Skip to main content
SpringerLink
Log in

Localization and Metastability

  • Chapter
  • First Online:
The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2085))

  • 1401 Accesses

Abstract

In this chapter, equipped with our previously obtained knowledge of exit and transition times in the limit of small noise amplitude \(\varepsilon \rightarrow 0\), we shall investigate the global asymptotic behavior of our jump diffusion process in the time scale in which transitions occur, i.e. in the scale given by \({\lambda }^{0}(\varepsilon ) =\nu (\frac{1} {\varepsilon } B_{\delta }^{c}(0)),\varepsilon,\delta > 0\). It turns out that in this time scale, the switching of the diffusion between neighborhoods of the stable solutions ϕ ± can be well described by a Markov chain jumping back and forth between two states with a characteristic Q-matrix determined by the quantities \(\frac{\mu ({(D_{0}^{\pm })}^{c})} {\mu (B_{\delta }^{c}(0))}\) as jumping rates.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. A. Araujo, A. Giné, On tails and domains of attraction of stable measures in Banach spaces. Trans. Am. Math. Soc. 1, 105–119 (1979)

    Article  Google Scholar 

  2. L. Arnold, Y. Kifer, Nonlinear diffusion approximation of the slow motion in multiscale systems (2001). Preprint

    Google Scholar 

  3. L. Arnold, Hasselmann’s program revisited: The analysis of stochasticity in deterministic climate models, in Stochastic Climate Models, ed. by P. Imkeller, J.-S. von Storch. Progress in Probability (Birkhauser, Basel, 2001)

    Google Scholar 

  4. D. Applebaum, J.-L. Wu, Stochastic partial differential equations driven by Lévy space-time white noise. Random Oper. Stoch. Equat. 3, 245–261 (2000)

    MathSciNet  Google Scholar 

  5. S. Albeverio, J.-L. Wu, T.-Sh. Zhang, Parabolic SPDEs driven by Poisson white noise. Stoch. Process. Appl. 74(1), 21–36 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Bovier, M. Eckhoff, V. Gayrard, M. Klein, Metastability in reversible diffusion processes I: Sharp asymptotics and exit times. J. Eur. Math. Soc. 6(4), 399–424 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation (Cambridge University Press, Cambridge, 1987)

    Book  MATH  Google Scholar 

  8. E. Saint Loubert Bie, Etude d’une EDPS conduite par un bruit Poissonien. Probab. Theor. Relat. Fields 111, 287–321 (1998)

    Article  MATH  Google Scholar 

  9. N. Bogolyubov, Y. Mitropolskii, Asymptotic Methods in the Theory of Non-linear Oscillations (Gordon and Breach, New York, 1961)

    MATH  Google Scholar 

  10. S. Brassesco, Some results on small random perturbations of an infinite-dimensional dynamical system. Stoch. Process. Appl. 38, 33–53 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Brezis, Functional Analysis (Masson, Paris, 1983)

    Google Scholar 

  12. L. Bo, Y. Wang, Stochastic Cahn-Hilliard partial differential equations with Lévy space-time white noise. Stoch. Dynam. 6, 229–244 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Cerrai, M. Freidlin, Approximation of quasi-potentials and exit problems for multidimensional RDEs with noise. Trans. Am. Math. Soc. 363(7), 3853–3892 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Caruana, P. Friz, H. Oberhauser, A (rough) pathwise approach to a class of nonlinear SPDEs. Ann. l’Institut Henri Poincaré/Analyse non linéaire 28, 27–46 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. T. Cazenave, A. Haraux, in An Introduction to Semilinear Evolution Equations. Oxford Lecture Series in Mathematics and Its Applications, vol. 13 (Oxford Science Publications, Oxford, 1998)

    Google Scholar 

  16. P.L. Chow, in Stochastic Partial Differential Equations. Applied Mathematics and Nonlinear Science Series (Chapman and Hall/CRC, New York, 2007)

    Google Scholar 

  17. I. Chueshov, Order-preserving skew product flows and nonautonomous parabolic systems. Acta Appl. Math. 66(1), 185–205 (2001)

