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Random acceleration is a fundamental stochastic process encountered in many applications. In the one-dimensional version of the process a particle is randomly accelerated according to the Langevin equation $\ddot{x}(t) = \sqrt{2D} \xi(t)$,…
We consider the motion of a particle along the geodesic lines of the Poincar\`e half-plane. The particle is specularly reflected when it hits randomly-distributed obstacles that are assumed to be motionless. This is the hyperbolic version…
We study closed systems of particles that are subject to stochastic forces in addition to the conservative forces. The stochastic equations of motion are set up in such a way that the energy is strictly conserved at all times. To ensure…
In the stochastic sandpile model on a graph, particles interact pairwise as follows: if two particles occupy the same vertex, they must each take an independent random walk step with some probability $0<p<1$ of not moving. These…
The cohesive collective motion (flocking, swarming) of autonomous agents is ubiquitously observed and exploited in both natural and man-made settings, thus, minimal models for its description are essential. In a model with continuous space…
We propose a geometric perspective to describe the motion of self-propelled particles moving at constant speed in d dimensions. We exploit the fact that the vector that conveys the direction of motion of the particle performs a random walk…
We analyze a class of continuous time random walks in $\mathbb R^d,d\geq 2,$ with uniformly distributed directions. The steps performed by these processes are distributed according to a generalized Dirichlet law. Given the number of changes…
A particle moves among the vertices of an $(m+1)$-gon which are labeled clockwise as $0,1,...,m$. The particle starts at 0 and thereafter at each step it moves to the adjacent vertex, going clockwise with a known probability $p$, or…
We consider a random two-phase process which we call a reset-return one. The particle starts its motion at the origin. The first, displacement, phase corresponds to a stochastic motion of a particle and is finished at a resetting event. The…
We describe all countable particle systems on $\mathbb{R}$ which have the following three properties: independence, Gaussianity and stationarity. More precisely, we consider particles on the real line starting at the points of a Poisson…
We present a simple model in dimension $d\geq 2$ for slowing particles in random media, where point particles move in straight lines among and inside spherical identical obstacles with Poisson distributed centres. When crossing an obstacle,…
A rotating continuum of particles attracted to each other by gravity may be modeled by the Euler-Poisson system. The existence of solutions is a very classical problem. Here it is proven that a curve of solutions exists, parametrized by the…
For a one-dimensional stationary system, we derive a third order equation of motion representing a first integral of the relativistic quantum Newton's law. We then integrate this equation in the constant potential case and calculate the…
We use Stokesian Dynamics simulations to study the microscopic motion of particles suspended in fluids passing through porous media. We construct model porous media with fixed spherical particles, and allow mobile ones to move through this…
In this paper we pose two fundamental ideas on the motion of an elementary particle supporting the internal "spin motion" or $\textit{Zitterbewegung}$ and a particle as concentrated energy. First, the particle moves randomly in a limited…
We investigate the one-dimensional telegraph random process in the presence of an elastic boundary at the origin. This process describes a finite-velocity random motion that alternates between two possible directions of motion (positive or…
We investigate the motion of a suspended non-Brownian sphere past a fixed cylindrical or spherical obstacle in the limit of zero Reynolds number for arbitrary particle-obstacle aspect ratios. We consider both a suspended sphere moving in a…
We introduce particle systems in one or more dimensions in which particles perform branching Brownian motion and the population size is kept constant equal to $N > 1$, through the following selection mechanism: at all times only the $N$…
Random flights in $\mathbb{R}^d,d\geq 2,$ with Dirichlet-distributed displacements and uniformly distributed orientation are analyzed. The explicit characteristic functions of the position $\underline{\bf X}_d(t),\,t>0,$ when the number of…
We consider a random trial-based telegraph process, which describes a motion on the real line with two constant velocities along opposite directions. At each epoch of the underlying counting process the new velocity is determined by the…