Rough flows and homogenization in stochastic turbulence

0.33 MB·

by  I. Bailleul

Rough flows and homogenization in stochastic
turbulence
I. BAILLEUL1 and R. CATELLIER2
Abstract. We provide in this work a tool-kit for the study of homogenisation of
6 random ordinary differential equations, under the form of a friendly-user black box
1 based on the tehcnology of rough flows. We illustrate the use of this setting on the
0 exampleofstochasticturbulence.
2

t
c Contents
O
1 Introduction 1
0
1 2 Tools for flows of random ODEs 3
2.1 The black box 3

]
R 2.2 How to use it 5
P 2.3 A toy example 7
. 3 A case study: Stochastic turbulence 9
h
3.1 Setting and result 9
t
a
3.2 Proof of homogenisation for stochastic turbulence 11
m
A Compactness results 23
[
B An elementary lemma 26

2
v
2
0
1 Introduction
9
1
0
. The history of averaging and homogenization problems for dynamical systems is
1
fairly long and has its roots in classical perturbative problems in mechanics, in the
0
6 19th centery. It has evolved in an impressive body of methods and tools used to
1 analyse a whole range of multiscale systems, such as (possibly random) transport
:
v equations with multiple time-scales [1, 2], or heat propagation in random media
i
X [3,4]. Thelatest developments ofOtto, Gloria&co[5]andArmstrong&co [6,7]on
homogenization forthesolutions ofHamilton-Jacobi equations useanddevelop deep
r
a results in partial differential equations. The present work deals with the transport
side of the story, in the line of the classical works of Kesten and Papanicolaou on
homogenization for random stochastic differential equations [8, 9, 10], and put them
in theflow of ideas and tools that have emerged in theearly 2000’s with rough paths
theory. Kelly and Melbourne [11, 12] have for instance shown recently how one can
use rough paths methods to investigate a fast-slow system of the form
1
x9 a x ,y b x ,y ,
ǫ ǫ ǫ ǫ ǫ
“ p q` ǫ p q
1I.B. thankstheU.B.O. for their hospitality.
2R.Catellier is supported by theLabex Lebesgue
AMSClassification: 60F05, 37H10
1
2
where the dynamics of the fast component y is autonomous and Anosov or axiom
ǫ
A, or even non-uniformly hyperbolic. We would like to put this result and other
homogenization results in the newly introduced setting of rough flows [13], that
encompasses a large part of the theory of rough differential equations, and unifies
it with the theory of stochastic flows. We provide for that purpose an easily usable
black box forthestudyof homogenisation ofrandomordinarydifferential equations,
under the form of a result
Convergence of finite dimensional marginals Moment/tightness bounds
‘
(for the driving vector fields)
Homogenisation
ùñ
for which no knowledge of the mechanics of rough flows is required. See Theorem 4
in section 2.2. As an illustration of use of this method in homogenization problems,
weshowinthepresentworkhowonecanget backandextendinaclean andefficient
way Kesten and Papanicolaou’ seminal result [8] on stochastic turbulence.
Thetheoryofroughflowsisbasedonthefollowingparadigm. Thekindofdynam-
ics weareaboutto considerare allgenerated bysome kindof time-dependentvector
fields,or drivers, thatgenerate flows bya deterministic continuous mechanism. Any
ordinary differential equation is naturally recast in this setting. The benefits of this
picture for the study of averaging and homogenization problems are obvious. If the
driversare randomanddependon someparameters, itsuffices that they converge in
law in thespace of drivers for their associated dynamics to converge in law, fromthe
continuity of the driver-to-flow map. Support theorems and large deviation results
are also automatically transported from the driver world to the flow world. The
rough flow setting somehow provides an optimized and friendly environment where
to apply ideas similar to those of rough paths theory, with the same benefits. As a
matter of fact, one can also study some homogenisation problems for random ordi-
nary differential equations from the latter point of view, such as done by Kelly and
Melbourne in their works [11, 12] on fast-slow systems, to the expense of working
with tensor products of Banach spaces and the involved subtleties. No such high
level technology is required in the elementary setting of rough drivers and rough
flows, which may then be easier to use [13]. More importantly, it has a dual version
on function spaces that can be used to study some hyperbolic partial differential
equations and seem beyond the scope of Lyons’ formulation of rough paths theory
[14].
