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This text provides a concise introduction, suitable for a one-semester special topics course, to the remarkable properties of Gaussian measures on both finite and infinite dimensional spaces. It begins with a brief resumé of probabilistic results in which Fourier analysis plays an essential role, and those results are then applied to derive a few basic facts about Gaussian measures on finite dimensional spaces. In anticipation of the analysis of Gaussian measures on infinite dimensional spaces, particular attention is given to those properties of Gaussian measures that are dimension independent, and Gaussian processes are constructed. The rest of the book is devoted to the study of Gaussian measures on Banach spaces. The perspective adopted is the one introduced by I. Segal and developed by L. Gross in which the Hilbert structure underlying the measure is emphasized. The contents of this book should be accessible to either undergraduate or graduate students who are interested in probability theory and have a solid background in Lebesgue integration theory and a familiarity with basic functional analysis. Although the focus is on Gaussian measures, the book introduces its readers to techniques and ideas that have applications in other contexts.

Geometry --- Mathematical analysis --- Operational research. Game theory --- Probability theory --- Mathematics --- analyse (wiskunde) --- waarschijnlijkheidstheorie --- stochastische analyse --- wiskunde --- kansrekening --- geometrie --- Gaussian measures. --- Mesures gaussianes

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Stochastic processes --- Mathematical statistics --- Probabiliteit--Theorie --- Probabiliteitstheorie --- Probabilité [Théorie de la ] --- Probabilités --- Statistieken --- Waarschijnlijkheid--Theorie --- Waarschijnlijkheidstheorie --- Gibbs' equation --- Perturbation (Mathematics) --- Inverse problems (Differential equations) --- Gaussian measures --- Isoperimetric inequalities --- Statistique mathématique --- Problèmes inverses (Equations différentielles) --- Probabilités --- Statistique mathématique --- Problèmes inverses (Equations différentielles) --- Statistiques --- Probabilities --- Congresses. --- Congrès --- Processus stochastiques --- Probabilités. --- Statistique mathématique. --- Probabilités - Congres --- Statistiques - Congres --- Statistique mathématique.

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Gaussian processes can be viewed as a  far-reaching infinite-dimensional extension of classical normal random variables. Their theory presents a powerful range of tools for probabilistic modelling in various academic and technical domains such as Statistics, Forecasting, Finance, Information Transmission, Machine Learning - to mention just a few. The objective of these Briefs is to present a quick and condensed treatment of the core theory that a reader must understand in order to make his own independent contributions. The primary intended readership are PhD/Masters students and researchers working in pure or applied mathematics. The first chapters introduce essentials of the classical theory of Gaussian processes and measures with the core notions of reproducing kernel, integral representation, isoperimetric property, large deviation principle. The brevity being a priority for teaching and learning purposes, certain technical details and proofs are omitted. The later chapters touch important recent issues not sufficiently reflected in the literature, such as small deviations, expansions, and quantization of processes. In university teaching, one can build a one-semester advanced course upon these Briefs.

Gaussian processes. --- Gaussian processes --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Gaussian measures. --- Isoperimetric inequalities. --- Large deviations. --- Deviations, Large --- Measures, Gaussian --- Mathematics. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Math --- Science --- Limit theorems (Probability theory) --- Statistics --- Geometry, Plane --- Inequalities (Mathematics) --- Measure theory --- Distribution (Probability theory) --- Stochastic processes --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities