getdoc71a6. 257KB Jun 04 2011 12:04:31 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y}

Vol. 14 (2009), Paper no. 80, pages 2328–2351. Journal URL

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

Asymptotic Growth Of Spatial Derivatives Of Isotropic

Flows

Holger van Bargen

∗

Abstract

It is known from the multiplicative ergodic theorem that the norm of the derivative of certain stochastic flows at a previously fixed point grows exponentially fast in time as the flows evolves. We prove that this is also true if one takes the supremum over a bounded set of initial points. We give an explicit bound for the exponential growth rate which is far different from the lower bound coming from the Multiplicative Ergodic Theorem.

Key words: stochastic flows, isotropic Brownian flows, isotropic Ornstein-Uhlenbeck flows, asymptotic behavior of derivatives.

AMS 2000 Subject Classification:Primary 60G15, 60G60, 60F99.

Submitted to EJP on April 23, 2009, final version accepted October 9, 2009.

∗_{Institut für Mathematik, MA 7-4, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, email:} van_barg@math.tu-berlin.de

1

Introduction

The evolution of the diameter of a bounded set under the evolution of a stochastic flow has been

studied since the 1990’s (see[5],[6],[11],[17], [12]and the survey article[16]to name just a few

references). It is known to be linearly growing in time if the flow has a positive Lyapunov exponent.

Of course the considered diameter links to the the supremum of_|φt(x)|ranging overxin a subset of

Rd_{. In the following we will consider the case where the flow is replaced by its spatial derivative. We}

emphasize that we consider the asymptotics in time (the spatial asymptotics for a fixed time horizon

have been considered in[8]in a very general setting and in[2]in the particular one treated here).

If the flow has a positive top exponent it is known that the growth is at least exponentially fast which is then true even for a singleton (this follows directly from the Multiplicative Ergodic Theorem). We will show in the case of an isotropic Brownian flow (IBF) or an isotropic Ornstein-Uhlenbeck flow

(IOUF) that sup_x|logD x_t

| grows at most linearly in time t where the supremum is taken over

x in a bounded subset ofRd _{no matter what the top Lyapunov exponent is. This shows that the}

growth of the norm of the derivative is indeed at most exponentially fast but it also gives some

insight into the distance of Dφt(x) to singularity by bounding sup0≤t≤T t−1infxlog

D x_t

from

below in the lim inf sense. This excludes super-exponential decay to singularity which might be of interest especially if the top exponent is negative. Exponential bounds on the growth of spatial

derivatives play a role in the proof of Pesin’s formula for stochastic flows (see[13]). It has also been

conjectured that this should yield a new proof of the fact that the diameter grows at most linearly in time (but there are much simpler proofs known for this - see the references given above). Despite the fact that we can come up with an upper bound for the exponential growth rate we make no claims about its optimality (and we conjecture that our bound is far from optimal).

1.1

Definition And Prerequisites

In this section we will recall the definition of an isotropic Brownian Flow (IBF) from [4] and an

Isotropic Ornstein-Uhlenbeck Flow (IOUF) from[2]. We will keep the convention of speaking of an

IOUF only if its driftcis not equal to zero. (see[2]or[3]for a discusion of this issue).

Definition 1.1(IBF and IOUF).

Let c > 0 and F(t,x,ω) be an isotropic Brownian field with a C4-covariance tensor i.e.

¬

Fi(_·,x),Fj(_·,y)¶

t =t bi j(x−y)where the function b(·) = bi j(·):R d

→Rd×d _{is four times} continu-ously differentiable with bounded derivatives up to order four and preserved by rigid motions (see[4]

or [7]). We define the semimartingale field V(t,x,ω) := F(t,x,ω)−c x t and an IOUF to be the solutionφ=φs,t(x,ω)of the Kunita-type stochastic differential equation (SDE)

φs,t(x) =x+ Z t

s

V(du,φs,u(x)) =x+ Z t

s

F(du,φs,u(x))−c Z t

s

φs,u(x)du. (1)

Note that the definition of b and[9, Theorem 3.1.3]imply

¬

∂lFi(·,x·),∂kFj(·,y·) ¶

t=− Z t

0

∂l∂kbi,j(xs−ys)ds. (2)

We call c the drift of φ and b the covariance tensor of φ. Note that we write x_t for φt(x) =

φ0,t(x) = φ0,t(x,ω) and x1t, . . . ,x d

t for its components. We will write the spatial derivatives as

∂ix j t :=

∂ ∂xiφ

j

0,t(x)and D x_t

:=

Dφ_t(x)

. The same notations are used for y_t =φ_0,t(y,ω)etc..

The assumptions on the smoothness of b ensure that the local characteristics (b,c·) are smooth

enough to guarantee the existence of a solution flow to (1) which can be shown to be of classC3,δ

for arbitrary 1> δ >0 (see[9, Theorem 3.4.1]). In fact one has to choose a modification to get the

mentioned smoothness (which we do without change of notation). IOUFs have been studied in[7]

and in[2]and we will recall some facts that we use later.

Lemma 1.2(some finite-dimensional marginals).

Letφbe an IOUF with drift c and covariance tensor b and let x,y∈Rd_{. Then we have}

1. φis a Brownian flow (i.e. it has independent increments) and its law is invariant under orthog-onal transformations.

2. The distance process_{|x_t− y_t|:t_∈R_{+}solves the SDE} |x_t−y_t|=_|x₋y_|+

Z t

0 p

2

1₋B_L(_|x_s−y_s|)

dW_s

+

Z t

0

(d−1)1−BN(|xs− ys|)

|x_s−y_s| −c|xs− ys|ds (3) for a standard Brownian motion(W_t)_t≥0. Therein B_L and B_N are the longitudinal and normal correlation functions of b respectively (see [4] or Lemma 1.3). There are constantsλ >0and

¯

σ >0such that the following is true.

(a) There is a standard Brownian motion (Wt)t≥0 such that we have a.s. for all t ≥ 0 that

|x_t−y_{t| ≤ |}x₋y_|eσ¯sup0≤s≤tWs+λt_.

(b) We have for each x,y_∈Rd_{, T >0and q≥1that}

E

–

sup

0≤t≤T|

x_t− y_t|q

™1/q

≤2_|x₋y_|e(λ+12qσ¯ 2_)T

. (4)

3. For x∈Rd _{the spatial derivative∂jx}i

t:=

∂ φi t(x)

d xj solves the following SDE. ∂jxti =δi j+

Z t

0 X

k

∂jxsk∂kFi ds,xs

−c

Z t

0

∂jxitds. (5)

We will use the symbolP_kas a shorthand forPd_k=1(also for multiple summation indeces).

Proof:[7, Proposition 7.1.1, Corollary 7.1.1 and p. 139]and[16, condition (H)]. ƒ

Observe that we writeE[X]qfor E[X]q

which is different fromE[Xq]. We will use this

conven-tion throughout the whole paper. The following lemma gives some insight into the structure of b

which is known since[18].

Lemma 1.3(local properties of b).

1. b can be written as bi,j(x) =

¨

(BL(|x|)−BN(|x|)) xixj

|x|2 : for x6=0

δi,jBL(0) =δi,jBN(0) =δi,j : else

where B_Land B_Nare the so-called longitudinal and transversal correlation functions defined by B_L(r) = bi,i(r e_i),r _≥

0and BN(r) = bi,i(r ej),r ≥ 0,i 6= j. BN and BL are bounded C4- functions from [0,∞) to

R_{with bounded derivatives up to order four. LettingβL,N :}=₋B′′_L,N(0)denote their right-hand derivatives at zero we have the Taylor-expansions BL,N(r) =1− 1₂βL,Nr2+O(r4)for r →+0.

