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A Perturbation Theory for Ergodic Markov Chains and Application to Numerical Approximations

Shardlow, T. and Stuart, A. M. (2000) A Perturbation Theory for Ergodic Markov Chains and Application to Numerical Approximations. SIAM Journal on Numerical Analysis, 37 (4). pp. 1120-1137. ISSN 0036-1429. doi:10.1137/S0036142998337235.

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Perturbations to Markov chains and Markov processes are considered. The unperturbed problem is assumed to be geometrically ergodic in the sense usually established through the use of Foster--Lyapunov drift conditions. The perturbations are assumed to be uniform, in a weak sense, on bounded time intervals. The long-time behavior of the perturbed chain is studied. Applications are given to numerical approximations of a randomly impulsed ODE, an Itô stochastic differential equation (SDE), and a parabolic stochastic partial differential equation (SPDE) subject to space-time Brownian noise. Existing perturbation theories for geometrically ergodic Markov chains are not readily applicable to these situations since they require very stringent hypotheses on the perturbations.

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Additional Information:© 2000 Society for Industrial and Applied Mathematics. Received by the editors April 13, 1998; accepted for publication (in revised form) September 16, 1998; published electronically March 23, 2000. Supported by the National Science Foundation under grant DMS-95-04879. We thank Peter Baxendale, Peter Glynn James Norris, and Neil O’Connell for helpful discussions and also the referees for a number of valuable comments.
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Subject Keywords:Markov chains, ergodic theory, numerical approximations, random impulses, stochastic differential equations, stochastic partial differential equations
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Andrew StuartJ45
Issue or Number:4
Classification Code:AMS subject classifications: 60J10, 60J27, 65U, 60H10, 60H15, 34A37
Record Number:CaltechAUTHORS:20170613-080747440
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78141
Deposited By: Tony Diaz
Deposited On:13 Jun 2017 16:09
Last Modified:15 Nov 2021 17:37

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