Ch.10 Feynman–Kac & the Dirichlet Problem

Section 1 / 8

The Feynman–Kac Formula and the Dirichlet Problem — Overview

The Feynman–Kac formula, named after the physicist and Nobel laureate Richard Feynman and the mathematician Mark Kac, establishes the announced connection between the diffusion equation and Brownian motion, or more generally between the diffusion equation and stochastic differential equations.

We motivate this connection by returning to the amoeba population from the previous chapter. As before, suppose that the population density $f(x)$ at time $0$ is known and that we want to model the population density $u(t,x)$ at time $t$ .

Instead of studying the density $u(t,x)$ directly, we focus on the behavior of a single amoeba and take a stochastic viewpoint. We make the following modelling assumptions:

The search movement of a single amoeba follows Brownian motion with diffusion coefficient $\sigma$ .
The amoebae move independently of one another and independently of their starting positions at time $0$ .

Choose an amoeba randomly from the total population at time $t$ . Let $X_t$ denote its position at time $t$ and $X_0$ its position at time $0$ . Both are random variables, and our model assumption gives

\boxed{ X_t=X_0+\sigma B_t. }

The population density can be written as

u(t,x) = \lim_{\Delta x\to0} \frac{1}{\Delta x} \Pp[x\le X_t<x+\Delta x].

Furthermore,

\begin{aligned} \Pp[x\le X_t<x+\Delta x] &= \Pp[x\le X_0+\sigma B_t<x+\Delta x]\\ &= \Pp[x+\sigma\widetilde B_t \le X_0 < x+\sigma\widetilde B_t+\Delta x], \end{aligned}

where

\widetilde B_t=-B_t

is again a Brownian motion.

The initial density condition gives

\Pp[a\le X_0<a+\Delta x] = f(a)\Delta x+o(\Delta x).

Conditioning on the Brownian motion $\widetilde B$ therefore gives

\boxed{ u(t,x) = \E\left[f(x+\sigma B_t)\right]. }

Using the density of the normal distribution, this becomes

\boxed{ u(t,x) = \int_{-\infty}^{\infty} f(x-y) \frac{1}{\sqrt{2\pi t\sigma^2}} \exp\left( -\frac{y^2}{2t\sigma^2} \right)\,dy. }

This agrees with the solution of the diffusion equation obtained in Theorem 9.1:

\boxed{ u_t(t,x) = \frac{\sigma^2}{2}u_{xx}(t,x), \qquad u(0,x)=f(x). }

Thus the stochastic approach to the amoeba population gives exactly the same result as the analytical approach when

\boxed{ \lambda=\frac{\sigma^2}{2}. }

Moreover, (10.1) shows that the solution of the diffusion equation can be written as the expectation of a functional of Brownian motion. This type of representation is called the Feynman–Kac formula.