"Invariance principle for variable speed random walks on trees" by Siva Athreya, Wolfgang Löhr et al.
 

Invariance principle for variable speed random walks on trees

Article Type

Research Article

Publication Title

Annals of Probability

Abstract

We consider stochastic processes on complete, locally compact tree-like metric spaces (T, r) on their "natural scale" with boundedly finite speed measure ν. Given a triple (T, r, ν) such a speed-ν motion on (T, r) can be characterized as the unique strong Markov process which if restricted to compact subtrees satisfies for all x,y ∈ T and all positive, bounded measurable f, Ex [∫0τy dsf (Xs ) ]= 2 ∫ T ν(dz)r ( y, c(x, y, z)) f (z) < ∞, where c(x, y, z) denotes the branch point generated by x, y, z. If (T, r) is a discrete tree, X is a continuous time nearest neighbor random walk which jumps from v to v' ~ v at rate 1/2 · (ν((v)) ·r(v,v'))-1. If (T, r) is pathconnected, X has continuous paths and equals the ν-Brownian motion which was recently constructed in [Trans. Amer. Math. Soc. 365 (2013) 3115-3150]. In this paper, we show that speed-νn motions on (Tn, rn) converge weakly in path space to the speed-ν motion on (T, r) provided that the underlying triples of metric measure spaces converge in the Gromov-Hausdorff-vague topology introduced [Stochastic Process. Appl. 126 (2016) 2527-2553].

First Page

625

Last Page

667

DOI

10.1214/15-AOP1071

Publication Date

3-1-2017

Comments

Open Access, Bronze, Green

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