报告题目：Stochastic generalized porous media equations over $\sigma$-finite measure spaces with non-continuous diffusivity function

报告人：武伟娜

报告时间：1月12日 9:00-10:00

报告地点：管理科研楼1308

摘要：In this talk, I will show you that stochastic porous media equations over $\sigma$-finite measure spaces $(E,\mathcal{B},\mu)$, driven by time-dependent multiplicative noise, with the Laplacian replaced by a self-adjoint transient Dirichlet operator $L$ and the diffusivity function given by a maximal monotone multi-valued function $\beta$ of polynomial growth, have a unique strong solution. This generalizes previous results in that we work on general measurable state spaces, allow non-continuous monotone functions $\beta$, for which, no further assumptions (as e.g. coercivity) are needed, but only that their multi-valued extensions are maximal monotone and of at most polynomial growth. Furthermore, an $L^p(\mu)$-It\^{o} formula in expectation is proved, which is not only crucial for the proof of our main result, but also of independent interest. The result in particular applies to fast diffusion stochastic porous media equations (in particular self-organized criticality models) and cases where $E$ is a manifold or a fractal, and to non-local operators $L$, as e.g. $L=-f(-\Delta)$, where $f$ is Bernstein function.

报告题目：Title: SVI solutions to stochastic nonlinear diffusion equations on general measure spaces

报告人：武伟娜

报告时间：1月12日 10:00-11:00

报告地点：管理科研楼1308

摘要：In this talk, I will show you a framework for the existence and uniqueness of stochastic variational inequality (SVI) 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. $f(\lambda)=\lambda^\alpha$, $0<\alpha<1$, or $f(\lambda)=(\lambda+1)^\alpha-1$, $1<\alpha<1$.