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].
The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. We use Dirichlet form methods to construct Brownian motion on any given locally compact R-tree (T,r) equipped with a Radon measure ν on (T,B(T)). We specify a criterion under which the Brownian motion is recurrent or transient. For compact recurrent R-trees we provide bounds on the mixing time.