Asymptotic first exit times of the Chafee-Infante equation with small heavy-tailed Lévy noise
Michael Hoegele (Universität Potsdam)
Peter Imkeller (Humboldt-Universität zu Berlin)
Abstract
This article studies the behavior of stochastic reaction-diffusion equations driven by additive regularly varying pure jump L'evy noise in the limit of small noise intensity. It is shown that the law of the suitably renormalized first exit times from the domain of attraction of a stable state converges to an exponential law of parameter 1 in a strong sense of Laplace transforms, including exponential moments. As a consequence, the expected exit times increase polynomially in the inverse intensity, in contrast to Gaussian perturbations, where this growth is known to be of exponential rate.
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 213-225
Publication Date: April 18, 2011
DOI: 10.1214/ECP.v16-1622
References
- Arnold, Ludwig. Hasselmann's program revisited: the analysis of stochasticity in deterministic climate models. Stochastic climate models (Chorin, 1999), 141--157, Progr. Probab., 49, Birkhäuser, Basel, 2001. MR1948294 (2003j:86007)
- Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation.Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1987. xx+491 pp. ISBN: 0-521-30787-2 MR0898871 (88i:26004)
- M. Claussen, L. A. Mysak, A. J. Weaver, M. Crucix, T. Fichefet, M.-F. Loutre, S. L. Weber, J. Alcamo, V. A. Alexeev, A. Berger, R. Calov, A. Ganopolski, H. Goosse, G. Lohmann, F. Lunkeit, I. I. Mokhov, V. Petoukhov, P. Stone, Z. Wang, Earth System Models of Intermediate Complexity: Closing the gap in the spectrum of climate system models, Climate Dynamics, 18, (2002), 579-586, DOI 10.1007/s00382-001-0200-1.
- A. Debussche, M. Högele, P. Imkeller. Metastability for the Chafee-Infante equation with small heavy-tailed LÃvy noise, (to appear), 2011.
- P. D. Ditlevsen, Observation of a stable noise induced millennial climate changes from an ice-core record}, Geophysical Research Letters, 26 (10) (1999), 1441-1444.
- P. D. Ditlevsen, Anomalous jumping in a double-well potential, Physical Review E, 60(1) (1999), 172-179, 1999.
- Eden, A.; Foias, C.; Nicolaenko, B.; Temam, R. Exponential attractors for dissipative evolution equations.RAM: Research in Applied Mathematics, 37. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. viii+183 pp. ISBN: 2-225-84306-8 MR1335230 (96i:34148)
- Faris, William G.; Jona-Lasinio, Giovanni. Large fluctuations for a nonlinear heat equation with noise. J. Phys. A 15 (1982), no. 10, 3025--3055. MR0684578 (84j:81073)
- V. V. Godovanchuk. Asymptotic probabilities of large deviations due to large jumps of a Markov process (English translation), Theory of Probab Appl. 26 (2) (1981-1982), 314-327.
- Henry, Daniel B. Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations. J. Differential Equations 59 (1985), no. 2, 165--205. MR0804887 (86m:58080)
- Henry. D. Geometric theory of semilinear parabolic equations, Lecture notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.
- Hult, Henrik; Lindskog, Filip. Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N.S.) 80(94) (2006), 121--140. MR2281910 (2008g:28016)
- Imkeller, Peter. Energy balance modelsâviewed from stochastic dynamics. Stochastic climate models (Chorin, 1999), 213--240, Progr. Probab., 49, Birkhäuser, Basel, 2001. MR1948298 (2003k:86014)
- Imkeller, Peter; Pavlyukevich, Ilya. Metastable behaviour of small noise Lévy-driven diffusions. ESAIM Probab. Stat. 12 (2008), 412--437. MR2437717 (2010b:60141)
- Imkeller, P.; Pavlyukevich, I. First exit times of SDEs driven by stable Lévy processes. Stochastic Process. Appl. 116 (2006), no. 4, 611--642. MR2205118 (2007e:60046)
- Peszat, S.; Zabczyk, J. Stochastic partial differential equations with Lévy noise.An evolution equation approach.Encyclopedia of Mathematics and its Applications, 113. Cambridge University Press, Cambridge, 2007. xii+419 pp. ISBN: 978-0-521-87989-7 MR2356959 (2009b:60200)
- Raugel. G. Global attractors in partial differential equations, Fiedler, Bernold (ed.), Handbook of dynamical systems. Vol 2, 885-982, North-Holland, Amsterdam, 2002.
- Walters, Peter. An introduction to ergodic theory.Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. ix+250 pp. ISBN: 0-387-90599-5 MR0648108 (84e:28017)

This work is licensed under a Creative Commons Attribution 3.0 License.