic flow of measurable maps
0 1 m
···
X : [0,T] Ω Rd Rd, which solves (1.1) for almost all initial x Rd and the push-forward
× × → ∈
(X (w, )) γ admits a density with respect to γ , which is in L1logL1.
t # d d
·
Whenthediffusioncoefficients satisfy theuniformellipticity, aclassical resultduetoStroock
and Varadhan [32] says that if the diffusion coefficients A , ,A are boundedcontinuous and
1 m
···
thedriftA isboundedBorelmeasurable,thentheweakuniquenessholds,thatistheuniqueness
0
inlaw of thediffusion. Thisresultwas strengthened byVeretennikov [33], sayingthat infact the
pathwise uniqueness holds. When A is not bounded, some conditions on diffusion coefficients
0
wereneeded. Inthecase wherethediffusion matrix a = (a )is theidentity, thedriftA in (1.1)
ij 0
can be quite singular: A Lp (Rd) with p > d+2 implies that the SDE (1.1) has the pathwise
0 ∈ loc
uniqueness (see Krylov-R¨ockner [20] for a more complete study); if the diffusion coefficients
1,2(d+1)
A , ,A are bounded continuous, under a Sobolev condition, namely, A W for
1 ··· m j ∈ loc
j = 1, ,m and A L2(d+1)(Rd), X. Zhang proved in [34] that the SDE (1.1) admits a
··· 0 ∈ loc
unique strong solution. Note that even in this uniformly non-degenerated case, if the diffusion
coefficients lose the continuity, there are counterexamples for which the weak uniqueness does
not hold, see [19, 31].
3
Finally we would like to mention that under weaker Sobolev type conditions, the connection
betweenweaksolutionsandFokker-Planck equationswasinvestigatedin[14,22],somenotionsof
“generalized solutions”, as well as the phenomenaof coalescence and splitting, were investigated
in [23, 24]. Stochastic transport equations were studied in [15, 36].
2 Lp estimate of the density
The purpose of this section is to derive a priori estimates for the density; we assume that
the coefficients A ,A ,...,A of SDE (1.1) are smooth with compact support in Rd. Then the
0 1 m
solution X , i.e., x X (x), is a stochastic flow of diffeomorphisms on Rd. Moreover SDE (1.1)
t t
7→
is equivalent to the following Stratonovich SDE
m
dX = A (X ) dwj +A˜ (X )dt, X = x, (2.1)
t j t ◦ t 0 t 0
j=1
X
where A˜ = A 1 m A and denotes the Lie derivative with respect to A.
0 0− 2 j=1LAj j LA
Let γ be the standard Gaussian measure on Rd, and γ = (X ) γ , γ˜ = (X−1) γ the
d P t t # d t t # d
push-forwardsof γ respectively by the flow X and its inverse flow X−1. To fix ideas, wedenote
d t t
by (Ω,F,P) the probability space on which the Brownian motion w is defined. Let K = dγt
t t dγd
and K˜ = dγ˜t be the densities with respect to γ . By Lemma 4.3.1 in [21], the Radon-Nikodym
t dγd d
derivative K˜ has the following explicit expression
t
m t t
K˜ (x) = exp δ(A )(X (x)) dwj δ(A˜ )(X (x))ds , (2.2)
t − j s ◦ s − 0 s
(cid:18) j=1Z0 Z0 (cid:19)
X
where δ(A ) denotes the divergence of A with respect to the Gaussian measure γ :
j j d
ϕ,A dγ = ϕδ(A )dγ , ϕ C1(Rd).
Rdh∇ ji d Rd j d ∈ c
Z Z
It is easy to see that K and K˜ are related to each other by the equality below:
t t
K (x) = K˜ X−1(x) −1. (2.3)
t t t
In fact, for any ψ C∞(Rd), we have (cid:2) (cid:0) (cid:1)(cid:3)
∈ c
ψ(x)dγ (x)= ψ X X−1(x) dγ (x)
d t t d
Rd Rd
Z Z
= ψ[(cid:2)X ((cid:0)y)]K˜ (y)(cid:1)d(cid:3)γ (y)= ψ(x)K˜ X−1(x) K (x)dγ (x),
t t d t t t d
Rd Rd
Z Z
(cid:0) (cid:1)
which leads to (2.3) due to the arbitrariness of ψ C∞(Rd). In the following we shall estimate
∈ c
the Lp(P γ ) norm of K .
