The Annals of Probability

Invariance principle for variable speed random walks on trees

Siva Athreya, Wolfgang Löhr, and Anita Winter

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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].

Article information

Source
Ann. Probab., Volume 45, Number 2 (2017), 625-667.

Dates
Received: April 2014
Revised: October 2015
First available in Project Euclid: 31 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1490947305

Digital Object Identifier
doi:10.1214/15-AOP1071

Mathematical Reviews number (MathSciNet)
MR3630284

Zentralblatt MATH identifier
06797077

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 60B05: Probability measures on topological spaces
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J27: Continuous-time Markov processes on discrete state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60B99: None of the above, but in this section 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]

Keywords
Brownian motion $\mathbb{R}$-tree Gromov–Hausdorff-vague topology convergence of Markov chains diffusions on metric measure trees speed measure Dirichlet form

Citation

Athreya, Siva; Löhr, Wolfgang; Winter, Anita. Invariance principle for variable speed random walks on trees. Ann. Probab. 45 (2017), no. 2, 625--667. doi:10.1214/15-AOP1071. https://projecteuclid.org/euclid.aop/1490947305


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