Analysis on Gaussian Spaces

The following sections are included:

• Dedication
• Preface
• Contents

Analysis of functions on finite dimensional Euclidean space ℝ^d with Lebesgue measure dx is one of the most important mathematical subjects. Researchers have been working to extend it to infinite dimension. In the absence of measure, there have been a great amount of works on nonlinear functional analysis, see for example, [Zeidler (1986, 1990a,b, 1985, 1988)].

Given a function f (of one, finite, or infinitely many variables), the Garsia- Rodemich-Rumsey inequality can be used to bound the increment |f(t) − f(s)| or rectangular increment of f by an integral with respect to the modulus of continuity of f. This type of inequality is powerful in establishing the sample path continuity and HÖlder continuity of a stochastic process. It is also useful in dealing with the uniform convergence in the absence of martingale structure. The first section concerns with the distance |f(t)−f(s)|. The second section deals with volume-metric like increment (rectangular increment) of the form |f(s₂, t₂) − f(s₁, t₂) − f(s₂, t₁) + f(s₁, t₂)|. The third section contains a simple application.

The following sections are included:

• Gaussian Measure in ℝ^d
• Some Inequalities Related to Gaussian Measure
• Brunn-Minkowski Inequality
• Hermite Polynomials
• Spectral Gap and Logarithmic Sobolev Inequalities
• Variance Identity and Inequality
• Correlation Inequality
• Hypercontractivity
• Hermite Polynomials in Physics and Hermite Functions
• Segal-Bargmann Space and Complex Hermite Polynomials
• Segal-Bargmann Transform

The following sections are included:

• Random Variables in Banach Space
• Abstract Wiener Space
• Canonical Wiener Space
• Right Tail Estimate
• Small Ball (Left Tail) Estimate

The following sections are included:

• Fock Space and Chaos Expansion
• Polarization
• Multiple Wiener-Itô Integrals
• Multiple Stratonovich Integrals
• Right Tail Estimate for Homogeneous Chaos
• Chaos Expansion of Exit Time and Skeleton of Wiener Functional

The following sections are included:

• Gross-Sobolev Derivatives
• Divergence Operator
• Regularity of Density of Wiener Functional
• Girsanov Transformation: Finite Dimension
• Girsanov-Ramer Theorem in Abstract Wiener Space
• Wick Product
• Wick Renormalization
• (Noncommutative) Composition of Wiener Functional
• Stop Brownian Motion at Anticipative Exit Time

The following sections are included:

• Complex Hypercontractivity
• Meyer's Inequality
• Multiplier Theorem
• Littlewood-Paley-Stein-Meyer Theory
• Meyer's Inequalities Revisited
• Interpolation Inequality
• Grothendieck Inequality

The following sections are included:

• General Nonlinear Wiener Functional
• Weak Convergence
• Representation of the Derivatives of the Density
• Random Variables in the q-th Wiener Chaos
• Uniform Estimation of Difference of Derivatives of Densities
• Density Convergence of Higher Rank Hermite Polynomials

The following sections are included:

• Local Time of Brownian Motion
• Chaos Expansion of Self-intersection Local Time
• Exponential Integrability
• Renormalization When d ≥ 3
• L²-Modulus of Continuity of Local Time of Brownian Motion

There are many papers and books on stochastic differential equations (sde). In this and next sections we are also concerned with stochastic differential equations. We view the solution of an sde as a nonlinear functional on the canonical Wiener space. We shall be concerned with existence and uniqueness, non-explosion, exponential integrability, some inequality associated with the marginal distribution, chaos expansion, and existence of density of the solution. In next chapter, we shall concentrate on the approximation of the solution (by time discretization scheme or by Wong-Zakai approximation).

In this section, we discuss two types of approximation of stochastic differential equations: time discretization schemes andWong-Zakai approximation. For time discretization schemes there are two types of convergence: strong convergence 𝔼 | − X_t|^p or weak convergence |𝔼 − 𝔼(X_t)|. If a rate is given, the number of terms to be included in the numerical scheme is different depending on that the rate is strong or weak. We shall present the proofs for both strong and weak convergence theorems for quite general schemes.

The following sections are included:

• Appendix A Appendix
  • Some Elementary Results from Analysis
  • Martingales
• Bibliography
• Index