Asymptotic Hölder regularity for the ellipsoid process

Abstract

We obtain an asymptotic Hölder estimate for functions satisfying a dynamic programming principle arising from a so-called ellipsoid process. By the ellipsoid process we mean a generalization of the random walk where the next step in the process is taken inside a given space dependent ellipsoid. This stochastic process is related to elliptic equations in non-divergence form with bounded and measurable coefficients, and the regularity estimate is stable as the step size of the process converges to zero. The proof, which requires certain control on the distortion and the measure of the ellipsoids but not continuity assumption, is based on the coupling method.

Hölder regularity for stochastic processes with bounded and measurable increments

“…Example 2.5 (Ellipsoid process). A particular case of the stochastic process considered in this paper is the so-called ellipsoid process (see [AP20]). This process arises when ν x is the uniform probability measure on E x \ B 1 , where E x denotes an ellipsoid centered at the origin such that…”

“…An asymptotic Hölder estimate was obtained in [AP20] under certain assumption on the ellipticity ratio of the ellipsoids. Now, Theorem 6.3 implies the Hölder estimate for u without any additional assumption and thus improves the result in [AP20].…”