I have the following homework question.
If $(W_t)_{t\geq 0}$ is a standard Wiener process, then $X_t=W_{1-t}$ is Markov for $t\in[0,1]$. Find its transition probabilities and the initial distribution.
My attempt at a solution: For $0\le s < t \le 1$ and $U\subset \mathbb{R}$,
$$ P_{s,t}(X_s, U) = P(X_t\in U | X_s) = P(W_{1-t}\in U | W_{1-s}). $$
I was going to try to add and subtract something to use the independence of increments (I think $X_t$ still has independent increments?) but I feel like I should be able to say something about this probability because $1-t<1-s$.
Also I wasn't able to add and subtract something that would be independent of $W_{1-s}$. I got $$ P(-(W_{1-s}-W_{1-t})\in U-W_{1-s} | W_{1-s}).$$