1. Space and time inversions of stochastic processes and Kelvin transform
- Author
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Larbi Alili, Loïc Chaumont, Tomasz Żak, Piotr Graczyk, Department of Mathematics, University of Warwick, Warwick Mathematics Institute (WMI), University of Warwick [Coventry]-University of Warwick [Coventry], Laboratoire Angevin de Recherche en Mathématiques (LAREMA), Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS), Technical university of Wroclaw, Panorisk, and Graczyk, Piotr
- Subjects
Spacetime ,Stochastic process ,General Mathematics ,Gaussian ,010102 general mathematics ,Mathematical analysis ,Probability (math.PR) ,Markov process ,Inversion (meteorology) ,[MATH] Mathematics [math] ,01 natural sciences ,law.invention ,010101 applied mathematics ,symbols.namesake ,60J45, 31C05 (Primary), 60J65, 60J60 (Secondary) ,law ,symbols ,FOS: Mathematics ,0101 mathematics ,[MATH]Mathematics [math] ,Mathematics - Probability ,Bessel function ,Brownian motion ,Kelvin transform ,Mathematics - Abstract
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.
- Published
- 2017
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