Differentiability of transition semigroup of generalized Ornstein-Uhlenbeck process: a probabilistic approach

Abstract

Let $P_s\varphi (x)=\mathbb{E}\varphi (X^x(s))$, be the transition semigroup on the space $B_b(E)$ of bounded measurable functions on a Banach space $E$, of the Markov family defined by the linear equation with additive noise $$dX(s)=\left(AX(s)+a\right)ds+BdW(s),X(0)=x\in E.$$ We give a simple probabilistic proof of the fact that null-controlla\-bility of the corresponding deterministic system $$dY(s)=\left(AY(s)+B\mathcal{U}(t)x)(s)\right)ds,Y(0)=x,$$ implies that for any $\varphi \in B_b(E)$, $P_t\varphi$ is infinitely many times Fréchet differentiable and that $$D^n{P}_t\varphi (x)[y_1,\dots ,y_n]=\mathbb{E}\varphi (X^x(t))(-1{)}^n{I}_t^n(y_1,\dots ,y_n),$$ where $I_t^n(y_1,\dots ,y_n)$ is the symmetric n-fold Itô integral of the controls $\mathcal{U}(t)y_1,\dots \mathcal{U}(t)y_n$.