Infinitesimal Operators of Markov Processes

Abstract

In 1931 A. Kolmogorov showed [9] that a wide class of one dimensional Markov processes can be described by the differential equation \[(1)\qquad \frac{{\partial u}}{{\partial t}} + a\frac{{\partial ^2 u}}{{\partial x^2 }} + b\frac{{\partial u}}{{\partial x}}.\]Are there one-dimensional Markov processes governed by equations of the type \[ (2)\qquad \frac{{\partial u}}{{\partial t}}\mathfrak{A}u, \] where $\mathfrak{A}$ is a differential operator of an order higher than 2? This question remained unsettled until 1954–1955 when it was completely solved by W. Feller. Feller showed that every one-dimensional Markov process with a continuous path function is described by (2), where $\mathfrak{A}$ is the generalized second derivative.The purely analytical method of Feller is essentially connected with the one-dimensional character of the problem. It is very difficult to extend this method to the case of two (and more) dimensions.In this paper, a method is developed which can be applied to n dimensions as well as...