×

zbMATH — the first resource for mathematics

Coalescing Brownian flows: a new approach. (English) Zbl 1345.60111
The paper under review provides a new state space and topology for the coalescing flow and its invariance principle for coalescing random walks. The authors provide the Schramm-Smirnov space of coalescing flows such as Arratia’s flow or the coalescing Brownian flow in Theorem 3.1, and prove an invariance principle for the convergence of scaled coalescing random walks on \(\mathbb{Z}\) towards Arratia’s flow under an optimal finite variance assumption in Theorem 4.1, and finally they illustrate the simplicity and flexibility for the coalescing random walks on the discrete Sierpinski gasket in Theorem 6.1. The essential idea is based on scaling limits of critical percolation by O. Schramm et al. [Ann. Probab. 39, No. 5, 1768–1814 (2011; Zbl 1231.60116)].
Section 2 provides the topological space of tubes with the metric induced from the usual Hausdorff metric on the compact subsets. Then the Schramm-Smirnov space \({\mathcal H}\) of coalescing flows is the subspace of hereditary closed subsets of the topological space of tubes. Theorem 2.7 characterizes the Schramm-Smirnov space \({\mathcal H}\) from Schramm-Smirnov’s fundamental results, which state that \({\mathcal H}\) with the induced topology of tubes is separable, closed and that the \(\sigma\)-field generated by \(\{H\in {\mathcal H}: T\in H\}\) for all \(T\) in a dense set of the space of tubes is the Borel \(\sigma\)-field on \({\mathcal H}\). Let \(C(t_0)\) be the set of continuous functions from \([t_0, \infty)\) to \(\mathbb{R}^d\) and define \[ \Pi = \cup_{t_0\in R} C(t_0) \times \{t_0\} \] as coalescing paths. Define a distance \(\rho\) on \(\Pi\) by \[ \rho ((\gamma_1, t_1), (\gamma_2, t_2)) = \sum_{k=1}^{\infty}2^{-k}\sup_{t\in [-k, k]} \min (\|\hat{\gamma}_1 (t) - \hat{\gamma}_2 (t)\|_2, 1), \] where \(\hat{\gamma}\) is the continuous function that extends \(\gamma\) to \(\mathbb{R}\) by setting \(\hat{\gamma} (t) = \gamma (t_0)\) for all \(t<t_0\). Thus \((\Pi, \rho)\) is a complete separable metric space. If \(\zeta \in \Pi\) is any collection of continuous functions with starting times, then define \(Cr(\zeta)\) to be the subset of all tubes traversed by \(\zeta\). If \(\zeta \in \Pi\) is compact, then \(Cr(\zeta) \in {\mathcal H}\) by Lemma 2.9. The same holds for \(\zeta_T\) under a slightly weaker assumption. The set of all probability measures on \({\mathcal H}\) is also compact under the topology of weak convergence, hence any sequence of probability measures on \({\mathcal H}\) automatically has a subsequential limit.
Section 3 is devoted to the main result Theorem 3.1 (Coalescing Brownian flow on \(\mathbb{R}\)). For the independent Brownian motions \((W_j(t))_{t\geq t_j}\) with \(W_j(t_j) = x_j\) on a common probability space, a random element \({\mathcal W}_n\in {\mathcal H}\) is defined by \[ {\mathcal W}_n = {\mathcal W}(z_1, \dots, z_n) = Cr(\{W_1^c, \dots, W_n^c\}), \] where \(W_1^c, \dots, W_n^c\) is a collection of \(n\) coalescing paths starting from \(z_1, \dots, z_n\), respectively. Theorem 3.1 states that the random variables \({\mathcal W}_n\in {\mathcal H}\) converge in distribution as \(n\to \infty\) to a random variable \({\mathcal W}_{\infty} \in {\mathcal H}\), whose law \(P_{\infty}\) does not depend on the dense countable set \(D\). The random variable \({\mathcal W}_{\infty} \in {\mathcal H}\) is called a coalescing Brownian flow on \(\mathbb{R}\). Theorem 3.4 uniquely characterizes \(P_{\infty}\) for the coalescing Brownian flow (or Arratia’s flow).
Section 4 shows the invariance principle for coalescing random walks in Theorem 4.1. Under diffusive scaling, the rescaled lattice \(\Gamma^{\eta}\) defines \({\mathcal W}^{\eta} = Cr(\Gamma^{\eta}) \in {\mathcal H}\). If \(E[\xi]=0, E[\xi^2]=\sigma^2\) and \(\xi\) is aperiodic, then \(P^{\eta} \to P_{\infty}\) as \(\eta \to 0\) weakly in \({\mathcal H}\) (Theorem 4.1). The proof consists of three steps: lower bound, uniform coming down from infinity, and upper bound.
Section 5 considers coalescing flows on the Sierpinski gasket. An existence result for a limit random variable on the Sierpinski gasket is given in Theorem 5.1, and the unique characterization is given in Theorem 5.4. The invariance principle for the coalescing Brownian flow on the Sierpinski gasket is proved in Theorem 6.1.
