Metastable behaviour of small noise Lévy-driven diffusions
ESAIM: Probability and Statistics, Tome 12 (2008), pp. 412-437.

We consider a dynamical system in $ℝ$ driven by a vector field $-{U}^{\text{'}}$, 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.

DOI : https://doi.org/10.1051/ps:2007051
Classification : 60E07,  60F10
Mots clés : Lévy process, jump diffusion, heavy tail, regular variation, metastability, extreme events, first exit time, large deviations
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author = {Imkeller, Peter and Pavlyukevich, Ilya},
title = {Metastable behaviour of small noise {L\'evy-driven} diffusions},
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Imkeller, Peter; Pavlyukevich, Ilya. Metastable behaviour of small noise Lévy-driven diffusions. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 412-437. doi : 10.1051/ps:2007051. http://www.numdam.org/articles/10.1051/ps:2007051/

[1] N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular variation, Encyclopedia of Mathematics and its applications 27. Cambridge University Press, Cambridge (1987). | MR 898871 | Zbl 0617.26001

[2] 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. | MR 2094397 | Zbl 1076.82045

[3] 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. | MR 2120991 | Zbl 1105.82025

[4] V.A. Buslov and K.A. Makarov, Life times and lower eigenvalues of an operator of small diffusion. Matematicheskie Zametki 51 (1992) 20-31. | MR 1165277 | Zbl 0755.34086

[5] S. Cerrai, Second order PDE's in finite and infinite dimension. A probabilistic approach. Lect. Notes Math. Springer, Berlin Heidelberg (2001). | MR 1840644 | Zbl 0983.60004

[6] A.V. Chechkin, V. Yu Gonchar, J. Klafter and R. Metzler, Barrier crossings of a Lévy flight. EPL 72 (2005) 348-354. | MR 2213557

[7] M.V. Day, On the exponential exit law in the small parameter exit problem. Stochastics 8 (1983) 297-323. | MR 693886 | Zbl 0504.60032

[8] P.D. Ditlevsen, Anomalous jumping in a double-well potential. Phys. Rev. E 60 (1999) 172-179.

[9] P.D. Ditlevsen, Observation of $\alpha$-stable noise induced millenial climate changes from an ice record. Geophysical Research Letters 26 (1999) 1441-1444.

[10] M.I. Freidlin and A.D. Wentzell, Random perturbations of dynamical systems, Grundlehren der Mathematischen Wissenschaften 260. Springer, New York, NY, second edition (1998). | MR 1652127 | Zbl 0522.60055

[11] 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. | MR 905332 | Zbl 0709.60058

[12] V.V. Godovanchuk, Asymptotic probabilities of large deviations due to large jumps of a Markov process. Theory Probab. Appl. 26 (1982) 314-327. | Zbl 0481.60037

[13] P. Imkeller and I. Pavlyukevich, First exit times of SDEs driven by stable Lévy processes. Stochastic Process. Appl. 116 (2006) 611-642. | MR 2205118 | Zbl 1104.60030

[14] O. Kallenberg, Foundations of modern probability. Probability and Its Applications. Springer, second edition (2002). | MR 1876169 | Zbl 0996.60001

[15] 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. | MR 813459 | Zbl 0592.60063

[16] 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. | MR 1443178 | Zbl 0912.58042

[17] P. Mathieu, Spectra, exit times and long time asymptotics in the zero-white-noise limit. Stoch. Stoch. Rep. 55 1-20 (1995). | MR 1382282 | Zbl 0886.60055

[18] Ph.E. Protter, Stochastic integration and differential equations, Applications of Mathematics 21. Springer, Berlin, second edition (2004). | MR 2020294 | Zbl 1041.60005

[19] 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. | MR 1972289 | Zbl 1075.60540

[20] A.D. Wentzell, Limit theorems on large deviations for Markov stochastic processes, Mathematics and Its Applications (Soviet Series) 38. Kluwer Academic Publishers, Dordrecht (1990). | MR 1135113 | Zbl 0743.60029

[21] M. Williams, Asymptotic exit time distributions. SIAM J. Appl. Math. 42 (1982) 149-154. | MR 646755 | Zbl 0482.60058

[22] Ai H. Xia, Weak convergence of jump processes, in Séminaire de Probabilités, XXVI, Lect. Notes Math. 1526 Springer, Berlin (1992) 32-46. | Numdam | MR 1231981 | Zbl 1126.60318

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