Cauchy's formulas for random walks in bounded domains

Journal Article · Fri Aug 01 04:00:00 EDT 2014 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4891299· OSTI ID:22306095

Mazzolo, Alain; Zoia, Andrea ^[1]

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1. CEA/Saclay, DEN/DANS/DM2S/SERMA/LTSD, 91191 Gif-sur-Yvette (France)

Cauchy's formula was originally established for random straight paths crossing a body B⊂R{sup n} and basically relates the average chord length through B to the ratio between the volume and the surface of the body itself. The original statement was later extended in the context of transport theory so as to cover the stochastic paths of Pearson random walks with exponentially distributed flight lengths traversing a bounded domain. Some heuristic arguments suggest that Cauchy's formula may also hold true for Pearson random walks with arbitrarily distributed flight lengths. For such a broad class of stochastic processes, we rigorously derive a generalized Cauchy's formula for the average length traveled by the walkers in the body, and show that this quantity depends indeed only on the ratio between the volume and the surface, provided that some constraints are imposed on the entrance step of the walker in B. Similar results are also obtained for the average number of collisions performed by the walker in B.

• OSTI ID: 22306095
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 8 Vol. 55; ISSN JMAPAQ; ISSN 0022-2488
• Publisher: American Institute of Physics (AIP)
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CAUCHY PROBLEM
GRAPH THEORY
RANDOMNESS
SET THEORY
STOCHASTIC PROCESSES
TRANSPORT THEORY