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SVI solutions to stochastic nonlinear diffusion equations on general measure spaces

Benjamin Gess, Michael Röckner, Weina Wu

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 $f(λ)=(λ+1)^α-1$, $0<α<1$.

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 being the generator of a transient Dirichlet form on a finite measure space and the initial value in , which is the dual space of an extended transient Dirichlet space. and replace the Laplace operator and , 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 being a manifold, a fractal or a graph. In addition, our results apply to operators , where is a Bernstein function, e.g. or , .
Paper Structure (10 sections, 7 theorems, 169 equations)

This paper contains 10 sections, 7 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Dirichlet spaces
  4. Some known results
  5. Assumptions and main results
  6. Applications
  7. Appendix
  8. Moreau-Yosida approximation of maximal monotone operators
  9. Lower semicontinuity of the energy potential funtional on $\mathcal{F}_e^*$
  10. Comparisons of different notions of solutions

Key Result

Lemma 2.1

(i) The map $\bar{L}:\mathcal{F}_e\rightarrow\mathcal{F}_e^*$ defined by (i.e. the Riesz isomorphism of $\mathcal{F}_e$ and $\mathcal{F}_e^*$ multiplied by (-1)) is the unique continuous linear extension of the map (ii) Let $u\in \mathcal{F}_e\cap L^{\frac{p}{p-1}}(\mu)$, $v\in \mathcal{F}_e^*\cap L^p(\mu)$. Then

Theorems & Definitions (33)

  • Lemma 2.1
  • Remark 3.1
  • Remark 3.2
  • Example 3.3
  • Example 3.4
  • Definition 3.5
  • Theorem 3.6
  • Remark 3.7
  • Proposition 3.8
  • proof
  • ...and 23 more