Accessibility, Martin boundary and minimal thinness for Feller processes in metric measure spaces

Kim, Panki; Song, Renming; Vondraček, Zoran

doi:10.4171/RMI/995

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Revista Matemática Iberoamericana

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Volume 34, Issue 2, 2018, pp. 541–592

DOI: 10.4171/RMI/995

Published online: 2018-05-28

Accessibility, Martin boundary and minimal thinness for Feller processes in metric measure spaces

Panki Kim^[1], Renming Song^[2] and Zoran Vondraček^[3]

(1) Seoul National University, Republic of Korea
(2) University of Illinois at Urbana-Champaign, USA and Nankai University, Tianjin, China
(3) University of Zagreb, Croatia and University of Illinois, Urbana, USA

In this paper we study the Martin boundary at infinity for a large class of purely discontinuous Feller processes in metric measure spaces. We show that if $\infty$ is accessible from an open set $D$, then there is only one Martin boundary point of $D$ associated with it, and this point is minimal. We also prove the analogous result for finite boundary points. As a consequence, we show that minimal thinness of a set is a local property.

Keywords: Martin boundary, Martin kernel, purely discontinuous Feller process, minimal thinness

Kim Panki, Song Renming, Vondraček Zoran: Accessibility, Martin boundary and minimal thinness for Feller processes in metric measure spaces. Rev. Mat. Iberoam. 34 (2018), 541-592. doi: 10.4171/RMI/995