BothβL andβN are strictly positive. Observe that our definition assumes the rigid-motion-part

of b to vanish (which is different from the definition in[10]where one has to assume α=1to be consistent with our notation).

2. The partial derivatives of b at0satisfy

∂k∂lbi,j(0) =1₂(βN−βL)(δkiδl j+δk jδl i)−βNδklδi j.

3. There is0<¯r<1and C >0such that for x∈Rd _with|x|<¯r we have |∂_k∂_lbi,j(x)₋∂_k∂_lbi,j(0)_{| ≤}C|x|2_.

Proof:[7, Proposition 1.2.2]. ƒ

Lemma 1.4(a lemma on real functions).

Let f,g:[0,∞) _→[0,∞)be increasing functions that are differentiable on (0,∞). Let further f be convex and g be concave. If we have for some t>0that f(t)_≥ g(t)and f′(t)_≥ g′(t)then we have for all s_≥t that f(s)_≥g(s).

Proof: easy undergraduate exercise ƒ

The following result is the main tool that allows for the estimation of suprema of the derivatives.

Observe thatr₊=r_∨0 denotes the positive part ofr_∈R.

Theorem 1.5(Chaining Growth Theorem).

Letψ:[0,_∞)_×Rd_×Ω_→Rd _{be a continuous random field satisfying}

1. There are A > 0 and B _≥ 0 such that for all k > 0 and bounded S _⊂ Rd _{we have}

lim sup_T→∞ 1_T log sup_x∈SP”_sup0

≤t≤T|ψt(x)|q>kT —

≤ −(k−B)

2

+ 2A2 .

2. There existΛ_≥0,σ >0,q0≥1and¯c>0such that for each x,y ∈Rd, T>0and even q≥q0

we have thatE”_sup0

≤t≤T|ψt(x)−ψt(y)|q —1/q

≤

¯

c|x−y|e(Λ+12qσ 2_)T

. ¯c may depend on q and d but neither on|x−y|nor on T .

LetΞbe a compact subset ofRd _{with box (see[16, page 19]for a definition; just note that a closed ball} inRd _{has box dimension d) dimension}∆>0. Then we have

lim sup_T→∞sup_0≤t≤Tsup_x∈ΞT1_|ψt(x)| ≤ K a.s. where for Λ0 :=

σ2_d

∆

€_d 2 −∆

Š

we put K :=

 



B+A

q

2∆

Λ +σ2_{∆ +}p_σ4_∆2_+2∆Λσ2 _{: if Λ≥Λ} 0

B+AÆ2∆_dd

−∆

€

Λ +1₂σ2_dŠ _{: otherwise}

Proof: A complete proof can be found in[16]as Theorem 5.1. Observe that the change of „q_≥1“

to „evenq≥q0“ does not alter the statement at all, because the assumptions above guarantee the

same assumptions forq≥1 with a change only in the value of ¯c. The fact that we allow ¯cto depend

on q does not play any role because the proof in [16] is perfectly valid with q-depending ¯c. We

will only briefly indicate what is involved in it. Choosing ε > 0 and a sufficiently small r0 > 0

one can coverΞwith at most eγT(∆+ε)subsets ofRd _{of diameter at most e}−γT _{< r0. Denote these}

subsets byΞ₁, . . . ,Ξ_eγT(∆+ε) For fixedT the assumption sup_0≤t≤Tsup_x∈Ξ 1

T|ψt(x)|> κimplies one of

the following to occur up to time T: one of theΞ_i gets diameter at least one or the center of aΞ_i

reaches a distance ofκT−1 from its original position. Hence we have forδ >0 that

P

–

sup

0≤t≤T

sup

x∈Ξ

|x_t|

T > κT+δ

™

≤S1+S2 (6)

withS1:=eγT(∆+ε)max x∈Ξ

P

–

sup

0≤t≤T|

xt| ≥κT−1 ™

andS₂:=eγT(∆+ε)max

i

P

–

sup

0≤t≤T

diamφt(Ξi)≥1

™

.

The chaining based result[16, Theorem 3.1]now shows that the two-point condition is sufficient

to control S₂ and the one-point-condition covers S₁. In this way one obtains that the sum over

T ∈N _{of the right hand side of (6) is finite for some γand κwhich completes the proof via the}

Borel-Cantelli-Lemma. The formula for the constant K is obtained via an optimization overγand

κ. ƒ

2

The Main Result

We are now ready to state the main result.

Theorem 2.1(exponential growth of spatial derivatives).

Let φ be an IOUF or an IBF with b and c as above and let Ξ be a compact subset of Rd _{with box} dimension∆>0. Then

lim sup

T→∞ ‚

sup

0≤t≤T

sup

x∈Ξ 1

T|log

D x_t

|

Œ

≤Ka.s. (7)

where K:=

 



B+A

q

2∆

Λ +σ2_{∆ +}p_σ4_∆2_+2∆Λσ2 _:ifΛ≥Λ 0

B+AÆ2∆_dd

−∆

€

Λ +1₂σ2_dŠ _{: otherwise}

forΛ₀:= σ_∆2d€d

2 −∆ Š

, A:=pβL, B:=

βL

2 +

|(d−1)βN−2c|

2 ,Λ:= Λ1∨Λ2∨Λ6 andσ:=σ1∨σ2∨σ6.

TheΛ_i andσi depend only on b and d and will be specified later.

Proof: This follows directly from Theorem 1.5 applied toψ:[0,_∞)_×Rd_×Ω_→R;

ψ_t(x,ω) := logD x_t

if we can verify the following lemmas. We will only use ψ in the

mean-ing given above from now on. Note that we choose the matrix norm to be the Frobenius norm

||(ai,j)1≤i,j≤d||:= ( P

i,ja 2 i,j)

1/2_{for its computational simplicity although the special choice of a norm}

Lemma 2.2(condition on the one-point motion).

We have for each bounded S⊂Rd _that

lim sup

T→∞

1

T log supx∈S

P

–

sup

0≤t≤T|

ψ_t(x)_|>kT

™

≤ −(k−B)

2

+

2A2 (8)

for A and B as given in Theorem 2.1.

Lemma 2.3(condition on the two-point motion).

We have for each x,y∈Rd_{, T >}0and even q≥q0:=

4pβL[2(d−1)βN−c−2βL]

128βL ∨

3that

E

–

sup

0≤t≤T|

ψt(x)−ψt(y)|q ™1/q

≤¯c_|x₋y_|pqe(Λ+12qσ 2_)T

(9)

forΛandσas given in Theorem 2.1 and¯c:=¯c1+¯c2+¯c6. The ¯ci are constants that depend on b and

d and will be specified later.

The proofs of these lemmas will be given in the next sections. Observe that ¯cdoes not enter into the

constantK, so we do not need to pay attention to get a small value for it.

3

Proof Of Lemma 2.2: The One-Point Condition

Before proving Lemma 2.2 we will need to establish some facts onD x_t

2

.

Lemma 3.1(SDE forD x_t 2

).

