[Paper Review] Iterated logarithm law for sizes of clusters in Arratia flow

A. A. Dorogovtsev, Alexander Gnedin+1 more|arXiv (Cornell University)|Oct 24, 2013

Stochastic processes and financial applications2 references3 citations

TL;DR

This paper establishes an iterated logarithm law for the size of the cluster containing the particle starting at 0 in the Arratia flow, a system of coalescing Brownian motions. Using stochastic analysis and Gaussian process inequalities, it proves that as time t approaches 0, the cluster size ν(t) satisfies limsup_{t→0+} ν(t)/(2√(t ln ln t⁻¹)) = 1 almost surely, extending to any Arratia flow with a Lipschitz drift via Girsanov-type arguments.

ABSTRACT

The asymptotics of sizes of clusters for the Arratia flow is considered, the Arratia flow being a system of coalescing Wiener processes starting from the real axis and independent before they meet. A cluster at time t is defined as a set of particles that have glued together not later than at t. The results obtained are remarked to hold for any Arratia flow with a Lipschitz drift.

Motivation & Objective

• To characterize the small-time asymptotic behavior of cluster sizes in the Arratia flow, where particles coalesce upon meeting.
• To extend known large-time cluster behavior to the small-time regime, particularly near t=0.
• To establish a precise almost sure limsup scaling law for cluster size ν(t) as t→0+.
• To prove that the result holds not only for the standard Arratia flow but also for any version with a Lipschitz drift.
• To use advanced tools from Gaussian process theory and concentration inequalities to derive the scaling limit.

Proposed method

• Define the cluster ν(t) as the Lebesgue measure of particles coalesced with the particle starting at 0 by time t.
• Use the identity P(ν(t) ≥ r) = P(τ[y(0), y(r)] ≤ t) = P(θ(r/√2) ≤ t), reducing the problem to hitting times of a Wiener process.
• Apply the Borel-Cantelli lemma to upper bound the limsup by analyzing the summability of tail probabilities over a dyadic sequence t_n = α^n.
• Use Sudakov's minoration inequality and Gaussian concentration to establish a lower bound on the expected supremum of a normalized Gaussian process on a discrete time grid.
• Leverage the Girsanov theorem to extend the result from the zero-drift Arratia flow to flows with Lipschitz drifts.
• Combine metric entropy estimates and tail bounds to show that the probability of the cluster size remaining below the threshold decays to zero, confirming the limsup equality.

Experimental results

Research questions

• RQ1 What is the exact asymptotic scaling of the cluster size ν(t) as t→0+ in the Arratia flow?
• RQ2 Does the iterated logarithm law ν(t)/(2√(t ln ln t⁻¹)) → 1 almost surely hold for the cluster size in the Arratia flow?
• RQ3 Can this scaling law be extended to Arratia flows with non-zero Lipschitz drifts?
• RQ4 How do the properties of hitting times of Brownian motion relate to the coalescence structure of the flow?
• RQ5 What role do Gaussian process inequalities and concentration of measure play in deriving the limsup result?

Key findings

• The limsup of ν(t)/(2√(t ln ln t⁻¹)) as t→0+ is almost surely equal to 1, establishing a precise almost sure scaling law.
• The upper bound is derived via the Borel-Cantelli lemma by showing the sum of probabilities P(ν(t_n)/(2√(t_n ln ln t_n⁻¹)) > 1+ε) is finite for any ε>0.
• The lower bound relies on Sudakov's minoration inequality applied to a normalized Gaussian process indexed over a discrete time grid.
• The expected supremum of the process on this grid grows like √(ln N) as N increases, and concentration inequalities ensure this dominates the threshold.
• The result extends to any Arratia flow with a Lipschitz drift, as such flows are absolutely continuous with respect to the zero-drift version via Girsanov's theorem.
• The proof structure confirms that the cluster size growth near t=0 is governed by the same logarithmic scaling as in the standard iterated logarithm law for Brownian motion.