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therefore the absolute value may be omitted. The

equation becomes

¼ x

þ 2

ﬃﬃﬃﬃﬃ

þ t ½19

Definition 8 The unique solution Z to

¼ x

þ 2

ﬃﬃﬃﬃﬃ

þ t ½20

is called ‘‘square -dimensional Bessel process’’

starting at x

; it is deno ted by BESQ



); for fine

properties of this process, see Revuz and Yor (1999,

ch. IX.3).

Since Z  0, we call -dimensional Bessel process

starting from x

the process X =

ﬃﬃﬃﬃ

. It is denoted

by BES



Remark 8 Let d  1. Let W = (W

, ..., W

)bea

classical d-dimensional Brownian motion. We set

= kW

k.(X

)

t0

is a d-dimensional Bessel proces s.

Remark 9 If >1, it is possible to see that

¼ W

  1

The Case with Distributional Drift

Pioneering work about diffusions with generalized

drift was presented by N I Portenko, but in the

framework of semimartingale processes. Recently,

some work was done characterizing solutions in the

class of the so-called Dirichlet proces ses, with some

motivations in random irregular environment.

A useful transformation in the theory of SDE is

the so-called ‘‘Zvonkin transformation.’’ Let (W

)be

an (F

)-classical Brownian motion. Let a (resp. b):

R !R (resp. C

) be locally bounded. We suppose

moreover a > 0. We fix x

2 R. Let (X

)

t0

be a

solution of

¼ x

bðX

Þds

aðX

ÞdW

½21

We set

ðxÞ¼

ðyÞdy

and we define h : R !R such that

hð0Þ¼0; h

¼ e



h is strictly increasing. We set

a(x) = (ah

)(h

1

(x)),

where h

1

is the inverse of h.WesetY

= h(X

Without entering into details, the classical Itoˆ

formula allows to show that (Y

) defines a solution of

aðY

ÞdW

¼ hðx

½22

Now, eqn [22] fulfills the requirements of the

Engelbert–Schmidt criterion so that it admits weak

existence and uniqueness in law. Consequently,

unless explosion, one can easily establish the same

well-posedness for [21].

Zvonkin transformation also allows to prove

strong existence and pathwise uniqueness results

for [21]; for instance, when



a has linear growth, and



y !

bðsÞ

ðsÞ

is a bounded function.

In fact, problem [22] satisfies pathwise uniquene ss

and strong existence since the coefficients are

Lipschitz with linear growth. Therefore, one can

deduce the same for [21].

Veretennikov generalized Zvonkin transformation

to the d-dimensional case in some cases which

include the case a = 1andb bounded Borel.

Zvonkin’s procedure suggests also to consider a

formal equation of the type

¼ dW

þ 

ðX

Þdt ½23

where  is only a continuous function and so b = 

is a Schwartz distribution;  could be, for instance,

the realization of an independent Brownian motion

of W. Therefore, eqn [23] is motivated by the study

of irregular random media. When  = 1, b = 

, SDE

[22], h

= e

2

still makes sense.

Using the Engelbert–Schmidt criterion, one can see

that problem [22] still admits weak existence and

uniqueness in the sense of distribution laws. If Y is a

solution of [22], X = h

1

(Y) provides a natural

candidate solution for [21]. R F Bass, Z-Q Chen and

F Flandoli, F Russo, and J Wolf investigated general-

ized SDEs as [23]: in particular, they made previous

reasoning rigorous, respectively, in the case of strong

and weak solutions, see Flandoli et al. (2003).

Connected Topics

We aim here at giving some ba sic references about

topics which are closely connected to SDEs.

Stochastic Partial Differential Equations (SPDEs)

If a SDE is a random perturbation of an ordinary

differential equation, an SPDE is a random

Stochastic Differential Equations 69

perturbation of a PDE. Several studies were

performed in the parabolic (evolution equation)

and hyperbolic case (wave equation). Most of the

work was done in the case of a fixed underlying

probability spaces. We only quote two basic

monographies which should be consu lted at first

before getting into the subject: the one of Walsh

(1986) and the one of Da Prato and Zabczyk

(1992).

However, it was possible to establish some results

about weak existence and uniqueness in law for

SPDEs. One possible tool was a generalization of

Girsanov theorem to the case of Gaussian spacetime

white noise. Weak existence for the stochastic

quantization equation was proved with the help of

infinite-dimensional Dirichlet forms by S Albeverio

and M Ro¨ ckner.

We also indicate a beautiful recent monography

by Da Prato (2004) which pays particular attention

to Kolmogorov equations with infinitely many

variables.

Numerical Approximations

Relevant work was done in numer ical approxima-

tion of solutions to SDEs and related approxima-

tions of solutions to linear parabolic equations

via Feynman–Kac probabilistic representation, see

Theorem 2). It seems that the stochastic simula tions

(of improved Monte Carlo type and related topics)

for solving deterministic problems are efficient when

the space dimension is greater than 4.

Malliavin Calculus

Malliavin calculus is a wide topic (see Malliavin

Calculus). Relevant applications of it concern

stochastic (ordinary and partial) differential equa-

tions. We only quote a monography of Nualart

(1995) on those applications. Two main objects

were studied.



