[Paper Review] Random walks with unbounded jumps among random conductances II: Conditional quenched CLT

Christophe Gallesco, Serguei Popov|arXiv (Cornell University)|Oct 1, 2012

Stochastic processes and statistical mechanics16 references3 citations

TL;DR

This paper establishes a quenched conditional invariance principle for one-dimensional random walks with unbounded jumps among i.i.d. random conductances, proving that under the condition of staying positive up to time $n$, the rescaled trajectory converges to a Brownian meander almost surely in the environment. The result extends uniform quenched invariance principles to conditioned paths and connects to the 3-dimensional Bessel process via hitting time and crossing time analysis.

ABSTRACT

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched conditional invariance principle for the random walk, under the condition that it remains positive until time $n$. As a corollary of this result, we study the effect of conditioning the random walk to exceed level $n$ before returning to 0 as $n o \infty$.

Motivation & Objective

• To establish a quenched conditional invariance principle for a one-dimensional random walk with unbounded jumps in a random conductance environment.
• To analyze the limiting behavior of the walk conditioned to remain positive up to time $n$ as $n \to \infty$.
• To extend the uniform quenched invariance principle from prior work to the case of path conditioning.
• To connect the quenched conditional law to the Brownian meander and the 3-dimensional Bessel process through hitting and crossing time analysis.

Proposed method

• Use of a quenched conditional probability measure $Q_{\omega}^n$ defined as $\mathbb{P}_\omega[\cdot \mid \Lambda_n]$, where $\Lambda_n = \{X_k > 0 \text{ for all } k=1,\dots,n\}$.
• Rescaling of the walk via $Z^n(t) = \sigma^{-1} n^{-1/2} X_{\lfloor nt \rfloor}$, with polygonal interpolation to define a continuous process on $[0,1]$.
• Application of the uniform quenched invariance principle from [15] as a key tool to control convergence speed and tightness.
• Use of the scaling property of Brownian motion and the transition density of the Brownian meander to analyze hitting and crossing times.
• Employment of the 3-dimensional Bessel process as the limiting process for the first passage time and maximum of the meander.
• Use of the functional limit theorem for the meander via convergence of finite-dimensional distributions and tightness arguments under the quenched measure.

Experimental results

Research questions

• RQ1 Does the rescaled trajectory of a one-dimensional random walk with unbounded jumps, among i.i.d. random conductances, converge to a Brownian meander under the quenched law conditioned to stay positive?
• RQ2 What is the limiting behavior of the quenched law of the walk when conditioned to exceed level $n$ before returning to 0 as $n \to \infty$?
• RQ3 How does the quenched invariance principle extend to conditional laws, particularly under positivity constraints?
• RQ4 Can the convergence to the Brownian meander be established with uniform estimates across environments satisfying ergodicity and moment conditions?

Key findings

• The quenched conditional law of the rescaled walk $Z^n$ converges weakly to the Brownian meander $W^+$ in the space $C[0,1]$ for almost every environment.
• The convergence holds under the condition that the walk remains positive up to time $n$, and the limiting process is the Brownian meander, which corresponds to a Brownian motion conditioned to stay positive.
• The limiting law of the hitting time of level $n$ before return to 0 is related to the 3-dimensional Bessel process, with the first passage time distribution matching that of the meander.
• The convergence is uniform in the sense that the speed of convergence is controlled uniformly over starting positions in a neighborhood of the origin, as established in the prior work [15].
• The probability of the walk remaining positive up to time $n$ decays like $n^{-1/2}$, consistent with the asymptotic behavior of the meander.
• The proof relies on tightness and convergence of finite-dimensional distributions, with the key step being the control of the quenched exit and hitting time probabilities using the 3-dimensional Bessel process.