Pinching and twisting Markov processes

Steven N. Evans, Richard B. Sowers

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)486-527
Number of pages42
JournalAnnals of Probability
Issue number1
StatePublished - Jan 2003


  • Excursion theory
  • Feller process
  • Intertwining
  • Markov function
  • Right process
  • Stratified space
  • Walsh's spider

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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