Speaker: Krzysztof Bogdan

Title: Nonlocal Neumann Boundary Conditions

Abstract: We all know that Markov processes can be used to describe evolutionary phenomena such as the transfer of heat or spreading of a population in a given area. Such movements can occur continuously or through jumps. Their statistical effects are described by solving parabolic and elliptic equations. Dirichlet conditions in such equations are related to absorption of particles. They are well studied, because the stopped process is easy to define. The more difficult and varied Neumann conditions, associated with reflection of particles, are the subject of important current research. In particular, the censored process [1], which we studied with Krzysztof Burdzy and Zhen-Qing Chen, is an instance of Neumann-type conditions. But there are more and they are challenging, because each boundary condition requires construction of the corresponding process [4], [5]. The topic is of much interest for PDEs [2], [3], too. We will mainly discuss specific jump Markov processes and non-local boundary Neumann conditions from [5] and some work in progress.

[1] K. Bogdan, K. Burdzy, Z.-Q. Chen, Censored stable processes, Probab. Theory Related Fields, 2003, 127(1), 89–152 

[2] S. Dipierro, X. Ros-Oton, E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 2017, 33(2), 377–416 

[3] K. Bogdan, T. Grzywny, K. Pietruska-Pałuba, A. Rutkowski, Extension and trace for nonlocal operators, J. Math. Pures Appl. (9), 2020, 137, 33–69 

[4] Z. Vondracek, A probabilistic approach to a non-local quadratic form and its connection to the Neumann boundary condition problem, Math. Nachr., 2021, 294(1), 177–194 

[5] K. Bogdan, M. Kunze, The fractional Laplacian with reflections, Potential Anal, 2023+