Stationary distributions for jump processes with memory

K. Burdzy; T. Kulczycki; R. L. Schilling

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 3, page 609-630
  • ISSN: 0246-0203

Abstract

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We analyze a jump processes Z with a jump measure determined by a “memory” process S . The state space of ( Z , S ) is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of ( Z , S ) is the product of the uniform probability measure and a Gaussian distribution.

How to cite

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Burdzy, K., Kulczycki, T., and Schilling, R. L.. "Stationary distributions for jump processes with memory." Annales de l'I.H.P. Probabilités et statistiques 48.3 (2012): 609-630. <http://eudml.org/doc/272069>.

@article{Burdzy2012,
abstract = {We analyze a jump processes $Z$ with a jump measure determined by a “memory” process $S$. The state space of $(Z,S)$ is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of $(Z,S)$ is the product of the uniform probability measure and a Gaussian distribution.},
author = {Burdzy, K., Kulczycki, T., Schilling, R. L.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stationary distribution; stable Lévy process; process with memory},
language = {eng},
number = {3},
pages = {609-630},
publisher = {Gauthier-Villars},
title = {Stationary distributions for jump processes with memory},
url = {http://eudml.org/doc/272069},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Burdzy, K.
AU - Kulczycki, T.
AU - Schilling, R. L.
TI - Stationary distributions for jump processes with memory
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 3
SP - 609
EP - 630
AB - We analyze a jump processes $Z$ with a jump measure determined by a “memory” process $S$. The state space of $(Z,S)$ is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of $(Z,S)$ is the product of the uniform probability measure and a Gaussian distribution.
LA - eng
KW - stationary distribution; stable Lévy process; process with memory
UR - http://eudml.org/doc/272069
ER -

References

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  1. [1] M. Barlow, A. Grigor’yan and T. Kumagai. Heat kernel upper bounds for jump processes and the first exit time. J. Reine Angew. Math.626 (2009) 135–157. Zbl1158.60039MR2492992
  2. [2] R. Bass, K. Burdzy, Z. Chen and M. Hairer. Stationary distributions for diffusions with inert drift. Probab. Theory Related Fields146 (2010) 1–47. Zbl1210.60058MR2550357
  3. [3] K. Burdzy, T. Kulczycki and R. Schilling. Stationary distributions for jump processes with inert drift. Preprint, 2010. Available at arXiv:1009.2347. Zbl1291.60170
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  5. [5] K. Burdzy and D. White. Markov processes with product-form stationary distribution. Electron. Commun. Probab.13 (2008) 614–627. Zbl1189.60139MR2461535
  6. [6] S. N. Ethier and T. G. Kurtz. Markov Processes: Characterization and Convergence. Wiley, New York, 1986. Zbl1089.60005MR838085
  7. [7] N. Ikeda, N. Nagasawa and S. Watanabe. A construction of Markov processes by piecing out. Proc. Japan Acad. Ser. A Math. Sci.42 (1966) 370–375. Zbl0178.53401MR202197
  8. [8] P.-A. Meyer. Renaissance, recollements, mélanges, ralentissement de processus de Markov. Ann. Inst. Fourier (Grenoble) 25 (1975) 464–497. Zbl0304.60041MR415784
  9. [9] Ya. G. Sinai. Topics in Ergodic Theory. Princeton Univ. Press, Princeton, NJ, 1994. Zbl0805.58005MR1258087

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