Fractional Langevin type equations for white noise distributions

In this paper, by applying the intertwining properties, we introduce the fractional powers of the number operator perturbed by generalized Gross Laplacians (infinite dimensional Laplacians), which are special types of the infinitesimal generators of generalized Mehler semigroups. By applying the intertwining properties and semigroup approach, we study the Langevin type equations associated with the infinite dimensional Laplacians and with white noise distributions as forcing terms. Then we investigate the unique solution of the fractional Langevin type equations associated with the Riemann-Liouville and Caputo time fractional derivatives, and the fractional power of the infinite dimensional Laplacians, for which we apply the intertwining properties again. For our purpose, we discuss the fractional integrals and fractional derivatives of white noise distribution valued functions.

Appearance-Based Sequential Robot Localization Using a Patchwise Approximation of a Descriptor Manifold

This paper addresses appearance-based robot localization in 2D with a sparse, lightweight map of the environment composed of descriptor–pose image pairs. Based on previous research in the field, we assume that image descriptors are samples of a low-dimensional Descriptor Manifold that is locally articulated by the camera pose. We propose a piecewise approximation of the geometry of such Descriptor Manifold through a tessellation of so-called Patches of Smooth Appearance Change (PSACs), which defines our appearance map. Upon this map, the presented robot localization method applies both a Gaussian Process Particle Filter (GPPF) to perform camera tracking and a Place Recognition (PR) technique for relocalization within the most likely PSACs according to the observed descriptor. A specific Gaussian Process (GP) is trained for each PSAC to regress a Gaussian distribution over the descriptor for any particle pose lying within that PSAC. The evaluation of the observed descriptor in this distribution gives us a likelihood, which is used as the weight for the particle. Besides, we model the impact of appearance variations on image descriptors as a white noise distribution within the GP formulation, ensuring adequate operation under lighting and scene appearance changes with respect to the conditions in which the map was constructed. A series of experiments with both real and synthetic images show that our method outperforms state-of-the-art appearance-based localization methods in terms of robustness and accuracy, with median errors below 0.3 m and 6∘.

White noise delta functions and infinite-dimensional Laplacians

In this paper, we introduce a new white noise delta function based on the Kubo–Yokoi delta function and an infinite-dimensional Brownian motion. We also give a white noise differential equation induced by the delta function through the Itô formula introducing a differential operator directed by the time derivative of the infinite-dimensional Brownian motion and an extension of the definition of the Volterra Laplacian. Moreover, we give an extension of the Itô formula for the white noise distribution of the infinite-dimensional Brownian motion.

The Hamiltonian path integrand for the charged particle in a constant magnetic field as white noise distribution

The concepts of Hamiltonian Feynman integrals in white noise analysis are used to realize as the first velocity-dependent potential of the Hamiltonian Feynman integrand for a charged particle in a constant magnetic field in coordinate space as a Hida distribution. For this purpose the velocity-dependent potential gives rise to a generalized Gauss kernel. Besides the propagators, the generating functionals are obtained.

ON IRREDUCIBILITY OF THE ENERGY REPRESENTATION OF THE GAUGE GROUP AND THE WHITE NOISE DISTRIBUTION THEORY

We consider the energy representation for the gauge group. The gauge group is the set of C∞-mappings from a compact Riemannian manifold to a semi-simple compact Lie group. In this paper, we obtain irreducibility of the energy representation of the gauge group for any dimension of M. To prove irreducibility for the energy representation, we use the fact that each operator from a space of test functionals to a space of generalized functionals is realized as a series of integral kernel operators, called the Fock expansion.