Let x_∈Rd _{and put Mt :=2}Rt

0 P

i,j,k∂kFi ds,xs

∂jxsk∂jxsi kD xsk

2. Then we have

1. (M_t)_t≥0is a continuous local martingale with

2(βL−βN)t+2(βL+βN)

Rt 0

P i,j,k,m

∂jxsk∂jxis∂mxks∂mxsi kD xsk

4 ds =〈M〉t ≤4βLt a.s. for all t >0and

hence a true martingale. 2. We have the SDED x_t

2

=d+2R₀tP_i,j,k∂kFi ds,xs

∂jxsk∂jxsi

+

(d₋1)βN+βL−2c Rt

0 D x_s

2

ds =d+Rt

0 D x_s

2

d Ms+

(d−1)βN+βL−2c Rt

0 D x_s

2

ds. 3. We haveDD x_·

2E

t=2(βL−βN) Rt

0 D x_s

4

ds +2(βN+βL)

Rt 0

P

i,j,k,m∂jxsk∂jxsi∂mxks∂mxsids.

4. ψt(x) =log D x_t

solves the SDE

ψt(x) = 1₂logd+1₂Mt+ h₍

d−1)βN+βL

2 −c i

Proof: Lemma 1.2 and Itô’s formula imply forD x_t 2

=P_i,j(∂jxit) 2_that

D x_t

2

=d+2X

i,j Z t

0

∂jxsid∂jxsi+ X

i,j ¬

∂jx_·i ¶

t. (10)

By Lemma 1.2 this is equal to

d+2X

i,j,k Z t

0

∂jxsi∂kFi ds,xs

∂jxsk−2c X

i,j Z t

0

(∂jxsi) 2_ds

+X

i,j Z t

0 X

k,l

∂_jx_sl∂_jx_skd¬∂_lFi d·,x_·

,∂_kFi d·,x_·¶ s

=d+2X

i,j,k Z t

0

∂kFi ds,xs

∂jxsi∂jxsk−2c Z t

0 D x_s

2

ds

− X

i,j,k,l Z t

0

∂jxsl∂jxsk∂k∂lbi,i(0)ds. (11)

Since we have by Lemma 1.3 thatP_i,j,k,lRt

0 ∂jx l

s∂jxsk∂k∂lbi,i(0)ds

=R₀tP_i,j,k,l∂jxsl∂jxsk

(βN−βL)δkiδl i−βNδkl

dswhich is nothing but

−

βL+ (d−1)βN Rt

0 D x_s

2

dswe also get from (11) that

D x_t

2

=d+2X

i,j,k Z t

0

∂kFi ds,xs

∂jxsi∂jxsk

−2c

Z t

0 D x_s

2

ds+

βL+ (d−1)βN

Z t

0 D x_s

2

ds

=d+2

Z t

0 D x_s

2

d M_s+

βL+ (d−1)βN−2c

Z t

0 D x_s

2

ds.

Thus 2. is proved. 4. follows from this and Itô’s formula sinceψt(x) =1₂log(

D x_t

2

). To prove 1.

and 3. we observe that by (2) and Lemma 1.3 we have that_〈M_〉t equals

−4 X

i,j,k,l,m,n Z t

0

∂jxsk∂jxsi∂mxsn∂mxsl

D x_s

4 ∂k∂nbi,l(0)ds

=2(βL−βN) Z t

0 X

i,j,l,m

(∂jxsi) 2_(∂

mxsn) 2

D x_s

4 ds

+2(βL−βN) Z t

0 X

i,j,k,m

∂_jx_sk∂_jx_si∂_mx_si∂_mxk_s

D x_s

4 ds

+4βN Z t

0 X

i,j,k,m

∂jxsk∂jxsi∂mxsk∂mxsi

D x_s

4 ds (12)

=2(βL−βN)t+2(βL+βN) Z t

0 X

i,j,k,m

∂jxsk∂jxsi∂mxsk∂mxsi

D x_s

This together with 2. proves 3. and 1. also follows from this and the next proposition. ƒ

Proposition 3.2(a simple estimate).

We have

P i,j,k,m

∂jxsk∂jxsi∂mxsk∂mxsi kD xsk

4

≤

1.

Proof: The proof is just the combination of the triangle inequality with Schwarz’ inequality. We leave the details to the reader. The reason why we state this fact as a proposition of its own is that we will

use it again. ƒ

Now we can turn to the proof of Lemma 2.2. Since we can writeM_t=W_〈M〉

t for a standard Brownian

Motion(Wt)t≥0we get with Lemma 3.1 that

ψt(x) =

1

2logd+

_(d

−1)βN+βL

2 −c

t+1

2

W_〈M〉t−

1 2〈M〉t

≤1

2logd+

_(d

−1)βN+βL

2 −c

t+1

20≤s≤sup4βLt

W_s−1

2s

(13)

d

=1

2logd+

₍

d−1)βN+βL

2 −c

t+pβLt sup 0≤s≤1

W_s−pβLts

where the latter means equality in distribution. Therefore we get for anyk>0 that

I:=P

–

sup

0≤t≤T

ψt(x)≥kT

™

(14)

≤P

–

logd 2 +0sup≤t≤T

¨

(d₋1)βN+βL

2 −c

t+pβLt sup

0≤s≤1

Ws−

p βLts)

«

≥kT

™

≤P

    

sup

0≤t≤T

    

(d−1)βN+βL−2c 2pβL

 

+

t+pt sup

0≤s≤1

Ws−

p βLts)

  

≥pk βL

T₋ logd 2pβL

    

.

Here we distinguish between two cases to treat (14). If(d−1)β_N+β_L−2c≤0 then we immediately

get (using 1₋Φ(t)_≤exp(₋1/2t2))

I_≤P

 

p T sup

0≤s≤1

W_s≥pk

βL

T₋ logd

2pβL

 =P

  sup

0≤s≤1

W_s≥ pk

βL

p

T₋ logd

2pβLT

 

=2



1−Φ 



kpT

p

βL

− logd

2pβLT



 

≤2e −1₂

 k2T

βL − klogd

βL + (logd)2

4βL T ‹

(15)

which gives lim sup_T→∞T1logP”_sup0

≤t≤Tψt(x)≥kT —

≤ −₂1_β Lk

2₌₋ 1 2βLk

2

+.

If

(d₋1)βN+βL

−2c>0 we get similarly to (15) that

lim sup_T→∞ 1_TlogP”_sup0

≤t≤Tψt(x)≥kT —

≤ −₂1_β L

h

k−(d−1)βN+βL−2c

2

i2

+.

We now only have to exclude the possibility that the modulus of the logarithm in

ψt(x) might become large due to a very small

D x_t

. Observe, that (13) implies

we get with a completely analogous computation lim sup_T→∞ 1

T logP

inf_0≤t≤Tψt(x)≤ −kT

≤ −_2β1

L

h

k− βL−(d−1)βN+2c

2

i2

+. This completes the proof of Lemma 2.2.

ƒ

4

Proof Of Lemma 2.3: The Two-Point Condition

4.1

General Estimates And Preparation

We now turn to the proof of Lemma 2.3 and start with a sketch of proof since it is rather technical. It consists of four steps.

1. Derivation of a SDE forψt(x)−ψt(y)involvingX

(i j)

t :=

∂jxsi kD xsk −

∂jysi kD ysk

.

