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

Abstract

Let $Z$ be a $H$-valued Ornstein--Uhlenbeck process, $b:[0,1]\times H\to H$ and $h:[0,1]\to H$ be a bounded, Borel measurable functions with $\Vert b{\Vert }_{\infty }\le 1$ then $\mathbb{E}\exp \alpha {\left|\underset{0}{\overset{1}{\int }}b(t,Z_t+h(t))-b(t,Z_t)\mathrm{d}t\right|}_H^2\le C$ holds, where the constant $C$ is an absolute constant and $\alpha >0$ depends only on the eigenvalues of the drift term of $Z$ and $\Vert h{\Vert }_{\infty }$, 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.