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Trapped dynamics widely appears in nature, e.g., the motion of particles in viscous cytoplasm. The famous continuous time random walk (CTRW) model with power law waiting time distribution ({\em having diverging first moment}) describes this…

Continuous Time Random Walks (CTRW) are widely used to coarse-grain the evolution of systems jumping from a metastable sub-set of their configuration space, or trap, to another via rare intermittent events. The multi-scaled behavior typical…

We show that the dynamics of supercooled liquids, analyzed from computer simulations of the binary mixture Lennard-Jones system, can be described in terms of a continuous time random walk (CTRW). The required discretization comes from…

The fluctuation-dissipation relation (FDR), a fundamental result of equilibrium statistical physics, ceases to be valid when a system is taken out of the equilibrium. A generalization of FDR has been theoretically proposed for…

It has been observed in numerous experiments, simulations, and various theoretical treatments that the spreading of particles can be modeled by the continuous-time random walk. We consider two well-known cases, i.e., Gaussian displacements…

The elephant random walk (ERW) is a microscopic, one-dimensional, discrete-time, non-Markovian random walk, which can lead to anomalous diffusion due to memory effects. In this study, I propose a multi-dimensional generalization in which…

We discuss a dynamic procedure that makes the fractional derivatives emerge in the time asymptotic limit of non-Poisson processes. We find that two-state fluctuations, with an inverse power-law distribution of waiting times, finite first…

The distribution of a population throughout the physiological age of the individuals is very relevant information in population studies. It has been modeled by the Langevin and the Fokker- Planck equations. A major problem with these…

Brain age prediction models have succeeded in predicting clinical outcomes in neurodegenerative diseases, but can struggle with tasks involving faster progressing diseases and low quality data. To enhance their performance, we employ a…

In this paper we investigate the porous medium equation with a fractional temporal derivative. We justify that the resulting equation emerges when we consider the waiting-time (or trapping) phenomenon that can happen in the medium. Our…

Let $(\xi_k,\eta_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $T:=(T_k)_{k\in\mathbb{N}}$ defined by…

We present a methodology for determining reservoir groundwater age and transit time probability distributions in a deterministic manner, considering advective-dispersive transport in steady velocity fields. In a first step, we propose to…

We consider the discrete time unitary dynamics given by a quantum walk on the lattice $\Z^d$ performed by a quantum particle with internal degree of freedom, called coin state, according to the following iterated rule: a unitary update of…

In this article, we present new random walk methods to solve flow and transport problems in unsaturated/saturated porous media, including coupled flow and transport processes in soils, heterogeneous systems modeled through random hydraulic…

Bias plays an important role in the enhancement of diffusion in periodic potentials. Using the continuous-time random walk in the presence of a bias, we provide a novel mechanism for the enhancement of diffusion in a random energy…

We consider a one-dimensional continuous time random walk (CTRW) on a fixed time interval $T$ where at each time step the walker waits a random time $\tau$, before performing a jump drawn from a symmetric continuous probability distribution…

We consider linear age-structured population equations with diffusion. Supposing maximal regularity of the diffusion operator, we characterize the generator and its spectral properties of the associated strongly continuous semigroup. In…

In this paper we are examining diffusion properties of stationary continuous-time Weierstrass walk (CTWW). We are showing it is a multi-phase representation of the L\'evy walk. The hierarchical spatial-temporal coupling, combined with…

Continuous time quantum walks (CTQW) do not necessarily perform better than their classical counterparts, the continuous time random walks (CTRW). For one special graph, where a recent analysis showed that in a particular direction of…

A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. The fundamental solution (for the…