Given a solution of an SDE, (X

), sufficient

conditions so that X

, t > 0, has a (smooth)

density p(t,  ). Small-time asymptotics of this

density, when t !0, and small-drift perturbati on

were performed, refining Freidlin–Ventsell large-

deviation estimates.



Coming back to SDE [11], one can conceive to

consider coefficients a, b nonadapted with respect

to the underlying filtration (F

). On the other

hand, the initial condition  may be anticipating,

that is, not F

-measurable. In that case, the Itoˆ

integral

a(s, X

)dW

is not defined. A replace-

ment tool is the so-called ‘‘Skorohod integral.’’

Rough Paths Approach

A very successful and significant research field is the

rough path theory. In the case of dimension d = 1,

Doss–Sussmann method allows to transform the

solution of an SDE into the solution of an ordinary

(random) differential equation. In particular, that

solution can be seen as depending (pathwise)

continuously from the driving Brownian motion

) with respect to the usual topology of C([0, T]).

Unless exceptions, this continuity does not hold in

case of general dimension d > 1. Rough paths

theory, introduced by T Lyons, allows to recover

somehow this lack of continuity and establishes a

true pathwise stochastic integration.

SDEs Driven by Non-semimartingales

At the moment, there is a very intense activity

towards SDEs driven by processes which are not

semimartingales. In this perspective, we list SDEs

driven by fractional Brownian motion with the help

of rough paths theory, using fractional and Young

type integrals and involving finite cubic variation

processes. Among the contributors in that area we

quote L Coutin, R Coviello, M Errami, M Gubinelli,

Z Qian, F Russo, P Vallois, and M Za¨ hle.

See also: Fractal Dimensions in Dynamics; Image

Processing: Mathematics; Interacting Stochastic Particle

Systems; Lagrangian Dispersion (Passive Scalar);

Malliavin Calculus; Path Integrals in Noncommutative

Geometry; Quantum Dynamical Semigroups; Quantum

Fields with Indefinite Metric: Non-Trivial Models; Random

Dynamical Systems; Random Walks in Random

Environments; Stochastic Hydrodynamics; Stochastic

Resonance.

Further Reading

Bally V, Gyongy I, and Pardoux E (1994) White noise driven

parabolic SPDEs with measurable drift. Journal of Functional

Analysis 120: 484–510.

Benachour S, Chassaing P, Roynette B, and Vallois P (1996)

Processus associe´s l’ e´quation des milieux poreux. Annali Scuola

Normale Superiore di Pisa, Classe di Scienze 23(4): 793–832.

Bouleau N and Le´pingle D (1994) Numerical Methods for

Stochastic Processes. New York: Wiley.

Da Prato G (2004) Kolmogorov Equations for Stochastic PDEs.

Basel: Birka¨ user.

Da Prato G and Zabczyk J (1992) Stochastic Equations in Infinite

Dimensions, Encyclopedia of Mathematics and its Applica-

tions, vol. 44. Cambridge: Cambridge University Press.

Flandoli F, Russo F, and Wolf J (2003) Some stochastic

differential equations with distributional drift. Part I. General

calculus. Osaka Journal of Mathematics 40(2): 493–542.

Fleming WH and Soner M (1993) Controlled Markov Processes

and Viscosity Solutions. New York: Springer.

Graham C, Kurtz Th G, Me´le´ard S, Protter Ph E, Pulvirenti M,

and Talay D (1996) Probabilistic models for nonlinear partial

70 Stochastic Differential Equations

differential equations. In: Talay and Tubaro L (eds.) Lectures

given at the 1st Session and Summer School held in

Montecatini Terme, 22–30 May 1995, Lecture notes in

Mathematics, vol. 1627. Springer-Verlag. Centro Internazio-

nale Matematico Estivo (C.I.M.E.), Florence, 1996.

Karatzas I and Shreve SE (1991) Brownian Motion and Stochastic

Calculus, 2nd edn. New York: Springer.

Kloeden PE and Platen E (1992) Numerical Solutions of

Stochastic Differential Equations. Berlin: Springer.

Lamberton D and Lapeyre B (1997) Introduction au Calcul

Stochastique et Applications la Finance. Paris: Collection

Ellipses.

Ma J and Yong J (1999) Forward–Backward Stochastic Differ-

ential Equations and Their Applications, Lecture Notes in

Mathematics, vol. 1702. Berlin: Springer.

Nualart D (1995) The Malliavin Calculus and Related Topics.

New York: Springer.

Øksendal B (2003) Stochastic Differential Equations. An Intro-

duction with Applications. Sixth edition, Universitext. Berlin:

Springer-Verlag.

Pardoux E (1998) Backward stochastic differential equations and

viscosity solutions of systems of semilinear parabolic and

elliptic PDEs of second order. Stochastic analysis and related

topics, VI (Geilo, 1996), 79–127, Progr. Probab., 42, Boston:

Birka¨user, 1998.

Protter P (1992) Stochastic Integration and Differential Equa-

tions. A new approach. Berlin: Springer.

