Lectures on Point-processes, Random measures, and stochastic geometry

Point-Processes

These lecture notes are based on the book « Random Measures, Point Processes and Stochastic Geometry« (F. Baccelli, B. Błaszczyszyn, M.K. Karray). These materials explore random structures embedded in Euclidean or more general topological spaces. The primary focus is on random measures, point processes, and stochastic geometry, which have become crucial tools in diverse fields, including natural sciences (cosmology, ecology, cell biology), engineering (material sciences, networks), and emerging fields like data science.

These notes, derived from the book, are designed to provide a clear and accessible path from fundamental definitions and properties to the application of these mathematical objects in concrete stochastic models. The book offers a rigorous and complete development of the theory, with all proofs included and special attention paid to measurability issues. A key strength of the book, and reflected in these notes, is the inclusion of elaborate examples (« models ») drawn from various fields, translated into clear mathematical terms, bridging theory and application. The material covers both structural results on stationary random measures and stochastic geometry objects and computational results on important parametric classes of point processes (especially Poisson and Determinantal).

The structure of the book is mirrored in these notes: Part I covers Foundations, Part II focuses on the Stationary Framework, and Part III explores Stochastic Geometry. To ensure a rigorous and self-contained treatment, the book provides in the appendix essential background material.

These lecture notes assume a basic familiarity with measure and probability theory. In these notes, we aim to provide clarifications, elaborations, additional examples, and guidance on applying these powerful tools to your own research and projects. We hope you find these lecture notes helpful in your journey to master random measures, point processes, and stochastic geometry!

I. Random Measures and Point Processes

1. Foundations

This lecture offers an in-depth exploration of random measures and point processes, which are foundational concepts in the study of stochastic processes. A point process can be viewed as a random object that takes values in locally finite configurations of points or, equivalently, counting measures. We extend this concept to the more general notion of a random measure, which is a random object that takes measures as possible realizations.

We begin by establishing the necessary framework, including key structures such as mean measures, Laplace transforms, and void probabilities. This foundational section sets the stage for understanding the fundamental properties and behaviors of random measures and point processes.

Next, we delve into the characterization of distributions. This part of the lecture explores powers and moment measures, providing a detailed examination of the Laplace transform and its role in characterizing distributions. We also address the concept of independence in the context of random measures, which is crucial for understanding the interactions between different random measures.

The lecture then introduces stochastic integrals, equipping readers with the tools to integrate with respect to random measures. We also discuss the vague topology on the space of measures, which is essential for understanding convergence properties.

Finally, we provide a detailed examination of point processes. This includes an exploration of simple point processes, methods for enumerating points, and the generating function. We also cover factorial powers and factorial moment measures, which are crucial for understanding higher-order properties of point processes.

This comprehensive approach ensures a thorough understanding of both random measures and point processes, equipping readers with the foundational knowledge needed to navigate this complex field.

Contents:
1 Framework
1.1 Basic Spaces and Structures
    1.2 Mean Measure, Laplace Transform and Void Probability
    1.3 Campbell’s Averaging Formula
    1.4 Powers and Moment Measures
    1.5 Polish and l.c.s.h. Spaces
2 Distribution Characterization
    2.1 Characterization via Generating Subclass
    2.2 Independence
    2.3 Laplace Transform Characterization
3 Stochastic Integral
4 Vague Topology on ¯M(G)
5 Point Processes
    5.1 Point Process vs. Random Measure
    5.2 Generating Function
    5.3 Simple Point Processes
    5.4 Enumeration of Points
    5.5 Factorial Powers and Factorial Moment Measures

2. Basic Models and Operations

In this comprehensive lecture, we delve into the essential concepts and operations related to random measures and point processes. We start with an in-depth examination of Poisson point processes, a fundamental element in stochastic processes, focusing on their Laplace transforms and characterizations.

  1. Following this, we explore various operations on random measures and point processes. Key operations such as superposition, thinning, and marking are discussed in detail, providing you with the tools necessary to manipulate and analyze these stochastic models effectively.

  2. We then move on to the construction of new models, introducing Cox, Gibbs, and cluster point processes. These models offer a versatile framework for representing and studying intricate random phenomena, enhancing your ability to model real-world situations.

