STOCHASTIC KOMATU-LOEWNER EVOLUTIONS

The present monograph on stochastic Komatu–Loewner evolutions (SKLEs) provides the first systematic extension of the Schramm–Loewner evolution (SLE) theory from a simply connected planar domain to multiply connected domains by using the Brownian motion with darning (BMD) that has arisen in a recent study of the boundary theory of symmetric Markov processes.

This volume is presented in an accessible manner for the interested researchers and graduate students. It also brings new insights into SLEs as special cases of SKLEs. Mathematically, it can be viewed as a powerful application of stochastic analysis via BMDs to complex analysis.

Contents:

• Preface
• About the Authors
• Multiply Connected Planar Domain and Brownian Motion
• Chordal Komatu–Loewner Differential Equation and BMD
• Komatu–Loewner Evolution (KLE)
• Stochastic Komatu–Loewner Evolution (SKLE)
• KLE and Its Transformation
• Appendix
• Notes
• Bibliography
• Index

Readership: Researchers and graduate students interested in the Schramm–Loewner evolution (SLE).

Key Features:

• This book gives a systematic extension of the mathematical theory of Schramm–Loewner evolution (SLE) to multiply connected planar domains, namely, domains allowing finitely many holes
• Topics are presented in a unified and mostly self-contained manner, accessible by interested researchers and graduate students
• Brownian motion with darning (BMD) and stochastic analysis are utilized in an essential way in the study of conformal mappings and Komatu–Loewner evolutions on multiply connected planar domains