Diffusion Processes and their Connection to Partial Differential Equations of Parabolic Type

Abstract

The theory of Markovian processes and especially the theory of diffusion processes is very deeply connected with the theory of partial differential equations of parabolic type. The problem which is to be treated in the lecture originated in the theory of diffusion processes but can be reformulated and solved as a special problem from the theory of partial differential equations of parabolic type. For better understanding of both the problem and its modification it is necessary to begin with some facts from the theory of diffusion processes. It is also necessary to show some aspects of the relations between the two theories. For this reason the first part of the lecture is devoted to this matter. However, the necessary definitions and statements can be mentioned only very briefly and apart from its historical background.