    Article  MathSciNet  Google Scholar 

  18. I.D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dynamical Systems (Acta Scientific Publishing House, Kharkov, 2002)

    Google Scholar 

  19. N. Chafee, E.F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type. Appl. Anal. 4, 17–37 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Chojnowska-Michalik, On processes of Ornstein-Uhlenbeck type in Hilbert spaces. Stochastics 21, 251–286 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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 c1omplexity: Closing the gap in the spectrum of climate system models. Clim. Dynam. 18, 579–586 (2002)

    Article  Google Scholar 

  22. J. Carr, R. Pego, Metastable patterns in solutions of u t  = u xx  − f(u). Comm. Pure. Appl. Math. 42(5), 523–574 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  23. P.D. Ditlevsen, Observation of a stable noise induced millennial climate changes from an ice-core record. Geophys. Res. Lett. 26(10), 1441–1444 (1999)

    Article  Google Scholar 

  24. P.D. Ditlevsen, Bifurcation structure and noise-assisted transitions in the Pleistocene glacial cycles. Paleoceanography 24, 3204–3215 (2009)

    Article  Google Scholar 

  25. P.D. Ditlevsen, M.S. Kristensen, K.K. Andersen, The recurrence time of Dansgaard-Oeschger events and limits to the possible periodic forcing. J. Clim. 18, 2594–2603 (2006)

    Article  Google Scholar 

  26. Z. Dong, Y. Xie, Ergodicity of linear SPDE driven by Lévy noise. J. Syst. Sci. Complex 23(1), 137–152 (2010)

    Article  MathSciNet  Google Scholar 

  27. G. DaPrato, J. Zabczyk, in Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, vol. 44 (Cambridge University Press, Cambridge, 1992)

    Google Scholar 

  28. A. Eden, C. Foias, B. Nicolaenko, R. Temam, Exponential Attractors for Dissipative Evolution Equations (Wiley/Massons, New York/Paris, 1994)

    MATH  Google Scholar 

  29. W.G. Faris, G. Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise. J. Phys. A 10, 3025–3055 (1982)

    Article  MathSciNet  Google Scholar 

  30. N. Fournier, Malliavin calculus for parabolic SPDEs with jumps. Stoch. Process. Appl. 87, 115–147 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. N. Fournier, Support theorem for solutions of a white noise driven SPDEs with temporal Poissonian jumps. Bernoulli 7, 165–190 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. M. Fuhrmann, M. Röckner, Generalized Mehler semigroups: the non-Gaussian case. Potential Anal. 12, 1–47 (2000)

    Article  MathSciNet  Google Scholar 

  33. M.I. Freidlin, Limit theorems for large deviations and reaction-diffusion equations. Ann. Probab. 13(3), 639–675 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  34. B. Fiedler, A. Scheel, Spatio-temporal dynamics of reaction-diffusion patterns. J. Phys. A Math. Gen. 15, 3025–3055 (1982)

    Article  Google Scholar 

  35. D. Filipović, S. Tappe, J. Teichmann, Jump-diffusions in Hilbert Spaces: Existence, stability and numerics. Stochastics 82(5), 475–520 (2010)

    MathSciNet  MATH  Google Scholar 

  36. D. Filipović, S. Tappe, J. Teichmann, Term structure models driven by Wiener processes and Poisson measures: Existence and positivity. SIAM J. Financ. Math. 1, 523–554 (2010)

    Article  MATH  Google Scholar 

  37. M.I. Freidlin, A.D. Ventsell, On small random perturbations of dynamical systems (English). Russ. Math. Surv. 25(1), 1–55 (1970)

    Article  MATH  Google Scholar 

  38. M.I. Freidlin, A.D. Ventsell, in Random Perturbations of Dynamical Systems, transl. from the Russian by J. Szuecs, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 260 (Springer, New York, 1998)

    Google Scholar 

  39. J. Gairing, C. Hein, P. Imkeller, I. Pavlyukevich, Dynamical models of climate time series and the rate of convergence of power variations. Mathematisches Forschungsinstitut Oberwolfach, Report, no. 40, 2011

    Google Scholar 

  40. L. Ganopolski, S. Rahmstorf, Rapid changes of glacial climate simulated in a coupled climate model. Nature 409, 153 (2001)