  3. The lecture also covers shot-noise processes, emphasizing their Laplace transforms, second-order moments, and U-statistics. These concepts are particularly relevant for applications in fields like telecommunications and finance.

  4. Finally, we address sigma-finite random measures, which broaden the scope of random measures beyond local finiteness. This discussion highlights the increased flexibility and applicability of sigma-finite measures in various stochastic modeling scenarios.

Contents:
1 Poisson Point Processes
    1.1 Laplace Transform
    1.2 Characterizations
2 Operations on Random Measures and Point Processes
    2.1 Superposition
    2.2 Thinning of Points
    2.3 Image of a Random Measure
    2.4 Independent Displacement of Points
    2.5 Independent Marking of Points
    2.6 Marked Random Measures
    2.7 Mixtures
3 Constructing New Models
    3.1 Cox Point Processes
    3.2 Gibbs Point Processes
    3.3 Cluster Point Processes
    3.4 Powers and Factorial Powers
4 Shot-Noise
    4.1 Laplace Transform
    4.2 Second Order Moments
    4.3 U-Statistics
5 Sigma-Finite Random Measures

3. Palm Theory

In this lecture we delve into the intricate and essential aspects of Palm theory, a cornerstone in the study of random measures and point processes. This lecture provides a comprehensive examination of Palm distributions, which are crucial for understanding the conditional properties of point processes given the presence of a point at a specific location.

  1. We begin with an in-depth analysis of Palm distributions, including the reduced Palm distribution, mixed Palm version, and local Palm probabilities. These foundational concepts are pivotal for characterizing the behavior and structure of point processes.
  2. The lecture then explores Palm distributions for specific models, such as Poisson point processes, Cox point processes, Gibbs point processes, and marked random measures. Each model is discussed in detail to illustrate the practical applications and theoretical implications of Palm theory in various contexts.
  3. Finally, we address higher-order Palm and reduced Palm distributions, extending the basic concepts to interactions involving multiple points.

Contents:
1 Palm Distributions
    1.1 Reduced Palm Distribution
    1.2 Mixed Palm Version
    1.3 Local Palm Probabilities
2 Palm Distributions for Particular Models
    2.1 Palm for Poisson Point Processes
    2.2 Palm for Cox Point Processes
    2.3 Palm Distribution of Gibbs Point Processes
    2.4 Palm Distribution for Marked Random Measures
3 Higher Order Palm and Reduced Palm
    3.1 Higher Order Palm
    3.2 Higher Order Reduced Palm

4. Transforms and Moment Measures

This lecture provides a rigorous introduction to transforms and moment measures as essential tools for characterizing random measures and point processes.

We begin with a detailed examination of characteristic functions and their associated cumulant and factorial cumulant measures, establishing the foundational concepts needed for subsequent analysis.

We then develop a range of both finite and infinite series expansions for the characteristic, Laplace, and generating functions, equipping attendees with a robust analytical framework.

Finally, we delve into the factorial moment expansion, highlighting its versatility through applications to marked point processes and shot-noise functions. This exploration reveals the intricate connections between the probabilistic nature of point processes and their analytical representations.

Contents:

  1. Characteristic Function
    1. Cumulant Measures
    2. Factorial Cumulant Measures
  2. Finite Series Transform Expansions
    1. Characteristic Function Expansion
    2. Laplace Transform Expansion
    3. Generating Function Expansion
  3. Infinite Series Transform Expansions
    1. Void Probability Expansion
    2. Symmetric Enumeration of Atoms of Finite Point Processes
    3. Janossy Measures
    4. Moment versus Janossy Measures
    5. Janossy versus Moment Measures
    6. Distribution of a Finite Point Process
    7. Order Statistics on R
  4. Factorial Moment Expansion
    1. Point Processes on R
    2. General Marked Point Processes
      1. Measurable Order
      2. Telescoping Formula
      3. Factorial Moment Expansion for Marked Point Processes
      4. Expansion Kernels
      5. Factorial Moment Expansion over Kernels
    3. Shot-Noise Functions

5. Determinantal and Permanental Point Processes

Access the detailed lecture notes in PDF format: Determinantal and Permanental Point Processes Lecture Notes
.

