Geometric Methods for Stochastic Dynamical Systems: A Concise Introduction

Abstract

There is growing interest in stochastic dynamics in the applied mathematics, interdis-ciplinary science, and artificial intelligence communities. This book is written primarilyfor applied mathematicians and scientists who may not have the necessary backgroundto go directly to advanced reference books or research literature in geometric methodsfor stochastic dynamics. Our goal is to present a few selected geometric methods for understanding solutionsof stochastic differential equations. Some necessary background is reviewed, while moreadvanced materials are referred to relevant literature. It is our hope that this book willprovide a guide to the reader in accessing advanced monographs and research literaturein stochastic dynamics. We have tried to strike a balance between mathematical precisionand accessibility for the readers of this book. For example, some proofs are presented,whereas some are outlined and others are directed to references. Some definitions arepresented in separate paragraphs starting with Definition, but others are introduced lessformally as they occur in the body of the text. This book may be used as a textbook or a reference for advanced doctoral students, postdoctoral fellows, researchers and practitioners in applied mathematics, machinelearning, artificial intelligence, engineering and applied science.