2 1. INTRODUCTION

possible to take a space-time white noise as random input to (1.1), while in higher

dimensions a non-degenerate spatial correlation is necessary [6, 7, 15].

For d = 1, 2, the stochastic wave equation driven by space-time white noise,

and noise that is white in time but spatially correlated, respectively, is now quite

well understood. We refer the reader to [3], [4], [5], [6], [15], [17], [19], [20], [21],

[24], [25],[28], [39] for a sample of articles on the subjects such as existence and

sample path regularity, study of the probability law of the solution, approximation

schemes and long-time existence.

For d = 3, the fundamental solution of the wave equation is the measure defined

by

(1.3) G(t) =

1

4πt

σt,

for any t 0, where σt denotes the uniform surface measure (with total mass

4πt2)

on the sphere of radius t ∈ [0, T ]. Hence, in the mild formulation of equation (1.1),

Walsh’s classical theory of stochastic integration developed in [38] does not apply.

In fact, this question motivated two different extensions of Walsh’s integral, given

in [7] and [9], respectively.

The Gaussian noise process F is first extended to a worthy martingale measure

M = (Mt(A), t ≥ 0, A ∈

Bb(Rd))

in the sense of [38], where

Bb(Rd)

denotes the

bounded Borel subsets of

Rd.

In [7], an extension of Walsh’s stochastic integral,

written

t

0

G(t − s, y) Z(s, y) M(ds, dy),

is proposed. This extension allows for a non-negative distribution G, a second-

order stationary process Z in the integrand, and requires, among other technical

properties, the integrability condition

(1.4)

T

0

ds

Rd

µ(dξ)

|FG(t)(ξ)|2

∞,

where µ = F−1Γ. Here, F denotes the Fourier transform. As is shown in Section

5 of [7], one can use this integral to obtain, in the case where the initial conditions

vanish, existence and uniqueness of a random field solution to (1.1), interpreted in

the mild form

u(t, x) =

t

0 R3

G(t − s, x − y)σ (u(s, y)) M(ds, dy)

+

t

0

ds [G(t − s) ∗ b(u(s, ·))](x).

In this framework, results on the regularity of the law of the solution to the sto-

chastic wave equation have been proved in [26] and [27] (see also [33]).

In [9], a new extension of Walsh’s stochastic integral based on a functional

approach is introduced. Neither the positivity of G nor the stationarity of Z are

required (see [9], Theorem 6). With this integral, the authors give a precise meaning

to the problem (1.1) with non vanishing initial conditions and coeﬃcient b ≡ 0 and

obtain existence and uniqueness of a solution (u(t), t ∈ [0, T ]) which is an

L2(R3)–

valued stochastic process (Theorem 9 in [9]). This is the choice of stochastic integral

that we will use in this paper to study the stochastic wave equation (1.1).