Orthogonal Polynomials in the Spectral Analysis of Markov Processes
Birth-Death Models and Diffusion
Encyclopedia of Mathematics and its Applications Series

In pioneering work in the 1950s, S. Karlin and J. McGregor showed that probabilistic aspects of certain Markov processes can be studied by analyzing orthogonal eigenfunctions of associated operators. In the decades since, many authors have extended and deepened this surprising connection between orthogonal polynomials and stochastic processes. This book gives a comprehensive analysis of the spectral representation of the most important one-dimensional Markov processes, namely discrete-time birth-death chains, birth-death processes and diffusion processes. It brings together the main results from the extensive literature on the topic with detailed examples and applications. Also featuring an introduction to the basic theory of orthogonal polynomials and a selection of exercises at the end of each chapter, it is suitable for graduate students with a solid background in stochastic processes as well as researchers in orthogonal polynomials and special functions who want to learn about applications of their work to probability.

1. Orthogonal polynomials; 2. Spectral representation of discrete-time birth-death chains; 3. Spectral representation of birth-death processes; 4. Spectral representation of diffusion processes; References; Index.

Manuel Domínguez de la Iglesia is Professor of Mathematics at the Instituto de Matemáticas of the Universidad Nacional Autónoma de México.