INTRODUCTION xv

Organization of this memoir. The first chapter of this memoir is purely an-

alytic. It provides the basic analytic framework for equation (3). Global estimates

for Green functions and Poisson kernels are presented. The emphasis is put on the

lattice structure on the set of solutions of (3) highlighted by Dynkin and Kuznetsov

in [DK98b]. Of fundamental importance is the detailed introduction of moderate

solutions, cr-moderate solutions and solutions of the UK-type. We conclude this

chapter with the definition of the rough trace of a solution which, except in the

case ud — 2", turns out not to be sufficient to classify the solutions of (3).

The second chapter focuses on the probabilistic point of view on equation (3).

A natural starting point is a brief overview of the well-known connections between

elliptic linear PDE's in D and Brownian motion in D. Most of this chapter however

is devoted to the links between the Brownian snake and equation (3). Of primary

importance is the construction of the correspondence between solutions and sto-

chastic boundary values. We establish a formula for the stochastic boundary values

of various classes of solutions: moderate solutions, cr-moderate solutions, and so-

lutions of the ur-type. Then, following [DK98b], we precisely define the singular

points of a solution on the boundary in terms of conditional Brownian motion, the

fine topologies on the boundary and the fine trace of a solution. We present various

properties involving these objects.

The third chapter provides a positive answer to Problem 2. The first part

is analytic. Using suitable capacities and the estimates about Green functions of

Chapter 1, we establish upper bounds for solutions UK of the type

uK{x) ^ C(x,K)cipd(K),

where C(x, K) is given explicitly. In the second part, using conditional Brownian

motion and fine properties of the Brownian snake, we find lower bounds for various

cr-moderate solutions.

This leads to the inequality

u ^

UY

© uv if tr(w) = (T, i/).

The fourth chapter is devoted to the reverse inequality

u ^ ur © uv if tr(w) = (T, v).

The proof makes a heavy use of the structure of the set of stochastic boundary

values of solutions studied in Chapter 2.

The fifth chapter consists in a synthesis of the above results, which leads to the

classification theorems stated above.

Further remarks. Note that, throughout this memoir, we use generic con-

stants. For instance, C(X) stands for a constant depending only on A.

Thanks. I would like to take this opportunity to thank my PhD supervisor

Jean-Frangois Le Gall. I am indebted to him for showing me the right directions to

follow and helping me to fill the gaps of a lot of proofs contained in this memoir.

I am very grateful to Eugene B. Dynkin for all the judicious remarks he made

concerning my work and for his constant support.

I am also indebted to Laurent Veron for the interest he showed in my results.