# CHAPTER 39

# DIFFUSION

A diffusion or a diffusion process is a solution to a stochastic differential equation. In this chapter, we will present some results about diffusion, in particular, Itô diffusion.

# 39.1 Basic Concepts and Facts

**Definition 39.1** (Compact Set). A set *K* ⊆ **R** is called a *compact set* if every open cover of *K* contains a finite subcover; that is, if {*V*_{i}}_{i I} is a collection of open sets such that

then there exists a finite subcollection {*V*_{ij}}_{j=1,2,…,n} such that

**Definition 39.2** (Closure). Let *E* ⊆ **R**. The closure of the set *E*, written as , is the smallest closed set in **R** that contains *E*.

**Definition 39.3** (Support). The support of a function *f* on **R** is the closure of the set

**Definition 39.4** (*C*^{1,2}-Function). A function *f* on **R** × **R** is called a *C*^{1,2}-*function* if *f*(*t, x*) is continuously differentiable on *t* (i.e., is continuous) and twice continuously differentiable on *x* (i.e., is continuous). ...

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