Lyons T and Qian Z (2002) Controlled Systems and Rough Paths.

Oxford: Oxford University Press.

Revuz D and Yor M (1999) Continuous Martingales and

Brownian Motion, Third edition. Berlin: Springer-Verlag.

Stroock D and Varadhan SRS (1979) Multidimensional Diffusion

Processes. Berlin–New York: Springer.

Walsh JB (1986) An introduction to stochastic partial differential

equations. In: Ecole d’e´te´ de probabilite´s de Saint–Flour, XIV –

1984, pp. 265–439. Lecture Notes in Mathematics. vol. 118,

Springer.

Yong J and Zhou XY (1999) Stochastic Controls: Hamiltonian

Systems and HJB Equations. New York: Springer.

Stochastic Hydrodynamics

B Ferrario, Universita

di Pavia, Pavia, Italy

Introduction

Mathematical models in hydrodynamics are intro-

duced to describe the motion of fluids. The basic

equations for Newtonian incompressible fluids are

the Euler and the Navier–Stokes equations, for

inviscid and viscous fluids, respectively. For a given

set of body forces acting on the fluid, these

nonlinear partial differential equations (PDEs)

model the evolution in time of the velocity and

pressure at each point of the fluid, given the initial

velocity and suitable boundary conditions (see

Partial Differential Equations: Some Examples).

The equations of hydrodynamics offer challenging

mathematical problems, like proving the existence

and uniqueness of solutions , determining their

regularity, their asymptotic behavior for large time,

and their stability. To gain some insight into the

behavior of fluids, stochastic analysis is introduced

into hydrodynamics. In fact, there are vari ous

attempts to describe turbulent regime (see Turbu-

lence Theories). But, analyzing individual solutions

that determine the flow at any time, for a give n

initial condition, is a desperate task, since the

dynamics in a turbulent regime is chaotic and highly

unstable. This is a particular chaotic motion with

some characteristic statistical properties (see Monin

and Yaglom (1987)). The aim of a statistical

description of turbulent flow is to single out some

relevant collective properties of the flow that,

hopefully, make it possible to grasp the salient

features of the dynamics. In this sense, stochastic

hydrodynamics is germane to the kinetic gas theory.

In the next section we shall review a typical topic of

stochastic hydrodynamics, the evolution of prob-

ability measures. Results on stationary probability

measures will be given in the subseque nt sections.

Another characteristic of turbulent flows is the lack

of space regularity of the velocity field. We shall

introduce in the section ‘‘The stochastic Navier–

Stokes equations’’ a stochastic model of turbulence,

which exhibits lack of regularity of the solutions.

The Euler equations are a singular limit of the

Navier–Stokes equations, since they are first order,

instead of second-order PDEs. It is little surprise if they

involve different mathematical techniques. A full sec-

tion will be devoted to a discussion of Euler equations

and another to the Navier–Stokes equations. Statistics

of an inviscid flow, when approximated by vortex

motion, will be described in the final section.

Statistical Solutions

Let u(t, x) be the fluid velocity at time t and point

x 2 D  R

; since the initial velocity is always

affected by experimental errors, it is reasonable to

assign a measure  determining the probability that

the initial velocity belongs to a Borel set  of the

space H of all admissible velocity fields u = u(x).

A spatial statistical solution is a family of

probability measures ( t, ), t  0, each supported

Stochastic Hydrodynamics 71

on the set H such that, given any Borel set  in H,

we have

Probfuðt; xÞ2g¼ðt; Þ; 8t > 0 ½1

with the initial condition (0, ) = (). The con-

struction and analysis of statistical solutions (t, )is

one of the crucial mathematical problems in

stochastic hyd rodynamics (see, e.g., Vishik and

Fursikov (1988)).

Hopf gave the first mathematical formulation of

the problem of describing turbulent flows by

statistical solutions. The first result on the existence

of statistical solutions is by Foias in 1973. Hopf

(1952) presented an equation in variational deriva-

tives satisfied by the characteristic functional (t, )

of the family of measures (t, ) associated with the

Navier–Stokes equations. The characteristic func-

tional (t, ) is the Fourier transform of the measure

(t, ):

ðt;Þ¼

ih;ui

ðt; duÞ½2

defined for any smooth test function .

We now derive the evolution equation for (t, ),

by assuming that the dynami cs takes place in the

phase space H and follows the nonlinear equation

¼ FðuÞ½3

If u

(t) is the solution started from v at time t = 0,

then its probability distribution is represented by

the time-evolved measure (t, ). Theref ore, we

have that

ih;ui

ðt; duÞ¼

ih;u

ðtÞi

ð0; dvÞ½4

Differentiating in time, we obtain

ðt;Þ¼

ih;u

ðtÞi

ih; Fðu

ðtÞÞið0; dvÞ

¼ i

ih;vi

h; FðvÞiðt; dvÞ½5

The last integral is uniquely determined by , since

the measure (t, ) is uniquely determined by (t, ).

We denote by (t, ) the last integral in [5]. The

evolution equation thus obtained for the character-

istic functional  is

ðt;Þ¼iðt;Þ; 8 ½6

This is called the Hopf equation associated with the

dynamical system [3].

