# Characterization of Time-homogeneous flows for conditional expectation

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 $$F_s^t = F_{s+\Delta}^{t+\Delta} ;\qquad (\forall t>s\geq 0)?$$

• It seems that we need some assumptions even to ensure that $F_s^t$ is well defined, since $f_t$ is typically not unique. – Nate Eldredge Mar 20 at 2:26
• I also don't see why $f_{t+s}$ can necessarily be written as a composition of some function with $f_t$, since $f_t$ need not be 1-1. – Nate Eldredge Mar 20 at 2:27
• If $F_s^t$ is instead defined by precomposition it may be more natural then, and defined almost everywhere. – AIM_BLB Mar 20 at 8:47