Random walks with unbounded jumps among random conductances II: conditional quenched CLT

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched conditional invariance principle for the random walk, under the condition that it remains positive until time n. As a corollary of this result, we study the effect of conditioning the random walk to exceed level n before returning to 0 as n →∞.We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we101253270FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO2009/51139–3; 2009/52379–8300886/2008–0; 472431/2009–9Belkin, B., An invariance principle for conditioned recurrent random walk attracted to a stable law (1972) Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 21, pp. 45-64. , MR0309200Bertoin, J., Doney, R.A., On conditioning a random walk to stay nonnegative (1994) Ann. Probab., 22 (4), pp. 2152-2167. , MR1331218Biane, Ph., Yor, M., Quelques précisions sur le méandre brownien (1988) Bull. Sci. Math., 112 (1-2), pp. 101-109. , MR942801Bolthausen, E., On a functional central limit theorem for random walks conditioned to stay positive (1976) Ann. Probability, 4 (3), pp. 480-485. , MR0415702Caravenna, F., A local limit theorem for random walks conditioned to stay positive (2005) Probab. Theory Related Fields, 133 (4), pp. 508-530. , MR2197112Caravenna, F., Chaumont, L., Invariance principles for random walks conditioned to stay positive (2008) Ann. Inst. Henri Poincaré Probab. Stat., 44 (1), pp. 170-190. , MR2451576Comets, F., Popov, S., Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards (2012) Ann. Inst. Henri Poincaré Probab. Stat., 48 (3), pp. 721-744. , MR2976561Comets, F., Popov, S., Schütz, G.M., Vachkovskaia, M., Quenched invariance principle for the Knudsen stochastic billiard in a random tube (2010) Ann. Probab., 38 (3), pp. 1019-1061. , MR2674993Comets, F., Popov, S., Schütz, G.M., Vachkovskaia, M., Billiards in a general domain with random reflections (2009) Arch. Ration. Mech. Anal., 191 (3), pp. 497-537. , MR2481068Comets, F., Popov, s., Schütz, G.M., Vachkovskaia, M., Knudsen gas in a finite random tube: transport diffusion and first passage properties (2010) J. Stat. Phys., 140 (5), pp. 948-984. , MR2673342Cooper, R.B., (1981) Introduction to queueing theory, , North-Holland Publishing Co., New York, second edition, ISBN 0-444-00379-7. MR636094Doyle, P.G., Snell, J.L., Random walks and electric networks (1984) Carus Mathematical Monographs, 22. , Mathematical Association of America, Washington, DC, ISBN 0-88385-024-9. MR920811Durrett, R., Conditioned limit theorems for some null recurrent Markov processes (1978) Ann. Probab., 6 (5), pp. 798-828. , MR503953Gallesco, C., Popov, s., Random walks with unbounded jumps among random conductances I: Uniform quenched CLT (2012) Electron. J. Probab., 17 (85), p. 22. , MR2988400Iglehart, D.L., Functional central limit theorems for random walks conditioned to stay positive (1974) Ann. Probability, 2, pp. 608-619. , MR0362499Imhof, J.-P., Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications (1984) J. Appl. Probab., 21 (3), pp. 500-510. , MR752015Mörters, P., Peres, Y., (2010) Brownian motion, , Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, ISBN 978-0-521-76018-8. With an appendix by Oded Schramm and Wendelin Werner. MR2604525Revuz, D., Yor, M., Continuous martingales and Brownian motion (1999) Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. , Springer-Verlag, Berlin, third edition, ISBN 3-540-64325-7. MR1725357Vatutin, V.A., Wachtel, V., Local probabilities for random walks conditioned to stay positive (2009) Probab. Theory Related Fields, 143 (1-2), pp. 177-217. , MR244912