Exact and Explicit Probability Densities for One-Sided Lévy Stable Distributions

K. A. Penson, K. Górska
2010 Physical Review Letters  
We study functions g_α(x) which are one-sided, heavy-tailed Levy stable probability distributions of index α, 0< α <1, of fundamental importance in random systems, for anomalous diffusion and fractional kinetics. We furnish exact and explicit expression for g_α(x), 0 ≤ x < ∞, satisfying ∫_0^∞ exp(-p x) g_α(x) dx = exp(-p^α), p>0, for all α = l/k < 1, with k and l positive integers. We reproduce all the known results given by k≤ 4 and present many new exact solutions for k > 4, all expressed in
more » ... erms of known functions. This will allow a 'fine-tuning' of α in order to adapt g_α(x) to a given experimental situation.
doi:10.1103/physrevlett.105.210604 pmid:21231282 fatcat:nu65new7dfawdc5cawhrh35nma