Abstract
The random walk (approximating the Wiener process in the area of normal deviations) and the Wiener process have, in general, different asymptotics of the large deviation probabilities. Therefore, special arrangements in the numerical procedures should be made to calculate solutions of various problems related to the large deviations for the Wiener process and for the corresponding PDE’s with a small parameter. Exit problem, wavefront propagation, stochastic resonance are among these problems. We calculate the action functional when the Wiener process is replaced by the random walk, find the relation between the small parameters which provide convergence to the solution of the problem with the Gaussian white noise as the perturbation, and calculate the correction term.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. I. Freidlin, “Sublimiting distributions and stabilization of solutions of parabolic equations with a small parameters,” Soviet Math, Dokl., vol. 18, no. 4, pp. 1114–1118, 1977.
M. I. Freidlin, Functional Integration and Partial Differential Equations, Princeton Univ. Press, 1985.
M. I. Freidlin, “Semi-linear PDE’s and limit theorems for large deviations,” Lectures Notes in Mathematics, 1527, Springer Verlag, 1991.
M. I. Freidlin, Markov Processes and Differential Equations: Asymptotic Problems, Birkhäuer, 1996.
M. I. Freidlin, “Quasi-deterministic approximation, metastability and stochastic resonance,” Physica D, vol. 137, pp. 333–352, 2000.
M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Second edition, Springer Verlag, 1998.
L. Gammaitoni, P. Hånggi, P. Jung, and F. Marchesooni, “Stochastic resonance,” Reviews in Modern Physics, vol. 70, no. 1, pp. 224–287, 2000
A. D. Wentzell, Limit Theorems on Large deviations for Markov Stochastic Processes, Kluwer, 1991.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Freidlin, M. (2003). Noise Sensitivity of Stochastic Resonance and Other Problems Related to Large Deviations. 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_4
Download citation
DOI: https://doi.org/10.1007/978-94-010-0179-3_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-3985-7
Online ISBN: 978-94-010-0179-3
eBook Packages: Springer Book Archive