2. Derivation of a SDE for(X(_ti j))q forq_≥2.

3. Obtaining estimates of the type required in Lemma 2.3 forX(_ti j). This is done for small_|x₋y_|

estimating the probability for very fast increase in the beginning and with a Grönwall type argument in the case this increase does not occur.

4. Obtaining the estimate forψt(x)−ψt(y)as integrated versions of the ones on theX(i j)using

the Burkholder-Davies-Gundy inequality. Observe that we have by Lemma 3.1

ψt(x)−ψt(y) = Z t

0 X

i,j,k

∂kFi ds,xs ∂jx

k s∂jxsi D x_s

2 −∂kF i _ds,y

s ∂jy

k s∂jysi D y_s

2

+(βL+βN)

2

Z t

0 X

i,j,k,m

∂_jy_sk∂_jy_si∂_my_sk∂_my_si

D y_s

4 −

∂_jx_sk∂_jx_si∂_mx_sk∂_mx_si

D x_s

4 ds

=: ˜M_t+A_t. (16)

To further analyze the latter we first prove the following lemma.

Lemma 4.1(general estimates forAt and ˜Mt).

WithM˜_t and A_t defined as above the following holds. 1. A.s. we have for all t_≥0that A_t≤(βN+βL)t.

2. Introducing the abbreviation˜bi,l_k,n(z):=∂k∂nbi,l(z)−∂k∂nbi,l(0)we can write for the quadratic

variation ofM˜

¬

˜

M¶

t=

βL+βN

2

Z t

0 X

i,k   

X

j

∂jxsi∂jxsk D x_s

2 −

∂jysi∂jysk D y_s

2

   2

ds

+2 X

i,j,k,l,m,n Z t

0

∂jxsi∂jxsk∂mysl∂mysn

D x_s

2 D y_s

2 ˜b i,l

k,n xs−ys

d e_≤˜c t

Proof: 1. is clear by definition ofA_t and Proposition 3.2. Clearly¬M˜¶

t equals

− X

i,j,k,l,m,n Z t

0

∂jxsk∂jxsi∂mxsn∂mxsl

D x_s

4 ∂k∂nbi,l(0)ds

+ X

i,j,k,l,m,n Z t

0

∂jxsk∂jxsi∂mysn∂mysl

D x_s

2 D y_s

2 ∂k∂nb i,l _x

s−ys

ds

+ X

i,j,k,l,m,n Z t

0

∂jysk∂jysi∂mxsn∂mxsl

D y_s

2 D x_s

2 ∂k∂nb i,l _y

s−xs

ds

− X

i,j,k,l,m,n Z t

0

∂_jy_sk∂_jy_si∂_my_sn∂_my_sl

D y_s

4 ∂k∂nb

i,l_(0)ds

=₋ X

i,j,k,l,m,n Z t

0 ₁

2(βN−βL)(δkiδnl+δklδni)−βNδknδil

 

∂jxsk∂jxsi∂mxsn∂mxsl

D x_s

4 +

∂jysk∂jysi∂mysn∂mysl

D y_s 4

 ds

+2 X

i,j,k,l,m,n Z t

0

∂_jxk_s∂_jx_si∂_my_sn∂_my_sl

D x_s

2 D y_s

2 ∂k∂nb i,l _x

s−ys

ds=:I I+I I I. (17)

Using I I = (βL−βN)t+

βL+βN

2 Rt

0 P

i,j,k,m

∂jxsk∂jxis∂mxsk∂mxis kD xsk

4 +

∂jysk∂jysi∂mysk∂mysi kD ysk

4 ds Proposition 3.2 now

yieldsI I≤(βL−βN)t+

(βN+βL)

2 2t =2βLtand sinceI I I≤2d

6_max

i,l,k,nsupz∈Rd∂k∂nbi,l(z)tis clear

the first part of 2. follows. To prove the second part we have to rearrange (17) using the symmetry

and isotropy of of b.

¬

˜

M¶

t

=₋ X

i,j,k,l,m,n Z t

0 



∂jxsi∂jxsk D x_s

2 −

∂jysi∂jysk D y_s

2



 



∂mxsn∂mxsl D x_s

2 −

∂mysn∂mysl D y_s

2



∂_k∂_nbi,l(0)ds (18)

+2 X

i,j,k,l,m,n Z t

0

∂jxsi∂jxsk∂mysl∂mysn

D x_s

2 D y_s

2 ˜b i,l

k,n xs−ys

Since (again by Lemma 1.3) we have∂k∂nbi,l(0) = 1₂(βN−βL)(δkiδnl+δklδni)−βNδknδil we get

thatI V equals

βL−βN

2

X

i,j,l,m 



(∂jxsi) 2 D x_s

2−

(∂jysi) 2 D y_s

2    

(∂mxsl) 2 D x_s

2 −

(∂mysl) 2 D y_s

2



ds

+ βL−βN

2 X i,j,k,m Z t 0  

∂jxsi∂jxsk D x_s

2 −

∂jysi∂jysk D y_s

2    

∂mxsi∂mxsk D x_s

2 −

∂mysi∂mysk

D y_s 2

 ds

+βN X i,j,k,m Z t 0  

∂jxsi∂jxsk D x_s

2 −

∂jysi∂jysk D y_s

2    

∂mxsk∂mxsi D x_s

2 −

∂mysk∂mysi D y_s

2



ds

=βL+βN

2 Z t 0 X i,k    X j

∂jxsi∂jxsk D x_s

2 −

∂jysi∂jysk D y_s

2    2

ds. (19)

This completes the proof of Lemma 4.1. ƒ

This Lemma shows that we have to control terms like ∂jxsi

kD xsk − ∂jysi kD ysk

. We postpone this until we will have derived the following estimate from Lemma 4.1.

Lemma 4.2(a priori bounds for theψ-estimation ).

1. We have for each x,y_∈Rd_{, T >0and q≥1that} E”_sup0

≤t≤T|ψt(x)−ψt(y)|q —1/q

≤(βN+βL)T+2e−

5 12p2π˜c

p

q+1pT . 2. We have for each x,y_∈Rd_{, T >0and q≥1that}

E”_sup0

≤t≤T|ψt(x)−ψt(y)|q —1/q

≤C¯₁e €

Λ1+12σ 2 1q

Š

T

withC¯1:=βN+βL+2e

1 4e−

5

12p2π˜c,Λ₁:=1/e andσ₁:=

p

2/e.

3. Let r > 0 be fixed. We have for any x,y _∈ Rd _{with |x − y| ≥ r, T > 0 and q ≥ 1 that} E”_sup0

≤t≤T|ψt(x)−ψt(y)|q —1/q

≤¯c1|x−y|e

€

Λ1+1₂σ12

Š

T _for

¯

c1:= 1_r h

β_N+β_L+2e−125+ 1 4ep2π˜c

i

andΛ₁ andσ1 as before.

