The Annals of Probability

Pinching and twisting Markov processes

Steven N. Evans and Richard B. Sowers

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Abstract

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.

Article information

Source
Ann. Probab., Volume 31, Number 1 (2003), 486-527.

Dates
First available in Project Euclid: 26 February 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1046294318

Digital Object Identifier
doi:10.1214/aop/1046294318

Mathematical Reviews number (MathSciNet)
MR1959800

Zentralblatt MATH identifier
1017.60084

Subjects
Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 60J40: Right processes 60J60: Diffusion processes [See also 58J65]

Keywords
Markov function intertwining right process Feller process excursion theory Walsh's spider stratified space

Citation

Evans, Steven N.; Sowers, Richard B. Pinching and twisting Markov processes. Ann. Probab. 31 (2003), no. 1, 486--527. doi:10.1214/aop/1046294318. https://projecteuclid.org/euclid.aop/1046294318


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  • UNIVERSITY OF CALIFORNIA, BERKELEY 367 EVANS HALL
  • BERKELEY, CALIFORNIA 94720-3860 E-MAIL: evans@stat.berkeley.edu WEB: http://www.stat.berkeley.edu/users/evans DEPARTMENT OF MATHEMATICS UNIVERSITY OF ILLINOIS
  • URBANA, ILLINOIS 60201 E-MAIL: r-sowers@uiuc.edu WEB: http://www.math.uiuc.edu/ r-sowers/