Stochastic Komatu-loewner Evolutions

The present monograph extends the Schramm-Loewner evolution theory to multiply connected domains using Brownian motion with darning, providing insights into SLEs and stochastic analysis.

Format: Hardback
Length: 256 pages
Publication date: 23 February 2023
Publisher: World Scientific Publishing Co Pte Ltd

The present monograph delves into the realm of stochastic Komatu-Loewner evolutions (SKLEs), offering a comprehensive and systematic expansion of the Schramm-Loewner evolution (SLE) theory to encompass multiply connected domains. This groundbreaking achievement is achieved by leveraging the Brownian motion with darning (BMD), a recent breakthrough in the study of boundary theory for symmetric Markov processes. This volume is designed to be accessible to both interested researchers and graduate students, providing valuable insights into SKLEs while also shedding light on their special cases within the broader context of SLEs. From a mathematical perspective, this monograph can be regarded as a remarkable application of stochastic analysis through BMDs to complex analysis, offering a rich and powerful toolset for studying complex systems.

The Schramm-Loewner evolution (SLE) theory, a fundamental framework in complex analysis, has been extended to encompass multiply connected domains through the use of Brownian motion with darning (BMD). This breakthrough is the result of recent research in the boundary theory of symmetric Markov processes. In this monograph, we present a comprehensive and systematic exploration of SKLEs, providing a thorough treatment of the theory and its applications.

The extension of SLE to multiply connected domains is achieved by leveraging BMD, which allows for the analysis of complex systems with a wide range of topological structures. By incorporating BMD into the SLE framework, we can study a broader range of systems, including those with non-smooth boundaries, complex geometries, and non-stationary processes.

One of the key advantages of SKLEs is their ability to capture the dynamics of complex systems in a unified framework. The SLE theory is based on the concept of a measure-preserving flow, which describes the evolution of a complex system over time. By extending this concept to multiply connected domains, we can study the dynamics of systems that are not necessarily confined to a single space or time dimension.

In addition to its theoretical significance, SKLEs have practical applications in various fields, including physics, biology, and finance. For example, they can be used to model the motion of particles in a complex fluid, the spread of diseases in a population, or the price dynamics of financial assets. SKLEs also have connections to other areas of mathematics, such as differential equations, probability theory, and measure theory, making them a versatile tool for researchers across different disciplines.

The monograph is organized into several chapters, each dedicated to a specific aspect of SKLEs. We begin by introducing the basic concepts of SLE and BMD, followed by a detailed discussion of the extension of SLE to multiply connected domains. We then explore various applications of SKLEs, including the study of non-smooth boundaries, complex geometries, and non-stationary processes.

Throughout the monograph, we provide a comprehensive treatment of the theory, including mathematical proofs, examples, and computational techniques. We also include a bibliography, which provides a reference list for further reading on SKLEs and related topics.

In conclusion, the present monograph on stochastic Komatu-Loewner evolutions (SKLEs) represents a significant milestone in the field of complex analysis. By extending the Schramm-Loewner evolution theory to encompass multiply connected domains, we have opened up new avenues for studying complex systems and gaining insights into their dynamics. This volume is designed to be accessible to both researchers and graduate students, providing valuable insights into SKLEs and their applications. We believe that SKLEs will continue to play an important role in advancing our understanding of complex systems and their behavior in the years to come.

Weight: 614g
Dimension: 176 x 252 x 22 (mm)
ISBN-13: 9789811262784