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 $\frac{\partial }{\partial t}u(t,x)=\frac{1}{2}\Delta u(t,x)+u\diamond \dot{W}(t,x)$, 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 $y\to \infty$ or when $t\to 0+$. 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 $y\to 0+$.