The following question is bothering me. I think it is probably known but I cannot find any reference...
Let $(X_t)$, $(Y_t)$, $(Z_t)$ denote 3 Feller processes with respective infinitesimal generators $A_X, A_Y, A_Z$.
Assume that $A_X = A_Y + A_Z$ (i.e. $X$ is a 'mix' of $Y$ and $Z$).
Moreover, at any time $t$, the law of $Y_t$ is stationary for the process $Z$.
How can it be proved that, if $X_0$ has the same law as $Y_0$, then $X_t$ and $Y_t$ have the same law at all time $t>0$ ?
Any argument/reference is welcome.
Thanks!
EDIT
@michael I don't think it is a relevant example. If I am not mistaken, Ornstein-Uhlenbeck processes have only one stationary measure.
Here, the basic framework is that the set $\mathcal{S}_Z$ of stationary distributions for $Z$ is not trivial and the law of $Y_t$ moves inside this set. Note that if the law of $Y_t$ does not change (i.e. $Y$ is stationary), then the result is immediate.
For example, you could take $Z$ to be a (totally asymmetric) exclusion process so that any Bernoulli product measure is stationary and take $Y$ a process that increase the probability of an occupied site as time goes by.
@Ilya The statement means that, at any time $t$, the law of $Y_t$ is a stationary distribution for the process $Z$.
@MJ73550 That would be too easy :-) Indeed, if $A_Y$ and $A_Z$ commute, then we can first run the dynamic of $Y$ up to time $t$ after that run $Z$ for another period of length $t$.
Here, we cannot assume such a strong property. At all time, both dynamics must be run concurrently yet I think that the result should still hold...