Section 2.1providesavery lightpresentation ofroughdriversandtheirassociated
flows;convergenceproblemsforflowsamountinthissettingtoconvergenceproblems
for their drivers – a philosophy shared by the martingale problem formulation of
stochasticdynamics,withthenoticeabledifferencethatwearehereinadeterministic
setting. Section 2.2 contains the above generic homogenisation result; it is proved
in Appendix A. An elementary deterministic example is given in section 2.3 as an
illustration of the mechanics at play in the rough driver/flow setting. The case of
homogenisation for stochastic turbulence is treated in section 3.
Notations. We shall use the sign for an inequality that holds up to a multiplica-
À
tive positive constant whose precise value is unimportant. The sign will be used
c
À
to indicate that this constant dependson a parameter c. Given a finite positive time
horizon T, we shall write D for s,t 0,T 2;s t . We shall use the sign
T
p q P r s ď |¨|
to denote any Euclidean norm on a finite dimensional space; its precise choice will
(
be unimportant.
3
Given a non-integer positive regularity index a, we shall denote by Ca the
‚
usual space of a-H¨older functions. Given 0 a 1, a 2-index map
1
Z with values in some space Ca2 Rd ăwill beăsaid to be a -H¨older
p tsq0ďsďtďT b p q 1
if
Z
ts Ca2
Z : sup b ;
Ctas1Cba2 “ 0 s t T ›t ›s a1 ă 8
ď ă ď ›| ´›|
we write Z P Ctas1››Cba2››.
An additive function of time V is a vector space valued function
ts 0 s t T
‚ p q ď ď ď
V of time such that V V V , for all 0 s u t T.
ts tu us
“ ` ď ď ď ď
Whenever convenient, we shall freely identify vector fields with first order
‚
differential operators, so that given two vector fields V ,V , the notation
1 2
V V will stand for the second order differential operator whose action on
1 2
smooth functions f is
V V f Df DV V D2f V ,V .
1 2 2 1 1 2
“ p q p q `p qp q
` ˘
Given f L Rd,Rd and σ in the unit ball of Rd, we define inductively a
8
‚ sequenceP∆m opf operaqtors on L Rd,Rd setting
σ 8
p q
∆ f f σ f x and ∆m 1f ∆ ∆mf .
σ σ` σ σ
p¨q “ p¨` q´ p q “ p q
` ˘
Implicit summation of repeated indices is used throughout, so aib means
i
‚
aib .
i i
We denote by Lk Ω the corresponding integrability spaces over some prob-
‚ ařbility space Ω,Fp,Pq .
p q
2 Tools for flows of random ODEs
The machinery of rough drivers and rough flows introduced in [13] provides a
very convenient setting for the study of convergence of flows and weak convergence
of random flows. Rather than giving the reader an account of the theory of rough
flows, we single out here part of it under the form of a friendly user black box that
requires no knowledge of the mechanics of rough flows. We refer the interested
reader to the work [13], and to Appendix A, for some more technical details.
2.1 The black box.
The starting point of this business is the elementary observation that if we are
given some smooth globally Lipschitz vector fields v ,...,,v on Rd, and some
1 ℓ
real-valued controls h1,...,hℓ on some time interval 0,T , then the solution flow
r s
ϕ of the controlled ordinary differential equation
ts 0 s t T
p q ď ď ď
z9 h9iv z
t t i t
“ p q
enjoys the Taylor expansion property
t u1
f ϕ f hi hi V f dhj dhk V V f O t s 2 (2.1)
˝ ts “ ` t´ s i ` u2 u1 j k ` | ´ |ą
ˆżs żs ˙
` ˘ ` ˘
4
for all smooth functions f. The notion of rough driver captures the essence of the
different terms that appear in this local description of the dynamics.
