Overview

This is a text on stochastic Finslerian geometry.The theory is rigorously presented and several applications in ecology, evolution and epidemiology are described. Amongst the various topics covered are the role of curvature in Finslerian diffusions, Nelson's stochastic mechanics, non-linear (Finslerian) filtering and entropy production. Two appendices deal with, respectively, the stochastic Hodge theory of Finslerian harmonic forms, and the theory of 2-dimensional Finsler spaces. The latter plays an important role in the applications described in the text. This volume should be of interest to probabilists, applied mathematicians, mathematical biologists and geometers. It can also be recommended as a supplementary graduate text.

Full Product Details

Author:   P.L. Antonelli ,  T.J. Zastawniak
Publisher:   Springer
Imprint:   Springer
Edition:   1999 ed.
Volume:   101
Dimensions:   Width: 15.50cm , Height: 1.40cm , Length: 23.50cm
Weight:   0.500kg
ISBN:  

9780792355113

ISBN 10:   0792355113
Pages:   205
Publication Date:   31 December 1998
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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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.

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