Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Maxwell equations in a continuum (dielectric)

medium:



@ð"EÞ

þ curl





¼ j

divð"EÞ¼

þ curl E ¼ 0

div B ¼ 0

½12

This system is supplied with initial conditions on the

fields B and E. On the other hand, boundary

conditions migh t be necessary when the equations

are restricted to a bounded domain. The latter

question, quite delicate, is postponed until next

section.

The MHD Coupling

For coupling systems [5] and [12], a threefold task is

in order.

On the one hand, the body force term in [5] needs

to be made precise, and this is completed by setting

f ¼j B þ f

ext

½13

The first term in the right-hand side is the Lorentz

force, consequence of the electric current j running

within the magnetic field B, a force that influences

the motion, along the velocity field u, of the

particles of the conducting fluid. The second term

is due to possible external forces. A typical case for

such forces is that of the gravity forces

ext

¼ g ½14

On the other hand, in order to be a mathemati-

cally closed system, the Maxwell system [12] needs

to be complemented by Ohm’s law, another type of

constitutive relation, like [10], that now relates the

current density j with the other fields. When dealing

with MHD phenomena, Ohm’s law most often

reads in the form

j ¼ E þ u BðÞ ½15

where  denotes the electric conductivity of the

fluid. The second term of [15] explicitly accounts for

the deviation of the lines of electric current by the

hydrodynamics flow. In some oversimplified situa-

tions, it can be neglected, leading to Ohm’s law in

the more usual form j = E, that is also valid for

solid media. Most of the times the term u B

contains cruci al information, and thus is not

neglected.

System [5]–[12] now reads



 u ru  u þrp ¼ j B þ f

ext

div u ¼ 0



@ð"EÞ

þ curl





¼ j

div E ¼



þ curl E ¼ 0

div B ¼ 0

j ¼ ðE þ u B Þ

½16

A third task is then in order.

Apart from the constitutive laws [10] and Ohm’s

law [15], the specificity of the Maxwell equations for

conducting fluids, as opposed to the same equations

written, for example, in the vacuum, resides in the

possible need for supplying the system with ad hoc

boundary conditions. Indeed, in their most general

form, the Maxwell equations are valid in the whole

physical space R

. On the other hand, as the goal here

is to simulate an MHD fluid that most often occupies

only a bounde d domain  in R

, there is the need to

adequately define the simulation domain.

A first possibility is to set the Max well equations

in the whole space, while solving the hydrodynamics

equation on the domain  occupied by the fluid.

Regarding only the Maxwell equations [12], this

seems to be the method of choice. But then there is

the need for an extension of Ohm’s law [15] outside

the fluid domain. Notice indeed that u appears in

[15]. In addition to this, the fact that the physical

confinement device for the fluid is then embedded in

the domain where the Maxwell equations are set

may be the source of various difficulties, as such a

device is often delicate to model and treat. There-

fore, alternative tracks may be follow ed.

A second possibility is to restrict the Maxwell

equation to a bounded domain. In turn, this option

divides in two: taking as the domain for the Max well

equations that occupied by the fluid, or choosing a

domain larger than . We cannot discuss this choice

without loss of generality, and refer the reader to the

literature (see e.g., Gerbeau et al. (2005 )). In either

situation, boundary con ditions are needed. We only

consider the former for the sake of brevity.

A standard choice for the bounda ry conditions for

[12] is the following:

E n

@

¼ k n

@

B  n

@

¼ q

½17

Magnetohydrodynamics 377

where k and q, respectively, are given vector and

scalar functions on the boundary.

A fact that needs to be emphasized is that it is not

so easy to design accurate boundary conditions, that

is, evaluations of k or g, especially because accurate

experimental measures of magnetic quantities are

often delicate to obtain, especially in industrial

environments.

A Commonly Used Simplified MHD Coupling

For the terrestrial MHD applications that are the

focus of the present article, a commonly used

assumption is to neglect the first term @("E)=@t,

often called the displacement current, in the

Maxwell–Ampe`re equation [6], that is the first

equation of [12] or the third of [16] above.

Then system [16] can be reorganized, eliminating

E and j, and leaving aside the Maxwell–Faraday

equation [8], Ohm ’s law [15],andtheMaxwell–

Coulomb equation [7]. The latter equations

amount to defining, respectively, E from B, j

from E and B ,and

from E. One is left with

the following system with the triple of unknown

fields (u, p, B)



u ru  u þrp ¼



curl B B þ f

ext

div u ¼ 0 ½18

þ curl



curl





¼ curl u BðÞ

div B ¼ 0

Correspondingly, the initial conditions are now

only on the pair (u, B ). Regarding the boundary

conditions on B, they can be derived from [17]

using, for example, a homogeneous Dirichlet bound-

ary condition on u:

curl B n

@

k n

@

B  n

@

¼ q

½19

Other simp lifications of system [16] can be

adopted, such as steady-state approximations. In

particular, it is often considered that electromagnetic

phenomena have characteristic times that are so

short in compar ison with the characteristic time

of hydrodynamics phenomena that the Maxwell

equations in their stationary form may be coupled to

the time-dependent hydrodynamics equations, such

as [5]. We refer to the ‘‘Furth er readi ng’’ section

for further information along these lines (see e.g.,

Gerbeau et al. (2005)).