Definition 1. Let 2 p 2 r 3 be given. A rough driver, with regularity indices
p and 2 r , is a faďmilyăV` ă , with V : V ,V , for some vector fields
p ` q ts 0 s t T ts “ ts ts
V , and V some second orderďdiďffeďrential operator, such that
ts ts
` ˘ ` ˘
(i) the vector field Vts is an additive function of time, with V P Ct1s{pCb2`r,
(ii) the second order differential operators
1
W : V V V ,
ts ts ts ts
“ ´ 2
are actually vector fields, and W P Ct2s{pCb1`r,
(iii) we have
V V V V V ,
ts tu us tu us
“ ` `
for any 0 s u t T.
ď ď ď ď
We define the norm of V to be
V : max V , W .
} } “ Ct1s{pCb2`r Ct2s{pCb1`r
´› › › › ¯
We simply talk of a rough drive›r w›hen its re›gul›arity indices are clear from the
context. We typically use rough drivers to give a local description of the dynamics
of a flow ϕ, under the form of a Taylor expansion formula
f ϕ f V f V f.
ts ts ts
˝ » ` `
IntheTaylor formula(2.1), the term hi hi V plays therole of V , whilethe term
t s i ts
´
t rdhj dhk V V has the role of V ; check that properties (i)-(iii) hold indeed
s s u r j k `ts ˘
f´or these two¯terms. More generally, given any sufficiently regular time-dependent
ş ş
vector field v on Rd, on can check that setting
t
t t u1
V : v du, V : v v du du (2.2)
ts “ u ts “ u2 u1 2 1
s s s
ż ż ż
defines a rough driver for which
t r
W v ,v du du ,
ts “ u2 u1 2 1
s s
ż ż
“ ‰
with Lie brackets of vector fields used here. Formula (2.2) defines the canonical lift
of a possibly time-dependent vector field v. As we shall use it later, remark here
that if V V,W 1 V2 stands for a rough driver with regularity indices p and
“ ` 2
2 r , and X stands for a 2-H¨older function with values in the space of C1 r vector
p ` q ` ˘p b`
fields on Rd, then the formula
1
V ,W V2 X X
ts ts ` 2 ts` t´ s
still defines a p,2 r -roug`h driver. ˘
p ` q
Definition 2. Let V be a bounded Lipschitz vector field on Rd; let also V be a rough
0
driver with regularity indices p and 2 r . A flow ϕ is said to solve
p ` q ts 0 s t T
the rough differential equation ď ď ď
` ˘
dϕ V ϕ dt V ϕ;dt
0
“ p q ` p q
5
ifthereexistsapossibly V ,V -dependentpositive constantδ suchthattheinequality
0
3
f ϕts f ` t ˘s V0f Vtsf Vtsf f C2`r t s p
˝ ´ `p ´ q ` ` À } } | ´ |
holds for a›››ll f P Cb2`!r, and all 0`ď s˘ď t ď T with t)´›››8s ď δ. Such flows are called
rough flows.
If V is the canonical lift of a C2 r time-dependent vector field v, its associated
b`
rough flow coincides with the classical flow generated by v. A robust well-posedness
result is provided by the next result, proved in [13].
Theorem 3. Assume p r 1. Then the differential equation on flows
3 ă ď
dϕ V ϕ dt V ϕ;dt
0
“ p q ` p q
has a unique solution flow; it takes values in the space of homeomorphisms of Rd,
and depends continuously on V and V in the topology of uniform convergence.
0
Moreover, if r 1, then the maps ϕ and their inverse have uniformly bounded
ts
ă
Cr-norms; if r 1, they have uniformly bounded Lipschitz norms.