Another way to analyze the evolution of measures

is through the moments; instead of the measure (t, )

describing the spatial statistical solution, we deal with

the moments of (t, ) of any order. For a nonlinear

dynamics [3], the moments equations are an infinite

chain of coupled equations, the so-called Friedman–

Keller equations.

A prominent role among statistical solutions is

played by stationary solutions. They contain all the

statistical information in the case of equilibrium in

time. We have that the characteristic functional of

an invariant measure is constant in time. Therefore,

ðt;Þ¼0

Bearing in mind equation [5], this is equivalent

to say that the signed measure h, F(v)i(t,dv) vani-

shes, for any test funct ion  and time t. Setting t = 0,

we obtain that an invariant measure  in the space

H satisfies the Liouville equation

hðvÞ; FðvÞidðvÞ¼0 ½7

for appropriate test funct ions . This equation is

also called the relation of infinitesimal invariance

and the measure  is said to be infinitesimally

invariant.

The stationary measures are natural candidates to

describe the statistical asymptotic behavior of the

system when t !1. Notice that, in a chaotic system

two motions that are arbitrarily close to one another at

t = 0 can evolve in completely different ways. So, to

describe satisfactorily the dynamics we take average

over a big number of experiments. This is the so-called

ensemble average. These averages are assumed to be

with respect to an invariant measure .Theinvariant

measures must exist and either they are unique or at

most one has physical meaning and enters in the

functional integral defining the ensemble average.

According to the ergodic principle (an assumption not

yet proved in hydrodynamics), ensemble averages

replace long-time averages: for every initial velocity

field v, except for a set of initial values negligible in

some sense, the time average of an observable tends,

as time goes to infinity, to the ensemble average

lim

T!1

ðu

ðtÞÞdt ¼

d ½8

However, it is extremely difficult to prove the

existence of stationary probability measures for the

Navier–Stokes equations solving directly equation

[7]. The situation is formally the same as in

equilibrium statistical mechanics, where the Liouville

equation is in fact solved, leading to the Boltzmann–

Gibbs distribution. However, the results in statistical

hydrodynamics are far from being satisfactory.

72 Stochastic Hydrodynamics

Recent studies to prove the existence of invariant

measures for the Navier–Stokes equations are based

on stochastic models (see the section ‘‘The stochastic

Navier–Stokes equations’’). On the other hand, for

the Euler equations it is possible to construct

formally invariant measures, by means of invariant

quantities of the classical motion (see the next

section).

Finally, we point out that there are techniques

using invariant measures to show some results for

the time evolution (e.g., the motion exists for almost

all initial values with respect to an invariant

measure).

The Euler Equations

We start recalling some basic facts on Euler

equations (see Incompressible Euler Equa tions:

Mathematical Theory).

The motion of an inviscid, incompressible, and

homogeneous fluid is described by the Euler

equations, which in Eulerian coordinates read as

þ u rðÞu þrp ¼ f

ru ¼ 0

u  n ¼ 0on@D

in D ½9

where, at time t  0 and position x 2 D, u = u(t, x)

is the vector veloc ity, p = p(t, x) the hydrodynamic

pressure. The units have been chosen so that the

mass density  = 1. r denotes the nabla vector

operator so

u r¼

j¼1

ru ¼

j¼1

rp ¼

; ...;



Finally, f denotes the external force. If the spatial

domain D has a boundary @D, then the velocity is

assumed to be tangent to the bounda ry (n denotes

the exterior normal vector to the boundary). Some

initial condition u

at time t = 0 is assigned.

When f = 0, there are invariant quantities for

system [9]. In the literature, there are many works

suggesting a Gaussian stationary statistics (see, e.g.,

the paper by Kraichnan (1980)). We consider

invariants that are quadratic in the velocity so as

to construct (formally) invariant measures of Gibbs

type: the energy

EðuÞ :¼

juj

and, only in the two-dimensional case (d = 2), the

enstrophy

SðuÞ :¼

jcurl uj

(with curl u=r

u @u

=@x

@u

=@x

for d =2).

It is natural to look for velocity fields in the

following function spaces: the space H

of finite

kinetic energy and the space H

of finite enstrophy.

Clearly, the admissible fields should also obey the

boundary conditions and divergence-free condition.

If P is the projection operator onto the space of

divergence-free vectors, and B is the bilinear form

B(u, v):= P [(u r)v], the Euler equations can be

given the structure of an evolution,

¼Bðu; u Þ½10

obtained by applying the projection operator P to

the first equation in [9]. The pressure disappears and

can be regarded as a Lagrange multiplier associated

with the divergence-free constraint ( ru = 0); it can

be fully recovered once the velocity field is known.

The dynamics is considered in the phase space of

divergence-free velocity vectors H (a large space

containing H

and H

), which is an infinite-

dimensional functional space. More precisely, iden-

tifying H

with its dual ( H

)

, we introduce the

Gelfand’s triplet

 H

ðH

¼ H

1

The space H



, with  = 1, 2, ..., are the usual

Sobolev spaces but with the additional divergence-

free and boundary conditions. For >0 noninteger,

the spaces H



are defined by interpolation, whereas

those with <0 by duality. As usual, regularity in

space is related to the spaces H



with higher

exponent . We have that H = [

2R



Invariance of E and S can be proved resorting to

eqn [9] and assuming that u is a smooth vector field.

