Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Stationary distributions for jump processes with memory

K. Burdzy, T. Kulczycki, and R. L. Schilling

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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.

Résumé

Nous proposons d’étudier un processus à sauts $Z$ avec une mesure de sauts déterminée par un processus $S$ représentant une “mémoire”. L’espace d’états de $(Z,S)$ est le produit Cartesien du cercle trigonométrique et de l’axe réel. Nous démontrons que la distribution stationnaire de $(Z,S)$ est la mesure produit d’une loi uniforme et d’une loi Gaussienne.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 3 (2012), 609-630.

Dates
First available in Project Euclid: 26 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1340714865

Digital Object Identifier
doi:10.1214/11-AIHP428

Mathematical Reviews number (MathSciNet)
MR2976556

Zentralblatt MATH identifier
1263.60072

Subjects
Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60G51: Processes with independent increments; Lévy processes 60J75: Jump processes 60J55: Local time and additive functionals

Keywords
Stationary distribution Stable Lévy process Process with memory

Citation

Burdzy, K.; Kulczycki, T.; Schilling, R. L. Stationary distributions for jump processes with memory. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 3, 609--630. doi:10.1214/11-AIHP428. https://projecteuclid.org/euclid.aihp/1340714865


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References

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