“
If B is an ℓ-dimensional Brownian motion and v ,...,v are C3 vector fields on
1 ℓ b
Rd, one can prove that setting
t u1
V Bi v , and V dBj dBk v v
ts “ ts i ts “ ˝ u2 ˝ u1 j k
ˆżs żs ˙
defines almost surely a rough driver with regularity indices p and 2 r , for any
p 2 r 3 with p r, and that the solution flow of the equationp ` q
ă ` ă 3 ă
dϕ V ϕ;dt
“ p q
coincides almost surely with the flow generated by the Stratonovich stochastic dif-
ferential equation
dx v x dBi.
t i t t
“ p q˝
See e.g. Lyons’ seminal paper [15].
2.2 How to use it.
Let then assume we are given a random ordinary differential equation
x9ǫ vǫ xǫ (2.3)
t t t
“
inRd, drivenby a randomtime-dependentg`lob˘ally Lipschitz vector field vtǫ, depend-
ing on a parameter ǫ, an element of 0,1 say. Onecan think for instance of the slow
p s
dynamics in a fast-slow system [12]
x9ǫ f xǫ,yǫ ,
t t t
“
1
y9ǫ `g yǫ ,˘
t “ ǫ t
driven by deterministic vector fields f,g, bu`t w˘here yǫ is random for instance, so we
0
have (2.3) with
vǫ f ,yǫ .
t t
p¨q “ ¨
We shall also reformulate in section 3 the stochastic turbulence dynamics in those
` ˘
terms. Fix a finite time horizon T and define, for 0 s t T, the canonical lift
ď ď ď
Vǫ of vǫ into a rough driver
t t u1
Vǫ : vǫ du, Vǫ : vǫ vǫ du du ,
ts “ u ts “ u2 u1 2 1
s s s
ż ż ż
6
and
t r
Wǫ : vǫ ,vǫ du du .
ts “ u2 u1 2 1
s s
ż ż
Denote by ϕǫ the random flow generat“ed by eq‰uation (2.3), so ϕǫ x is, for any
ts
p q
0 s t T, the value at time t of the solution to equation (2.3) started from x
ď ď ď
at time s. This flow is also the solution flow of the equation
dϕǫ Vǫ ϕǫ;dt .
“
Given 0 r 1, denote by Cr the space of r-H¨older continuous functions from
ă ă 0 ` ˘
Rd to itself that are at finite Crp-dqistance from the identity, and write Diffr for the
0
space of Cr-homeomorphisms with Cr-inverse, for which both the homeomp qorphism
and its inverse are at finite Cr-distance from the identity. Thefollowing convergence
result, proved in Appendix A, is an elementary
Convergence of finite dimensional marginals Moment/tightness bounds
‘
(for the driving vector fields)
Homogenisation
ùñ
result. Theexponentainthestatementistobethoughtofasabigpositiveconstant.
Theorem 4. Let some positive finite exponents p,r;a , with r 1, be given such
p q ă
that
1 d
0 2 r .
ă 1 1 ´ ă ´ a
p ´ 2a
Assume that for each 0 s t T, and each y Rd, the random variables
‚ ď ď ď P
Vǫ y and Wǫ y converge weakly as ǫ goes to 0.
