Invariance principle for variable speed random walks on trees

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, {equation}{eabstract}Eˣ[∫^{τy}₀dsf(Xₛ)]=2∫Tν(dz)r(y,c(x,y,z))f(z)<∞,{equation} 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(\)· r(v,v'))⁻¹. If $(T,r)$ is path-connected, 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 in [Stochastic Process. Appl. 126 (2016) 2527–2553].