Abstract
We study the sample path regularity of the solution of a stochastic wave equa-
tion in spatial dimension d = 3. The driving noise is white in time and with
a spatially homogeneous covariance defined as a product of a Riesz kernel and a
smooth function. We prove that at any fixed time, a.s., the sample paths in the
spatial variable belong to certain fractional Sobolev spaces. In addition, for any
fixed x
R3,
the sample paths in time are older continuous functions. Further,
we obtain joint older continuity in the time and space variables. Our results rely
on a detailed analysis of properties of the stochastic integral used in the rigourous
formulation of the s.p.d.e., as introduced by Dalang and Mueller (2003). Sharp
results on one- and two-dimensional space and time increments of generalized Riesz
potentials are a crucial ingredient in the analysis of the problem. For spatial co-
variances given by Riesz kernels, we show that the older exponents that we obtain
are optimal.
Received by the editor 19.12.2005.
2000 Mathematics Subject Classification. Primary 60H15; Secondary 60J45, 35R60, 35L05.
Key words and phrases. Stochastic partial differential equations, sample path regularity,
spatially homogeneous random noise, wave equation.
Partially supported by the Swiss National Foundation for Scientific Research.
Partially supported by the grant BFM2003-01345 from Direcci´ on General de Investigaci´on,
Ministerio de Ciencia y Tecnolog´ ıa, Spain.
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