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].Comment: 45 page