1. INTRODUCTION 5
the Dirac delta–function (see (2.2)), and the study of the corresponding smoothed
equation, to which we can successfully apply standard techniques of stochastic cal-
culus. Secondly, at the level of the smoothed equation, increments of stochastic
integrals, whether in space or in time, initially expressed in terms of increments of
the fundamental solution, can be reexpressed in terms of increments of the covari-
ance function of the noise. Using the semigroup property of Riesz potentials (see for
instance ), we implement the ideas concerning fractional derivatives described
above (see Lemma 2.6). With these results, we are able to obtain bounds on one
and two dimensional increments, in space and in time, of certain generalized Riesz
potentials of a smoothed version of the fundamental solution of the wave equation.
The sharp character of these estimates leads to the optimality of our results.
The paper is organized as follows. In Chapter 2, we define the smoothings Gn
of the fundamental solution G and prove some of their basic properties. Then we
describe precisely the type of stochastic noise we are considering in the paper and
prove the above mentioned fractional derivative properties.
In Chapter 3, we study the path properties of the indefinite stochastic integral
introduced in . Briefly stated, we prove that if the sample paths of the stochastic
integrand belong to some fractional Sobolev space with a fixed order of differentia-
bility, then the stochastic integral inherits the same property with a related order
of differentiability (Theorem 3.1). This fact, together with Sobolev’s embeddings
of increments in time of the stochastic integral (Theorem 3.8),
complete the analysis.
Chapter 4 is devoted to the study of equation (1.1) itself. The idea is to transfer
the properties of the stochastic integral obtained in Chapter 3 to the solution of
the equation. First, in Section 4.1, we give a more general version of existence and
uniqueness of a solution and its properties than in , allowing non vanishing initial
conditions and an additive non-linearity b. We also show how the
the solution depend on properties of the initial conditions (see Theorem 4.1).
Next, in Section 4.2, we go beyond the Lq–norm in the space variable. We see
in Theorem 4.6 how the assumptions on the initial conditions and on
F imply that
the fractional Sobolev norm in the space variable of the solution of equation (1.1) is
finite. A first step is to analyse the effect of the initial conditions on the fractional
Sobolev norm of the solution (Lemma 4.4). The main analysis is carried out at
the level of the s.p.d.e. driven by the smoothed kernel Gn, and then transferred
to the solution of equation (1.1) by means of an approximation result proved in
Proposition 4.3 and Fatou’s lemma. Using the Sobolev embeddings, we obtain the
H¨ older continuity property in the space variable of the sample paths.
In Section 4.3, we prove regularity in time using the classical approach based
on Kolmogorov’s continuity criterion. We fix a bounded domain D ⊂ R3 and we
first study the H¨ older continuity in time, uniformly in x ∈ D, of the contribution of
the initial conditions (Lemma 4.9). Secondly, we find upper bounds for the
norm of increments in time of the solution of (1.1), uniformly in x ∈ D (Theorem
4.10). We finish with the joint H¨ older continuity in space and in time stated in
Theorem 4.11. In particular, these results are non-trivial even for the deterministic
inhomogeneous three-dimensional wave equation (see Corollary 4.12).
In Chapter 5, we check the sharpness of the results proved in Chapter 4 by
considering the simplest case consisting of an equation with vanishing initial con-
ditions and coeﬃcient b, and a constant coeﬃcient σ. In this case, the solution