In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area) A(T ), covered by the process in the time interval [0, T ]. The Laplace transform of A(T ) follows as a consequence. The main proof involves a double Laplace transform of A(T ) and is based on excursion theory and local time for Browni...

What is easier to notice is that, since the electron we are interested in is itself also delocalized (and not a point charge q), the total repulsion is a double integral, where we integrate over the space of the first electron the repulsion felt by the fractional charge at any point defined by the equation above that involves integrating over the space of the second electron. The result is just...