The paper ends with an interesting remark: since the tube topology is weaker than the path topology, it would be interesting to find examples of events which are measurable in the path topology but whose images under \(\hat{Cr}\) are not measurable in the tube topology.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J65 Brownian motion
60F17 Functional limit theorems; invariance principles
60G50 Sums of independent random variables; random walks
82C21 Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] Aizenman, M. and Burchard, A. (1999). Hölder regularity and dimension bounds for random curves. Duke Math. J. 99 419-453. · Zbl 0944.60022
[2] Angel, O., Berestycki, N. and Limic, V. (2012). Global divergence of spatial coalescents. Probab. Theory Related Fields 152 625-679. · Zbl 1271.92022
[3] Arratia, R. (1981). Coalescing Brownian motions and the voter model on \(\mathbb{Z}\). Unpublished partial manuscript. Available from .
[4] Arratia, R. A. (1979). Coalescing Brownian motions on the line. Ph.D. Thesis, Univ. Wisconsin, Madison.
[5] Barlow, M. T. (1998). Diffusions on fractals. In Lectures on Probability Theory and Statistics ( Saint-Flour , 1995). Lecture Notes in Math. 1690 1-121. Springer, Berlin. · Zbl 0916.60069
[6] Barlow, M. T. and Perkins, E. A. (1988). Brownian motion on the Sierpiński gasket. Probab. Theory Related Fields 79 543-623. · Zbl 0635.60090
[7] Belhaouari, S., Mountford, T., Sun, R. and Valle, G. (2006). Convergence results and sharp estimates for the voter model interfaces. Electron. J. Probab. 11 768-801 (electronic). · Zbl 1113.60092
[8] Berestycki, N., Garban, C. and Sen, A. (2015). A new approach to coalescing Brownian flows II: Black noise property. In preparation.
[9] Camia, F. and Newman, C. M. (2006). Two-dimensional critical percolation: The full scaling limit. Comm. Math. Phys. 268 1-38. · Zbl 1117.60086
[10] Coletti, C. F., Fontes, L. R. G. and Dias, E. S. (2009). Scaling limit for a drainage network model. J. Appl. Probab. 46 1184-1197. · Zbl 1186.60104
[11] Evans, S. N., Morris, B. and Sen, A. (2013). Coalescing systems of non-Brownian particles. Probab. Theory Related Fields 156 307-342. · Zbl 1277.60172
[12] Fontes, L. R. G., Isopi, M., Newman, C. M. and Ravishankar, K. (2004). The Brownian web: Characterization and convergence. Ann. Probab. 32 2857-2883. · Zbl 1105.60075
[13] Garban, C., Pete, G. and Schramm, O. (2013). Pivotal, cluster, and interface measures for critical planar percolation. J. Amer. Math. Soc. 26 939-1024. · Zbl 1276.60111
[14] Garban, C., Pete, G. and Schramm, O. (2013). The scaling limits of near-critical and dynamical percolation. Preprint. Available at . arXiv:1305.5526 · Zbl 1276.60111
[15] Jones, O. D. (1996). Transition probabilities for the simple random walk on the Sierpiński graph. Stochastic Process. Appl. 61 45-69. · Zbl 0853.60058
[16] Le Jan, Y. (2006). New developments in stochastic dynamics. In International Congress of Mathematicians. Vol. III 649-667. Eur. Math. Soc., Zürich. · Zbl 1099.60044
[17] Le Jan, Y. and Lemaire, S. (2004). Products of Beta matrices and sticky flows. Probab. Theory Related Fields 130 109-134. · Zbl 1063.60107
[18] Le Jan, Y. and Raimond, O. (2004). Flows, coalescence and noise. Ann. Probab. 32 1247-1315. · Zbl 1065.60066
[19] Le Jan, Y. and Raimond, O. (2004). Sticky flows on the circle and their noises. Probab. Theory Related Fields 129 63-82. · Zbl 1070.60089
[20] Newman, C. M., Ravishankar, K. and Sun, R. (2005). Convergence of coalescing nonsimple random walks to the Brownian web. Electron. J. Probab. 10 21-60. · Zbl 1067.60099
[21] Norris, J. and Turner, A. (2012). Hastings-Levitov aggregation in the small-particle limit. Comm. Math. Phys. 316 809-841. · Zbl 1259.82026
[22] Sarkar, A. and Sun, R. (2013). Brownian web in the scaling limit of supercritical oriented percolation in dimension \(1+1\). Electron. J. Probab. 18 no. 21, 23. · Zbl 1290.60107
[23] Schramm, O. and Smirnov, S. (2011). On the scaling limits of planar percolation. Ann. Probab. 39 1768-1814. · Zbl 1231.60116
[24] Spitzer, F. (1976). Principles of Random Walk , 2nd ed. Springer, New York. · Zbl 0359.60003
[25] Tóth, B. and Werner, W. (1998). The true self-repelling motion. Probab. Theory Related Fields 111 375-452. · Zbl 0912.60056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.