Invariance principle for variable speed random walks on trees

Article Type

Research Article

Publication Title

Annals of Probability


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].

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