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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
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