The Mathematical Nature of the

Equations

With a view to unde rstand the mathematical

nature of systems [16] and [18], we first briefly

recall some mathematical facts concerning hydro-

dynamics, before focusing on the coupling with

electromagnetics.

Regarding the incompressible Navier–Stokes

equation, we recall that the state of the art of the

mathematical knowledge heavily depends on the

dimension of the ambient space. In dimension 2,

solutions are unique and regular (they are said to be

strong), for regular enough data of course. Unfortu-

nately, as the focus is here on MHD and electro-

magnetism is fundamentally a three-di mensional

phenomenon, only the three-dimensional case for

the Navier–Stokes equation is relevant. Now, in the

context of the Navier–Stokes equations alone, only

the existence of weak solutions for large times, and

the existence and uniqueness of strong solutions for

small times are known. Whether or not there exists a

unique strong solution for all time (of course again

for sufficiently regular data) is an open problem, of

outstanding difficulty, (see Temam 1995).

In the coupled setting examined here, there is no

reason to expect a better situation. At best, one may

hope for the same situation as that for the

uncoupled case (Navier–Stokes equations alone).

Regarding the existence and uniqueness of solutions,

a commonly used strategy is that of regularization:

the Cauchy problem is studied for regularized data,

and then one passes to the limit in the regulariza-

tion. In this latter step, the linear terms cause no

difficulty, since they pass to the limit only using

weak convergence. On the other hand, the main

concern is always the treatment of the nonlinear

terms, which require strong convergence. Here, for

the Navier–Stokes equation in the MHD setting, the

additional difficulty stems from the presence of

the nonlinear term j B on the right-hand side. The

mathematical treatment of this nonlinear term calls

for a compactness argument, which in turn requires

obtaining some information on the fields j and B,

and their derivatives, from the Maxwell equations.

In this respect, the situation is radically different for

system [16] and for system [18]. Likewise, these

two systems behave differently regarding the other

nonlinear term of electromagnetic nature, namely

u B in Ohm’s law, or curl(u B) on the right-

hand side of the equation in B, respectively.

The Hyperbolic Variant

Due to the presence of the Maxwell equations [12]

in their general form, that is a hy perbolic form,

378 Magnetohydrodynamics

system [16] is indeed very difficult, from the

standpoint of mathematical analysis.

In order to realize this, it suffices to recall that the

first step in the proof of the existence of solution to

such a system of equations is to write down an

a priori energy estimate. It is a simple manipulation

on [16] to show that, formally, a solution to [16]

satisfies



juj

þ 



jruj



ðj BÞu ½20

multiplying the Navier–Stokes equation by u and

integrating over the domain , while, on the other

hand,



"jEj





jBj

¼



j  E ½21

multiplying the Maxwell–Ampe` re equation by E,

the Maxwell–Faraday equation by (1=)B, integrat-

ing over , and summing up the two. Next, the

right-hand side of [21] can be modified, accounting

for Ohm’s law:



"jEj





jBj

¼





jj j





ðj BÞu ½22

Summing up [20] and [22] yields the energy

estimate:



juj

þ "jEj



jBj







jjj

þ 



jruj

¼ 0 ½23

Notice that, in the above, we set the external forces

and all boundary conditions to zero, for the sake of

simplicity.

Estimate [23] clearly indicates that we dispose of

([0, T], L

()) bounds on the vector fields E and

B together with an L

([0, T]  ) bound on the

current j, and with the (classical) L

([0, T],

()) \L

([0, T], H

()) bounds on the velocity

u. In addition, div B and, when assuming 

bounded, div E are bounded in L

([0, T]  ).

Unfortunately, these bounds do not allow for

passing to the limit in the nonlinear term j B on

the right-hand side of the Navier–Stokes equation.

In addition, there seems to be no way of deriving

further e nergy estimates on system [16] that would

provide with more a priori regularity on the fields

E, B,andj.Todate,system[16] presents an

unsolved mathematical difficulty.

The Parabolic Variant

On the other hand, system [18] is radically different

in mathematical nature, because the Maxwell

equations then reduce to a parabolic-type equation.

