On Recurrent and Transient Sets of Inhomogeneous Symmetric Random Walks
Giambattista Giacomin, Universitè Paris 7 and Laboratoire de Probabilités et Modèles Aléatoires C.N.R.S.
Gustavo Posta, Politecnico di Milano
Abstract
We consider a continuous time random walk on the d-dimensional
lattice Zd: 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.
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