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 $f$, \[ \mathbb{E}^x [ \int^{τ_y}_0\mathrm{d}s\, f(X_s) ] = 2\int_Tν(\mathrm{d}z)\, r(y,c(x,y,z))f(z) < \infty, \] 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 $v^{\prime }\sim v$ at rate $\frac{1}{2}\cdot (\nu (\{v\})\cdot r(v,v^{\prime }){)}^{-1}$. 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].

45 pages

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• A Unified Framework for Generalizing the Gromov-Hausdorff Metric
• Invariance principles for random walks in random environment on trees
• Anomalous scaling regime for one-dimensional Mott variable-range hopping
• Scaling limit of linearly edge-reinforced random walks on critical Galton-Watson trees