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 to one dimension. It is based on considering the probability properties of path functions. We prove by this method that every continuous Markov process in a space of arbitrary dimension is governed by (2), where $\mathcal{A}$ is a generalized elliptical differential operator of the second order. Our method can be applied not only to continuous Markov processes, but also to any Markov processes with path functions continuous on the right. In Feller’s papers it is assumed that the semi-group associated with the Markov process under consideration takes continuous functions into continuous ones. The class of processes specified by this condition is very important. We call them Feller processes. However, our method can be applied to a wider class of processes which we call strong Markov processes. By the conventional definition of a Markov process it is required that the development of the process after the time t does not depend on its previous history if the position at the time t is known. We say that a process is a strong Markov process if this condition is fulfilled not only for any constant t, but also for a random t which “does not depend on the future”. All Feller processes are strong Markov processes, however, the converse is not true. In the introductory § 1 a brief exposition of basic facts about Markov and strong Markov processes and associated semi-groups of operators is given. In this we use the results of [5] and In § 2 the main method of calculating infinitesimal operators corresponding to Markov processes is developed. Feller processes are treated in § 3. The operator $\mathfrak{A}$ which is constructed in this section, plays the same role in the general case as the differential operator in the right-hand side of (1) for diffusion processes, or the matrix of the densities of transition probabilities for processes with denumerably many states. In order to determine a Markov process, some lateral conditions (boundary conditions in the case of diffusion processes, probability distribution at the moment of condensation of jumps in the case of a process with denumerably many states) are needed in addition to the operator 2. In terms of semi-groups this means that generally the infinitesimal operator A of the Markov process is a contraction of the operator $\mathfrak{A}$ (see Theorem 4). However, Theorem 5 shows that $A = \mathfrak{A}$ if the state space is compact. In § 4 Feller processes with continuous path functions are treated. It is shown that in this case the operator $\mathfrak{A}$ is a natural generalization of an elliptical differential operator of the second order. A more detailed description of one-dimensional continuous Feller processes is given, and it is proved that the operator $\mathfrak{A}$ in this case is identical to Feller’s generalized second derivative. At the same time the probability meaning of the two functions, which appear in the definition of this derivative, comes to light. In § 5 some discontinuous processes are considered. Some results contained in this paper have been published in [7] without detailed proofs.