ts ts
p q p q
Assume further that there is an integer k 3 for which the positive quantity
1
‚ ě
a a a
Vǫ y Wǫ y ∆k1Vǫ y dσ
tsp q tsp q dy σ tsp q dy
żRd$&››|t´s|p1 ››L2apΩq`››|t´s|p2 ››LapΩq,. `RdˆijBp0,1q›› |t´s|p1 ››L2apΩq |σ|p2`rqa`d
› › › › › ›
› › › › › › a
%› › › › - ›∆k1´1Wǫ ›y dσ
σ tsp q dy
`RdˆijBp0,1q›› |t´s|p2 ››LapΩq |σ|p1`rqa`d
› ›
is bounded above by a finite constant indepe››ndent of ǫ. ››
Then for every pair of regularity indices p,2 r , with p 2 r 3, and
1 1 1 1
p ` q ă ` ă
d 1 1 1 1
r r , and ,
1
ă ´ a 3 ă p ă p ´ 2a
1
there exists a random rough driver V, with regularity indices p and 2 r , whose
1 1
p ` q
associated random flow
dϕ V ϕ;dt
“ p q
is the weak limit in C 0,T ,Diffr1 of the random flows ϕǫ generated by the dy-
0
r s
p q
namics (2.3). ´ ¯
The above ǫ-uniform moment bound is actually a sufficient condition for tight-
ness in the space of drivers with regularity indices p and 2 r . Note that the
1 1
p ` q
convergence and moment assumptions are about the vector fields Vǫ and Wǫ that
generate the dynamics, while the conclusion is on the dynamics itself. The possibil-
ity totransferaweak convergence resulton theroughdriverstothedynamicscomes
from the continuity of the solution map, given as a conclusions in Theorem 3. Note
7
also that we work here with vector fields V,W that are in particular bounded, as
required by the definition of a Cα function, for a non-integer regularity exponent α.
In applications, one may have firstto localize the dynamics in a big ball of radius R,
use Theorem 4, and remove the localization in a second step. This is what we shall
bedoing inour study of stochastic turbulencein section 3. Let us note herethatthe
results proved by Kelly and Melbourne [11, 12] in their study of fast-slow systems
with a chaotic fast component can actually be rephrased exactly in the terms of
Theorem 4, so one can get back their conclusions from the point of view developed
here.
2.3 A toy example.
Before applyingTheorem 4 inthe setting of stochastic turbulence, we illustrate in
thissectiononanelementaryandinterestingtoyexamplethefundamentalcontinuity
property of the solution map to an equation
dϕ V ϕ;dt ,
“ p q
inadeterministicsetting. Inthisexample,weconstructafamilyVǫ ofroughdrivers,
obtained asthecanonical liftof asmooth ǫ-dependentvector fieldontheplane, such
that its first level Vǫ converges to 0 in a strong sense while the flow ϕǫ associated
with Vǫ does not converge to the identity. This shows the crucial influence of the
second level object Vǫ on the dynamics generated by Vǫ. We work in R2 C ; set
»
v x : if x eif x t,
t p q
p q “ p q
for some C3 non-zero phase f, so that its canonical lift V V,1 V2 W as a
b “ 2 `
rough driver has first level
` ˘
V x eif x t eif x s.
ts p q p q
p q “ ´
Given 2 p 3, we define a space/time rescaled rough driver Vǫ, with regularity
ď ă
indices p and 1, setting
Vtǫs x : ǫVtǫ´2sǫ´2 ǫ2x , Wtǫs x : ǫ4 Wtǫ´2sǫ´2 ǫ2x ;
p q “ p q “
´ ¯
this is the canonical lift of the ǫ`-dep˘endent vector field ` ˘
vtǫpxq :“ 1ǫ vpǫ2xq“ ifpǫǫ2xqeifpǫ2xqt.
Theorem 5. The rough driver Vǫ converges as a p,1 -rough drivers to the pure
p q
second level rough driver
1
V x : 0, t2 s2 f 0 ∇f 0 .
t,s
p q “ ´4 ´ p qp qp q
ˆ ˙
` ˘
As a corollary, the solution flow ϕǫ to the equation
x9ǫ vǫ xǫ
t t t
“ p q
converges to the elementary flow generated by the ordinary differential equation
1
x9 f 0 ∇f 0
t
“ ´2 p qp qp q
with constant vector field.
8
Proof – Weshallprovetheclaimasadirectconsequenceofthefollowingelementary
estimate
Dℓ eif t eif s Tℓ t s γ, (2.4)
p¨q p¨q f,γ 1
´ ď | ´ |
that holds for all t›ime´s 0 s t T¯›8 , every exponent 0 γ 1, and any
› 1›
› ď ď ď ›ă 8 ă ď
derivative index 0 ℓ 3, as shown by interpolating two trivial bounds.
