## The Annals of Probability

### Invariance principle for variable speed random walks on trees

#### 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 $f$, $$\label{eabstract}\mathbb{E}^{x}[\int^{\tau_{y}}_{0}\mathrm{d}sf(X_{s})]=2\int_{T}\nu(\mathrm{d}z)r(y,c(x,y,z))f(z)<\infty,$$ 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
Revised: October 2015
First available in Project Euclid: 31 March 2017

https://projecteuclid.org/euclid.aop/1490947305

Digital Object Identifier
doi:10.1214/15-AOP1071

Mathematical Reviews number (MathSciNet)
MR3630284

Zentralblatt MATH identifier
06797077

#### 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

#### References

• [1] Abraham, R., Delmas, J.-F. and Hoscheit, P. (2013). A note on the Gromov–Hausdorff–Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Probab. 18 no. 14, 21.
• [2] Abraham, R., Delmas, J.-F. and Hoscheit, P. (2014). Exit times for an increasing Lévy tree-valued process. Probab. Theory Related Fields 159 357–403.
• [3] Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990). London Mathematical Society Lecture Note Series 167 23–70. Cambridge Univ. Press, Cambridge.
• [4] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
• [5] Athreya, S., Eckhoff, M. and Winter, A. (2013). Brownian motion on $\mathbb{R}$-trees. Trans. Amer. Math. Soc. 365 3115–3150.
• [6] Athreya, S., Löhr, W. and Winter, A. (2016). The gap between Gromov-vague and Gromov–Hausdorff-vague topology. Stochastic Process. Appl. 126 2527–2553.
• [7] Barlow, M. T. and Kumagai, T. (2006). Random walk on the incipient infinite cluster on trees. Illinois J. Math. 50 33–65 (electronic).
• [8] Bhamidi, S., Evans, S. N., Peled, R. and Ralph, P. (2008). Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan. In Probability and Statistics: Essays in Honor of David A. Freedman. Inst. Math. Stat. Collect. 2 42–75. IMS, Beachwood, OH.
• [9] Bogachev, V. I. (2007). Measure Theory. Vol. I, II. Springer, Berlin.
• [10] Burago, D., Burago, Y. and Ivanov, S. (2001). A Course in Metric Geometry. Graduate Studies in Mathematics 33. Amer. Math. Soc., Providence, RI.
• [11] Croydon, D. (2008). Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree. Ann. Inst. Henri Poincaré Probab. Stat. 44 987–1019.
• [12] Croydon, D. A. (2010). Scaling limits for simple random walks on random ordered graph trees. Adv. in Appl. Probab. 42 528–558.
• [13] Duquesne, T. (2009). Continuum random trees and branching processes with immigration. Stochastic Process. Appl. 119 99–129.
• [14] Evans, S. N. (2008). Probability and Real Trees. Lecture Notes in Math. 1920. Springer, Berlin.
• [15] Evans, S. N., Pitman, J. and Winter, A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 81–126.
• [16] Evans, S. N. and Winter, A. (2006). Subtree prune and regraft: A reversible real tree-valued Markov process. Ann. Probab. 34 918–961.
• [17] Fukaya, K. (1987). Collapsing of Riemannian manifolds and eigenvalues of Laplace operator. Invent. Math. 87 517–547.
• [18] Fukushima, M., Oshima, Y. and Takeda, M. (2011). Dirichlet Forms and Symmetric Markov Processes, extended ed. de Gruyter Studies in Mathematics 19. de Gruyter, Berlin.
• [19] Greven, A., Pfaffelhuber, P. and Winter, A. (2009). Convergence in distribution of random metric measure spaces ($\Lambda$-coalescent measure trees). Probab. Theory Related Fields 145 285–322.
• [20] Gromov, M. (2007). Metric Structures for Riemannian and Non-Riemannian Spaces, English ed. Modern Birkhäuser Classics. Birkhäuser, Boston, MA.
• [21] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
• [22] Kesten, H. (1986). Subdiffusive behavior of random walk on a random cluster. Ann. Inst. Henri Poincaré Probab. Stat. 22 425–487.
• [23] Kigami, J. (1995). Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal. 128 48–86.
• [24] Krebs, W. B. (1995). Brownian motion on the continuum tree. Probab. Theory Related Fields 101 421–433.
• [25] Kuwae, K. and Shioya, T. (2003). Convergence of spectral structures: A functional analytic theory and its applications to spectral geometry. Comm. Anal. Geom. 11 599–673.
• [26] Löhr, W. (2013). Equivalence of Gromov–Prohorov- and Gromov’s $\underline{\square}\sb\lambda$-metric on the space of metric measure spaces. Electron. Commun. Probab. 18 no. 17, 10.
• [27] Löhr, W., Voisin, G. and Winter, A. (2015). Convergence of bi-measure $\mathbb{R}$-trees and the pruning process. Ann. Inst. Henri Poincaré Probab. Stat. 51 1342–1368.
• [28] Mosco, U. (1969). Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3 510–585.
• [29] Mosco, U. (1994). Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123 368–421.
• [30] Schweinsberg, J. (2000). A necessary and sufficient condition for the $\Lambda\n$-coalescent to come down from infinity. Electron. Commun. Probab. 5 1–11 (electronic).
• [31] Stone, C. (1963). Limit theorems for random walks, birth and death processes, and diffusion processes. Illinois J. Math. 7 638–660.