Langevin equation with super-heavy-tailed noise

Abstract

We extend the Langevin approach to a class of driving noises whose generating processes have independent increments with super-heavy-tailed distributions. The time-dependent generalized Fokker-Planck equation that corresponds to the first-order Langevin equation driven by such a noise is derived and solved exactly. This noise generates two probabilistic states of the system, survived and absorbed, that are equivalent to those for a classical particle in an absorbing medium. The connection between the rate of absorption and the super-heavy-tailed distribution of the increments is established analytically. A numerical scheme for the simulation of the Langevin equation with super-heavy-tailed noise is developed and used to verify our theoretical results.

12 pages, 3 figures

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• Continuous-time random walk theory of superslow diffusion
• Continuous-time random walk with a superheavy-tailed distribution of waiting times
• Asymptotic solutions of decoupled continuous-time random walks with superheavy-tailed waiting time and heavy-tailed jump length distributions
• Limiting distributions of continuous-time random walks with superheavy-tailed waiting times
• Statistics of bounded processes driven by Poisson white noise
• Probability distribution function for systems driven by superheavy-tailed noise