×

The first exit problem of reaction-diffusion equations for small multiplicative Lévy noise. (English) Zbl 1423.60100

Summary: This article studies the dynamics of a nonlinear dissipative reaction-diffusion equation with well-separated stable states which is perturbed by infinitedimensional multiplicative Lévy noise with a regularly varying component at intensity \(\varepsilon>0\). The main results establish the precise asymptotics of the first exit times and locus of the solution \(X^\varepsilon\) from the domain of attraction of a deterministic stable state, in the limit as \(\varepsilon\to0\). In contrast to the exponential growth for respective Gaussian perturbations the exit times grow essentially as a power function of the noise intensity as \(\varepsilon\to 0\) with the exponent given as the tail index \(-\alpha\), \(\alpha > 0\), of the Lévy measure, analogously to the case of additive noise in [A. Debussche et al., The dynamics of nonlinear reaction-diffusion equations with small Lévy noise. Cham: Springer (2013; Zbl 1321.60004)]. In this article we substantially improve their quadratic estimate of the small jump dynamics and derive a new exponential estimate of the stochastic convolution for stochastic Lévy integrals with bounded jumps based on the recent pathwise Burkholder-Davis-Gundy inequality by P. Siorpaes [Bernoulli 24, No. 4B, 3222–3245 (2018; Zbl 1407.60059)]. This allows to cover perturbations with general tail index \(\alpha > 0\), multiplicative noise and perturbations of the linear heat equation. In addition, our convergence results are probabilistically strongest possible. Finally, we infer the metastable convergence of the process on the common time scale \(t/\varepsilon^\alpha\) to a Markov chain switching between the stable states of the deterministic dynamical system.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G51 Processes with independent increments; Lévy processes
60G52 Stable stochastic processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
35K05 Heat equation
35K91 Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian
PDFBibTeX XMLCite
Full Text: arXiv Link

References:

[1] D. Applebaum.Lévy processes and stochastic calculus, volume 116 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition (2009). ISBN 978-0-521-73865-1 · Zbl 1200.60001
[2] N. Berglund and B. Gentz. On the noise-induced passage through an unstable periodic orbit. I. Two-level model.J. Statist. Phys.114(5-6), 1577-1618 (2004) · Zbl 1072.82018
[3] N. Berglund and B. Gentz.The Eyring-Kramers law for potentials with nonquadratic saddles.Markov Process. Related Fields16(3), 549-598 (2010) · Zbl 1234.60076
[4] N. Berglund and B. Gentz. Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers’ law and beyond.Electron. J. Probab.18, no. 24, 58 (2013) · Zbl 1285.60060
[5] N. H. Bingham, C. M. Goldie and J. L. Teugels.Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1987). ISBN 0-521-30787-2 · Zbl 0617.26001
[6] A. Bovier, M. Eckhoff, V. Gayrard and M. Klein. Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times.J. Eur. Math. Soc. (JEMS)6(4), 399-424 (2004) · Zbl 1076.82045
[7] A. Bovier, V. Gayrard and M. Klein. Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues.J. Eur. Math. Soc. (JEMS)7(1), 69-99 (2005) · Zbl 1105.82025
[8] S. Brassesco. Some results on small random perturbations of an infinite-dimensional dynamical system.Stochastic Process. Appl.38(1), 33-53 (1991) · Zbl 0734.60068
[9] S. Brassesco. Unpredictability of an exit time.Stochastic Process. Appl.63(1), 55-65 (1996) · Zbl 0908.60059
