Scaling limit theorem for transient random walk in random environment

Abstract

We construct a sequence of transient random walks in random environments and prove that by proper scaling, it converges to a diffusion process with drifted Brownian potential. To this end, we prove a counterpart of convergence for transient random walk in non-random environment, which is interesting itself.

Keywords

Random walk / random environment / diffusion process / Brownian motion with drift

References

[1] Borisov I S, Nikitina N N. The distribution of the number of crossings of a strip by paths of the simplest random walks and of a Wiener process with drift. Theory Probab Appl, 2012, 56: 126–132

[2] Brox T. A one-dimensional diffusion process in a Wiener medium. Ann Probab, 1986, 14: 1206–1218

[3] Comets F, Gantert N, Zeitouni O. Quenched, annealed and functional large deviations for one-dimensional random walk in random environment. Probab Theory Related Fields, 2000, 118: 65–114

[4] Dembo A, Peres Y, Zeitouni O. Tail estimates for one-dimensional random walk in random environment. Comm Math Phys, 1996, 181: 667–683

[5] Durrett R. Probability: Theory and Examples. 3rd ed. Belmont: Brooks/Cole-Thomson Learning, 2004

[6] Ethier S N, Kurtz T G. Markov Processes: Characterization and Convergence. 2nd ed. Wiley Ser Probab Stat. Hoboken: Wiley, 2005

[7] Greven A, den Hollander F. Large deviations for a random walk in random environment. Ann Probab, 1994, 22: 1381–1428

[8] Hu Y, Shi Z, Yor M. Rates of convergence of diffusions with drifted Brownian potentials. Trans Amer Math Soc, 1999, 351: 3915–3934

[9] Kawazu K, Tanaka H. A diffusion process in a Brownian environment with drift. J Math Soc Japan, 1997, 49: 189–211

[10] Kesten H, Kozlov M V, Spitzer F. A limit law for random walk in random environment. Compos Math, 1975, 30: 145–168

[11] Kurtz T G. Approximation of Population Processes. Philadelphia: SIAM, 1981

[12] Schumacher S. Diffusions with random coefficients. Contemp Math, 1985, 41: 351–356

[13] Seignourel P. Discrete schemes for processes in random media. Probab Theory Related Fields, 2000, 118: 293–322

[14] Sinai Y G. The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab Appl, 1982, 27: 256–268

[15] Stroock DW, Varadhan S R S. Multidimensional Diffusion Processes. Berlin: Springer, 2009

[16] Taleb M. Large deviations for a Brownian motion in a drifted Brownian potential. Ann Probab, 2001, 29: 1173–1204

[17] Tanaka H. Diffusion processes in random environments. In: Proceedings of the International Congress of Mathematicians, Vol 2. Basel: Birkhüuser, 1995, 1047–1054

[18] Zeitouni O. Random walks in random environment. In: Tavaré S, Zeitouni O, eds. Lectures on Probability Theory and Statistics. Lecture Notes in Math, Vol 1837. Berlin: Springer, 2004, 190–312

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