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

Let $P_s\phi (x)=\mathbb{E}\phi (X^x(s))$, 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 $$dX(s)=(AX(s)+a)ds+BdW(s),X(0)=x\in E\mathrm{.}$$ We give a simple probabilistic proof of the fact that null-controlla-bility of the corresponding deterministic system $$dY(s)=(AY(s)+B\mathcal{U}(t)x)(s))ds,Y(0)=x,$$ implies that for any $\phi \in B_b(E)$, $P_t\phi$ is infinitely many times Fréchet differentiable and that $$D^n{P}_t\phi (x)[y_1,\dots ,y_n]=\mathbb{E}\phi (X^x(t))(-1{)}^n{I}_t^n(y_1,\dots ,y_n),$$ where $I_t^n(y_1,\dots ,y_n)$ is the symmetric n-fold Itô integral of the controls $\mathcal{U}(t)y_1,\dots \mathcal{U}(t)y_n$.