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

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

2005-01-19

arxiv.org/abs/cond-mat/0501445v1

J. Phys. A: Math. Gen. 38, 4097 (2005).

Physics

Condensed Matter

Statistical Mechanics

15 pages, two figures (included)

Scientific paper

10.1088/0305-4470/38/19/004

The area swept out under a one-dimensional Brownian motion till its first-passage time is analysed using a backward Fokker-Planck technique. We obtain an exact expression of 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 the area-perimeter generating functions in a class of discrete polygons. We also provide a succinct derivation for the distribution of the maximum displacement observed till the first-passage time.

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