Proof: Once again observe that by the triangle inequality

E

–

sup

0≤t≤T|

ψt(x)−ψt(y)|q ™1/q

≤E

–

sup

0≤t≤T

Aq_t

™1/q

+E

–

sup

0≤t≤T

˜

Mq_t

™1/q

≤(βN+βL)T+E –

sup

0≤t≤T

˜

M_tq

™1/q

=:(βN+βL)T+V I. (20)

Since we can write ˜M_t=W_〈M˜_〉

t and

¬

˜

M¶_t≤˜c t a.s. we get using

E”_sup0

≤t≤T M˜ q t

—

≤E”_sup0

≤t≤˜c TW q t

—

= (˜c T)q2

Æ 2

π

Γq+1

2

2 2

q+1 2

that Stirling’s formula for the Gamma function implies

V I _≤(2π)−

1 2q₂

q+1 2q _Γ

_q+1

2

1

qp

˜

c T_≤2e−125p2π˜c

p

q+1pT. (21)

Thus 1. follows. For the proof of 2. it is sufficient to observe that since for any t _≥ 0 we have

t≤et/eandpt≤1/4+t we also get

(βN+βL)T+2e−

5 12p2π˜c

p

q+1pT_≤ (βN+βL)e

T e +2e−

5

12p2π˜ce

p

q+1pT e ≤ (βN+βL)e

T e +2e−

5

12p2π˜ce

1/4+(q+1)T e ≤

h

βN+βL+2e

1 4e−

5 12p2π˜c

i

eq+E1T. (22)

This proofs 2. and 3. follows from this by using |x−y|

r ≤1. ƒ

Since we can now control the moments of ψt(x)−ψt(y) provided x and y are not too close to

each other we introduce the following stopping time. Let ¯r be chosen according to Lemma 1.3 and

˜r _≤¯r to be specified later. Remember that we assumed ¯r _≤ 1. We now define for x,y _∈Rd _with

|x− y| ≤r the stopping time

τ:=inf

t>0{|xt−yt| ≥˜r}

and assumer<˜rin the following (which ensuresτ >0 a.s.).

4.2

Derivation Of Formula H

We now proceed to work on ∂jxsi

kD xsk − ∂jysi kD ysk

by proving the following proposition on ∂jxsi

kD xsk

.

Proposition 4.3(SDE for the direction of the derivative ).

We have the SDE ∂jxsi D x_s

=δ

i j

p d +

Z t

0 X

k

∂kFi ds,xs ∂jx

k s D x_s

−

Z t

0 X

k,l,m

∂mFk ds,xs ∂lx

m

s ∂lxsk∂jxis

D x_s 3

+

₁

4−

d

2

βN−

1 4βL

Z t

0

∂jxsi D x_s

ds−βN+βL

2

Z t

0 X

k,l

∂jxsk∂lxsi∂lxsk

D x_s

3 ds

+3

4(βN+βL)

Z t

0 X

k,l,m,n

∂_lx_sk∂_lx_sn∂_mx_sk∂_mx_sn∂_jx_si

D x_s

5 ds. (23)

Proof: Since by Itô’s formula

∂jxsi D x_s

=δ

i j

p d +

Z t

0

d(∂jx_·i)s D x_s

−1

2

Z t

0

∂jxsi D x_s

3d D x_·

2 s

−1

2

Z t

0

d

D

∂jx_·i, D x_·

2E s

D x_s

3 +

3 8

Z t

0

∂jxsi D x_s

5d D

D x_·

2E

we only have to note that by Lemmas 1.2 and 3.1

D

∂jx_·i, D x_·

2E t=−2

Z t

0 X

k,l,m,n

∂jxsk∂mxsl∂mxsn∂k∂lbi,n(0)ds

=(βL−βN) Z t

0

∂jxsi D x_s

2

ds+ (βL+βN) Z t

0 X

k,m

∂jxsk∂mxsi∂mxskds (25)

which combined with Lemmas 1.2 and 3.1 and put into (24) yields

∂jxit D x_t

=δ i j p d + Z t 0 X k

∂kFi ds,xs ∂jx

k s D x_s

−c

Z t

0

∂jxsi D x_s

ds − Z t 0 X k,l,m

∂mFk ds,xs ∂lx

m

s ∂lxsk∂jxsi

D x_s 3

−(d−1)βN+βL−2c

2

Z t

0

∂jxsi D x_s

ds

+βN−βL

2

Z t

0

∂jxis D x_s

ds₋ βN+βL

2

Z t

0 X

k,l

∂jxsk∂lxsi∂lxsk

D x_s

3 ds

+3

4(βL−βN)

Z t

0

∂_jx_si

D x_s

ds

+3

4(βN+βL)

Z t

0 X

k,l,m,n

∂lxsk∂lxsn∂mxsk∂mxsn∂jxsi

D x_s

5 ds. (26)

This proves Proposition 4.3. ƒ

Of course the latter implies

∂jxsi D x_s

− ∂jy

i s D y_s

= Z t 0 X k

∂kFi ds,xs ∂jx

k s D x_s

−∂kFi ds,ys ∂jy

k s D y_s

! − Z t 0 X k,l,m 

∂_mFk ds,x_s ∂lx

m

s ∂lxsk∂jxsi

D x_s

3 −∂mF

k _ds,y s

∂ly m

s ∂lysk∂jysi

D y_s 3   + ₁ 4− d 2

βN−

1 4βL

Z t

0

∂jxis D x_s

− ∂jy

i s D y_s

!

ds

−βN+βL

2 Z t 0 X k,l  

∂jxsk∂lxsi∂lxsk

D x_s

3 −

∂jysk∂lysi∂lysk

D y_s 3  ds +3

4(βN+βL)

Z t 0 X k,l,m,n  

∂lxsk∂lxsn∂mxsk∂mxsn∂jxsi

D x_s

5 −

∂lysk∂lysn∂mysk∂mysn∂jysi

D y_s 5



Letting

V I I_t:=

Z t

0 X

k

∂kFi ds,xs ∂jx

k s D x_s

−∂kFi ds,ys ∂jy

k s D y_s

!

, (28)

V I I I_t:=

Z t

0 X

k,l,m 

∂_mFk ds,x_s ∂lx

m

s ∂lxsk∂jxsi

D x_s

3 −∂mF

k _ds,y s

∂ly m

s ∂lysk∂jysi

D y_s 3



 (29)

we have to compute the cross variations to apply Itô’s formula for powers to (27).

〈V I I_〉t =₋ Z t 0 X k,l  

∂jxsk∂jxsl D x_s

2 +

∂jysk∂jysl D y_s

2



∂_k∂_lbi,i(0)ds

+

Z t

0 X

k,l

∂jxsk∂jysl

D x_s D y_s

+ ∂jy

k s∂jxsl

D y_s D x_s

!

∂k∂lbi,i xs−ys

ds

=₋X

k,l Z t

0

∂jxsk D x_s

− ∂jy

k s D y_s

!

∂jxsl D x_s

− ∂jy

l s D y_s

!

(βN−βL)δkiδl i−βNδkl ds +2 Z t 0 X k,l

∂jxsk∂jysl

D x_s D y_s

”

∂k∂lbi,i xs−ys

−∂k∂lbi,i(0) —

ds

=(βL−βN) Z t

0

∂jxsi D x_s

− ∂jy

i s D y_s

!2

ds+βN Z t

0 X

k

∂jxsk D x_s

− ∂jy

k s D y_s

!2 ds +2 Z t 0 X k,l

∂jxsk∂jysl

D x_s D y_s

˜_bi,i

k,l xs−ys

ds (30)

and similarly

〈V I I I_〉t=βL+βN

2 Z t 0 X k,n   X l

∂lxsn∂lxsk∂jxsi

D x_s

3 −

∂lysn∂lysk∂jysi

D y_s 3   2 ds

+ βL−βN

2

Z t

0

∂jxsi D x_s

− ∂jy

i s D y_s

!2

ds (31)

+2 X

k,l,m,n,p,r Z t

0

∂lxsm∂lxsk∂jxsi∂pysr∂pysn∂jysi

D x_s

3 D y_s

3 ˜b k,n

r,m xs−ys

as well as

V I I_·,V I I I_· t

=βL−βN

2

Z t

0

∂jxsi D x_s

− ∂jy

i s D y_s

!2

ds

+βL+βN

2

Z t

0 X

k,l

∂jxsk D x_s

− ∂jy

k s D y_s

!  