This lecture offers a comprehensive exploration of Determinantal and Permanental Point Processes (DPPs/PPPs), powerful tools for modeling spatial data with complex dependencies. We explore key aspects of these processes in a structured manner:

  1. Determinantal Point Process Basics: A precise formulation of DPPs, including their defining kernels and background measures, along with fundamental properties, such as
    thinning, simplicity, and restrictions, with illustrative examples including the Poisson process.
  2. Kernel Indistinguishability and Uniqueness: The concept of μ-indistinguishable kernels and the conditions that ensure a DPP/PPP is uniquely determined by its kernel and background measure.
  3. Generating Function and Laplace Transform: Derivation of expressions for the generating function and Laplace transform, essential for analyzing distributional properties.
  4. Existence with Regular Kernels: Definition of regular kernels and detailing general assumptions on the kernels for the existence of the determinantal point processes, and review examples of canonical point processes.
  5. α-Determinantal Point Processes: Introduction of a broader class of point processes including DPPs and PPPs and discuss conditions for their construction via superpositions.
  6. Palm Distributions: Derivation of The palm version of a Determinantal Point Processes
  7. Stationary Determinantal Point Processes: Discussion of Ginibre Determinantal Point Process & shift-invariant kernel
  8. Discrete Determinantal Point Processes: Detailed description of Characterization & Regularity.

The lecture content is organized as follows:

  • 1 Determinantal Point Process Basics
    • Definition and Basic Properties
    • Indistinguishable Kernels
    • Uniqueness of the Distribution
    • Generating Function and Laplace Transform
    • Inequalities for Moment Measures
  • 2 Existence of Determinantal Point Processes with Regular Kernels
    • Canonical Determinantal Point Processes
    • Integral Operator: Essentials
    • Canonical Version of a Kernel
    • Regular Kernels
  • 3 α-Determinantal Point Processes
    • Definition and Basic Properties
    • Uniqueness of the Distribution of an α-Determinantal Point Process
    • Generating Function and Laplace Transform
    • Permanental Point Process as Cox Point Process
    • Existence of α-Determinantal Point Processes
  • 4 Laplace Transform and Janossy Measures Revisited
    • Laplace Transform as Operator Determinant
    • Janossy Measures of α-Determinantal Point Processes; α ∈ {1/m : m ∈ N∗}
    • Janossy Measures of α-Determinantal Point Processes; α ∈ {−1/m : m ∈ N∗}
  • 5 Palm Distributions of Determinantal Point Processes
  • 6 Stationary Determinantal Point Processes on Rd
    • Ginibre Determinantal Point Process
    • Shift-invariant kernel
  • 7 Discrete Determinantal Point Processes
    • Characterization
    • Regularity
    • Janossy Measures
    • Palm Version

II. Stationary Random Measures and Point Processes

6. Palm Theory in the Stationary Framework

Access the detailed lecture notes in PDF format: Palm Theory in the Stationary Framework Lecture Notes.

This lecture focuses on Stationary Random Measures and Point Processes, emphasizing the significance of Palm Theory within the Stationary Framework. The content is structured to provide a comprehensive understanding of the foundational concepts and advanced applications of Palm probabilities.

  1. The first section introduces Palm Probabilities in the Stationary Framework, covering essential topics such as the Shift Operator, Flow, and Compatibility. We delve into the Palm Probability of a Random Measure, the Campbell-Little-Mecke-Matthes Theorem, and the Mass Transport Formula, culminating in Mecke’s Invariance Theorem.
  2. The second section presents the Palm Inversion Formula, exploring concepts like Voronoi Tessellation, the Inversion Formula, and the distinction between Typical and Zero Cells. We also discuss the Particular Case of the Line and Renewal Processes, followed by the Direct and Inverse Construction of Palm Theory.
  3. The final section examines Further Properties of Palm Probabilities, including Independence, Superposition, and Neveu’s Exchange Formula. We also cover the Alternative Version of Neveu’s Exchange Theorem, the Holroyd-Peres Representation of Palm Probability, and the Reduced Second Moment Measure.

