If we consider a one dimensional continuous-time random walk problem, starting at the origin. Assume the distribution is symmetric. We let a walk end the first time it returns back to 0 (this includes a step that goes past zero onto the other side). Then X, our variable of interest, is the furthest point from the origin reached on a given walk.
If we know the probability distribution of X, then given that we find ourselves at point Y, what is the new conditional probability distribution of X.
More explicitly, the probability distribution X|Y, where X is the furthest absolute value reached before going back to 0 and Y, the starting point, is any point, 0 or otherwise.
I would also be happy to know of a solution for an identical problem with an integer walk, if the former is not accessible.