[Paper Review] Branching random walk with a random environment in time

Chunmao Huang, Quansheng Liu|arXiv (Cornell University)|Jul 29, 2014

Stochastic processes and statistical mechanics34 references10 citations

TL;DR

This paper studies a branching random walk on ℝ with a time-dependent random environment, establishing large deviation principles, central and local limit theorems for the empirical measure of particle positions, and a law of large numbers for the extreme positions (rightmost and leftmost particles). The key contribution is the almost sure convergence of normalized extreme positions to explicit deterministic limits under supercriticality and ergodicity assumptions.

ABSTRACT

We consider a branching random walk on $\mathbb{R}$ with a stationary and ergodic environment $ξ=(ξ_n)$ indexed by time $n\in\mathbb{N}$. Let $Z_n$ be the counting measure of particles of generation $n$. For the case where the corresponding branching process $\{Z_n(\mathbb{R})\}$ $ (n\in\mathbb{N})$ is supercritical, we establish large deviation principles, central limit theorems and a local limit theorem for the sequence of counting measures $\{Z_n\}$, and prove that the position $R_n$ (resp. $L_n$) of rightmost (resp. leftmost) particles of generation $n$ satisfies a law of large numbers.

Motivation & Objective

• To analyze the asymptotic behavior of the counting measure $ Z_n $ in a branching random walk with a time-homogeneous random environment.
• To establish a large deviation principle for the rescaled empirical measure $ Z_n(nullet) $ under quenched and annealed laws.
• To derive central and local limit theorems for the normalized empirical measure $ \mathbb{E}_\xi[Z_n(\bullet)/Z_n(\mathbb{R})] $ and its annealed counterpart.
• To prove a law of large numbers for the rightmost and leftmost particle positions $ R_n $ and $ L_n $, showing their normalized positions converge almost surely to explicit limits.
• To extend classical results from i.i.d. environments to the case of stationary, ergodic time-dependent environments.

Proposed method

• Apply the Gärtner-Ellis theorem to derive a large deviation principle for the quenched mean $ \mathbb{E}_\xi Z_n(n\bullet) $, using the convergence of the free energy $ \frac{1}{n}\log \tilde{Z}_n(t) $ to an explicit limit.
• Use the martingale convergence of $ Z_n(\mathbb{R})/P_n $ to $ W $, where $ P_n = \prod_{i=0}^{n-1} m_i $, and assume $ \mathbb{E}\log m_0 > 0 $ and $ \mathbb{E}\frac{N}{m_0}\log^+ N < \infty $ for non-degeneracy.
• Establish a quenched central limit theorem for $ \mathbb{E}_\xi[Z_n(-\infty, b_n x + a_n]/Z_n(\mathbb{R}) \mid Z_n(\mathbb{R})>0] \to \Phi(x) $ a.s., with $ a_n = \sum_{i=0}^{n-1} \mu_i $, $ b_n = (\sum_{i=0}^{n-1} \sigma_i^2)^{1/2} $.
• Prove the annealed central limit theorem by conditioning on survival and using the dominated convergence theorem to control the difference between quenched and annealed limits.
• Analyze the extreme positions $ R_n $ and $ L_n $ via the convergence of $ \frac{1}{n} \log \tilde{Z}_n(t) $ and the asymptotic behavior of the moment generating function.
• Use the fact that $ \mathbb{E}_\xi Z_n(n\bullet) $ satisfies a large deviation principle and that $ \tilde{Z}_n(t) $ grows exponentially with rate determined by the Lyapunov exponent of the environment.

Experimental results

Research questions

• RQ1 Does the empirical measure $ Z_n(n\bullet) $ satisfy a large deviation principle in a branching random walk with a random environment in time?
• RQ2 How do the normalized positions of the rightmost and leftmost particles behave asymptotically?
• RQ3 Do central limit theorems hold for the quenched and annealed distributions of particle positions, and what are their scaling limits?
• RQ4 What is the almost sure limit of $ R_n/n $ and $ L_n/n $, and how does it depend on the environment?
• RQ5 Can the classical results for branching random walks in i.i.d. environments be extended to the case of stationary, ergodic time-dependent environments?

Key findings

• The sequence $ \{Z_n(n\bullet)\} $ satisfies a large deviation principle under the quenched law, with the rate function derived via the Gärtner-Ellis theorem.
• The free energy $ \frac{1}{n}\log \tilde{Z}_n(t) $ converges almost surely to an explicit limit depending on the environment's moment generating function.
• The normalized rightmost and leftmost particle positions satisfy a law of large numbers: $ \frac{R_n}{n} \to \rho $ and $ \frac{L_n}{n} \to \ell $ a.s., where $ \rho $ and $ \ell $ are explicitly determined by the environment's moment generating function.
• The quenched and annealed distributions of particle positions satisfy central limit theorems: $ \mathbb{E}_\xi[Z_n(-\infty, b_n x + a_n]/Z_n(\mathbb{R}) \mid Z_n(\mathbb{R})>0] \to \Phi(x) $ a.s., and the annealed version converges in distribution to the standard normal CDF.
• The local limit theorem holds under appropriate non-lattice and moment conditions, ensuring the density of particle positions converges to a Gaussian density after centering and scaling.
• The convergence of the quenched mean $ \mathbb{E}_\xi Z_n(n\bullet) $ to a deterministic limit is essential for the large deviation result, and the condition $ \mathbb{E}\frac{N}{m_0}\log^+ N < \infty $ ensures non-degeneracy of the limit martingale $ W $.