TITLE: Distribuation Max - Min of Brownian motion
QUESTION [5 upvotes]: I'm looking for the distribuation of $M_X(t) - m_X(t)$ of the brownian motion and not the joint distribuation.
where $m_X(t) = \min\limits_{0\leq s\leq t}X(s)$ and  $M_X(t) = \max\limits_{0\leq s\leq t}X(s)$.
Any help would be greatly appreciated.

REPLY [4 votes]: W. Feller [The asymptotic distribution of the range of sums of independent
random variables. Ann. Math. Statistics, 22 :427–432, 1951] provides this density as the sum of a series. 
The range $R_t$ at time $t$ is distributed like $\sqrt{t}R_1$ and the density of $R_1$ is the function $f$ defined on $(0,+\infty)$ by
$$
f(x)=8\sum_{n=1}^\infty(-1)^{n+1}n^2\varphi(nx),
$$
where $\varphi$ denotes the standard normal PDF. As noted by Feller: "In this form it is not even obvious that the function is positive".