Tianyuan Mathematical Center
in Southwest China

Potential theory of Dirichlet forms degenerate at the boundary

[Math. Dept.]

November 10, 2020  09:30-10:30

腾讯会议（线上）

ID: 455 133 487     Password: 23579

SPEAKER

Renming Song (University of Illinois at Urbana-Champaign & Sichuan University)

ABSTRACT

In this talk I will present some recent results on the potential theory of Markov processes with jump kernels degenerate at the boundary of its state space $R^d_+$, the upper half space  of $R^d$. The jump kernel is of the form $J^D(x, )=|x-y|^{-d-\alpha}B(x, y)$, where $\alpha\in (0, 2)$ and $B(x, y)$, which involves three parameters $\beta_1, \beta_2$ and $\beta_2$, tends to 0 when $x$  or $y$ tends to the boundary. We assume that the killing function is of the form $\kappa(x)=cx_d^{-\alpha}$. Our main results are sharp two-sided estimates on the Green functions of these processes. As applications of the green function estimates, we show that the boundary Harnack principle holds for certain region of the parameters and fails in the other region. This talk is based on two joint papers with Panki Kim and Zoran Vondracek.

SUPPORTED BY

Tianyuan Mathematical Center in Southwest China

School of Mathematics, Sichuan University

VIDEO