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# Invariance principle for variable speed random walks on trees

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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 $$f$$, $\mathbb{E}^x [ \int^{\tau_y}_0\mathrm{d}s\, f(X_s) ] = 2\int_T\nu(\mathrm{d}z)\, r(y,c(x,y,z))f(z) < \infty,$ 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].

### Author and article information

###### Journal
2014-04-24
2017-04-01
###### Article
10.1214/15-AOP1071
1404.6290
ccda8772-862f-4a3f-9a8f-457f97048aa6