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

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the value at (i

, i

) of the associated function is

(i

, i

) = i

The proof that the SOS model with the boundary

condition () has a roughening transition is a highly

nontrivial result due to Fro¨ hlich and Spencer. When

 is small enough, the fluctuations of  are of order

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ln L

(in a cubic box of side L).

Moreover, other interface models, with additional

conditions on the allowed microscopic interfaces,

are exactly solva ble. The BCSOS (body-centered

SOS) model, introduced by van Beijeren, belongs to

this class. It is, in fact, the first model for which the

existence of a roughening transition has been

proved. More recently, also the TISOS (triangular

Ising SOS) model, introduced by Blo¨ te and Hilhorst

and further studied by Nienhuis and co-workers, has

been considered in this context.

The interested reader can find more information

and references, concerning the subject of this

section, in the review article by Abraham (1986).

Wetting Phenomena

Next we consider the Ising model over a plane

horizontal substrate (also called a wall) and study

the difference of surface tensions which governs the

wetting properties of this system.

We first describe the approach developed by

Fro¨ hli ch and Pfister (1987) and briefly report some

results of their study. We consider the model on the

semi-infinite lattice

¼fi 2 Z

: i

 0g½12

A magnetic field, K  0, is added on the boundary

sites, i

= 0, which describes the interaction with the

substrate, supposed to occupy the complementary

region LnL

We constrain the model in the finite box 

=\L

with  as above, and impose the value of the spins

outside. The Hamiltonian becomes



ð



jÞ¼J

hi;ji\

6¼;

ðiÞðjÞK

i2

¼0

ðiÞ½13

Here 



represents the configuration inside 

, the

pairs hi, ji are contained in L

,and(i) =(i) when

i 62 

, the configuration  being the given boundary

condition (see Figure 2). The corresponding parti-

tion function is denoted by Z

w

(

Since there are two pure phases in the model, we

must consider two surface free energies, or surfaces

tensions, 

wþ

and 

w

, between the wall and the

positive or negative phase present in the bulk. They

are defined through the choice of the boundary

condition, (i) = 1or(i) = 1, for all i 2L

. Let us

consider first the case of the () boundary condition.

The surface free energy contribution per unit area

due to the presence of the wall, when we have the

negative phase in the bulk, is



w

ð; KÞ

¼ lim

lim



L

w

ð



ðÞ

1=2

½14

The division by Z



()

1=2

allows us to subtract from

the total free energy, ln Z

w

(

), the bulk term and

all boundary terms which are not related to the

presence of the wall. The existence of limit [14]

follows from correlation inequalities, and we have



w

 0.

One can prove, as well, the existence of the Gibbs

state 

w

of the semi-infinite system, associated to

the () boundary condition. This state is the limit of

the finite volume Gibbs measures 



(



j())

defined by the Ham iltonian [13]. It describes the

local equilibrium properties of the system near

the wall, when deep inside the bulk the system is

in the negative phase. Similar definitions give the

surface tension 

wþ

and the Gibbs state 

wþ

corresponding to the boundary condition  (i) = 1,

for all i 2 

We remark that the states 

wþ

and 

w

are invariant

by translations parallel to the plane i

= 0, and

introduce the magnetizations, m

w

(z) = 

w

((z)),

where z denotes the site (0, 0, z), m

w

= m

w

(0), and

–

Figure 2 Boundary conditions for the cubic lattice. Above, the

box  with the () and (step) boundary conditions. Below, the

box 

and the wall W with the (w ) boundary conditions.

Statistical Mechanics of Interfaces 59

similarly m

wþ

(z)andm

wþ

. Their connection with

the surface free energies is given by the formula



w

ð; KÞ

wþ

ð; KÞ

ðm

wþ

ð; sÞm

w

ð; sÞÞds ½15

We mention in the following theorem some

results of Fro¨ hlich and Pfister’s study. Here 



is,

as before, the usual surface tension between the

two pure phases of the system, for a horizontal

interface.

Theorem 3 With the above definitions, we have



w

ð; KÞ

wþ

ð; KÞ



ðÞ½16

wþ

ð; KÞm

w

ð; KÞ0 ½17

and the difference in [17] is a monotone decreasing

function of the parameter K. Moreover, if m

wþ

= m

w

then the Gibbs states 

wþ

and 

w

coincide.

The proof is a subtle application of correlation

inequalities. Since, from Theorem 3, the integrand in

eqn [15] is a positive and decreasing function, the

difference  = 

w

 

wþ

is a monotone increasing

and concave (and hence continuous) function of the

parameter K. On the other hand, one can prove that

 = 



,ifK  J. This justifies the following

definition:

ðÞ¼minfK:ð; KÞ¼

þ

ðÞg ½18

In the thermodynamic description of wetting, the

partial-wetting regime is characterized by the strict

inequality in [16]. Equivalently, by K < K

(). We

must have then m

wþ

6¼ m

w

, because of eqn [15].

This shows that, in the case of partial wetting, 

wþ

and 

w

are different Gibbs states.

The complete-wetting regime is characterized by the

equality in [16],thatis,byK  K

(). Then, we have

wþ

= m

w

, and taking into account the last statement

in Theorem 3,also

wþ

= 

w

. This last result implies

that there is only one Gibbs state. Thus, complete

wetting correspond s to unicity of the Gibbs state.