ď ď
Working with T Tǫ 2, and since DℓVǫ x ǫ2ℓ 1 DℓVǫ xǫ2 , it
1 “ ´ t,sp q “ ` tǫ´2,sǫ´2
already follows from (2.4) that ´ ¯
` ˘
DℓVǫ Tℓǫ1 2γ t s γ,
ts ´
À | ´ |
› ›8
so, indeed, we have › ›
› ›
Vtǫ,s C3
sup } } 0
0 s t T t s γ ÝÑǫÑ0
ď ď ď | ´ |
if one chooses 0 γ 1.
ă ă 2
To deal with Wǫ, note first that an integration by parts gives for W the decom-
position
1 t 1 t
W x : D v V x D V v x dr DV x V x DV x v x dr
t,s x r rs x r,s r ts t,s rs r
p q “2 p q´ p q “ 2 p q p q´ p q p q
żs ´ ¯ żs
1 t 1
DV x V x if x reifpxqr seifpxqs ∇f x ieifpxqr ∇f x ieifpxqr dr;
ts t,s
“2 p q p q´ p q ´ 2 p q ` p q
żs ´ ¯ ´ ¯
x∇fpxq,vrpxqy
so one can write loooooooooooooooooooooomoooooooooooooooooooooon
1
W x t2 s2 f x ∇f x R x
t,s ts
p q “ ´4p ´ q p q p q` p q
with
1 s t
R x DV x V x eif x sf x ∇f x ∇f x e2if x r dr
t,s ts ts p q p q
p q “2 p q p q` 2 p q p q´ p q
s
ż ´ ¯
1 t
f x ∇f¯x re2if x rdr.
p q
` p q p q2
s
ż
Hence
Wtǫ,s x ǫ4Wtǫ´2,sǫ´2 xǫ2
p q “ p q
1
t2 s2 b ǫ2x ∇b ǫ2x Rǫ x ,
“ ´4p ´ q p q p q` tsp q
where
Rtǫs x : ǫ4Rtǫ´2sǫ´2 xǫ2 .
p q “ p q
The scaling in ǫ between space and time gives to convergence
Rtǫ,s C2
sup } } 0,
t s 2γ Ñ
0 s t T
ď ď ď | ´ |
which is enough to conclude that Wǫ converges in the same space to
1
t,s,x t2 s2 f 0 ∇f 0 .
p q Ñ ´4p ´ q p q p q
⊲
9
3 A case study: Stochastic turbulence
We show in this section how one can use the black box provided by Theorem 4
to reprove and improve in a simple way Kesten and Papanicolaou’ seminal result
on stochastic turbulence [8]. The object of interest here is the dynamics of a par-
ticle subject to a random velocity field that is a small perturbation of a constant
deterministic velocity. Precisely, consider the random ordinary differential equation
x9 v ǫF x ,
t t
“ ` p q
withinitial condition x fixed, wherev is adeterministic non-zeromean velocity and
0
F is asufficiently regular centered, stationary, randomfield; precise assumptionsare
given below. To investigate the fluctuations of x around its typical value, one looks
‚
at the dynamics of the recentered and time-rescaled process
xǫt : xǫ´2t ǫ´2tv,
“ ´
and prove that the continuous random processes xǫ converge in law, as ǫ
t 0 t 1
decreasesto0, toaBrownianmotion withsomeconstantďdrďiftbandsomecovariance
` ˘
σ σ, both given explicitly in terms of the statistics of F. We actually use Theorem
˚
4 to prove a similar result for flows directly.
