On short-time asymptotics of one-dimensional Harris flows

TL;DR

This paper investigates the short-time behavior of one-dimensional Harris flows, revealing how their stochastic properties relate to associated Gaussian processes and establishing bounds on their deviations using advanced probabilistic techniques.

Contribution

It provides new asymptotic estimates for Harris flows based on the regularity of the associated Gaussian process, employing Gaussian measure concentration and Slepian's inequality.

Findings

Coupling of flows with Gaussian processes when the latter is continuous

Uniform $o( ext{ln ln } t^{-1})$ difference in the continuous case

$O( ext{ln ln } t^{-1})$ estimate when the Gaussian process lacks continuity

Abstract

We study the short-time asymptotical behavior of stochastic flows on \mathbb{R} in the \sup-norm. The results are stated in terms of a Gaussian process associated with the covariation of the flow. In case the Gaussian process has a continuous version the two processes can be coupled in such a way that the difference is uniformly $o(\ln\ln t^{-1})$. In case it has no continuous version, an $O(\ln\ln t^{-1})$ estimate is obtained under mild regularity assumptions. The main tools are Gaussian measure concentration and a martingale version of the Slepian comparison principle.