Sliusarenko O. Relaxation processes and barrier crossing in stochastical systems with Levy statistics and non-Markovian correlations

The thesis is devoted to studying of stochastical systems with anomalous relaxation processes paying major attention to the problem of particle escape from the potential well due to the external random force with Levy probability distribution law or fractional Gaussian noise. By using the numerical simulation method of generalized Langevin equations integration the data for mean escape times and escape times probability density functions is obtained. Within the constant flux approximation generalization and Wilemski-Fixmann approach the analytical results for these systems are received and compared with the numerical ones. Both similarities and differences with classical Brownian motion are underlined. Also, a generalized Fokker-Planck equation for linear Gaussian stochastical systems for arbitrary correlation function of the noise is derived, the probability density function is constructed and some sample systems exhibiting generalized Brownian motion are considered.