Titlebook: Stochastic Processes and Applications; Diffusion Processes, Grigorios‘A. Pavliotis Textbook 2014 Springer Science+Business Media, LLC, part

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Diffusion Processes,ct. 2.1, we present various examples of Markov processes in discrete and continuous time. In Sect. 2.2, we give the precise definition of a Markov process and we derive the fundamental equation in the theory of Markov processes, the Chapman–Kolmogorov equation. In Sect. 2.3, we introduce the concept

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Introduction to Stochastic Differential Equations,an white noise, defined formally as the derivative of Brownian motion. In Sect. 3.1, we introduce SDEs. In Sect. 3.2, we introduce the Itô and Stratonovich stochastic integrals. In Sect. 3.3, we present the concept of a solution to an SDE. The generator, Itô’s formula, and the connection with the Fo

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Modeling with Stochastic Differential Equations,e obtain the Stratonovich stochastic equation. This is usually called the Wong–Zakai theorem. In this section, we derive the limiting Stratonovich SDE for a particular class of regularization of the white noise process using singular perturbation theory for Markov processes. In particular, we consid

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The Langevin Equation,the main properties of the corresponding Fokker–Planck equation. In Sect. 6.2 we give an elementary introduction to the theories of hypoellipticity and hypocoercivity. In Sect. 6.3, we calculate the spectrum of the generator and Fokker–Planck operators for the Langevin equation in a harmonic potenti

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Exit Problems for Diffusion Processes and Applications,he boundary of the domain. We then use this formalism to study the problem of Brownian motion in a bistable potential. Applications such as stochastic resonance and the modeling of Brownian motors are also presented. In Sect. 7.1, we motivate the techniques that we will develop in this chapter by lo

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Exit Problems for Diffusion Processes and Applications, process from a domain. We then use this formalism in Sect. 7.3 to calculate the escape rate of a Brownian particle from a potential well. The phenomenon of stochastic resonance is investigated in Sect. 7.4. Brownian motors are studied in Sect. 7.5. Bibliographical remarks and exercises can be found in Sects. 7.6 and 7.7, respectively.