    Article  Google Scholar 

  41. M. Hairer, Rough stochastic PDEs. Comm. Pure Appl. Math. 64(11), 1547–1585 (2011)

    MathSciNet  MATH  Google Scholar 

  42. M. Hairer, Solving the KPZ equation. Ann. Math. (2013) (to appear)

    Google Scholar 

  43. J.K. Hale, in Infinite-Dimensional Dynamical Systems. Geometric Dynamics (Rio de Janeiro, 1981). Lecture Notes in Mathematics, vol. 1007 (Springer, Berlin, 1983)

    Google Scholar 

  44. D.L. Hartmann, Global Physical Climatology. International Geophysics Series, vol. 56 (Academic, San Diego, 1994)

    Google Scholar 

  45. K. Hasselmann, Stochastic climate models: Part I. Theory. Tellus 28, 473–485 (1976)

    Article  Google Scholar 

  46. E. Hausenblas, Existence, uniqueness and regularity of parabolic SPDEs driven by Poisson random measure. Electron. J. Probab. 10, 1496–1546 (electronic) (2005)

    Google Scholar 

  47. E. Hausenblas, SPDEs driven by Poisson random measure with non-Lipschitz coefficients: Existence results. Probab. Theor. Relat. Fields 137, 161–200 (2006)

    Article  MathSciNet  Google Scholar 

  48. D. Henry, Geometric Theory of Semilinear Parabolic Equations (Springer, Berlin, 1983)

    Google Scholar 

  49. D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations. J. Differ. Equat. 59, 165–205 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  50. C. Hein, P. Imkeller, I. Pavlyukevich, Limit theorems for p-variations of solutions of SDEs driven by additive stable Lévy noise and model selection for paleo-climatic data. Interdiscip. Math. Sci. 8, 137–150 (2009)

    Google Scholar 

  51. H. Hulk, F. Lindskog, Regular variation for measures on metric spaces. Publi. l’Institut de Mathématiques Nouvelle Sér. 80, 94 (2006)

    Google Scholar 

  52. M. Högele, Metastability of the Chafee-Infante equation with small heavy-tailed Lévy noise - a conceptual climate model. Ph.D. thesis, Humboldt-Universität zu Berlin, 2011

    Google Scholar 

  53. M. Hairer, M.D. Ryser, H. Weber, Triviality of the 2d stochastic Allen-Cahn equation. Electron. J. Probab. 17(39), 14 (2012)

    Google Scholar 

  54. P. Imkeller, A. Monahan, Conceptual stochastic climate models. Stoch. Dynam. 2, 311–326 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  55. P. Imkeller, Energy balance models: Viewed from stochastic dynamics. Prog. Probab. 49, 213–240 (2001)

    MathSciNet  Google Scholar 

  56. P. Imkeller, I. Pavlyukevich, First exit times of SDEs driven by stable Lévy processes. Stoch. Process. Appl. 116(4), 611–642 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  57. P. Imkeller, I. Pavlyukevich, Lévy flights: Transitions and meta-stability. J. Phys. A Math. Gen. 39, L237–L246 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  58. P. Imkeller, I. Pavlyukevich, Metastable behaviour of small noise Lévy-driven diffusions. ESAIM Probab. Stat. 12, 412–437 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  59. Working Group 2 IPCC-Report. Glossary working group 2. Fourth Assesment Report of the Intergouvernmental Panel on Climate Change (2010), www.ipcc.ch/pdf/glossary/ar4-wg2.pdf

  60. O. Kallenberg, Foundations of Modern Probability, 2nd edn. (2002) (Springer, New York, 1997)

    Google Scholar 

  61. R.Z. Khasminskii, A limit theorem for solutions of differential equations with random right-hand side. Theor. Probab. Appl. 11, 390–406 (1966)

    Article  Google Scholar 

  62. A. Klenke, Wahrscheinlichkeitstheorie, vol. 2. korrigierte Auflage (2008) (Springer, New York, 2005)

    Google Scholar 

  63. C. Knoche, SPDEs in infinite dimension with Poisson noise. C. R. Acad. Sci. Paris Ser. I 339, 647–652 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  64. P. Kotelenez, Stochastic Ordinary and Stochastic Partial Differential Equations, Transitions from Microscopic to Macroscopic Dynamics (Springer, New York, 2008)

