Controlled diffusion processes, a special class of Markov processes in continuous time and space, are considered in this monograph. A short review of basic aspects of ergodic theory of Markov processes is given in Chapter 1 based on measure preserving transformations in order to model notions like “time averages equal ensemble averages”. The basic theory of diffusion processes follows then in Chapter 2. Controlled diffusions in the case of observed state and non-degenerate diffusion matrix are treated in Chapter 3. Constrained and multi-objective problems followed by singularly perturbed ergodic control problems involving two time-scales are studied in Chapter 4, and switching diffusion is considered in Chapter 5. In order to treat the problem of degeneracy, in Chapter 6 an abstract framework of ergodic control of martingale problems is developed. Here, a criterion for stationary distributions is generalized to controlled martingale problems on a Polish space. Controlled diffusions with degeneracy are treated in Chapter 7. Chapter 8 studies non-degenerate diffusions under partial observation. In the Epilogue some open problems are sketched. Finally, in the Appendix some results about second order elliptic equations are given. Furthermore, many bibliographical notes are included.
Assuming good knowledge in analysis, probability theory and stochastic processes, a careful and comprehensive treatment of ergodic control of diffusion processes is provided.