Lévy random walks in a semi-bounded domain: a perturbative approach

Details Lévy random walks in a semi-bounded domain: a perturbative approach Lévy random walks in a semi-bounded domain: a perturbative approach

2011-10-10

arxiv.org/abs/1110.1955v1

Physics

Condensed Matter

Statistical Mechanics

19 pages, 2 figures

Scientific paper

We study the statistics of a one-dimensional L\'evy random walks of index 0< \alpha \leq 2 in a semi-bounded domain. We construct a solution of the associated fractional Fokker-Planck equation with non-local boundary conditions using a perturbative expansion in \epsilon = 2 - \alpha << 1. This perturbation theory around the Brownian motion (corresponding to \alpha = 2) follows a method proposed in [A. Zoia, A. Rosso, M. Kardar, Phys. Rev. E 76, 021116 (2007)], which is implemented here on a process which is continuous both in space and time. We apply this method to obtain an explicit analytical expression, exact at lowest non trivial order O(\epsilon), for two physically relevant quantities: (i) the distribution of the maximal displacement of a L\'evy random walk on a fixed time interval, and (ii) the probability distribution function of the position of a L\'evy random walk in the presence of an absorbing wall at the origin.

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