The probability of a Brownian particle traveling a distance $L$ before returning to its point-of-origin What's the probability that a Brownian particle diffusing along a one-dimensional interval returns to its point of origin before traveling a distance $L$?
We know that in the limit of a random walk on a discrete lattice of points, a random walker starting at some position $(X = 1)$ has a probability of reaching $L$ before reaching $0$ of $p(L)=\frac{1}{L}$ [Feller, W. An Introduction to Probability Theory and Its Applications (chapter 5). John Wiley and Sons (1958)].  What happens in the limit of a continuous Brownian motion process?
 A: You seem to be asking three different questions. 


*

*A Brownian motion returns to its starting position instantly, so the 
answer to the question in the title is "zero". 

*The question is reversed in the first paragraph of post, the answer now is "one".

*Finally, if you start a Brownian motion at position 1, the chance that it will 
hit $L>1$ before hitting zero is $1/L$, just like for the random walk.  
A: Fix $n\in\mathbb N$ and consider the discrete walk given by the Brownian particle's passages at positions $kL/n$ for $-n\le k\le n$, where once the particle has reached a position $kL/n$ the next passage is recorded when it reaches either $(k-1)L/n$ or $(k+1)L/n$, but not when it reaches $kL/n$ again. This is a simple discrete one-dimensional random walk. After the first step away from the origin, the particle has probability $1/n$ of reaching $\pm L$ before returning to the origin in the discrete walk, and this is an upper bound for the probability of the Brownian motion reaching $\pm L$ before returning to the origin. Since we can choose $n$ freely, it follows that the latter probability is $0$.
