Random Walk, Laplace Equation, Poisson Equation, Brownian Motion, Maxwell Boltzmann.
1944 Kakutani proved that the solution of Laplace equation could be expressed in terms of Brownian motion. The solution of the corresponding Laplace φ(x) is the expected value of φ at the first exit point from U of the Brownian motion starting at x ∈ U. Furthermore, For a large class of