The same manipulations as above, in ord er to

establish a priori estimates on the solution of [18],

now lead to



juj



jBj







curl







þ



jruj

¼ 0 ½24

which, together with the divergence-free constraint

on B, yields L

([0, T], L

()) \L

([0, T], H

())

bounds on both the velocity u and the magnetic

field B. These bounds now allow for passing to the

limit in the terms curl B B and curl(u B) on the

right-hand side of the equations. This being estab-

lished, the rest of the mathematical analysis is

straightforward, and a theorem of existence and

uniqueness of solutions can be pro ved. Like in the

case of the Navier–Stokes equations alone, we have

(in dimension 3) the existence of a global-in-time

weak solution (i.e., for any T, u and B both

([0, T], L

()) \L

([0, T], H

()) satisfying the

divergence-free constraint). No uniqueness of this

weak solution is known. On the other hand, for

sufficiently regular data, we have the existence of a

local-in-time strong solution (i.e., for T sufficiently

small, u and B both L

([0, T], H

()) \L

([0, T],

()), and uniqueness of this strong solution in

the class of weak solutions as long as it exists. We

refer to Sermange and Temam, (1983) and Gerbeau

et al. (2005).

At this stage, it i s to be remarked that there i s a

formal simil arity, at first sight at least, between

the parabolic form of the Maxwell equations,

namely

þ curl curl B ¼ curl h

div B ¼ 0

½25

and the incompressible Navier–Stokes equation [5].

Note that indeed the curl operator in the first

equation of [25] can be replaced by (minus) the

Laplacian operator , since div B = 0. Actually,

this formal similarity cannot be translated into

mathematical arguments, simply because there is

no press ure in [25]. In other terms, the divergence-

free constraint div B = 0 simply propagates in time in

[25] (note that the right-hand side curl h is also

Magnetohydrodynamics 379

divergence-free by construction), while on the other

hand div u = 0 is enforced as a constraint in [5], the

pressure playing the role of a Lagrange multiplier

that adjusts itself in time in order to allow for u to

be divergence-free.

Of course, as in the purely hydrodynamics case,

much more can be said on the equati ons than simply

establishing the existence and uniqueness of solu-

tions. For instance, the long time limit of the

solutions can be studied, etc.... For this and other

issues, we refer to the ‘‘Furt her readin g’’ section

(Duvaut and Lions 1972a, b, Sermange and Temam

1983, Gerbeau et al. 2005).

Numerical Issues

We concentrate again on system [18]. It is illustra-

tive to mention that this system, when written in

nondimensional variables, reads

þ u ru 

u þrp ¼ S curl B B þ f

ext

div u ¼ 0

mag

curl ðcurl BÞ¼curlðu BÞ

div B ¼ 0

where S is the coupling parameter, Re is the

(hydrodynamic) Reynolds number, and Re

mag

denotes the magnetic Reynolds number.

As expected, the numerical simulation of a system

such as [18] superposes the difficulties of the

hydrodynamics simulation of incompressible viscous

fluids, and those faced when simula ting the para-

bolic form of the Maxwell equations. Therefore, the

goal is to efficiently combine the techniques

employed to overcome either of them.

For incompressible fluid mechanics, the method

of choice is the finite-element method for the

discretization of differential ope rators i n space . A

typical discretization of eqn [5], called the ‘‘mixed’’

finite-element method, makes use of a pair of finite

elements, one for the velocity, and one for the

pressure. Other possibilities exist, that amount

more or less in eliminating one unknown in a

first stage and calculating the second one as a

postprocessing task. The mixed formulation in the

pair of unknowns (u, p) is however the most

employed method to date, at least in the present

setting. The finite-element space for the velocity is

taken richer than that for the pressure: a possibility

is, for example, to take the degree of the finite

element for the velocity equal to the degree of the

finite element for the pressure plus one. The

heuristics for this is the fact that the velocity is

derived twice in [5] while the pressure is only

derived once. Of course, a mathematical ground

for this is available, and a key issue is the ‘‘inf–

sup’’ condition (also compatibility condition, or

stability condition) that dictates the possible choice

for finite-elements pairs, so that problem [5] is well

posed at the discrete level. Typically, Q2finite

elements for the velocity can be combined with

(continuous) Q1 finite elements for the pressure.

An alternative choice is to ignore the inf–sup

condition, adopting, for example, Q1 finite ele-

ments for both fields u and p, but this requires for

a so-cal led stabilized formulation of [5] at the

discrete level. The ‘‘Further reading’’ s ecti on

provides details on the broad variety of techniques

available in the field: Quarteroni and Valli (1997),

Gerbeau et al. (2005).

On the other hand, the parabolic equation on B in

[18] may be disc retized with the same finite elements

as those used for the velocity. The enforcement of

the divergence constraint div B = 0 at the discrete

level deserves some attention. Recall indeed that

at the continuous level the divergence-free constr aint

is spontaneously propagated by the equation. At

the discrete level, a crucial role in this respect is

played by the weak formulation of the parabolic

equation and an ad hoc account for the boundary

condition [17].

For the sake of compl eteness, let us mention that

an alternative strategy to the use of the finite

elements that have been mentioned above (and that

are called Lagrangian finite elements), is to use

‘‘edge elements.’’ In some sense, the use of such

elements simplifies the treatment of the boundary

conditions [17], since they are very well adapted to

their mathematical nature.

