First-passage and first-arrival problems in continuous-time random walks: Beyond the diffusion approximation.

Some exact solutions of the first-passage and first-arrival problems for the continuous-time random-walk model are obtained. On the basis of these exact solutions, the following has been revealed. First, for some jump-length distributions with a finite variance, the approximate solutions obtained in the diffusion approximation can differ significantly from the exact solutions.

First-passage behavior of the random-barrier model.

The previously proposed transport equation for the random-barrier model, which is the diffusion equation with resetting to positions visited in the past, is used here to calculate the first-passage times. The results obtained are compared with those obtained using the normal diffusion equation with an effective diffusion coefficient. It is shown that, under certain conditions, the equation with the effective diffusion coefficient can greatly overestimate the time of the first passage.

Non-Markovian edge-based compartmental modeling.

A method is proposed for generalizing the equations obtained within the framework of the edge-based compartmental modeling approach for the case of non-Poissonian transmission and recovery processes. It is confirmed that non-Markovian systems of equations obtained in this manner, which describe the spread of epidemic diseases, can be represented as Markovian systems of equations. The application of the proposed method in particular types of edge-based compartmental models is considered.

Subordinated stochastic processes with aged operational time.

In this paper, subordinated stochastic processes are considered, where the renewal process acting as the operational time. It is assumed that the observation of the process begins at a certain time after the start of the renewal process. A recurrence formula was derived for calculating the multipoint probability density functions of the aged renewal process.

Continuous-time random walk under time-dependent resetting.

Continuous-time random walks of a particle that is randomly reset to an initial position are considered. The distribution of the waiting time between the reset events is represented as a sum of an arbitrary number of exponentials. The governing equation of this stochastic process is established.

Reaction-subdiffusion equations for the A<=>B reaction.

We consider a linear reversible isomerization reaction A <=> B under subdiffusion described by continuous time random walks (CTRW). The reactants' transformations take place independently on the motion and are described by constant rates. We show that the form of the ensuing system of mesoscopic reaction-subdiffusion equations is unusual: the equation for time derivative of say A(x,t) contains the terms depending not only on DeltaA , but also on DeltaB .