d t
×
We rewrite the density (2.2) with the Itˆo integral:
m t t 1 m
K˜ (x) = exp δ(A )(X (x))dwj δ(A )+δ(A˜ ) (X (x))ds . (2.4)
t − j s s − 2 LAj j 0 s
(cid:18) j=1Z0 Z0 (cid:20) j=1 (cid:21) (cid:19)
X X
4
Lemma 2.1. We have
m m m
1 1 1
δ(A )+δ(A˜ )= δ(A )+ A 2+ A ,( A )∗ , (2.5)
2 LAj j 0 0 2 | j| 2 h∇ j ∇ j i
j=1 j=1 j=1
X X X
where , denotes the inner product of Rd Rd and ( A )∗ the transpose of A .
j j
h· ·i ⊗ ∇ ∇
Proof. Let A be a C2 vector field on Rd. From the expression
d ∂Ak
δ(A) = x Ak ,
k
− ∂x
k=1(cid:18) k(cid:19)
X
we get
d ∂Ak ∂2Ak
δ(A) = AℓAkδ +Aℓx Aℓ . (2.6)
A kℓ k
L ∂x − ∂x ∂x
ℓ,k=1(cid:18) ℓ ℓ k(cid:19)
X
Note that
∂ ∂Ak ∂Ak ∂Aℓ ∂2Ak
Aℓ = +Aℓ .
∂x ∂x ∂x ∂x ∂x ∂x
k(cid:18) ℓ (cid:19) ℓ k k ℓ
Thus, by means of (2.6), we obtain
δ(A) = A2+δ( A)+ A,( A)∗ . (2.7)
A A
L | | L h∇ ∇ i
Recall that δ(A˜ ) = δ(A ) 1 m δ( A ). Hence, replacing A by A in (2.7) and summing
0 0 − 2 j=1 LAj j j
over j, gives formula (2.5). (cid:3)
P
We can now prove the following key estimate.
Theorem 2.2. For p > 1,
m p−1
p(2p−1)
kKtkLp(P×γd) ≤ Rdexp pt 2|δ(A0)|+ |Aj|2+|∇Aj|2+2(p−1)|δ(Aj)|2 dγd .
(cid:20)Z (cid:18) h Xj=1(cid:0) (cid:1)i(cid:19) (cid:21)
(2.8)
Proof. Using relation (2.3), we have
E[Kp(x)]dγ (x) = E K˜ X−1(x) −pdγ (x)
t d t t d
Rd Rd
Z Z
(cid:2) (cid:0) (cid:1)(cid:3)
= E K˜ (y) −pK˜ (y)dγ (y)
t t d
Rd
Z
(cid:2) (cid:3)
= E K˜ (x) −p+1 dγ (x). (2.9)
t d
Rd
Z
(cid:2)(cid:0) (cid:1) (cid:3)
To simplify the notation, denote the right hand side of (2.5) by Φ. Then K˜ (x) rewrites as
t
m t t
K˜ (x) = exp δ(A )(X (x))dwj Φ(X (x))ds .
t − j s s − s
(cid:18) j=1Z0 Z0 (cid:19)
X
5
Fixing an arbitrary r > 0, we get
m t t
K˜ (x) −r = exp r δ(A )(X (x))dwj +r Φ(X (x))ds
t j s s s
(cid:18) j=1Z0 Z0 (cid:19)
(cid:0) (cid:1) X
m t m t
= exp r δ(A )(X (x))dwj r2 δ(A )(X (x)) 2ds
j s s − j s
(cid:18) j=1Z0 j=1Z0 (cid:19)
X X (cid:12) (cid:12)
t m (cid:12) (cid:12)
exp r2 δ(A )2+rΦ (X (x))ds .
