Related papers: From pseudo-random walk to pseudo-Brownian motion:…
A random walk (or a Wiener process), possibly with drift, is observed in a noisy or delayed fashion. The problem considered in this paper is to estimate the first time \tau the random walk reaches a given level. Specifically, the p-moment…
In this paper we study some aspects of search for an immobile target by a swarm of N non-communicating, randomly moving searchers (numbered by the index k, k = 1, 2,..., N), which all start their random motion simultaneously at the same…
A random walk with counterbalanced steps is a process of partial sums $\check S(n)=\check X_1+ \cdots + \check X_n$ whose steps $\check X_n$ are given recursively as follows. For each $n\geq 2$, with a fixed probability $p$, $\check X_n$ is…
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails…
In this work we consider a stochastic movement process with random resets to the origin followed by a random residence time there before the walker restarts its motion. First, we study the transport properties of the walker, we derive an…
We study the mean traversal time for a class of random walks on Newman-Watts small-world networks, in which steps around the edge of the network occur with a transition rate F that is different from the rate f for steps across small-world…
A random walk scheme, consisting of alternating phases of regular Brownian motion and L\'evy walks, is proposed as a model for run-and-tumble bacterial motion. Within the continuous-time random walk approach we obtain the long-time and…
For drifted Brownian motion $X(t)= x - \mu t + B_t \ (\mu >0)$ starting from $x>0,$ we study the joint distribution of the first-passage time below zero, $\tau(x),$ and the first-passage area, $A(x),$ swept out by $X$ till the time…
Spontaneous collapse models use non-linear stochastic modifications of the Schroedinger equation to suppress superpositions of eigenstates of the measured observable and drive the state to an eigenstate. It was recently demonstrated that…
Consider an arbitrary transient random walk on $\Z^d$ with $d\in\N$. Pick $\alpha\in[0,\infty)$ and let $L_n(\alpha)$ be the spatial sum of the $\alpha$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range,…
We investigate the hitting times of random walks on graphs, where a hitting time is defined as the number of steps required for a random walker to move from one node to another. While much of the existing literature focuses on calculating…
Quantum random walks have been much studied recently, largely due to their highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum random walk on the line: the use of multiple…
We consider a walker moving in a one-dimensional interval with absorbing boundaries under the effect of Markovian resettings to the initial position. The walker's motion follows a random walk characterized by a general waiting time…
We present some new results about the distribution of a random walk whose independent steps follow a $q-$Gaussian distribution with exponent $\frac{1}{1-q}; q \in \mathbb{R}$. In the case $q>1$ we show that a stochastic representation of…
We propose diffusion-like equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which,…
The mean-squared displacement (MSD) is an averaged quantity widely used to assess anomalous diffusion. In many cases, such as molecular motors with finite processivity, dynamics of the system of interest produce trajectories of varying…
A matrix random walk is a stochastic process of the form $B_k = (I+A_1)\cdots(I+A_k)$ where $A_j$ are independent ``step'' matrices in $\mathrm{M}_N(\mathbb{C})$. With the right entry-covariance, a rescaled matrix random walk converges to…
The Arcsine laws of Brownian motion are a collection of results describing three different statistical quantities of one-dimensional Brownian motion: the time at which the process reaches its maximum position, the total time the process…
Let (S_n)_{n\in\N} be a Z-valued random walk with increments from the domain of attraction of some \alpha-stable law and let (\xi(i))_{i\in\Z} be a sequence of iid random variables. We want to investigate U-statistics indexed by the random…
We investigate a tight binding quantum walk on a graph. Repeated stroboscopic measurements of the position of the particle yield a measured "trajectory", and a combination of classical and quantum mechanical properties for the walk are…