Positivity of the density for the stochastic wave equation in two spatial dimensions

¹ Université Pierre et Marie Curie, Laboratoire de Probabilités, 175/179 rue du Chevaleret, 75013 Paris, France; and Université René Descartes, 45 rue des Saints Pères, 75006 Paris, France; mcm@ccr.jussieu.fr.
² Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain; sanz@mat.ub.es.

Received: 8 January 2002

Abstract

We consider the random vector , where t > 0, x₁,...,x_d are distinct points of and u denotes the stochastic process solution to a stochastic wave equation driven by a noise white in time and correlated in space. In a recent paper by Millet and Sanz–Solé [10], sufficient conditions are given ensuring existence and smoothness of density for . We study here the positivity of such density. Using techniques developped in [1] (see also [9]) based on Analysis on an abstract Wiener space, we characterize the set of points where the density is positive and we prove that, under suitable assumptions, this set is .