Note also that, in the vein of what is done for

purely hydrodynamics flow simulations, stabilized

finite-elements techniques have been developed for

the MHD system [18], that allow for a discretization

of the three unknown fields (u, p, B) over the same

finite elements, for example, Q1.

When coupling the two discrete formulations for

simulating the whole system [18], two main strate-

gies can be adopted: one can either treat each of the

two equations separately, independently describing

the propagation of u and B

forward in time, or one

can address directly the coupled system of equa-

tions, describing the propagation of u and B in

parallel.

The first option aims in particular at obtaining in

the end small algebraic systems. An instance of such

380 Magnetohydrodynamics

a segregated algorithm reads, formally and setting

all constants to unity for simplicity,

nþ1

 u

t

þ u

ru

nþ1

 u

nþ1

þrp

nþ1

¼ curl B

B

þ f

ext

div u

nþ1

¼0

nþ1

 B

t

þcurl curlB

nþ1

¼ curl u

B

nþ1



divB

nþ1

¼0

½26

At each time step, the two independent subsystems

are solved, providing with u

nþ1

and B

nþ1

for the

next time step. The difficulty is that it is not

possible, with such segregated algorithms, to repro-

duce the energy estimate [24] at the discrete level.

Note that, at the continuous level, the estimate [24]

is based upon a proper cancelation of the term



(j B)u present on the two right-hand sides.

Such a cancelation basically stems for a nonli near

interplay that cannot be present in a segregated

iteration. Consequently, some spurious energy is

created in the system simply by an inadequate

iteration between the two equations. More precisely,

the scheme obtained is at best only conditionally

stable, that is, stable for small enough time steps, a

condition that might be prohibitive when it is

needed to simulate the MHD coupling over large

times.

On the other ha nd, the other option consists in

attacking the full system [18] directly:

nþ1

 u

t

þ u

ru

nþ1

 u

nþ1

þrp

nþ1

¼curl B

nþ1

B

þ f

ext

div u

nþ1

¼0

nþ1

 B

t

þ curl curlB

nþ1

¼curl u

nþ1

B



div B

nþ1

¼0

½27

Note that B

nþ1

is present in the equation yielding

nþ1

, while conversely u

nþ1

is present in that

yielding B

nþ1

. Then the coupled system admi ts at

the discrete level an energy estimate analogous to

the energy estimate [24], and the scheme is much

more stable than the previous one, and even

unconditionally stable. The price to pay is that the

system is, at the algebraic level, of very large size.

Being sparse, it may however be treated, for

example, via a GMRES-type iterative solver.

Let us make a final remark on these numerical

issues. In the whole generality, the numerical

simulation of viscous fluids raises the question of

large Reynolds numbers, that is, the question of the

difficulties encountered in the numerical approxi-

mation for viscosities  small with respect to the

other dimensionalized parameters of the problem

(density, velocity, and dimension of the domain).

For such small viscosities, the flow becomes

turbulent rather than laminar, and the broad

range of length and energy scales in the flow turns

out to be too difficult to capture numerically. A

commonly used technique that is resorted to in

such difficult cases is the turbulence modeling.

Schematically, an averaged, or homogenized, model

is derived on t he basis of the Navier–Stokes

equation, with the help of simplifying hypotheses,

for example, in the form of closure relations. The

quality of the simulation of the averaged model,

and its relation to the true flow, heavily depends on

these s implifying assumptions, which are in turn

based upon a very deep understanding on the

various physical phenomena at play. In the context

of MHD flows, the situation is not clear, regarding

such assumptions. It seems that there are no well-

established models for turbulent MHD to date, at

least from a rigorous viewpoint. In the absence of

those, only a direct simulation of the Navier–Stokes

equation seems possible.

The Industrial Production of Aluminium

A prototypical example of an application of MHD

to the industrial context is the production of

aluminum in electrolysis cells. The numerical simu-

lation of the process involves the simulation of the

evolution of two layers of nonmiscible incomp res-

sible viscous fluids, separated by an interface, and

covered by a free surface. A schematic description of

an industrial cell indeed is the following. An electric

current of 10

A, or more, runs through two

horizontal layers of conducting fluids: a bath of

aluminum oxide above, and a layer of liquid

aluminum below. The aluminum is produced by

the reduction of the aluminum oxide, a reaction that

only occurs at a temperature where aluminum is

liquid. The high magnetic field induced by such a

huge current produces in turn high Lorentz forces

that influence the motion of either fluid . A key issue

in the modeling, as well as in the technological

control of the cell, is to understand the motion of

the interface separating the two fluids. In a rough

picture, this interface may be seen as a mobile

Magnetohydrodynamics 381

cathode, moving below a fixed anode. The equa-

tions describing the interior of the cell are basically

of the type [18], with an important modification

though: one needs to account for the presence of

two fluids. They read:

