I have what should be a very simple questions for Brownian motion experts…
Let $[a,b]$ be a given time interval. Let $f(x)$ be the probability that a linear Brownian motion with initial value $x$ at time $t=0$ has a zero in the interval $[a,b]$. I want to argue that $f(x)$ is maximal for $x=0$. This seems intuitively clear but I cannot figure out a simple proof of this (other than writing the exact expression for $f$ which is a double integral and analyzing its variations via long calculations). I would be interested, in order of preference, by such a simple proof or by a reference where this lemma is proven.
Thanks!
Best Answer
For any other initial x, construct a coupling between BMs started at x and 0, where the processes move in opposite directions until they meet (if they do), then they stick together afterwards. Then the answer is apparent.