Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Large deviation estimates for some nonlocal equations. General bounds and applications


Authors: Cristina Brändle and Emmanuel Chasseigne
Journal: Trans. Amer. Math. Soc. 365 (2013), 3437-3476
MSC (2010): Primary 47G20, 60F10; Secondary 35A35, 49L25
DOI: https://doi.org/10.1090/S0002-9947-2013-05629-2
Published electronically: February 7, 2013
MathSciNet review: 3042591
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Large deviation estimates for the following linear parabolic equation are studied:

$\displaystyle \frac {\partial u}{\partial t}=\mathop {\rm Tr}\Big ( a(x)D^2u\Big ) + b(x)\cdot D u+ \mathcal {L}[u](x), $

where $ \mathcal {L}[u]$ is a nonlocal Lévy-type term associated to a Lévy measure $ \mu $ (which may be singular at the origin):

$\displaystyle \mathcal {L}[u](x)=\int _{\mathbb{R}^N} \Big \{(u(x+y)-u(x)-(D u(x)\cdot y) 1\!\!{\rm I}_{\{\vert y\vert<1\}} (y)\Big \}\mathrm {d}\mu (y)\,. $

Assuming only that some negative exponential integrates with respect to the tail of $ \mu $, it is shown that given an initial data, solutions defined in a bounded domain converge exponentially fast to the solution of the problem defined in the whole space. The exact rate, which depends strongly on the decay of $ \mu $ at infinity, is also estimated.

References [Enhancements On Off] (What's this?)