[10] Z. Brzeźniak, E. Hausenblas and P. A. Razafimandimby.Stochastic reactiondiffusion equations driven by jump processes.Potential Anal.49(1), 131-201 (2018) · Zbl 1398.60077
[11] A. Budhiraja, J. Chen and P. Dupuis. Large deviations for stochastic partial differential equations driven by a Poisson random measure.Stochastic Process. Appl. 123(2), 523-560 (2013) · Zbl 1259.60065
[12] A. Budhiraja, P. Dupuis and A. Ganguly. Moderate deviation principles for stochastic differential equations with jumps.Ann. Probab.44(3), 1723-1775 (2016) · Zbl 1346.60026
[13] A. Budhiraja and P. Nyquist.Large deviations for multidimensional statedependent shot-noise processes.J. Appl. Probab.52(4), 1097-1114 (2015) · Zbl 1334.60028
[14] N. Chafee and E. F. Infante. A bifurcation problem for a nonlinear partial differential equation of parabolic type.Applicable Anal.4, 17-37 (1974/75) · Zbl 0296.35046
[15] M. V. Day. On the exponential exit law in the small parameter exit problem. Stochastics8(4), 297-323 (1983) · Zbl 0504.60032
[16] M. V. Day. Exit cycling for the van der Pol oscillator and quasipotential calculations.J. Dynam. Differential Equations8(4), 573-601 (1996) · Zbl 0866.60055
[17] A. Debussche, M. Högele and P. Imkeller. Asymptotic first exit times of the ChafeeInfante equation with small heavy-tailed Lévy noise.Electron. Commun. Probab. 16, 213-225 (2011) · Zbl 1233.60037
[18] A. Debussche, M. Högele and P. Imkeller.The dynamics of nonlinear reactiondiffusion equations with small Lévy noise, volume 2085 ofLecture Notes in Mathematics. Springer, Cham (2013). ISBN 978-3-319-00827-1; 978-3-319-00828-8 · Zbl 1321.60004
[19] A. Dembo and O. Zeitouni.Large deviations techniques and applications, volume 38 ofApplications of Mathematics (New York). Springer-Verlag, New York, second edition (1998). ISBN 0-387-98406-2 · Zbl 0896.60013
[20] J.-D. Deuschel and D. W. Stroock.Large deviations, volume 137 ofPure and Applied Mathematics. Academic Press, Inc., Boston, MA (1989). ISBN 0-12-213150-9 · Zbl 0705.60029
[21] W. G. Faris and G. Jona-Lasinio. Large fluctuations for a nonlinear heat equation with noise.J. Phys. A15(10), 3025-3055 (1982) · Zbl 0496.60060
[22] M. I. Freidlin. Random perturbations of reaction-diffusion equations: the quasideterministic approximation.Trans. Amer. Math. Soc.305(2), 665-697 (1988) · Zbl 0673.35049
[23] M. I. Freidlin and A. D. Wentzell. Small random perturbations of dynamical systems.Uspehi Mat. Nauk25(1 (151)), 3-55 (1970) · Zbl 0297.34053
[24] M. I. Freidlin and A. D. Wentzell.Random perturbations of dynamical systems, volume 260 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, second edition (1998). ISBN 0-387-98362-7 · Zbl 0935.60038
[25] A. Galves, E. Olivieri and M. E. Vares. Metastability for a class of dynamical systems subject to small random perturbations.Ann. Probab.15(4), 1288-1305 (1987) · Zbl 0709.60058
[26] V. V. Godovan’čuk. Asymptotic behavior of probabilities of large deviations arising from the large jumps of a Markov process.Teor. Veroyatnost. i Primenen.26(2), 321-334 (1981) · Zbl 0462.60029
[27] J. K. Hale. Dynamics of a scalar parabolic equation.Canad. Appl. Math. Quart. 5(3), 209-305 (1997) · Zbl 0924.35002
[28] D. Henry.Geometric theory of semilinear parabolic equations, volume 840 ofLecture Notes in Mathematics. Springer-Verlag, Berlin-New York (1981). ISBN 3-54010557-3 · Zbl 0456.35001
[29] D. B. Henry. Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations.J. Differential Equations59(2), 165-205 (1985) · Zbl 0572.58012
[30] M. Högele and I. Pavlyukevich. The exit problem from a neighborhood of the global attractor for dynamical systems perturbed by heavy-tailed Lévy processes.Stoch. Anal. Appl.32(1), 163-190 (2014) · Zbl 1296.60150
