This book is a comprehensive exploration of the interplay between Stochastic Analysis, Geometry, and Partial Differential Equations (PDEs). It aims to investigate the influence of geometry on diffusions induced by underlying structures, such as Riemannian or sub-Riemannian geometries, and examine the implications for solving problems in PDEs, mathematical finance, and related fields. The book aims to unify the relationships between PDEs, nonholonomic geometry, and stochastic processes, focusing on a specific condition shared by these areas known as the bracket-generating condition or Hörmander's condition. The main objectives of the book are:
1. To unify the relationship between PDEs, nonholonomic geometry, and stochastic processes by examining the common condition imposed on vector fields in both fields.
2. To explore diffusions induced by underlying geometry, whether Riemannian or sub-Riemannian, and study how curvature affects the diffusion of Brownian movement along curves.
3. To compute heat kernels and fundamental solutions for various operators, using stochastic methods, and analyze their properties.
4. To investigate the dynamics of elliptic and sub-elliptic diffusions on different geometric structures and their applications.
5. To explore the connections between sub-elliptic differential systems and sub-Riemannian geometry.
6. To analyze the dynamics of LC-circuits using variational approaches and establish their relationship with stochastic analysis and geometric analysis.
The intended audience for this book includes researchers and practitioners in mathematics, physics, and engineering, who are interested in stochastic techniques applied to geometry and PDEs, as well as their applications in mathematical finance and electrical circuits.
Author(s): Ovidiu Calin
Edition: 1
Publisher: World Scientific
Year: 2023
Language: English
Pages: 556
Tags: Stochastic Calculus, Brownian Motion, Stochastic Geometry, Hypoelliptic Operators, Heat Kernels, (Sub-)Elliptic Diffusion, LC-Circuits
Contents
Preface
List of Notations and Symbols
1 Topics of Stochastic Calculus
1.1 Prerequisites of Stochastic Calculus
1.2 Exponentially Stopped Brownian Motion
1.3 The operator T
1.4 Uniformly Stopped Brownian Motion
1.5 Erlang Stopped Brownian Motion
1.6 Inversion Formula for T
1.7 Cameron-Martin’s Formula
1.8 Interpretation of Cameron-Martin’s Formula
1.9 The Law of BAt
1.10 Ornstein-Uhlenbeck Process
1.11 Lévy’s Area
1.12 Characteristic Function for the Lévi Area
1.13 Asymmetric Area Process
1.14 Integrated Geometric Brownian Motion
1.15 Lamperti-Type Properties
1.16 Reflected Brownian Motion as Bessel Process
1.17 The Heat Semigroup
1.18 The Linear Flow
1.19 The Stochastic Flow
1.20 Summary
1.21 Exercises
2 Stochastic Geometry in Euclidean Space
2.1 Brownian Motion on a Line
2.2 Directional Brownian Motion
2.3 Rotations
2.4 Reflections in a Line
2.5 Projections
2.6 Distance from a Brownian Motion to a Line
2.7 Central Projection
2.8 Distance Between Brownian Motions in Rn
2.9 Euclidean Invariance of Brownian Motion
2.10 Some Hyperbolic Integrals
2.11 The Product of 2 Brownian Motions
2.12 Inner Product of Brownian Motions
2.13 The Process Yt = W2t + B2t + 2cWtBt
2.14 The Length of a Median
2.15 The Length of a Triangle Side
2.16 The Brownian Chord
2.17 Brownian Triangles
2.17.1 Area of a Brownian Triangle
2.17.2 Sum of Squares of Sides
2.18 Brownian Motion on Curves
2.18.1 Brownian Motion on the unit circle
2.18.2 Brownian Motion on an ellipse
2.18.3 Brownian Motion on a hyperbola
2.18.4 Scaled Brownian Motion
2.18.5 Brownian Motion on Plane Curves
2.19 Extrinsic Theory of Curves
2.20 Dynamic Brownian Motion on a Curve
2.21 Summary
2.22 Exercises
3 Hypoelliptic Operators
3.1 Elliptic Operators
3.2 Sum of Squares Laplacian
3.3 Hypoelliptic Operators
3.4 Hörmander’s Theorem
3.5 Non Bracket-generating Vector Fields
3.6 Some Applications of Hörmander’s Theorem
3.7 Elements of Sub-Riemannian Geometry
3.8 Examples of Distributions
3.9 Examples of Nonholonomic Systems
3.10 Summary
3.11 Exercises