j s
× | |
(cid:18)Z0 (cid:16) Xj=1 (cid:17) (cid:19)
By Cauchy-Schwarz’s inequality,
m t m t 1/2
E K˜ (x) −r Eexp 2r δ(A )(X (x))dwj 2r2 δ(A )(X (x)) 2ds
t ≤ j s s − j s
(cid:20) (cid:18) j=1Z0 j=1Z0 (cid:19)(cid:21)
(cid:2)(cid:0) (cid:1) (cid:3) X X (cid:12) (cid:12)
t m (cid:12) 1/2 (cid:12)
Eexp 2r2 δ(A )2 +2rΦ (X (x))ds
j s
× | |
(cid:20) (cid:18)Z0 (cid:16) Xj=1 (cid:17) (cid:19)(cid:21)
t m 1/2
= Eexp 2r2 δ(A )2+2rΦ (X (x))ds , (2.10)
j s
| |
(cid:20) (cid:18)Z0 (cid:16) Xj=1 (cid:17) (cid:19)(cid:21)
since the first term on the right hand side of the inequality in (2.10) is the expectation of a
martingale. Let
m
Φ˜ =2r δ(A ) +r A 2+ A 2+2r δ(A )2 .
r 0 j j j
| | | | |∇ | | |
j=1
X(cid:0) (cid:1)
Then by (2.10), along with the definition of Φ and Cauchy-Schwarz’s inequality, we obtain
t 1/2
E K˜ (x) −r dγ Eexp Φ˜ (X (x))ds dγ . (2.11)
t d r s d
ZRd ≤ (cid:20)ZRd (cid:18)Z0 (cid:19) (cid:21)
(cid:2)(cid:0) (cid:1) (cid:3)
Following the idea of A.B. Cruzeiro ([6] Corollary 2.2, see also Theorem 7.3 in [8]) and by
Jensen’s inequality,
t t ds 1 t
exp Φ˜ (X (x))ds = exp tΦ˜ (X (x)) etΦ˜r(Xs(x))ds.
r s r s
t ≤ t
(cid:18)Z0 (cid:19) (cid:18)Z0 (cid:19) Z0
Define I(t) = sup E[Kp(x)]dγ . Integrating on both sides of the above inequality and
0≤s≤t Rd t d
by H¨older’s inequality,
R
t 1 t
Eexp Φ˜ (X (x))ds dγ (x) E etΦ˜r(Xs(x))dγ (x)ds
r s d d
ZRd (cid:18)Z0 (cid:19) ≤ t Z0 ZRd
1 t
= E etΦ˜r(y)K (y)dγ (y)ds
s d
t Z0 ZRd
1 t
≤ t Z0 etΦ˜r Lq(γd)kKskLp(P×γd)ds
etΦ˜r(cid:13)(cid:13) (cid:13)(cid:13)I(t)1/p,
≤ Lq(γd)
(cid:13) (cid:13)
(cid:13) (cid:13)
6
where q is the conjugate number of p. Thus it follows from (2.11) that
E K˜ (x) −r dγ (x) etΦ˜r 1/2 I(t)1/2p. (2.12)
Rd t d ≤ Lq(γd)
Z
(cid:2)(cid:0) (cid:1) (cid:3) (cid:13) (cid:13)
(cid:13) (cid:13)
Taking r = p 1 in the above estimate and by (2.9), we obtain
−
E[Kp(x)]dγ (x) etΦ˜p−1 1/2 I(t)1/2p.
Rd t d ≤ Lq(γd)
Z
(cid:13) (cid:13)
(cid:13) (cid:13)
Thus we have I(t) etΦ˜p−1 1/2 I(t)1/2p. Solving this inequality for I(t) gives
≤ Lq(γd)
(cid:13) (cid:13)
(cid:13) (cid:13) p−1
E[Kp(x)]dγ (x) I(t) exp pt Φ˜ (x) dγ (x) 2p−1.
Rd t d ≤ ≤ Rd p 1 p−1 d
Z (cid:20)Z (cid:18) − (cid:19) (cid:21)
Now the desired estimate follows from the definition of Φ˜ . (cid:3)
p−1
Corollary 2.3. For any p > 1,
1
m 2p+1
kK˜tkLp(P×γd) ≤  Rdexp(p+1)t 2|δ(A0)|+ |Aj|2+|∇Aj|2+2p|δ(Aj)|2 dγd .
