  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 $\mathcal{F}_e^*$, which is the dual space of an extended
transient Dirichlet space. $L$ and $\mathcal{F}_e^*$ 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 $f(\lambda)=(\lambda+1)^\alpha-1$, $0<\alpha<1$.