  • 1. O. ALVAREZ, A. TOURIN, Viscosity solutions of nonlinear integro-differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 3, 293-317. MR 1395674 (97d:35023)
  • 2. D. APPLEBAUM, Lévy processes and stochastic calculus. Second edition. Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009. MR 2512800 (2010m:60002)
  • 3. G. BARLES, Solutions de viscosité des équations de Hamilton-Jacobi. Mathématiques & Applications 17, Springer-Verlag, Paris, 1994. MR 1613876 (2000b:49054)
  • 4. G. BARLES, An approach of deterministic control problems with unbounded data. Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), no. 4, 235-258. MR 1067774 (91h:49035)
  • 5. G. BARLES, P. SOUGANIDIS, A remark on the asymptotic behavior of the solution of the KPP equation. C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 7, 679-684. MR 1300069 (95k:35105)
  • 6. G. BARLES, CH. DAHER, M. ROMANO, Convergence of numerical schemes for parabolic equations arising in finance theory, Math. Models Methods Appl. Sci. 5 (1995), no. 1, 125-143. MR 1315000 (95m:90031)
  • 7. G. BARLES, E. CHASSEIGNE, C. IMBERT, Dirichlet boundary conditions for second order elliptic non-linear integro-differential equations, Indiana Univ. Math. J. 57 (2008), no. 1, 213-246. MR 2400256 (2009b:35088)
  • 8. G. BARLES, C. IMBERT, Second-order elliptic integro-differential equations: viscosity solutions' theory revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), no. 3, 567-585. MR 2422079 (2009c:35102)
  • 9. G. BARLES, S. MIRRAHIMI, B. PERTHAME, Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result, Methods Appl. Anal. 16 (2009), no. 3, 321-340. MR 2650800 (2011g:35017)
  • 10. G. BARLES, L.C. EVANS, P.E. SOUGANIDIS, Wavefront propagation for reaction-diffusion systems of PDE, Duke Math. J. 61 (1990), no. 3, 835-858. MR 1084462 (91k:35119)
  • 11. J. BERTOIN, Lévy processes. Cambridge Tracts in Mathematics 121, Cambridge University Press, Cambridge, 1996. MR 1406564 (98e:60117)
  • 12. C. BRäNDLE, E. CHASSEIGNE, Large deviations estimates for some non-local equations. Fast decaying kernels and explicit bounds. Nonlinear Analysis 71 (2009), 5572-5586. MR 2560225 (2010k:60099)
  • 13. C. BRäNDLE, E. CHASSEIGNE, R. FERREIRA, Unbounded solutions of the nonlocal heat equation, Commun. Pure Appl. Anal. 10 (2011), no. 6, 1663-1686. MR 2805332 (2012e:35257)
  • 14. X. CABRé, J.-M. ROQUEJOFFRE, Front propagation in Fisher-KPP equations with fractional diffusion, C. R. Math. Acad. Sci. Paris 347 (2009), no. 23-24, 1361-1366. MR 2588782 (2010j:35577)
  • 15. L. CAFFARELLI, L. SILVESTRE, An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245-1260. MR 2354493 (2009k:35096)
  • 16. L. CAFFARELLI, L. SILVESTRE, Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62 (2009), no. 5, 597-638. MR 2494809 (2010d:35376)
  • 17. P. CARR, H. GEMAN, D. MADAN, M. YOR, The Fine Structure of Asset Returns: An Empirical Investigation, Journal of Business 75 (2002), 2, 305-332.
  • 18. E. CHASSEIGNE, M. CHAVES, J.D. ROSSI, Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl. (9) 86 (2006), no. 3, 271-291. MR 2257732 (2007e:35279)
  • 19. E. CHASSEIGNE, The Dirichlet problem for some nonlocal diffusion equations. Differential Integral Equations. 20 (2007), no. 12, 1389-1404. MR 2377023 (2009a:35106)
  • 20. R. CONT, P. TANKOV, Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series (2004). MR 2042661 (2004m:91004)
  • 21. J. COVILLE, L. DUPAIGNE, On a non-local equation arising in population dynamics. Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 4, 727-755. MR 2345778 (2008i:35139)
  • 22. H. CRAMéR, Sur un nouveau théorème limite de la théorie des probabilités. Actualités Scientifiques et industrielles 736 (1938), 5-23.
  • 23. M.G. CRANDALL, P.-L. LIONS, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.277 (1983), no. 1, 1-42. MR 690039 (85g:35029)
  • 24. M.D. DONSKER, S.R.S. VARADHAN, Asymptotic evaluation of certain Markov process expectations for large time, Comm. Pure Appl. Math. 28 (1975), 1-47. MR 0386024 (52:6883)
  • 25. L.C. EVANS, H. ISHII, A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities, Ann. Inst. H. Poincaré 2 (1985) no. 1, 1-20. MR 781589 (86f:35183)
  • 26. L.C. EVANS, P.E. SOUGANIDIS, A PDE approach to geometric optics for certain semilinear parabolic equations, Indiana Univ. Math. J. 38 (1989), no. 1, 141-172. MR 982575 (90f:35093)
  • 27. W.H. FLEMING, Exit probabilities and optimal stochastic control, Applied Math. and Opt. 4 (1978), 329-346. MR 512217 (80h:60100)
  • 28. J. FENG, T.G. KURTZ, Large deviations for stochastic processes, Mathematical Surveys and Monographs, 131. American Mathematical Society, Providence, RI, 2006. MR 2260560 (2009g:60034)
  • 29. M. FREIDLIN, Functional integration and partial differential equations. Annals of Mathematics Studies 109. Princeton University Press, Princeton, NJ, 1985. MR 833742 (87g:60066)
  • 30. M. FREIDLIN, Limit theorems for large deviations and reaction-diffusion equations, Ann. Probab. 13 (1985), no. 3, 639-675. MR 799415 (87a:35104)
  • 31. F. DEN HOLLANDER, Large deviations. Fields Institute Monographs, 14. American Mathematical Society, Providence, RI, 2000. MR 1739680 (2001f:60028)
  • 32. A. KOLMOGOROFF, I. PETROVSKII, N. PISKOUNOV, Etude de l'équation de la quantité de la matière et son application à un problème biologique. Moscow Univ. Math. Bull. 1 (1937), 1-25.
  • 33. P.-L. LIONS, Generalized solutions of Hamilton-Jacobi equations. Research Notes in Mathematics, 69. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 667669 (84a:49038)
  • 34. B. ØKSENDAL, A. SULEM, Applied stochastic control of jump diffusions. Universitext. Springer-Verlag, Berlin, 2005. MR 2109687 (2005i:93003)
  • 35. B. PERTHAME, Transport equations in biology. Frontiers in Mathematics. Birkhaüser Verlag, Basel, 2007. MR 2270822 (2007j:35004)
  • 36. R.T. ROCKAFELLAR, Convex analysis. Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683 (43:445)
  • 37. S.R.S. VARADHAN, Large deviations, Ann. Probab. 36 (2008), no. 2, 397-419. MR 2393987 (2009d:60070)
  • 38. A.D WENTZELL, M.I FREIDLIN, Random Perturbations of Dynamical Systems. Springer, New-York, 1984. MR 722136 (85a:60064)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 47G20, 60F10, 35A35, 49L25

Retrieve articles in all journals with MSC (2010): 47G20, 60F10, 35A35, 49L25


Additional Information

Cristina Brändle
Affiliation: Departamento de Matemáticas, Universidad Carlos III Madrid, Avda. de la Universidad, 30, 28911 Leganés, Spain
Email: cbrandle@math.uc3m.es

Emmanuel Chasseigne
Affiliation: Laboratoire de Mathématiques et Physique Théorique, UMR7350, Université F. Rabelais - Tours, Parc de Grandmont, 37200 Tours, France – and – Fédération de Recherche Denis Poisson - FR2964 - Université d’Orléans & Université F. Rabelais - Tours
Email: echasseigne@univ-tours.fr

DOI: https://doi.org/10.1090/S0002-9947-2013-05629-2
Keywords: Nonlocal diffusion, large deviation, Hamilton-Jacobi equation, Lévy operators.
Received by editor(s): April 14, 2010
Received by editor(s) in revised form: May 12, 2011
Published electronically: February 7, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society