# CHAPTER 40

# THE FEYNMAN-KAC FORMULA

The solutions to certain second-order partial differential equations can be represented as expectations of stochastic functionals. In fact, the Feynman-Kac formula establishes a link between certain partial differential equations and stochastic processes. In this chapter, we present the Feynman-Kac formula.

# 40.1 Basic Concepts and Facts

**Definition 40.1** (Dirichlet Problem). Let *D* = (*a, b*) be an interval. A Dirichlet problem is an ordinary differential equation of the form

(40.1a)

(40.1b)

where *b*, σ, *f, a*, and ϕ are given functions, and ∂*D* is the boundary of of *D*. The function *u* is called a solution to the Dirichlet problem.

**Definition 40.2** (Cauchy-Dirichlet Problem). Let *Q* = (0, *T*) × *D*, where *D* = (*a, b*) is an interval and *T* > 0. Let ∂_{p}*Q* = ([0, *T*] × {*a, b*}) ∪ ({*T*} × *D*). A Cauchy-Dirichlet problem is a partial differential equation of the form

(40.2a)

(40.2b)

where *b*, σ, *f, a*, and ϕ are given functions. The function *u* is called a solution to the Cauchy-Dirichlet problem.

**Definition 40.3** (Cauchy Problem). Let *Q* = (0, *T*) × **R**, where *T* > 0. A Cauchy problem ...

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