1Introduction [email protected] STOCHASTICFLOWSOFDIFFEOMORPHISMSFORONE-DIMENSIONALSDEWITHDISCONTINUOUSDRIFT

in PROBABILITY

STOCHASTIC FLOWS OF DIFFEOMORPHISMS FOR ONE-DIMENSIONAL SDE WITH DISCONTINUOUS DRIFT

STEFANO ATTANASIO

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy email: [email protected]

SubmittedJuly 2,2009, accepted in final formMay 24,2010 AMS 2000 Subject classification: 60H10

Keywords: Stochastic flows, Local time

Abstract

The existence of a stochastic flow of classC^1,α, forα <¹₂, for a 1-dimensional SDE will be proved under mild conditions on the regularity of the drift. The diffusion coefficient is assumed constant for simplicity, while the drift is an autonomous BV function with distributional derivative bounded from above or from below. To reach this result the continuity of the local time with respect to the initial datum will also be proved.

1 Introduction

The problem of the existence and smoothness of the stochastic flow under conditions of low reg- ularity of the coefficients has been much studied. Apart from the intrinsic interest of the problem, there is an interest due to the range of possible applications of these results to PDE theory. For example, in[3]the uniqueness of the stochastic linear transport equation with Hölder continuous drift was proved, through new results about stochastic flows of classC^1,α. Because of the greater regularity of stochastic flows compared to deterministic flows, there are cases in which a PDE ad- mits infinitely many solutions in the deterministic case, but it becomes well posed if it is perturbed by a stochastic noise. In addition in[2]it is proved that in some cases, through a zero-noise limit, it is possible to find a criterion to select one particular solution.

We consider an equation of the form

d X_t^x=b(X_t^x)d t+dW_t

X^x(0) =x (1)

A complete proof of existence of the stochastic flows of classC^1,αis known only whenbis Hölder continuous and bounded, see[3]. In the 1-dimensional case, an important example that deals with discontinuousbhas been studied in[5]. Moreover there are preliminary results in[6]. The class of bounded variation (BV) fields bemerges from these works as a natural candidate for the flow property, although only a few partial properties have been proved. Moreover, BV fields are the most general class considered also in the deterministic literature, see[1]: in any dimension,

213

whenb∈BVand the negative part of the distributional divergence ofbis bounded, a generalized notion of flow exists and is unique.

The aim of this work is to give a complete proof of existence of the stochastic flows of classC^1,α, for α < ¹₂, in dimension one, whenb∈BV and the positive or the negative part of the distributional derivative of bis bounded. This result, although restricted to the 1-dimensional case, is stronger than the deterministic one both because we accept a bound on b⁰from any side, and because we construct a flow of classC^1,α, not only a generalized flow.

The partial results of the paper[6]suggest the problem whether b∈BV is sufficient. We cannot reach this result without a one-side control on b⁰. The fact that a similar assumption is imposed in[1]is maybe an indication that it is not possible to avoid it.

2 Flow of homeomorphisms and known results

All results contained in this paper will be proved under the following hypothesis:

1. b∈BV(R)andb=b¹−b², withb¹andb²increasing and bounded functions 2. b¹∈W^1,∞orb²∈W^1,∞

We will first suppose b¹ ∈W^1,∞. Under this hypothesis we will prove that the local time of the stochastic differential equations (SDE) solutions is Hölder continuous with respect to the initial data. Thanks to this result we will prove the existence of the stochastic flow of class C^1,α, for α < ¹₂. At the end of the paper, using standard facts on the backward equations, we will show that the results proved hold if we replace the hypothesis b¹∈W^1,∞with the hypothesisb²∈W^1,∞. A result about one-dimension stochastic flows, under the hypothesis b ∈ BV, was given in[6]. There it was first proved the non-coalescence property through an elegant proof different from the one proposed in these notes. Then, the flow continuity was proved, and this property, together with the continuity of the flow of the backward equation, implies the homeomorphic property of the flow. However the proof of the continuity of the flow appears to be incomplete. Indeed, in order to apply Kolmogorov’s lemma, the following inequality is shown:

E

– sup

0≤s≤t(X_s∧τ^x −X_s∧τ^y )²

™

≤n(x−y)²

whereτ is a stopping time depending on x and y. This doesn’t appear sufficient to apply Kol- mogorov’s lemma.

