SVI solutions to stochastic nonlinear diffusion equations on general measure spaces
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914664403697664 |
|---|---|
| author | Gess, Benjamin Röckner, Michael Wu, Weina |
| author_facet | Gess, Benjamin Röckner, Michael Wu, Weina |
| contents | 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 $f(λ)=(λ+1)^α-1$, $0<α<1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_01479 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | SVI solutions to stochastic nonlinear diffusion equations on general measure spaces Gess, Benjamin Röckner, Michael Wu, Weina Probability 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 $f(λ)=(λ+1)^α-1$, $0<α<1$. |
| title | SVI solutions to stochastic nonlinear diffusion equations on general measure spaces |
| topic | Probability |
| url | https://arxiv.org/abs/2402.01479 |