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, 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]. older exponents are also useful when
addressing questions of probabilistic potential theory [8]. 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 older continuity, both in time and space, of the solution to (1.1), and
check the optimality of the 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
Previous Page Next Page