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Intermittency in a catalytic random medium. (English) Zbl 1117.60065

The parabolic Anderson equation is \[ \frac{\partial}{\partial t}u(x,t)=\kappa\Delta u(x,t)+\xi(x,t)u(x,t), \;\;x\in \mathbb{Z}^d, t\geq 0, \] where the \(u\)-field is \(R\)-valued, \(\kappa\in (0,\infty)\), \(\Delta\) is the discrete Laplacian, \(\xi=\{\xi(x,\cdot): x\in \mathbb{Z}^d\}\) is an \(\mathbb{R}\)-valued random field. In the paper \(\xi\) is given by \(\xi(x,t)=\gamma\sum_k\delta_{Y_k(t)}(x)\), where \(\gamma\in (0,\infty)\) and \(\{Y_k(\cdot): k\in \mathbb{N}\}\) is a collection of independent continuous-time simple random walks with diffusion constant \(\rho\in (0,\infty)\) starting from a Poisson random field with intensity \(\nu\in (0,\infty).\) In the paper the annealed Lyapunov exponents of \(u\) are computed and their dependence on the parametres \(d, k, \gamma, \rho, \nu\) is studied.

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
60F10 Large deviations
35B40 Asymptotic behavior of solutions to PDEs
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References:

[1] Biskup, M. and König, W. (2001). Long-time tails in the parabolic Anderson model with bounded potential. Ann. Probab. 29 636–682. · Zbl 1018.60093 · doi:10.1214/aop/1008956688
[2] Biskup, M. and König, W. (2001). Screening effect due to heavy lower tails in the one-dimensional parabolic Anderson model. J. Statist. Phys. 102 1253–1270. · Zbl 1174.82333 · doi:10.1023/A:1004840328675
[3] Carmona, R. A., Koralov, L. and Molchanov, S. A. (2001). Asymptotics for the almost-sure Lyapunov exponent for the solution of the parabolic Anderson problem. Random Oper. Stochastic Equations 9 77–86. · Zbl 0972.60050 · doi:10.1515/rose.2001.9.1.77
[4] Carmona, R. A. and Molchanov, S. A. (1994). Parabolic Anderson Problem and Intermittency . Amer. Math. Soc., Providence, RI. · Zbl 0925.35074
[5] Carmona, R. A., Molchanov, S. A. and Viens, F. (1996). Sharp upper bound on the almost-sure exponential behavior of a stochastic partial differential equation. Random Oper. Stochastic Equations 4 43–49. · Zbl 0849.60062 · doi:10.1515/rose.1996.4.1.43
[6] Cranston, M., Mountford, T. S. and Shiga, T. (2002). Lyapunov exponents for the parabolic Anderson model. Acta Math. Univ. Comenian. 71 163–188. · Zbl 1046.60057
[7] Cranston, M., Mountford, T. S. and Shiga, T. (2005). Lyapunov exponents for the parabolic Anderson model with Lévy noise. Probab. Theory Related Fields 132 321–355. · Zbl 1082.60057 · doi:10.1007/s00440-004-0346-y
[8] Dawson, D. A. and Fleischmann, K. (2000). Catalytic and mutually catalytic branching. In Infinite Dimensional Stochastic Analysis (Ph. Clément, F. den Hollander, J. van Neerven and B. de Pagter, eds.) 145–170. Royal Netherlands Academy of Arts and Sciences, Amsterdam. · Zbl 0983.92036
[9] Dawson, D. A. and Gärtner, J. (1987). Large deviations from the McKean–Vlasov limit for weakly interacting diffusions. Stochastics 20 247–308. · Zbl 0613.60021 · doi:10.1080/17442508708833446
[10] Donsker, M. D. and Varadhan, S. R. S. (1983). Asymptotics for the polaron. Comm. Pure Appl. Math. 36 505–528. · Zbl 0538.60081 · doi:10.1002/cpa.3160360408
[11] Gärtner, J. and Heydenreich, M. (2006). Annealed asymptotics for the parabolic Anderson model with a moving catalyst. Stochastic Process. Appl. 116 1511–1529. · Zbl 1102.60058 · doi:10.1016/j.spa.2006.04.002
[12] Gärtner, J. and den Hollander, F. (1999). Correlation structure of intermittency in the parabolic Anderson model. Probab. Theory Related Fields 114 1–54. · Zbl 0951.60069 · doi:10.1007/s004400050220
[13] Gärtner, J., den Hollander, F. and Maillard, G. (2006). Intermittency on catalysts: Symmetric exclusion. EURANDOM Report 2006-015. Ann. Probab. · Zbl 1129.60061
[14] Gärtner J. and König, W. (2000). Moment asymptotics for the continuous parabolic Anderson model. Ann. Appl. Probab. 10 192–217. · Zbl 1171.60359 · doi:10.1214/aoap/1019737669
[15] Gärtner, J. and König, W. (2005). The parabolic Anderson model. In Interacting Stochastic Systems (J. D. Deuschel and A. Greven, eds.) 153–179. Springer, Berlin. · Zbl 1111.82011
[16] Gärtner, J., König, W. and Molchanov, S. A. (2000). Almost sure asymptotics for the continuous parabolic Anderson model. Probab. Theory Related Fields 118 547–573. · Zbl 0972.60056 · doi:10.1007/s004400000096
[17] Gärtner, J., König, W. and Molchanov, S. A. (2006). Geometric characterization of intermittency in the parabolic Anderson model. Ann. Probab. · Zbl 1126.60091 · doi:10.1214/009117906000000764
[18] Gärtner, J. and Molchanov, S. A. (1990). Parabolic problems for the Anderson Hamiltonian. I. Intermittency and related topics. Comm. Math. Phys. 132 613–655. · Zbl 0711.60055 · doi:10.1007/BF02156540
[19] Gärtner, J. and Molchanov, S. A. (1998). Parabolic problems for the Anderson Hamiltonian. II. Second-order asymptotics and structure of high peaks. Probab. Theory Related Fields 111 17–55. · Zbl 0909.60040 · doi:10.1007/s004400050161
[20] Gärtner, J. and Molchanov, S. A. (2000). Moment asymptotics and Lifshitz tails for the parabolic Anderson model. In Stochastic Models (L. Gorostiza, ed.). Proceedings of the International Conference in Honour of D. A. Dawson. CMS Conf. Proc. 26 141–157. Amer. Math. Soc., Providence, RI. · Zbl 0959.60051
[21] Greven, A. and den Hollander, F. (2005). Phase transitions for the long-time behavior of interacting diffusions. EURANDOM Report 2005-002. Ann. Probab. · Zbl 1126.60085 · doi:10.1214/009117906000001060
[22] den Hollander, F. (2000). Large Deviations . Amer. Math. Soc., Providence, RI. · Zbl 0949.60001
[23] Kesten, H. and Sidoravicius, V. (2003). Branching random walk with catalysts. Electron. J. Probab. 8 1–51. · Zbl 1064.60196
[24] Klenke, A. (2000). A review on spatial catalytic branching. In Stochastic Models . CMS Conf. Proc. 26 245–263. Amer. Math. Soc., Providence, RI. · Zbl 0956.60096
[25] Lieb, E. H. (1977). Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57 93–105. · Zbl 0369.35022
[26] Molchanov, S. A. (1994). Lectures on random media. Ecole d ’ Eté de Probabilités de St. Flour XXII–1992 242–411. Lecture Notes in Math. 1581 . Springer, Berlin. · Zbl 0814.60093
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