4 Heat Kernels with Applications
4.1 Kolmogorov Operators
4.2 Options on a Geometric Moving Average
4.3 The Grushin Operator
4.4 Financial Interpretation of Grushin Diffusion
4.5 The Generalized Grushin Operator
4.5.1 The Case of One Missing Direction
4.5.2 The Case of Two Missing Directions
4.5.3 The Case of m Missing Directions
4.6 The Exponential-Grushin Operator
4.7 The Exponential-Kolmogorov Operator
4.8 Options on Arithmetic Moving Average
4.9 The Heisenberg Group
4.10 The Heat Kernel for Heisenberg Laplacian
4.11 The Heat Kernel for Casimir Operator
4.12 The Nonsymmetric Heisenberg Group
4.13 The Saddle Laplacian
4.14 Operators with Potential
4.14.1 The Cumulative Expectation
4.14.2 The Linear Potential
4.14.3 The Quadratic Potential
4.14.4 The Inverse Quadratic Potential
4.14.5 The Combo Potential
4.14.6 The Exponential Potential
4.14.7 Other Operators
4.14.8 Brownian Bridge
4.15 Deduction of Formulas
4.16 Summary
4.17 Exercises
5 Fundamental Solutions
5.1 Definition and Construction
5.2 The Euclidean Laplacian
5.3 A Fundamental Solution for ΔH
5.4 Exponential Grushin Operator
5.5 Bessel Process with Drift in R3
5.6 The Heisenberg Laplacian
5.7 The Grushin Laplacian
5.8 The Saddle Laplacian
5.9 Green Functions and the Resolvent
5.10 Summary
5.11 Exercises
6 Elliptic Diffusions
6.1 Differential Structure Induced by a Diffusion
6.1.1 The Diffusion Metric
6.1.2 The Length of a Curve
6.1.3 Geodesics and Their Role
6.1.4 Killing Vector Fields
6.1.5 The Dispersion Induced n-form
6.2 Brownian Motion on Manifolds
6.2.1 Gradient Vector Fields
6.2.2 Divergence of Vector Fields
6.2.3 The Laplace-Beltrami Operator
6.2.4 Brownian Motions on (M, g)
6.3 One-dimensional Diffusions
6.4 Multidimensional Diffusions
6.4.1 n-Dimensional Hyperbolic Diffusion
6.4.2 Two-Dimensional Hyperbolic Diffusion
6.4.3 Financial Interpretation
6.4.4 Hyperbolic Space Form
6.4.5 The Sphere Sn
6.4.6 Conformally Flat Manifolds
6.4.7 Brownian Motions on Surfaces
6.4.8 Diffusions and Brownian Motions
6.5 Stokes’ Formula for Brownian Motions
6.5.1 Escape Probability for Brownian Motions
6.5.2 Entrance Probability of a Brownian Motion
6.5.3 Stokes’ Formula for Ito Diffusions
6.5.4 The Divergence Theorem
6.5.5 The Riemannian Action
6.5.6 The Eiconal Equation
6.5.7 Computing the Flux
6.6 Bessel Processes on Riemannian Manifolds
6.7 Radial Processes on Model Manifolds
6.8 Potential Induced Diffusions
6.9 Bessel Process with Drift in R3
6.10 The Norm of n Processes
6.11 Central Projection on Sn
6.12 The Skew-product Representation
6.13 The h-Laplacian
6.14 Stochastically Complete Manifolds
6.14.1 Geodesic Completeness
6.14.2 Stochastic Completeness
6.14.3 The Cauchy Problem
6.14.4 The Volume Test
6.15 Maximum Integral Principles
6.15.1 The Boltzmann Distribution
6.15.2 Action Weighted Integrals
6.16 The Role of the Exponential Map
6.17 Brownian Motion in Geodesic Coordinates
6.18 Summary
6.19 Exercises
7 Sub-Elliptic Diffusions
7.1 Heisenberg Diffusion
7.2 The Joint Distribution of (Rt, St)
7.3 Koranyi Process
7.4 Grushin Diffusion
7.4.1 The Two-Dimensional Case
7.4.2 The n-Dimensional Case
7.5 Martinet Diffusion
7.6 A Step 3 Grushin Diffusion
7.7 The Engel Diffusion
7.8 General Sub-elliptic Diffusions
7.9 Nonholonomic Diffusions
7.10 A Sub-elliptic Diffusion
7.11 The Knife Edge Diffusion
7.12 Nonholonomic Hyperbolic Diffusion
7.13 The Rolling Coin Diffusion
7.14 Summary
7.15 Exercises
8 Systems of Sub-Elliptic Differential Equations
8.1 Poincaré’s Lemma
8.2 Definition of Sub-elliptic Systems
8.3 Nonholonomic systems of equations
8.4 Uniqueness and Smoothness
8.5 Solutions Existence
8.5.1 The Heisenberg Distribution
8.5.2 Nilpotent Distributions
8.5.3 Heisenberg-type Distributions
8.6 The Engel Distribution
8.7 Grushin-Type Distributions
8.8 Summary
8.9 Exercises
9 Applications to LC-Circuits
9.1 The LC-circuit
9.2 The Lagrangian Formalism
9.3 The Hamiltonian Formalism
9.4 The Hamilton-Jacobi Equation
9.5 Complex Lagrangian Mechanics
9.6 A Stochastic Approach
9.7 Summary
9.8 Exercises
Epilogue
Bibliography
Index
