An ergodic theorem of a parabolic Anderson model driven by Lévy noise

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.

Keywords

Parabolic Anderson model / ergodic theorem / invariant measure / Lévy noise / self-duality

References

[1.] Aldous D. Stopping times and tightness. Ann Probab, 1978, 6 2 335-340

[2.] Anderson P. W. Absence of diffusion in certain random lattices. Phys Rev, 1958, 109, 1492-1505

[3.] Carmona R. A., Molchanov S. A. Parabolic Anderson Problem and Intermittency. Mem Amer Math Soc, 108, 1994, Providence: Amer Math Soc.

[4.] 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

[5.] Cox J. T., Greven A. Ergodic theorems for infinite systems of locally interacting diffusions. Ann Probab, 1994, 22, 833-853

[6.] Cox J. T., Klenke A., Perkins E. A. Convergence to equilibrium and linear systems duality. Stochastic Models (Ottawa, ON, 1998), CMS Conf Proc, 26, 2001, Providence: Amer Math Soc 41-66

[7.] Cranston M., Mountford T. Lyapunov exponent for the parabolic Anderson model in ℝ^d. J Funct Anal, 2006, 236, 78-119

[8.] Cranston M., Mountford T., Shiga T. Lyapunov exponent for the parabolic Anderson model. Acta Math Univ Comenian (NS), 2002, 71 2 163-188

[9.] Cranston M., Mountford T., Shiga T. Lyapunov exponent for the parabolic Anderson model with Lévy noise. Probab Theory Relat Fields, 2005, 132, 321-355

[10.] Ethier S. N., Kurtz T. G. Markov Processes: Characterization and Convergence, 1986, New York: John Wiley and Sons.

[11.] Furuoya T., Shiga T. Sample Lyapunov exponent for a class of linear Markovian systems over ℤ^d. Osaka J Math, 1998, 35, 35-72

[12.] Gärtner J., König W. Deuschel J. D., Greven A. The Parabolic Anderson Model. Stochastic Interacting Systems, 2005, Berlin: Springer-Verlag 153-179

[13.] Gärtner J., Molchanov S. A. Parabolic problems for the Anderson model. I. Intermittency and related topics. Comm Math Phys, 1990, 132, 613-655

[14.] Granston M., Gauthier D., Mountford T. S. On the large deviations for parabolic Anderson Model. Probab Theory Relat Fields, 2010, 147, 349-378

[15.] Greven A., den Hollander F. Phase transitions for the long-time behaviour of interacting diffusions. Ann Probab, 2007, 35, 1250-1306

[16.] Ikeda N., Watanabe S. Stochastic Differential Equations and Diffusion processes, 1981, Amsterdam: North-Holland.

[17.] König W., Lacoin H., Moeters P., Sidorova N. A two cities theorem for Parabolic Anderson Model. Ann Probab, 2009, 37, 347-392

[18.] Leschke H., Müller P., Warzel S. Deuschel J. D., Greven A. A survey of rigorous results on random Schrödinger operators for amorphons solids. Stochastic Interacting Systems, 2005, Berlin: Springer-Verlag.

[19.] Liggett T. M. A characterization of the invariant measures for an infinite particle system with interactions. Trans Amer Math Soc, 1973, 179, 433-453

[20.] 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

[21.] Liggett T. M. Interacting Particle Systems, 1985, Berlin: Springer-Verlag

[22.] 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

[23.] 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

[24.] 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

[25.] 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)

[26.] Protter P. E. Stochastic Integration and Differential Equations, 2005, 2nd ed, Berlin: Springer.

[27.] Sato K. Lévy Processes and Infinitely Divisible Distributions, 1999, Cambridge: Cambridge University Press.

[28.] Sawyer S. Isotropic random walks in a tree. ZWahrsch verw Gebiete, 1978, 42, 279-292

[29.] Shiga T. An interacting system in population genetics. J Math Kyoto Univ, 1980, 20, 213-242

[30.] Shiga T. An interacting system in population genetics II. J Math Kyoto Univ, 1980, 20, 723-733

[31.] Shiga T. Ergodic theorems and exponential decay of sample paths for certain interacting diffusion systems. Osaka J Math, 1992, 29, 789-807

[32.] Stroock D.W., Varadhan S. R. S. Multidimensional Diffusion Processes, 1979, Berlin: Springer.

[33.] Yang F. X. Some problems on long-time behaviour of interacting diffusions, 2010, Beijing: Peking University.