Suppose $f: \mathcal{P}(X) \to \mathcal{P}(X)$ is a function that satisfies, for every set $A,B \subseteq X$ $(C_1): f(\emptyset) = \emptyset$ $(C_2): A \subseteq f(A)$ $(C_3): f(A \cup B) =f(A) \cup f(B)$ $(C_4): f(f(A)) = f(A)$ Prove that there exists a unique t...

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 Malliavi...

Let $\widetilde{B}$ be a Brownian Motion under the measure $\mathbb{P}$. Let $\theta$ be a stochastic process fulfilling the Novikov's condition and $Z_\theta$ the relative Radon–Nikodym derivative for which it holds $\mathbb{Q}(d\omega)=\mathbb{P}(d\omega)Z_\theta$. Then by Girsanov, $B_t = \wi...