The stochastic porous media equation in $\R^d$

Abstract

Existence and uniqueness of solutions to the stochastic porous media equation $dX-\Dψ(X) dt=XdW$ in $\rr^d$ are studied. Here, $W$ is a Wiener process, $ψ$ is a maximal monotone graph in $\rr\times\rr$ such that $ψ(r)\le C|r|^m$, $\ff r\in\rr$, $W$ is a coloured Wiener process. In this general case the dimension is restricted to $d\ge 3$, the main reason being the absence of a convenient multiplier result in the space $\calh=\{φ\in\mathcal{S}'(\rr^d);\ |ξ|(\calfφ)(ξ)\in L^2(\rr^d)\}$, for $d\le2$. When $ψ$ is Lipschitz, the well-posedness, however, holds for all dimensions on the classical Sobolev space $H^{-1}(\rr^d)$. If $ψ(r)r\geρ|r|^{m+1}$ and $m=\frac{d-2}{d+2}$, we prove the finite time extinction with strictly positive probability.