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We give exact and explicit expressions of mean first-passage times for random walks in a rectangular domain, in both cases of reflecting boundary conditions and periodic boundary conditions. The situations with one or two absorbing targets…

We study exploration properties of a random walk on a network. For a fully connected network we find that the problem can be mapped to the well known coupon collector problem, thus allowing us to estimate form of $P(S,t)$: the distribution…

We consider a random walk on the fully-connected lattice with $N$ sites and study the time evolution of the number of distinct sites $s$ visited by the walker on a subset with $n$ sites. A record value $v$ is obtained for $s$ at a record…

We study the time until first occurrence, the first-passage time, of rare density fluctuations in diffusive systems. We approach the problem using a model consisting of many independent random walkers on a lattice. The existence of spatial…

We study the probability density function (PDF) of the cover time $t_c$ of a finite interval of size $L$, by $N$ independent one-dimensional Brownian motions, each with diffusion constant $D$. The cover time $t_c$ is the minimum time needed…

We consider a population of $N$ labeled random walkers moving on a substrate, and an excitation jumping among the walkers upon contact. The label $\mathcal{X}(t)$ of the walker carrying the excitation at time $t$ can be viewed as a…

Exploiting the coherent medium approximation, random walk among sites distributed randomly in space is investigated when the jump rate depends on the distance between two adjacent sites. In one dimension, it is shown that when the jump rate…

We study a symmetric random walk (RW) in one spatial dimension in environment, formed by several zones of finite width, where the probability of transition between two neighboring points and corresponding diffusion coefficient are…

We study the probability that a random walk started inside a subgraph of a larger graph exits that subgraph (or, equivalently, hits the exterior boundary of the subgraph). Considering the chance a random walk started in the subgraph never…

We study records generated by Brownian particles in one dimension. Specifically, we investigate an ordinary random walk and define the record as the maximal position of the walk. We compare the record of an individual random walk with the…

The presence of temporal correlations in random movement trajectories is a widespread phenomenon across biological, chemical and physical systems. The ubiquity of persistent and anti-persistent motion in many natural and synthetic systems…

In empirical studies of random walks, continuous trajectories of animals or individuals are usually sampled over a finite number of points in space and time. It is however unclear how this partial observation affects the measured…

Consider a one dimensional diffusion process on the diffusion interval $I$ originated in $x_0\in I$. Let $a(t)$ and $b(t)$ be two continuous functions of $t$, $t>t_0$ with bounded derivatives and with $a(t)<b(t)$ and $a(t),b(t)\in I$,…

Continuous time random walk models with decoupled waiting time density are studied. When the spatial one jump probability density belongs to the Levy distribution type and the total time transition is exponential a generalized…

Nearest neighbor random walks in the quarter plane that are absorbed when reaching the boundary are studied. The cases of positive and zero drift are considered. Absorption probabilities at a given time and at a given site are made…

The cover time is defined as the time needed for a random walker to visit every site of a confined domain. Here, we focus on persistent random walks, which provide a minimal model of random walks with short range memory. We derive the exact…

We consider the distribution of the duration time, the time elapsed since it began, of a diffusion process given its present position, under the assumption that the process began at the origin. For unbiased diffusion, the distribution does…

Let $R_n=\max_{0\leq j\leq n}S_j-S_n$ be a random walk $S_n$ reflected in its maximum. Except in the trivial case when $P(X\ge0)=1$, $R_n$ will pass over a horizontal boundary of any height in a finite time, with probability 1. We extend…

We study the passage (translocation) of a self-avoiding polymer through a membrane pore in two dimensions. In particular, we numerically measure the probability distribution Q(T) of the translocation time T, and the distribution P(s,t) of…

The probability distribution of the number $s$ of distinct sites visited up to time $t$ by a random walk on the fully-connected lattice with $N$ sites is first obtained by solving the eigenvalue problem associated with the discrete master…