Gaussian upper density estimates for spatially homogeneous SPDEs

2012/06/14 by Lluís Quer-Sardanyons, Quer-Sardanyons, Lluis
Economics, Econometrics and Finance · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1206.3328

openalex publication_date 2012/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a general class of SPDEs in ℝ^d driven by a Gaussian spatially homogeneous noise which is white in time. We provide sufficient conditions on the coefficients and the spectral measure associated to the noise ensuring that the density of the corresponding mild solution admits an upper estimate of Gaussian type. The proof is based on the formula for the density arising from the integration-by-parts formula of the Malliavin calculus. Our result applies to the stochastic heat equation with any space dimension and the stochastic wave equation with d∈ \1,2,3\. In these particular cases, the condition on the spectral measure turns out to be optimal.

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