Title:

Some properties of a class of stochastic heat equations

We study stochastic heat equations of the forms $[\partial_t u\sL u]\d t\d x=\lambda\int_\R\sigma(u,h)\tilde{N}(\d t,\d x,\d h),$ and $[\partial_t u\sL u]\d t\d x=\lambda\int_{\R^d}\sigma(u,h)N(\d t,\d x,\d h)$. Here, $u(0,x)=u_0(x)$ is a nonrandom initial function, $N$ a Poisson random measure with its intensity $\d t\d x\nu(\d h)$ and $\nu(\d h)$ a L\'{e}vy measure; $\tilde{N}$ is the compensated Poisson random measure and $\sL$ a generator of a L\'{e}vy process. The function $\sigma:\R\rightarrow\R$ is Lipschitz continuous and $\lambda>0$ the noise level. The above discontinuous noise driven equations are not always easy to handle. They are discontinuous analogues of the equation introduced in \cite{Foondun} and also more general than those considered in \cite{Saint}. We do not only compare the growth moments of the two equations with each other but also compare them with growth moments of the class of equations studied in \cite{Foondun}. Some of our results are significant generalisations of those given in \cite{Saint} while the rest are completely new. Second and first growth moments properties and estimates were obtained under some linear growth conditions on $\sigma$. We also consider $\sL:=(\Delta)^{\alpha/2}$, the generator of $\alpha$stable processes and use some explicit bounds on its corresponding fractional heat kernel to obtain more precise results. We also show that when the solutions satisfy some nonlinear growth conditions on $\sigma$, the solutions cease to exist for both compensated and noncompensated noise terms for different conditions on the initial function $u_0(x)$. We consider also fractional heat equations of the form $ \partial_t u(t,x)=(\Delta)^{\alpha/2}u(t,x)+\lambda\sigma(u(t,x)\dot{F}(t,x),\,\, \text{for}\,\, x\in\R^d,\,t>0,\,\alpha\in(1,2),$ where $\dot{F}$ denotes the Gaussian coloured noise. Under suitable assumptions, we show that the second moment $\Eu(t,x)^2$ of the solution grows exponentially with time. In particular we give an affirmative answer to the open problem posed in \cite{Conus3}: given $u_0$ a positive function on a set of positive measure, does $\sup_{x\in\R^d}\Eu(t,x)^2$ grow exponentially with time? Consequently we give the precise growth rate with respect to the parameter $\lambda$.
