Abstract
Diffusion models arising in analysis of real world systems are typically far too complex for exact solution, or even meaningful simulation. The purpose of this paper is to develop foundations for model reduction, and new modeling tech niques for diffusion models. Based on the main assumption of V-uniform ergodicity of the diffusion process it is shown that real eigenfunctions provide a decomposition of the state space into so-called metastable sets. We give a novel definition of metastability via exit rates which seems to be promising for a algorithmic identification of metastable sets even for large scale systems.
Partially supported by the DFG Research Center “Mathematics for key technologies” (FZT 86) in Berlin, and within DFG priority program “Analysis, Modeling and Simulation of Multiscale Problems” (SPP 1095).
Supported in part by NSF Grant ECS 99 72957
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References
S. Balaji and S. P. Meyn, “Multiplicative ergodicity and large deviations for an irreducible Markov chain,” Stochastic Process. Appl., vol. 90, no. 1, pp. 123–144, 2000.
A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein, Metastability in reversible diffusion processes I Sharp asymptotic s for capacities and exit times, Preprint. Weierstrass-Institut für Angewandte Analysis und Stochastik, Berlin, 2000.
E. G. Coffman, Jr. and V. Misra, “Network resilience: Exploring cascading failures,” in Proceedings of the Fortieth Annual Allerton Conference on Communication, Control, and Computing, October 2–4, 2002, New York, NY, Plenum Press, 2002.
G. E. Crooks and D. Chandler, “Efficient transition path sampling for nonequilibirum stochastic dynamics,” Phys. Rev. E, vol. 64, pp. 26–109, 2001.
M. I. Freidlin, “Quasi-deterministic approximation, metastability and stochastic resonance,” Physica D, vol. 137, pp. 333–352, 2000.
P. W. Glynn and H. Thorisson, “Two-sided taboo limits for Markov processes and associated perfect simulation,” Stochastic Process. Appl., vol. 91, no. 1, pp. 1–20, 2001.
W. Huisinga, S. Meyn, and C. Schütte, “metastability in Markovian and molecular systems,” Ann. Appl. Probab., To appear. 2003. Available via http://www.math.fu-berlin.de/~biocomp.
I. Kontoyiannis and S. P. Meyn, “Spectral theory and limit theorems for geometrically ergodic Markov processes,” submitted for publication, 2001. Also presented at the 2001 INFORMS Applied Probability Conference, NY, July, 2001.
I. Kontoyiannis and S. P. Meyn, “Spectral theory and limit theorems for geometrically ergodic Markov processes, Part II: Empirical measures & unbounded functionals,” submitted for publication, 2001.
S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag, London, 1993.
S. P. Meyn and R. L. Tweedie, “Stability of Markovian processes II: Continuous time processes and sampled chains,” Adv. Appl. Probab., vol. 25, pp. 487–517, 1993.
E. Nummelin, General Irreducible Markov Chains and Nonnegative Operators. Cambridge University Press, Cambridge, 1984.
L. Rey-Bellet and L. E. Thomas, “Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators,” Comm. Math. Phys., vol. 215, no. 1, pp. 1–24, 2000.
C. Schiitte, A. Fischer, W. Huisinga, and P. Deuflhard. “A direct approach to conformational dynamics based on hybrid Monte Carlo,” J. Comput. Phys., Special Issue on Computational Biophysics, vol. 151, pp. 146–168, 1999.
C. Schiitte, W. Huisinga, and P. Deuflhard, “Transfer operator approach to conformational dynamics in biomolecular systems,” in B. Fiedler, editor, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. Springer, 2001.
E. Seneta and D. Vere-Jones, “On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states,” J. Appl Probability, vol. 3, pp. 403–434, 1966.
R. L. Tweedie, “Quasi-stationary distributions for Markov chains on a general state space,” J. Appl. Probability, vol. 11, pp. 726–741, 1974.
E. Weinan, W. Ren, and E. Vanden Eijnden, “Optimal paths for spatially extended metastable systems driven by noise,” Preprint. Available via http://math.nyu.edu/~eve2, 2002.
E. Weinan, W. Ren, and E. Vanden Eijnden, “Probing multi-scale energy landscapes using the string method,” Preprint. Available via http://math.nyu.edu/~eve2, 2002.
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Schütte, C., Huisinga, W., Meyn, S. (2003). Metastability of Diffusion Processes. In: Namachchivaya, N.S., Lin, Y.K. (eds) IUTAM Symposium on Nonlinear Stochastic Dynamics. Solid Mechanics and Its Applications, vol 110. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0179-3_6
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DOI: https://doi.org/10.1007/978-94-010-0179-3_6
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