For instance,

EðuðtÞÞ ¼

juj

dx ¼

u 

¼

u ½ðu rÞudx 

u rp dx

Stochastic Hydrodynamics 73

By integrating by parts and bearing in mind the

divergence-free condition and the boundary condi-

tion, we conclude that

EðuÞ¼0

In the same way, the invariance of S can be proved.

As a consequence, the following Gibbs measures

which are defined on the space H



ðduÞ¼

EðuÞ



ðduÞ¼

SðuÞ

½11

are heuristically invariant in time. In [11], Z. are the

partition functions, that is, they are normalization

constants needed to guarantee that 

and 

are

genuine probability measures (e.g., Z

E(u)

du).

Actually, these measures  solve the Liouville

equation

hðuÞ; Bðu; uÞidðuÞ¼0 ½12

for any test function , cylindrical, infinitely differ-

entiable, bounded, and with bounded derivatives.

On the other hand, the (global and not only

infinitesimal) invariance means that if there exists a

global flow in time which is well defined in a phase

space of full measure , then the measure  is invariant

under this dynamics. The measures 

and 

are

centered Gaussian measures whose support is in a

space larger than H

, as can be proved by standard

methods in the theory of Gaussian measures on

infinite-dimensional spaces. By the very definition, 

is a cylindrical measure in H

and 

is cylindrical in

. Then the support of 

is any Hilbert space

H such

that H



H is a Hilbert–Schmidt embedding, and the

support of 

is any space

H such that H



H is a

Hilbert–Schmidt embedding. When the spatial dimen-

sion d is 2, supp(

)= \

<1



and supp(

)= \

<0



.Whend is 3, supp(

)= \

<3=2



Moreover, 

) = 

) = 0, that is, the space

of finite energy H

is negligible with respect to these

measures. Let us show this property for the

‘‘enstrophy measure’’ 

when d = 2. Let {e

}

j = 1

a complete orthonormal system in H

. Hence, for

u =

, we have kuk

and kuk



(with 0 <

 

 and 

 j as

j !1). Keeping in mind its definition, the measure



can be considered as a measure on the space of

the sequences {u

}

and written as an infinite product

of one-dimensional centered Gaussian measures



ðduÞ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2

1

ð

=2Þju

½13

The energy is

EðuÞ¼

and the renormalized energy is

: E : ðuÞ¼





ðduÞ



Since, as can be easily shown

(:E :(u))



(du)

<1,:E :(u) is finite for 

-almost every u. On the

contrary, since



(du) =



1

= þ1,

E(u) is infinite for 

-almost every u.

We also note in passing that, for any >0 and

>

:E:ðuÞ

SðuÞ

du < 1

so that



ðÞ;ðÞ

ðduÞ¼

:EðuÞ:SðuÞ

du ½14

is a probability measure, which is infinitesimally

invariant for the Euler flow.

Since the space of finite-energy velocity is negligible

with respect to these measures, it is necessary to

replace the classical solutions having finite energy with

generalized solutions. This is not an easy task in the

three-dimensional case, whereas some results have

been proved for the two-dimensional problem, where

the following existence result holds. Let us analyze

the quadratic term B(u, u) = P[(u r)u].(u r)u can

be rewritten as r(u  u), taking in account the

divergence-free condition. Trivially, we have that

r(u  u) = r(u  u  : u  u :), where :u  u : =

u  u; 

(du): We consider the quadratic expres-

sion (u  u  : u  u : ). This is integrable with respect

to the measure 

in the sense that

ku  u : u  u : k

"



ðduÞ < 1½15

for any ">0. We remark that this property is

similar to the integrability of the renormalized

energy, which is a quadratic expression as well.

This implies that the H

1"

-norm of r(u  u)is

integrable with respect to the measure 

. There-

fore, B(u, u) is defined for 

-a.e. u.

Now, let us replace eqn [10] with a system of infinite

equations for all the components u

with respect to

the orthonormal basis {e

}

, obtained by taking the

scalar product with e

of both sides of eqn [10]:

¼ B

ðu; uÞ; j ¼ 1; 2; ... ½16

74 Stochastic Hydrodynamics

Each component B

(u, u) is defined for 

-a.e. u.

These estimates lead to define a weak solution (see

Albeverio and Cruzeiro (1990)):

Theorem 1 Let d = 2. There exists a flow U(t, !)

defined on a probability space (, F, P) with values

in H

"1

for any ">0, U(  , !) 2 C(R, H

"1

) P-a.e.

!, such that for each component U

we have

ðt;!Þ

¼ U

ð0;!Þþ

ðUð s;!Þ; Uðs;!ÞÞds;

P a:e:!; 8t 2 R

Moreover, the measure 

is invariant under this

flow.

