PUB - Publikationen an der Universität Bielefeld

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, B, mu) and the initial value in F-e*. which is the dual space of an extended transient Dirichlet space. L and 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 applies to cases with E being a manifold, a fractal, or a graph. In addition, our results apply to operators -f(-L), where is a Bernstein function, e.g., f(lambda) = lambda(alpha) or f(lambda) = (lambda + 1)(alpha) - 1, 0 < alpha < 1.