On the area under a continuous time Brownian motion till its first-passage time

The area swept out under a one-dimensional Brownian motion till its first-passage time is analysed using a Fokker-Planck technique. We obtain an exact expression for the area distribution for the zero drift case, and provide various asymptotic results for the non-zero drift case, emphasising the critical nature of the behaviour in the limit of vanishing drift. The results offer important insights into the asymptotic behaviour of a number of discrete models. We also provide a succinct derivation for the distribution of the maximum displacement observed during a first-passage. PACS: 02.50.-r, 05.10.Gg, 05.40.Jc Short title: Area under a Brownian motion Email: [email protected]