Stochastic Equations and Differential Geometry

Frontmatter

Chapter 1. Functions and Measures in Linear Spaces

Abstract

In this chapter we introduce the main notions of differential calculus for functions of infinite-dimensional arguments defined in a region of a Banach space. Next, we consider measures in Banach spaces and develop a theory of measure differentiation along vector fields.

Ya. I. Belopolskaya, Yu. L. Dalecky

Chapter 2. Functions and Measures on Smooth Manifolds

Abstract

In this chapter we give some necessary preliminaries concerning differential geometric objects; namely, smooth manifolds, vector bundles and their connections. These notions are used afterwards to construct invariant differential operators in sections of vector bundles. The final topic of this chapter is the theory of smooth measures on manifolds.

Ya. I. Belopolskaya, Yu. L. Dalecky

Chapter 3. Stochastic Equations in Banach Spaces

Abstract

In this chapter, we set down a probability framework which will be needed in the sequel. We describe probabilistic machinery and, especially, Ito stochastic analysis, or stochastic calculus, in Banach spaces with smooth norms. We have tried to make the exposition detailed enough and adjusted to our future needs while dealing with smooth Banach manifolds.

Ya. I. Belopolskaya, Yu. L. Dalecky

Chapter 4. Stochastic Equations on Smooth Manifolds

Abstract

In this chapter, we shall give an invariant construction of stochastic differential equations on smooth manifolds equipped with linear connections and investigate their solutions. We shall pay special attention to the case in which a manifold possesses a vector bundle total space structure and a stochastic equation is compatible with this structure. Finally, we shall construct formal differential extensions of stochastic equations and prove that the solutions of the equations on the considered manifold are smooth with respect to the initial values under some assumptions.

Ya. I. Belopolskaya, Yu. L. Dalecky

Chapter 5. Kolmogorov Equations

Abstract

In this chapter, we deal with linear evolution families of operators acting on spaces of functions and measures defined on a manifold, which are generated by solutions of stochastic differential equations, studied in the previous chapter. We derive both backward and forward Kolmogorov equations and study the Cauchy problem for them.

Ya. I. Belopolskaya, Yu. L. Dalecky

Chapter 6. Diffusion Processes on Lie Groups and Principal Fibre Bundles

Abstract

Let X be a smooth Banach manifold and G a group of smooth transformations of X . Sometimes we shall denote such a pair by (X, G). In this chapter, we shall deal with stochastic processes on X compatible in a certain sense with G-actions. Particular attention will be paid to the case X = G and X = P, where P is a principal fibre bundle over a certain manifold Y with G the structural group of p : P → Y. The most interesting in those two cases are equations with invariant (under actions of the group G) coefficients.

Ya. I. Belopolskaya, Yu. L. Dalecky

Backmatter