# 澳门蒲京娱乐场

Density of parabolic Anderson random variable

I will present some results on  the shape estimates of the 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.  I mainly  concern with  the asymptotic behavior of    $\rho(t,x; y)$ when $y\rightarrow \infty$ 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 data is positive, then $\rho(t,x;y)$ is supported on the positive half line $y\in [0, \infty)$  and in this case $\rho(t,x; 0+)=0$   and I will give  n an upper bound for $\rho(t,x; y)$ when $y\rightarrow 0+$.