A four-dimensional random motion at finite speed

We consider the random motion of a particle that moves with constant finite speed in the space ℝ 4 and, at Poisson-distributed times, changes its direction with uniform law on the unit four-sphere. For the particle's position, X (t) = (X 1(t), X 2(t), X 3(t), X 4(t)), t > 0, we obtain the explicit forms of the conditional characteristic functions and conditional distributions when the number of changes of directions is fixed. From this we derive the explicit probability law, f( x , t), x ∈ ℝ 4, t ≥ 0, of X (t). We also show that, under the Kac condition on the speed of the motion and the intensity of the switching Poisson process, the density, p( x ,t), of the absolutely continuous component of f( x ,t) tends to the transition density of the four-dimensional Brownian motion with zero drift and infinitesimal variance σ2 = ½.

Stochastic velocity motions and processes with random time

The aim of this paper is to analyze a class of random processes which models the motion of a particle on the real line with random velocity and subject to the action of friction. The speed randomly changes when a Poissonian event occurs. We study the characteristic and moment generating functions of the position reached by the particle at timet> 0. We are able to derive the explicit probability distributions in a few cases. The moments are also widely analyzed. For the random motions having an explicit density law, further interesting probabilistic interpretations emerge if we consider randomly varying time. Essentially, we consider two different types of random time, namely Bessel and gamma times, which contain, as particular cases, some important probability distributions (e.g. Gaussian, exponential). For the random processes built by means of these compositions, we derive the probability distributions for a fixed number of Poisson events. Some remarks on possible extensions to random motions in higher spaces are proposed. We focus our attention on the persistent planar random motion.

Unimodal and bimodal random motions of independent exponential steps

We consider random walks that arise from the repetition of independent, statistically identical steps, whose nature may be arbitrary. Such unimodal motions appear in a variety of contexts, including particle propagation, cell motility, swimming of micro-organisms, animal motion and foraging strategies. Building on general frameworks, we focus on the case where step duration is exponentially distributed. We explore systematically unimodal processes whose steps are ballistic, diffusive, cyclic or governed by rotational diffusion, and give the exact propagator in Fourier-Laplace domain, from which the moments and the diffusion coefficient are obtained. We also address bimodal processes, where two kinds of step are taken in turn, and show that the mean square displacement, the quantity of prime importance in experiments, is simply related to those of unimodal motions.

Denoising-enhancing images on elastic manifolds

The conflicting demands for simultaneous low-pass and high-pass processing, required in image denoising and enhancement, still present an outstanding challenge, although a great deal of progress has been made by means of adaptive diffusion-type algorithms. To further advance such processing methods and algorithms, we introduce a family of second-order (in time) partial differential equations. These equations describe the motion of a thin elastic sheet in a damping environment. They are also derived by a variational approach in the context of image processing. The new operator enables better edge preservation in denoising applications by offering an adaptive lowpass filter, which preserves high-frequency components in the pass-band better than the adaptive diffusion filter, while offering slower error propagation across edges. We explore the action of this powerful operator in the context of image processing and exploit for this purpose the wealth of knowledge accumulated in physics and mathematics about the action and behavior of this operator. The resulting methods are further generalized for color and/or texture image processing, by embedding images in multidimensional manifolds. A specific application of the proposed new approach to superresolution is outlined.

A Note on Planar Random Motion at Finite Speed

A simple derivation of the explicit form of the transition density of a planar random motion at finite speed, based on some specific properties of the wave propagation on the plane R 2, is given.