Z (cid:20) j=1 (cid:21)
X(cid:0) (cid:1)
   (2.13)
Proof. Similar to (2.12), we have for r > 0,
E K˜ (x) r dγ (x) etΦ˜r 1/2 I(t)1/2p, (2.14)
Rd t d ≤ Lq(γd)
Z
(cid:2)(cid:0) (cid:1) (cid:3) (cid:13) (cid:13)
(cid:13) (cid:13)
where Φ˜ and I(t) are defined as above. Since I(t) etΦ˜p−1 p/(2p−1), by taking r = p 1, we
r ≤ Lq(γd) −
get
(cid:13) (cid:13)
(cid:13) (cid:13)
E K˜ (x) p−1 dγ (x) etΦ˜p−1 p/(2p−1)
Rd t d ≤ Lq(γd)
Z (cid:2)(cid:0) (cid:1) (cid:3) (cid:13) m (cid:13) p−1
= exp pt 2δ(A ) (cid:13)+ (cid:13)A 2+ A 2+2(p 1)δ(A )2 dγ 2p−1.
0 j j j d
Rd | | | | |∇ | − | |
(cid:20)Z (cid:18) h Xj=1(cid:0) (cid:1)i(cid:19) (cid:21)
Replacing p by p+1 in the last inequality gives the claimed estimate. (cid:3)
3 Absolute continuity under flows generated by SDEs
Now assume that the coefficients A in SDE (1.1) are continuous and of linear growth. Then
j
it is well known that SDE (1.1) has a weak solution of infinite life time. In order to apply the
resultsoftheprecedingsection,weshallregularizethevectorfieldsusingtheOrnstein-Uhlenbeck
semigroup P on Rd:
ε ε>0
{ }
P A(x) = A e−εx+ 1 e−2εy dγ (y).
ε d
Rd −
Z
(cid:0) p (cid:1)
We have the following simple properties.
Lemma 3.1. Assume that A is continuous and A(x) C(1+ x q) for some q 0. Then
| | ≤ | | ≥
7
(i) there is C > 0 independent of ε, such that
q
P A(x) C (1+ x q), for all x Rd;
ε q
| | ≤ | | ∈
(ii) P A converges uniformly to A on any compact subset as ε 0.
ε
→
Proof. (i) Note that e−εx+√1 e−2εy x + y and that there exists a constant C > 0
− ≤ | | | |
such that (x + y )q C(x q + y q). Using the growth condition on A, we have for some
| | | | (cid:12)≤ | | | | (cid:12)
constant C > 0 (depen(cid:12)ding on q), (cid:12)
P A(x) A e−εx+ 1 e−2εy dγ (y)
ε d
| | ≤ Rd −
Z
(cid:12) (cid:0) p (cid:1)(cid:12)
C (cid:12) 1+ x q + y q dγ (y)(cid:12) C 1+ x q +M
d q
≤ Rd | | | | ≤ | |
Z
(cid:0) (cid:1) (cid:0) (cid:1)
where M = y qdγ (y). Changing the constant yields (i).
q Rd| | d
(ii) Fix R > 0 and x in the closed ball B(R) of radius R, centered at 0. Let R > R be
1
R
arbitrary. We have
P A(x) A(x) A e−εx+ 1 e−2εy A(x) dγ (y)
ε d
| − | ≤ Rd − −
Z
(cid:12) (cid:0) p (cid:1) (cid:12)
= (cid:12) + A e−εx+ 1 (cid:12) e−2εy A(x) dγ (y)
d
− −
(cid:18)ZB(R1) ZB(R1)c(cid:19)
(cid:12) (cid:0) p (cid:1) (cid:12)
=:I1+I2. (cid:12) (cid:12) (3.1)
By the growth condition on A, for some constant C > 0, independent of ε, we have
q
I A e−εx+ 1 e−2εy + A(x) dγ (y)
2 d
≤ − | |
ZB(R1)c(cid:16)(cid:12) (cid:0) p (cid:1)(cid:12) (cid:17)
C (cid:12) 1+Rq + y q dγ (y),(cid:12)
q d
≤ | |
ZB(R1)c
(cid:0) (cid:1)
where the last term tends to 0 as R + . For given η > 0, we may take R large enough
1 1
→ ∞
such that I < η. Then there exists ε > 0 such that for ε< ε and y R ,
2 R1 R1 | |≤ 1
e−εx+ 1 e−2εy e−εR+ 1 e−2εR R .
