The Annals of Probability

Invariant measure for random walks on ergodic environments on a strip

Dmitry Dolgopyat and Ilya Goldsheid

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Abstract

Environment viewed from the particle is a powerful method of analyzing random walks (RW) in random environment (RE). It is well known that in this setting the environment process is a Markov chain on the set of environments. We study the fundamental question of existence of the density of the invariant measure of this Markov chain with respect to the measure on the set of environments for RW on a strip. We first describe all positive subexponentially growing solutions of the corresponding invariant density equation in the deterministic setting and then derive necessary and sufficient conditions for the existence of the density when the environment is ergodic in both the transient and the recurrent regimes. We also provide applications of our analysis to the question of positive and null recurrence, the study of the Green functions and to random walks on orbits of a dynamical system.

Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 2494-2528.

Dates
Received: December 2016
Revised: May 2018
First available in Project Euclid: 4 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1562205714

Digital Object Identifier
doi:10.1214/18-AOP1313

Mathematical Reviews number (MathSciNet)
MR3980926

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60J05: Discrete-time Markov processes on general state spaces 82C44: Dynamics of disordered systems (random Ising systems, etc.)

Keywords
RWRE random walks on a strip invariant measure

Citation

Dolgopyat, Dmitry; Goldsheid, Ilya. Invariant measure for random walks on ergodic environments on a strip. Ann. Probab. 47 (2019), no. 4, 2494--2528. doi:10.1214/18-AOP1313. https://projecteuclid.org/euclid.aop/1562205714


