Path Integrals for Stochastic Processes

An Introduction

This book provides an introductory albeit solid presentation of path integration techniques as applied to the field of stochastic processes. The subject began with the work of Wiener during the 1920's, corresponding to a sum over random trajectories, anticipating by two decades Feynman's famous work on the path integral representation of quantum mechanics. However, the true trigger for the application of these techniques within nonequilibrium statistical mechanics and stochastic processes was the work of Onsager and Machlup in the early 1950's. The last quarter of the 20th century has witnessed a growing interest in this technique and its application in several branches of research, even outside physics (for instance, in economy).

The aim of this book is to offer a brief but complete presentation of the path integral approach to stochastic processes. It could be used as an advanced textbook for graduate students and even ambitious undergraduates in physics. It describes how to apply these techniques for both Markov and non-Markov processes. The path expansion (or semiclassical approximation) is discussed and adapted to the stochastic context. Also, some examples of nonlinear transformations and some applications are discussed, as well as examples of rather unusual applications. An extensive bibliography is included. The book is detailed enough to capture the interest of the curious reader, and complete enough to provide a solid background to explore the research literature and start exploiting the learned material in real situations.

Contents:

• Stochastic Processes: A Short Tour
• The Path Integral for a Markov Stochastic Process
• Generalized Path Expansion Scheme I
• Space-Time Transformation I
• Generalized Path Expansion Scheme II
• Space-Time Transformation II
• Non-Markov Processes: Colored Noise Case
• Non-Markov Processes: Non-Gaussian Case
• Non-Markov Processes: Nonlinear Cases
• Fractional Diffusion Process
• Feynman–Kac Formula, the Influence Functional
• Other Diffusion-Like Problems
• What was Left Out

Readership: Advanced undergraduate and graduate students, researchers interested in stochastic analysis and statistical physics.
Key Features:

• Offers an introductory presentation of path integral techniques focused on the realm of stochastic processes
• Presents the application of these techniques to the analysis of non-Markov and/or non-Gaussian process, as well as fractional motions discussed only in specialized articles, presented in a clear and didactic way
• Most useful to become acquainted with these stochastic techniques for its application in real situations