Jan 25, 2017
This book I’m reading, Lawler’s Random Walk and the Heat Equation, describes a really nice way to solve the Dirichlet problem for the heat equation - in other words, given a fixed temperature at each boundary point of the relevant domain, a way to solve for the equilibrium temperature distribution in the interior.
The idea is very simple: for each point x in the interior, set a particle undergoing Brownian motion there. If we then ask which boundary point the particle hits first, we get a probability distribution over boundary points. Then the equilibrium temperature at x is the expected value of the temperature of the first boundary point the particle hits.