In this case, we also have lim m

w

(z) = m



, when

z !1, because this is always true for m

wþ

(z). This

indicates that we are in the positive phase of the

system although we have used the () bounda ry

condition, so that the bulk ne gative phase cannot

reach the wall anymore. The film of positive phase,

which wets the wall completely, has an infinite

thickness with respect to the unit lattice spacing, in

the thermodynamic limit.

When  = 1, only a few particular ground con-

figurations contribute to the partition functions,

such as the configuration (i) = 1 for the partition

function Z

w

, etc., and we obtain  = 2K and





= 2J. For nonzero but low temperatures, the

small perturbations of these ground states have to be

considered, a problem that can be treated by the

method of cluster expansions. In fact, the corre-

sponding defects can be described by closed con-

tours as in the case of pure phases.

Theorem 4 For K < J, the functions 

w

(, K)

and 

wþ

(, K) are analytic at low temperatures,

that is, provided that  (J  K)  c

, where c

is a

given constant. Moreover, m

wþ

(z) and m

w

(z) tend,

respectively, to m



and to m



, when z !1,

exponentially fast.

The last statement in Theorem 4 tells us that the

wall affects only a layer of finite thickness (with

respect to the lattice spacing). From a macroscopic

point of view, the negative phase reaches the wall,

and we are in the partial-wetting regime. Indeed, a

strict inequality holds in [16].

Thus, for K < J there is always partial wetting at

low temperatures. Then the following question arises:

Question 2 Is there a situation of complete wetting

at higher temperatures? It is understood here that K

takes a fixed value, characteristic of the substrate,

such that 0 < K < J.

This is known to be the case in dimension d = 2,

where the exact value of K

() can be obtained

from Abraham’s solution of the model:

cosh 2K

¼ cosh 2J  e

2J

sinh 2J

Then complete wetting occurs for  in the interval



<  

(K), where 

is the critical inverse

temperature and 

(K) is the solution of K

() = K.

The case d = 2 has been reviewed in Abraham

(1986).

To our knowledge, the above question remains an

open problem for the Ising model in dimension

d = 3. The problem has, however, been solved for

the simpler case of a SOS interface model. In this

case, a nice and rather brief proof of the following

result has been given by Chalker (1982): one has

wþ

= m

w

, and hence complete wetting, if

2ðJ  KÞ<  lnð1  e

8J

It is very plausible that a similar statement is valid

for the semi-infinite Ising model and, also that

Chalker’s method could play a role for extending the

proof to this case, provided an additional assum p-

tion is made. Namely, that  is sufficiently large,

and hence J  K small enough, in order to insure the

convergence of the cluster expansions and to be able

to use them.

60 Statistical Mechanics of Interfaces

Equilibrium Crystals

The shape of an equilibrium crystal is obtained,

according to thermodynamics, by minimizing the

surface free energy between the crystal and the

medium, for a fixed volume of the crystal phase.

Given the orientation-dependent surface tension

(n), the solution to this variational problem,

known under the name of Wulff constru ction, is

the following set:

W¼fx 2 R

: x  n  ðnÞ for all ng½19

Notice that the problem is scale invariant, so that if we

solve it for a given volume of the crystal, we get the

solution for other volumes by an appropriate scaling.

We notice also that the symmetry (n) = (n)isnot

required for the validity of formula [19].Inthepresent

case, (n) is obviously a symmetric function, but

nonsymmetric situations are also physically interesting

and appear, for instance, in the case of a drop on a wall

discussed in the last section.

The surface tension in the Ising model between

the positive and negative phases has been defined in

eqn [7]. In the two-dimensional case, this function

(n) has (as shown by Abraham) an exact expression

in terms of some Onsager’s function. It follows (as

explained in Miracle-Sole (1999)) that the Wulff

shape W, in the plane (x

, x

), is given by

cosh x

þ cosh x

 cosh

2J= sinh 2J

This shape reduces to the empty set for   

, since

the critical 

satisfies sinh 2J

= 1. For >

,itis

a strictly convex set with smooth boundary.

In the three-dimens ional case, only certain inter-

face models can be exactly solved (see the section

‘‘Gibbs states an d interfa ces’’). Consi der the Ising

model at zero temperature. The ground configura-

tions have only one defect, the microscopic interface

, imposed by the bounda ry condition (, n). Then,

from eqn [9], we may write

ð nÞ¼ lim

ðE



ðnÞ

1



ðnÞÞ ½20

where E



= 2Jjj is the energy (all  ha ve the

same minimal area) and N



the number of

ground states. Every such  has the property of

being cut only once by all straight lines orthogo-

nal to the diagonal plane i

þ i

= 0, provided

that n

> 0, for k = 1, 2, 3. Each  can then be

described by an integer function defined on a

triangular plane lattice, the projection of the

cubic lattice L on the diagonal plane. The model

defined by t his set of admissible microscopic

interfaces is preci sely the T ISOS model. A sim ilar

definition can be given for the BCSOS model that

describes the ground configurations on the body-

centered cubic lattic e.

