[Paper Review] On The Existence of an Invariant Measure for Isotropic Diffusions in Random Environment

Benjamin Fehrman|arXiv (Cornell University)|Apr 21, 2014

Advanced Mathematical Modeling in Engineering2 references4 citations

TL;DR

This paper establishes the existence of a unique invariant measure for isotropic diffusions in random environments on $\mathbb{R}^d$ ($d \geq 3$) that are small perturbations of Brownian motion, under stationarity, finite-range dependence, and restricted isotropy. The key result is the convergence of the process law to a unique invariant measure $\pi$ absolutely continuous with respect to the underlying probability measure $\mathbb{P}$, enabling a general homogenization result for locally measurable initial data.

ABSTRACT

The results of this paper build upon those first obtained by Sznitman and Zeitouni in [11]. We establish, for spacial dimensions greater than two, the existence of a unique invariant measure for isotropic diffusions in random environment which are small perturbations of Brownian motion. Furthermore, we establish a general homogenization result for initial data which are locally measurable with respect to the coefficients.

Motivation & Objective

• To establish the existence of a unique invariant measure for isotropic diffusions in random environments on $\mathbb{R}^d$ for $d \geq 3$.
• To extend the results of Sznitman and Zeitouni [11] to continuous-time diffusions with small perturbations from Brownian motion.
• To prove a general homogenization result for initial data that are locally measurable with respect to the random coefficients.
• To characterize the invariant measure $\pi$ as the limit of solutions to the backward Kolmogorov equation along an exponentially increasing time scale.

Proposed method

• Analyzes the long-time behavior of solutions $u(x,t,\omega)$ to the backward Kolmogorov equation with initial data $f_E(x,\omega) = \mathbf{1}_E(\tau_x\omega)$, where $E \in \mathcal{F}$.
• Identifies the invariant measure $\pi(E)$ as the limit $\lim_{n \to \infty} \mathbb{E}[u_E(0, L_n^2, \omega)]$ along a sequence $L_n^2$ of exponentially increasing time scales.
• Uses coupling techniques from [11] to show that, with high probability, the diffusion process couples to a deterministic Brownian motion at large space and time scales.
• Applies the ergodicity of the environment group $\{\tau_x\}_{x \in \mathbb{R}^d}$ to prove mutual absolute continuity of $\pi$ and $\mathbb{P}$, and that $\pi$ is an ergodic measure for the canonical Markov process on $\Omega$.
• Establishes almost sure convergence $\lim_{t \to \infty} u_E(0,t,\omega) = \pi(E)$ for a class of sets $E$ whose indicator functions satisfy a finite-range dependence condition.
• Derives homogenization results for equations with oscillating right-hand sides by relating solutions to the invariant measure via time integration of the backward equation.

Experimental results

Research questions

• RQ1 Does a unique invariant measure exist for isotropic diffusions in random environments on $\mathbb{R}^d$ for $d \geq 3$ under stationarity and finite-range dependence?
• RQ2 Can the invariant measure $\pi$ be characterized as the limit of the process law along a sequence of increasing time scales?
• RQ3 Is the invariant measure $\pi$ absolutely continuous with respect to the underlying probability measure $\mathbb{P}$, and mutually absolutely continuous if the environment group is ergodic?
• RQ4 Does the convergence of the process law to $\pi$ hold almost surely for a class of initial data satisfying local measurability?
• RQ5 Can the homogenization of parabolic and elliptic equations with oscillating coefficients be established using the invariant measure $\pi$?

Key findings

• A unique invariant measure $\pi$ exists on $(\Omega, \mathcal{F})$ that is absolutely continuous with respect to $\mathbb{P}$, satisfying $\pi(E) = \int_\Omega P_t(\omega, E) \, d\pi$ for all $t \geq 0$ and $E \in \mathcal{F}$.
• If the environment group $\{\tau_x\}_{x \in \mathbb{R}^d}$ is ergodic, then $\pi$ is mutually absolutely continuous with respect to $\mathbb{P}$ and defines an ergodic probability measure on $\Omega$.
• The invariant measure $\pi(E)$ is identified as the limit $\lim_{n \to \infty} \mathbb{E}[u_E(0, L_n^2, \omega)]$ along an exponentially increasing time scale $L_n^2$, with convergence established in Propositions 3.10–3.12.
• For a class of sets $E$ satisfying a finite-range dependence condition, almost sure convergence $\lim_{t \to \infty} u_E(0,t,\omega) = \pi(E)$ holds on a full-probability subset depending on $E$, as shown in Proposition 4.3.
• Homogenization of the parabolic equation with oscillating right-hand side $f(x/\epsilon, \omega)$ holds locally uniformly in space and time, with $u^\epsilon(x,t,\omega) \to t \overline{\pi}(f)$ as $\epsilon \to 0$, where $\overline{\pi}(f) = \int_\Omega f(0,\omega) \, d\pi$, as per Theorem 5.2.
• Homogenization of the elliptic equation with oscillating right-hand side $f(x/\epsilon, \omega)$ holds locally uniformly in space, with $u^\epsilon(x,\omega) \to \overline{\pi}(f)$ as $\epsilon \to 0$, where $\overline{\pi}(f) = \int_\Omega f(0,\omega) \, d\pi$, as per Theorem 5.3.