Isoperimetry and Rough Path Regularity

TL;DR

This paper investigates the regularity of stochastic process paths using generalized variation norms, extending classical results to Gaussian rough paths and solutions of rough differential equations, with implications for path regularity and integrability.

Contribution

It extends variation norm results to Gaussian rough paths and solutions of rough differential equations, providing new regularity and integrability insights without Gaussian or Markovian assumptions.

Findings

Established Gaussian integrability of ψ-variation for rough paths.

Extended variation regularity results to processes in abstract metric spaces.

Identified ψ-variation as a robust property for solutions to rough differential equations.

Abstract

Optimal sample path properties of stochastic processes often involve generalized H\"{o}lder- or variation norms. Following a classical result of Taylor, the exact variation of Brownian motion is measured in terms of $\psi (x) \equiv $ $x^{2}/\log \log (1/x) $ near $0+$. Such $\psi $-variation results extend to classes of processes with values in abstract metric spaces. (No Gaussian or Markovian properties are assumed.) To establish integrability properties of the $\psi $-variation we turn to a large class of Gaussian rough paths (e.g. Brownian motion and L\'{e}vy's area viewed as a process in a Lie group) and prove Gaussian integrability properties using Borell's inequality on abstract Wiener spaces. The interest in such results is that they are compatible with rough path theory and yield certain sharp regularity and integrability properties (for iterated Stratonovich integrals, for example) which would be difficult to obtain otherwise. At last, $\psi $-variation is identified as robust regularity property of solutions to (random) rough differential equations beyond semimartingales.