The lecture content is organized as follows:

  1. Palm Probabilities in the Stationary Framework
    • Stationary Framework
    • Shift Operator and Stationarity
    • Flow and Compatibility
    • Palm Probability of a Random Measure
    • Campbell-Little-Mecke-Matthes Theorem
    • Mass Transport Formula
    • Mecke’s Invariance Theorem
  2. Palm Inversion Formula
    • Voronoi Tessellation
    • Inversion Formula
    • Typical versus Zero Cell
    • Particular Case of the Line
    • Renewal Processes
    • Direct and Inverse Construction of Palm Theory
    • Direct Construction: From Stationary to Palm Probability
    • Inverse Construction: From Palm to Stationary Probability
  3. Further Properties of Palm Probabilities
    • Independence
    • Superposition
    • Neveu’s Exchange Formula
    • Alternative Version of Neveu’s Exchange Theorem
    • Holroyd-Peres Representation of Palm Probability
    • Reduced Second Moment Measure

7. Marks in the Stationary Framework

Access the detailed lecture notes in PDF format: Marks in the Stationary Framework Lecture Notes.

This lecture, titled « Stationary Random Measures and Point Processes: Marks in the Stationary Framework, » explores the fascinating intersection of point processes, random measures, and Palm theory within a stationary framework. We delve into how to incorporate additional information, represented as « marks, » into these models.

  1. We begin by establishing the foundations of stationary marked random measures. This includes defining stationarity, introducing the shift operator, exploring compatibility with the flow, constructing stationary marked point processes, and considering the PASTA property in this more general setting.
  2. Next, we investigate marks that take values in general measurable spaces, a setting beyond the usual locally compact Hausdorff spaces. Key topics include examining marks generated by compatible stochastic processes, understanding the « shadowing property« , presenting a generalized Campbell-Little-Mecke-Matthes theorem applicable in this context, exploring mark-dependent thinning, and analyzing various transformations of stationary point processes based on their marks.
  3. Finally, we apply Palm theory to the specific case of stationary marked random measures. This involves analyzing the Palm distribution of the mark, exploring the Palm distributions of the marked random measures, and developing the concept of the Palm probability conditional on the mark to bridge the gap between the point process and the mark.

Table of Contents:

  • 1. Stationary Marked Random Measures
    • Stationary Marked Point Processes
    • Extension of PASTA to Rd
  • 2. Marks in a General Measurable Space
    • Selected Marks and Conditioning
    • Transformations of Stationary Point Processes Based on Marks
  • 3. Palm Theory for Stationary Marked Random Measures
    • Palm Distribution of the Mark
    • Palm Distributions of Marked Random Measures
    • Palm Probability Conditional on the Mark

8. Ergodicity

Access the detailed lecture notes in PDF format: Ergodicity Lecture Notes.

Dive into the fascinating world of Stationary Random Measures and Point Processes with this lecture on Ergodicity. This lecture delves into the theoretical foundations and practical implications of ergodicity in the context of these stochastic models. The structure of the lecture is as follows:

  1. We start by the necessary background and Motivation for studying ergodicity in the context of stationary random measures and point processes.
  2. Then we present a review of the Birkhoff’s Pointwise Ergodic Theorem, a cornerstone of ergodic theory, essential for understanding the subsequent developments.
  3. This section focuses on Ergodic Theorems for Random Measures, introducing the concept of Ergodicity of Random Measures and then demonstrating the Ergodic Theorem for Random Measures. We also cover the concept of Cross-Ergodicity.
  4. Finally, we explore the Ergodicity of Marked Random Measures, extending the ergodic framework to processes where each point carries additional information or « mark ».

Table of Contents:

  1. Motivation
  2. Birkhoff’s Pointwise Ergodic Theorem
  3. Ergodic Theorems for Random Measures
    • Ergodicity of Random Measures
    • Ergodic Theorem for Random Measures
    • Cross-Ergodicity
  4. Ergodicity of Marked Random Measures

III. Stochastic Geometry

9. Framework for Stochastic Geometry

Access the detailed lecture notes in PDF format: Framework for Stochastic Geometry – Lecture Notes.

