Mean Value Sets and Kakutani-Feynman-Kac formulas for Differential Operators with Rough Coefficients

Abstract

Eigenfunctions of Schrodinger operators L := −divA∇ + V are observed to localise to a number of coupled subregions of the domain U ⊂ Rd when the coefficients A, potential V are irregular, or when the domain has a complicated geometry. These situations provide a mathematical model for the Anderson localisation phenomenon in physics, which is the observation that particles in disordered materials will diffuse very slowly or effectively not at all outside of specific subregions. Filoche, Mayboroda and collaborators have introduced the solution v to the elliptic boundary value problem Lv = 1 on U with v = 0 on ∂U to describe these localisation subregions. That is, an eigenfunction of L that corresponds to a small eigenvalue λ > 0 must decay at least as fast as an exponential function outside the superlevel set {x ∈ U : v(x) > 1/λ}. Semigroup theory is used to identify a diffusion process X associated to the operator L and domain U. If we suppose that the disordered material occupies the region U and has properties best described by A and V , then paths of X are used to model particle movement in the medium. For the first domain exit time τU := inf{t > 0 : Xt ̸∈ U}, the Kakutani-Feynman-Kac formula v(x) = Ex[τU] shows that v should be understood to describe domain exit time statistics for the process given that it starts at x ∈ U. This links eigenfunction localisation with localisation of the diffusion process. However, most proofs for such formulas appeal to the Ito theory which requires A, V and U to be taken sufficiently regular so that these approaches do not align with the assumption of irregularity underlying Anderson localisation. There is an alternative method to prove the probabilistic formula for Laplace’s equation with prescribed boundary data which instead appeals to geometry and measure theory. We observe that this argument is based on the fact that the spatial statistics of Brownian motion are described by Euclidean balls, the mean value sets of the Laplace operator. We present the work of Caffarelli, and Blank and Hao which guarantees the existence of mean value sets for divA(x)∇ assuming only that A is elliptic, symmetric, measurable and bounded. Using these ideas, we begin here the development of a general theory that connects mean value sets with the diffusion process, with the hope that this geometric approach is useful to the localisation theory. This document not only builds the idea that the mean value sets play the theoretical role of describing the space structure of the underlying diffusion, but also applies the sets to the problem of building a random walk to approximate paths of the process. We present the work of Etore and Lejay which provides a random walk scheme for the one dimensional process generated by d/dx (a(x) d/dx ) where a need not be regular. By combining their one dimensional approach with the mean value sets we suggest a novel geometric way to construct a random walk approximation to diffusions associated to divergence form operators in all dimensions d ≥ 1, and regardless of coefficient regularity. Show more