Stochastic porous media equation on general measure spaces with increasing Lipschitz nonlinearities

We prove the existence and uniqueness of probabilistically strong solutions to stochastic porous media equations driven by time-dependent multiplicative noise on a general measure space $(E,?\left(E\right),\mu )$, and the Laplacian replaced by a negative definite self-adjoint operator $L$. In the case of Lipschitz nonlinearities $\Psi$, we in particular generalize previous results for open $E?{\mathbb{R}}^d$ and $L=$Laplacian to fractional Laplacians. We also generalize known results on general measure spaces, where we succeeded in dropping the transience assumption on $L$, in extending the set of allowed initial data and in avoiding the restriction to superlinear behavior of $\Psi$ at infinity for $L^2\left(\mu \right)$-initial data.