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

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.