Related papers: Probability Law For the Euclidean Distance Between…
Random motions on the line and on the plane with space-varying velocities are considered and analyzed in this paper. On the line we investigate symmetric and asymmetric telegraph processes with space-dependent velocities and we are able to…
We consider the Markov random flight $\bold X(t)$ in the Euclidean space $\Bbb R^m, \; m\ge 2,$ starting from the origin $\bold 0\in\Bbb R^m$ that, at Poisson-paced times, changes its direction at random according to arbitrary distribution…
We consider a random variable expressed as the Euclidean distance between an arbitrary point and a random variable uniformly distributed in a closed and bounded set of a three-dimensional Euclidean space. Four cases are considered for this…
We use direct numerical simulations to calculate the joint probability density function of the relative distance $R$ and relative radial velocity component $V_R$ for a pair of heavy inertial particles suspended in homogeneous and isotropic…
Let $p\_n$ be the probability for a planar Poisson-Voronoi cell to have exactly $n$ sides. We construct the asymptotic expansion of $\log p\_n$ up to terms that vanish as $n\to\infty$. We show that {\it two independent biased random walks}…
When we represent a network of sensors in Euclidean space by a graph, there are two distances between any two nodes that we may consider. One of them is the Euclidean distance. The other is the distance between the two nodes in the graph,…
A random flight on a plane with non-isotropic displacements at the moments of direction changes is considered. In the case of exponentially distributed flight lengths a Gaussian limit theorem is proved for the position of a particle in the…
Given any $\gamma>0$ and for $\eta=\{\eta_v\}_{v\in \mathbb Z^2}$ denoting a sample of the two-dimensional discrete Gaussian free field on $\mathbb Z^2$ pinned at the origin, we consider the random walk on~$\mathbb Z^2$ among random…
In this paper, we provide an application to the random distance-$t$ walk in finite planes and derive asymptotic formulas (as $q \to \infty$) for the probability of return to start point after $\ell$ steps based on the "vertical"…
Let K be a compact subset of ${\mathbb R}^n$. We choose at random with uniform law a point at distance $\epsilon$ of K and start a Brownian motion (BM) from this point. We study the probability that this BM hits K for the first time at a…
Consider a sequence of independent random isometries of Euclidean space with a previously fixed probability law. Apply these isometries successively to the origin and consider the sequence of random points that we obtain this way. We prove…
We are studying the motion of a random walker in two and three dimensional continuum with uniformly distributed jump-length. This is different from conventional Lavy flight. In 2D and 3D continuum, a random walker can move in any direction,…
We consider a one-dimensional Brownian motion of fixed duration $T$. Using a path-integral technique, we compute exactly the probability distribution of the difference $\tau=t_{\min}-t_{\max}$ between the time $t_{\min}$ of the global…
In this paper we study random flights in R^d with displacements possessing Dirichlet distributions of two different types and uniformly oriented. The randomization of the number of displacements has the form of a generalized Poisson process…
A random motion on the Poincar\'e half-plane is studied. A particle runs on the geodesic lines changing direction at Poisson-paced times. The hyperbolic distance is analyzed, also in the case where returns to the starting point are…
We consider the distribution of free path lengths, or the distance between consecutive bounces of random particles, in an n-dimensional rectangular box. If each particle travels a distance R, then, as R tends to infinity the free path…
A general theory is provided delivering convergence of maximal cyclically monotone mappings containing the supports of coupling measures of sequences of pairs of possibly random probability measures on Euclidean space. The theory is based…
Consider a walk in the plane made of $n$ unit steps, with directions chosen independently and uniformly at random at each step. Rayleigh's theorem asserts that the probability for such a walk to end at a distance less than 1 from its…
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…
We compute the joint distribution of relative velocities and separations of identical inertial particles suspended in randomly mixing and turbulent flows. Our results are obtained by matching asymptotic forms of the distribution. The method…