Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré
Dominique Bakry,Dominique Bakry,Patrick Cattiaux,Patrick Cattiaux,Arnaud Guillin,Arnaud Guillin +5 more
TLDR
In this paper, the relationship between two classical approaches for quantitative ergodic properties, Lyapunov type controls and functional inequalities (of Poincare type), is studied. And explicit examples for diffusion processes are studied.About:
This article is published in Journal of Functional Analysis.The article was published on 2008-02-01 and is currently open access. It has received 253 citations till now. The article focuses on the topics: Lyapunov equation & Lyapunov exponent.read more
Citations
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Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations
TL;DR: In this article, the Fokker-Planck Equation is modelled with Stochastic Differential Equations (SDE) and the Langevin Equation (LDE).
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The Markov chain Monte Carlo revolution
TL;DR: The use of simulation for high dimensional intractable computations has revolutionized applied mathematics and design, improving and understanding the new tools leads to fascinating mathematics.
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Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations
TL;DR: Bakhtin and Mattingly as discussed by the authors proved unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris' theorem.
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A simple proof of the Poincaré inequality for a large class of probability measures
TL;DR: In this paper, a simple and direct proof of the existence of a spectral gap under some Lyapunov type condition which is satisfied in particular by log-concave probability measures on ρ √ R n was given.
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Nonasymptotic convergence analysis for the unadjusted Langevin algorithm
Alain Durmus,Eric Moulines +1 more
TL;DR: In this article, a sampling technique based on the Euler discretization of the Langevin stochastic differential equation is studied, and for both constant and decreasing step sizes, non-asymptotic bounds for the convergence to stationarity in both total variation and Wasserstein distances are obtained.
References
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Markov Chains and Stochastic Stability
Sean P. Meyn,Richard L. Tweedie +1 more
TL;DR: This second edition reflects the same discipline and style that marked out the original and helped it to become a classic: proofs are rigorous and concise, the range of applications is broad and knowledgeable, and key ideas are accessible to practitioners with limited mathematical background.
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Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes
Sean P. Meyn,Richard L. Tweedie +1 more
TL;DR: In this paper, the authors developed criteria for continuous-parameter Markovian processes on general state spaces, based on Foster-Lyapunov inequalities for the extended generator, and applied the criteria to several specific processes, including linear stochastic systems under nonlinear feedback, work-modulated queues, general release storage processes and risk processes.
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The Poincaré inequality for vector fields satisfying Hörmander’s condition
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Stability of Markovian processes II: continuous-time processes and sampled chains
Sean P. Meyn,Richard L. Tweedie +1 more
TL;DR: In this paper, the authors extend the results of Meyn and Tweedie (1992b) from discrete-time parameter to continuous-parameter Markovian processes evolving on a topological space, and prove connections between these and standard probabilistic recurrence concepts.
Related Papers (5)
Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes
Sean P. Meyn,Richard L. Tweedie +1 more