This lecture provides a comprehensive introduction to the mathematical tools underpinning Stochastic Geometry. It begins by establishing a rigorous foundation in the space of closed sets, essential for defining and characterizing random geometric structures. The core concept of a Random Closed Set (RCS) is then introduced, focusing on its characterization through the powerful Capacity Functional. We proceed to explore Set Processes and their connection to RCSs, leading to a detailed analysis of Stationary RCSs and Set Processes. A significant portion of the lecture is devoted to understanding the fundamental characteristics of Random Closed Sets, including the capacity functional, coverage functions, and their properties. Finally, the lecture culminates in a focused discussion on the specific characteristics of Stationary Random Closed Sets such as volume fraction, reduced covariance function, and contact distribution function, providing the necessary framework for analyzing spatial patterns and dependencies in stochastic geometric models.

Table of Contents:

  1. Space of Closed Sets
  2. Random Closed Sets (RCS)
    1. The Capacity Functional
    2. Set Processes
    3. Stationary RCS and Set Process
    4. Characteristics of Random Closed Set
    5. Characteristics of Stationary Random Closed Set

10. Coverage and Germ-Grain Models

Access the detailed lecture notes in PDF format: Coverage and Germ-Grain Models – Lecture Notes.

This lecture provides a detailed exploration of coverage models and germ-grain models within the framework of stationary random measures and point processes. These models are fundamental tools for understanding irregular geometric patterns in various fields, from material and biological sciences to wireless communications. The lecture begins by formally defining coverage models as unions of random sets and establishes key properties, focusing on their capacity functionals. It then delves into germ-grain models, which are constructed from point processes marked by some random sets called grains. A thorough analysis of germ-grain constructions is presented, including stationarity conditions and derivations of mean measures. Furthermore, the lecture discusses an inverse construction, which allows for the derivation of stationary marked point processes from stationary set processes. Finally, the lecture showcases concrete examples of specialized coverage models, including hard-core coverage models (with non-overlapping grains) and shot-noise coverage models, highlighting their applications and distinct characteristics.

The lecture is structured as follows:

  1. Coverage Model
  2. Germ-Grain Model
    • Germ-Grain Construction
    • Inverse Construction
  3. Further Examples
    • Hard-Core Coverage Models
    • Shot-Noise Coverage Models

11. Line Processes and Tessellations

Access the detailed lecture notes in PDF format: Line Processes and Tessellations – Lecture Notes.

This lecture provides an introduction to line processes and tessellations, fundamental concepts in stochastic geometry. It explores the fundamental concepts and properties of these geometric structures, aiming to provide a solid foundation for further study and application. The lecture is structured as follows:

  1. Line Processes in R²: This section formally defines line processes in the two-dimensional Euclidean space. It introduces essential tools and concepts, starting with a bijective parameterization of lines using polar coordinates. It further discusses stationary line processes, characterizing the conditions under which the underlying point process is invariant under cylinder shears. Finally, it covers the construction of line measures and their properties, providing the foundation for understanding more complex models.
  2. Planar Tessellations: This section shifts focus to planar tessellations, which are partitions of the plane into cells. It explores two key examples: the Voronoi tessellation, constructed from a point process, and the Crofton tessellation, generated by a line process. The lecture investigates their properties and the relationships between the point processes associated with their vertices, edges, and cells.

This lecture offers a comprehensive overview of line processes and tessellations, bridging the gap between theoretical concepts and practical applications in various fields.

Table of Contents:

  1. Line Processes in R²
    1. Parameterization of Lines in R²
    2. Stationary Line Processes
    3. Associated Random Measures
  2. Planar Tessellations
    1. Voronoi Tessellation
    2. Crofton Tessellation

12. Complements

Access the detailed lecture notes in PDF format: Complements – Lecture Notes.

This lecture explores a key property of Poisson point processes on l.c.s.h. spaces: the Strong Markov Property. It focuses on providing a rigorous definition of stopping sets and stating the Strong Markov Property. This material is particularly relevant for those studying Poisson point processes on general spaces.

