Abstract
In this paper, we study an ergodic theorem of a parabolic Andersen model driven by Lévy noise. Under the assumption that A = (a(i, j)) i,j∈S is symmetric with respect to a σ-finite measure gp, we obtain the long-time convergence to an invariant probability measure ν h starting from a bounded nonnegative A-harmonic function h based on self-duality property. Furthermore, under some mild conditions, we obtain the one to one correspondence between the bounded nonnegative A-harmonic functions and the extremal invariant probability measures with finite second moment of the nonnegative solution of the parabolic Anderson model driven by Lévy noise, which is an extension of the result of Y. Liu and F. X. Yang.
Similar content being viewed by others
References
Aldous D. Stopping times and tightness. Ann Probab, 1978, 6(2): 335–340
Anderson P W. Absence of diffusion in certain random lattices. Phys Rev, 1958, 109: 1492–1505
Carmona R A, Molchanov S A. Parabolic Anderson Problem and Intermittency. Mem Amer Math Soc, 108. Providence: Amer Math Soc, 1994
Cox J T, Fleischmann K, Greven A. Comparison of interacting diffusions and an application to their ergodic theory. Probab Theory Relat Fields, 1996, 92: 513–528
Cox J T, Greven A. Ergodic theorems for infinite systems of locally interacting diffusions. Ann Probab, 1994, 22: 833–853
Cox J T, Klenke A, Perkins E A. Convergence to equilibrium and linear systems duality. In: Stochastic Models (Ottawa, ON, 1998), CMS Conf Proc, 26. Providence: Amer Math Soc, 2001, 41–66
Cranston M, Mountford T. Lyapunov exponent for the parabolic Anderson model in ℝd. J Funct Anal, 2006, 236: 78–119
Cranston M, Mountford T, Shiga T. Lyapunov exponent for the parabolic Anderson model. Acta Math Univ Comenian (NS), 2002, 71(2): 163–188
Cranston M, Mountford T, Shiga T. Lyapunov exponent for the parabolic Anderson model with Lévy noise. Probab Theory Relat Fields, 2005, 132: 321–355
Ethier S N, Kurtz T G. Markov Processes: Characterization and Convergence. New York: John Wiley and Sons, 1986
Furuoya T, Shiga T. Sample Lyapunov exponent for a class of linear Markovian systems over ℤd. Osaka J Math, 1998, 35: 35–72
Gärtner J, König W. The Parabolic Anderson Model. In: Deuschel J D, Greven A, eds. Stochastic Interacting Systems. Berlin: Springer-Verlag, 2005, 153–179
Gärtner J, Molchanov S A. Parabolic problems for the Anderson model. I. Intermittency and related topics. Comm Math Phys, 1990, 132: 613–655
Granston M, Gauthier D, Mountford T S. On the large deviations for parabolic Anderson Model. Probab Theory Relat Fields, 2010, 147: 349–378
Greven A, den Hollander F. Phase transitions for the long-time behaviour of interacting diffusions. Ann Probab, 2007, 35: 1250–1306
Ikeda N, Watanabe S. Stochastic Differential Equations and Diffusion processes. Amsterdam: North-Holland, 1981
König W, Lacoin H, Moeters P, Sidorova N. A two cities theorem for Parabolic Anderson Model. Ann Probab, 2009, 37: 347–392
Leschke H, Müller P, Warzel S. A survey of rigorous results on random Schrödinger operators for amorphons solids. In: Deuschel J D, Greven A, eds. Stochastic Interacting Systems. Berlin: Springer-Verlag, 2005
Liggett T M. A characterization of the invariant measures for an infinite particle system with interactions. Trans Amer Math Soc, 1973, 179: 433–453
Liggett T M. A characterization of the invariant measures for an infinite particle system with interactions II. Trans Amer Math Soc, 1974, 198: 201–213
Liggett T M. Interacting Particle Systems. Berlin: Springer-Verlag, 1985
Liggett T M, Spitzer F. Ergodic theorems for coupled random walks and other systems with locally interacting components. Z Wahrsch verw Gebiete, 1981, 56: 443–468
Liu Y. An application of the compound Poisson process to the stochastic differential equations with jumps. In: Some Problems in Stochastic Analysis, the Research Report of Postdoctor at AMSS, CAS. 2001
Liu Y, Gong G L. Approximation of homogeneous Lévy processes by compound Poisson processes on separable metric groups. Acta Math Sci, 1998, 18: 221–227 (in Chinese)
Liu Y, Yang F X. Some ergodic theorems of a parabolic Anderson model. Preprint. Or see: Some ergodic theorems of linear systems of interacting diffusion. Acta Math Sin (Engl Ser) (to appear)
Protter P E. Stochastic Integration and Differential Equations. 2nd ed. Berlin: Springer, 2005
Sato K. Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press, 1999
Sawyer S. Isotropic random walks in a tree. ZWahrsch verw Gebiete, 1978, 42: 279–292
Shiga T. An interacting system in population genetics. J Math Kyoto Univ, 1980, 20: 213–242
Shiga T. An interacting system in population genetics II. J Math Kyoto Univ, 1980, 20: 723–733
Shiga T. Ergodic theorems and exponential decay of sample paths for certain interacting diffusion systems. Osaka J Math, 1992, 29: 789–807
Stroock DW, Varadhan S R S. Multidimensional Diffusion Processes. Berlin: Springer, 1979
Yang F X. Some problems on long-time behaviour of interacting diffusions. Ph D Thesis. Beijing: Peking University, 2010 (in Chinese)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, Y., Wu, J., Yang, F. et al. An ergodic theorem of a parabolic Anderson model driven by Lévy noise. Front. Math. China 6, 1147–1183 (2011). https://doi.org/10.1007/s11464-011-0124-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-011-0124-y