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References

  • [1] Alili, S. (1999). Asymptotic behaviour for random walks in random environments. J. Appl. Probab. 36 334–349.
  • [2] Berger, N., Cohen, M. and Rosenthal, R. (2016). Local limit theorem and equivalence of dynamic and static points of view for certain ballistic random walks in i.i.d. environments. Ann. Probab. 44 2889–2979.
  • [3] Bolthausen, E. and Goldsheid, I. (2000). Recurrence and transience of random walks in random environments on a strip. Comm. Math. Phys. 214 429–447.
  • [4] Bolthausen, E. and Goldsheid, I. (2008). Lingering random walks in random environment on a strip. Comm. Math. Phys. 278 253–288.
  • [5] Bolthausen, E. and Sznitman, A.-S. (2002). Ten Lectures on Random Media. DMV Seminar 32. Birkhäuser, Basel.
  • [6] Bolthausen, E. and Sznitman, A.-S. (2002). On the static and dynamic points of view for certain random walks in random environment. Methods Appl. Anal. 9 345–375.
  • [7] Brémont, J. (2009). One-dimensional finite range random walk in random medium and invariant measure equation. Ann. Inst. Henri Poincaré Probab. Stat. 45 70–103.
  • [8] Deuschel, J.-D., Guo, X. and Ramírez, A. F. (2018). Quenched invariance principle for random walk in time-dependent balanced random environment. Ann. Inst. Henri Poincaré Probab. Stat. 54 363–384.
  • [9] Dolgopyat, D. and Goldsheid, I. Constructive approach to limit theorems for recurrent diffusive random walks on a strip. Submitted.
  • [10] Dolgopyat, D. and Goldsheid, I. (2012). Quenched limit theorems for nearest neighbour random walks in 1D random environment. Comm. Math. Phys. 315 241–277.
  • [11] Dolgopyat, D. and Goldsheid, I. (2013). Limit theorems for random walks on a strip in subdiffusive regimes. Nonlinearity 26 1743–1782.
  • [12] Dolgopyat, D. and Goldsheid, I. (2018). Central limit theorem for recurrent random walks on a strip with bounded potential. Nonlinearity 31 3381–3412.
  • [13] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics 31. Cambridge Univ. Press, Cambridge.
  • [14] Enriquez, N., Sabot, C., Tournier, L. and Zindy, O. (2013). Quenched limits for the fluctuations of transient random walks in random environment on $\mathbb{Z}^{1}$. Ann. Appl. Probab. 23 1148–1187.
  • [15] Goldsheid, I. Ya. (2008). Linear and sub-linear growth and the CLT for hitting times of a random walk in random environment on a strip. Probab. Theory Related Fields 141 471–511.
  • [16] Hong, W. and Zhang, M. (2012). Branching structure for the transient random walk on a strip in a random environment. Available at arXiv:1204.1104v1.
  • [17] Kaloshin, V. Yu. and Sinai, Ya. G. (2000). Nonsymmetric simple random walks along orbits of ergodic automorphisms. In On Dobrushin’s Way. From Probability Theory to Statistical Physics. Amer. Math. Soc. Transl. Ser. 2 198 109–115. Amer. Math. Soc., Providence, RI.
  • [18] Kaloshin, V. Yu. and Sinai, Ya. G. (2000). Simple random walks along orbits of Anosov diffeomorphisms. Proc. Steklov Inst. Math. 228 224–233.
  • [19] Kesten, H. (1975). Sums of stationary sequences cannot grow slower than linearly. Proc. Amer. Math. Soc. 49 205–211.
  • [20] Kesten, H., Kozlov, M. V. and Spitzer, F. (1975). A limit law for random walk in a random environment. Compos. Math. 30 145–168.
  • [21] Kozlov, S. M. (1978). Averaging of random structures. Dokl. Akad. Nauk SSSR 241 1016–1019. Translation in: Sov. Math., Dokl. 19 (1978) 950–954.
  • [22] Kozlov, S. M. (1985). The method of averaging and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40 61–120. Translation in: Russian Math. Surveys 40 (1985) 73–145.
  • [23] Lawler, G. F. (1982). Weak convergence of a random walk in a random environment. Comm. Math. Phys. 87 81–87.
  • [24] Leskelä, L. and Stenlund, M. (2011). A local limit theorem for a transient chaotic walk in a frozen environment. Stochastic Process. Appl. 121 2818–2838.
  • [25] Menshikov, M., Popov, S. and Wade, A. (2017). Non-Homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems. Cambridge Tracts in Mathematics 209. Cambridge Univ. Press, Cambridge.
  • [26] Papanicolaou, G. C. and Varadhan, S. R. S. (1982). Diffusions with random coefficients. In Statistics and Probability: Essays in Honor of C. R. Rao (G. Kallianpur, P. R. Krishnajah and J. K. Gosh, eds.) 547–552. North-Holland, Amsterdam.
  • [27] Peterson, J. and Samorodnitsky, G. (2012). Weak weak quenched limits for the path-valued processes of hitting times and positions of a transient, one-dimensional random walk in a random environment. ALEA Lat. Am. J. Probab. Math. Stat. 9 531–569.
  • [28] Peterson, J. and Samorodnitsky, G. (2013). Weak quenched limiting distributions for transient one-dimensional random walk in a random environment. Ann. Inst. Henri Poincaré Probab. Stat. 49 722–752.
  • [29] Rassoul-Agha, F. (2003). The point of view of the particle on the law of large numbers for random walks in a mixing random environment. Ann. Probab. 31 1441–1463.
  • [30] Roitershtein, A. (2008). Transient random walks on a strip in a random environment. Ann. Probab. 36 2354–2387.
  • [31] Sabot, C. (2013). Random Dirichlet environment viewed from the particle in dimension ${d\geq3}$. Ann. Probab. 41 722–743.
  • [32] Sinaĭ, Ya. G. (1982). The limit behavior of a one-dimensional random walk in a random environment. Theory Probab. Appl. 27 256–268.
  • [33] Sinai, Ya. G. (1999). Simple random walks on tori. J. Stat. Phys. 94 695–708.
  • [34] Zeitouni, O. (2004). Random walks in random environment. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1837 189–312. Springer, Berlin.