Anomalous diffusion and nonlinear relaxation phenomena in stochastic models of interdisciplinary physics

The study of nonlinear dynamical systems in the presence of both Gaussian and non-Gaussian noise sources is the topic of this research work. In particular, after shortly present new theoretical results for statistical characteristics in the framework of Markovian theory, we analyse four different physical systems in the presence of Levy noise source. (a) The residence time problem of a particle subject to a non-Gaussian noise source in arbitrary potential profile was analyzed and the exact analytical results for the statistical characteristics of the residence time for anomalous diffusion in the form of Levy flights in fully unstable potential profile was obtained. Noise enhanced stability phenomenon was found in the system investigated. (b) The correlation time of the particle coordinate as a function of the height of potential barrier, the position of potential wells and noise intensity was investigated in the case of confined steady-state Levy flights with Levy index alpha=1, that is Cauchy noise, in the symmetric bistable quartic potential. (c) The stationary spectral characteristics of superdiffusion of Levy flights in one-dimensional confinement potential profiles were investigated both theoretically and numerically. Specifically, for Cauchy stable noise we calculated the steady-state probability density function for an infinitely deep rectangular potential well and for a symmetric steep potential well. (d) For two-dimensional diffusion the general Kolmogorov equation for the joint probability density function of particle coordinates was obtained by functional methods directly from two Langevin equations with statistically independent non-Gaussian noise sources. We compared the properties of Brownian diffusion and Levy flights in parabolic potential with radial symmetry. Afterwards, we analyzed the nonlinear relaxation in the presence of Gaussian noise for the stochastic switching dynamics of the memristors. We have studied three different models. (a) We started from consideration of the simplest model of resistive switching. (b) Further, the charge-controlled and the current-controlled ideal Chua memristors with external Gaussian noise were investigated. For both cases we have obtained exact analytical expressions for the probability density function of the charge flowing through the memristor and of the memristance. (c) Moreover, we proposed a stochastic macroscopic model of a memristor, based on a generalization of known approaches and experimental results. Steady-state concentration of defects for different boundary conditions was found. Also we analysed how the concentration of defects is changed with time under arbitrary values of external voltage, noise intensity, effective diffusion coefficient and other parameters. An examination of the results was performed, the possible implications of this work and the future development of this study were outlined.