Optimal Gaussian density estimates for a class of stochastic equations with additive noise

2009/12/18 by Nualart, David, Quer-Sardanyons, Lluis
#60H07 #60H15 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0912.3707

Abstract

In this note, we establish optimal lower and upper Gaussian bounds for the density of the solution to a class of stochastic integral equations driven by an additive spatially homogeneous Gaussian random field. The proof is based on the techniques of the Malliavin calculus and a density formula obtained by Nourdin and Viens. Then, the main result is applied to the mild solution of a general class of SPDEs driven by a Gaussian noise which is white in time and has a spatially homogeneous correlation. In particular, this covers the case of the stochastic heat and wave equations in ℝ^d with d≥ 1 and d≤ 3, respectively. The upper and lower Gaussian bounds have the same form and are given in terms of the variance of the stochastic integral term in the mild form of the equation.

Related

• Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations
• Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension
• Gaussian upper density estimates for spatially homogeneous SPDEs
• Gaussian estimates of the density for systems of non-linear stochastic heat equations
• Existence and smoothness of the density for spatially homogeneous SPDEs