1 1
− ≤ − ≤
Note that (cid:12) p (cid:12) p
(cid:12) (cid:12)
e−εx+ 1 e−2εy x εR+√2εR , for x R, y R .
1 1
− − ≤ | | ≤ | | ≤
Since A is unifo(cid:12)(cid:12)rmly copntinuous on B(R(cid:12)(cid:12)1), there exits ε0 ≤ εR1 such that
A(e−εx+ 1 e−2εy) A(x) η for all y B(R ), ε ε .
1 0
− − ≤ ∈ ≤
As a result, the(cid:12)term I pη. Therefore by (3.1(cid:12)), for any ε ε ,
(cid:12) 1 (cid:12) 0
≤ ≤
sup P A(x) A(x) 2η.
ε
| − | ≤
|x|≤R
The result follows from the arbitrariness of η > 0. (cid:3)
The vector field P A is smooth on Rd but does not have compact support. We introduce
ε
cut-off functions ϕ C∞(Rd,[0,1]) satisfying
ε ∈ c
1 1
ϕ (x) = 1 if x , ϕ (x) = 0 if x +2 and ϕ 1.
ε ε ε ∞
| |≤ ε | |≥ ε k∇ k ≤
8
Set
Aε =ϕ P A , j = 0,1,...,m.
j ε ε j
Now consider the Itˆo SDE (1.1) with A being replaced by Aε (j = 0,1,...,m), and denote the
j j
corresponding terms by adding the superscript ε, e.g. Xε, Kε, etc.
t t
Inthesequel,weshallgiveauniformestimatetoKε. Tothisend,weneedsomepreparations
t
in the spirit of Malliavin calculus [28]. For a vector field A on Rd and p > 1, we say that
A Dp(γ ) if A Lp(γ ) and if there exists A: Rd Rd Rd in Lp(γ ) such that for any
∈ 1 d ∈ d ∇ → ⊗ d
v Rd,
∈
A(x)(v) = ∂ A:= lim A(x+ηv)−A(x) holds in Lp′(γ ) for any p′ < p.
v d
∇ η→0 η
For such A Dp(γ ), the divergence δ(A) Lp(γ ) exists and the following relations hold:
∈ 1 d ∈ d
P A= e−εP ( A), δ(P A) = eεP (δ(A)). (3.2)
ε ε ε ε
∇ ∇
If A Lp(γ ), then P A Dp(γ ) and lim P A A = 0.
∈ d ε ∈ 1 d ε→0k ε − kLp
Lemma 3.2. Assume the vector field A Dp(γ ) with p > 1, and denote by Aε = ϕ P A. Then
∈ 1 d ε ε
for ε ]0,1],
∈
δ(Aε) P A +eδ(A) ,
ε
| |≤ | | | |
Aε 2 P A2 ,
ε(cid:0) (cid:1)
| | ≤ | |
Aε 2 P 2 A2 + A2 ,
ε(cid:0) (cid:1)
|∇ | ≤ | | |∇ |
δ(Aε)2 P 2 A2 +e2 δ(A)2 .
ε(cid:2) (cid:0) (cid:1)(cid:3)
| | ≤ | | | |
Proof. Notethataccordingto(3.2),δ(Aε) =(cid:2) (cid:0)δ(ϕ P A) = ϕ eε(cid:1)P(cid:3)δ(A) ϕ ,P A ,fromwhere
ε ε ε ε ε ε
−h∇ i
the first inequality follows. In the same way, the other results are obtained. (cid:3)
Applying Theorem 2.2 to Kε with p = 2, we have
t
m 1/6
kKtεkL2(P×γd) ≤ Rdexp 2t 2|δ(Aε0)|+ |Aεj|2+|∇Aεj|2+2|δ(Aεj)|2 dγd . (3.3)
(cid:20)Z (cid:18) h Xj=1(cid:0) (cid:1)i(cid:19) (cid:21)
By Lemma 3.2,
m
2 δ(Aε) + Aε 2+ Aε 2+2δ(Aε)2
| 0 | | j| |∇ j| | j |
j=1
X(cid:0) m (cid:1)
P 2A +2eδ(A ) + 7A 2+2 A 2+4e2 δ(A )2 .
