Asymptotic Hölder regularity for the ellipsoid process

¹ Department of Mathematics, University of Genoa, Via Dodecaneso 35, 16146 Genova, Italy.
² Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35, 40014 Jyväskylä, Finland.

^* Corresponding author: arroyo@dima.unige.it

Received: 15 October 2019
Accepted: 20 May 2020

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.