Quantitative Homogenization for Log-Normal Coefficients via Malliavin Calculus: The One-Dimensional Case

Abstract

The quantitative analysis of stochastic homogenization problems has been a very active field in the last 15 years. Whereas the first results were motivated by applied questions (namely, the numerical approximation of homogenized coefficients), the more recent achievements in the field are much more analytically driven and focus on the subtle interplay between partial differential equation analysis (in particular, elliptic regularity theory) and probability (concentration, stochastic cancellations, and scaling limits). The aim of this article is threefold. First, we provide a complete and self-contained analysis for the popular example of log-normal coefficients with possibly fat tails in dimension $d=1$ , establishing new results on the accuracy of the two-scale expansion and characterizing fluctuations (in the perspective of uncertainty quantification). Second, we work in a context where explicit formulas allow us to bypass analytical difficulties and therefore mostly focus on the probabilistic side of the theory. Last, the one-dimensional setting gives intuition on the available results in higher dimensions (provided the results are correctly reformulated), to which we give precise entries to the recent literature.