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

AbstractConsider the following general type of perturbed stochastic partial differential equations: Luεt,x=εαuεt,xḞ(t,x)+βuεt,x,(t,x)∈R+×Rd,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 pt,xε(y),t>0,x∈Rd, 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 2ε2logpt,xε(y), as ε↓0. This kind of estimates is also called Varadhan–Léandre estimates