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Finite Markov Chains Coupled to General Markov Processes and An Application to Metastability I

In: Stochastic Analysis, Filtering, and Stochastic Optimization

Author

Listed:
  • Thomas G. Kurtz

    (University of Wisconsin-Madison)

  • Jason Swanson

    (University of Central Florida)

Abstract

We consider a diffusion given by a small noise perturbation of a dynamical system driven by a potential function with a finite number of local minima. The classical results of Freidlin and Wentzell show that the time this diffusion spends in the domain of attraction of one of these local minima is approximately exponentially distributed and hence the diffusion should behave approximately like aMarkov chain on the local minima.By thework ofBovier and collaborators, the local minimacan be associated with the small eigenvalues of the diffusion generator. Applying a Markov mapping theorem, we use the eigenfunctions of the generator to couple this diffusion to a Markov chain whose generator has eigenvalues equal to the eigenvalues of the diffusion generator that are associated with the local minima and establish explicit formulas for conditional probabilities associatedwith this coupling. The fundamental question then becomes to relate the coupled Markov chain to the approximateMarkov chain suggested by the results of Freidlin and Wentzel.

Suggested Citation

  • Thomas G. Kurtz & Jason Swanson, 2022. "Finite Markov Chains Coupled to General Markov Processes and An Application to Metastability I," Springer Books, in: George Yin & Thaleia Zariphopoulou (ed.), Stochastic Analysis, Filtering, and Stochastic Optimization, pages 293-307, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-98519-6_12
    DOI: 10.1007/978-3-030-98519-6_12
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