Note that, as a standard consequence of the pathwise uniqueness we have that ∀h > 0, a.s., X_t^x+h≥X_t^x. Moreover, assumingb¹∈W^1,∞, we have

(X_t^x+h−X_t^x)−(X_s^x+h−X_s^x) = Zt

s

b(X_r^x+h)−b(X_r^x)d r≤ Z t

s

kD b¹k∞(X_r^x+h−X_r^x)d r From this inequality, using Gronwall’s lemma we obtain:

(X_t^x+h−X_t^x)≤e^{(t−s)kD b1k∞}(X_s^x+h−X_s^x)

This fact, together with the proof of the non-coalescence property, contained in[6], is sufficient to prove the existence of a stochastic flow of homeomorphisms. Therefore the proof of the home- omorphic property contained in[6]can be corrected easily under our hypothesis 1 and 2.

However we are interested in stronger results about smoothness of the flow. In particular we are interested in the smoothness of the inverse flow, which is a basic ingredient, for instance, in the analysis of stochastic transport equations. While the homeomorphic property implies only the continuity of the inverse flow, we will prove that the inverse flow is of classC^1,αforα <¹₂. Notation

Throughout the paper we will assume given a stochastic basis with a 1-dimensional Brownian motion(Ω,(Ft)t≥0,F,P,W_t). Moreover, for each 0≤s< t we denote byFs,t the completedσ- algebra generated byW_u−W_r fors≤ r≤u≤ t. We will use the following notation: L=kbk_∞, K=kD b¹k_∞,b^∗=b¹+b²,L^∗=kb^∗k_∞.

We will denote byX_t^x the unique solution of the stochastic differential equation (1).

3 Local-time continuity with respect to the initial data

Definition 3.1. Let X_t^x be the unique solution of equation (1), and let a ∈R. We will denote by L^a_t(X^x)its local time at a, i.e. the continuous and increasing process such that

|X_t^x−a|=|x−a|+ Z t

0

sgn(X_s^x−a)d X_s^x+L^a_t(X^x) Further details about local time can be found in[8].

Remark 1. Recall the following inequality which is used, for example, to prove the continuity with respect to(a,t)of the local time: LetX_t =X₀+A_t+M_t be a continuous semimartingale, where M_t is a continuous local martingale, vanishing in 0 andA_t is a continuous process with bounded variation, vanishing in 0. Suppose that sup_t≤T|M_t| ∨sup_t≤T|A_t| ≤K. Then it holds:

E





Z t

0

1{a<X_s≤b}d〈M〉s

p

≤C_p,K|a−b|^p

Theorem 3.2. There exists a modification of L^a_t(X^x)which is jointly continuous in(a,t,x),and it is Hölder continuous in(a,x),of orderα,forα <¹₂.

Proof. Define an increasing sequence of stopping times as follows:

T_n:=inf

t≥0 : |W_t| ∨t L≥n We haveT_n↑ ∞a.s. Denote byX_T

nthe unique stopped solution of equation (1). It is sufficient to prove the theorem forL^a_t∧T

n(X^x). Note that∀t≥0|X_t∧T

n−x| ≤2n, andX_T

nsatisfies the hypothesis of remark 1. By definition of L^a_t(X^x)it follows:

L^a_t∧T

n(X^x) =|X_t∧T^x

n−a| − |x−a| − Zt∧T_n

0

sgn(X_s^x−a)b(X_s^x)ds− Z t∧T_n

0

sgn(X_s^x−a)dW_s Thanks to the inequality |X_t^x−X_t^y| ≤ e^{K t}|x−y|, which holds a.s. the first term on the right hand admits a modification jointly continuous in (a,t,x), and lipschitz continuous in (a,x). In particular it is Hölder continuous in(a,x), of orderα, forα < ¹₂. The second term is obviously continuous in(a,t,x)and Hölder continuous in(a,x), of orderα, forα < ¹₂. We now prove that the third one admits a modification jointly continuous in(a,t,x), and Hölder continuous in(a,x),

of orderα, forα < ¹₂. We will apply Kolmogorov’s lemma to the spaceC([0,∞)), endowed with the sup norm. It will be useful to apply remark 1.

