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We prove quantitative estimates for the stochastic homogenization of linear uniformly elliptic equations in nondivergence form. Under strong independence assumptions on the coefficients, we obtain optimal estimates on the subquadratic growth of the correctors with stretched exponential-type bounds in probability. Like the theory of Gloria and Otto \cite{GO1,GO2} for divergence form equations, the arguments rely on nonlinear concentration inequalities combined with certain estimates on the Green's functions and derivative bounds on the correctors. We obtain these analytic estimates by developing a $C^{1,1}$ regularity theory down to microscopic scale, which is of independent interest and is inspired by the $C^{0,1}$ theory introduced in the divergence form case by the first author and Smart \cite{AS2}., Comment: 49 pages, revised version to appear in Arch Ration Mech Anal

We prove stochastic homogenization for a general class of coercive, nonconvex Hamilton–Jacobi equations in one space dimension. Some properties of the effective Hamiltonian arising in the nonconvex case are also discussed.

We prove explicit estimates for the error in random homogenization of degenerate, second-order Hamilton-Jacobi equations, assuming the coefficients satisfy a finite range of dependence. In particular, we obtain an algebraic rate of convergence with overwhelming probability under certain structural conditions on the Hamiltonian., Comment: 54 pages

We present comparison principles, Lipschitz estimates and study state constraints problems for degenerate, second-order Hamilton-Jacobi equations., 35 pages, minor revision

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-Amp\`ere equation., Comment: 40 pages. This version correctors some minor mistakes in the published version of the article, which are described in Section 1.5. Compared to v4, a typo has been fixed in (4.4) and (4.5)

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of $\mathbb{R}^n$, and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragm\'en-Lindel\"of result as well as a principle of positive singularities in certain Lipschitz domains., Comment: 41 pages, 2 figures