Laplace and the origin of the Ornstein-Uhlenbeck process

Abstract

It is shown that a second-order partial differential equation, found by Laplace in 1810, is the Fokker-Planck equation for a one-dimensional Ornstein-Uhlenbeck process. It is argued that Laplace's reasoning, although not rigorous, can be entirely justified by using the modern theory of weak convergence of stochastic processes. The solutions to the differential equation found by Laplace and others, using expansions in terms of Hermite polynomials, are discussed.