Let(a,x)∈R²and(b,y)∈R². We will consider only the case b>aandy>x. In the other cases the following estimates are similar. It holds:

E



sup

t≥0

Z t∧Tn

0

sgn(X_s^x−a)b(X_s^x)−sgn(X_s^y−b)b(X_s^y)ds

p



≤C_pE



sup

t≥0

Z t∧Tn

0

sgn(X_s^x−a)b(X_s^x)−sgn(X_s^x−b)b(X_s^x)ds

p



+C_pE



sup

t≥0

Zt∧Tn

0

sgn(X_s^x−b)b(X_s^x)−sgn(X_s^y−b)b(X_s^y)ds

p



≤C_pE





2L Z T_n

0

1a≤X_s^x<bds

p

+C_pE





Z T_n

0

|b(X_s^x)(sgn(X_s^x−b)−sgn(X_s^y−b))|ds

p



+C_pE





ZT_n

0

|sgn(X_s^y−b)(b(X_s^x)−b(X_s^y))|ds

p

≤C_pE





2L Z T_n

0

1a≤X_s^x<bds

p



+C_pE





2L Z T_n

0

1X_s^x≤b<Xs^yds

p

+C_pE





ZT_n

0

|b(X_s^x)−b(X_s^y)|ds

p



≤C_pE





2L Z T_n

0

1a≤X_s∧Tn^x <bds

p

+C_pE





2L ZT_n

0

1b−e^{K T}(y−x)<X_s∧Tn^x ≤bds

p



+C_pE





ZT_n

0

b^∗(X_s^y)−b^∗(X_s^x)ds

p



≤C_p,nL^p|a−b|^p+C_p,nL^p|x−y|^p+C_pE





Z T_n

0

b^∗(X_s^y)−b^∗(X_s^x)ds

p



We need to estimate the last term to apply Kolmogorov’s lemma: defineh=e^{K T}(y−x); letf be such that f⁰⁰(r) = b^∗(hr+h)−b^∗(hr), and f⁰(r) =−L^∗+Rr

−∞f⁰⁰(s)ds. Note that f⁰⁰(r)≥0∀r, and thatR+∞

−∞ f⁰⁰(s)ds=limr→+∞b^∗(r)−b^∗(−r)≤2L^∗. So we have|f⁰(r)| ≤L^∗∀r∈R. Using Itô formula and the boundness of f⁰ we will obtain theL^p-boundness of ¹_hRT_n

0 f⁰⁰_Xx s

h

ds. Indeed we have

1 2

Z T_n

0

b^∗(X_s^y)−b^∗(X_s^x)ds

≤1 2

Z T_n

0

b^∗(X_s^x+h)−b^∗(X_s^x)ds=1 2

Z T_n

0

f⁰⁰ X_s^x

h

ds

≤h² f

‚X_T^x

n

h

Œ

−f

x h

‹

+h

ZT_n

0

f⁰ X_s^x

h

b(X_s^x)ds

+h

Z T_n

0

f⁰ X_s^x

h

dW_s

≤L^∗h X_T

n−x

+L^∗h Z T_n

0

|b(X_s^x)|ds+h

Z T_n

0

f⁰ X_s^x

h

dW_s

≤L^∗h(2n+n) +h

ZT_n

0

f⁰ X_s^x

h

dW_s Thus we have:

E





Z T_n

0

b^∗(X_s^y)−b^∗(X_s^x)ds

p

≤h^pC_n,p⁰





1+E







ZT_n

0

f⁰

X_s^x h

² ds

p 2











≤h^pC_n,p⁰⁰ Therefore, thanks to Kolmogorov’s lemma we have proved that

Z t∧T_n

0

sgn(X_s^x−a)b(X_s^x)ds

admits a modification jointly continuous in(a,t,x), and Hölder continuous in(a,x), of orderα, forα <¹₂.

