Existence and Exponential Mixing of Infinite White $\alpha$-Stable Systems with Unbounded Interactions

Lihu Xu (Technische Universität Berlin)
Boguslaw Zegarlinski (Imperial College London)

Abstract


We study an infinite white $\alpha$-stable systems with unbounded interactions, and prove the existence of a solution by Galerkin approximation and an exponential mixing property by an $\alpha$-stable version of gradient bounds.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1994-2018

Publication Date: December 2, 2010

DOI: 10.1214/EJP.v15-831

References

  1. Albeverio, S.; Kondratiev, Yu. G.; Röckner, M. Uniqueness of the stochastic dynamics for continuous spin systems on a lattice. J. Funct. Anal. 133 (1995), no. 1, 10--20. MR1351639 (96i:31006)
  2. Albeverio, S.; Kondratiev, Yu. G.; Tsikalenko, T. V. Stochastic dynamics for quantum lattice systems and stochastic quantization. I. Ergodicity.Translated by the authors. Random Oper. Stochastic Equations 2 (1994), no. 2, 103--139. MR1293068 (95i:82008)
  3. Albeverio, S.; Mandrekar, V.; Rüdiger, B. Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise. Stochastic Process. Appl. 119 (2009), no. 3, 835--863. MR2499860 (2010e:60113)
  4. Albeverio, Sergio; Rüdiger, Barbara; Wu, Jiang-Lun. Invariant measures and symmetry property of Lévy type operators. Potential Anal. 13 (2000), no. 2, 147--168. MR1782254 (2001i:60138)
  5. Albeverio, Sergio; Wu, Jiang-Lun; Zhang, Tu-Sheng. Parabolic SPDEs driven by Poisson white noise. Stochastic Process. Appl. 74 (1998), no. 1, 21--36. MR1624076 (99c:60124)
  6. Bakry, Dominique; Émery, Michel. Inégalités de Sobolev pour un semi-groupe symétrique.(French) [Sobolev inequalities for a symmetric semigroup] C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 8, 411--413. MR0808640 (86k:60141)
  7. Bass, Richard F.; Chen, Zhen-Qing. Systems of equations driven by stable processes. Probab. Theory Related Fields 134 (2006), no. 2, 175--214. MR2222382 (2007k:60164)
  8. Bertoin, Jean. Lévy processes.Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0 MR1406564 (98e:60117)
  9. Bichteler, Klaus. Stochastic integration with jumps.Encyclopedia of Mathematics and its Applications, 89. Cambridge University Press, Cambridge, 2002. xiv+501 pp. ISBN: 0-521-81129-5 MR1906715 (2003d:60002)
  10. Spin glasses.Edited by Erwin Bolthausen and Anton Bovier.Lecture Notes in Mathematics, 1900. Springer, Berlin, 2007. x+179 pp. ISBN: 978-3-540-40902-1; 3-540-40902-5 MR2309595 (2007k:82149)
  11. Da Prato, Giuseppe; Zabczyk, Jerzy. Stochastic equations in infinite dimensions.Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. xviii+454 pp. ISBN: 0-521-38529-6 MR1207136 (95g:60073)
  12. Giuseppe Da Prato and Jerzy Zabczyk, Ergodicity for infinite-dimensional systems, London Mathematical Society Lecture Note Series, vol. 229, Cambridge University Press, Cambridge, 1996.
  13. Don Dawson, Stochastic population dynamics, (2009),http://www.math.ubc.ca/~db5d/SummerSchool09/lectures-dd/lecture14.pdf
  14. Folland, Gerald B. Real analysis.Modern techniques and their applications.Second edition.Pure and Applied Mathematics (New York). A Wiley-Interscience Publication.John Wiley & Sons, Inc., New York, 1999. xvi+386 pp. ISBN: 0-471-31716-0 MR1681462 (2000c:00001)
  15. Funaki, Tadahisa; Xie, Bin. A stochastic heat equation with the distributions of Lévy processes as its invariant measures. Stochastic Process. Appl. 119 (2009), no. 2, 307--326. MR2493992 (2010g:60153)
  16. Greven, A.; den Hollander, F. Phase transitions for the long-time behavior of interacting diffusions. Ann. Probab. 35 (2007), no. 4, 1250--1306. MR2330971 (2008h:60405)
