SVI solutions to stochastic nonlinear diffusion equations on general measure spaces

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 \((E,\mathcal {B},\mu )\) $(E,\mathcal{B},\mu )$ and the initial value in \(\mathcal {F}_e^*\) ${\mathcal{F}}_e^{\ast }$ , which is the dual space of an extended transient Dirichlet space. L and \(\mathcal {F}_e^*\) ${\mathcal{F}}_e^{\ast }$ replace the Laplace operator \(\Delta \) $\mathrm{\Delta }$ and \(H^{-1}\) $H^{-1}$ , respectively, in the classical case. This framework includes stochastic fast diffusion equations, stochastic fractional fast diffusion equations, the Zhang model, and applies to cases with E being a manifold, a fractal, or a graph. In addition, our results apply to operators \(-f(-L)\) $-f(-L)$ , where f is a Bernstein function, e.g., \(f(\lambda )=\lambda ^\alpha \) $f(\lambda )={\lambda }^{\alpha }$ or \(f(\lambda )=(\lambda +1)^\alpha -1\) $f(\lambda )={(\lambda +1)}^{\alpha }-1$ , \(0<\alpha <1\) $0<\alpha <1$ .