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

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 $\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$ 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,wheretheconstant$C$isanabsoluteconstantand$\alpha>0$ depends only on the eigenvalues of the drift term of $Z$ and $\mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}$, 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.