A Probability Approach to the Heat Equation
J. L. Doob · Transactions of the American Mathematical Society · 1955
DOOB 0. Introduction.Let u be a function whose domain is some open set in a Euclidean space.Then, under various conditions on u, there is a well defined first boundary value (Dirichlet) problem, involving an analysis of the boundary points of the domain, and a more or less closely associated problem of the existence of boundary limits of u.The evaluation of u in terms of the boundary function thus obtained leads back to the Dirichlet problem.These problems have been studied most intensively for harmonic functions, and lead to problems in potential theory and to the theory of subharmonic and superharmonic functions.To the probabilist, the most natural way to study these problems is by means of stochastic processes of diffusion type.This means that the probabilist lumps together parabolic and elliptic partial differential equations.At first this may seem unnatural to the analyst, since the general Dirichlet problem is not ordinarily studied for parabolic equations, but we shall see that, in fact, the probability approach makes the setting, and solution, of the Dirichlet problem just as natural for parabolic as for elliptic equations.In a previous paper, [2], an elliptic equation, Laplace's equation, was studied from a probability point of view.In the present paper we study a parabolic equation, the heat equation, from this point of view.Just as in the case of Laplace's equation, we find that the key questions are tied up with the