4 - Spectral Representation of Diffusion Processes

Summary

This chapter is devoted to the spectral analysis of one-dimensional diffusion processes, which are the most basic and important continuous-time Markov processes where now the state space is a continuous interval contained in the real line. Diffusion processes are characterized by an infinitesimal operator which is a second-order differential operator with drift and diffusion coefficients. A spectral representation of the transition probability density of the process is obtained in terms of the orthogonal eigenfunctions of the corresponding infinitesimal operator, for which a Sturm–Liouville problem with certain boundary conditions will be solved. An analysis of the behavior of these boundary points will also be made. An extensive collection of examples related to special functions and orthogonal polynomials is provided, including the Brownian motion with drift and scaling, the Orstein–Uhlenbeck process, a population growth model, the Wright–Fisher model, the Jacobi diffusion model and the Bessel process, among others. Finally, the concept of quasi-stationary distributions is studied, for which the spectral representation plays an important role.