Electronic Communications in Probability

A local limit theorem for random walks in balanced environments

Mikko Stenlund

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Central limit theorems for random walks in quenched random environments have attracted plenty of attention in the past years. More recently still, finer local limit theorems - yielding a Gaussian density multiplied by a highly oscillatory modulating factor - for such models have been obtained. In the one-dimensional nearest-neighbor case with i.i.d. transition probabilities, local limits of uniformly elliptic ballistic walks are now well understood. We complete the picture by proving a similar result for the only recurrent case, namely the balanced one, in which such a walk is diffusive. The method of proof is, out of necessity, entirely different from the ballistic case.

Article information

Electron. Commun. Probab., Volume 18 (2013), paper no. 19, 13 pp.

Accepted: 8 March 2013
First available in Project Euclid: 7 June 2016

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Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments
Secondary: 60F15: Strong theorems 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50] 82D30: Random media, disordered materials (including liquid crystals and spin glasses) 35K15: Initial value problems for second-order parabolic equations

Balanced random environment local limit theorem Nash inequality

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Stenlund, Mikko. A local limit theorem for random walks in balanced environments. Electron. Commun. Probab. 18 (2013), paper no. 19, 13 pp. doi:10.1214/ECP.v18-2336. https://projecteuclid.org/euclid.ecp/1465315558

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