Say $S$ is a subordinator (an almost surely increasing Levy Process). My understanding from sources like Subordination of a Levy process when the "subordinator" is not nondecreasing is that for any Levy process, such as the wiener process $W$, $W(S(t))$ is also a Levy process.
Now say $X$ is white noise. Is the stochastic process $X(S(t))$ also white noise? What about the process $X(X(t))$? Does this change if $X$ is non-stationary?