The Annals of Probability

Pinching and twisting Markov processes

Steven N. Evans and Richard B. Sowers

Full-text: Open access


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

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

First available in Project Euclid: 26 February 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

Export citation


  • [1] BARLOW, M. T. and EVANS, S. N. (2002). Markov processes on vermiculated state spaces. Random Walks and Geometry (V. Kaimanovich, ed.). de Gruy ter, Berlin.
  • [2] BARLOW, M. T., PITMAN, J. and YOR, M. (1989). On Walsh's Brownian motions. Séminaire de Probabilités XXXII. Lecture Notes in Math. 1613 30-36. Springer, Berlin.
  • [3] BARLOW, M. T., ÉMERY, M., KNIGHT, F. B., SONG, S. and YOR, M. (1998). Author d'un théorème de Tsirelson sur les filtrations Browniennes et non Browniennes. Séminaire de Probabilités XXXII. Lecture Notes in Math. 1686 264-305. Springer, Berlin.
  • [4] BASS, R. F. and BURDZY, K. (2000). Fiber Brownian motion and the "hot spots" problem. Duke Math. J. 105 25-58.
  • [5] BERTOIN, J. (1996). Lévy processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press.
  • [6] BIANE, P. (1995). Interwining of Markov semi-groups, some examples. Séminaire de Probabilités XXIX. Lecture Notes in Math. 1613 30-36. Springer, Berlin.
  • [7] CARMONA, P., PETIT, F. and YOR, M. (1998). Beta-gamma random variables and intertwining relations between certain Markov processes. Rev. Mat. Iberoamericana 14 311-367.
  • [8] DELLACHERIE, C. and MEy ER, P.-A. (1978). Probabilities and Potential. North-Holland, Amsterdam.
  • [9] ETHIER, S. N. and KURTZ, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [10] FREIDLIN, M. and WENTZELL, A. D. (1994). Random perturbations of Hamiltonian sy stems. Mem. Amer. Math. Soc. 109 viii+82.
  • [11] KURTZ, T. G. (1998). Martingale problems for conditional distributions of Markov processes. Electronic J. Probab. 3(9).
  • [12] ROGERS, L. C. G. and PITMAN, J. W. (1981). Markov functions. Ann. Probab. 9 573-582.
  • [13] ROGERS, L. C. G. and WILLIAMS, D. (1978). Diffusions, Markov Processes, and Martingales: Itô Calculus 2. Wiley, Chichester.
  • [14] ROGERS, L. C. G. and WILLIAMS, D. (2000). Diffusions, Markov Processes, and Martingales 1: Foundations, 2nd ed. Cambridge Univ. Press.
  • [15] SHARPE, M. (1988). General Theory of Markov Processes. Academic Press, San Diego.
  • [16] SOWERS, R. B. (2002). Stochastic averaging with a flattened Hamiltonian: A Markov process on a stratified space (a whiskered sphere). Trans. Amer. Soc. 354 853-900.
  • [17] TSIRELSON, B. (1997). Triple points: From non-Brownian filtrations to harmonic measures. Geom. Funct. Anal. 7 1096-1142.
  • [18] WALSH, J. B. (1978). A diffusion with discontinuous local time. Temps Locaux. Astérisque 52-53 37-45.
  • [19] YOR, M. (1989). Une extension markovienne de l'algébre des lois béta-gamma. C. R. Acad. Sci. Paris Sér. I Math. 308 257-260.
  • URBANA, ILLINOIS 60201 E-MAIL: WEB: r-sowers/