SVI solutions to stochastic nonlinear diffusion equations on general measure spaces

Abstract

We establish a framework for the existence and uniqueness of solutions to stochastic nonlinear (possibly multi-valued) diffusion equations driven by multiplicative noise, with the drift operator $L$ being the generator of a transient Dirichlet form on a finite measure space $(E,\mathcal{B},\mu )$ and the initial value in ${\mathcal{F}}_e^{\ast }$, which is the dual space of an extended transient Dirichlet space. $L$ and ${\mathcal{F}}_e^{\ast }$ replace the Laplace operator $\Delta$ and $H^{-1}$, respectively, in the classical case. This framework includes stochastic fast diffusion equations, stochastic fractional fast diffusion equations, the Zhang model, and apply to cases with $E$ being a manifold, a fractal or a graph. In addition, our results apply to operators $-f(-L)$, where $f$ is a Bernstein function, e.g. $f(\lambda )={\lambda }^{\alpha }$ or $f(\lambda )=(\lambda +1{)}^{\alpha }-1$, $0<\alpha <1$.

26 pages