Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\mathbb{R}^d\rightarrow \mathbb{R}$ such that $$ f_t(Y_t)=\mathbb{E}[\phi(X_t)|\sigma(Y_t)]. $$

From this, we can define the flow $F_s^t:\mathbb{R} \rightarrow \mathbb{R}$ such that $$ F_s^t\circ f_t (y) \triangleq f_{t+s}(y). $$

My question is, when is this flow time-homogeneous. That is, is there a characterization of the square-integrable processes $X_t,Y_t$, for which there exists $\Delta>0$ satisfying $$ F_s^t = F_{s+\Delta}^{t+\Delta} ;\qquad (\forall t>s\geq 0)? $$