Quantitative Stochastic Homogenization of Elliptic Equations in Nondivergence Form

Abstract

Abstract. We introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form. The main application is an algebraic error estimate, asserting that deviations from the homogenized limit are at most proportional to a power of the microscopic length scale, assuming a finite range of dependence. The results are new even for linear equations. The arguments rely on a new geometric quantity which is controlled in part by adapting elements of the regularity theory for the Monge-Ampère equation.

Optimal Quantitative Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form

“…There are then two main analytic ingredients we need to conclude: first, an estimate on the decay of the Green's function for the heterogenous operator (note that, in contrast to the divergence form case, there is no useful deterministic bound on the decay of the Green's function); and (ii) a higher-order regularity theory asserting that, with high P-probability, solutions of our random equation are more regular than the deterministic regularity theory would predict. We prove each of these estimates by using the suboptimal (but algebraic) quantitative homogenization result of [4]: we show that, since solutions are close to those of the homogenized equation on large scales, we may "borrow" the estimates from the constant-coefficient equation. This is an idea that was introduced in the context of stochastic homogenization for divergence form equations by the first author and Smart [5] (see also [17,3]) and goes back to work of Avellaneda and Lin [6,7] in the case of periodic coefficients.…”

“…Therefore, the estimate (1.4) is a quantitative homogenization result for this particular problem which asserts that the speed of homogenization is O(E(ε)). Moreover, it is well-known that estimating the speed of homogenization for the Dirichlet problem is essentially equivalent to obtaining estimates on the approximate correctors (see [15,7,11,4]). Indeed, the estimate (1.4) can be transferred without any loss of exponent to an estimate on the speed of homogenization of the Dirichlet problem.…”

“…Some of our main results rely on concentration inequalities (stated in Section 2.2) which require a stronger independence assumption than finite range of dependence (P2) which was used in [4]. Namely, we require that P is the pushforward of another probability measure which has a product space structure.…”

Stochastic Homogenization of Linear Elliptic Equations: Higher-Order Error Estimates in Weak Norms Via Second-Order Correctors

“…A more probabilistic viewpoint of stochastic homogenization of linear elliptic equations may be found e. g. in [21]. For the case of fully nonlinear elliptic equations, we refer to the works of Caffarelli and Souganidis [9] and Armstrong and Smart [3].…”

“…Note that further regularity properties of random elliptic operators have been explored in numerous previous works: Marahrens and the fourth author [20] have established a C 0,α regularity theory on large scales. Armstrong and Smart [3] developed a large-scale Lipschitz regularity theory; motivated by their work, Gloria, Neukamm, and the fourth author [15] have established a C 1,α theory. In a recent work of the third and the fourth author [12], higher-order homogenization correctors have been introduced for the first time in the setting of stochastic homogenization to develop a large-scale C k,α regularity theory; however, the focus being on the regularity result, the estimates on the higher-order correctors in [12] are non-optimal in the case of fast decorrelation.…”