The General Diffusion Operator and Positivity Preserving Semi-Groups in One Dimension

The main purpose of this paper is to give an abstract definition of a diffusion process and to derive the most general form of a diffusion operator in one dimension. The latter is a generalized differential operator which includes, as a special case, the operator appearing on the right of the classical diffusion equation (1.1) (also when the coefficients vanish or are singular). It is noteworthy that we get our operator essentially from the sole requirement that the resolvent be positivity-preserving. Thus the diffusion equation (1.1) is now characterized as the only differential equation of the form ut = Au which preserves positivity. We begin (Section 2) by defining processes, that is, semi-groups of transformations of positive measures. ft is shown that, and why, only such semigroups qualify which are adjoints to semi-groups on continuous functions. This fact is of fundamental importance for probability theory (and, in various disguises, for many applications of semi-groups to analysis). Section 4 contains a discussion of this and of the relation of our approach to the more established theory of semi-groups based on the strong topology. When our transformations are measure preserving (that is, when probability measures are transformed into probability measures) then our processes reduce to stochastic processes of Markovian character. However, in the general case, probabilities are replaced by arbitrary measures. These may be interpreted as variable expectations; the physicist would speak of diffusion with creation or destruction of masses. In Section 3 we define diffusion processes by the property that their infinitesimal generator is of local character. A Markov process is a diffusion process if, and only if, the path functions are continuous. This shows that our definition refers to a natural class of processes.2 Our problem now reduces to finding the most general form of an infinitesimal generator A of local character of a semi-group { Tt } which transforms non-negative continuous functions into non-negative continuous functions. (No restrictions on the norm 1f Tt 11 are imposed.) It is clear that our problem is not affected by topological transformations of the x-axis or, what amounts to the same, by the choice of the scale. However,