Metastable behaviour of small noise Lévy-Driven diffusions
Peter Imkeller; Ilya Pavlyukevich
ESAIM: Probability and Statistics (2008)
- Volume: 12, page 412-437
- ISSN: 1292-8100
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topImkeller, Peter, and Pavlyukevich, Ilya. "Metastable behaviour of small noise Lévy-Driven diffusions." ESAIM: Probability and Statistics 12 (2008): 412-437. <http://eudml.org/doc/250429>.
@article{Imkeller2008,
abstract = {
We consider a dynamical system in $\mathbb\{R\}$ driven by a vector field -U', where U is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a Lévy noise of small intensity and such that the heaviest tail of its Lévy measure is regularly varying. We show that the perturbed dynamical system exhibits metastable behaviour i.e. on a proper time scale it reminds of a Markov jump process taking values in the local minima of the potential U. Due to the heavy-tail
nature of the random perturbation, the results differ strongly from the well studied purely Gaussian case.
},
author = {Imkeller, Peter, Pavlyukevich, Ilya},
journal = {ESAIM: Probability and Statistics},
keywords = {Lévy process; jump diffusion; heavy tail; regular variation; metastability;
extreme events; first exit time; large deviations; extreme events},
language = {eng},
month = {7},
pages = {412-437},
publisher = {EDP Sciences},
title = {Metastable behaviour of small noise Lévy-Driven diffusions},
url = {http://eudml.org/doc/250429},
volume = {12},
year = {2008},
}
TY - JOUR
AU - Imkeller, Peter
AU - Pavlyukevich, Ilya
TI - Metastable behaviour of small noise Lévy-Driven diffusions
JO - ESAIM: Probability and Statistics
DA - 2008/7//
PB - EDP Sciences
VL - 12
SP - 412
EP - 437
AB -
We consider a dynamical system in $\mathbb{R}$ driven by a vector field -U', where U is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a Lévy noise of small intensity and such that the heaviest tail of its Lévy measure is regularly varying. We show that the perturbed dynamical system exhibits metastable behaviour i.e. on a proper time scale it reminds of a Markov jump process taking values in the local minima of the potential U. Due to the heavy-tail
nature of the random perturbation, the results differ strongly from the well studied purely Gaussian case.
LA - eng
KW - Lévy process; jump diffusion; heavy tail; regular variation; metastability;
extreme events; first exit time; large deviations; extreme events
UR - http://eudml.org/doc/250429
ER -
References
top- N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular variation, Encyclopedia of Mathematics and its applications27. Cambridge University Press, Cambridge (1987).
- A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein, Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times. Eur. Math. Soc.6 (2004) 399–424.
- A. Bovier, V. Gayrard and M. Klein, Metastability in reversible diffusion processes II: Precise asymptotics for small eigenvalues. Eur. Math. Soc.7 (2005) 69–99.
- V.A. Buslov and K.A. Makarov, Life times and lower eigenvalues of an operator of small diffusion. Matematicheskie Zametki51 (1992) 20–31.
- S. Cerrai, Second order PDE's in finite and infinite dimension. A probabilistic approach. Lect. Notes Math. Springer, Berlin Heidelberg (2001).
- A.V. Chechkin, V.Yu Gonchar, J. Klafter and R. Metzler, Barrier crossings of a Lévy flight. EPL72 (2005) 348–354.
- M.V. Day, On the exponential exit law in the small parameter exit problem. Stochastics8 (1983) 297–323.
- P.D. Ditlevsen, Anomalous jumping in a double-well potential. Phys. Rev. E60 (1999) 172–179.
- P.D. Ditlevsen, Observation of α-stable noise induced millenial climate changes from an ice record. Geophysical Research Letters26 (1999) 1441–1444.
- M.I. Freidlin and A.D. Wentzell, Random perturbations of dynamical systems, Grundlehren der Mathematischen Wissenschaften260. Springer, New York, NY, second edition (1998).
- A. Galves, E. Olivieri and M.E. Vares, Metastability for a class of dynamical systems subject to small random perturbations. Ann. Probab.15 (1987) 1288–1305.
- V.V. Godovanchuk, Asymptotic probabilities of large deviations due to large jumps of a Markov process. Theory Probab. Appl.26 (1982) 314–327.
- P. Imkeller and I. Pavlyukevich, First exit times of SDEs driven by stable Lévy processes. Stochastic Process. Appl.116 (2006) 611–642.
- O. Kallenberg, Foundations of modern probability. Probability and Its Applications. Springer, second edition (2002).
- C. Kipnis and C.M. Newman, The metastable behavior of infrequently observed, weakly random, one-dimensional diffusion processes. SIAM J. Appl. Math.45 (1985) 972–982.
- V.N. Kolokol'tsov and K.A. Makarov, Asymptotic spectral analysis of a small diffusion operator and the life times of the corresponding diffusion process. Russian J. Math. Phys.4 (1996) 341–360.
- P. Mathieu, Spectra, exit times and long time asymptotics in the zero-white-noise limit. Stoch. Stoch. Rep.55 1–20 (1995).
- Ph.E. Protter, Stochastic integration and differential equations, Applications of Mathematics21. Springer, Berlin, second edition (2004).
- G. Samorodnitsky and M. Grigoriu, Tails of solutions of certain nonlinear stochastic differential equations driven by heavy tailed Lévy motions. Stoch. Process. Appl.105 (2003) 69–97.
- A.D. Wentzell, Limit theorems on large deviations for Markov stochastic processes, Mathematics and Its Applications (Soviet Series) 38. Kluwer Academic Publishers, Dordrecht (1990).
- M. Williams, Asymptotic exit time distributions. SIAM J. Appl. Math.42 (1982) 149–154.
- Ai H. Xia, Weak convergence of jump processes, in Séminaire de Probabilités, XXVI, Lect. Notes Math.1526 Springer, Berlin (1992) 32–46.
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