We point out that uniqueness is an open problem

also for d = 2. But already in the classical analysis of

the Euler equations in a bounded domain, unique-

ness for initial velocity of finite energy is not known.

Working with the measure 

is even worse,

especially when d = 3, because its support is a larger

space within which more irregular velocity vectors

live. The more irregular the spaces where the flow

lives, the more difficult is to handle the nonlinear

term B(u, u).

On the other hand, for d = 1, the mathematical

analysis is much easier. For instance, it can be

proved (see Robert (2003)) that the one-dimensional

inviscid Burgers equation on the line



¼ 0 ½17

has intrinsic invariant statistical solution, given by a

class of Le´vy’s processes with negative jumps.

The Stochastic Navier–Stokes Equations

The Navier–Stokes equations describe advection

with velocity u and diffusion with kinematic

viscosity >0(see Viscous Incompressible Fluids:

Mathematical Theory)

 u þðu rÞu þrp ¼ f

ru ¼ 0

u ¼ 0on@D

in D ½18

where  is the Laplace operator. Nonslip boundary

conditions are assumed. Although the Euler equa-

tions [9] are formally obtained from [18] by setting

 = 0, the presence of the second-order operator

 makes the analysis needed to prove the

existence, uniqueness, and regularity of solutions

easier than for the Euler equations. However, at

variance with the Euler equations, the Navier–

Stokes equations do not possess invariants, since

the viscosity dissipates energy. Hence, it is difficult

to find explicit expressions of invariant measures for

the deterministic Navier–Stokes equations, except

the trivial invariant measures concentrated on a

stationary solution. However, as soon as a stochastic

force is introduced in these equations, it is possible

to have nontrivial invariant measures. It is impos-

sible to review here the wide liter ature concerning

the stochastic Navier–Stokes equations and we

confine ourselves to make some remarks. Most

results are concerned with proving the existence

and/or uniqueness of an invariant measure , with-

out giving an explicit representation, apart some

attempts like Gallavotti (2002) , where a formal

representation of stationary distributions is given in

terms of functional integrals. Some properties of the

not explicit invariant measures are given like, for

instance, estimates of moments, exponential conver-

gence of the statistical solution for large time.

Stochastic forces can enter in the Navier–Stokes

equations in different ways. We can con sider

randomness in the forci ng term, so that the force f

in [18] has a deterministic component which

represents its mean varying slowly and a stochastic

one, which accounts small fluctuations around the

mean and varying very rapidl y. Alternatively, since

the molecules are not rigidly connected to one

another in the fluid, they are subjected to fluctua-

tions. A complete descript ion of fluctuations relating

the microscopic and macroscopic motion is not

achieved at present. However, we shall introduce

some models for which rigorous mathematical

results can be proved.

The first part of this section concerns the Navier–

Stokes equations with noise n:

 u þðu rÞu þrp ¼ n

ru ¼ 0

½19

for which invariant measures exist, one of which can

be ergodic provided that the noise is suitably chosen.

In the second part, a Navier–Stokes-type stochastic

system is described, which has irregular solutions, as

expected in turbulence.

Let us introduce the stochastic Navier–Stokes

equations with time white noise. The first equation

in [19] is an Itoˆ equation:

u þ½u þðu rÞu þrp¼@

w ½20

Here w = w

ð1Þ

, ..., w

ðdÞ

is a Brownian motion, that

is, its time derivative n = @w=@t is a Gaussian

Stochastic Hydrodynamics 75

stochastic field with zero mean and correlation

function given by

E½n

ðjÞ

ðt; xÞn

ðkÞ

ðt

; x

Þ

¼ 

qðx  x

Þðt  t

Þ½21

for j, k = 1, ..., d.

We shall use the differential form for the Itoˆ

equation [20] always understood in the integral

form

uðtÞuð0Þþ

½uðsÞþðuð sÞrÞuðsÞ

þrpðsÞds ¼ wðtÞ½22

Modeling perturbations by a white noise process

represents the first step to understand how a random

perturbation acts in the mathematical equations,

rather than a good physical or numerical model. The

first results are in a paper by Bensoussan and

Temam (1973).

Obviously, the regularity of the solutions depends

on the spatial covariance q of the noise.

Let us consider the following cases.



q = : the noise is white also in space.

An invariant measure is known explicitly. Indeed,

assume periodic boundary conditions on the square

(d = 2) or the cube (d = 3) D, which makes the

spatial domain a torus. In this case, the Euler and

Navier–Stokes equations are set in the same func-

tional spaces. The generator of the stochastic

Navier–Stokes equations [20] corresponds to the

sum of the generator of the Euler equations [9] and

of the stochastic Stokes equations

u ¼½u rpþ@

ru ¼ 0

½23

Since the first equation in [23] is linear in the

unknown velocity u, the Stokes system has a unique

invariant measure which is a centered Gaussian

measure. In particular, when the noise is a space-

time white noise and d = 2, this is the invariant

measure [14] of the enstrophy:



ð0Þ;ð2Þ

ðduÞ¼

2SðuÞ

On a bidimensional torus, it is proved that this

measure is not only infinitesimally invariant, but

also global ly invariant for a unique flow [20]

defined for 

(0), (2)

-a.e. initial velocity. We recall

that initial velocities of finite energy are negligible

with respect to the measure 

(0), (2)



q more regular than above, that is, the noise is

colored in space.

