Large Deviations for Processes in Random Environments with Jumps
Abstract
A deterministic walk in a random environment can be understood as a general random process with finite-range dependence that starts repeating a loop once it reaches a site it has visited before. Such process lacks the Markov property. We study the exponential decay of the probabilities that the walk will reach sites located far away from the origin. We also study a similar problem for the continuous analogue: the process that is a solution to an ODE with random coefficients. In this second model the environment also has ``teleports'' which are the regions from where the process can make discontinuous jumps.
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Pages: 2406-2438
Publication Date: November 23, 2011
DOI: 10.1214/EJP.v16-962
References
- Aldous, David J. Self-intersections of 1-dimensional random walks. Probab. Theory Relat. Fields 72 (1986), no. 4, 559-587. MR0847386 (88a:60125)
- Asselah, Amine. Large deviations estimates for self-intersection local times for simple random walk in $\Bbb Z\sp 3$. Probab. Theory Related Fields 141 (2008), no. 1-2, 19-45. MR2372964 (2009c:60261)
- Bunimovich, Leonid A. Deterministic walks in random environments. Microscopic chaos and transport in many-particle systems. Phys. D 187 (2004), no. 1-4, 20-29. MR2046689 (2005f:82110)
- Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications. Corrected reprint of the second (1998) edition. Stochastic Modelling and Applied Probability, 38. Springer-Verlag, Berlin, 2010. xvi+396 pp. ISBN: 978-3-642-03310-0 MR2571413 (2011b:60094)
- Grimmett, Geoffrey R. Stochastic pin-ball. Random walks and discrete potential theory (Cortona, 1997), 205-213, Sympos. Math., XXXIX, Cambridge Univ. Press, Cambridge, 1999. MR1802432 (2002a:60158)
- Kosygina, Elena; Rezakhanlou, Fraydoun; Varadhan, S. R. S. Stochastic homogenization of Hamilton-Jacobi-Bellman equations. Comm. Pure Appl. Math. 59 (2006), no. 10, 1489-1521. MR2248897 (2009d:35017)
- Rassoul-Agha, Firas. Large deviations for random walks in a mixing random environment and other (non-Markov) random walks. Comm. Pure Appl. Math. 57 (2004), no. 9, 1178-1196. MR2059678 (2005b:60067)
- Rassoul-Agha, Firas; Seppäläinen, Timo. Process-level quenched large deviations for random walk in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), no. 1, 214-242. MR2779403 (2012a:60283)
- Rassoul-Agha, Firas; Seppäläinen, Timo; Yilmaz, Attila. Quenched free energy and large deviations for random walks in random potentials. Submitted
- Rezakhanlou, Fraydoun; Tarver, James E. Homogenization for stochastic Hamilton-Jacobi equations. Arch. Ration. Mech. Anal. 151 (2000), no. 4, 277-309. MR1756906 (2001f:35047)
- Rosenbluth, Jeffrey Quenched large deviations for multidimensional random walk in random environment: a variational formula. Ph.D. thesis, New York University. (2006) arXiv:0804.1444v1
- Shen, Lian. On ballistic diffusions in random environment. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 5, 839-876. MR1997215 (2005d:60130)
- Steele, J. Michael. Probability theory and combinatorial optimization. CBMS-NSF Regional Conference Series in Applied Mathematics, 69. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. viii+159 pp. ISBN: 0-89871-380-3 MR1422018 (99d:60002)
- Sznitman, Alain-Sol. Brownian motion, obstacles and random media. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. xvi+353 pp. ISBN: 3-540-64554-3 MR1717054 (2001h:60147)
- Sznitman, Alain-Sol; Zerner, Martin. A law of large numbers for random walks in random environment. Ann. Probab. 27 (1999), no. 4, 1851-1869. MR1742891 (2001f:60116)
- Varadhan, S. R. S. Random walks in a random environment. Proc. Indian Acad. Sci. Math. Sci. 114 (2004), no. 4, 309-318. MR2067696 (2005h:60312)
- Yilmaz, Atilla. Large deviations for random walk in a space-time product environment. Ann. Probab. 37 (2009), no. 1, 189-205. MR2489163 (2010b:60278)
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