# Intuition for using Brownian motion to solve Laplace and heat equations

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?

• Can you please give a reference?
– Deep
Commented Apr 18, 2017 at 5:14
• @Deep I'm hesitant to recommend any books because I don't feel like any of them give particularly good explanations. I think Steele's Stochastic Calculus and Financial Applications gets the closest. You could also try Bass's Stochastic Processes if you want a little more rigor. Commented Apr 18, 2017 at 13:43
• Random speculation: I wonder if the averaging property of the Laplace equation could be connected to the Brownian motion.
– d_b
Commented Jul 4, 2021 at 3:29

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.