From a macroscopic point of view, the roughness

or the rigidity of an interface should be apparent

when considering the shape of the equilibrium

crystal associated with the system. A typical equili-

brium crystal at low temperatures has smooth plane

facets linked by rounded edges and corners. The

area of a particular facet decreases as the tempera-

ture is raised and the facet finally disappears at a

temperature characteristic of its orientation. It can

be argued that the disappearance of the facet

corresponds to the roughening transition of the

interface whose orientation is the same as that of the

considered facet.

The exactly solvable interface models mentioned

above, for which the function (n) has been

computed, are interesting examples of this behavior,

and provide a valuable informat ion on several

aspects of the roughening transition. This subject

has been reviewed by Abraham (1986), van Beijeren

and Nolden (1987),andKotecky (1989).

For example, we show in Figure 3 the shape

predicted by the TISOS model (one-eighth of the

shape because of the condition n

> 0). In this

model, the interfaces orthogonal to the three

coordinate axes are rigid at low temperatures.

For the three-dimensional Ising model at positive

temperatures, the description of the microscopic

interface, for any orientation n, appears as a very

difficult problem. It has been possib le, however,

to analyze the interfaces which are very near to

the particular orientations n

, discussed in the

Figure 3 Cubic equilibrium crystal shown in a projection

parallel to the (1,1,1) direction. The three regions (1, 2, and 3)

indicate the facets and the remaining area represents a curved

part of the crystal surface.

Statistical Mechanics of Interfaces 61

section ‘‘Gibbs states and interfa ces.’’ This analys is

allows us to determine the shape of the facets in a

rigorous way.

We first observe that the appearance of a facet

in the equilibrium crystal shape is related, according

to the Wulff construction, to the existence of a

discontinuity in the derivative of the surface

tension with respect to the orientation. More

precisely, assume that the surface tension satisfies

the convexity co ndition of Theorem 1, and let this

function (n) = (, ) be expressed in terms of the

spherical coordinates of n, the vector n

being taken

as the x

-axis. A facet orthogonal to n

appears in

the Wulff shape if and only if the derivative

@ (, )=@ is discontinuous at the point  = 0,

for all . The facet F@W consists of the points

x 2 R

belonging to the plane x

= (n

) and such

that, for all  between 0 and 2,

cos  þ x

sin   @ ð; Þ=@j

¼0

½21

The step free energy is expected to play an

important role in the facet formation. It is defined

as the free energy associated with the introduction

of a step of height 1 on the interface, and can be

regarded as an order parameter for the roughening

transition. Let  be a parallelepiped as in the section

‘‘Pure phases an d surface tension, ’’ and intr oduce

the (step, m) boundary conditions (see Figure 2),

associated to the unit vectors m = ( cos , sin ) 2

,by

ðiÞ¼

1ifi > 0orifi

¼ 0and

þ i

 0

1 otherwise

½22

Then, the step free energy per unit length for a step

orthogonal to m (with m

> 0) on the horizontal

interface, is



step

ðÞ

¼ lim

lim



cos 

L

step;m

ðÞ

;n

ðÞ

½23

A first result concerning this point was obtained

by Bricmont and co-workers, by proving a correla-

tion inequality which establish 

step

(0) as a lower

bound to the one-sided derivative @ (,0)=@ at

 = 0

(the inequality extends also to  6¼ 0). Thus,

when 

step

> 0, a facet is expected.

Using the perturbation theory of the horizontal

interface, it is possible to also study the microscopic

interfaces associated with the (step, m) boundary

conditions. When considering these configurations,

the step may be viewed as an additional defect on

the rigid inte rface described in the section ‘‘Pure

phases and surface tension .’’ It is, in fact , a long wall

going from one side to the other side of the box .

The step structure at low temperatures can then be

analyzed with the help of a new cluster expansion.

As a consequence of this analysis, we have the

following theorem.

Theorem 5 If the temperature is low enough, that

is, if J  c

, where c

is a given constant, then the

step free energy, 

step

(), exists, is strictly positive,

and extends by positive homogeneity to a strictly

convex function. Moreover, 

step

() is an analytic

function of  = e

2J

, for which an explicit conver-

gent series expansion can be found.

Using the above results on the step structure,

similar methods allow us to evaluate the increment

in surface tension of an interface tilted by a very

small angle  with respect to the rigid horizontal

interface. This increment can be expressed in terms

of the step free energy, and one obtains the

following relation.

Theorem 6 For J  c

, we have

@ ð; Þ=@j

¼0

¼ 

step

ðÞ½24

This relation, together with eqn [21], implies that

one obtains the shape of the facet by means of the

two-dimensional Wulff construction applied to the

step free energy. The reader will find a detailed

discussion on these points, as well as the proofs of

Theorems 5 and 6,inMiracle-Sole (1995).

From the properties of 

step

stated in Theorem 5,

it follows that the Wulff equilibrium crystal presents

well-defined boundary lines, smooth and without

straight segments, between a rounded part of the

crystal surface and the facets parallel to the three

main lattice planes.

It is expected, but not proved, that at a higher

temperature, but before reaching the critical

temperature, the facets associated with the Ising

model undergo a roughening transition. It is then

natural to believe that the equality [24] is true for

any  larger than 

, allowing us to determine the

facet shape from eqns [21] and [24], and that for

  

, both sides in this equality vanish, and

thus, the disappearance of the facet is involved.

