We consider a dynamical system in $\mathbb{R}$ 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.

Keywords: Lévy process, jump diffusion, heavy tail, regular variation, metastability, extreme events, first exit time, large deviations

@article{PS_2008__12__412_0, author = {Imkeller, Peter and Pavlyukevich, Ilya}, title = {Metastable behaviour of small noise {L\'evy-driven} diffusions}, journal = {ESAIM: Probability and Statistics}, pages = {412--437}, publisher = {EDP-Sciences}, volume = {12}, year = {2008}, doi = {10.1051/ps:2007051}, mrnumber = {2437717}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2007051/} }

TY - JOUR AU - Imkeller, Peter AU - Pavlyukevich, Ilya TI - Metastable behaviour of small noise Lévy-driven diffusions JO - ESAIM: Probability and Statistics PY - 2008 SP - 412 EP - 437 VL - 12 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2007051/ DO - 10.1051/ps:2007051 LA - en ID - PS_2008__12__412_0 ER -

%0 Journal Article %A Imkeller, Peter %A Pavlyukevich, Ilya %T Metastable behaviour of small noise Lévy-driven diffusions %J ESAIM: Probability and Statistics %D 2008 %P 412-437 %V 12 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2007051/ %R 10.1051/ps:2007051 %G en %F PS_2008__12__412_0

Imkeller, Peter; Pavlyukevich, Ilya. Metastable behaviour of small noise Lévy-driven diffusions. ESAIM: Probability and Statistics, Volume 12 (2008), pp. 412-437. doi : 10.1051/ps:2007051. http://www.numdam.org/articles/10.1051/ps:2007051/

[1] Regular variation, Encyclopedia of Mathematics and its applications 27. Cambridge University Press, Cambridge (1987). | MR | Zbl

, and ,[2] Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times. Eur. Math. Soc. 6 (2004) 399-424. | MR | Zbl

, , , and ,[3] Metastability in reversible diffusion processes II: Precise asymptotics for small eigenvalues. Eur. Math. Soc. 7 (2005) 69-99. | MR | Zbl

, and ,[4] Life times and lower eigenvalues of an operator of small diffusion. Matematicheskie Zametki 51 (1992) 20-31. | MR | Zbl

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

,[6] Barrier crossings of a Lévy flight. EPL 72 (2005) 348-354. | MR

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

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

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

,[10] Random perturbations of dynamical systems, Grundlehren der Mathematischen Wissenschaften 260. Springer, New York, NY, second edition (1998). | MR | Zbl

and ,[11] Metastability for a class of dynamical systems subject to small random perturbations. Ann. Probab. 15 (1987) 1288-1305. | MR | Zbl

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

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

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

,[15] The metastable behavior of infrequently observed, weakly random, one-dimensional diffusion processes. SIAM J. Appl. Math. 45 (1985) 972-982. | MR | Zbl

and ,[16] 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 | Zbl

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

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

,[19] Tails of solutions of certain nonlinear stochastic differential equations driven by heavy tailed Lévy motions. Stoch. Process. Appl. 105 (2003) 69-97. | MR | Zbl

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

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

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

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