This book seeks to bridge the gap between the parlance, the models, and even the notations used by physicists and those used by mathematicians when it comes to the topic of probability and stochastic processes. The opening four chapters elucidate the basic concepts of probability, including probability spaces and measures, random variables, and limit theorems. Here, the focus is mainly on models and ideas rather than the mathematical tools. The discussion of limit theorems serves as a gateway to extensive coverage of the theory of stochastic processes, including, for example, stationarity and ergodicity, Poisson and Wiener processes and their trajectories, other Markov processes, jump-diffusion processes, stochastic calculus, and stochastic differential equations. All these conceptual tools then converge in a dynamical theory of Brownian motion that compares the Einstein-Smoluchowski and Ornstein-Uhlenbeck approaches, highlighting the most important ideas that finally led to a connection between the Schrödinger equation and diffusion processes along the lines of Nelson's stochastic mechanics. A series of appendices cover particular details and calculations, and offer concise treatments of particular thought-provoking topics.
Nicola Cufaro Petroni is a theoretical physicist and Associate Professor of Probability and Mathematical Statistics at the University of Bari (Italy). He is the author of over 80 publications in international journals and on various research topics: dynamics and control of stochastic processes; stochastic mechanics; entanglement of quantum states; foundations of quantum mechanics; Lévy processes and applications to physical systems; quantitative finance and Monte Carlo simulations; option pricing with jump-diffusion processes; control of the dynamics of charged particle beams in accelerators; neural networks and their applications; and recognition and classification of acoustic signals. He has taught a variety of courses in Probability and Theoretical Physics, including Probability and Statistics, Econophysics, Probabilistic Methods in Finance, and Mathematical Methods of Physics. He currently teaches Probabilistic Methods of Physics for the Master's degree in Physics.
Part 1: Probability.- Chapter 1. Probability spaces.- Chapter 2. Distributions.- Chapter 3. Random variables.- Chapter 4. Limit theorems.- Part 2: Stochastic Processes.- Chapter 5. General notions.- Chapter 6. Heuristic de?nitions.- Chapter 7. Markovianity.- Chapter 8. An outline of stochastic calculus.- Part 3: Physical modeling.- Chapter 9. Dynamical theory of Brownian motion.- Chapter 10. Stochastic mechanics.- Part 4: Appendices .- A Consistency (Sect. 2.3.4).- B Inequalities (Sect. 3.3.2).- C Bertrand's paradox (Sect. 3.5.1).- D Lp spaces of rv's (Sect. 4.1).- E Moments and cumulants (Sect. 4.2.1).- F Binomial limit theorems (Sect. 4.3).- G Non uniform point processes (Sect 6.1.1).- H Stochastic calculus paradoxes (Sect. 6.4.2).- I Pseudo-Markovian processes (Sect. 7.1.2).- J Fractional Brownian motion (Sect. 7.1.10) .- K Ornstein-Uhlenbeck equations (Sect. 7.2.4).- L Stratonovich integral (Sect. 8.2.2).- M Stochastic bridges (Sect. 10.2).- N Kinematics of Gaussian di?usions (Sect. 10.3.1).- O Substantial operators (Sect. 10.3.3).- P Constant di?usion coe?cients (Sect. 10.4).