Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations

Abstract

In this paper we establish lower and upper Gaussian bounds for the solutions to the heat and wave equations driven by an additive Gaussian noise, using the techniques of Malliavin calculus and recent density estimates obtained by Nourdin and Viens in [17]. In particular, we deal with the one-dimensional stochastic heat equation in [0, 1] driven by the space-time white noise, and the stochastic heat and wave equations in R d ( d ≥ 1 and d ≤ 3 , respectively) driven by a Gaussian noise which is white in time and has a general spatially homogeneous correlation.