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Space and time inversions of stochastic processes and Kelvin transform
Author(s) -
Alili L.,
Chaumont L.,
Graczyk P.,
Żak T.
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201700152
Subject(s) - mathematics , bessel process , bessel function , inversion (geology) , brownian motion , harmonic function , mathematical analysis , gaussian , statistical physics , pure mathematics , quantum mechanics , physics , classical orthogonal polynomials , paleontology , statistics , gegenbauer polynomials , structural basin , orthogonal polynomials , biology
Let X be a standard Markov process. We prove that a space inversion property of X implies the existence of a Kelvin transform of X ‐harmonic, excessive and operator‐harmonic functions and that the inversion property is inherited by Doob h ‐transforms. We determine new classes of processes having space inversion properties amongst transient processes satisfying the time inversion property. For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly. We treat in details the examples of free scaled power Bessel processes, non‐colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion.