The stochastic porous media equation in ${\mathbb{R}}^d$

Existence and uniqueness of solutions to the stochastic porous media equation $dX-\text{D}\psi (X)dt=XdW$ in ${\text{rr}}^d$ are studied. Here, $W$ is a Wiener process, $\psi$ is a maximal monotone graph in $\text{rr}\times \text{rr}$ such that $\psi®\le C|r|^m$,$\ff r\in\rr$,$W$ 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 $\calh={\varphi\in$\mathcal{S}$'(\rr^d);\ |\xi|(\calf\varphi)(\xi)\in L^2(\rr^d)}$,for$d\le2$\mathrm{.}When$\psi$ is Lipschitz, the well-posedness, however, holds for all dimensions on the classical Sobolev space $H^{-1}({\text{rr}}^d)$. If $\psi (r)r\ge \rho \mathrm{\mid }r{\mathrm{\mid }}^{m+1}$ and $m=\frac{d-2}{d+2}$, we prove the finite time extinction with strictly positive probability.