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Increasingly larger data sets of processes in space and time ask for statistical models and methods that can cope with such data. We show that the solution of a stochastic advection-diffusion partial differential equation provides a…
We propose a generalization of the classical M/M/1 queue process. The resulting model is derived by applying fractional derivative operators to a system of difference-differential equations. This generalization includes both non-Markovian…
This paper addresses the nonparametric estimation of the drift function over a compact domain for a time-homogeneous diffusion process, based on high-frequency discrete observations from $N$ independent trajectories. We propose a neural…
We implement Bayesian model selection and parameter estimation for the case of fractional Brownian motion with measurement noise and a constant drift. The approach is tested on artificial trajectories and shown to make estimates that match…
This paper is concerned with analysis of coupled fractional reaction-diffusion equations. It provides analytical comparison for the fractional and regular reaction-diffusion systems. As an example, the reaction-diffusion model with cubic…
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…
We focus on variational inference in dynamical systems where the discrete time transition function (or evolution rule) is modelled by a Gaussian process. The dominant approach so far has been to use a factorised posterior distribution,…
We show that the generalized diffusion coefficient of a subdiffusive intermittent map is a fractal function of control parameters. A modified continuous time random walk theory yields its coarse functional form and correctly describes a…
In this paper continuous time random walk models approximating fractional space-time diffusion processes are studied. Stochastic processes associated with the considered equations represent time-changed processes, where the time-change…
We review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics. The problems in continuum mechanics concern mathematical…
Fractional Brownian motion is a Gaussian stochastic process with long-range correlations in time; it has been shown to be a useful model of anomalous diffusion. Here, we investigate the effects of mutual interactions in an ensemble of…
Fractional Brownian motion (fBm) is an important scale-invariant Gaussian non-Markovian process with stationary increments, which serves as a prototypical example of a system with long-range temporal correlations and anomalous diffusion.…
We consider a fractional Ornstein-Uhlenbeck process involving a stochastic forcing term in the drift, as a solution of a linear stochastic differential equation driven by a fractional Brownian motion. For such process we specify mean and…
This paper deals with fractional-order controlled systems and fractional-order controllers in the frequency domain. The mathematical description by fractional transfer functions and properties of these systems are presented. The new ways…
The linear fractional stable motion (LFSM) extends the fractional Brownian motion (fBm) by considering $\alpha$-stable increments. We propose a method to forecast future increments of the LFSM from past discrete-time observations, using the…
Diffusion is a fundamental physical phenomenon with critical applications in fields such as metallurgy, cell biology, and population dynamics. While standard diffusion is well-understood, anomalous diffusion often requires complex non-local…
Fractional, anomalous diffusion in space-periodic potentials is investigated. The analytical solution for the effective, fractional diffusion coefficient in an arbitrary periodic potential is obtained in closed form in terms of two…
We study a Bayesian approach to nonparametric estimation of the periodic drift function of a one-dimensional diffusion from continuous-time data. Rewriting the likelihood in terms of local time of the process, and specifying a Gaussian…
Based on a novel dynamic Whittle likelihood approximation for locally stationary processes, a Bayesian nonparametric approach to estimating the time-varying spectral density is proposed. This dynamic frequency-domain based likelihood…
In this paper we present a study of anomalous diffusion using a Fokker-Planck description with fractional velocity derivatives. The distribution functions are found using numerical means for varying degree of fractionality observing the…