Asymptotics of the density of parabolic Anderson random fields

Abstract

We investigate the sharp density $\rho(t,x; y)$ of the solution $u(t,x)$ to stochastic partial differential equation $\frac{\partial }{\partial t} u(t,x)=\frac12 \Delta u(t,x)+u\diamond \dot W(t,x)$, where $\dot W$ is a general Gaussian noise and $\diamond$ denotes the Wick product. We mainly concern with the asymptotic behavior of $\rho(t,x; y)$ when $y\rightarrow \infty$ or when $t\to0+$. Both upper and lower bounds are obtained and these two bounds match each other modulo some multiplicative constants. If the initial datum is positive, then $\rho(t,x;y)$ is supported on the positive half line $y\in [0, \infty)$ and in this case we show that $\rho(t,x; 0+)=0$ and obtain an upper bound for $\rho(t,x; y)$ when $y\rightarrow 0+$.