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 .. Given a triple (T,r,.) such a speed-. 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[..y0dsf(Xs)]=2.T.(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 12.(.({v}).r(v,v.)).1. If (T,r) is path-connected, X has continuous paths and equals the .-Brownian motion which was recently constructed in [Trans. Amer. Math. Soc. 365 (2013) 3115.3150]. In this paper, we show that speed-.n motions on (Tn,rn) converge weakly in path space to the speed-. 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].