Generalized Wiener Process and Kolmogorov's Equation for Diffusion induced by Non-Gaussian Noise Source

We show that the increments of generalized Wiener process, useful to describe non-Gaussian white noise sources, have the properties of infinitely divisible random processes. Using functional approach and the new correlation formula for non-Gaussian white noise we derive directly from Langevin equation, with such a random source, the Kolmogorov's equation for Markovian non-Gaussian process. From this equation we obtain the Fokker-Planck equation for nonlinear system driven by white Gaussian noise, the Kolmogorov-Feller equation for discontinuous Markovian processes, and the fractional Fokker-Planck equation for anomalous diffusion. The stationary probability distributions for some simple cases of anomalous diffusion are derived.Comment: 8 pages. in press, Fluctuation and Noise Letters, 200

Critical Phenomena and Diffusion in Complex Systems

Editorial of the International Conference on Critical Phenomena and Diffusion in Complex Systems held on 5--7 December, 2006 in Nizhniy Novgorod State University, Russia and was dedicated to the memory and 80th anniversary of Professor Askold N. Malakhov.Comment: 4 pages, to appear in International Journal of Bifurcation and Chao

Noise Enhanced Stability

The noise can stabilize a fluctuating or a periodically driven metastable state in such a way that the system remains in this state for a longer time than in the absence of white noise. This is the noise enhanced stability phenomenon, observed experimentally and numerically in different physical systems. After shortly reviewing all the physical systems where the phenomenon was observed, the theoretical approaches used to explain the effect are presented. Specifically the conditions to observe the effect: (a) in systems with periodical driving force, and (b) in random dichotomous driving force, are discussed. In case (b) we review the analytical results concerning the mean first passage time and the nonlinear relaxation time as a function of the white noise intensity, the parameters of the potential barrier, and of the dichotomous noise.Comment: 18 pages, 6 figures, in press Acta Physica Polonica (2004

Exact Results for Spectra of Overdamped Brownian Motion in Fixed and Randomly Switching Potentials

The exact formulae for spectra of equilibrium diffusion in a fixed bistable piecewise linear potential and in a randomly flipping monostable potential are derived. Our results are valid for arbitrary intensity of driving white Gaussian noise and arbitrary parameters of potential profiles. We find: (i) an exponentially rapid narrowing of the spectrum with increasing height of the potential barrier, for fixed bistable potential; (ii) a nonlinear phenomenon, which manifests in the narrowing of the spectrum with increasing mean rate of flippings, and (iii) a nonmonotonic behaviour of the spectrum at zero frequency, as a function of the mean rate of switchings, for randomly switching potential. The last feature is a new characterization of resonant activation phenomenon.Comment: in press in Acta Physica Polonica, vol. 35 (4), 200

L\'evy flights versus L\'evy walks in bounded domains

L\'evy flights and L\'evy walks serve as two paradigms of random walks resembling common features but also bearing fundamental differences. One of the main dissimilarities are discontinuity versus continuity of their trajectories and infinite versus finite propagation velocity. In consequence, well developed theory of L\'evy flights is associated with their pathological physical properties, which in turn are resolved by the concept of L\'evy walks. Here, we explore L\'evy flights and L\'evy walks models on bounded domains examining their differences and analogies. We investigate analytically and numerically whether and under which conditions both approaches yield similar results in terms of selected statistical observables characterizing the motion: the survival probability, mean first passage time and stationary PDFs. It is demonstrated that similarity of models is affected by the type of boundary conditions and value of the stability index defining asymptotics of the jump length distribution.Comment: 15 pages, 13 figure

Noise Enhanced Stability in Fluctuating Metastable States

We derive general equations for the nonlinear relaxation time of Brownian diffusion in randomly switching potential with a sink. For piece-wise linear dichotomously fluctuating potential with metastable state, we obtain the exact average lifetime as a function of the potential parameters and the noise intensity. Our result is valid for arbitrary white noise intensity and for arbitrary fluctuation rate of the potential. We find noise enhanced stability phenomenon in the system investigated: the average lifetime of the metastable state is greater than the time obtained in the absence of additive white noise. We obtain the parameter region of the fluctuating potential where the effect can be observed. The system investigated also exhibits a maximum of the lifetime as a function of the fluctuation rate of the potential.Comment: 7 pages, 5 figures, to appear in Phys. Rev. E vol. 69 (6),200

L\'{e}vy flights in inhomogeneous environments

We study the long time asymptotics of probability density functions (pdfs) of L\'{e}vy flights in different confining potentials. For that we use two models: Langevin - driven and (L\'{e}vy - Schr\"odinger) semigroup - driven dynamics. It turns out that the semigroup modeling provides much stronger confining properties than the standard Langevin one. Since contractive semigroups set a link between L\'{e}vy flights and fractional (pseudo-differential) Hamiltonian systems, we can use the latter to control the long - time asymptotics of the pertinent pdfs. To do so, we need to impose suitable restrictions upon the Hamiltonian and its potential. That provides verifiable criteria for an invariant pdf to be actually an asymptotic pdf of the semigroup-driven jump-type process. For computational and visualization purposes our observations are exemplified for the Cauchy driver and its response to external polynomial potentials (referring to L\'{e}vy oscillators), with respect to both dynamical mechanisms.Comment: Major revisio

Heavy-tailed targets and (ab)normal asymptotics in diffusive motion

We investigate temporal behavior of probability density functions (pdfs) of paradigmatic jump-type and continuous processes that, under confining regimes, share common heavy-tailed asymptotic (target) pdfs. Namely, we have shown that under suitable confinement conditions, the ordinary Fokker-Planck equation may generate non-Gaussian heavy-tailed pdfs (like e.g. Cauchy or more general L\'evy stable distribution) in its long time asymptotics. For diffusion-type processes, our main focus is on their transient regimes and specifically the crossover features, when initially infinite number of the pdf moments drops down to a few or none at all. The time-dependence of the variance (if in existence), $\sim t^{\gamma}$ with $0<\gamma <2$, in principle may be interpreted as a signature of sub-, normal or super-diffusive behavior under confining conditions; the exponent $\gamma$ is generically well defined in substantial periods of time. However, there is no indication of any universal time rate hierarchy, due to a proper choice of the driver and/or external potential.Comment: Major revisio

Linear and nonlinear approximations for periodically driven bistable systems

We analyze periodically driven bistable systems by two different approaches. The first approach is a linearization of the stochastic Langevin equation of our system by the response on small external force. The second one is based on the Gaussian approximation of the kinetic equations for the cumulants. We obtain with the first approach the signal power amplification and output signal-to-noise ratio for a model piece-wise linear bistable potential and compare with the results of linear response approximation. By using the second approach to a bistable quartic potential, we obtain the set of nonlinear differential equations for the first and the second cumulants