∂lxsk∂lxsi∂jxsi

D x_s

3 −

∂lysk∂lysi∂jysi

D y_s 3



ds

+ X

k,l,m,n Z t

0 



∂_jx_sk∂_my_sn∂_my_sl∂_jy_si

D x_s

D y_s

3 +

∂_jy_sk∂_mx_sn∂_mx_sl∂_jx_si

D y_s

D x_s

3



˜bi,l

k,n xs−ys

ds. (32)

The combination of (30), (31) and (32) with (27) and Itô’s formula now yields forX(_ti j):= ∂jxit

kD xtk − ∂jyti

kD ytk

Proposition 4.4(formula H).

We have for q≥q0 that X(_ti j)q

=q

Z t

0

X(i j)q−1

s

X

k

∂kFi ds,xs

∂jx

k s

D x_s

−∂kFi ds,ys

∂jy

k s

D y_s

! −q Z t 0

X_s(i j)q−1X k,l,m



∂_mFk ds,x_s ∂lx

m s ∂lxsk∂jxsi

D x_s

3 −∂mF k

ds,ys

∂ly

m s ∂lysk∂jysi

D y_s

3   + _β

L−βN

4 q

2

−d−1

2 βNq− 1 2βLq

Zt

0

X_s(i j)qds

+q(q−1) 2 βN

Zt

0

X_s(i j)q−2X k

X_s(k j)2ds

−qβN+βL 2

Zt

0

X(i j)q−1

s

X

k,l

 

∂jxks∂lxsi∂lxsk

D x_s

3 −

∂jysk∂lysi∂lysk

D y_s

3

 ds

+q(q₋1)βN+βL 4

Zt

0

X(i j)q−2

s

X

k,n

 

X

l

∂lxsn∂lxsk∂jxsi

D x_s

3 −

∂lysn∂lysk∂jysi

D y_s

3   2 ds

−q(q₋1)βN+βL 2

Zt

0

X(i j)q−2

s

X

k,l X(k j)

s

 

∂lxsk∂lxsi∂jxis

D x_s

3 −

∂lysk∂lysi∂jysi

D y_s

3  ds +3

4q(βN+βL)

Zt

0

X(i j)q−1

s

X

k,l,m,n

∂lxsk∂lxsn∂mxsk∂mxsn∂jxsi

D x_s

5

∂lysk∂lysn∂mysk∂mysn∂jysi

D y_s

5

!

ds

+q(q−1)

Zt

0

X(_si j)q−2X k,l

∂jxsk∂jysl

D x_s

D y_s

˜_bi,i k,l xs−ys

ds

−q(q₋1)

Zt

0

X(i j)q−2

s

X

k,l,m,n

 

∂jxsk∂mysn∂mysl∂jysi

D x_s

D y_s

3 +

∂jysk∂mxsn∂mxsl∂jxis

D y_s

D x_s

3

 ˜b

i,l

k,n xs−ys

ds

+q(q₋1)

Zt

0

X(i j)q−2

s

X

k,l,m,n,p,r

∂lxms∂lxsk∂jxsi∂pysr∂pysn∂jysi

D x_s

3 D y_s

3 ˜b

k,n r,m xs−ys

ds. (33)

Proof: There is nothing left to show. ƒ

Proposition 4.4 will be useful to estimate the expectation in Lemma 2.3 on the event _{T _≤ τ}

but since this requires some additional preparations we will first consider the reversed case in the following intermezzo.

4.3

Treating Small

_|

x

₋

y

_|

And Large

T

Since we obviously have from Schwarz’ inequality that

E

–

sup

0≤t≤T|

ψt(x)−ψt(y)|q11{T≥τ} ™1 q ≤E – sup 0≤t≤T|

ψt(x)−ψt(y)|2q ™1

2q

P[T≥τ]

1

2q ₍₃₄₎

it seems reasonable to compute some useful estimate for the tails of τ. We will also immediatly

specify conditions on r and ˜r. Assume first with Lemma 1.3 that ˜r <¯r is small enough to ensure

that for any 0_≤r_≤˜rwe have

p

2

1₋BL(r)

≤2pβLr and(d−1)

1₋B_N(r)

r −c r≤

2(d−1)βN−c

Lemma 4.5(tails ofτ).

If we assume q_≥q₀ then we have for T _≤ log

˜

r

|x−y| 128βLq ∧

log ˜r |x−y|

4pβL[2(d−1)βN−c−2βL] = log

˜

r

|x−y| 128βLq that

P_[T≥τ]≤

2 ˜

r2q|x−y|

2q

∧_˜r24q|x− y|

4q_.

Proof: LetXt=|x− y|+

Rt 02

p

βLXsdWs+ Rt

0

2(d−1)βN−c

Xsdsi.e.

X_t:=_|x₋y_|exp

n

2pβLWs+

2(d₋1)βN−c−2βL

t

o

for some BM(W_t)_t≥0. Then we may start

with (see (3) and (35))

P_{[T ≥τ] =}P

–

sup

0≤t≤T|

x_t−y_{t| ≥}˜r

™

≤P

–

sup

0≤t≤T

X_t≥˜r

™

(36)

=P



 sup 0≤t≤T

W_t≥ 1

2pβL

log ˜r

|x−y|−

2(d₋1)βN−2βL−c

+T

 

≤P

   sup

0≤t≤1

W_t≥

log ˜r

|x−y|

2pβLT

−

2(d−1)β_N−2β_L−c

+

p T

  =:I X.

Let first 2(d₋1)βN−2βL−c≤0. Then we have using 1−Φ(t)≤e−1/2t

2

that

I X _≤2

  1−Φ

  

log ˜r

|x−y|

2pβ_LT

  

  ≤2e

−1₈



log ˜r

|x−y|

‹2

βL T

=2

|x₋ y_|

˜r

_8β1

L Tlog

˜

r

|x−y|

≤ 2

˜

r2q|x−y|

2q ₍₃₇₎

forT ≤ log

˜

r

|x−y|

16βLq (remember|x−y|<r<

˜

r ). Let now 2(d−1)βN−2βL−c>0. In this case we can

use forT_≤ log

˜

r

|x−y|

4pβL[2(d−1)βN−c−2βL]∧

log ˜r |x−y| 128βLq that

I X =P

   sup

0≤t≤1

W_t≥

log ˜r

|x−y|

2pβLT

−

2(d−1)β_N−2β_L−cp

T

  

≤P

   sup

0≤t≤1

W_t≥

log ˜r

|x−y|

4pβLT

  =2

  1−Φ

  

log ˜r

|x−y|

4pβLT

  

  

≤2 exp

¨

−1

2 1 16βLT

log ˜r

|x−y|

2«

=2

|x− y|

˜r

log ˜r

|x−y|

32βL T

≤2|x−y|

2q

˜

r2q ∧2

|x₋y_|4q

˜

r4q . (38)

Proposition 4.7(expectation of zero-mean-martingales on rare events).

