| Abstract: |
| Let $X$ be a Banach space-valued Ornstein-Uhlenbeck process driven by a possibly degenerate Brownian Motion and let $P_t$ stand for the transition operator of the process. We provide a simple probabilistic proof of the fact that null-controllability of the corresponding deterministic system implies infinite Fr\`echet differentiability of $P_tf$ for every bounded Borel function $f$. Moreover, using the Girsanov theorem we provide a pointwise formula for all the derivatives of $P_tf$ in terms of multiple stochastic integrals. Applications to the analysis of transition semigroups in finite and infinite dimensions are given. In particular we consider the heat equation with space-time white noise boundary conditions. |
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