Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Ergodicity for multidimensional jump diffusions with position dependent jump rate

Eva Löcherbach and Victor Rabiet

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Abstract

We consider a jump type diffusion $X=(X_{t})_{t}$ with infinitesimal generator given by

\[L\psi(x)=\frac{1}{2}\sum_{1\le i,j\le d}a_{ij}(x)\frac{\partial^{2}\psi(x)}{\partial x_{i}\,\partial x_{j}}+g(x)\nabla\psi(x)+\int_{\mathbb{R}^{d}}(\psi (x+c(z,x))-\psi(x))\gamma(z,x)\mu(\mathrm{d}z),\] where $\mu$ is of infinite total mass. We prove Harris recurrence of $X$ using a regeneration scheme which is entirely based on the jumps of the process. Moreover we state explicit conditions in terms of the coefficients of the process allowing to control the speed of convergence to equilibrium in terms of deviation inequalities for integrable additive functionals.

Résumé

On considère une diffusion $X=(X_{t})_{t}$, avec des sauts, correspondant au générateur infinitésimal suivant :

\[L\psi(x)=\frac{1}{2}\sum_{1\le i,j\le d}a_{ij}(x)\frac{\partial^{2}\psi(x)}{\partial x_{i}\,\partial x_{j}}+g(x)\nabla\psi(x)+\int_{\mathbb{R}^{d}}(\psi (x+c(z,x))-\psi(x))\gamma(z,x)\mu(\mathrm{d}z)\] où $\mu$ est de masse totale infinie. On prouve ici la récurrence au sens de Harris de $X$ en utilisant un schéma de régénération entièrement basé sur les sauts du processus. De plus, on donnera des conditions explicites en terme de coefficients du processus $X$ permettant de contrôler la vitesse de convergence à l’équilibre en terme d’inégalités de déviations pour des fonctionnelles additives intégrables.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1136-1163.

Dates
Received: 21 April 2015
Revised: 1 March 2016
Accepted: 1 March 2016
First available in Project Euclid: 21 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1500624033

Digital Object Identifier
doi:10.1214/16-AIHP750

Mathematical Reviews number (MathSciNet)
MR3689963

Zentralblatt MATH identifier
1372.60115

Subjects
Primary: 60J55: Local time and additive functionals 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60F10: Large deviations 62M05: Markov processes: estimation

Keywords
Diffusions with jumps Harris recurrence Nummelin splitting Continuous time Markov processes Additive functionals

Citation

Löcherbach, Eva; Rabiet, Victor. Ergodicity for multidimensional jump diffusions with position dependent jump rate. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1136--1163. doi:10.1214/16-AIHP750. https://projecteuclid.org/euclid.aihp/1500624033


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