Some new examples of Markov processes which enjoy the time-inversion property
\textit{S. Watanabe} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 31, 115--124 (1975; Zbl 0286.60035)] established the problem of finding time-depending homeomorphisms \(g(t,.)\) for the state spaces \(E\) and \(E'\) of 2 conservative diffusions \((X(t),P)\) and \((X'(t),P')\) in such a way that \((g(t,X(1/t)),P)\) and \((X'(t),P')\) are equivalent. As an important tool to find some answers Watanabe used M. G. Krein's spectral theory of the string. The present authors have modified the time-inversion problem from above to the following one: For a given homogeneous Markov process \((X(t),P.)\) in \(\mathbb{R}^n\) find conditions for the associated Markov semigroup under which the process \((tX(1/t),P.)\) is also a homogeneous Markov process. Using quite different methods some answers are given. The impressive set of very different examples contains, e.g., the examples of Bessel processes considered by Watanabe and several new ones as Dunkl processes and their radial parts.
scientific article; zbMATH DE number 2164757
scientific article