@ðuÞ

þ divðu uÞdivððru þðruÞ

ÞÞ

¼rp þ g þ



curl B B

divu ¼0

@

þ divð uÞ¼0

þ curl



curlB



¼curlðu BÞ

divB ¼0

½28

where g denotes the gravity field, we recall, and are

supplied with the boundary conditions

u ¼0



curlB n

@

¼k n

@

B:n

@

¼q

½29

As opposed to [18], the density  in [28] is no longer

the constant

, but is only piecewise constant, that is,

constant in each (moving) subdomain occupied by

each fluid. Likewise, the viscosity , and the con-

ductivity  are taken constant in each fluid, but with

different values from one fluid to the other. While the

density and the viscosity are only slightly different, the

conductivity varies from many orders of magnitude, a

discrepancy which ends up in some numerical stiffness

of the equations. On the other hand, the permeability

 can be considered as constant throughout the

domain, within a good level of approximation.

Mathematically, system [28] is an order of magni-

tude more difficult than [18]. We refer to Lions

(1996) and Gerbeau and LeBris (1997) for some

mathematical ingredients. A first major difficulty

stems from the fact that the domain occupied by the

fluids is no longer fixed. Notice that this difficulty

already arises when simulating the MHD of one

conducting fluid with a free surface. A second major

difficulty is the discontinuity of the physical para-

meters at the interface, which causes a loss of

regularity at the interface for the solution fields. The

best result known to date is the existence of a global-

in-time weak solution to [28]. Both mathematical

difficulties above of course have significant numerical

counterparts. A notable issue in such a simulation is

how to handle the motion of the free interface, while

ensuring that each fluid remains of constant mass (or

volume) throughout the simulation. One of the most

efficient method in such a context, introduced three

decades ago, is the arbitrary-Lagrangian Eulerian

(ALE) method. We refer to Brackbill and Pracht

(1973) and Gerbeau et al. (2003a, b, 2005).

Apart from the direct numerical attack of system

[28], which carries significant analytical and geome-

trical nonlinearities, there is the possibility, in

particular in the industrial context, to derive a set

of linearized equations at the vicinity of some

equilibrium configuration of the system. This track

has been extensively followed in the past and

provides information that efficient ly complement

those provided by the much more satisfactory, but

also more costly, nonlinear approach.

See also: Compressible Flows: Mathematical Theory;

Computational Methods in General Relativity: The Theory;

Fluid Mechanics: Numerical Methods; Newtonian Fluids

and Thermohydraulics; Partial Differential Equations:

Some Examples; Stability of Flows; Symmetric

Hyperbolic Systems and Shock Waves; Topological Knot

Theory and Macroscopic Physics.

Further Reading

Bossavit A (1998) Computational Electromagnetism. Variational

Formulations, Complementarity, Edge elements. San Diego:

Academic Press.

Brackbill JU and Pracht WE (1973) An implicit, almost

lagrangian algorithm for magnetohydrodynamics. Journal of

Computational Physics 13: 455–4882.

Davidson PA (2001) An Introduction to Magnetohydrodynamics.

Cambridge: Cambridge University Press.

Descloux J, Flueck M, and Romerio MV (1991) Modelling for

instabilities in Hall He´roult cells: mathematical and numerical

aspects. Magnetohydrodyn. Process Metall. 107–110.

Duvaut G and Lions J-L (1972a) Les ine´quations en me´canique et

en physique. Paris: Dunod.

Duvaut G and Lions J-L (1972b) Ine´quations en thermoe´lasticite´

et magne´tohydrodynamique. Archives for Rational and

Mechanical Analysis 46: 241–279.

Gerbeau JF and LeBris C (1997) Existence of solution for a

density-dependent magnetohydrodynamic equation. Advances

in Differential Equations 2(3): 427–452.

Gerbeau JF, LeBris C, and Lelie` vre T (2003a) Modeling and

simulation of the industrial production of aluminium: the

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Malliavin Calculus

A B Cruzeiro, University of Lisbon, Lisbon, Portugal

Introduction

Malliavin calculus was initiated in 1976 with the

work by P Malliavin (1978) and is essentially an

infinite-dimensional differential calculus on the

Wiener space. Its initial goal was to give conditions

ensuring that the law of a random variable has a

density with respect to Lebesgue measure as well as

estimates for this density and its derivatives. When

the random variables are solutions of stochastic

differential equations (SDEs), these densities are heat

kernels and Malliavin used Ho¨ rmander-type

assumptions on the corresponding operators, thus

providing a probabili stic proof of a Ho¨ rmander-type

theorem for hypoelliptic operators.

The theory was much developed in the 1980s by

Stroock, Bismut, and Watanabe, among others (the

reader is referred to Nualart (1995) and Malliavin

(1997)). In recent years, Malliavin calculus had

great success in probabilistic numerical methods,

mainly in the field of stochastic finance (Malliavin

and Thalmaier 2005). However, the theory has also

been applied to other fie lds of mathematics and

physics, notably in statistical mechanics and statistical

hydrodynamics (see Stochastic Hydrodynamics). In

addition, one should remember that Wiener measure

can be viewed as an ‘‘imaginary time’’ (but well-

defined) counterpart of Feynman’s ‘‘measure’’ for

quantum systems. A stochastic calculus of variations

for Wiener functionals could not be irrelevant to the

path-integral approach to quantum theory.