To complete the proof we have to show thatRt∧T_n

0 sgn(X_s^x−a)dW_sadmits a modification jointly continuous in(a,t,x), and Hölder continuous in(a,x), of orderα, forα <¹₂. We have:

E



sup

t≥0

Zt∧T_n

0

sgn(X_s^x−a)−sgn(X_s^y−b)dW s

p



≤C_pE





 Z T_n

0

|sgn(X_s^x−a)−sgn(X_s^y−b)|²ds

!^p₂





≤C_pE





 Z T_n

0

|sgn(X_s^x−a)−sgn(X_s^x−b)|²ds

!^p₂





+C_pE





 Z T_n

0

|sgn(X_s^x−b)−sgn(X_s^y−b)|²ds

!^p₂





≤C_pE





 Z T_n

0

1a≤X_s^x<bds

!^p₂



+C_pE





 ZT_n

0

1b−e^{K T}(y−x)<X_s^x≤bds

!^p₂





≤C_p,n|a−b|^p2+C_p,n|x−y|^2p This inequalities and Kolmogorov’s lemma prove thatRt∧Tn

0 sgn(X_s^x−a)dW_sadmits a modification jointly continuous in(a,t,x), and Hölder continuous in(a,x), of orderα, forα <¹₂. The proof is complete.

From now on we will consider only the continuous version ofL^a_t(X^x).

Corollary 3.3. Let T≥0and x≤y. Then the process s→ sup

t∈[0,T] sup

u∈[x,y]sup

a∈R

L^a_t+s(X^u)−L^a_t(X^u) (2) is continuous.

Proof. Note that∀s≤t, and∀u∈Rit holds|X_s^u−u| ≤L t+sup_s≤t|W_s|.

Thusa6∈[u−L t−sup_s≤t|W_s|,u+L t+sup_s≤t|W_s|]impliesL^a_t(X^u) =0. Denote byA_sthe random compact set[x−L(T+s)−sup_r≤T+s|W_r|,y+L(T+s)+sup_r≤T+s|W_r|]. Note that a.s.s<rimplies A_s⊂A_r. Thus∀s≤rit holds:

sup

t∈[0,T] sup

u∈[x,y]sup

a∈R

L^a_t+s(X^u)−L^a_t(X^u) = sup

t∈[0,T] sup

u∈[x,y]sup

a∈Ar

L^a_t+s(X^u)−L^a_t(X^u)

Thanks to this equality and to the compactness of[0,T]×[x,y]×A_r, the continuity of the process (2) is proved on the interval[0,r]. Because of the arbitrariness ofrthe claim is proved.

4 Existence of the stochastic flow of diffeomorphisms

We will now prove the non-coalescence property of the solutions of equation (1). This result has been already proved in[6]under more general hypothesis. However, for the sake of completeness we will give a proof based on the continuity of L^a_t(X^x). The following lemma, which appears in [8], and in[7]with a complete proof, will be useful.

Lemma 4.1. Let X be a continuous semimartingale, and denote by〈X〉t, its quadratic variation. Let f :R⁺×R×Ω→Rbe a bounded measurable function. Then a.s.,∀t≥0

Zt

0

f(s,X_s,·)d〈X〉t = Z

R

d a Z t

0

f(s,a,·)d L_s^a(X) Proposition 4.2. ∀x∈R,∀h>0,and T≥0,a.s. X_T^x+h−X_T^x>0.

Proof. Fixx∈R,h>0 andT≥0. From corollary 3.3, it follows that the processs→sup_t∈[0,T]sup_a∈RL^a_t+s(X^x)−

L^a_t(X^x), is continuous and vanishing in 0. Therefore∀"∈(0, 1)a.s. existss_",x(ω)>0 such that

s ∈[0,s_",x(ω)]implies L^a_t+s(X^x)−L^a_t(X^x) < _2L∗(1+")^" ∀a∈ R and t ∈[0,T]. So, a.s. it holds