However, the condition that the temperature is

low enough is needed in the proofs of Theorems 5

and 6.

See also: Dimer Problems; Phase Transitions in

Continuous Systems; Phase Transition Dynamics;

Two-Dimensional Ising Model; Wulff Droplets.

62 Statistical Mechanics of Interfaces

Further Reading

Abraham DB (1986) Surface structures and phase transitions. In:

Domb C and Lebowitz JL (eds.) Critical Phenomena, vol. 10,

pp. 1–74. London: Academic Press.

Chalker JT (1982) The pinning of an interface by a planar defect.

Journal of Physics A: Mathematical and General 15:

L481–L485.

Fro

hlich J and Pfister CE (1987) Semi-infinite Ising model. I.

Communications in Mathematical Physics 109: 493–523.

Fro¨ hlich J and Pfister CE (1987) Semi-infinite Ising model. II.

Communication in Mathematical Physics 112: 51–74.

Gallavotti G (1999) Statistical Mechanics: A Short Treatise.

Berlin: Springer.

Kotecky R (1989) Statistical mechanics of interfaces and

equilibrium crystal shapes. In: Simon B, Truman A, and

Davies IM (eds.) IX International Congress of Mathematical

Physics, pp. 148–163. Bristol: Adam Hilger.

Lebowitz JL and Pfister CE (1981) Surface tension and phase

coexistence. Physical Review Letters 46: 1031–1033.

Messager A, Miracle-Sole S, and Ruiz J (1992) Convexity

properties of the surface tension and equilibrium crystals.

Journal of Statistical Physics 67: 449–470.

Miracle-Sole S (1995) Surface tension, step free energy and facets

in the equilibrium crystal shape. Journal of Statistical Physics

79: 183–214.

Miracle-Sole S (1999) Facet shapes in a Wulff crystal. In: Miracle-

Sole S, Ruiz J, and Zagrebnov V (eds.) Mathematical Results

in Statistical Mechanics, pp. 83–101. Singapore: World

Scientific.

van Beijeren H and Nolden I (1987) The roughening transition.

In: Schommers W and von Blackenhagen P (eds.) Topics in

Current Physics, vol. 43. pp. 259–300. Berlin: Springer.

Widom B (1991) Interfacial phenomena. In: Hansen JP, Levesque D,

and Zinn-Justin J (eds.) Liquid, Freezing and the Glass

Transition, pp. 506–546. Amsterdam: North-Holland.

Wolf PE, Balibar S, and Gallet F (1983) Experimental observa-

tions of a third roughening transition in hcp

He crystals.

Physical Review Letters 51: 1366–1369.

Stochastic Differential Equations

F Russo, Universite

Paris 13, Villetaneuse, France

Introduction

Stochastic differential equ ations (SDEs) appear

today as a modeling tool in several sciences as

telecommunications, economics, finance, biology,

and quantum field theory.

An SDE is essentially a classical differential

equation which is perturbed by a random noise.

When nothing else is specified, SDE means in fact

ordinary SDE; in that case it corresponds to the

perturbation of an ordinary differential equation.

Stochastic partial differential equations (SPDEs) are

obtained as random perturbation of partial differ-

ential equations (PDEs).

One of the most important differe nce between

deterministic and stochastic ordinary differential

equations is described by the so-called Peano type

phenomenon. A classical differential equation with

continuous and linear growth coefficients admits

global existence but not uniqueness as classical

calculus text books illustrate studying equations of

the type

ðtÞ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

XðtÞ

; Xð0Þ¼0

However, if one perturbs the right member of the

equality with an additive Gaussian white noise (

)

(even with very small intensity), then the problem

becomes well stated. A similar phenomenon happens

with linear PDEs of evolution type perturbed with a

spacetime white noise.

SDEs constitute a vast subject and account for an

incredible amount of relevant contributions. We try

to orientate the reader about the main axes trying to

indicate references to the different subfields. We will

prefer to refer to monographs when available,

instead of articles.

Motivation and Preliminaries

In the whole article T will be a strictly positive real

number. Let us consider continuous functions

b : R

 R

, a : R

 R

dm

and x

2 R

We consider a differential problem of the following

type:

¼ bðt; X

¼x

½1

Let (, F, P) be a complete probability space.

Suppose that previous equation is perturbed by a

random noise (

)

t0

. Because of modeling reasons it

could be reasonable to suppose (

)

t0

satisfying the

following properties.

1. It is a family of independent random variables

(r.v.’s)

2. (

)

t0

is ‘‘stationary’’, that is, for any posit ive

integer n, positive reals h, t

, t

, ..., t

, the law of

(

þh

, ..., 

þh

) does not depend on h.

Stochastic Differential Equations 63

More precisely we perturb eqn [1] as follows:

¼ bðt; X

Þþaðt; X

Þ

¼ x

½2

We suppose for a moment that d = m = 1. In reality no

reasonable real-valued process (

)

t0

fulfilling pre-

vious assumptions exists. In particular, if process (

)

exists (resp. (

) exists and each 

is a square-

integrable r.v.), then the process cannot have contin-

uous paths (resp. it cannot be measurable with respect

to   R

). However, suppose that such a process

exists; we set B



ds. In that case, properties (1)

and (2) can be translated into the following on (B

(P1) It has independent increments, which means

that for any t

, ..., t

, h  0, B

þh

 B

þh

, ...,

þh

 B

n1

þh

are independent r.v.’s.

