Let $X$ and $Y$ be two standard Brownian motions with mean $0$ and variance $1$, both started at zero. If we know that \begin{align} X_n &= Y_n, \end{align} for some $n>0$, can we say something about \begin{align} &X_t - Y_t, \text{ or } \tag{1}\\ &|X_t - Y_t|, \text{ or } \tag{2}\\ &\max |X_t - Y_t | . \tag{3} \end{align} when $0 \le t \le n$.
I tried to find the expected square difference $$ E\left( |X_t - Y_t|^2 \Big| X_n = Y_n \right), $$ without success. I tried to view the problem as a random walk, thought about excursion theory, still without success.
I need information on either (1), (2) or (3) in other to bound some quantity in another problem.