Fundamentals of Finslerian Diffusion with Applications

The erratic motion of pollen grains and other tiny particles suspended in liquid is known as Brownian motion, after its discoverer, Robert Brown, a botanist who worked in 1828, in London. He turned over the problem of why this motion occurred to physicists who were investigating kinetic theory and thermodynamics; at a time when the existence of molecules had yet to be established. In 1900, Henri Poincare lectured on this topic to the 1900 International Congress of Physicists, in Paris [Wic95]. At this time, Louis Bachelier, a thesis student of Poincare, made a monumental breakthrough with his Theory of Stock Market Fluctuations, which is still studied today, [Co064]. Norbert Wiener (1923), who was first to formulate a rigorous concept of the Brownian path, is most often cited by mathematicians as the father of the subject, while physicists will cite A. Einstein (1905) and M. Smoluchowski. Both considered Markov diffusions and realized that Brownian behaviour nd could be formulated in terms of parabolic 2 order linear p. d. e. 'so Further­ more, from this perspective, the covariance of changes in position could be allowed to depend on the position itself, according to the invariant form of the diffusion introduced by Kolmogorov in 1937, [KoI37]. Thus, any time­ homogeneous Markov diffusion could be written in terms of the Laplacian, intrinsically given by the symbol (covariance) of the p. d. e. , plus a drift vec­ tor. The theory was further advanced in 1949, when K.

Table of Contents:
1 Finsler Spaces.- 1.1 The Tangent and Cotangent Bundle.- 1.2 Fiber Bundles.- 1.3 Frame Bundles and Linear Connections.- 1.4 Tensor Fields.- 1.5 Linear Connections.- 1.6 Torsion and Curvature of a Linear Connection.- 1.7 Parallelism.- 1.8 The Levi-Cività Connection on a Riemannian Manifold.- 1.9 Geodesics, Stability and the Orthonormal Frame Bundle.- 1.10 Finsler Space and Metric.- 1.11 Finsler Tensor Fields.- 1.12 Nonlinear Connections.- 1.13 Affine Connections on the Finsler Bundle.- 1.14 Finsler Connections.- 1.15 Torsions and Curvatures of a Finsler Connection.- 1.16 Metrical Finsler Connections. The Cartan Connection.- 2 Introduction to Stochastic Calculus on Manifolds.- 2.1 Preliminaries.- 2.2 Itô’s Stochastic Integral.- 2.3 Ito Processes. Itô Formula.- 2.4 Stratonovich Integrals.- 2.5 Stochastic Differential Equations on Manifolds.- 3 Stochastic Development on Finsler Spaces.- 3.1 Riemannian Stochastic Development.- 3.2 Rolling Finsler Manifolds Along Smooth Curves and Diffusions.- 3.3 Finslerian Stochastic Development.- 3.4 Radial Behaviour.- 4 Volterra-Hamilton Systems of Finsler.- 4.1 Berwald Connections and Berwald Spaces.- 4.2 Volterra-Hamilton Systems and Ecology.- 4.3 Wagnerian Geometry and Volterra-Hamilton Systems.- 4.4 Random Perturbations of Finslerian Volterra-Hamilton Systems.- 4.5 Random Perturbations of Riemannian Volterra-Hamilton Systems.- 4.6 Noise in Conformally Minkowski Systems.- 4.7 Canalization of Growth and Development with Noise.- 4.8 Noisy Systems in Chemical Ecology and Epidemiology.- 4.9 Riemannian Nonlinear Filtering.- 4.10 Conformai Signals and Geometry of Filters.- 4.11 Riemannian Filtering of Starfish Predation.- 5 Finslerian Diffusion and Curvature.- 5.1 Cartan’s Lemma in Berwald Spaces.- 5.2 Quadratic Dispersion.- 5.3Finslerian Development and Curvature.- 5.4 Finsleriam Filtering and Quadratic Dispersion.- 5.5 Entropy Production and Quadratic Dispersion.- 6 Diffusion on the Tangent and Indicatrix Bundles.- 6.1 Slit Tangent Bundle as Riemannian Manifold.- 6.2 hv-Development as Riemannian Development with Drift.- 6.3 Indicatrized Finslerian Stochastic Development.- 6.4 Indicatrized hv-Development Viewed as Riemannian.- A Diffusion and Laplacian on the Base Space.- A.1 Finslerian Isotropic Transport Process.- A.2 Central Limit Theorem.- A.3 Laplacian, Harmonic Forms and Hodge Decomposition.- B Two-Dimensional Constant Berwald Spaces.- B.1 Berwald’s Famous Theorem.- B.2 Standard Coordinate Representation.