CHAPTER 1

Introduction

This paper studies the stochastic wave equation in spatial dimension d = 3:

∂2

∂t2

− ∆ u(t, x) = σ

(

u(t, x)

)

˙

F (t, x) + b

(

u(t, x)

)

, (1.1)

u(0, x) = v0(x),

∂

∂t

u(0, x) = ˜0(x), v

where t ∈ ]0, T ] for some fixed T 0, x ∈

R3

and ∆ denotes the Laplacian on

R3.

The coeﬃcients σ and b are Lipschitz continuous functions, the noise process

˙

F is

the formal derivative of a Gaussian random field, white in time and correlated in

space. More precisely, for any d ≥ 1, let D(Rd+1) be the space of Schwartz test

functions (see [31]) and let Γ be a non-negative and non-negative definite tempered

measure on

Rd.

Then, on probability space, there exists a Gaussian process

F =

(

F (ϕ), ϕ ∈

D(Rd+1)

)some

with mean zero and covariance functional given by

(1.2) E

(

F (ϕ)F (ψ)

)

=

R+

ds

Rd

Γ(dx) (ϕ(s) ∗

˜(s))(x),

ψ

where

˜(s)(x)

ψ = ψ(s)(−x).

We are interested in solutions to (1.1) which are random fields, that is, real-

valued processes (u(t, x), (t, x) ∈ [0, T ] ×

R3),

that are well defined for every fixed

(t, x) ∈ [0, T ] ×

R3.

The main objective of this paper is to study sample path

regularity properties of the solution to (1.1).

Sample path regularity, and, more precisely, H¨ older continuity, is a key property

that is needed early on in any fine study of a random field. For instance, establishing

properties of the probability law of the solution often requires a priori information

about sample path regularity [23, 33]. H¨ older exponents are also useful when

addressing questions of probabilistic potential theory [8]. H¨ older continuity results

are also needed when developping numerical approximation schemes, in particular

to obtain their rates of convergence (see for instance [13]). In this paper, we shall

study the H¨ older continuity, both in time and space, of the solution to (1.1), and

check the optimality of the H¨ older exponents that we obtain.

The first issue, however, is to give a rigourous formulation of the Cauchy prob-

lem (1.1). For this, different approaches are possible. However, in all of them the

fundamental solution associated to the wave operator L =

∂2

∂t2

− ∆ naturally plays

an important role. Since its singularity increases with the spatial dimension d, the

diﬃculties in studying regularity of the solutions of the stochastic wave equation

increase accordingly. Moreover, keeping the requirement of obtaining random field

solutions amounts to adjusting the roughness of the noise to the degeneracy of the

differential operator which defines the equation. It is only for d = 1 that it is

1