Pinching and twisting Markov processes

Evans, Steven N. ; Sowers, Richard B.

Ann. Probab., Tome 31 (2003) no. 1, p. 486-527 / Harvested from Project Euclid

Résumé

We develop a technique for "partially collapsing'' one Markov process to produce another. The state space of the new Markov process is obtained by a pinching operation that identifies points of the original state space via an equivalence relationship. To ensure that the new process is Markovian we need to introduce a randomized twist according to an appropriate probability kernel. Informally, this twist randomizes over the uncollapsed region of the state space when the process leaves the collapsed region. The "Markovianity'' of the new process is ensured by suitable intertwining relationships between the semigroup of the original process and the pinching and twisting operations. We construct the new Markov process, identify its resolvent and transition function and, under some natural assumptions, exhibit a core for its generator. We also investigate its excursion decomposition. We apply our theory to a number of examples, including Walsh's spider and a process similar to one introduced by Sowers in studying stochastic averaging.

Détail

Publié le : 2003-01-14
Classification:  Markov function,  intertwining,  right process,  Feller process,  excursion theory,  Walsh's spider,  stratified space,  60J35,  60J40,  60J60