(P2) It has stationary increments, which means that

for any t

, ..., t

, h  0, the law of (B

þh



þh

,...,B

þh

B

n1

þh

) does not depend on h.

On the other hand, it is natural to require that

(C1) B

= 0 a.s.,

(C2) it is a continuous process, that is, it has

continuous paths a.s.

Equation [2] should be rewritten in some integral form

¼ X

bðs; X

Þds

aðs; X

ÞdB

½3

Clearly the paths of process (B

) cann ot be differ-

entiable, so one has to give meaning to integral

a(s, X

)dB

. This will be intended in the ‘‘Itoˆ’’

sense, see considerations below.

An important result of probability theory says

that a stochastic process (B

) fulfill ing properties P1,

P2 and C1, C2 is esse ntially a ‘‘Brownian motion’’.

More precisely, there are real constants b,  such

that B

= bt þ W

, where (W

) is a classical Brow-

nian motion defined below.

Definition 1

(i) A (continuous) stochastic process (W

) is called

classical ‘‘Brownian motion’’ if W

= 0 a.s.,

it has independent increments and the law of

 W

is a Gaussian N(0, t  s) r.v.

(ii) A m-dimensional Brownian motion is a vector

, ..., W

) of independent classical Brow-

nian motions.

Let (F

)

t0

be a filtration fulfilling the usual

conditions, see (Karatzas and Shreve (1991, section 1.1).

There one can find basic concepts of the theory of

stochastic processes as the concept of adapted,

progressively measurable proces s. An adapted pro-

cess is also said to be nonanticipating towards the

filtration (F

) which represents the state of the

information at each time t. A process (X

) is said to

be adapted if for any t, X

is F

-measurable. The

notion of progressively measurable process is a slight

refinement of the notion of adapted process.

Definition 2

(i) A (continuous) (F

) adapted process (W

) is called

(classical) (F

)-Brownian motion if W

= 0, if

for any s < tW

 W

is an N(0, t  s ) distrib-

uted r.v. which is independent of F

(ii) An (F

)-m-dimensional Brownian motion is a

vector (W

, ..., W

)of(F

)-classical indepen-

dent Brownian motions.

From now on, we will consider a probability

space (, F, P) equipped with a filtration (F

)

t0

fulfilling the usual conditions. From now on all the

considered filtrations will have that property.

Let W = (W

)

t0

be an (F

)

t0

-m-dimensional clas-

sical Brownian motion. In Karatzas and Shreve (1991,

chapter 3) and Revuz and Yor (1999, chapter 4), one

introduces the notion of stochastic Itoˆintegral

announced before. Let Y = (Y

, ..., Y

)beaprogres-

sively measurable m-dimensional process such that

ds < 1, then the Itoˆintegral

is well

defined. In particular the indefinite integral



an (F

)-progressively measurable continuous process.

If Y is an R

dm

matrix-valued process, the integral

is componentwise defined and it will be a

vector in R

. The analogous of differential calculus in

the framework of stochastic processes is Itoˆ calculus,

see again Karatzas and Shreve (1991, chapter 3) and

Revuz and Yor (1999, chapter 4). Important tools are

the concept of quadratic variation [X]ofastochastic

process when it exists. For instance, the quadratic

variation [W]

of a classical Brownian motion equals t.

If M

,then[M]

ds. One cele-

brated theorem of P Le´vy states the following: if (M

)

defines a continuous (F

)-local martingale such that

]  t,thenM is an (F

)-classical Brownian motion.

That theorem is called the ‘‘Le´vy characterization

theorem of Brownian motion.’’ Itoˆ formula constitutes

the natural generalization of fundamental theorem of

differential calculus to the stochastic calculus. Another

significant tool is Girsanov theorem; it states essen-

tially the following: suppose that the following so-

called ‘‘Novikov condition’’ is verified:

E exp



< 1

64 Stochastic Differential Equations

Then the process

= W

ds, t 2 [0, T]is

again an m-dimensional (F

)-classical Brownian

motion under a new probability measure Q on

(, F

) defined by

dQ ¼ dP exp





Let  be an F

-measurable r.v., for instance,  

x 2 R

. We are interested in the SDE

¼aðt; X

ÞdW

þ bðt; X

Þdt

¼

½4

Definition 3 A progressively measurable process

)

t2[0, T]

is said to be solution of [4] if a.s.

¼ Z þ

aðt; X

ÞdW

bðt; X

Þdt

8t 2½0; T

½5

provided that the right-hand side member makes

sense. In particular, such a solution is continuous.

The function a (resp. b) is called the diffusion (drift)

coefficient of the SDE. a and b may sometimes be

allowed to be random; however, this dependence

has to be progressively measurable. Clearly, we can

define the notion of solution (X

)

t0

on the whole

positive real axis.

We remark that those equations are called Itoˆ

SDEs. A solution of previous equation is named

diffusion process.

