Related papers: Asymptotic behaviour for critical slowing-down ran…

We consider a recent model of random walk that recursively grows the network on which it evolves, namely the Tree Builder Random Walk (TBRW). We introduce a bias $\rho \in (0,\infty)$ towards the root, and exhibit a phase transition for…

This paper is not (or at least not only) about human infant mortality. In line with reliability theory, "infant" will refer here to the time interval following birth during which the mortality (or failure) rate decreases. This definition…

We study the exact asymptotics for the distribution of the first time $\tau_x$ a L\'evy process $X_t$ crosses a negative level $-x$. We prove that $\mathbf P(\tau_x>t)\sim V(x)\mathbf P(X_t\ge 0)/t$ as $t\to\infty$ for a certain function…

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…

We consider trap models on Z^d, namely continuous time Markov jump process on Z^d with embedded chain given by a generic discrete time random walk, and whose mean waiting time at x is given by tau_x, with tau = (tau_x, x in Z^d) a family of…

We first study a model, introduced recently in \cite{ES}, of a critical branching random walk in an IID random environment on the $d$-dimensional integer lattice. The walker performs critical (0-2) branching at a lattice point if and only…

Recently, Takayasu and Tretyakov [Phys. Rev. Lett. {\bf 68}, 3060 (1992)], studied branching annihilating random walks (BAW) with $n=1$-5 offspring. These models exhibit a continuous phase transition to an absorbing state. For odd $n$ the…

Using the continuous-time random walk (CTRW) approach, we study the phenomenon of relaxation of two-state systems whose elements evolve according to a dichotomous process. Two characteristics of relaxation, the probability density function…

We study a one-dimensional sluggish random walk with space-dependent transition probabilities between nearest-neighbour lattice sites. Motivated by trap models of slow dynamics, we consider a model in which the trap depth increases…

Let $(Z_n)$ be a supercritical branching process in an independent and identically distributed random environment $\xi$. We show the exact decay rate of the probability $\mathbb{P}(Z_n=j | Z_0 = k)$ as $n \to \infty$, for each $j \geq k,$…

Given a supercritical branching random walk $\{Z_n\}_{n\geq 0}$ on $\mathbb{R}$, let $Z_n([y,\infty))$ be the number of particles located in $[y,\infty)\subset\mathbb{R}$ at generation $n$. Let $m$ be the mean of the offspring law of…

We consider the process $\{x-N(t):t\geq 0\}$, where $x\in\mathbb{R}_+$ and $\{N(t):t\geq 0\}$ is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of $(\tau(x),A(x))$ where $\tau(x)$…

In this paper, we introduce a variation of the elephant random walk whose steps are polynomially decaying. At each time $k$, the walker's step size is $k^{-\gamma}$ with $\gamma>0$. We investigate effects of the step size exponent $\gamma$…

Several phase transitions for excited random walks on the integers are known to be characterized by a certain drift parameter delta. For recurrence/transience the critical threshold is |delta|=1, for ballisticity it is |delta|=2 and for…

The dynamics of populations is frequently subject to intrinsic noise. At the same time unknown interaction networks or rate constants can present quenched uncertainty. Existing approaches often involve repeated sampling of the quenched…

We consider a discrete-time quantum walk $W_{t,\kappa}$ at time $t$ on a graph with joined half lines $\mathbb{J}_\kappa$, which is composed of $\kappa$ half lines with the same origin. Our analysis is based on a reduction of the walk on a…

For many stochastic processes, the probability $S(t)$ of not-having reached a target in unbounded space up to time $t$ follows a slow algebraic decay at long times, $S(t)\sim S_0/t^\theta$. This is typically the case of symmetric compact…

Consider a simple random walk on the integers with the following transition mechanism. At each site $x$, the probability of jumping to the right is $\omega(x)\in[\frac12,1)$, until the first time the process jumps to the left from site $x$,…

In this paper we investigate the asymptotic properties of the wait-first and jump-first L\'evy walk with rest, which is a generalization of standard jump-first and jump-first L\'evy walk that assumes each waiting time in the model is a sum…

Random multiplicative processes $w_t =\lambda_1 \lambda_2 ... \lambda_t$ (with < \lambda_j > 0 ) lead, in the presence of a boundary constraint, to a distribution $P(w_t)$ in the form of a power law $w_t^{-(1+\mu)}$. We provide a simple and…