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Cauchy's formulas for random walks in bounded domains

### Abstract

Cauchy's formula was originally established for random straight paths crossing a body 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 travelled 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.

© 2014 AIP Publishing LLC

Received 03 March 2014
Accepted 14 July 2014
Published online 01 August 2014

Article outline:

I. INTRODUCTION
II. A CAUCHY'S FORMULA FOR THE AVERAGE NUMBER OF COLLISIONS
III. AVERAGE TRAVELLED LENGTH
A. The term *P* _{0}
B. Proof that *p* _{ in } = *p* _{ out }
C. The term
D. The term
IV. CONCLUSIONS