1,720,987 research outputs found
Random Processes in Hyperbolic Spaces. Hyperbolic Brownian Motion and Processes with Finite Velocity in the Hyperbolic Plane.
No full textNodal area distribution for arithmetic random waves
Full text linkWe obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on (three-dimensional ``arithmetic random waves"). We prove that, as the multiplicity of the eigenspace goes to infinity, the nodal area converges to a universal, non-Gaussian distribution. Universality follows from the equidistribution of lattice points on the sphere. Our arguments rely on the Wiener chaos expansion of the nodal area: we show that, analogous to the two-dimensional case addressed by Marinucci et al., [Geom. Funct. Anal. 26 (2016), pp. 926-960] the fluctuations are dominated by the fourth-order chaotic component. The proof builds upon recent results from Benatar and Maffiucci [Int. Math. Res. Not. IMRN (to appear)] that establish an upper bound for the number of nondegenerate correlations of lattice points on the sphere. We finally discuss higher-dimensional extensions of our result
HITTING SPHERES ON HYPERBOLIC SPACES
Full text linkFor a hyperbolic Brownian motion on the Poincare half-plane H-2, starting from a point z = (eta, alpha) inside a hyperbolic disc U of radius (eta) over bar, we obtain the probability of hitting the boundary partial derivative U at the point ((eta) over bar, (alpha) over bar). For (eta) over bar -> infinity we derive the asymptotic Cauchy hitting distribution on partial derivative H-2. In particular, it follows that the hyperbolic Brownian motion starting at (x, y) is an element of H-2 "hits" the boundary of H-2 at a point which is Cauchy distributed with scale parameter y' = y/(x(2) + y(2)) and position parameter x' = x/(x(2) + y(2)). For small values of eta and (eta) over bar. we obtain the classical Euclidean Poisson kernel. The exit probabilities from a hyperbolic annulus in H-2 of radii eta(1) and eta(2) are derived and the transient behavior of hyperbolic Brownian motion is considered. Similar probabilities are calculated also for a Brownian motion on the surface of the three-dimensional sphere. For the hyperbolic half-space H-n we obtain, with a proof based on the method of separation of variables, the Poisson kernel of a ball. For small domains in H-n we obtain the n-dimensional Euclidean Poisson kernel. The exit probabilities from an annulus are derived also in the n-dimensional case
Travelling randomly in the Poincaré half-plane with a Pythagorean compass
Full text linkA random motion on the Poincaré 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 admitted. The main results concern the mean hyperbolic distance (and also the conditional mean distance) in all versions of the motion envisaged. Also an analogous motion on orthogonal circles of the sphere is examined and the evolution of the mean distance from the starting point is investigated
Joint distribution of the process and its sojourn time on the positive half-line for pseudo-processes governed by high-order heat equation.
Full text linkConsider the high-order heat-type equation ∂u/∂t = ±∂Nu/∂xN for an integer N > 2 and introduce the related Markov pseudo-process (X(t))t≥0. In this paper, we study the sojourn time T(t) in the interval [0, +∞) up to a fixed time t for this pseudo-process. We provide explicit expressions for the joint distribution of the couple (T(t),X(t))
Cascades of particles moving at finite velocity in hyperbolic spaces
No full textA branching process of particles moving at finite velocity over the geodesic lines of the hyperbolic space (Poincaré half-plane and Poincaré disk) is examined. Each particle can split into two particles only once at Poisson spaced times and deviates orthogonally when splitted. At time t , after N (t ) Poisson events, there are N (t ) + 1 particles moving along different geodesic lines. We are able to obtain the exact expression of the mean hyperbolic distance of the center of mass of the cloud of particles. We derive such mean hyperbolic distance from two different and independent ways and we study the behavior of the relevant expression as t increases and for different values of the parameters c (hyperbolic velocity of motion) and λ (rate of reproduction). The mean hyperbolic distance of each moving particle is also examined and a useful representation, as the distance of a randomly stopped particle moving over the main geodesic line, is presented
On the most visited sites of planar Brownian motion
Full text linkLet (Bt: t ≥ 0) be a planar Brownian motion and define a family of gauge functions φα(s) = log(1/s) -α for α > 0. If α < 1 we show that almost surely there exists a point x in the plane such that H φα ({t ≥ 0: Bt = x}) > 0, but if α > 1 almost surely H φα ({t ≥ 0: Bt = x}) = 0 simultaneously for all x ∈ R 2. This resolves a longstanding open problem posed by S. J. Taylor in 1986
Two Point Function for Critical Points of a Random Plane Wave
Full text linkRandom plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemannian manifolds. This is known to be true on average. In the present paper we discuss one of important geometric observable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius
ANGULAR PROCESSES RELATED TO CAUCHY RANDOM WALKS
Full text linkWe study the angular process related to random walks in the Euclidean and in the non-Euclidean space where steps are Cauchy distributed. This leads to different types of nonlinear transformations of Cauchy random variables which preserve the Cauchy density. We give the explicit form of these distributions for all combinations of the scale and the location parameters. Continued fractions involving Cauchy random variables are analyzed. It is shown that the n-stage random variables are still Cauchy distributed with parameters related to Fibonacci numbers. This permits us to show the convergence in distribution of the sequence to the golden ratio
No repulsion between critical points for planar Gaussian random fields
Full text linkWe study the behaviour of the point process of critical points of isotropic stationary Gaussian fields. We compute the main term in the asymptotic expansion of the two-point correlation function near the diagonal. Our main result implies that for a ‘generic’ field the critical points neither repel nor attract each other. Our analysis also allows to study how the short-range behaviour of critical points depends on their index
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