Kantorovich distance in the martingale CLT and quantitative homogenization of parabolic equations with random coefficients

2012/03/15 by Jean-Christophe Mourrat, Mourrat, Jean-Christophe
Mathematics · #35B27 #35K05 #60F05 #60G44 #60K37 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #math.AP #math.PR #msc:35B27 #msc:35K05 #msc:60F05 #msc:60G44 #msc:60K37

paper · pdf · doi:10.48550/arxiv.1203.3417

25 pages

arxiv created 2012/03/15 · arxiv updated 2012/03/16

Abstract

The article begins with a quantitative version of the martingale central limit theorem, in terms of the Kantorovich distance. This result is then used in the study of the homogenization of discrete parabolic equations with random i.i.d. coefficients. For smooth initial condition, the rescaled solution of such an equation, once averaged over the randomness, is shown to converge polynomially fast to the solution of the homogenized equation, with an explicit exponent depending only on the dimension. Polynomial rate of homogenization for the averaged heat kernel, with an explicit exponent, is then derived. Similar results for elliptic equations are also presented.

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