The Lipschitz Case

The most natural framework for studyi ng the

existence and uniqueness for SDEs appears when

the coefficients are Lipschitz.

A function  : [0, T]  R

! R

is said to have

‘‘polynomial growth’’ (with respect to x uniformly in

t), if for some n there is a constant C > 0 with

sup

t2½0;T

kðt; xÞk  Cð1 þkxk

Þ½6

The same function is said to have ‘‘linear growth’’ if

[6] holds with n = 1. A function  : R

 R

said to be ‘‘locally Lipschitz’’ (with respect to

x uniformly in t), if for every t 2 [0, T], K > 0,



j[0, T][K, K]

is Lipsch itz (with respect to x uniformly

with respect to t).

Let a : R

 R

dm

! R

, b : R

 R

! R

,be

Borel functions,  an R

-valued r.v. F

-measurable

and (W

)

t0

be a m-dimensional (F

)-Brownian

motion.

Classical fixed-point theorems allow to establish

the following classical result.

Theorem 1 We suppose a and b locally Lipschitz

with linear growth. Let  be a square-integrable r.v.

that is F

-measurable. Then [4] has a unique

solution X. Moreover,

E sup

tT

< 1

Remark 1

(i) Equation [4] can be settle d similarly by putting

initial con dition x at some time s. In that case

the problem is again well stated. If   x is a

deterministic point of R

, then we will often

denote by X

s, x

the solution of that problem.

(ii) If the coefficients are only locally Lipschitz, the

equation may be solved until a stopping time. If

d = 1, it is possible to state necessary and sufficient

conditions for nonexplosion (Feller test).

(iii) The theorem above admits several generaliza-

tions. For instance, the Brownian motion can be

replaced by general semimartingales, (possibly

with jumps as Le´vy processes) .

An important role of diffusion processes is the fact

that they provide probabilistic representatio n to

PDEs of parabolic (and even elliptic) type. We will

only mention here the parabolic framework.

We denote A(t, x) = a(t, x)a(t, x)



, where  means

transposition for matrices. (t, x) !A(t, x) = (A

(t, x))

is a d  d matrix-valued function. Let us consider also

continuous functions k : [0, T]  R

, g : [0, T] 

with polynomial growth or non-negative.

Given a solution of [4], we can associate its

generator (L

, t 2 [0, T]) setting

f ðxÞ¼

i;j¼1

ðt; xÞ@

f ðxÞþbðt; xÞrf ðxÞ

Feynman–Kac theorem is stated below and it

provides probabilistic representation of an asso-

ciated parabolic linear PDEs.

Theorem 2 Suppose there is a function v : [0, T[ 

continuous with polynomial growth of

class C

1, 2

([0, T]  R

) satisfying the following

Cauchy problem:

ð@

v þ L

Þv  kv ¼ g

vðT; xÞ¼f ðxÞ

½7

Then

vðs; xÞ¼EfðX

Þexp 

kð; X



Þd





gðt; X

Þexp 

kð; X



Þd





Stochastic Differential Equations 65

for (s, x) 2 [0, T]  R

, where X = X

s, x

. In particu-

lar, such a solution is unique.

Remark 2

(i) In order to obtain ‘‘classical solutions’’ of the

above Cauchy problem, one needs some condi-

tions. It is the case, for instance, when the

following ellipticity condition holds on A:

9c > 0; 8ðt;xÞ2½0;TR

; 8ð

;...;

Þ2R

i;j

ðt;xÞ



c

i¼1

j

½8

In the degenerate case, it is possible to deal with

viscosity solutions, in the sense of P L Lions.

This theorem establishes an important link

between deterministic PDEs and SDEs.

(ii) A natural generalization of Feynman–Kac theo-

rem comes from the system of forward–backward

SDEs in the sense of Pardoux and Peng.

(iii) Other types of probabilistic representation do

appear in stochastic control theory through the

so-called verification theorems, see for instance,

Fleming and Soner (1993) and Yong and Zhou

(1999). In that case, the (nonlinear) Hamilton–

Jacobi–Bellmann deterministic equation is

represented by a controlled SDE.

(iv) Another bridge between nonlinear PDEs and

diffusions can be provided in the framework of

interacting particle systems with chaos propaga-

tion, see Graham et al. (1996) for a survey on

those problems. Among the most significant

nonlinear PDEs investigated probabilistically, we

quote the case of porous media equations. For

instance, for a positive integer m, a solution to

u ¼

ðu

2mþ1

Þ½9

can be represented by a (nonlinear) diffusion of

the type, see Benachour et al. (19 96),

d X

¼ u

ðs; X

ÞdW

uðt; Þ ¼ law density of X

½10

Different Notions of Solutions

Let a and b as at the beginning of the previous

section. Let (, F, P) be a probability space, a

filtration (F

)

t0

fulfilling the usual conditions, an

)

t  0

-cl assical Bro wnian motion (W

)

t  0

. Let  be

an F

-me asurable r.v. In the section ‘‘Mot ivation

and preliminar ies,’’ we defined the notion of so lu-

tion of the following equation:

d X

¼ bðt; X

Þdt þ aðt; X

ÞdW

¼ 

½11

This equation will be denoted by E(a, b)(withoutinitial

condition). However, as we will see, the general

concept of solution of an SDE is more sophisticated

and subtle than in the deterministic case. We distin-

guish several variants of existence and uniqueness.