  1. Strong Markov Property of Poisson Point Process: This section provides a formal definition of stopping sets with respect to a point process. It then presents the Strong Markov Property for Poisson point processes on l.c.s.h. spaces, explaining how it extends the usual Markov Property to cases where conditioning occurs on a random stopping set. Finally, it illustrates the application of this property by analyzing the distances from a fixed point to ordered points.

Table of Contents

  1. Strong Markov Property of Poisson Point Process

IV. Appendix

13. Transforms of Random Variables


Access the complete lecture notes in PDF format: Transforms of Random Variables – Lecture Notes.

This lecture provides a comprehensive exploration of transforms used to characterize and analyze random variables and random vectors. It is structured into three key sections, each building upon the previous one to offer a robust foundation for further study in probability and stochastic processes.

  1. The lecture begins by examining random variables, diving into the theoretical aspects and practical applications of the characteristic function and generating function. Emphasis is placed on understanding how these transforms relate to the moments and cumulants of a distribution, providing powerful tools for extracting distributional information. Key concepts covered include differentiability, Taylor expansions, and uniqueness theorems associated with these transforms. The subtle interplay between ordinary moments and factorial moments is also explored.
  2. The focus then shifts to nonnegative random variables, where the Laplace transform is introduced as a central analytical technique. This section mirrors the structure of the previous one, detailing the properties of the Laplace transform, its relationship to the moments of the random variable, and an alternative cumulant definition. Understanding the Laplace transform is essential for analyzing waiting times, reliability, and other important quantities in probability and queueing theory.
  3. Finally, the lecture extends the concepts to random vectors. It begins by defining multivariate characteristic functions and generating functions and their relationship to mixed moments. The discussion extends to nonnegative random vectors and defines Multivariate Laplace Transform. Throughout this section, the intricate relationships between moments and cumulants in the multivariate setting are explored, providing a means for relating the joint behavior of multiple random variables.

Table of Contents:

  1. Random Variables
    1. Characteristic Function
    2. Generating Function
    3. Moments Versus Factorial Moments
    4. Ordinary Cumulants
    5. Factorial Cumulants
      1. Cumulants Versus Moments
  2. Nonnegative Random Variables
    1. Laplace Transform
    2. Cumulants via the Laplace Transform
  3. Random Vectors
    1. Moments from Transforms
      1. Characteristic Function
      2. Generating Function
    2. Cumulants
      1. Ordinary Cumulants
      2. Factorial cumulants
      3. Cumulants Versus Moments
    3. Nonnegative Random Vectors
      1. Laplace Transform
      2. Multivariate Cumulants via the Laplace Transform

14. Useful Results in Measure Theory

Access the complete lecture notes in PDF format: Useful Results in Measure Theory – Lecture Notes.

This lecture provides a comprehensive compendium of essential results from measure theory and probability theory, serving as a foundational resource for advanced studies. The lecture is structured as follows:

  1. Basic Results: Explores fundamental measure-theoretic and probabilistic results, including properties of negligible sets in product spaces, the inclusion-exclusion principle, and basic facts about symmetric measures.
  2. Support of a Measure: Delves into the concept of the support of a measure, characterizing the region where the measure is concentrated, and explores its properties in relation to negligible sets and product measures.
  3. Functional Monotone Class Theorems: Presents the Monotone Class Theorem for Nonnegative Functions, a powerful tool for demonstrating that a class of functions contains all nonnegative measurable functions.
  4. Mixture and Disintegration of Measures: Provides a comprehensive treatment of mixture and disintegration of measures, defining measure kernels, probability kernels, the Measure Mixture Theorem, and the Measure Disintegration Theorem.
  5. Power and Factorial Powers of Measures: Introduces power measures and factorial power measures, derived from a base measure, with a focus on counting measures and their behavior in superpositions.
  6. Equality of Measures: Presents a crucial uniqueness result establishing conditions under which two measures on a product space are guaranteed to be equal based on the equality of integrals of product functions.
  7. Symmetric Complex Gaussian Random Variables: Introduces symmetric complex Gaussian random variables and vectors, defining them, exploring their properties, Wick’s formula, and generated Hilbert subspaces.

