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We study the quenched behaviour of a perturbed version of the simple symmetric random walk on the set of integers. The random walker moves symmetrically with an exception of some randomly chosen sites where we impose a random drift. We show that if the gaps between the marked sites are i.i.d. and regularly varying with a sufficiently small index, then there is no strong quenched limit laws for the position of the random walker. As a consequence we study the quenched limit laws in the context of weak convergence of random measures.

We investigate a branching random walk where the displacements are independent from the branching mechanism and have a stretched exponential distribution. We describe the positions of the particles in the vicinity of the rightmost particle in terms of point process convergence. As a consequence we give a~new limit theorem for the position of the rightmost particle. Our methods rely on providing precise large deviations for sums of i.i.d. random variables with stretched exponential distributions outside the so-called one big jump regime., 31 pages

We study a class of kinetic-type differential equations $\partial \phi_t/\partial t+\phi_t=\widehat{\mathcal{Q}}\phi_t$, where $\widehat{\mathcal{Q}}$ is an inhomogeneous smoothing transform and, for every $t\geq 0$, $\phi_t$ is the Fourier--Stieltjes transform of a probability measure. We show that under mild assumptions on $\widehat{\mathcal{Q}}$ the above differential equation possesses a unique solution and represent this solution as the characteristic function of a certain stochastic process associated with the continuous time branching random walk pertaining to $\widehat{\mathcal{Q}}$. Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as $t\to\infty$., Comment: 24 pages, to appear in Stochastic Processes and Their Applications

We study the asymptotics of the $k$-regular self-similar fragmentation process. For $\alpha > 0$ and an integer $k \geq 2$, this is the Markov process $(I_t)_{t \geq 0}$ in which each $I_t$ is a union of open subsets of $[0,1)$, and independently each subinterval of $I_t$ of size $u$ breaks into $k$ equally sized pieces at rate $u^\alpha$. Let $k^{ - m_t}$ and $k^{ - M_t}$ be the respective sizes of the largest and smallest fragments in $I_t$. By relating $(I_t)_{t \geq 0}$ to a branching random walk, we find that there exist explicit deterministic functions $g(t)$ and $h(t)$ such that $|m_t - g(t)| \leq 1$ and $|M_t - h(t)| \leq 1$ for all sufficiently large $t$. Furthermore, for each $n$, we study the final time at which fragments of size $k^{-n}$ exist. In particular, by relating our branching random walk to a certain point process, we show that, after suitable rescaling, the laws of these times converge to a Gumbel distribution as $n \to \infty$., Comment: 29 pages, 1 figure

We study the one-dimensional branching random walk in the case when the step size distribution has a stretched exponential tail, and, in particular, no finite exponential moments. The tail of the step size $X$ decays as $\mathbb{P}[X \geq t] \sim a \exp(-\lambda t^r)$ for some constants $a, \lambda > 0$ where $r \in (0,1)$. We give a detailed description of the asymptotic behaviour of the position of the rightmost particle, proving almost-sure limit theorems, convergence in law and some integral tests. The limit theorems reveal interesting differences betweens the two regimes $ r \in (0, 2/3)$ and $ r \in (2/3, 1)$, with yet different limits in the boundary case $r = 2/3$., Comment: 30 pages

A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk $(X_n)_{n\in \mathbb{N}\cup\{0\}}$ in a sparse random environment $(S_k,\lambda_k)_{k\in\mathbb{Z}}$ is a nearest neighbor random walk on $\mathbb{Z}$ that jumps to the left or to the right with probability $1/2$ from every point of $\mathbb{Z}\setminus \{\ldots,S_{-1},S_0=0,S_1,\ldots\}$ and jumps to the right (left) with the random probability $\lambda_{k+1}$ ($1-\lambda_{k+1}$) from the point $S_k$, $k\in\mathbb{Z}$. Assuming that $(S_k-S_{k-1},\lambda_k)_{k\in\mathbb{Z}}$ are independent copies of a random vector $(\xi,\lambda)\in \mathbb{N}\times (0,1)$ and the mean $\mathbb{E}\xi$ is finite (moderate sparsity) we obtain stable limit laws for $X_n$, properly normalized and centered, as $n\to\infty$. While the case $\xi\leq M$ a.s.\ for some deterministic $M>0$ (weak sparsity) was analyzed by Matzavinos et al., the case $\mathbb{E} \xi=\infty$ (strong sparsity) will be analyzed in a forthcoming paper., Comment: submitted, 42 pages