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 , , , , , , , , ,
, ,,  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
(1.3) G(t) =
for any t 0, where σt denotes the uniform surface measure (with total mass
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  does not apply.
In fact, this question motivated two different extensions of Walsh’s integral, given
in  and , respectively.
The Gaussian noise process F is first extended to a worthy martingale measure
M = (Mt(A), t ≥ 0, A ∈
in the sense of , where
bounded Borel subsets of
In , an extension of Walsh’s stochastic integral,
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
where µ = F−1Γ. Here, F denotes the Fourier transform. As is shown in Section
5 of , 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) =
G(t − s, x − y)σ (u(s, y)) M(ds, dy)
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  and  (see also ).
In , 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 , 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
valued stochastic process (Theorem 9 in ). This is the choice of stochastic integral
that we will use in this paper to study the stochastic wave equation (1.1).