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Feynman path integrals, suggested heuristically by Feynman in the 40s, have become the basis of much of contemporary physics, from non-relativistic quantum mechanics to quantum fields, including gauge fields, gravitation, cosmology. Recently ideas based on Feynman path integrals have also played an important role in areas of mathematics like low-dimensional topology and differential geometry, algebraic geometry, infinite-dimensional analysis and geometry, and number theory. The 2nd edition of LNM 523 is based on the two first authors' mathematical approach of this theory presented in its 1st edition in 1976. To take care of the many developments since then, an entire new chapter on the current forefront of research has been added. Except for this new chapter and the correction of a few misprints, the basic material and presentation of the first edition has been maintained. At the end of each chapter the reader will also find notes with further bibliographical information.

Feynman integrals. --- Feynman, Intégrales de --- Feynman integrals --- Calculus --- Mathematical Theory --- Atomic Physics --- Physics --- Mathematics --- Physical Sciences & Mathematics --- Mathematics. --- Functional analysis. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Integral equations. --- Measure theory. --- Operator theory. --- Probabilities. --- Integral Equations. --- Measure and Integration. --- Functional Analysis. --- Operator Theory. --- Probability Theory and Stochastic Processes. --- Global Analysis and Analysis on Manifolds. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Functional analysis --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Equations, Integral --- Functional equations --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Functional calculus --- Calculus of variations --- Integral equations --- Math --- Science --- Feynman diagrams --- Multiple integrals --- Distribution (Probability theory. --- Global analysis. --- Global analysis (Mathematics) --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities

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Differential geometry. Global analysis --- Operator theory --- Functional analysis --- Operational research. Game theory --- Mathematical physics --- Computer. Automation --- computergestuurd meten --- differentiaalvergelijkingen --- analyse (wiskunde) --- differentiaal geometrie --- stochastische analyse --- functies (wiskunde) --- kansrekening

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Feynman path integrals, suggested heuristically by Feynman in the 40s, have become the basis of much of contemporary physics, from non-relativistic quantum mechanics to quantum fields, including gauge fields, gravitation, cosmology. Recently ideas based on Feynman path integrals have also played an important role in areas of mathematics like low-dimensional topology and differential geometry, algebraic geometry, infinite-dimensional analysis and geometry, and number theory. The 2nd edition of LNM 523 is based on the two first authors' mathematical approach of this theory presented in its 1st edition in 1976. To take care of the many developments since then, an entire new chapter on the current forefront of research has been added. Except for this new chapter and the correction of a few misprints, the basic material and presentation of the first edition has been maintained. At the end of each chapter the reader will also find notes with further bibliographical information.

Differential geometry. Global analysis --- Operator theory --- Functional analysis --- Operational research. Game theory --- Mathematical physics --- Computer. Automation --- computergestuurd meten --- differentiaalvergelijkingen --- analyse (wiskunde) --- differentiaal geometrie --- stochastische analyse --- functies (wiskunde) --- kansrekening

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Sergio Albeverio gave important contributions to many fields ranging from Physics to Mathematics, while creating new research areas from their interplay. Some of them are presented in this Volume that grew out of the Random Transformations and Invariance in Stochastic Dynamics Workshop held in Verona in 2019. To understand the theory of thermo- and fluid-dynamics, statistical mechanics, quantum mechanics and quantum field theory, Albeverio and his collaborators developed stochastic theories having strong interplays with operator theory and functional analysis. His contribution to the theory of (non Gaussian)-SPDEs, the related theory of (pseudo-)differential operators, and ergodic theory had several impacts to solve problems related, among other topics, to thermo- and fluid dynamics. His scientific works in the theory of interacting particles and its extension to configuration spaces lead, e.g., to the solution of open problems in statistical mechanics and quantum field theory. Together with Raphael Hoegh Krohn he introduced the theory of infinite dimensional Dirichlet forms, which nowadays is used in many different contexts, and new methods in the theory of Feynman path integration. He did not fear to further develop different methods in Mathematics, like, e.g., the theory of non-standard analysis and p-adic numbers.

Mathematical analysis --- Operational research. Game theory --- Probability theory --- Mathematics --- Mathematical physics --- analyse (wiskunde) --- waarschijnlijkheidstheorie --- stochastische analyse --- wiskunde --- fysica --- kansrekening