1. INTRODUCTION 3

We consider the particular case of a covariance measure Γ that is absolutely

continuous with respect to Lebesgue measure, with density given by

(1.5) f(x) = ϕ(x) kβ (x),

where ϕ is a smooth positive function and kβ denotes the Riesz kernel kβ (x) = |x|−β,

with β ∈ ]0, 2[ (see Assumption 2.4). Riesz kernels are a natural class of correlation

functions and are already present in previous work on the stochastic heat and wave

equations, for instance in [6], [7], [15], [19]. They provide examples where condition

(1.4) is satisfied: for these covariances, (1.4) is equivalent to the condition 0 β 2

(see Example 2.5).

Related questions for an equation that is second order in time but with frac-

tional Laplacian in any spatial dimension d and general covariance measure Γ have

been considered in [10], in the setting of an

L2–theory

(see [30]). The results there

are shown to be optimal in time. We adopt here a similar strategy, but we work in

an

Lq

–framework (see [16]), for any q ≥ 2. Indeed, the particular structure of the

wave equation in dimension d = 3 makes it possible to go beyond the Hilbert space

setting and to obtain sharp results, both in time and space.

The main result of the paper is Theorem 4.11, stating joint H¨older-continuity

in (t, x) of the solution to (1.1), together with the analysis of the optimality of the

exponents studied in Chapter 5. The optimal H¨ older exponent is the same for the

time and space variables: this is an intrinsic property of the d’Alembert operator.

Moreover, this result shows how the driving noise

˙

F contributes to the roughness

of the sample paths, since it expresses the optimal H¨ older exponent in terms of

the parameters β and δ appearing in Assumption 2.4 on the covariance of

˙

F (see

Section 2.2).

Notice that for the stochastic heat equation with Lipschitz coeﬃcients in any

spatial dimension d ≥ 1, joint H¨ older-continuity in (t, x) of the sample paths of

the solution has been established in [32] (see also [38]). Unlike the stochastic

wave equation, the H¨ older exponent in the time variable is half that for the spatial

variable. This is also an intrinsic property of the heat operator. However, it turns

out that effect of the driving noise

˙

F on the regularity in the spatial variable is the

same for both equations (see Theorem 4.11 and Remark 4.8). Similar problems for

non-Lipschitz coeﬃcients have been recently tackled in [22].

We should point out that despite the similarities just mentioned, establishing

regularity results for the solution of the stochastic wave equation requires funda-

mentally different methods than those for the stochastic heat equation. Indeed,

taking for simplicity b ≡ 0 and vanishing initial conditions, equation (1.1), written

in integral form, becomes

u(t, x) =

t

0 Rd

G(t − s, x − v)Z(s, v)M(ds, dv),

where Z(s, v) = σ(u(s, v)). A spatial increment of the solution is

u(t, x) − u(t, y) =

t

0 Rd

(G(t − s, x − v) − G(t − s, y − v))Z(s, v)M(ds, dv).