Logarithmic asymptotics of the densities of SPDEs driven by spatially correlated noise

Abstract: We consider the family of stochastic partial differential equations indexed by a parameter $\eps\in(0,1]$, \begin{equation*} Lu^{\eps}(t,x) = \eps\sigma(u^\eps(t,x))\dot{F}(t,x)+b(u^\eps(t,x)), \end{equation*} $(t,x)\in(0,T]\times\Rd$ with suitable initial conditions. In this equation, $L$ is a second-order partial differential operator with constant coefficients, $\sigma$ and $b$ are smooth functions and $\dot{F}$ is a Gaussian noise, white in time and with a stationary correlation in space. Let $p^\eps_{t,x}$ denote the density of the law of $u^\eps(t,x)$ at a fixed point $(t,x)\in(0,T]\times\Rd$. We study the existence of $\lim_{\eps\downarrow 0} \eps^2\log p^\eps_{t,x}(y)$ for a fixed $y\in\R$. The results apply to a class of stochastic wave equations with $d\in{1,2,3}$ and to a class of stochastic heat equations with $d\ge1$.