Definition 4 (Strong existence). We will say that

equation E(a, b) admits strong existenc e if the

following holds. Given any probability space

(, F, P), a filtration (F

)

t0

,an(F

)

t0

-Brownian

motion (W

)

t0

,anF

-measurable and square-

integrable r.v. , there is a process (X

)

t0

solution

to E(a, b) with X

=  a.s.

Definition 5 (Pathwise uniqueness). We will say

that equation E(a, b) admits pathwise uniqueness if

the following property is fulfilled. Let (, F, P)bea

probability space, a filtration (F

)

t0

,an(F

)

t0

Brownian motion (W

)

t0

. If two processes X,

X are

two solutions such that X

a.s., then X and

coincide.

Definition 6 (Existence in law or weak existence).

Let  be a probability law on R

. We will say that

E(a, b; ) admits weak existence if there is a

probability space (, F, P), a filtration (F

)

t0

,an

)

t0

-Brownian motion (W

)

t0

, and a process

)

t0

solution of E(a, b)with being the law of X

We say that E(a, b) admits weak existence if

E(a, b; ) admits weak existence for every .

Definition 7 (Uniqueness in law). Let  be a

probability law on R

. We say that E(a, b; )hasa

unique solution in law if the following holds. We

consider an arbitrary probab ility space (, F, P) and

a filtration (F

)

t0

on it; we consider also another

probability space (

,

P) equipped with another

filtration (

)

t0

; we consider an (F

)

t0

-Brownian

motion (W

)

t0

, and an (

)

t0

-Brownian motion

(

)

t0

; we suppose having a process (X

)

t0

(resp. a

process (

)

t0

) solution of E(a, b) on the first (resp.

on the second) probability space such that both the

law of X

and

are identical to . Then X and

must have the same law as r.v. with values in

E = C(R

) (or C[0, T]).

We say that E (a, b) has a unique solution in law if

E(a, b; ) has a unique solution in law for every .

There are important theorems which establish

bridges among the preceding notions. One of the

most celebrated is the following.

Proposition 1 (Yamada–Watanabe). Consider the

equation E(a, b).

(i) Pathwise uniqueness implies uniqueness in law.

(ii) Weak existence and pathwise uniqueness imply

strong existence.

66 Stochastic Differential Equations

A version can be stated for E(a, b; ) where  is a

fixed probability law.

Remark 3

(i) If a and b are locally Lipschitz with linear

growth, Theorem [1] implies that E(a, b) admits

strong existence and pathwise uniqueness.

(ii) If a and b are only locally Lipschitz, then

pathwise uniqueness is fulfill ed.

Existence and Uniqueness in Law

A way to create weak solutions of E(1, b) when

(t, x) !b(t, x) is Borel with linear growth is the

Girsanov theorem. Suppose d = 1 for simplicity. Let

us consider an (F

)-classical Brownian motion (X

We set

¼ X



bðs; X

Þds

Under some suitable probability Q,(W

)isan(F

classical Brownian motion. Therefore, (X

) provides

a solution to E(1, b; 

We continue with an example where E(a, b) does

not admit pathwise uniqueness, even though it

admits uniqueness in law.

Example 1 We consider the stochastic equation

signðX

ÞdW

½12

with

signðxÞ¼

1ifx  0

1ifx < 0



It corresponds to E(a, b; 

) with b = 0and

a(x) = sign(x).

If (W

)

t0

is an (F

)-classical Brownian motion,

then (X

)

t0

is (F

)

t0

-continuous local martingale

vanishing at zero such that [X]

 t. According to

Le´vy characterization theorem stated earlier, X is an

)

t0

-classical Brownian motion. This shows in

particular that E(a, b; 

) admits uniqueness in law.

In the sequel, we will show that E(a, b; 

) also

admits weak existence.

Let now (, F, P) be a probability space, an

)

t0

-classical Brownian motion with respect to a

filtration and (X

)

t0

such that [12] is verified. Then

= X

can also be shown to be a solution.

Therefore, E(a, b; 

) does not admit pathwise

uniqueness.

We continue stat ing a result true in the multi-

dimensional case.

Proposition 3 (Stroo ck–Varadhan). Let  be a

probability on R

such that

kxk

ðdxÞ < þ1 ½13

for a certain m > 1. We suppose that a, b are

continuous with linear growth. Then E(a, b; )

admits weak existence.

From now on, a function  : [0, T]  R

will

be said Ho¨ lder-continuous if it is Ho¨ lder-continuous

in the space variable x 2 R

uniformly with respect

to the time variable t 2 [0, T].

Stroock and Varadhan (1979) also provide the

following result, which is an easy consequence of

their theorem 7.2.1.

Proposition 4 We suppose a, bbothHo¨ lder-

continuous, bounded such that condition; [8] is

fulfilled. Then SDE E(a, b; ) admits weak uniqueness.

Remark 4

(i) The Ho¨ lder condition and [8] in Proposition 4

may be relaxed and replaced with the solva-

bility of a Cauchy problem of a parabolic PDE

with suitable terminal value.

