Stochastic porous media equations in R^d

Existence and uniqueness of solutions to the stochastic porous media equation dX-δψ(X)dt=XdW in Rd are studied. Here, W is a Wiener process, ψ is a maximal monotone graph in R×R such that ψ(r)≤C|r|^m, ∀r∈R. In this general case, the dimension is restricted to d≥3, the main reason being the absence of a convenient multiplier result in the space H={ϕ∈S'(Rd);|ξ|(Fϕ)(ξ)∈L2(Rd)}, for d≤2. When ψ is Lipschitz, the well-posedness, however, holds for all dimensions on the classical Sobolev space H-1(Rd). If ψ(r)r≥ρ|r|^m+1 and m=d-2d+2, we prove the finite time extinction with strictly positive probability.