Fluctuations in multiplicative systems with jumps

Abstract

Fluctuation properties of the Langevin equation including a multiplicative, power-law noise and a quadratic potential are discussed. The noise has the Levy stable distribution. If this distribution is truncated, the covariance can be derived in the limit of large time; it falls exponentially. Covariance in the stable case, studied for the Cauchy distribution, exhibits a weakly stretched exponential shape and can be approximated by the simple exponential. The dependence of that function on system parameters is determined. Then we consider a dynamics which involves the above process and obey the generalised Langevin equation, the same as for Gaussian case. The resulting distributions possess power-law tails - that fall similarly to those for the driving noise - whereas central parts can assume the Gaussian shape. Moreover, a process with the covariance 1/t at large time is constructed and the corresponding dynamical equation solved. Diffusion properties of systems for both covariances are discussed.

12 pages, 7 figures

References in corpus (5)

• Verhulst model with Levy white noise excitation
• Generic Multifractality in Exponentials of Long Memory Processes
• Analytical results for long time behavior in anomalous diffusion
• Anomalous diffusion in systems driven by the stable Levy noise with a finite noise relaxation time and inertia
• Stretched-Gaussian asymptotics of the truncated Lévy flights for the diffusion in nonhomogeneous media

Cited by in corpus (6)

• Anomalous diffusion: A basic mechanism for the evolution of inhomogeneous systems
• Langevin equation with fluctuating diffusivity: a two-state model
• Langevin dynamics for Lévy walk with memory
• The Fluctuation-Dissipation Relations: Growth, Diffusion, and Beyond
• Bistable generalised Langevin dynamics driven by correlated noise possessing a long jump distribution: barrier crossing and stochastic resonance
• Dynamical behavior of a nonlocal Fokker-Planck equation for a stochastic system with tempered stable noise