Electronic Communications in Probability

On Recurrent and Transient Sets of Inhomogeneous Symmetric Random Walks

Giambattista Giacomin and Gustavo Posta

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We consider a continuous time random walk on the $d$-dimensional lattice $\mathbb{Z}^d$: the jump rates are time dependent, but symmetric and strongly elliptic with ellipticity constants independent of time. We investigate the implications  of heat kernel estimates on recurrence-transience  properties of the walk and we give conditions for recurrence as well as for transience: we give applications of these conditions  and discuss them in relation with the (optimal) Wiener test available in the time independent context. Our approach relies on estimates on the time spent by the walk in a set and on a 0-1 law. We show also that, still via heat kernel estimates, one can avoid using a 0-1 law, achieving this way quantitative estimates on more general hitting probabilities.

Article information

Electron. Commun. Probab., Volume 6 (2001), paper no. 4, 39-53.

Accepted: 18 January 2001
First available in Project Euclid: 19 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J75: Jump processes 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Inhomogeneous Symmetric Random Walks Heat Kernel Estimates Recurrence-Transience Hitting Probabilities Wiener test Paley-Zygmund inequality

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Giacomin, Giambattista; Posta, Gustavo. On Recurrent and Transient Sets of Inhomogeneous Symmetric Random Walks. Electron. Commun. Probab. 6 (2001), paper no. 4, 39--53. doi:10.1214/ECP.v6-1033. https://projecteuclid.org/euclid.ecp/1461097549

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