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 $f$, \begin{equation}\label{eabstract}\mathbb{E}^{x}[\int^{\tau_{y}}_{0}\mathrm{d}sf(X_{s})]=2\int_{T}\nu(\mathrm{d}z)r(y,c(x,y,z))f(z)<\infty,\end{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 $v'\sim v$ at rate $\frac{1}{2}\cdot(\nu(\{v\})\cdot r(v,v'))^{-1}$. If $(T,r)$ is path-connected, $X$ has continuous paths and equals the $\nu$-Brownian motion which was recently constructed in [Trans. Amer. Math. Soc. 365 (2013) 3115–3150]. 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 in [Stochastic Process. Appl. 126 (2016) 2527–2553].