∀t∈[0,T]:

r∈[t,tinf+s_",x(ω)](X_r^x+h−X_r^x)−(X_t^x+h−X_t^x)≥

Z t+s",x(ω)

t

b^∗(X_u^x)−b^∗(X_u^x+ (X_t^x+h−X_t^x))du

= Z

R

”b^∗(a)−b^∗€ a+€

X_t^x+h−X_t^xŠŠ—

×h L^a_t+s

",x(ω)(X^x)−L^a_t(X^x)i

d a

≥ −kL^·_t+s

",x(ω)(X^x)−L^·_t(X^x)k∞× kb^∗(·)−b^∗€

·+€

X_t^x+h−X_t^xŠŠ

k1

≥ −

sup

a∈R

L^a_t+s

",x(ω)(X^x)−L^a_t(X^x)

×2L^∗€

X_t^x+h−X_t^xŠ

≥ − "

1+"(X_t^x+h−X_t^x)

This inequality implies:

r∈[t,t+sinf",x(ω)](X_r^x+h−X_r^x)≥(X_t^x+h−X_t^x)

1+" (3)

In the same way we obtain:

sup

r∈[t,t+s",x(ω)](X_r^x+h−X_r^x)≤(1+")(X_t^x+h−X_t^x)≤(X_t^x+h−X_t^x)

1−" (4)

In particular a.s.N_",T,x(ω):=d_s ^T

",x(ω)e<∞, and thus a.s. we have

X_T^x+h−X_T^x≥h 1

1+"

N_",T,x(ω)

>0

Remark 2. Using corollary 3.3, with the same argument of the preceding proof, it is possible to prove that given an interval [x,y],∀" >0 a.s. exists s_",x,y(ω)>0 such thats ∈[0,s_",x,y(ω)]

impliesL^a_t+s(X^u)−L^a_t(X^u)< _2L∗(1+")^" ∀a∈R,t∈[0,T], andu∈[x,y]. This fact will be used in the next theorem, which is crucial to prove the existence of a flow of classC^1,α.

Theorem 4.3. Let x,y,t∈Rsuch that x< y,and t≥0.Then, a.s.

u∈[infx,y]exp

‚Z

R

L^a_t(X^u)D b(d a)

Œ

≤

‚X_t^y−X_t^x y−x

Œ

≤ sup

u∈[x,y]

exp

‚Z

R

L^a_t(X^u)D b(d a)

Œ

Proof. Step 1. Fix x < y and t ≥ 0. Fix " > 0, and lets_",x,y(ω) be defined as in remark 2.

Moreover defineN_",t,x,y(ω):=d_s ^t

",x,y(ω)e. Define ,∀i∈N,t_i(ω) = (i×s_",x,y(ω))∧t. Obviously we

have, a.s., thati≥N",t,x,y(ω)impliest_i(ω) =t. Define g:Ω×R⁺→R⁺as:

g(h) = sup

r∈[0,h]

sup

z∈[x,y]

sup

s∈[0,t]

sup

a∈R|L_s^r+a(X^z)−L_s^a(X^z)|

Note that, with the same reasoning used in corollary 3.3 and remark 2, it is possible to show that g is a.s. continuous, increasing and vanishing in 0. Let z,w ∈[x,y]such thatz <w. Then it holds:

ln

‚X_t^w−X^z_t w−z

Œ

= Z t

0

b(X_s^w)−b(X_s^z) X_s^w−X_s^z ds=

X∞

i=0

Z t_i+1

t_i

b(X_s^w)−b(X_s^z)

X_s^w−X_s^z ds:=I₀^",z,w

Observe that in the last summation a.s. only a finite number of terms are different from 0. Define:

I₁^",z,w:=

X∞

i=0

Z t_i+1

t_i

b(X_s^w)−b(X_s^z) X^w_t

i−X^z_t

i

ds

I₂^",z,w:=

X∞

i=0

Zt_i+1

t_i

1 X^w_t

i−X^z_t

i

– b

‚ X_s^z+

‚X^w_t

i−X^z_t

i

1+"

ŒŒ

−b(X_s^z)

™ ds

I₃^",z,w:= X∞

i=0

Z t_i+1

t_i

1 X_t^w

i −X_t^z

i

– b^∗

‚ X_s^z+

‚X_t^w

i −X^z_t

i

1−"