As soon as the forcing term is more regular in space,

the Navier–Stokes system has a solution of finite

energy. These are solutions close to those of the

deterministic equation. Techniques similar to those

used to prove the existence and/or uniqueness of

solutions for the deterministic equations work also

in the stochastic case with an additive noise (or even

a multiplicative noise) to get weak or strong

solutions. Global existence in the space H

is proved

for d = 2, 3 and uniqueness only for d = 2, as is the

case for the deterministic Navier–Stokes equations.

The interesting feature is that by adding a noise

which acts on all the components with respect to a

Hilbert basis (or at least on many components), the

stochastic Navier–Stokes system has a unique

invariant measure, which is ergodi c. This is proved

for the spatial dimension d = 2. By means of the

Krylov–Bogoliubov’s method, existence of at least

an invariant measure is proved by compactness of a

family of averaged meas ures; the limit measures are

stationary measures. But, when many modes are

perturbed by a noise, there is a mixing effect on the

dynamics, avoiding existence of many stationary

measures. For the spatial dimension d = 2, the best

result in this context is in Hairer and Mattingly

(2004), where the noise acts on very few modes. For

the spatial dimension d = 3, the result in Da Prato

and Debussche (2003) shows the existence of an

invariant measure; even if there is no uniqueness of

the solutions (as in the deterministic case), by a

selection principle, they construct a transition

semigroup, which has a unique invariant measure,

ergodic and strongly mixing.

Mathematical proofs are given for very different

noises. (The reader is urged to consult, among the

others, the papers by E, Mattingly and Sinai; Flandoli

and Maslowski; Mikulevicius and Rozovskii; Vishik

and Fursikov. The latter authors study also statistical

solutions in two and three dimensions. For a kick noise

n =

(t  k)q

(x)inequations[19], there are results

for d = 2 by Bricmont, Kupiainen and Lefevere; Kuksin

and Shirikyan.)

We conclude that, as far as invariant measures

and their ergodicity are concerned, the stochastic

Navier–Stokes equations have richer results than the

deterministic Navier–Stokes equations. It is appeal-

ing to investigate the limit as the intensity of the

noise goes to zero, so as to recover the deterministic

equation. Now, think of equation [19] with a noise

"n, for n fixed and " !0. Due to the sensitive

dependence on initial conditions, even a small noise

may have important effects on the dynamics. A

conjecture by Kolmogorov is that the unique

invariant measure 

tends, when " !0, to a specific

measure, the so-called Kolmogorov measure, which

76 Stochastic Hydrodynamics

would enter into the ergodic principle. This is a

difficult problem, not yet solved.

We also mention the analysis of the inviscid limit.

Kuksin (2004) showed that the solution u



of the

two-dimensional stochastic Navier–Stokes equations

u þ u rðÞu þrp ¼

ﬃﬃﬃ



n; 0 < 1 ½24

on the torus converg es in distribution to a stationary

solution of the Euler equations. Here n is a random

force white in time and smooth in space. More

precisely, for each subsequence u



lim



lim

T!1



ðT þ tÞ¼UðtÞ½25

and almost every trajectory of the nontrivial limit

process U solves the Euler equations [9] without the

forcing term. Moreover, the process U keeps

memory of some features of the noise force n, since

the mean values of the enstrophy and of the energy

of U depend on the noise n.

We now present the second part on stochastic

models for viscou s fluids. In his 1884 paper,

Reynolds introduced the decomposition of turbulent

flow into mean and fluctuating flows. The equations

obtained are difficult to study. We shall show now a

tractable model for a one-dimensional problem

(d = 1) with a suitable model of fluctuations.

Decompose the velocity field into the sum of a

mean flow

u and a fluctuation 

u ¼

u þ 

The fluctuation is assumed to be highly irregular; it

is reasonable to mod el it by a stochastic process. If

we choose

 ¼ b

where b is a given velocity field and dw=dt is white

noise, then the motion of the fluid is governed by a

stochastic equation of Itoˆ type. Indeed, the Navier–

Stokes equations are ba lance equations of linear

momentum:

¼ u rp ½26

where Du=Dt is the material time derivative along

the trajecto ry of a particle which is at time t in

position x(t) movi ng with velocity u (so

u(x(t )) = (dx=dt)(t)):

uðt; xðtÞÞ ¼

þðu rÞu ½27

According to the mathematical model for the

fluctuation, we have

dxðtÞ¼

uðt; xðtÞÞdt þ bðxðtÞÞdwðtÞ½28

Therefore, D

u is computed by means of Itoˆ’s

formula

uðt; xðtÞÞ ¼

dt þ

k¼1

ðtÞ

k;s¼1

dt ½29

This leads to the stochastic Navier–Stokes-type

equations (we neglect the overline symbol)

u þ½u þðu rÞu þrp þ

Qud t

¼ðb rÞu dwðtÞ

ru ¼ 0

½30

where Q is the second-order differential operator

given by the last term in [29].

