Langevin Equation in Heterogeneous Landscapes: How to Choose the Interpretation.

The Langevin equation is a common tool to model diffusion at a single-particle level. In nonhomogeneous environments, such as aqueous two-phase systems or biological condensates with different diffusion coefficients in different phases, the solution to a Langevin equation is not unique unless the interpretation of stochastic integrals involved is selected. We analyze the diffusion of particles in such systems and evaluate the mean, the mean square displacement, and the distribution of particles, as well as the variance of the time-averaged mean-square displacements.

Fractional Brownian motion with fluctuating diffusivities.

Despite the success of fractional Brownian motion (fBm) in modeling systems that exhibit anomalous diffusion due to temporal correlations, recent experimental and theoretical studies highlight the necessity for a more comprehensive approach of a generalization that incorporates heterogeneities in either the tracers or the environment. This work presents a modification of Lévy's representation of fBm for the case in which the generalized diffusion coefficient is a stochastic process. We derive analytical expressions for the autocovariance function and both ensemble- and time-averaged mean squared displacements.

Convergence to a Gaussian by Narrowing of Central Peak in Brownian yet Non-Gaussian Diffusion in Disordered Environments.

In usual diffusion, the concentration profile, starting from an initial distribution showing sharp features, first gets smooth and then converges to a Gaussian. By considering several examples, we show that the art of convergence to a Gaussian in diffusion in disordered media with infinite contrast may be strikingly different: sharp features of initial distribution do not smooth out at long times. This peculiarity of the strong disorder may be of importance for diagnostics of disorder in complex, e.

Large deviations in continuous-time random walks.

We discuss large deviation properties of continuous-time random walks (CTRWs) and present a general expression for the large deviation rate in CTRWs in terms of the corresponding rates for the distributions of steps' lengths and waiting times. In the case of Gaussian distribution of steps' lengths the general expression reduces to a sequence of two Legendre transformations applied to the cumulant generating function of waiting times. The discussion of several examples (Bernoulli and Gaussian random walks with exponentially distributed waiting times, Gaussian random walks with one-sided Lévy and Pareto-distributed waiting times) reveals interesting general properties of such large deviations.