Invariance principle for variable speed random walks on trees

Abstract

We consider stochastic processes on complete, locally compact tree-like metric spaces \((T,r)\) on their "natural scale" with boundedly finite speed measure \(\nu\). Given a triple \((T,r,\nu)\) such a speed-\(\nu\) motion on \((T,r)\) can be characterized as the unique strong Markov process which if restricted to compact subtrees satisfies for all \(x,y\in T\) and all positive, bounded measurable \(f\), \[ \mathbb{E}^x [ \int^{\tau_y}_0\mathrm{d}s\, f(X_s) ] = 2\int_T\nu(\mathrm{d}z)\, r(y,c(x,y,z))f(z) < \infty, \] 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 \(v'\sim v\) at rate \(\tfrac{1}{2}\cdot (\nu(\{v\})\cdot r(v,v'))^{-1}\). If \((T,r)\) is path-connected, \(X\) has continuous paths and equals the \(\nu\)-Brownian motion which was recently constructed in [AthreyaEckhoffWinter2013]. In this paper we show that speed-\(\nu_n\) motions on \((T_n,r_n)\) converge weakly in path space to the speed-\(\nu\) motion on \((T,r)\) provided that the underlying triples of metric measure spaces converge in the Gromov-Hausdorff-vague topology introduced recently in [AthreyaLohrWinter2016].