Stochastic Models, Information Theory, and Lie Groups, Volume 1

This chapter is an overview of the sorts of problems that can be addressed using the methods from this book. It also discusses the major differences between mathematical modeling and mathematics, and reviews some basic terminology that is used throughout the book. The appendix provides a much more in-depth review of engineering mathematics. This book is meant to be self-contained in the sense that only prior knowledge of college-level calculus, linear algebra, and differential equations is assumed. Therefore, if it is read sequentially and something does not make sense, then the appendix most likely contains the missing piece of knowledge. Standard references on classical mathematics used in engineering and physics include [2, 5], which also can be consulted to fill in any missing background.

Even after consulting the appendix and the cited references, some of the concepts presented toward the end of each chapter may be difficult to grasp on the first reading. That is okay. To a large extent, it should be possible to skip over some of the more difficult concepts in any given chapter, and still understand the fundamental ideas in subsequent chapters. In order to focus the reader on the most important ideas in each chapter, the equations that are necessary to successfully navigate through later chapters are circumscribed with a box. This also makes it easier to refer back to key equations.

In this chapter the Gaussian distribution is defined and its properties are explored. The chapter starts with the definition of a Gaussian distribution on the real line. In the process of exploring the properties of the Gaussian on the line, the Fourier transform and heat equation are introduced, and their relationship to the Gaussian is developed. The Gaussian distribution in multiple dimensions is defined, as are clipped and folded versions of this distribution. Some concepts from probability and statistics such as mean, variance, marginalization, and conditioning of probability densities are introduced in a concrete way using the Gaussian as the primary example. The properties of the Gaussian distribution are fundamental to understanding the concept of white noise, which is the driving process for all of the stochastic processes studied in this book. The main things to take away from this chapter are: To become familiar with the Gaussian distribution and its properties, and to be comfortable in performing integrals involving multi-dimensional Gaussians; To become acquainted with the concepts of mean, covariance, and informationtheoretic entropy; To understand how to marginalize and convolve probability densities, to compute conditional densities, and to fold and clip Gaussians; To observe that there is a relationship between Gaussian distributions and the heat equation; To know where to begin if presented with a diffusion equation, the symmetries of which are desired.

This chapter serves as an introduction to concepts from elementary probability theory and information theory in the concrete context of the real line and multi-dimensional Euclidean space. The probabilistic concepts of mean, variance, expected value, marginalization, conditioning, and conditional expectation are reviewed. In this part of the presentation there is some overlap with the previous chapter, which has some pedagogical benefit. There will be no mention of Borel measurability, σ-algebras, filtrations, or martingales, as these are treated in numerous other books on probability theory and stochastic processes such as [1, 14, 15, 32, 27, 48]. The presentation here, while drawing from these excellent works, will be restricted only to those topics that are required either in the mathematical and computational modeling of stochastic physical systems, or the determination of properties of solutions to the equations in these models. Basic concepts of information theory are addressed such as measures of distance, or “divergence,” between probability density functions, and the properties of “information” and entropy. All pdfs treated here will be differentiable functions on Rⁿ. Therefore the entropy and information measures addressed in this chapter are those that are referred to in the literature as the “differential” or “continuous” version.

The chapter begins with Section 4.1 in which motivational examples of random walks and stochastic phenomena in nature are presented. In Section 4.2 the concept of random processes is introduced in a more precise way. In Section 4.3 the concept of a Gaussian and Markov random process is developed. In Section 4.4 the important special case of white noise is defined. White noise is the driving force for all of the stochastic processes studied in this book. Other sections in this chapter define Itô and Stratonovich stochastic differential equations (SDEs), their properties and corresponding Fokker–Planck equations, which describe how probability densities associated with SDEs evolve over time. In particular, Section 4.7 examines the Fokker–Planck equation for a particular kind of SDE called an Ornstein–Uhlenbeck process. And Section 4.8 examines how SDEs and Fokker–Planck equations change their appearance when different coordinate systems are used. The main points that the reader should take away from this chapter are: Whereas a deterministic system of ordinary differential equations that satisfies certain conditions (i.e., the Lipschitz conditions) are guaranteed to have a unique solution for any given initial conditions, when random noise is introduced the resulting “stochastic differential equation” will not produce repeatable solutions. It is the ensemble behavior of the sample paths obtained from numerically solving a stochastic differential equation many times that is important. This ensemble behavior can be described either as a stochastic integral (of which there are two main types, called Itˆo and Stratonovich), or by using a partial differential equation akin to the diffusion equations studied in Chapter 2, which is called the Fokker–Planck (or forward Kolmogorov) equation. Two different forms of the Fokker–Planck equation exist, corresponding to the interpretation of the solution of a given SDE as being either an Itˆo or Stratonovich integral, and an analytical apparatus exists for converting between these forms. Multi-dimensional SDEs in Rn can be written in Cartesian or curvilinear coordinates, but care must be taken when converting between coordinate systems because the usual rules of multivariable calculus do not apply in some situations.

