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

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  • $\begingroup$ It seems that we need some assumptions even to ensure that $F_s^t$ is well defined, since $f_t$ is typically not unique. $\endgroup$ – Nate Eldredge Mar 20 at 2:26
  • $\begingroup$ 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. $\endgroup$ – Nate Eldredge Mar 20 at 2:27
  • $\begingroup$ If $F_s^t$ is instead defined by precomposition it may be more natural then, and defined almost everywhere. $\endgroup$ – AIM_BLB Mar 20 at 8:47

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