Related papers: Semi-Markov approach to continuous time random wal…

There is a well-established theory linking certain semi-Markov chains and continuous-time random walks to time-fractional equations and anomalous diffusion. In this work, we go beyond the semi-Markov framework by considering some…

The purpose of this tutorial is to introduce the main concepts behind normal and anomalous diffusion. Starting from simple, but well known experiments, a series of mathematical modeling tools are introduced, and the relation between them is…

Continuous-time random walks are generalisations of random walks frequently used to account for the consistent observations that many molecules in living cells undergo anomalous diffusion, i.e. subdiffusion. Here, we describe the…

We study the so-called elephant random walk (ERW) which is a non-Markovian discrete-time random walk on $\mathbb{Z}$ with unbounded memory which exhibits a phase transition from diffusive to superdiffusive behaviour. We prove a law of large…

Recently, in ["The coin-turning walk and its scaling limit", Electronic Journal of Probability, 25 (2020)], the ``coin-turning walk'' was introduced on ${\mathbb Z}$. It is a non-Markovian process where the steps form a (possibly)…

We introduce a model, in which a particle performs a continuous time random walk (CTRW) coupled to an environment with Ising dynamics. The particle shows locally varying diffusivity determined by the geometrical properties of the underlying…

Graph-limit theory focuses on the convergence of sequences of graphs when the number of nodes becomes arbitrarily large. This framework defines a continuous version of graphs allowing for the study of dynamical systems on very large graphs,…

Analyses of serially-sampled data often begin with the assumption that the observations represent discrete samples from a latent continuous-time stochastic process. The continuous-time Markov chain (CTMC) is one such generative model whose…

Background: This study is mainly motivated by the need of understanding how the diffusion behaviour of a biomolecule (or even of a larger object) is affected by other moving macromolecules, organelles, and so on, inside a living cell,…

We provide necessary and sufficient conditions for explosion and implosion of birth-and-death (non-Markov) continuous-time random walks. In other words, we obtain conditions for $\infty$ to be accessible and for it to be an entrance point.…

The proliferation and migration dichotomy of the tumor cell invasion is examined within a two-component continuous time random walk (CTRW) model. The balance equations for the cancer cells of two phenotypes with random switching between…

We study continuous time random walks (CTRW) with power law distribution of waiting times under resetting which brings the walker back to the origin, with a power-law distribution of times between the resetting events. Two situations are…

Here, a new two-dimensional process, discrete in time and space, that yields the results of both a random walk and a quantum random walk, is introduced. This model describes the population distribution of four coin states |1>,-|1>, |0> -|0>…

The motion of self-propelled particles is modeled as a persistent random walk. An analytical framework is developed that allows the derivation of exact expressions for the time evolution of arbitrary moments of the persistent walk's…

We study one-dimensional discrete as well as continuous time random walks, either with a fixed number of steps (for discrete time) $n$ or on a fixed time interval $T$ (for continuous time). In both cases, we focus on symmetric probability…

We introduce a new class of asymmetric random walks on the one-dimensional infinite lattice. In this walk the direction of the jumps (positive or negative) is determined by a discrete-time renewal process which is independent of the jumps.…

We study a new technique for the asymptotic analysis of heavy-tailed systems conditioned on large deviations events. We illustrate our approach in the context of ruin events of multidimensional regularly varying random walks. Our approach…

We study a limit behavior of a sequence of Markov processes (or Markov chains) such that their distributions outside of any neighborhood of a "singular" point attract to some probability law. In any neighborhood of this point the behavior…

New algorithms for construction of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes with finite phase spaces are presented. These algorithms are based on a special technique of sequential…

We study a one-dimensional Markov modulated random walk with jumps. It is assumed that amplitudes of jumps as well as a chosen velocity regime are random and depend on a time spent by the process at a previous state of the underlying Markov…