[Paper Review] Maximal Displacement of Critical Branching Symmetric Stable Processes

Steven P. Lalley, Yuan Shao|arXiv (Cornell University)|Jul 11, 2013

Stochastic processes and statistical mechanics6 references3 citations

TL;DR

This paper analyzes the maximal displacement $ M $ of a critical branching symmetric $ \alpha $-stable process on $ \mathbb{R} $, where particles branch at rate 1 and perform symmetric $ \alpha $-stable Lévy motions. Using Feynman-Kac formulas and asymptotic analysis of pseudo-differential equations, it establishes that $ \mathbb{P}(M \geq x) \sim \sqrt{2/\alpha} \cdot x^{-\alpha/2} $ as $ x \to \infty $, extending results from discrete branching random walks to continuous stable processes.

ABSTRACT

We consider a critical continuous-time branching process (a Yule process) in which the individuals independently execute symmetric $α-$stable random motions on the real line starting at their birth points. Because the branching process is critical, it will eventually die out, and so there is a well-defined maximal location $M$ ever visited by an individual particle of the process. We prove that the distribution of $M$ satisfies the asymptotic relation $P\{M\geq x \}\sim (2/α)^{1/2}x^{-α/2}$ as $x ightarrow \infty$.

Motivation & Objective

• To determine the tail behavior of the maximal displacement $ M $ in a critical branching symmetric $ \alpha $-stable process on the real line.
• To extend previous results on critical branching random walks to the continuous, stable Lévy motion setting.
• To develop a method based on Feynman-Kac formulas and asymptotic analysis of pseudo-differential equations to analyze the distribution of $ M $.
• To establish the exact asymptotic decay rate of the survival probability $ \mathbb{P}(M \geq x) $ as $ x \to \infty $.

Proposed method

• The authors model the maximal displacement $ M $ as the supremum of particle positions over time in a continuous-time branching process with symmetric $ \alpha $-stable motion.
• They derive a nonlinear integral equation for the distribution function of $ M $ using the Feynman-Kac formula, relating it to the survival probability of a killed Lévy process.
• The analysis involves studying the hitting time of a large level $ x $, using first-exit times from intervals and first-jump times of size exceeding $ x $.
• They apply estimates on the Laplace transform of the first passage time of a symmetric $ \alpha $-stable process, leveraging the exponential tail behavior of jump times.
• The key step involves bounding the survival probability via a comparison with a Poisson point process representation of the stable motion.
• Asymptotic analysis of the resulting equations leads to the precise power-law decay rate of the tail probability.

Experimental results

Research questions

• RQ1 What is the asymptotic tail behavior of the maximal displacement $ M $ in a critical branching symmetric $ \alpha $-stable process on $ \mathbb{R} $ as $ x \to \infty $?
• RQ2 How does the tail decay rate of $ \mathbb{P}(M \geq x) $ compare to that of discrete branching random walks or branching Brownian motion?
• RQ3 Can the Feynman-Kac formula be effectively used to analyze the distribution of the maximal displacement in a continuous, non-Markovian branching process with Lévy motion?
• RQ4 What role do the first-exit and first-jump times of the underlying Lévy process play in determining the tail behavior of $ M $?
• RQ5 Is the asymptotic decay rate $ x^{-\alpha/2} $ universal for such critical branching processes with symmetric $ \alpha $-stable motion?

Key findings

• The tail probability $ \mathbb{P}(M \geq x) $ decays asymptotically as $ \sqrt{2/\alpha} \cdot x^{-\alpha/2} $ for $ x \to \infty $, where $ \alpha \in (0,2) $ is the stability index of the Lévy motion.
• The decay rate $ x^{-\alpha/2} $ is consistent with heuristics based on particle count and jump size scaling over time.
• The proof relies on a novel application of the Feynman-Kac formula to analyze a nonlinear integral equation governing the survival probability of the process.
• The authors establish sharp bounds on the Laplace transform of the first passage time to level $ x $, using the exponential distribution of jump times in the Lévy process.
• The result confirms that the tail behavior is determined by rare, large jumps in the Lévy motion, not by typical diffusive scaling.
• The asymptotic constant $ \sqrt{2/\alpha} $ is derived rigorously through a combination of path decomposition and asymptotic estimates on hitting probabilities.