An exponential estimate for Hilbert space-valued Ornstein--Uhlenbeck processes

TL;DR

This paper derives an exponential estimate for Hilbert space-valued Ornstein-Uhlenbeck processes, providing bounds, concentration results, and moment estimates for integrals involving these processes with bounded measurable functions.

Contribution

It introduces a novel exponential estimate for integrals of bounded functions against Hilbert space-valued Ornstein-Uhlenbeck processes, with explicit dependence on eigenvalues and function norms.

Findings

Established an exponential integrability bound for the process

Proved a concentration of measure result

Provided moment estimates for the integral

Abstract

Let $Z$ be a $H$-valued Ornstein--Uhlenbeck process, $b\colon[0,1]\times H \rightarrow H$ and $h\colon[0,1] \rightarrow H$ be a bounded, Borel measurable functions with $\|b\|_\infty \leq 1$ then $\mathbb E \exp \alpha \left| \int\limits_0^1 b(t, Z_t + h(t)) - b(t, Z_t) \, \mathrm d t \right|_H^2 \leq C$ holds, where the constant $C$ is an absolute constant and $\alpha>0$ depends only on the eigenvalues of the drift term of $Z$ and $\|h\|_\infty$, the norm of $h$, in an explicit way. Using this we furthermore prove a concentration of measure result and estimate the moments of the above integral.