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We study the escape probability problem in random walks over graphs. Given vertices, $s,t,$ and $p$, the problem asks for the probability that a random walk starting at $s$ will hit $t$ before hitting $p$. Such probabilities can be…

Let $S=(S_n)$ be an oscillatory random walk on the integer lattice $\mathbb{Z}$ with i.i.d. increments. Let $V_{{\rm d}}(x)$ be the renewal function of the strictly descending ladder height process for $S$. We obtain several sufficient…

The jump processes W(t) on [0,\infty[ with transitions w -> alpha w at rate b*w^beta (0 =< alpha =< 1, b>0, beta>0) are considered. Their moments are shown to decay not faster than algebraically for t -> \infty, and an equilibrium…

Place an obstacle with probability $1-p$ independently at each vertex of $\mathbb Z^d$, and run a simple random walk until hitting one of the obstacles. For $d\geq 2$ and $p$ strictly above the critical threshold for site percolation, we…

We study the relationship between tournaments and random walks. This connection was first observed by Erd\H{o}s and Moser. Winston and Kleitman came close to showing that $S_n=\Theta(4^n/n^{5/2})$. Building on this, and works by Tak\'acs,…

The L\'evy walk process for the lower interval of the time of flight distribution ($\alpha<1$) and with finite resting time between consecutive flights is discussed. The motion is restricted to a region bounded by two absorbing barriers and…

The inverse first passage time problem asks whether, for a Brownian motion $B$ and a nonnegative random variable $\zeta$, there exists a time-varying barrier $b$ such that $\mathbb{P}\{B_s>b(s),0\leq s\leq t\}=\mathbb{P}\{\zeta>t\}$. We…

We have investigated the random walk problem in a finite system and studied the crossover induced in the the persistence probability scales by the system size.Analytical and numerical work show that the scaling function is an exponentially…

We consider a one dimensional asymmetric random walk whose jumps are identical, independent and drawn from a distribution \phi(\eta) displaying asymmetric power law tails (i.e. \phi(\eta) \sim c/\eta^{\alpha +1} for large positive jumps and…

We investigate the first passage time beyond a barrier located at $b\geq0$ of a random walk with independent and identically distributed jumps, starting from $x_0=0$. The walk is subject to stochastic resetting, meaning that after each step…

We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes…

We consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy tailed steps, the limiting…

Let $Z_{n}$ be the number of individuals in a subcritical BPRE evolving in the environment generated by iid probability distributions. Let $X$ be the logarithm of the expected offspring size per individual given the environment. Assuming…

For the moving average process $X_n=\rho \xi_{n-1}+\xi_n$, $n\in\mathbb{N}$, where $\rho\in\mathbb{R}$ and $(\xi_i)_{i\ge -1}$ is an i.i.d. sequence of normally distributed random variables, we study the persistence probabilities…

Let (Z_n)_{n\in\N_0} be a d-dimensional random walk in random scenery, i.e., Z_n=\sum_{k=0}^{n-1}Y_{S_k} with (S_k)_{k\in\N_0} a random walk in Z^d and (Y_z)_{z\in Z^d} an i.i.d. scenery, independent of the walk. We assume that the random…

We give a complete and unified description -- under some stability assumptions -- of the functional scaling limits associated with some persistent random walks for which the recurrent or transient type is studied in [1]. As a result, we…

In this article, we consider additive functionals $\zeta_t = \int_0^t f(X_s)\mathrm{d} s$ of a c\`adl\`ag Markov process $(X_t)_{t\geq 0}$ on $\mathbb{R}$. Under some general conditions on the process $(X_t)_{t\geq 0}$ and on the function…

Consider a stochastic process that behaves as a $d$-dimensional simple and symmetric random walk, except that, with a certain fixed probability, at each step, it chooses instead to jump to a given site with probability proportional to the…

Be $X_t$ a random walk. We study its span $S$, i.e. the size of the domain visited up to time $t$. We want to know the probability that $S$ reaches $1$ for the first time, as well as the density of the span given $t$. Analytical results are…

Particle hopping is a common feature in heterogeneous media. We explore such motion by using the widely applicable formalism of the continuous time random walk and focus on the statistics of rare events. Numerous experiments have shown that…