Another field of application worth mentioning is

the study of representations of stochastic oscillatory

integrals with quadratic phase function and their

stationary phase estimation. For this, complexifica-

tion of the Wiener space must be properly defined

(Malliavin and Taniguchi (1997)).

In order to give a flavor of what Malliavin

calculus is all about, let us consider a second-order

differential operator in R

of the form

A¼

i;j¼1

i;j

with smooth bounded coefficients and such that

the matrix a is symmetric and non-negative, admit-

ting a square root . The corresponding Cauchy

value problem consists in finding a smooth solution

u(t, x)of

¼Au; uð0;:Þ¼ð:Þ½1

Then there exists a transition probability function

p(t, x, .) such that

uðt; xÞ¼

ðyÞpðt; x; dyÞ

When p(t, x,dy) = p(t , x, y)dy, the function p is the

heat kernel associated to the operator A, and

from eqn [1] one may deduce Focker–Planck’s

equation for p.

Since Kolmogorov we know that it is possible to

associate with such a second-order operator a stochas-

tic family of curves like a deterministic flow is

associated with a vector field. This stochastic family

is a Markov process, 

(t), which is adapted to the

increasing family P



,  2 [0, 1], of sigma-fields gener-

ated by the past events, that is, u() 2P



for every .

Itoˆ calculus allows us to write the SDE

satisfied by :

dðtÞ¼ð

ðtÞÞdWðtÞþbð

ðtÞÞdt;

ð0Þ¼x ½2

where W(t) stands for R

-valued Brownian motion

(see Stochastic Differential Equations). The n p is the

image of the Wiener measure  (the law of

Brownian motion), namely p(t,x,.)=  

1

(t)(.)

and we have the representation

uðt; xÞ¼E



ðð

ðtÞÞÞ

Malliavin Calculus 383

The following criterion for absolute continu ity of

measures in finite dimensions holds:

Lemma If  is a probability measure on R

and,

for every f 2 C

f d



 c

kf k

where c

, i = 1, ..., d, are constants, then  is abso-

lutely continuous with respect to Lebesgue measure.

Now one can think about Wiener measure as an

infinite (actually continuous) product of finite-

dimensional Gaussian measures. Considering the

toy model of the above-mentionned situation in

one dimension, we replace Wiener measure by

d(x) = (1=

ﬃﬃﬃﬃﬃﬃ

2

x

dx and look at the process at

a fixed time as a function g on R. In order to apply

the lemma and study the law of g, one would write

ðf

 gÞd ¼

ðf  gÞ

d

and then integrate by parts to obtain

(f  g) d.A

simple computation shows that (x ) = (g

þ xg

)=(g

)

and, in particular, that the nondegeneracy of the

derivative of g plays a role in the existence of the

density.

To work with functionals on the Wiener space,

one needs an infinite-dimensional calculus. Of

course, other (Gateaux, Fre´chet) calculi on infinite-

dimensional settings are already available but the

typical functionals we are dealing with, solutions of

SDEs, are not continuous with respect to the

underlying topology, nor even defined at every

point, but only almost everywhere. Malliavin calcu-

lus, as a Sobolev differential calculus, requires very

little regularity, given that there is no Sobolev

imbedding theory in infinite dimensions.

Differential Calculus on the

Wiener Space

We restrict ourselves to the classical Wiener space,

although the theory may be developed in abstract

Wiener spaces, in the sense of Gross. For a

description of this theory as well as of Segal’s

model developed in the 1950s for the needs of

quantum field theory, the reader is referred to

Malliavin (1997).

Let H be the Cameron–Martin space,

H= {h : [0, 1] ! R

such that

h is square integrable

and h(t) =

h()d}, which is a separable Hilbert

space with scalar product <h

, h

> =

().

()d. The classical Wiener measure will be

denoted by ; it is realized on the Banach space X

of continuous paths on the time interval [0,1]

starting from zero at time zero, a space where H is

densely imbedded. In finite dimensions, Lebesgue

measure can be characterized by its invariance under

the group of translations. In infinite dimensions

there is no Lebesgue measure and this invariance

must be replaced by quasi-invariance for transla-

tions of Wiener measures (Cameron–Martin admis-

sible shifts). We recall that, if h 2H, Cameron–

Martin theorem states that



ðFð! þ hÞÞ ¼E



Fð!Þexp

hðÞd!ðÞ





hðÞj

d



where d! denotes Itoˆ integration.