  17. Guionnet, A.; Zegarlinski, B. Lectures on logarithmic Sobolev inequalities. Séminaire de Probabilités, XXXVI, 1--134, Lecture Notes in Math., 1801, Springer, Berlin, 2003. MR1971582 (2004b:60226)
  18. Hairer, Martin; Mattingly, Jonathan C. Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. of Math. (2) 164 (2006), no. 3, 993--1032. MR2259251 (2008a:37095)
  19. M. Hutzenthaler and A. Wakolbinger, Ergodic behavior of locally regulated branching populations, Ann. Appl. Probab. 17 (2007), no. 2, 474--501.
  20. Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes.North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. xiv+464 pp. ISBN: 0-444-86172-6 MR0637061 (84b:60080)
  21. Carlo Marinelli and Michael R?ckner, Well-posedness and asymptotic behavior for stochastic reaction-diffusion equations with multiplicative poisson noise, arXiv:0903.3299v2.
  22. Øksendal, Bernt. Stochastic partial differential equations driven by multi-parameter white noise of Lévy processes. Quart. Appl. Math. 66 (2008), no. 3, 521--537. MR2445527 (2009i:60128)
  23. Robert Olkiewicz, Lihu Xu, and Boguslaw Zegarlinski, Nonlinear problems in infinite interacting particle systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11 (2008), no. 2, 179--211.
  24. 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)
  25. Peszat, Szymon; Zabczyk, Jerzy. Stochastic heat and wave equations driven by an impulsive noise. Stochastic partial differential equations and applications—VII, 229--242, Lect. Notes Pure Appl. Math., 245, Chapman & Hall/CRC, Boca Raton, FL, 2006. MR2227232 (2007h:60056)
  26. Pistorius, Martijn R. An excursion-theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected Lévy processes. Séminaire de Probabilités XL, 287--307, Lecture Notes in Math., 1899, Springer, Berlin, 2007. MR2409012 (2009e:60105)
  27. E. Priola and J. Zabczyk, Structural properties of semilinear spdes driven by cylindrical stable processes, Probab. Theory Related Fields (2009), (to appear).
  28. Priola, Enrico; Zabczyk, Jerzy. Densities for Ornstein-Uhlenbeck processes with jumps. Bull. Lond. Math. Soc. 41 (2009), no. 1, 41--50. MR2481987 (2010f:60241)
  29. Rogers, L. C. G.; Williams, David. Diffusions, Markov processes, and martingales. Vol. 1.Foundations.Reprint of the second (1994) edition.Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. xx+386 pp. ISBN: 0-521-77594-9 MR1796539 (2001g:60188)
  30. Marco Romito and Lihu Xu, Ergodicity of the 3d stochastic navier-stokes equations driven by mildly degenerate noise, 2009, preprint.
  31. Rost, Hermann. Skorokhod's theorem for general Markov processes. Transactions of the Sixth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes (Tech. Univ. Prague, Prague, 1971; dedicated to the memory of Antonín Špaček), pp. 755--764. Academia, Prague, 1973. MR0356248 (50 #8719)
  32. Lihu Xu, Stochastic population dynamics, Lecture notes based on the lectures given by Don Dawson at summer school of UBC in 2009 (pp 44, still being writing in progress).
  33. Lihu Xu and Boguslaw Zegarlinski, Ergodicity of the finite and infinite dimensional $\alpha$-stable systems, Stoch. Anal. Appl. 27 (2009), no.~4, 797--824.
  34. Xu, Lihu; Zegarliński, Bogusław. Ergodicity of the finite and infinite dimensional $\alpha$-stable systems. Stoch. Anal. Appl. 27 (2009), no. 4, 797--824. MR2541378 (2010k:60180)
  35. Boguslaw Zegarlinski, The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice, Comm. Math. Phys. 175 (1996), no.~2, 401--432.
Bookmark and Share


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