Table of Contents:

  1. Basic Results
  2. Support of a Measure
  3. Functional Monotone Class Theorems
  4. Mixture and Disintegration of Measures
    1. Mixture of Measures
    2. Disintegration of Measures
  5. Power and Factorial Powers of Measures
  6. Equality of Measures
  7. Symmetric complex Gaussian random variables

15. Useful Results in Algebra

Access the complete lecture notes in PDF format: Useful Results in Algebra – Lecture Notes.

This lecture provides a curated collection of algebraic results essential for advanced analysis. It is structured into two main sections, focusing on matrices and power series composition.

  1. The first section, dedicated to matrices, delves into fundamental properties, determinants, and related inequalities, with a strong emphasis on complex matrices. Key topics include:
    • A comprehensive overview of important inequalities like Hadamard’s inequality, Fischer’s inequality, and Koteljanskii’s inequality for Hermitian nonnegative-definite matrices.
    • In-depth introduction to the concept of the Schur complement, including its formula, inverse, and applications.
    • Various formulas for the diagonal expansion of the determinant.
    • An introduction to the α-determinant, a generalization of the standard determinant, encompassing both the determinant and the permanent.
  2. The second section focuses on the composition of power series, establishing conditions for the validity of substituting one power series into another. It presents a crucial lemma outlining the existence and convergence of the resulting power series.

Table of Contents:

  1. Matrices
    • Inequalities
    • Schur Complement
    • Diagonal Expansion of the Determinant
    • α-Determinant of a Matrix
  2. Power Series Composition

16. Useful Results in Functional Analysis


Access the complete lecture notes in PDF format: Useful Results in Functional Analysis – Lecture Notes.

This lecture presents a compilation of useful results in Functional Analysis. It focuses particularly on integral operators and the Fredholm determinant, providing a rigorous treatment suitable for researchers and advanced students in the fields of mathematics and theoretical physics.

  1. Integral Operators: This section initiates a study of integral operators, a distinguished class of linear operators acting on L² spaces. These operators, defined via integration against a kernel, are central to various analytical frameworks, including the study of differential equations and spectral theory.
    Our approach begins with a review of the requisite properties of L² spaces, establishing the foundation for a rigorous treatment of integral operators and their characteristics. We will then explore key properties of these operators, including boundedness, compactness, and the nature of their adjoints, culminating in an analysis of integral operators with Hermitian kernels.
  2. Operator Determinant: This section introduces the Fredholm determinant, a sophisticated generalization of the determinant concept to the realm of trace class operators on separable Hilbert spaces. We will carefully develop the Fredholm determinant using the framework of antisymmetric tensor products, providing a rigorous construction rooted in functional analysis. This development culminates in Lidskii’s theorem, establishing a fundamental connection between the Fredholm determinant and the eigenvalues of the operator.

Table of Contents

  1. Integral Operators
    1. L2 Space Properties
    2. Linear Operators
    3. Integral Operator Basics
    4. Trace Class Operators
    5. Hilbert-Schmidt Operators
    6. Restriction of Integral Operators
    7. Canonical and Pre-Canonical Kernels
    8. Continuous Kernels
    9. Nonnegative-Definite Property
  2. Operator Determinant
    1. Fredholm Determinant
    2. Expansion of Integral Operator’s Determinant

Book: Point-processes, Random measures, and stochastic geometry

F. Baccelli, B. Błaszczyszyn, M.K. Karray (2020). « Random Measures, Point Processes and Stochastic Geometry. » [ HAL ]

Our Work in the Scientific Community

I’m pleased that Paul Keeler (hpaulkeeler.com) references my point process resources at mohamedkadhem.com/point-processes, particularly highlighting the Palm calculus aspects related to telecommunication network modeling using stochastic geometry. This mention appears on this page of his website. It’s great to see this work getting visibility in the community!

Acknowledgements

The authors thank Oliver Diaz-Espinosa, Mayank Manjrekar, James Murphy, Eliza O’Reilly, Pierre Bernhard, Philippe Sarotte, and Ottmar Cronie for their comments on early versions of this manuscript.

Last Updated on 4 heures by Mohamed Kadhem KARRAY

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