Continuous Parameter Markov Processes and Stochastic Differential Equations

Rabi Bhattacharya is Professor of Mathematics at The University of Arizona. He is a Fellow of the Institute of Mathematical Statistics and a recipient of the U.S. Senior Scientist Humboldt Award and of a Guggenheim Fellowship. He has made significant contributions to the theory and application of Markov processes, and more recently, nonparametric statistical inference on manifolds. He has served on editorial boards of many international journals and has published several research monographs and graduate texts on probability and statistics.Edward C. Waymire is Emeritus Professor of Mathematics at Oregon State University. He received a PhD in mathematics from the University of Arizona in the theory of interacting particle systems. His primary research concerns applications of probability and stochastic processes to problems of contemporary applied mathematics pertaining to various types of flows, dispersion, and random disorder. He is a formerchief editor of the Annals of Applied Probability, and past president of the Bernoulli Society for Mathematical Statistics and Probability.

“This book is rich in content and logically rigorous, making it an excellent reference for studying Markov processes and stochastic differential equations. After reading it, it can give everyone a clearer and deeper understanding of this field, which is very beneficial for those who are engaged in or interested in researching in this field.” (Jiankang Liu, zbMATH 1555.60001, 2025)

• 1. A review of Martingaels, stopping times and the Markov property.- 2. Semigroup theory and Markov processes.-3. Regularity of Markov process sample paths.- 4. Continuous parameter jump Markov processes.- 5. Processes with independent increments.- 6. The stochastic integral.- 7. Construction of difficusions as solutions of stochastic differential equations.- 8. Itô's Lemma.- 9. Cameron-Martin-Girsanov theorem.- 10. Support of nonsingular diffusions.- 11. Transience and recurrence of multidimensional diffusions.- 12. Criteria for explosion.- 13. Absorption, reflection and other transformations of Markov processes.- 14. The speed of convergence to equilibrium of discrete parameter Markov processes and Diffusions.- 15. Probabilistic representation of solutions to certain PDEs.- 16. Probabilistic solution of the classical Dirichlet problem.- 17. The functional Central Limit Theorem for ergodic Markov processes.- 18. Asymptotic stability for singular diffusions.- 19. Stochastic integrals with L2-Martingales.- 20. Local time for Brownian motion.- 21. Construction of one dimensional diffusions by Semigroups.- 22. Eigenfunction expansions of transition probabilities for one-dimensional diffusions.- 23. Special Topic: The Martingale Problem.- 24. Special topic: multiphase homogenization for transport in periodic media.- 25. Special topic: skew random walk and skew Brownian motion.- 26. Special topic: piecewise deterministic Markov processes in population biology.- A. The Hille-Yosida theorem and closed graph theorem.- References.- Related textbooks and monographs.