As the process xǫ solves the random ordinary differential equation
x9ǫ ǫ 1F xǫ ǫ 2tv , (3.1)
t ´ t ´
“ `
the flow genetared by the latter dynamic`s is also ass˘ociated with the ”rough driver”
1
Vǫ Vǫ,Wǫ Vǫ 2 , (3.2)
“ ` 2p q
´ ¯
where
1 t u t
Vǫ x : F x v du : vǫ x du
t,sp q “ ǫ ` ǫ2 “ up q
żs ´ ¯ żs
and
1 t u1 1 t u1
Wǫ x : Vǫ ,Vǫ vǫ ,vǫ du du ,
t,sp q “ 2 du2 du1 “ 2 u2 u1 2 1
s s s s
ż ż ż ż
thatiscanonically associatedwi“ththespac‰e/timerescal“eddynam‰ ics,equation (3.1).
We put here quotation marks around ”rough driver” as V and W only satisfy the
algebraic conditions defining a rough driver, and not all of the analytic conditions
since they are a priori unbounded. This is the very reason why we shall later
proceed in a two step process for the analysis of the homogenisation phenomenon,
by firstlocalizing the dynamics in a ball of arbitrary radius, homogenising, and then
removing the localisation.
3.1 Setting and result
Let F be an almost surely continuous Rd-valued random field on Rd, defined on
some probability space Ω,F,P . Given a measurable subset Λ of Rd, define the
p q
σ-algebra generated by F on Λ by
G : σ F x ;x Λ F.
Λ
“ p q P Ă
We define the correlation coefficient`of F on two˘measurable subsets Λ1 and Λ2 of
Rd by
α G ,G : sup P A A P A P A .
Λ1 Λ2 “ A1PGΛ1,A2PGΛ2 ˇ p 1X 2q´ p 1q p 2qˇ
` ˘ ˇ ˇ
ˇ ˇ
10
The mixing rate of F is defined as the function
α u : sup α G ,G
p q “ Λ1,Λ2 B Λ1 Λ2
δ Λ1,Λ2P u ` ˘
p qě
for any non-negative u, and where
δ Λ ,Λ : inf λ λ .
1 2 1 2
p q “ λ1 Λ1,λ2 Λ2| ´ |
P P
We make the following Assumptions on the random field F.
(i) The random field F is centered and stationary.
(ii) It takes values in C3 Rd,Rd , and
bp q
3
E sup DkF x 2a0
p q ă 8
«k 0 x 1 ff
ÿ“ | |ď ˇ ˇ
for some integrability exponentˇa 3ˇ d .
0
ą p _ q
(iii) We also have
`8α u κdu
p q ă `8
ż0
for some exponent κ 0, 1 1 1 .
P 3 ^ d ´ a0
The parameters a and κ w`ill be fixed t˘hroughout; we fix them once and for
0
all. One can find in the Appendix of the work [8] of Kesten and Papanicolaou two
interesting classes of examples of random fields satisfying the above assumptions,
some Gaussian vector fields, and vector fields constructed from some side Poisson
process.
A last piece of notation is needed to state our main result. For any two points
x,y of Rd, set
C x,y : E F x F y uv du, (3.3)
p q “ R p qb p ` q
ż ” ı
and note that it is a function of y x , since F is stationary. This covariance
function is C2 r, for any 0 u p1, ´undqer the above assumptions on F. One can
`
then define a Brownian moătionďV in the space of C2 r vector fields on Rd, with
`
covariance C, and use the results of [13] to define a C1 r time-dependent random
`
vector field Wtpssq on Rd by the formula
t u1
s
Wtp,sqpxq “ s s V˝du2,V˝du1 pxq
ż ż ” ı
at each point x of Rd; we use Stratonovich integration here. This can be done in
such a way that the formula V,W s 1V2 defines almost surely a rough driver
p q ` 2
with regularity indices p and 2 r , for any 2 p 2 r 3. Note that the
`p ` q ˘ ă ă ` ă
integral
1
b : 8E DuvF F 0 D0F F uv du
“ 2 p q´ p q
ż0 ” ı
is also well-defined as a consequ`ence of˘the deca`y assu˘mption (iii) on F, and define
a rough driver V, with regularity indices p and 2 r , setting
p ` q
1
Vts :“ Vts,Wtpssq` 2 Vt2s`pt´sqb .
ˆ ˙