By using the backward fractional Fokker-Planck equation we investigate the barrier crossing event in the presence of Levy noise. After shortly review recent results obtained with different approaches on the time characteristics of the barrier crossin
g, we derive a general differential equation useful to calculate the nonlinear relaxation time. We obtain analytically the nonlinear relaxation time for free Levy flights and a closed expression in quadrature of the same characteristics for cubic potential.
The functional method to derive the fractional Fokker-Planck equation for probability distribution from the Langevin equation with Levy stable noise is proposed. For the Cauchy stable noise we obtain the exact stationary probability density function
of Levy flights in different smooth potential profiles. We find confinement of the particle in the superdiffusion motion with a bimodal stationary distribution for all the anharmonic symmetric monostable potentials investigated. The stationary probability density functions show power-law tails, which ensure finiteness of the variance. By reviewing recent results on these statistical characteristics, the peculiarities of Levy flights in comparison with ordinary Brownian motion are discussed.
Transient properties of different physical systems with metastable states perturbed by external white noise have been investigated. Two noise-induced phenomena, namely the noise enhanced stability and the resonant activation, are theoretically predic
ted in a piece-wise linear fluctuating potential with a metastable state. The enhancement of the lifetime of metastable states due to the noise, and the suppression of noise through resonant activation phenomenon will be reviewed in models of interdisciplinary physics: (i) dynamics of an overdamped Josephson junction; (ii) transient regime of the noisy FitzHugh-Nagumo model; (iii) population dynamics.
