Intuition for using Brownian motion to solve Laplace and heat equations

Question

I'm a mathematics student with zero physical intuition. In my course, we learned that Brownian motion can be used to construct the solutions to certain PDEs, including Laplace's equation and the heat equation.

For Laplace's equation, to find the solution $f(x)$ in a bounded domain $U$ with given boundary conditions on $\partial U$, you can start a Brownian motion at $x$ and take the expectation over the exit distribution (that is, the value of the boundary data at the point the BM first hits $\partial U$).

For the heat equation and similar PDE, you can use the Feynman-Kac formula.

Both these equations have physical interpretations, as the equation for a steady-state and the equation for heat. Further, you can think of Brownian motion as diffusion. In light of this (and the fact that Feynman was a physicist...) it seems there should be some physical, heuristic reasons why these solution methods work. The proofs I learned are not particularly enlightening.

What is this physical intuition for these solution formulas?

Answer 1

This is a hard question to answer, because “intuition” is subjective.

However, if you think about the definition of Brownian Motion (and its implications) and for example the Laplace Equation, you should be able to see a general correspondence.

For Brownian Motion, the state at $(t+1)$ only depends on the state at time $t$. From this you know that the variance ($2^{\mathrm{nd}}$ moment) is zero.

But that's also exactly what the Laplace Equation states (for discrete systems).

In fact, this correspondence is a specific example of a larger class of stochastic processes (Weiner process) that can be used to solve a variety of PDE's (Ito calculus).