Why does the Malliavin derivative of a Markovian semigroup being strong Feller imply the semigroup strong Feller?

Question

I am reading "Ergodicity Results for the Stochastic Navier–Stokes Equations: An Introduction" by Arnaud Debussche

In it, he claims that for $\varphi \in B_b(H)$, a bounded measurable function on a Hilbert space $H$ if $DP_t\varphi$ is Lipschitz then $P_t$ is Strong Feller. ($D$ is the Malliavin derivative) Why is this true?

I think this shows that $DP_t$ is strong Feller - it maps bounded functions to continuous bounded functions. But why does that show that $P_t$ is strong Feller? Does $DP_t$ being strong Feller mean that $P_t$ is strong Feller?

Answer 1

Actually it's not that bad:

$$|DP_t\varphi(x)\cdot h|=|\lim_{\varepsilon \to 0}\frac{P_t\varphi(x+\varepsilon h)-P_t\varphi(x)}{\varepsilon}|\le Const$$

So

$$\lim_{\varepsilon \to 0}|P_t\varphi(x+\varepsilon h)-P_t\varphi(x)|=0$$