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
Δf = 0
where
Δ is the
Laplacian operator
d2/
dx12 + ... +
d2/
dxn2
in
n dimensions.
So, given boundary conditions (that is, desired values of f along some hypersurface surrounding a region of Rn), 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)=EZ.
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.