This chapter consists of a variety of topics in geometry. The approach to geometry that is taken in this chapter and throughout this book is one in which the objects of interest are described as being embedded¹ in Euclidean space. There are two natural ways to describe such embedded objects: (1) parametrically and (2) implicitly. The vector-valued functions x = x(t) and x = x(u, v) are respectively parametric descriptions of curves and surfaces when \(X \varepsilon\mathbb{R}^3\). For example, \(x(\Psi ) = [\cos \Psi , \sin \Psi , 0]^T\) for ψ ∈ [0, 2π) is a parametric description of a unit circle in \(\mathbb{R}^3\), and \(x(\phi ,\theta ) = [\cos \phi \sin {\rm \theta },\sin \phi \sin \theta {\rm ,}\cos {\rm \theta }]^T\) for φ ∈ [0, 2π) and θ ∈ [0, π] is a parametric description of a unit sphere in \(\mathbb{R}^3\). Parametric descriptions are not unique. For example, \(x(t) = [{\rm 2t/(1 + t}^{\rm 2} {\rm ), (1 } - {\rm t}^{\rm 2} {\rm )/(1 + t}^{\rm 2} {\rm ), 0]}^{\rm T}\) for \(t \varepsilon\mathbb{R}\) describes the same unit circle as the one mentioned above.²

This chapter introduces differential forms, exterior differentiation, and multi-vectors in a concrete and explicit way by restricting the discussion to ℝⁿ. This is extended to more general settings later. Roughly speaking, differential forms generalize and unify the concepts of the contour integral, curl, element of surface area, divergence, and volume element that are used in statements of Stokes’ theorem and the divergence theorem. At first it may seem unnecessary to learn yet another new mathematical construction. The trouble is that without an appropriate extension of the concept of the cross product, it is difficult and messy to try to extend the theorems of vector calculus to higher dimensions, and to non-Euclidean spaces. As was illustrated in Chapter 1 in the context of heat and fluid flow problems, these theorems play a central role. Likewise, in probability flow problems involving stochastic differential equations and their associated Fokker–Planck equations, these theorems play a role in assessing how much probability density flows past a given surface. Since the problems of interest (such as the stochastic cart in Figure 1.1) will involve stochastic flows on Lie groups, understanding how to extend Stokes’ theorem and the divergence theorem to these generalized settings will be useful. The first step in achieving this goal is to understand differential forms in ℝⁿ.

This chapter extends the review of geometrical ideas from the previous chapters to include geometrical objects in higher dimensions. These include hyper-surfaces and “ghyper-polyhedra” (or polytopes) in ℝⁿ. A parametric description of an m-dimensional embedded manifold¹ in an n-dimensional Euclidean space is of the form x = x(q) where x ε ℝⁿ and q ε ℝ^m with m ≤ n. If m = n-1, then this is called a hyper-surface. An implicit description of an m-dimensional embedded manifold in ℝⁿ is a system of constraint equations of the form φ_i(x) = 0 for i = 1,..., n-m. In the context of engineering applications, the two most important differences between the study of two-dimensional surfaces in ℝ³ and m-dimensional embedded manifolds in ℝⁿ are: (1) there is no crossproduct operation for ℝⁿ; and (2) if m ≪ n, it can be more convenient to leave behind Rn and describe the manifold intrinsically. For these reasons, modern mathematical concepts such as differential forms and coordinate-free differential geometry can be quite powerful.

This chapter extends the discussion of stochastic differential equations and Fokker–Planck equations on Euclidean space initiated in Chapter 4 to the case of processes that evolve on a Riemannian manifold. The manifold either can be embedded in ℝⁿ or can be an abstract manifold with Riemannian metric defined in coordinates. Section 8.1 formulates SDEs and Fokker–Planck equations in a coordinate patch. Section 8.2 formulates SDEs for an implicitly defined embedded manifold using Cartesian coordinates in the ambient space. Section 8.3 focuses on Stratonovich SDEs on manifolds. The subtleties involved in the conversion between Itô and Stratonovich formulations are explained. Section 8.4 explores entropy inequalities on manifolds. In Section 8.5 the following examples are used to illustrate the general methodology: (1) Brownian motion on the sphere and (2) the stochastic kinematic cart described in Chapter 1. Section 8.6 discusses methods for solving Fokker–Planck equations on manifolds. Exercises involving numerical implementations are provided at the end of the chapter. The main points to take away from this chapter are: SDEs and Fokker–Planck equations can be formulated for stochastic processes in any coordinate patch of a manifold in a way that is very similar to the case of Rn; Stochastic processes on embedded manifolds can also be formulated extrinsically, i.e., using an implicit description of the manifold as a system of constraint equations; In some cases Fokker–Planck equations can be solved using separation of variables; Practical examples of this theory include Brownian motion on the sphere and the kinematic cart with noise.

This volume presented the fundamentals of probability, parts of information theory, differential geometry, and stochastic processes at a level that is connected with physical modeling. The emphasis has been on reporting results that can be readily implemented as simple computer programs, though detailed numerical analysis has not been addressed. In this way it is hoped that a potentially useful language for describing physical problems from various engineering and scientific fields has been made accessible to a wider audience. Not only the terminology and concepts, but also the results of the theorems presented serve the goal of efficient physical description. In this context, efficiency means that the essence of any stochastic phenomenon drawn from a broad set of such phenomena can be captured with relatively simple equations in few variables. And these equations can be solved either analytically or numerically in a way that requires minimal calculations (either by human or computer). This goal is somewhat different than that of most books on stochastic processes. A common goal in other books is to train students of mathematics to learn how to prove theorems. While the ability to prove a theorem is at the center of a pure mathematician’s skill set, the results that are spun off during that process sometimes need reinterpretation and restatement in less precise (but more accessible) language in order to be used by practitioners. In other words, rather than stating results in the classical definition–theorem–proof style aimed at pure mathematicians, this book is intended for mathematical modelers including engineers, computational biologists, physical scientists, numerical analysts, and applied and computational mathematicians.