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

Chaleyat-Maurel, Mireille; Sanz-Solé, Marta

ESAIM: Probability and Statistics, Tome 7 (2003), pp. 89-114.

We consider the random vector $u(t,\underline{x})=(u(t,x_1),\cdots ,u(t,x_d))$, where $t>0,x_1,\cdots ,x_d$ are distinct points of ${ℝ}^2$ 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 $u(t,\underline{x})$. 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 $y\in {ℝ}^d$ where the density is positive and we prove that, under suitable assumptions, this set is ${ℝ}^d$.

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