ŒŒ

−b^∗

‚ X_s^z+

‚X^w_t

i−X^z_t

i

1+"

ŒŒ™

ds Note that the following estimate holds:

I₂^",z,w−I₃^",z,w≤I₁^",z,w≤I₂^",z,w+I₃^",z,w (5)

Finally define the random set:

Ai,",z,w:=

− 1 1−"

X_t^w

i −X^z_t

i

,− 1 1+"

X_t^w

i −X^z_t

i

and

ρ^Ai,",z,w(a):=







1−"² 2"

X_t^w

i −X_t^z

i

×1Ai,",z,w





(a) The following properties are immediate: ρ^Ai,",z,w ≥0,R

Rρ^Ai,",z,w(a)d a =1, and has support con- tained in[−_1−"¹ (w−z)e^{K t}, 0]. We will use the following notation: ˇρ^Ai,",z,w(a) =ρ^Ai,",z,w(−a). Similarly we define:

Bi,",z,w:=

− 1 1+"

X_t^w

i −X^z_t

i

, 0

and

ρi,",z,w^B (a):=







1+"

X^w_t

i −X^z_t

1

×1Bi,",z,w





(a)

ρi,",z,w^B satisfies properties similar to those of ρ^Ai,",z,w. In particular its support is contained in [−_1+"¹ (w−z)e^{K t}, 0].

Step 2.Thanks to estimates (3) and (4) we have:

|I₀^",z,w−I₁^",z,w|=

X∞

i=0

Z t_i+1

t_i

b(X_s^w)−b(X_s^z)

X_s^w−X_s^z − b(X_s^w)−b(X^z_s) X^w_t

i −X^z_t

i

ds

≤"

X∞

i=0

Z t_i+1

t_i

b(X_s^w)−b(X_s^z) X_t^w

i −X_t^z

i

ds

Using the decompositionb^∗=2b¹−b, and the relation, which holds forα≤β,|b(β)−b(α)| ≤ b^∗(β)−b^∗(α), we obtain:

|I₀^",z,w−I₁^",z,w| ≤2"

X∞

i=0

Z ^ti+1

t_i

b¹(X_s^w)−b¹(X^z_s) X^w_t

i−X_t^z

i

ds

!

−"

X∞

i=0

Z ^ti+1

t_i

b(X_s^w)−b(X_s^z) X^w_t

i−X_t^z

i

ds

!

≤ 2"

1−"K t−"I₁^",z,w (6)

Step 3.From the occupation time formula we have:

I₃^",z,w=

X∞

i=0

Z

R

1 (X_t^w

i −X^z_t

i)b^∗

‚ a+

‚X_t^w

i −X^z_t

i

1+"

ŒŒ

−b^∗

‚ a+

‚X_t^w

i −X^z_t

i

1−"

ŒŒ

×(L^a_t

i+1(X^z)−L^a_t

i(X^z))d a

=

X∞

i=0

Z

R

2"

1−"²

D€

b^∗∗ρ^Ai,",z,w

Š(a)×(L^a_t

i+1(X^z)−L^a_t

i(X^z))d a

= 2"

1−"²

X∞

i=0

Z

R

ρˇ^Ai,",z,w∗(L^·_t

i+1(X^z)−L^·_t

i(X^z))(a)

D b^∗(d a)

≤ 2"

1−"²

X∞

i=0

Z

R

h

ρˇ^Ai,",z,w∗ L^·_t

i+1(X^z)−L^·_t

i(X^z) (a)−

L^a_t

i+1(X^z)−L^a_t

i(X^z)i

×D b^∗(d a) +

2"

1−"² Z

R

L^a_t(X^z)D b^∗(d a)

≤ 2"

1−"²

Z

R

L^a_t(X^z)D b^∗(d a)

+4L^∗ 2"

1−"²

g 1

1−"(w−z)e^{K t}

N",t,x,y (7)

Step 4.It holds:

I₂^",z,w=

X∞

i=0

Z

R

1 (X_t^w

i −X^z_t

i)

– b

‚ a+

‚X^w_t

i−X^z_t

i

1+"

ŒŒ

−b(a)

