x INTRODUCTIO N

Besides, if K is a compact subset of dD, there exists a compact set K' such that

TR(uK) = (K',0) and capd(KAK') = 0,

where

KAKf

=

(K\Kf)

U

(Kf\K)

(see Section 1.3.8). More specifically, in the

subcritical case (d — 2), if a solution of (3) has a rough trace equal to (K, i/), then

u =

UK

0 uv.

(See Theorem 1.29). Our aim is to generalize this type of representation to dimen-

sions d ^ 3.

Probabilistic approach. Many analytic results concerning linear elliptic (or

parabolic) PDE's have a probabilistic counterpart (see [Dur]). We need more

elaborate tools than superdiffusions in order to tackle nonlinear PDE's of the type

(2), namely superdiffusions and the Brownian snake. The work of Watanabe [Wa]

in 1968 already suggests that there exist deep links between the measure-valued

processes now called superprocesses and nonlinear PDE's of the type (2). In the

early nineties, Dynkin reduced the problem of the classification of the solutions

of (2) to the description of a certain class of functionals of superdiffusions (see in

particular [Dy93]).

{L,ip)-superprocess. If L is a second order elliptic differential elliptic operator

and ip belongs to a large class of monotone increasing convex functions including in

particular u \—

ua,

1 a ^ 2, one can construct a superdiffusion called the (L, ip)-

superprocess. The (L, ^-superprocess is as a measure-valued branching process

which can be obtained by a passage to the limit from branching particle systems.

In this approximation, the spatial motion of the particles is a L-diffusion and their

branching mechanism is determined by I/J (see [LG99] and [Dy02], Chapter 4).

When L — A A and

I/J(U)

=

2u2,

which is our case of interest in this thesis, this

superprocess is super-Brownian motion.

Exit measures and the nonlinear Dirichlet problem. Given a (L, I/J)-superprocess

and a bounded smooth domain D, one can construct a random measure

ZD

on dD

called the exit measure from D. Roughly speaking, it is obtained by freezing the

particles of the cloud represented by the superprocess when they hit dD. If / is a

continuous nonnegative function on dD, and P^ stands for the law of the (L, ip)-

superprocess started at the Dirac measure 5X, x G D, then the unique nonnegative

solution to the nonlinear Dirichlet problem

Lu = ip(u) in D

U\dD = f

is given by

u(x) = -logE^ (exp(-

ZDJ

)) , xeD.

(See [Dy91]). This is the key to the connection with partial differential equations.

The Brownian snake. We restrict ourselves to the specific case L — A A, ip(u) =

2u2.

Super-Brownian motion can be represented in terms of the path-valued process

called the Brownian snake introduced by Le Gall in 1993 (see [LG93a] and [LG99]).

We use Le Gall's approach which is more trajectorial and maybe more tractable

but also less general. The definition of exit measures of super-Brownian motion is

made easier by the use of the Brownian snake. If Nx is the excursion measure from