Quasi-sure convergence theorem in $p$-variation distance for stochastic differential equations

TL;DR

This paper proves that dyadic approximations of Brownian motion converge in $p$-variation distance quasi-surely, using capacity estimates and rough path analysis, advancing the understanding of stochastic differential equations.

Contribution

It establishes a quasi-sure convergence theorem in $p$-variation distance for Brownian motion approximations, linking capacity estimates with rough path theory.

Findings

Dyadic approximations converge in $p$-variation quasi-surely.

Capacity estimates are used to measure convergence outside slim sets.

Provides a framework for quasi-sure analysis of Wiener functionals.

Abstract

In this paper by calculating carefully the capacities (defined by high order Sobolev norms on the Wiener space) for some functions of Brownian motion, we show that the dyadic approximations of the sample paths of the Brownian motion converge in the $p$-variation distance to the Brownian motion except for a slim set (i.e. except for a zero subset with respect to the capacity on the Wiener space of any order). This presents a way for studying quasi-sure properties of Wiener functionals by means of the rough path analysis.