Convolution-Like Structures, Differential Operators and Diffusion Processes

ISBN-13: 9783031052958 / Angielski / Miękka / 2022

Rúben Sousa;Manuel Guerra;Semyon Yakubovich

Opis

T​his book provides an introduction to recent developments in the theory of generalized harmonic analysis and its applications. It is well known that convolutions, differential operators and diffusion processes are interconnected: the ordinary convolution commutes with the Laplacian, and the law of Brownian motion has a convolution semigroup property with respect to the ordinary convolution. Seeking to generalize this useful connection, and also motivated by its probabilistic applications, the book focuses on the following question: given a diffusion process X_t on a metric space E, can we construct a convolution-like operator * on the space of probability measures on E with respect to which the law of X_t has the *-convolution semigroup property? A detailed analysis highlights the connection between the construction of convolution-like structures and disciplines such as stochastic processes, ordinary and partial differential equations, spectral theory, special functions and integral transforms.The book will be valuable for graduate students and researchers interested in the intersections between harmonic analysis, probability theory and differential equations.

Spis treści

- 1. Introduction. - 2. Preliminaries. - 3. The Whittaker Convolution. - 4. Generalized Convolutions for Sturm-Liouville Operators. - 5. Convolution-Like Structures on Multidimensional Spaces.