Pinching and twisting Markov processes
The authors give a far-reaching generalization of \textit{J. B. Walsh}'s spider process [in: Temps locaux. Astérisque 52-53 (1978; Zbl 0385.60063)]. The latter deals initially with reflecting Brownian motion on \(n\) copies of \(\mathbb R_+\); the process evolves this way until it reaches zero on some of the copies, at which time it moves with some probability (independently of the previous evolution) to the zero point of some (other or the same) copy. In the rigorous treatment, the state space of the process is obtained by identifying the \(n\) zero points of all the copies. In the paper under review the authors begin with a general Markov process and construct a new one whose state space is obtained by a pinching operation that identifies points of the original state space via an equivalence relation. To ensure that the new process is Markovian they introduce a randomized twist which, as well as the pinching, must be in a certain agreement with the semigroup of the original process. A description of the resolvent, transition function, and a core of the generator of the new process are given. An excursion theory is developed. The paper contains many examples with explicit calculations.
scientific article; zbMATH DE number 1906129
scientific article