ε 0 0 j j j
≤ | | | | | | |∇ | | |
(cid:20) j=1 (cid:21)
X(cid:0) (cid:1)
We deduce from Jensen’s inequality and the invariance of γ under the action of the semigroup
d
P that
ε
m 1/6
kKtεkL2(P×γd) ≤ Rdexp 4t |A0|+e|δ(A0)|+ 4|Aj|2+|∇Aj|2+2e2|δ(Aj)|2 dγd
(cid:20)Z (cid:18) h Xj=1(cid:0) (cid:1)i(cid:19) (cid:21)
(3.4)
for any ε 1. According to (3.4), we consider the following conditions.
≤
Assumptions (H):
9
(A1) For j = 1,...,m, A Dq(γ ), A is continuous and δ(A ) exists.
j ∈∩q≥1 1 d 0 0
(A2) The vector fields A ,A ,...,A have linear growth.
0 1 m
(A3) There exists λ > 0 such that
0
m
exp λ δ(A ) + δ(A )2 dγ < + .
0 0 j d
Rd | | | | ∞
Z (cid:20) (cid:18) j=1 (cid:19)(cid:21)
X
(A4) There exists λ > 0 such that
0
m
exp λ A 2 dγ < + .
0 j d
Rd |∇ | ∞
Z (cid:18) j=1 (cid:19)
X
Note that by Sobolev’s embedding theorem, the diffusion coefficients A ,...,A admit
1 m
H¨older continuous versions. In what follows, we consider these continuous versions. It is clear
that under the conditions (A2)–(A4), there exists T > 0 small enough, such that
0
m 1/6
Λ := exp 4T A +eδ(A ) + 4A 2+ A 2+2e2 δ(A )2 dγ < . (3.5)
T0 Rd 0 | 0| | 0 | | j| |∇ j| | j | d ∞
(cid:20)Z (cid:18) h Xj=1(cid:0) (cid:1)i(cid:19) (cid:21)
In this case, for t [0,T ],
0
∈
sup kKtεkL2(P×γd) ≤ ΛT0. (3.6)
0<ε≤1
Theorem 3.3. Let T > 0 be given. Under (A1)–(A4) in Assumptions (H), there are two
positive constants C and C , independent of ε, such that
1 2
sup E Kε logKε dγ 2(C T)1/2Λ +C TΛ2 , for all t [0,T].
0<ε≤1 Rd t| t| d ≤ 1 T0 2 T0 ∈
Z
Proof. We follow the arguments of Proposition 4.4 in [12]. By (2.3) and (2.4), we have
m t t
Kε(Xε(x)) = K˜ε(x) −1 = exp δ(Aε)(Xε(x))dwj + Φ (Xε(x))ds ,
t t t j s s ε s
(cid:18)j=1Z0 Z0 (cid:19)
(cid:2) (cid:3) X
where
m m
1 1
Φ = δ(Aε)+ Aε 2+ Aε,( Aε)∗ .
ε 0 2 | j| 2 h∇ j ∇ j i
j=1 j=1
X X
Thus
E Kε logKε dγ = E logKε(Xε(x)) dγ (x)
Rd t| t| d Rd t t d
Z Z
m t (cid:12) (cid:12) t
E δ(Aε)(Xε(cid:12)(x))dwj dγ (x(cid:12))+E Φ (Xε(x))ds dγ (x)
≤ ZRd(cid:12)j=1Z0 j s s(cid:12) d ZRd(cid:12)Z0 ε s (cid:12) d
(cid:12)X (cid:12) (cid:12) (cid:12)
=:I +I(cid:12) . (cid:12) (cid:12) (cid:12) (3.7)
1 (cid:12)2 (cid:12) (cid:12) (cid:12)
Using Burkholder’s inequality, we get
m t m t 1/2
E δ(Aε)(Xε(x))dwj 2E δ(Aε)(Xε(x)) 2ds .
j s s ≤ j s
(cid:12)j=1Z0 (cid:12) (cid:20)(cid:18)j=1Z0 (cid:19) (cid:21)
(cid:12)X (cid:12) X (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)
10