    Google Scholar 

  65. G. Kallianpur, V. Perez-Abreu, Stochastic evolution equations driven by nuclear space valued martingales. Appl. Math. Optim. 3, 237–272 (1988)

    Article  MathSciNet  Google Scholar 

  66. N.V. Krylov, B.L. Rozovskii, Stochastic evolution equations, in Stochastic Differential Equations: Theory and Applications. A Volume in Honor of Prof. Boris L. Rozovskii, ed. by P. Baxendale et al. Interdisciplinary Mathematical Sciences, vol. 2 (World Scientific, Hackensack, 2007), pp. 1–69

    Google Scholar 

  67. D. Khoshnevisan, F. Rassoul-Agha, R. Dalang, C. Mueller, D. Nualart, Y. Xiao, in A minicourse on SPDE. Lecture Notes in Mathematics (Springer, New York, 2008)

    Google Scholar 

  68. E.N. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  69. L.E. Lisiecki, M.E. Raymo, A Pliocene-Pleistocene stack of 57 globally distributed benthic δ 18 records. Paleoceanography 20, PA 1003 (2005)

    Google Scholar 

  70. L.R.M. Maas, A simple model for the three-dimensional, thermally and wind-driven ocean circulation. Tellus 46A, 671–680 (1994)

    Google Scholar 

  71. F. Martinelli, E. Olivieri, E. Scoppola, Small random perturbations of finite- and infinite-dimensional dynamical systems: Unpredictability of exit times. J. Stat. Phys. 55(3/4), 477–504 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  72. C. Marinelli, C. Prévôt, M. Röckner, Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise. J. Funct. Anal. 258(2), 616–649 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  73. A.J. Majda, I. Timofeyev, E. Vanden Eijnden, Models for stochastic climate prediction. Proc. Natl. Acad. Sci. USA 96, 14687–14691 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  74. C. Mueller, The heat equation with Lévy noise. Stoch. Process. Appl. 74, 133–151 (1998)

    Article  MathSciNet  Google Scholar 

  75. L. Mytnik, Stochastic partial differential equations driven by stable noise. Probab. Theor. Relat. Fields 123, 157–201 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  76. Members NGRIP, Ngrip data. Nature 431, 147–151 (2004)

    Article  Google Scholar 

  77. E. Pardoux, Equations aux dérivés partielles stochastiques de type monotone (French). Séminaire sur les Equations aux Dérivées Partielles (1974–1975), III, Exp. No. 2, 10 pp., 1975

    Google Scholar 

  78. J.R. Petit, I. Basile, A. Leruyet, D. Raynaud, J. Jouzel, M. Stievenard, V.Y. Lipenkov, N.I. Barkov, B.B. Kudryashov, M. Davis, E. Saltzmann, V. Kotlyakov, Four climate cycles in Vostok ice core. Nature 387, 359–360 (1997)

    Article  Google Scholar 

  79. J.R. Petit, J. Jouzel, D. Raynaud, N.I. Barkov, J.-M. Barnola, I. Basile, M. Benders, J. Chapellaz, M. Davis, G. Delayque, M. Delmotte, V.M. Kotlyakov, M. Legrand, V.Y. Lipenkov, C. Lorius, L. Pépin, C. Ritz, E. Saltzmann, M. Stievenard, Climate and atmospheric history of the past 420,000 years from the vostok ice core, Antarctica. Nature 399, 429–436 (1999)

    Article  Google Scholar 

  80. C. Prévôt, M. Röckner, A Concise Course on Stochastic Partial Differential Equations (Springer, Berlin, 2007)

    MATH  Google Scholar 

  81. C. Prévôt, Existence, uniqueness and regularity w.r.t. the initial condition of mild solutions of SPDEs driven by Poisson noise. Infin. Dimens. Anal. Quant. Probab. Relat. Top. 13(1), 133–163 (2010)

