Minimum of two Wiener process

Question

I came to this question that if we have $Z(t)=\min\{ X(t),Y(t) \}$, where $X(t)$ and $Y(t)$ are two independent Wiener processes, can we say that $Z(t)$ is also a Wiener process?

What if $X(t)=\min\{ X(t-a)+t,Y(t-a)+d\}$?

Thanks!

Answer 1

For starters, $Z(t)$ is not even a Gaussian process, so the answer is a resounding no.