[Paper Review] A new proof of Liggett's theorem for non-interacting Brownian motions

Xinxin Chen, Christophe Garban+1 more|arXiv (Cornell University)|Dec 7, 2020

Stochastic processes and financial applications3 references4 citations

TL;DR

This paper presents a new, direct proof of Liggett's theorem for non-interacting drifted Brownian motions on the real line, bypassing the classical Choquet-Deny convolution equation. By analyzing the time-evolution of point processes and using probabilistic estimates on particle displacements, the authors identify the invariant measures as Cox processes with intensity $ Z_{ au}e^{-2\lambda x} + Y_{\tau} $, confirming the Poissonian structure without relying on harmonic analysis tools.

ABSTRACT

In this note, we give a new proof of Liggett's theorem on the invariant measures of independent particle systems from [Lig78] in the particular case of independent drifted Brownian motions. This particular case has received a lot of attention recently due to its applications for the analysis of the local extrema of discrete Gaussian free field. The novelty of our proof is that it identifies directly the expected Poisson Point Process with exponential intensity without relying on the Choquet-Deny convolution equation $μ* P=μ$ ([ChoquetDeny60,Deny60]).

Motivation & Objective

• To provide a new, direct proof of Liggett’s theorem for non-interacting drifted Brownian motions on $\mathbb{R}$, avoiding reliance on the Choquet-Deny convolution equation.
• To characterize the invariant point processes under the dynamics of independent drifted Brownian motions in continuous time.
• To establish that the invariant measures are distributed as Cox processes with intensity $ Z_{\infty}e^{-2\lambda x} + Y_{\infty} $, where $ Z_{\infty}, Y_{\infty} \geq 0 $ are random variables.
• To develop a method applicable to more general Markov kernels beyond Brownian motion, using pathwise probabilistic estimates.
• To lay a foundation for extending such fixed-point characterizations to more complex systems, such as branching Brownian motion, where Liggett’s original proof fails.

Proposed method

• The proof avoids the Choquet-Deny equation by directly analyzing the time-evolution of the point process $ \theta_t $ under independent drifted Brownian motions.
• It uses a decomposition of the expected number of particles in a compact set $ [-K_f, K_f] $ into three regions: left, central, and right of the drift center.
• For each region, the method bounds the expected number of particles that land in $ [-K_f, K_f] $ after time $ t $, using transition density estimates of Brownian motion with drift.
• Key estimates compare the transition probabilities at times $ t $ and $ s = t \pm t^{1/2} $, showing exponential separation in likelihood for particles far from the drift center.
• The proof applies concentration bounds and stochastic domination via the local finiteness of $ \theta $, showing that the expected number of particles in distant regions converges to zero in probability.
• The argument relies on comparing $ \mathbb{P}(|x + B_s - \lambda s| \leq K_f) $ and $ \mathbb{P}(|x + B_t - \lambda t| \leq K_f) $ for $ |x| \geq \lambda t \pm t^{2/3} $, establishing a lower bound on the ratio that grows as $ \exp(\lambda t^{1/6}/4) $.

Experimental results

Research questions

• RQ1 Can the invariant measures of non-interacting drifted Brownian motions be characterized without invoking the Choquet-Deny convolution equation?
• RQ2 What is the precise structure of the invariant point processes under continuous-time independent Brownian motion with drift?
• RQ3 How can one identify the intensity measure of the invariant process directly through pathwise probabilistic analysis?
• RQ4 Can this method be extended to more general Markov kernels on $ \mathbb{R}^d $, especially those arising in stochastic processes with long-range dependence?
• RQ5 Why does Liggett’s original proof fail in settings like branching Brownian motion, and can this new approach overcome that limitation?

Key findings

• The invariant measures for independent drifted Brownian motions on $ \mathbb{R} $ are exactly the Cox processes with random intensity measure $ Z_{\infty}e^{-2\lambda x} + Y_{\infty} $, where $ Z_{\infty}, Y_{\infty} \geq 0 $ are non-negative random variables.
• The proof establishes this result without using the Choquet-Deny equation, instead relying on direct probabilistic estimates on particle displacements and convergence in probability.
• The expected number of particles in any compact set $ [-K_f, K_f] $ that originate from positions far from the drift center converges to zero in probability as time $ t \to \infty $.
• For particles located at $ x $ with $ |x| \geq \lambda t + t^{2/3} $, the probability of hitting $ [-K_f, K_f] $ after time $ t $ decays exponentially when compared to a slightly earlier or later time $ s = t \pm t^{1/2} $, with a ratio bounded below by $ \frac{1}{2}e^{\lambda t^{1/6}/4} $.
• The method shows that the contribution from the left and right tails of the initial point process becomes negligible over time, implying that the invariant measure is determined by the central region and its asymptotic behavior.
• The approach is robust enough to be extendable to other Markov kernels beyond Brownian motion, as demonstrated by the authors’ intention to apply it to branching Brownian motion.