# Cauchy's formulas for random walks in bounded domains

## Abstract

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.

- Authors:

- CEA/Saclay, DEN/DANS/DM2S/SERMA/LTSD, 91191 Gif-sur-Yvette (France)
- CEA/Saclay, DEN/DANS/DM2S/SERMA/LTSD, 91191 Gif-sur-Yvette and CNRS - Université Paris-Sud, LPTMS, UMR8626, 91405 Orsay Cedex (France)

- Publication Date:

- OSTI Identifier:
- 22306095

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CAUCHY PROBLEM; GRAPH THEORY; RANDOMNESS; SET THEORY; STOCHASTIC PROCESSES; TRANSPORT THEORY

### Citation Formats

```
Mazzolo, Alain, E-mail: alain.mazzolo@cea.fr, Zoia, Andrea, E-mail: andrea.zoia@cea.fr, and Mulatier, Clélia de, E-mail: clelia.demulatier@cea.fr.
```*Cauchy's formulas for random walks in bounded domains*. United States: N. p., 2014.
Web. doi:10.1063/1.4891299.

```
Mazzolo, Alain, E-mail: alain.mazzolo@cea.fr, Zoia, Andrea, E-mail: andrea.zoia@cea.fr, & Mulatier, Clélia de, E-mail: clelia.demulatier@cea.fr.
```*Cauchy's formulas for random walks in bounded domains*. United States. doi:10.1063/1.4891299.

```
Mazzolo, Alain, E-mail: alain.mazzolo@cea.fr, Zoia, Andrea, E-mail: andrea.zoia@cea.fr, and Mulatier, Clélia de, E-mail: clelia.demulatier@cea.fr. Fri .
"Cauchy's formulas for random walks in bounded domains". United States.
doi:10.1063/1.4891299.
```

```
@article{osti_22306095,
```

title = {Cauchy's formulas for random walks in bounded domains},

author = {Mazzolo, Alain, E-mail: alain.mazzolo@cea.fr and Zoia, Andrea, E-mail: andrea.zoia@cea.fr and Mulatier, Clélia de, E-mail: clelia.demulatier@cea.fr},

abstractNote = {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.},

doi = {10.1063/1.4891299},

journal = {Journal of Mathematical Physics},

number = 8,

volume = 55,

place = {United States},

year = {Fri Aug 01 00:00:00 EDT 2014},

month = {Fri Aug 01 00:00:00 EDT 2014}

}