Related papers: Conservative and semiconservative random walks: re…
Conservative and semiconservative random walks in $\mathbb{Z}^d$ were introduced and studied in [V.M. Abramov, J. Theor. Probab. (2017). https://doi.org/10.1007/s10959-017-0747-3]. In the present paper, we extend these concepts for random…
Recently, in ["The coin-turning walk and its scaling limit", Electronic Journal of Probability, 25 (2020)], the ``coin-turning walk'' was introduced on ${\mathbb Z}$. It is a non-Markovian process where the steps form a (possibly)…
We discuss the question of recurrence for persistent, or Newtonian, random walks in Z^2, i.e., random walks whose transition probabilities depend both on the walker's position and incoming direction. We use results by Toth and Schmidt-Conze…
We define a random walk on the set of primitive points of $\mathbb{Z}^d$. We prove that for walks generated by measures satisfying mild conditions these walks are recurrent in a strong sense. That is, we show that the associated Markov…
We consider a walker that at each step keeps the same direction with a probabilitythat depends on the time already spent in the direction the walker is currently moving. In this paper, we study some asymptotic properties of this persistent…
Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(\xi_y,y\in\mathbb Z)$ are two independent sequences of i.i.d. random variables. We suppose that the distributions of…
Simple random walks on a partially directed version of $\mathbb{Z}^2$ are considered. More precisely, vertical edges between neighbouring vertices of $\mathbb{Z}^2$ can be traversed in both directions (they are undirected) while horizontal…
We study branching random walks in random i.i.d. environment in $\Z^d, d \geq 1$. For this model, the population size cannot decrease, and a natural definition of recurrence is introduced. We prove a dichotomy for recurrence/transience,…
We consider two continuous-time generalizations of conservative random walks introduced in [J.Englander and S.Volkov (2022)], an orthogonal and a spherically-symmetrical one; the latter model is known as {\em random flights}. For both…
A parametric family of two-dimensional random walks $\mathbf{S}_t(a)$ $=\big(S_t^{(1)}(a),$ $S_t^{(2)}(a)\big)$ in the main quarter plane is studied. The components $S_t^{(1)}(a)$ and $S_t^{(2)}(a)$ are assumed to be correlated in the way…
In this note, we prove without using Fourier analysis that the symmetric square integrable random walks in $\Z^{2}$ are recurrent.
We survey distributional properties of $\mathbb{R}^d$-valued cocycles of finite measure preserving ergodic transformations (or, equivalently, of stationary random walks in $\mathbb{R}^d$) which determine recurrence or transience.
We consider $d$ random walks $\big(S_n^{(j)}\big)_{n\in\mathbb{N}}$, $1\leq j \leq d$, in the same random environment $\omega$ in $\mathbb{Z}$, and a recurrent simple random walk $(Z_n)_{n\in\mathbb{N}}$ on $\mathbb{Z}$. We assume that,…
A cyclic random walk is a random walk whose transition probabilities/rates can be written as a superposition of the empirical measures of a family of finite cycles. This identifies a convex set of models. We discuss the problem of…
In this paper, we are interested in the asymptotic behaviour of the sequence of processes $(W_n(s,t))_{s,t\in[0,1]}$ with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(1_{\{\xi_{S_k}\leq s\}}-s\big) \end{equation*} where…
In this paper, we introduce random walks with absorbing states on simplicial complexes. Given a simplicial complex of dimension $d$, a random walk with an absorbing state is defined which relates to the spectrum of the $k$-dimensional…
Simple random walks on various types of partially horizontally oriented regular lattices are considered. The horizontal orientations of the lattices can be of various types (deterministic or random) and depending on the nature of the…
A celebrated result in probability theory is that a simple symmetric random walk on the $d$-dimensional lattice $\mathbb{Z}^d$ is recurrent for $d=1,2$ and transient for $d\geq 3$. In this note, we derive a closed-form expression, in terms…
In this paper we introduce the notion of Random Walk in Changing Environment - a random walk in which each step is performed in a different graph on the same set of vertices, or more generally, a weighted random walk on the same vertex and…
We consider a model for random walks on random environments (RWRE) with random subset of Z^d as the vertices, and uniform transition probabilities on 2d points (two "coordinate nearest points" in each of the d coordinate directions). We…