[31] M. Högele and I. Pavlyukevich. Metastability in a class of hyperbolic dynamical systems perturbed by heavy-tailed Lévy type noise.Stoch. Dyn.15(3), 1550019, 26 (2015) · Zbl 1316.60095
[32] H. Hult and F. Lindskog. Regular variation for measures on metric spaces.Publ. Inst. Math. (Beograd) (N.S.)80(94), 121-140 (2006) · Zbl 1164.28005
[33] P. Imkeller and I. Pavlyukevich. First exit times of SDEs driven by stable Lévy processes.Stochastic Process. Appl.116(4), 611-642 (2006) · Zbl 1104.60030
[34] P. Imkeller and I. Pavlyukevich. Metastable behaviour of small noise Lévy-driven diffusions.ESAIM Probab. Stat.12, 412-437 (2008) · Zbl 1205.60113
[35] C. Kipnis and C. M. Newman. The metastable behavior of infrequently observed, weakly random, one-dimensional diffusion processes.SIAM J. Appl. Math.45(6), 972-982 (1985) · Zbl 0592.60063
[36] V. N. Kolokol’tsov.Semiclassical analysis for diffusions and stochastic processes, volume 1724 ofLecture Notes in Mathematics. Springer-Verlag, Berlin (2000). ISBN 3-540-66972-8 · Zbl 0951.60001
[37] 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(3), 341-360 (1996) · Zbl 0912.58042
[38] H. A. Kramers. Brownian motion in a field of force and the diffusion model of chemical reactions.Physica7, 284-304 (1940) · Zbl 0061.46405
[39] C. Marinelli, C. Prévôt and M. Röckner. Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise.J. Funct. Anal. 258(2), 616-649 (2010) · Zbl 1186.60060
[40] C. Marinelli and M. Röckner. On uniqueness of mild solutions for dissipative stochastic evolution equations.Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13(3), 363-376 (2010a) · Zbl 1207.60046
[41] C. Marinelli and M. Röckner. Well-posedness and asymptotic behavior for stochastic reaction-diffusion equations with multiplicative Poisson noise.Electron. J. Probab.15, no. 49, 1528-1555 (2010b) · Zbl 1225.60108
[42] I. Pavlyukevich. First exit times of solutions of stochastic differential equations driven by multiplicative Lévy noise with heavy tails.Stoch. Dyn.11(2-3), 495- 519 (2011) · Zbl 1235.60069
[43] A. Pazy.Semigroups of linear operators and applications to partial differential equations, volume 44 ofApplied Mathematical Sciences. Springer-Verlag, New York (1983). ISBN 0-387-90845-5 · Zbl 0516.47023
[44] S. Peszat and J. Zabczyk.Stochastic partial differential equations with Lévy noise. An evolution equation approach, volume 113 ofEncyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2007). ISBN 978-0521-87989-7 · Zbl 1205.60122
[45] P. E. Protter.Stochastic integration and differential equations, volume 21 ofApplications of Mathematics (New York). Springer-Verlag, Berlin, second edition (2004). ISBN 3-540-00313-4 · Zbl 1041.60005
[46] G. Raugel.Global attractors in partial differential equations.InHandbook of dynamical systems, Vol. 2, pages 885-982. North-Holland, Amsterdam (2002) · Zbl 1005.35001
[47] C. Rocha. Examples of attractors in scalar reaction-diffusion equations.J. Differential Equations73(1), 178-195 (1988) · Zbl 0652.35061
[48] E. Salavati and B. Z. Zangeneh. A maximal inequality forpth power of stochastic convolution integrals.J. Inequal. Appl.pages Paper No. 155, 16 (2016) · Zbl 1338.60151
[49] K. Sato.Lévy processes and infinitely divisible distributions, volume 68 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1999). ISBN 0-521-55302-4 · Zbl 0973.60001
[50] P. Siorpaes.Applications of pathwise Burkholder-Davis-Gundy inequalities. Bernoulli24(4B), 3222-3245 (2018) · Zbl 1407.60059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.