[Paper Review] From the Sinai's walk to the Brox diffusion using bilinear forms

Carlos G. Pacheco|arXiv (Cornell University)|May 10, 2016

Stochastic processes and statistical mechanics15 references3 citations

TL;DR

This paper establishes a rigorous connection between Sinai's random walk in a random environment and the Brox diffusion using Dirichlet forms and generator convergence. By scaling the environment and showing the generator of the Sinai walk converges to that of the Brox diffusion almost surely, it demonstrates that both processes arise from the same microscopic model under different local conditions, with the Brox diffusion emerging as the limit when the environment is rescaled appropriately.

ABSTRACT

Using the generators, we establish a connection between the Sinai's random walk and the so-called Brox process. We first find the Dirichlet form of the Brox diffusion, and then prove that it is the limit of the Dirichlet form of the Sinai's random walk. This also gives a natural way to connect between the Brox diffusion and the Brownian motion.

Motivation & Objective

• To establish a direct link between Sinai's random walk and the Brox diffusion using stochastic generators and Dirichlet forms.
• To show that the generator of the Sinai walk converges to the generator of the Brox diffusion under a specific scaling of the environment.
• To demonstrate that the Brox diffusion and Brownian motion emerge from the same microscopic system with different local specifications.
• To provide a natural framework for understanding the convergence of discrete random walks to diffusive processes in random media.

Proposed method

• Use of bilinear forms and Dirichlet forms to compare the infinitesimal generators of the Sinai walk and Brox diffusion.
• Rescaling of the environment via $ p_n(x) = \frac{1}{2} + \frac{q(x)}{n^{1/4}} $, where $ q(x) $ is a Bernoulli random variable.
• Application of the mean value theorem for integrals to express the generator of the Sinai walk in terms of discrete differences.
• Use of stochastic integral theory to analyze the limit of the drift term involving $ q_n(x) $, which scales as $ \sqrt{\Delta_n} $.
• Proof of $ L^2 $-convergence of the generator $ L^{(n)} $ to the limit generator $ \frac{1}{2}f'' - \frac{1}{2}f' dW $, followed by almost sure convergence along a subsequence.
• Use of uniform convergence of monotone functions on compact sets (Theorem 11) to support convergence arguments in the proof.

Experimental results

Research questions

• RQ1 How can the Brox diffusion be rigorously connected to Sinai's random walk through generator convergence?
• RQ2 What scaling of the random environment leads to the Brox diffusion as the limit of Sinai walks?
• RQ3 In what sense do the Brox diffusion and Brownian motion arise from the same underlying microscopic model?
• RQ4 Can the Dirichlet form of the Brox diffusion be explicitly derived and shown to be the limit of the Dirichlet forms of the Sinai walks?

Key findings

• The generator of the Sinai walk converges in $ L^2 $ to $ \frac{1}{2}f'' - \frac{1}{2}f' dW $, which is the generator of the Brox diffusion.
• Almost sure convergence of the generator holds along a subsequence for almost every trajectory of the environment $ W $.
• The Dirichlet form of the Brox diffusion is identified as the limit of the Dirichlet forms of the Sinai walks under the specified scaling.
• When the variance of the environment increments scales as $ \Delta^{1/2} $, the limit generator corresponds to the Brox diffusion; if it scales slower, the limit is Brownian motion.
• The convergence result shows that both Brownian motion and Brox diffusion are limits of the same class of discrete processes under different local specifications of the environment.
• The method provides a unified framework to understand the emergence of diffusive behavior in random media via generator convergence.