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This book presents various results and techniques from the theory of stochastic processes that are useful in the study of stochastic problems in the natural sciences. The main focus is analytical methods, although numerical methods and statistical inference methodologies for studying diffusion processes are also presented. The goal is the development of techniques that are applicable to a wide variety of stochastic models that appear in physics, chemistry and other natural sciences. Applications such as stochastic resonance, Brownian motion in periodic potentials and Brownian motors are studied and the connection between diffusion processes and time-dependent statistical mechanics is elucidated.

The book contains a large number of illustrations, examples, and exercises. It will be useful for graduate-level courses on stochastic processes for students in applied mathematics, physics and engineering. Many of the topics covered in this book (reversible diffusions, convergence to equilibrium for diffusion processes, inference methods for stochastic differential equations, derivation of the generalized Langevin equation, exit time problems) cannot be easily found in textbook form and will be useful to both researchers and students interested in the applications of stochastic processes.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction to Stochastic Processes

Abstract
In this chapter, we present some basic results from the theory of stochastic processes and investigate the properties of some standard continuous-time stochastic processes. In Sect. 1.1, we give the definition of a stochastic process. In Sect. 1.2, we present some properties of stationary stochastic processes. In Sect. 1.3, we introduce Brownian motion and study some of its properties. Various examples of stochastic processes in continuous time are presented in Sect. 1.4. The Karhunen–Loève expansion, one of the most useful tools for representing stochastic processes and random fields, is presented in Sect. 1.5. Further discussion and bibliographical comments are presented in Sect. 1.6. Section 1.7 contains exercises.
Grigorios A. Pavliotis

Chapter 2. Diffusion Processes

Abstract
In this chapter, we study some of the basic properties of Markov stochastic processes, and in particular, the properties of diffusion processes. In Sect. 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 of the generator of a Markov process. In Sect. 2.4, we study ergodic Markov processes. In Sect. 2.5, we introduce diffusion processes, and we derive the forward and backward Kolmogorov equations. Discussion and bibliographical remarks are presented in Sect. 2.6, and exercises can be found in Sect. 2.7.
Grigorios A. Pavliotis

Chapter 3. Introduction to Stochastic Differential Equations

Abstract
In this chapter, we study diffusion processes at the level of paths. In particular, we study stochastic differential equations (SDEs) driven by Gaussian 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 Fokker–Planck equation are covered in Sect. 3.4. Examples of SDEs are presented in Sect. 3.5. The Lamperti transformation and Girsanov’s theorem are discussed briefly in Sect. 3.6. Linear SDEs are studied in Sect. 3.7. Bibliographical remarks and exercises can be found in Sects. 3.8 and 3.9, respectively.
Grigorios A. Pavliotis

Chapter 4. The Fokker–Planck Equation

Abstract
In Chap. 2, we derived the backward and forward (Fokker–Planck) Kolmogorov equations. The Fokker–Planck equation enables us to calculate the transition probability density, which we can use to calculate the expectation value of observables of a diffusion process. In this chapter, we study various properties of this equation such as existence and uniqueness of solutions, long-time asymptotics, boundary conditions, and spectral properties of the Fokker–Planck operator. We also study in some detail various examples of diffusion processes and of the associated Fokker–Planck equation. We will restrict attention to time-homogeneous diffusion processes, for which the drift and diffusion coefficients do not depend on time.
Grigorios A. Pavliotis

Chapter 5. Modeling with Stochastic Differential Equations

Abstract
When the white noise in a stochastic differential equation is approximated by a smoother process, then in the limit as we remove the regularization, we 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 consider colored noise, which we model as a Gaussian stationary diffusion process, i.e., the Ornstein–Uhlenbeck process.
Grigorios A. Pavliotis

Chapter 6. The Langevin Equation

Abstract
In this chapter, we study the Langevin equation and the associated Fokker –Planck equation. In Sect. 6.1, we introduce the equation and study some of 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 potential. In Sect. 6.4, we study Hermite polynomial expansions of solutions to the Fokker–Planck equation. In Sect. 6.5, we study the overdamped and underdamped limits for the Langevin equation. In Sect. 6.6, we study the problem of Brownian motion in a periodic potential. Bibliographical remarks and exercises can be found in Sects. 6.7 and 6.8, respectively.
Grigorios A. Pavliotis

Chapter 7. Exit Problems for Diffusion Processes and Applications

Abstract
In this chapter, we develop techniques for calculating the statistics of the time that it takes for a diffusion process in a bounded domain to reach the 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 looking at the problem of Brownian motion in bistable potentials. In Sect. 7.2, we obtain a boundary value problem for the mean exit time of a diffusion 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.
Grigorios A. Pavliotis

Chapter 8. Derivation of the Langevin Equation

Abstract
In this chapter, we derive the Langevin equation from a simple mechanical model for a small system (which we will refer to as a Brownian particle) that is in contact with a thermal reservoir that is at thermodynamic equilibrium at time t = 0. The full dynamics, Brownian particle plus thermal reservoir, are assumed to be Hamiltonian. The derivation proceeds in three steps. First, we derive a closed stochastic integrodifferential equation for the dynamics of the Brownian particle, the generalized Langevin equation (GLE). In the second step, we approximate the GLE by a finite-dimensional Markovian equation in an extended phase space. Finally, we use singular perturbation theory for Markov processes to derive the Langevin equation, under the assumption of rapidly decorrelating noise. This derivation provides a partial justification for the use of stochastic differential equations, in particular the Langevin equation, in the modeling of physical systems.
Grigorios A. Pavliotis

Chapter 9. Linear Response Theory for Diffusion Processes

Abstract
In this chapter, we study the effect of a weak external forcing on a system at equilibrium. The forcing moves the system away from equilibrium, and we are interested in understanding the response of the system to this forcing. We study this problem for ergodic diffusion processes using perturbation theory. In particular, we develop linear response theory. The analysis of weakly perturbed systems leads to fundamental results such as the fluctuation–dissipation theorem and to the Green–Kubo formula, which enables us to calculate transport coefficients.
Grigorios A. Pavliotis

Backmatter

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