Rigorous math ematical results for the above

equations have been proved for the one-dimensional

case, that is, the Bur gers equations on the line.

Given an initial velocity of finite energy u

2 H

there exists a unique solution u 2 C([0, T]; H

) \

(0, T; H

) (P-a.s.). But it can be shown that for a

more regular initial velocity there is no higher

regularity of the solution of eqn [30],ifb 6¼ 0. This

means that these stochastic Burgers equations

cannot have too regular nontrivial solutions, as

expected in turbulent motion.

Statistics of Vortices and Bidimensional

Turbulence

Onsager (1949) proposed to investigate bidimen-

sional turbulent flows, extending in a rigorous way

to hydrodynamics the statistical mechanics approach

of Boltzmann. If we are interested in flows of finite

energy, the results of the section ‘‘The Euler

equations’’ provide no answer to the problem.

Another way to proceed is by approximating the

Euler equations in a suitable way. Actually, in a

two-dimensional turbulent flow, there appears a

large-scale organization leading to coherent struc-

tures. These are hy drodynamical vortices, whose

dynamics is governed by the Euler equations.

Onsager suggested to approximate the continuous

Euler equations by a great (but finite) number of

point vortices. This leads to a finite-dimensional

Hamiltonian system, to which the methods of

statistical mechanic s can be successfully applied. Of

course, the crucial point is to pass to the limit, to

Stochastic Hydrodynamics 77

recover the continuous system. But there are many

different ways to approximate a continuous vorticity

by a cloud of point vortices and different approx-

imations may lead to very different statistical

equilibrium states.

We present here the approach presented in Lions

(1997). To get an idea of a completely different

approximation, see, for example, Robert (2003).

Let D be a bounded open smooth simply

connected subset of R

. Then there exists a function

(the stream function) such that u = r

and

= 0. Given the velocity u, we recover the stream

function by means of the vorticity ! = curl u =  ,

so (x) =

g(x, y)!(y)dy (here g is the Green’s

function of the Laplacian  and x, y are points in

D). The Euler equations can be written as

þ u r! ¼ 0

! ¼ curl u

½31

Consider now a solution given by vorticity concen-

trated in a finite number N of points:

! ¼

i¼1





ðtÞ

½32

Here the vortex intensities 

are real values and

(t) are distinct points in D for i = 1, ..., N.

According to the Euler equations, these points evolve

as follows (see also Marchioro and Pulvirenti (1994)):

ðtÞ¼r

l¼1; l6¼j



gðx

ðtÞ; x

ðtÞÞ

þ 

gðx

Þ; j ¼ 1; ...; N ½33

where

g is related to the Green’s function g. This is a

Hamiltonian system in D

. Hereafter, we shall

suppose that the vortex intensities are the same

(

=  8i), so that the Hamiltonian is

Hðx

; ...; x

Þ¼

l; j¼1; l6¼j

gðx

; x

Þþ

j¼1

gðx

½34

By means of H, we define the canonical Gibbs

measure



ðdx

dx

ZðNÞ



Hðx

;...; x

dx

½35

where Z(N) is the partition function. If Z(N) < 1,

then 

is a well-defined probability measure on D

and, by construction, it is an invariant measure for

system [33]. We can prove that Z(N) is finite for

 2 (8=N,4), so that it is natural to choose as

a scaling

N = . Hence,



ðdx

dx

ZðNÞ

ð=NÞH

dx

½36

is consid ered for 8< 0, or >0 with

N >=4.

Bearing in mind the Onsager approach to approx-

imate the turbulent Euler motion by means of point

vortices, we are interested in the limit as N goes to

þ1, for  fixed in (8, þ1). It turns out that,

when the number of point vortices becomes very

large, their statistical behavior corresponds to a very

large number of independent particles moving in a

mean force field that they create.

More precisely, consider  = 1=N,

 = . The

empirical measure

i¼1



ðtÞ

describing the vorticity, weakly converges to a

probability density  and each correlation function



ðx

; ; x

Þ¼

jþ1



ZðNÞ

ð=NÞH

for j ¼ 1; ...; N  1 ½37

weakly converges to 

i = 1

 =

i = 1

(x

The equation satisfied by , also called the mean-

field equation, is

ðxÞ¼

UðxÞ

UðyÞ

;

with UðxÞ¼

gðx; yÞðyÞdy ½ 38

The relation between U and  can also be written as

U =  in D, U = 0on@D. We point out that

u = r

U is a stationary solution of the Euler

equations. Indeed, ! = U =  and  is a function

of U, let us say  = F (U). This gives that

r! = rUF

(U) and thus the term u r! in the

Euler equation [31] vanishes.

It can be proved that there exists a solut ion of the

mean-field equation when   0 or when <0 and

D is simply connected. Uniqueness is known in some

cases, for instance, when D is a bounded open

smooth simply connected domain and the velocity is

assumed tangent to the boundary.

There are numerical evidences of this approxima-

tion approach (see references in Lions (1997)

referring to the periodic case). They show that for

78 Stochastic Hydrodynamics