™

×h L^a_t

i+1(X^z)−L^a_t

i(X^z)i d a

= X∞

i=0

Z

R

D€

b∗ρi,",z,w^B

Š(a)×h L^a_t

i+1(X^z)−L^a_t

i(X^z)i d a

= X∞

i=0

Z

R

h L^·_t

i+1(X^z)−L^·_t

i(X^z)i

∗ρˇ^Bi,",z,w

(a)D b(d a) From this equality it follows:

|I₂^",z,w−

Z

R

L^a_t(X^z)D b(d a)|

=

X∞

i=0

Z

R

nh L^·_t

i+1(X^z)−L^·_t

i(X^z)i

∗ρˇi,",z,w^B (a)−h L^a_t

i+1(X^z)−L^a_t

i(X^z)io

D b(d a)

≤4L^∗g 1

1+"(w−z)e^{K t}

N_",t,x,y (8)

Step 5. From (5), and from (6), (7) and (8), follows that, assuming " ∈(0,¹

2), there exists a constantM dependent only onx,y,t, and on the constantsK, LandL^∗, such that

|I₀^",z,w−

Z

R

L^a_t(X^z)D b(d a)| ≤M"(1+ sup

u∈[x,y]sup

a∈R

L^a_t(X^u)) +M g(M(w−z))N",t,x,y (9)

We call∆a finite partition of[x,y]if, for somen∈N,∆ ={x=z₀<z₁<· · ·<z_i<z_i+1<· · ·<

z_n=y}. Moreover we define|∆|:=max_zi∈∆\y(z_i+1−z_i). We denote byΛthe set of finite partition of[x,y]. Obviously it holds:

sup∆∈Λ min

z_i∈∆\y

‚X^z_tⁱ⁺¹−X^z_tⁱ z_i+1−z_i

Œ

≤

‚X_t^y−X_t^x y−x

Œ

≤ inf

∆∈Λ max

z_i∈∆\y

‚X_t^zi+1−X^z_tⁱ z_i+1−z_i

Œ

(10)

Using (9), we have,∀"∈(0,¹₂):

∆∈Λinf max

z_i∈∆\y

‚X^z_tⁱ⁺¹−X_t^zi z_i+1−z_i

Œ

≤ inf

∆∈Λ max

z_i∈∆\yexp

‚Z

R

L^a_t(X^zi)D b(d a)

Œ

×exp

‚

M"(1+ sup

u∈[x,y]

sup

a∈R

L^a_t(X^u)) +M g(M|∆|)N_",t,x,y

Œ

≤ sup

u∈[x,y]exp

‚Z

R

L^a_t(X^u)D b(d a)

Œ

×exp

‚

M"(1+ sup

u∈[x,y]sup

a∈R

L^a_t(X^u))

Œ

With the same reasoning we obtain:

sup

∆∈Λ inf

z_i∈∆\y

‚X^z_tⁱ⁺¹−X^z_tⁱ z_i+1−z_i

Œ

≥ inf

z∈[x,y]exp

‚Z

R

L^a_t(X^z)D b(d a)

Œ

×exp

‚

−M"(1+ sup

u∈[x,y]

sup

a∈R

L^a_t(X^u))

Œ

Thanks to the arbitrariness of", and thanks to relation (10) the proof is complete.

Using the preceding theorem we can now prove the existence of the stochastic flow of classC^1,α. All results we have proved refers to solutions starting fromt=0. This choice was made to simplify notations. In the next theorem we will treat the general case. So, we will consider solutions of the following equation:

d X^s,x_t =b(X_t^s,x)d t+dW_t

X^s,x(s) =x (11)

Obviously forX^0,x_t theorem 4.3 holds.

Theorem 4.4. Assume conditions 1 and 2 of section 2, and let T > 0. Then there exists a map (s,t,x,ω)→φs,t(x)(ω)defined for0≤s≤t≤T , x∈R,ω∈Ωwith values inR, such that

1. given any0≤s≤T , x ∈Rthe process X^s,x= (X^s,x_t s≤t≤T)defined as X^s,x_t =φs,t(x)is a continuousFs,t−measurable solution of equation (11).