# Asymptotic H\"older Regularity for the Ellipsoid Process

## TL;DR

This paper establishes an asymptotic H"older regularity estimate for functions governed by a dynamic programming principle linked to an ellipsoid process, a generalization of random walks within space-dependent ellipsoids, related to elliptic PDEs.

## Contribution

It introduces a new regularity estimate for solutions of a dynamic programming principle associated with ellipsoid processes, extending understanding of elliptic equations with measurable coefficients.

## Key findings

- Proves asymptotic H"older regularity for the ellipsoid process
- Establishes stability of the regularity estimate as step size approaches zero
- Uses coupling method without requiring continuity of coefficients

## Abstract

We obtain an asymptotic H\"older 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.