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) (x),
where ϕ is a smooth positive function and denotes the Riesz kernel (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
(see [30]). The results there
are shown to be optimal in time. We adopt here a similar strategy, but we work in
–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 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 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 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 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) =
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) =
0 Rd
(G(t s, x v) G(t s, y v))Z(s, v)M(ds, dv).
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