Properties of the density for a three dimensional stochastic wave equation

2008/02/12 by Sanz-Solé, Marta
#35L05 #60G17 #60G60 #60H07 #60H15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0802.1607

Abstract

We consider a stochastic wave equation in space dimension three driven by a noise white in time and with an absolutely continuous correlation measure given by the product of a smooth function and a Riesz kernel. Let p_t,x(y) be the density of the law of the solution u(t,x) of such an equation at points (t,x)∈]0,T]× \IR³. We prove that the mapping (t,x)↦ p_t,x(y) owns the same regularity as the sample paths of the process \u(t,x), (t,x)∈]0,T]× \mathbbR³\ established Dalang and Sanz-Solé [Memoirs of the AMS, to appear]. The proof relies on Malliavin calculus and more explicitely, Watanabe's integration by parts formula and estimates derived form it.

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