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).
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