Piecewise deterministic Markov processes and their invariant measures

被引：21

作者：

Durmus, Alain ^{[1
]}

Guillin, Arnaud ^{[2
]}

Monmarche, Pierre ^{[3
,4
]}

机构：

[1] Univ Paris Saclay, Ctr Borelli, CNRS, ENS Paris Saclay, F-91190 Gif Sur Yvette, France

[2] Univ Clermont Auvergne, Lab Math Blaise Pascal, CNRS, UMR 6620, Ave Landais, F-63177 Aubiere, France

[3] Sorbonne Univ, Lab Jacques Louis Lions, 4 Pl Jussieu, F-75005 Paris, France

[4] Sorbonne Univ, Lab Chim Theor, 4 Pl Jussieu, F-75005 Paris, France

来源：

ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES | 2021年 / 57卷 / 03期

关键词：

PDMP; Generator; Synchronous coupling; Invariant measure; Bouncy particle sampler; ERGODICITY; SIMULATION; STABILITY;

D O I：

10.1214/20-AIHP1125

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

摘要：

Piecewise Deterministic Markov Processes (PDMPs) are studied in a general framework. First, different constructions are proven to be equivalent. Second, we introduce a coupling between two PDMPs following the same differential flow which implies quantitative bounds on the total variation between the marginal distributions of the two processes. Finally two results are established regarding the invariant measures of PDMPs. A practical condition to show that a probability measure is invariant for the associated PDMP semi-group is presented. In a second time, a bound on the invariant probability measures in V-norm of two PDMPs following the same differential flow is established. This last result is then applied to study the asymptotic bias of some non-exact PDMP MCMC methods.

引用

页码：1442 / 1475

页数：34

共 25 条

[1]

Azais Romain, 2014, ESAIM Proceedings, V44, P276, DOI 10.1051/proc/201444017

[2] Qualitative properties of certain piecewise deterministic Markov processes [J].

Benaim, Michel ;

Le Borgne, Stephane ;

Malrieu, Florent ;

Zitt, Pierre-Andre .

ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, 2015, 51 (03) :1040-1075

[3]

Bertsekas D. P., 1996, Neuro-Dynamic Programming

[4] The Bouncy Particle Sampler: A Nonreversible Rejection-Free Markov Chain Monte Carlo Method [J].

Bouchard-Cote, Alexandre ;

Vollmer, Sebastian J. ;

Doucet, Arnaud .

JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION, 2018, 113 (522) :855-867

[5]

Bouin E., 2020, Pure Appl. Anal, V2, P203

[6] CONFINEMENT BY BIASED VELOCITY JUMPS: AGGREGATION OF ESCHERICHIA COLI [J].

Calvez, Vincent ;

Raoul, Gael ;

Schmeiser, Christian .

[7] Stability and ergodicity of piecewise deterministic Markov processes [J].

Costa, O. L. V. ;

Dufour, F. .

SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 2008, 47 (02) :1053-1077

[8]

Davis M. H., 2018, Markov Models Optimization, DOI [10.1007/978-1-4899-4483-2, DOI 10.1007/978-1-4899-4483-2]

[9] EXPONENTIAL ERGODICITY OF THE BOUNCY PARTICLE SAMPLER [J].

Deligiannidis, George ;

Bouchard-Cote, Alexandre ;

Doucet, Arnaud .

ANNALS OF STATISTICS, 2019, 47 (03) :1268-1287

[10]

Douc R., 2018, Markov chains, DOI DOI 10.1007/978-3-319-97704-1

← 1 2 3 →