For a cylindrical ‘‘test’’ functional F(!) =

f (!(

), ..., !(

)), where f 2 C

)and0





 1, the derivative operator is

defined by



Fð!Þ¼

k¼1

<

f ð!ð

Þ; ...;!ð

ÞÞ ½3

This operator is closed in W

2, 1

(X; R), the comple-

tion of the space of cylindrical functionals with

respect to the Sobolev norm

jjFjj

2;1

¼ E



jjFjj

þ E





d

Define F to be H-differentiable at ! 2 X when there

exists a linear operator rF(!)suchthat,forallh 2H,

Fð! þ hÞFð!Þ¼hrFð!Þ; hiþoðjjhjj

as jjhjj ! 0

Then D



disintegrates the derivative in the sense that

Fð!ÞhrFð!Þ; hi¼



Fð!Þ:

hðÞd ½4

Higher (r)-order derivatives, as r-linear functionals,

can be considered as well in suitable Sobolev spaces.

Denote by  the L



adjoint of the operator r, that

is, for a process u : X !Hin the domain of , the

divergence (u) is characterized by



ðFðuÞÞ ¼ E





uðÞd



½5

For an elementary process u of the form

u() =

( ^ 

), where the F

are smooth ran-

dom variables and the sum is finite, the divergence is

ðu Þ¼

!ð

Þ



d

384 Malliavin Calculus

The characterization of the domain of  is delicate,

since both terms in this last expression are not

independently closable. It can be shown that

1, 2

(X; H) is in the domain of  and that the

following ‘‘energy’’ identity holds:



ððuÞÞ

¼ E



jjujj

þ E









d d

Notice that when u is adapted to P



,Cameron–

Martin–Girsanov theorem implies that the divergence

coincides with Itoˆ stochastic integral

u()d!()

and, in this adapted case, the last term of the energy

identity vanishes. We recover the well-known Itoˆ

isometry which is at the foundation of the construction

of this integral. When the process is not adapted, the

divergence turns out to coincide with a generalization

of Itoˆ integral, first defined by Skorohod.

The relation [5] is an integration-by-parts formula

with respect to the Wiener measure , one of the

basic ingredients of Malliavin calculus. This formula

is easily generalized when the base measure is

absolutely continuous with respect to .

Considering all functionals of the form

P(!) = Q(!(

), ..., !(

)) with Q a polynomial on

, the Wiener chaos of order n, C

, is defined as

= P

n1

,whereP

denote the polynomials

on X of degree n. The Wiener-chaos decomposition



(X) =

n = 0

holds. Denoting by 

the ortho-

gonal projection onto the chaos of order n,wehave



nþ1



; h

ðrF; h

hiÞ

The derivative D

corresponds to the annihilation

operator A(u) and the divergence (u) to the creation

operator A

(u) on bosonic Fock spaces.

An important result, known as the Clark–

Bismut–Ocone formula, states that any functional

F 2 W

1, 2

(X; R) can be represented as

F ¼ E



ðFÞþ



ðD



FÞd!ðÞ

where E



denotes the conditional expectation with

respect to the events prior to time  (or, for short,

the past P



of ).

The Ornstein–Uhlenbeck generator (or minus

number operator) is defined by LF = rF.On

cylindrical functionals F(!) = f (!(

), ..., !(

)), it

has the form

LFð!Þ¼

i;j



^ 

i;j

f ð!ð

Þ; ...;!ð

ÞÞ



!ð

Þ@

f ð!ð

Þ; ...;!ð

ÞÞ

where i,j denote multi-dimensional (d) indexes.

As a multiplicative operator on the Wiener- chaos

decomposition LF = 

n

F. It is the generator

of a positive -self-adjoint semigroup, the Ornstein–

Uhlenbeck semigroup, formally given by

F =

nt



F. Another familiar representation

of this semigroup is Mehler formula,

Fð!Þ¼E



F e

t

! þ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð1  e

2t





dðÞ



Considering the map X ! R

, ! ! (!(

), ...,

!(

)), the image of this operator is the Ornstein–

Uhlenbeck generator (corresponding to the Langevin

equation) on R

with Euclidean metric defined by

the matrix 

^ 

The fundamental theorem concerning existence of

the density laws of Wiener functionals is the following:

Theorem Let F be an R

-valued Wiener functional

such that F

and LF

belong to L



for every

i = 1, ..., d. If the covariance matrix

hrF

; rF

is almost surely invertible, then the law of F is

absolutely continuous with respect to the Lebesgue

measure on R

Under more regularity assumptions, smoothness

of the density is also derived. On the other hand, the

integrability assumptions on L can be replaced by

integrability of the second derivatives, due to Kre´e–

Meyer inequalities on the Wien er space.

We remark that, although equivalent, the initial

formulation (Malliavin 1978) of Malliavin calculus

was different, relying on the construction of the

two-parameter process associated to L and on its

properties. In the early 1980s, the theory was

elaborated, the main applications being the study

of heat kernels (cf., e.g., Stroock (1981), Ikeda and

Watanabe (1989), and Bismut (1984)). Starting from

an SDE [2], it is possible to apply these techniq ues to

obtain existence and smoothness of the transition

probability function p(t, x, y) if the vector fields



(@=@x

) together with their Lie brackets

generate the tangent space for ‘‘sufficientely many’’

(in terms of probability) paths. These results shed a

new light on Ho¨ rmander theorem for partial

differential equations.