(ii) In the case d = 1, if a, b are bounded and just

Borel with [8] for x on each compact, then

E(a, b; ) admits weak existence and uniqueness

in law. See Stroock and Varadhan (1979,

exercises 7.3.2 and 7.3.3).

(iii) If d = 2, the same holds as at previous point

provided that moreover a does not depend on

time.

We proceed with some more specifically unidi-

mensional material stating some results from

K J Engelbert and W Schmidt, who furnished

necessary an d sufficient conditions for weak exis-

tence and uniqueness in law of SDEs.

For a Borel function  : R !R, we first define

ZðÞ¼fx 2 Rjð xÞ¼0g

then we define the set I(

) as the set of real numbers

x such that

xþ"

x"



ðyÞ

¼1; 8">0

Proposition 5 (Engelbert–Schmidt criterion). Sup-

pose that a : R !R, that is, does not depend on time

and we consider the equation without drift E(a, 0).

(i) E(a,0) admits weak existence (without explo-

sion) if and only if

IðaÞZða Þ½14

Stochastic Differential Equations 67

(ii) E(a,0) admits weak existence and uniqueness in

law if and only if

IðaÞ¼Zða Þ½15

Remark 5

(i) If a is continuous then, [14] is always verified.

Indeed, if a(x) 6¼ 0, there is ">0 such that

jaðyÞj > 0; 8y 2½x  "; x þ "

Therefore, x cannot belong to I(a).

(ii) Equation [14] is verified also for some discon-

tinuous functions as, for inst ance, a(x) =

sign(x). This confirm s what was affirmed

previously, that is, the weak existence (and

uniqueness in law) for E(a, 0).

(iii) If a(x) = 1

{0}

(x), [14] is not verified.

(iv) If a(x) = jxj



,   1=2, then

ZðaÞ¼Iða Þ¼f0g

So there is at most one solution in law for

E(a, 0).

(v) The proof is technical and makes u se of

Le´ vy character isation theor em of Brownian

motion.

Results on Pathwise Uniqueness

Proposition 6 (Yamada–Watanabe). Let a, b : R



R !R and consider again E(a, b). Suppose b

globally Lipschitz and h : R

strictly increas-

ing continuous such that

(i) h(0) = 0;

(ii)

(1/h

)(y)dy = 1, 8">0; and

(iii) ja(t, x)  a(t, y )jh(x  y).

Then pathwise uniqueness is verified.

Remark 6

(i) In Proposition 6, one typical choice is

h(u) = u



, >1=2.

(ii) Pathwise uniqueness for E(a, b) holds therefore

if b is globally Lipschitz and a is Ho¨ lder-

continuous with paramete r equal to 1/2.

Corollary 1 Suppose that the assumptions of

Proposition 6 are verified and a, b continuous with

linear growth. Then E(a, b; ) admits strong exis-

tence and pathwise uniqueness, whenever  verifies

condition [13].

Proof It follows from Propositions 6 and 3

together with Proposition 1 (ii).

Remark 7 Suppose d = 1. Pathwise uniqueness for

E(a, b) also holds under the following assumptions.

(i) a, b are bounded, a is time independent and

a  const. > 0, h as in Proposition 6. This result

has an analogous form in the case of spacetime

white noise driven SPDEs of parabolic type, as

proved by Bally, Gyongy, and Pardoux in 1994.

(ii) a independent on time, b bounded and a 

const. > 0; moreover, ja(x)  a( y )j

jf (y) 

f (x)j and f is increasing and bounded.

For illustration we provide some significant

examples.

Example 2



; t  0 ½16

We set a(x) = jxj



,0<<1. This is equation

E(a, 0) with a(x) = jxj



. According to Engelbert–

Schmidt notations, we have Z(a) = {0}. Moreover

(i) If   1=2, then I(a) = {0}.

(ii) If <1=2 then I(a) = ;.

Therefore, according to Proposition 5, E(a,0) admits

weak existence. On the other hand, if   1=2,



 y



jhðjx  yjÞ ½17

where h(z) = z



. According to Proposition 6, [16]

admits pathwise uniqueness and by Corollary [1],

also strong existence. The unique solution is X  0.

If <1=2, X  0 is always a solution. This is not

the only one; even uniqueness in law is not true.

Example 3 Let a (x) =

ﬃﬃﬃﬃﬃﬃ

jxj

, b Lipschitz. Then

E(a, b) admits strong existence and pathwise unique-

ness. In fact, a is Ho¨ lder-continuous with parameter

1/2 and the second item of Remark 6 applies; so

pathwise uniqueness holds. Strong existence is a

consequence of Propositions 3 and 1 (ii).

An interesting particular case is provided by the

following equation. Let x

, ,   0, k 2 R. The

following equation admits strong existence and

pathwise uniqueness.

¼ x

þ 

ﬃﬃﬃﬃﬃﬃﬃﬃ

ð  kX

Þds

t 2½0; T½18

Equation [18] is widely used in mathematical finance

and it constitutes the model of Cox–Ingersoll–Ross:

the solution of the mentioned equation represents the

short interest rate.

Consider now the particular case where k = 0,

 = 2. According to some comparison theorem for

SDEs, the solution Z is always non-negative and

68 Stochastic Differential Equations