SVI solutions to stochastic nonlinear diffusion equations on general measure spaces

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  • Authors: Benjamin Gess, Michael Röckner, Weina Wu
  • First public date: 2024-02-02
  • arXiv: 2402.01479
  • Preprint year: 2024
  • Status: Published
  • Publication type: Journal article
  • Publication year: 2024
  • Journal: Journal of evolution equations, 24 (2024) 4, p. 94
  • DOI: 10.1007/s00028-024-01023-z

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 $(E,\mathcal{B},μ)$ 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 $Δ$ 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. $f(λ)=λ^α$ or $f(λ)=(λ+1)^α-1$, $0<α<1$.

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BibTeX

@article{arxiv240201479,
  title = {SVI solutions to stochastic nonlinear diffusion equations on general measure spaces},
  author = {Benjamin Gess and Michael Röckner and Weina Wu},
  year = {2024},
  journal = {Journal of evolution equations, 24 (2024) 4, p. 94},
  doi = {10.1007/s00028-024-01023-z},
  eprint = {2402.01479},
  archivePrefix = {arXiv},
  primaryClass = {math.PR}
}

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