[Paper Review] Stochastic Homogenization for Some Nonlinear Integro-Differential Equations

Russell W. Schwab|arXiv (Cornell University)|Jan 31, 2011

Advanced Mathematical Modeling in Engineering24 references4 citations

TL;DR

This paper establishes stochastic homogenization for a broad class of fully nonlinear integro-differential equations with stationary ergodic random coefficients, proving almost everywhere uniform convergence of solutions to a deterministic, translation-invariant effective equation. The key contribution is the identification of the effective equation and the rigorous justification of homogenization via a comparison principle with measurable ingredients, extending periodic homogenization results to the random setting.

ABSTRACT

In this note we extend to the random, stationary ergodic setting previous results of periodic homogenization for a particular family of nonlinear nonlocal "elliptic" equations with oscillatory coefficients. Such equations include, but are not limited to Bellman equations and the Isaacs equations for the control and differential games of some pure jump processes. The existence of an effective equation and convergence the solutions of the family of the original equations is obtained. Even in the linear case of the equations contained herein the results appear to be new.

Motivation & Objective

• To extend periodic homogenization results for nonlinear integro-differential equations to the general stationary ergodic setting.
• To establish the existence and identification of a deterministic, translation-invariant effective equation governing the limit of solutions.
• To prove almost everywhere in ω, uniform-in-x convergence of solutions u^ε to the solution of the effective equation.
• To validate the homogenization framework by proving a 'comparison with measurable ingredients' result, essential for nonlocal elliptic equations.
• To generalize prior results from periodic to random environments, particularly for equations arising in optimal control and stochastic differential games with jump processes.

Proposed method

• The analysis relies on viscosity solution theory for fully nonlinear nonlocal elliptic equations with random coefficients.
• A comparison principle with measurable ingredients is established as a key technical tool, ensuring convergence under minimal assumptions.
• The effective operator F̄ is identified via ergodicity and stationarity of the coefficient family K^αβ and f^αβ under a group of transformations τ_x.
• The proof uses a penalization method and regularity estimates for the obstacle problem to control oscillations in the solution family u^ε.
• Lemmas on translation invariance, monotonicity, and Hölder continuity of obstacle solutions are used to derive uniform bounds.
• The convergence is shown via a blow-up argument and the use of maximal functions to control the measure of 'bad' sets in the domain.

Experimental results

Research questions

• RQ1 Does the solution family u^ε of a fully nonlinear integro-differential equation with stationary ergodic coefficients converge uniformly to a deterministic, translation-invariant effective equation?
• RQ2 Can the comparison principle with measurable ingredients be established for nonlocal elliptic equations in the stochastic setting, as required for homogenization?
• RQ3 What is the structure of the effective equation F̄(ū, x) = 0, and how is it related to the original random operator F?
• RQ4 How does the convergence of u^ε to ū behave in terms of almost sure convergence and uniformity in x?
• RQ5 To what extent do the results generalize known periodic homogenization results to the random, ergodic case?

Key findings

• The effective equation F̄(ū, x) = 0 is deterministic, translation-invariant, and fully nonlinear, arising from the ergodic average of the original random operator.
• Solutions u^ε converge almost surely in ω and uniformly in x to the solution ū of the effective equation on compact subsets of D.
• The convergence rate is quantified via a function m(η)/ε, where m(η) → 0 as η → 0, indicating the measure of 'bad' sets shrinks with ε.
• The comparison principle with measurable ingredients is proven as a key technical result, enabling the homogenization argument.
• The obstacle problem for the nonlocal operator is shown to be uniformly Hölder continuous with exponent γ depending on n, λ, Λ, σ, and the domain A.
• The translation and monotonicity properties of obstacle solutions are rigorously established and used to propagate regularity and comparison across scales.