Behaviour of the density in perturbed SPDE's with spatially correlated noise

Abstract

Consider the following general type of perturbed stochastic partial differential equations: Lu ε t,x=εα u ε t,x F ̇ (t,x)+β u ε t,x , (t,x)∈ R +× R d, ε>0, with null initial conditions, L a second-order partial differential operator and F a Gaussian noise, white in time and correlated in space. In a previous work we proved the existence of smooth density p t,x ε(y), t>0, x∈ R d, for the law of the solution of above-mentioned equation. In this paper we study the logarithmic estimates for this density, that means to establish the behaviour of 2 ε 2log p t, x ε ( y), as ε↓0. This kind of estimates is also called Varadhan–Léandre estimates.