We consider stochastic processes on complete, locally compact tree-like metric spaces (T,r)(T,r) on their “natural scale” with boundedly finite speed measure νν. Given a triple (T,r,ν)(T,r,ν) such a speed-νν motion on (T,r)(T,r) can be characterized as the unique strong Markov process which if restricted to compact subtrees satisfies for all x,y∈Tx,y∈T and all positive, bounded measurable ff, Ex[∫τy0dsf(Xs)]=2∫Tν(dz)r(y,c(x,y,z))f(z)<∞, Ex[∫0τydsf(Xs)]=2∫Tν(dz)r(y,c(x,y,z))f(z)<∞, where c(x,y,z)c(x,y,z) denotes the branch point generated by x,y,zx,y,z. If (T,r)(T,r) is a discrete tree, XX is a continuous time nearest neighbor random walk which jumps from vv to v′∼vv′∼v at rate 12⋅(ν( v )⋅r(v,v′))−112⋅(ν( v )⋅r(v,v′))−1. If (T,r)(T,r) is path-connected, XX has continuous paths and equals the νν-Brownian motion which was recently constructed in [Trans. Amer. Math. Soc. 365 (2013) 3115–3150]. In this paper, we show that speed-νnνn motions on (Tn,rn)(Tn,rn) converge weakly in path space to the speed-νν motion on (T,r)(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].
In this subsection we relate our invariance principle to the one obtain earlier in @cite_20 . We first recall the excursion representation of a rooted compact measure @math -tree. We denote by the set of continuous excursions on @math . From each excursion @math , we can define a measure @math -tree in the following way: @math is a pseudo-distance on @math , @math are said to be equivalent, @math , if @math , the image of the projection @math endowed with the push forward of @math (again denoted @math ), i.e. @math , is a rooted compact @math -tree. We endow this space with the probability measure @math which is the push forward of the Lebesgue measure on @math . We denote by @math the resulting glue function'', which sends an excursion to a rooted probability measure @math -tree.
@cite_20
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Consider a family of random ordered graph trees (T-n)(n >= 1), where T-n has n vertices. It has previously been established that if the associated search-depth processes converge to the normalised Brownian excursion when rescaled appropriately as n -> infinity, then the simple random walks on the graph trees have the Brownian motion on the Brownian continuum random tree as their scaling limit. Here, this result is extended to demonstrate the existence of a diffusion scaling limit whenever the volume measure on the limiting real tree is nonatomic, supported on the leaves of the limiting tree, and satisfies a polynomial lower bound for the volume of balls. Furthermore, as an application of this generalisation, it is established that the simple random walks on a family of Galton-Watson trees with a critical infinite variance offspring distribution, conditioned on the total number of offspring, can be resealed to converge to the Brownian motion on a related alpha-stable tree.