Stochastic processes and potential theory: the fundamentals,Potential theory of stochastic processes with jump kernels degenera

Title: Stochastic processes and potential theory: the fundamentals

Abstract: The goal of this talk is to present some fundamentals of stochastic processes and the probabilistic potential theory. I will first introduce some basic notions related to time-continuous stochastic processes and the concept of a martingale. In the main part of the talk, I will discuss in some detail the multidimensional Brownian motion. The discussion will include the strong Markov property, the Brownian semigroup, the infinitesimal generator, the corresponding Dirichlet form, and the potential operator. By use of Dynkin's formula I will give the probabilistic definition of a harmonic function and the interpretation of the Poisson kernel.  Then I will describe two fundamental results on positive harmonic functions. This part of the talk will end with the killed version of Brownian motion. In the last part of the talk I will focus on the isotropic stable process and compare its potential theory to the one of Brownain motion. Although there are many similarities, we will see some fundamental differences.

Title: Potential theory of stochastic processes with jump kernels degenerate at the boundary

Abstract: The goal of this talk is to present some recent results on jump processes in the upper half-space of $\mathbb{R}^d$ whose jump kernels are of the form $J(x,y)=\mathcal{B}(x,y)|x-y|^{-d-\alpha}$, where $)=\mathcal{B}( (x,y)$ may decay at the boundary or explode at the boundary of the state space. This is in sharp contrast with the most of the previous results in the literature where $)=\mathcal{B}( (x,y)$ was assumed to be bounded between two positive constants -- a sort of a uniformly elliptic condition. I will first give a brief overview of those previous results including the killed and the censored stable process. Then I will present examples of jump processes whose jump kernels are degenerate at the boundary -- these examples served as the motivation for the general theory of such processes. The main part of the talk will be devoted to the setting and potential-theoretic results of processes with jump kernels exploding at the boundary. I will discuss the boundary Harnack principle and sharp two-sided Green function estimates. Comparison with the corresponding results for processes with jump kernels decaying at the boundary will be also considered.

(Joint work with Panki Kim and Renming Song)