For i,j∈ {1, . . . ,d}we defineMˇ_t=_Mˇ_t(i j):=Rt 0X

(i j)q−1

s d V I It−

Rt

0X

(i j)q−1

s d V I I It. Then we have 1. Mˇ_t is a zero-mean-martingale and we have any t_≥0that¬Mˇ¶

t≤ˇc t a.s. forˇc:=4(d

2_+2d4₊

d6)max_k,l,m,nsup_z∈Rd∂k∂lbm,n(z). 2. For all0_≤t_≤T and q_≥q₀we get that

E

”

ˇ

M_t11_{T≤τ}

—

≤ |x−y|2q¯c3 ˜r2qe

(Λ3+1₂σ23q+σ 2 4q

2_)T

=: f(T)for¯c3:= p

2ˇc∨

q

ˇ

c

128βL ∨1,

Λ₃:= 1

2e,σ3:=

q

64βL

e ∨

(128βLˇc)

1 4 andσ

4:=

p

256βL.

Proof: 1. is clear from the definition of ˇM except for the value of ˇc. This value follows from (30), (31) and (32) since¬Mˇ¶_{t ≤ 〈}V I I_〉i+_〈V I I I_〉t+2_{| 〈}V I I,V I I I_〉t|. For the proof of 2. first assume

T_≤ log

˜

r

|x−y|

128βLq. From Lemma 4.5 we know that

P_{[T > τ]≤ |x−y|}4q 2 ˜

r4q. This combined with Schwarz’

inequality and 1. yields

E

”

ˇ

Mt11{T≤τ}

—

=

E

”

ˇ

M_t11_{T>τ}

—

≤

q

E”¬_Mˇ¶

T

—p

P[T> τ]

≤

p

2ˇc T|x−y|

2q

˜r2q ≤

p

2ˇc|x− y|

2q

˜r2q e T

2e (42)

which proves Proposition 4.7 in this case. We always have as above that

E

”

ˇ

Mt11{T≤τ}

—

≤

p ˇ

c T=:g(T)and so we can conclude for the sameT0 as before

g(T0) =

È

ˇ

c

128βL

r

log ˜r |x₋y_| ≤

È

ˇ

c

128βL

_˜r

|x₋y_|

1

2e

≤¯c3

_˜r

|x₋ y_|

σ2₃

128βL

≤¯c3

_˜r

|x₋y_|

Λ₃+σ2₃q+σ2₄q2

128βL q |x−y|

˜

r

2q

=_|x− y|2q ¯c3 ˜r2qe

(Λ3+12σ 2 3q+σ

2 4q

2_)T

0_{=: f(T}

0) (43) so now (with Lemma 1.4) we only have to establish the inequality g′(T0)≤ f′(T0)to complete the proof of Proposition 4.7. The proof of this is as follows.

g′(T₀) =1 2

s

128βLqˇc

log ˜r

|x−y|

≤ 1 2

p

128βLqˇc (44)

≤

|x₋y_|

˜r

2

Λ₃+1 2σ

2 3q+σ

2 4q

2 ˜r |x₋y_|

Λ₃+1₂σ2₃q+σ₄2q2

128βL

= f′(T₀).

4.4

Evaluation Of Formula H

Now we are prepared to prove the following key result. Lemma 4.8(first estimate forτafter T).

We have for even q≥q0if we assume˜r≤ p1₂ the following.

1. For h(t):=E

h

(max_i,jX(i j)q_t ∨ |xt−yt|q)11{t≤τ}

i

the estimate holds

h(t)_{≤ |}x₋y_|q2¯c3d

2_q ˜

r2q

e(Λ3+(λ∨1₂σ23)q+( 1 2σ¯

2_∨σ2 4)q

2_+d2_κ

q)t

Λ₃+ (λ∨1₂σ2₃)q+ (1₂σ¯2_∨σ2

4)q2+d2κq

(45)

whereinκq behaves like a polynomial of degree2(w.r.t. growth of its modulus) in q and equals

|βL−βN

4 q 2

− d−₂1βNq−1₂βLq|

+_{C d6+

h

9

4(βN+βL) +2C

i

d4+”3₂(βN+βL) +C

—

d2+βN

2 d}q 2 −¦C d6+”2C₋3₂(βN+βL)

—

d4+C d2+1₂βNd

©

q. 2.

E

max

i,j X (i j)q

t 11{T≤τ}

≤h(t)_≤¯c5˜r−2q|x−y|qe(Λ5q+σ 2 5q

2_)t

(46)

for¯c₅:=sup_q≥3

2¯c3d2q

Λ3+(λ∨1₂σ23)q+( 1 2σ¯

2_∨σ2 4)q2+d2κq

,

Λ₅:= Λ₃+λ∨1₂σ₃2−nC d8+”2C−3₂(βN+βL)

—

d6+C d4−|βN−βL|

2 d 2o_and

σ2 5 :=

1 2σ¯

2_∨σ2 4+

n

C d8+h9

4(βN+βL) +2C

i

d6+”3

2(βN+βL) +C

—

d4

+βN

2 d

3₊|βL−βN|

4 d 2o_.

Proof: 2. follows obviously from 1. so we only have to prove this. Since the treatment of Proposi-tion 4.4 leads to rather huge formulas we restrict ourselves to indicate how to computaProposi-tion is done omitting terms when treating similar ones. We first multiply (33) with 11_{t≤τ}then we take

expec-tations and apply Fubini’s theorem. Recalling the short-hand ˜bi_k,_,j_l(z):=∂k∂lbi,j(z)−∂k∂lbi,j(0)we

get

E

h

X(i j)q_t 11_{t≤τ}

i

=qE

h

11_{t≤τ}Mˇ_t(i j)

i

+. . .₋qβN+βL

2

Z t

0

E”_X(i j)q−1

s

X

k,l





∂jxsk∂lxsi∂lxsk

D x_s

3 −

∂jysk∂lysi∂lysk

D y_s

3



11_{t≤τ} 

ds+. . .

+q(q₋1)

Z t

0 E



X_s(i j)q−2 X

k,l

∂jxsk∂jysl

D x_s

D y_s

˜_bi,i

k,l xs−ys

11_{t≤τ}



Applying the triangle inequality and using Lemma 1.3 we get

E

h

X(i j)q_t 11_{t≤τ}i_≤qE

h

11_{t≤τ}Mˇ_t(i j)

i

+. . .+qβN+βL

2 Z t 0 E • X

(i j)q−1

s

X

k,l

∂jxks∂lxsi∂lxsk

D x_s

3 −

∂jysk∂lysi∂lysk

D y_s

3

11_{t≤τ}



ds+. . .

