Yes, it's true!
Brownian motion (or, less mathematically, a drunkard's walk) can actually be used to solve a Dirichelet equation, a difficult PDE.

The Dirichelet equation is

**Δ**f =

where

**Δ** is the

Laplacian operator

**d**^{2}/

**d**`x`_{1}^{2} + ... +

**d**^{2}/

**d**`x`_{n}^{2}
in

`n` dimensions.

So, given boundary conditions (that is, desired values of f along some hypersurface surrounding a region of **R**^{n}), we want to find an f with those values on the boundary which satisfies **Δf**=0 *inside* the region. It is known that the solution to such a problem is *unique*; thus, it doesn't matter what technique we use to solve it.

It turns out that the following procedure actually solves Dirichelet's equation, producing a value f(`z`) for any `z` inside the region.

- Start Brownian motion B(
`t`)+`z` from `z`.
- Almost surely the motion will hit the boundary; let
`s` be the *first* time at which B(`s`)+`z` is at the boundary. Then the (known) boundary value Z=f(B(`s`)+`z`) is a random variable.
*DEFINE* f(`z`)=**E**Z.

Along the boundary,

`s`=0, so clearly this f agrees with the desired boundary values. Inside the region, a moment's consideration shows that f has the following property:

*For any small sphere S about *`z` that fits inside the region, f(`z`) is the average of f on S.

It turns out that this property is characteristic of

harmonic functions, as solutions of Dirichelet's equation are known! So our construction of f is indeed a

solution. And since the solution is unique, f is

*the* solution of the PDE.

More advanced techniques of stochastic analysis let us solve more complex PDEs involving a **Δ** term.