Quasi-Sure Analysis

Quasi-sure analysis is a refinement of classical

probability theory and, generally speaking, replaces

the fact that, due to Sobolev imbedding theorems,

functions in finite dimensions belonging to Sobolev

Malliavin Calculus 385

classes are in fact smooth. We work in classical

probability up to sets of probability zero; in quasi-

sure analysis negligible sets are smaller and are those

of capacity zero. This is the class of sets which are

not charged by any measure of finite energy.

Under a nondegenera te map, Wie ner measure and

more general Gaussian measures may be disinte-

grated through a co-area formula. This principle,

developed by Malliavin and co-authors (cf.

Malliavin (1997) and references therein), implies

that a property which is true quasi-surely will also

hold true almost surely under conditioning by such

a map. One can use this principle to study

finer properties of SDEs. It was also used in

M P Malliavin and P Malliavin (1990) to transfer

properties from path to loop groups (see Measure on

Loop Spaces). A pinned Brownian motion, for

example, is well defined in quasi-sure analysis. It is

possible to treat anticipative problems using quasi-

sure analysis by solving the adapted problem after

restriction of the solution to the finite-codimensional

manifold which describes the anticipativity. These

methods have also been applied to the computation

of Lyapunov exponents of stochastic dynamical

systems (Imkeller 1998). With a geometry of finite-

codimensional manifolds of Wiener spaces well

established, it is reasonable to think about applica-

tions to cases where such submanifolds correspond to

level surfaces of invariant quantities for infinite-

dimensional dynamical systems (cf. Cipriano (1999)

for an example of such a situation in hydrodynamics).

The (p, r)-capacity of an open subset O of the

Wiener space is defined by

cap

p;r

ðOÞ¼inffkk

; 0; 1 -a:s: on Og

and, for a general set B, cap

p, r

(B) = inf {cap

p, r

(O):

B  O, O open}. A set is said to be slim if all its

(p, r)-capacities are zero. For  2 W

, the space of

functionals with every Malliavin derivative belong-

ing to all L



, there exists a redefinition of ,

denoted by 



, which is smooth and defined on the

complement of a slim set.

Following Airault and Malliavin (1988), let G 2

(X; R

) be of maximal rank and nondegenerate

in the sense that the inverse of

ðdet Þ

ð!Þ¼detðhr

ð!Þ; r

ð!ÞiÞ

belongs to W

. Then for every functional G 2 W

the measures   

1

and (G)  

1

are absolutely

continuous with respect to Lebesgue measure on R

and have C

Radon–Nikodym derivatives. If

ðÞ¼

d  

1

d

and 

ðÞ¼

dðGÞ

1

d

the function  ! 

()=() will be smooth in the

open set O= { : () > 0}.

For every  2O, it is possible to define (up to slim

sets) a submani fold of the Wiener space of codimen-

sion d, S



= (



)

1

(), as well as a measure 

satisfying





d

ð!Þ¼E

ð!Þ¼

ðGÞ¼



ðÞ

ðÞ

for every G 2 W

. This measure does not charge

slim sets.

The area measure @ on the submanifold S



defined by



d@¼ðÞ



ð!Þ detðhr

ð!Þ;

r

ð!ÞiÞ

1=2

d

ð!Þ

The following co-area formula on the Wiener

space

f ðð!ÞÞFð!Þðdet Þð!Þdð!Þ

f ðÞ





ð!Þd@ð!Þd

was proved in Airault and Malliavin (1988).

Calculus of Variations in a

Non-Euclidean Setting

Let M be a d-dimensional compact Riemannian

manifold with metric ds

i, j

i,j

. The

Laplace–Beltrami operator is expressed in the local

chart by



¼ g

i;j

 g

i;j



i;j

where 

i, j

are the Christoffel symbols associated

with the Levi-Civita connection. The corresponding

Brownian motion p

is locally expressed as a

solution of the SDE:

ðtÞ¼a

i;j

ðpðtÞÞdW

ðtÞ

j;k



j;k

ðpðtÞÞdt

with p(0) = m

2 M and where a =

ﬃﬃﬃ

. Its law on

the space of paths P(M) = {p : [0, 1] ! M, p contin-

uous, p(0) = m

} will be denoted by .

How can we develop differential calculus and

geometry on the space P(M)? An infinite-dimensional

local chart approach is delicate, due to the difficulty

of finding an atlas in which the changes of charts

preserve the measures. A possibility, developed in

Cruzeiro and Malliavin (1996),consistsinreplacing

the local chart approach by the Cartan-like metho-

dology of moving frames. The canonical moving

386 Malliavin Calculus