XIV

INTRODUCTION

Our results. The main objective of this work is to give a positive answer to

the questions raised in Problem 1 and Problem 2, in the particular case of equation

Au — u2. In order to use the tools provided by the Brownian snake approach, we

have restricted ourselves to the equation Au =

u2

instead of trying to tackle the

general equation Lu — ip(u). For technical reasons, we shall consider only domains

D that are bounded and of class

C4.

However, we are reasonably confident as far as

the generalization of our results and some of our methods to more general equations

and domains is concerned.

Equivalence between UY and capd(T). We emphasize above that UY = 0 if and

only if cap^(r) = 0. We investigate more closely the connections between ur and

cap

a

(r).

RESULT

0.1 (See Chapter 2, Section 3.3.1). There exist two positive continuous

functions C\ and C2 defined on D such that

Ci(x)capd(T) ^ UY{X) ^ C2{x)capd(T)

for every Borel set Y C dD and every x G D.

We give specific possible values for C\ and C2 in Theorem 3.15 (for C2) and

Proposition 3.16 (for C\).

ur = ur and r = r. We prove the following theorem which provides a positive

answer to Problem 2.

RESULT 0.2 (See Chapter 3, Theorem 3.1). For every Borel set T C dD, we

have

ur = ur-

In particular, ur is a-moderate.

It immediately follows from this theorem that the two fine topologies defined

by Dynkin and Kuznetsov coincide.

RESULT

0.3 (See Chapter 3, Corollary 3.5). We have the identity

T = T.

Classification results and probabilistic representation. We now give a detailed

statement of our main theorem.

RESULT 0.4 (See Chapter 5, Theorem 5.1). Let d^ 3 be an integer, let D be a

bounded domain of class C4 in M.d and let u be a nonnegative solution of Au — 4u2

in D such that

tr(u) = (T,v).

Then

u —

UY

® uv.

In probabilistic terms, for every x G D,

u(x) = Nx

(£D

n r ^ 0) + Nx (l£D

n r = 0

(1 - exp(-^))) .

A nonnegative solution is uniquely determined by its fine trace.

As a consequence of this theorem, we give a positive answer to Problem 1.

RESULT 0.5 (See Chapter 5, Theorem 5.2). All nonnegative solutions of Au =

u2

in a bounded domain of class

C4

in

Rd,

d^ 3, are a-moderate.

Note that Theorem 0.5 is deduced from Theorem 0.4 even though a direct proof

of Theorem 0.5 would imply Theorem 0.4.