Evolution Equations with General Fractional Derivatives as Models of Anomalous Diffusion 

Abstract: In this talk, we discuss a class of anomalous diffusion models in form of evolution equations involving the general time-fractional derivatives with Sonin kernels. Under suitable conditions, these equations can be embedded within the framework of the classical continuous-time random walk model, with the waiting-time probability density function determined by the corresponding Sonin kernels.

Then we focus on an important particular case of this model in form of a fractional diffusion equation involving the general time-fractional derivatives with Sonin kernels. In particular, a concise formula for the mean squared displacement of the diffusing particles governed by this equation as well as an interpretation of its fundamental solution as a spatial probability density function evolving in time are discussed.  

Time-Fractional Diffusion: The Role of Dimension in Decay and Harnack Estimates 

Abstract: I will focus on two types of estimates for time-fractional diffusion equations: large-time decay estimates and Harnack estimates. Particular emphasis will be placed on the role of the spatial dimension. We will see that, in contrast to the classical parabolic case, in the time-fractional setting the spatial dimension may determine the very form of an estimate and even whether such an estimate can hold at all.

Fractional Monge–Ampère equations

Abstract: A relatively long-standing open problem in the fractional and nonlocal PDE communities is to find the adequate, most appropriate version of a fractional, nonlocal Monge–Ampère equation. We will discuss several possible definitions and their properties, and present some recent progress.