+q(q₋1)C

Z t

0 E



X_s(i j)q−2 X

k,l

∂_jxk

s∂jysl

D x_s D y_s

|x_s−y_s|211_{t≤τ}



ds (48)

We recall the inequality _|Qai−

Q

bi| ≤

P

i|ai−bi| ≤ dmaxi|ai −bi| (valid for|ai| ∨ |bi| ≤ 1),

use that the expectation of a positive random variable is growing in the domain of integration and conclude that the latter is less or equal to

qE

h

11_{t≤τ}Mˇ_t(i j)

i

+qβN+βL

2

Z t

0 3d2E

max

i,j X (i j)q

s ∨ |xs−ys|q

11_{s≤τ}

ds

+q(q−1)C d2

Z t

0 E

max

i,j X (i j)q

s ∨ |xs−ys|q

11_{s≤τ}

ds+. . . (49)

Summing up all the terms we suppressed we get thatE

h

X(i j)q_t 11_{t≤τ}

i

is at most

qE

h

11_{t≤τ}Mˇ (i j) t i +

βL−βN

4 q

2

−d−1 2 βNq−

1 2βLq

Z t 0 E max

i,j X (i j)q

s ∨ |xs−ys|q

11_{s≤τ}

ds

+q(q−1) 2 βNd

Z t

0 E

max

i,j X (i j)q

s ∨ |xs− ys|q

11_{s≤τ}

ds

+qβN+βL

2 3d 2 Z t 0 E max

i,j X (i j)q

s ∨ |xs−ys|q

11_{s≤τ}

ds

+q(q−1)βN+βL 4 9d

4 Z t 0 E max

i,j X (i j)q

s ∨ |xs−ys|q

11_{s≤τ}

ds

+q(q₋1)βN+βL 2 3d

2 Z t 0 E max

i,j X (i j)q

s ∨ |xs−ys|q

11_{s≤τ}

ds

+3

4q(βN+βL)5d 4 Z t 0 E max

i,j X (i j)q

s ∨ |xs− ys|q

11_{s≤τ}

ds

+q(q−1)(d2+2d4+d6)C

Z t

0 E

max

i,j X (i j)q

s ∨ |xs−ys|q

11_{s≤τ}

ds

=qE

h

11_{t≤τ}Mˇ (i j) t

i

+κq

Z t

0 E

max

i,j X (i j)q

s ∨ |xs−ys|q

11_{s≤τ}

This enables us to conclude with Proposition 4.7 and Lemma 1.2 E

max

i,j X (i j)q

t 11{t≤τ}∨ |xt− yt|q11{t≤τ}

≤X

i,j

E

h

X_t(i j)q11_{t≤τ}11{t≤τ}

i

+E_|xt−yt|q_≤E_|xt−yt|q

+d2qX i,j

E

h

11_{t≤τ}Mˇ_t(i j)

i

+d2κq

Z t

0 E

max

i,j X (i j)q

s ∨ |xs−ys|q

11_{s≤τ}

ds

≤d2q_|x₋y_|2q ¯c3

˜

r2qe (Λ3+12σ

2

3q+σ24q2)t₊₂q_|x−y|q_e(λq+ 1 2q

2_σ¯2_)t

+d2κ_q Z t

0 E

max

i,j X (i j)q

s ∨ |xs−ys|q

11_{s≤τ}

ds

≤d2κq

Z t

0 E

max

i,j X (i j)q

s ∨ |xs−ys|q

11_{s≤τ}

ds

+_|x−y|q

‚

2¯c₃d2q

˜r2q

Œ

e(Λ3+(Λ∨1₂σ23)q+( 1 2σ¯

2_∨σ2 4)q

2_)t

(50) since we assumed ˜r ≤ 2−1/2. This now implies via Grönwall’s inequality (see[15, I.§2.1] for an appropriate version) that

h(t)_{≤ |}x−y|q2¯c3d 2_q ˜

r2q

e(Λ3+(λ∨1₂σ23)q+( 1 2σ¯

2_∨σ2

4)q2+d2κq)t

Λ₃+ (λ∨1₂σ2 3)q+ (

1 2σ¯

2_∨σ2

4)q2+d2κq

. (51)

The proof is complete. ƒ

The next lemma will be the last ingredient to the proof of Lemma 2.3. Lemma 4.9(second estimate forτafterT).

We have for even q_≥q₀that

E”_sup0

≤t≤T|ψt(x)−ψt(y)|q11{T≤τ}

—1

q

≤¯c₆pq_|x₋ y_|e(Λ6+1₂σ22q)T

withΛ₆= Λ₅+1

e,σ6:=

p

2σ5and ¯

c6:=supq≥3 ¯

c

1

q

5 ˜

r2

p

2d2C

1

q q q−

1

2pβL+βN+ p

C d3C

1

q q q−

1

2p₂+2d4(βN+βL) (Λ5q+σ52q2)

1

q

!

. Therein C

1

q

q is the largest positive zero

of the Hermite polynomial of order2q and can be estimated via C

1

q q ≤ k

p

4q+1 for some constant k>0(see[1]and[14]).

Proof: By the triangle inequality and the Burkholder-Davies-Gundy inequality, Lemma 4.1 and Jensen’s inequality we have

E

–

sup 0≤t≤T|

ψt(x)−ψt(y)|q11{T≤τ}

™1

q

≤E

–

sup 0≤t≤T

˜

M_tq11_{T≤τ}

™1

q

+E

–

sup 0≤t≤T

Aq_t11_{T≤τ}

™1

q

≤C

1

q qE

•

〈M_〉 q

2

T∧τ

˜1

q

+E”_Aq

T∧τ

—1

≤C

1

q q

r

βL+βN

2 E



 

Z T∧τ

0

4d4

max

i,j X (i j)

t ∨ |xt−yt|

2

ds

!q

2



 

1

q

+C

1

q q

p 2E





 C d

6

Z T∧τ

0

max

i,j X (i j)

t ∨ |xt−yt|

2

ds

!q

2



 

1

q

+ βN+βL

2 E



 4d4 Z T∧τ

0

max

i,j X (i j)

t ∨ |xt−yt|ds

!q 

1

q

≤p2d2C

1

q q

p

βL+βNT q−2

2q _E



 Z T

0

max

i,j X (i j)

t ∨ |xt−yt|

q

11_{t≤τ}ds





1

q

+pC d3C

1

q q

p 2T

q−2 2q _E



 Z T

0

max

i,j X (i j)

t ∨ |xt− yt|

q

11_{t≤τ}ds





1

q

+2d4(β_N+β_L)T q−1

q _E



 Z T

0

max

i,j X (i j)

t ∨ |xt−yt|

q

11_{t≤τ}ds





1

q

. (52)

Another application of Fubini’s theorem now yields that the latter is less or equal to (remember that we chose evenq_≥3)

p 2d2C

1

q q

p

βL+βNT q−2

2q ₊p_{C d}3_C

1

q q

p 2T

q−2

2q _+2d4_(β

N+βL)T q−1

q

RT

0 h(s)ds

1

q

≤

p 2d2C

1

q q

p

βL+βN+

p

C d3C

1

q q

p

2+2d4(βN+βL)

¯

c

1

q

5|x−y|e

(Λ₅+1_e+σ2₅q)T

(Λ5q+σ25q2) 1

q_˜r2 .

The proof is complete. ƒ

We now fix ˜r _≤ ¯r _∧ 2−1/2 subject to (35) and r _≤ e−1˜r and conclude that since sup_0≤t≤T|ψt(x) − ψt(y)|q is the sum of the terms sup0≤t≤T|ψt(x) − ψt(y)|q11{T≤τ} and

sup_0≤t≤T|ψt(x)− ψt(y)|q11{T>τ} another triangle inequality with Lemmas 4.2, 4.6 and 4.9

completes the proof of Lemma 2.3 and hence of Theorem 2.1. ƒ

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