    Google Scholar 

  82. E. Priola, L. Xu, J. Zabcyk, Exponential mixing for some SPDEs with Lévy noise. Stoch. Dynam. 11(2–3), 521–534 (2011)

    Article  MATH  Google Scholar 

  83. S. Peszat, J. Zabczyk, Stochastic heat and wave equations driven by an impulsive noise, in Stochastic Partial Differential Equations and Applications VII, ed. by G. Da Prato, L. Tubaro. Lecture Notes in Pure and Applied Mathematics, vol. 245 (Chapman and Hall/CRC, Boca Raton, 2006), pp. 229–242

    Google Scholar 

  84. S. Peszat, J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise (an Evolution Equation Approach) (Cambridge University Press, Cambridge, 2007)

    Book  MATH  Google Scholar 

  85. E. Priola, J. Zabcyk, Structural properties of semilinear SPDEs driven by cylindrical stable processes. Probab. Theor. Relat. Fields 29(2), 97–137 (2010)

    Google Scholar 

  86. S. Rahmstorf, Timing of abrupt climate change: A precise clock. Geophys. Res. Lett. 30(10), 1510 (2003)

    Google Scholar 

  87. G. Raugel, Global attractors in partial differential equations, in Handbook of Dynamical Systems, vol. 2, ed. by B. Fiedler (Elsevier, Amsterdam, 2002), pp. 885–982

    Google Scholar 

  88. J.C. Robinson, in Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2001)

    Google Scholar 

  89. M. Röckner, T. Zhang, Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles. Potential Anal. 3, 255–279 (2007)

    Article  Google Scholar 

  90. H.M. Stommel, Thermohaline convection with two stable regimes of flow. Tellus 13, 224–230 (1961)

    Article  Google Scholar 

  91. S. Stolze, Stochastic equations in Hilbert space with Lévy noise and their applications in finance. Diplomarbeit Universität Bielefeld, 2005

    Google Scholar 

  92. S.G. Sell, Y. You, in Dynamics of Evolutionary Equations. Applied Mathematical Sciences, vol. 143 (Springer, Berlin, 2002)

    Google Scholar 

  93. R. Temam, in Dynamical Systems in Physics and Applications. Springer Texts in Applied Mathematics (Springer, Berlin, 1992)

    Google Scholar 

  94. T. Wakasa, Exact eigenvalues and eigenfunctions associated with linearization of Chafee-Infante equation. Funkcialaj Ekvacioj 49, 321–336 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  95. J.B. Walsh, A stochastic model of neural response. Adv. Appl. Probab. 13, 231–281 (1981)

    Article  MATH  Google Scholar 

  96. J.B. Walsh, An introduction to stochastic partial differential equations, in Ecole d’été de probabilité de Saint-Flour XIV - 1984. Lecture Notes in Mathematics, vol. 1180 (Springer, Berlin, 1986), pp. 265–437

    Google Scholar 

  97. J.-L. Wu, Lévy white noise, elliptic SPDEs and Euclidian random fields, in Recent Development in Stochastic Dynamics and Stochastic Analysis. Interdisciplinary Mathematical Sciences, vol. 8 (2010), pp. 251–268

    Google Scholar 

  98. A.H. Xia, Weak convergence of jump processes, in Séminaire de Probabiliés XXVI. Lecture Notes in Mathematics, vol. 1526 (Springer, Berlin, 1992), pp. 32–46

    Google Scholar 

  99. Y. Xie, Poincaré inequality for linear SPDEs driven by Lévy noise. Stoch. Process. Appl. 120(10), 1950–1965 (2010)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Antenne de Bretagne, Ecole Normale Supérieure Cachan, Bruz, France

    Arnaud Debussche

  2. Institut für Mathematik, Universität Potsdam, Potsdam, Germany

    Michael Högele

  3. Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Peter Imkeller

Authors
  1. Arnaud Debussche
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Högele
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peter Imkeller
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Debussche, A., Högele, M., Imkeller, P. (2013). Localization and Metastability. In: The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise. Lecture Notes in Mathematics, vol 2085. Springer, Cham. https://doi.org/10.1007/978-3-319-00828-8_7

Download citation

Publish with us

Policies and ethics

Search

Navigation