On short-time asymptotics of one-dimensional Harris flows

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.