A Probability Approach to the Heat Equation

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 properties of the solutions along certain probability paths, and that the sub and super functions, introduced by Perron in his study of the Dirichlet problem for Laplace's equation, are fundamental tools. Just as the discontinuous subharmonic functions cause some difficulty in the one study, so the discontinuous subparabolic functions cause difficulty in the present one. In both cases, the discontinuous functions are, however, well behaved on certain probability paths. As would be expected, a detailed approach to these classical partial differential equations from a modern probability point of view is essentially a real variable approach. In the case of Laplace's equation, the real variable approach to the Dirichlet problem, boundary value problems, and potential theory, is well known, although much of the work is of very recent date. Since there are almost no results in this direction available in the literature of the heat equation, the logical order of development in this paper differs from that in [2]. However the methods and results are surprisingly similar, and some proofs are omitted or only sketched here, since they are too similar to those