The Infinitesimal Operators of a Class of Markov Processes

The basic purpose of the paper is to obtain a general form of infinitesimal operators for a class of Markov processes in $n$-dimensional space. It is shown that if transition probabilities satisfy certain rather general conditions, then an infinitesimal operator of the process is a natural generalization of the operator which was considered for the one-dimensional case by K. Ito in [6]. At the same time K. Ito’s demand of “stochastic differentiability” of the process is replaced by other conditions which, as we see it, can be verified more simply.
All results in the paper are formulated in terms of transition probabilities; the connection of these results with properties of path functions is not discussed.
In the first part of the paper the form of the infinitesimal operator is ascertained for the one-dimensional case. Then, certain consequences which follow from it are obtained. In particular, the conditions which must be imposed on transition probabilities for the process to be of special form are ascertained. Finally, the case of an $n$-dimensional process is discussed.