On a Class of Exponential-type Operators and their Limit Semigroups

The paper is mainly focused upon the study of a class of second order degenerate elliptic operators on unbounded intervals. The starting points were some preceding papers where we showed, among other things that the iterates of Sz\'{a}sz-Mirakjan operators, Baskakov operators and Post-Widder operators converge to C_0-semigroups of positive operators acting on suitable weighted spaces of continuous functions on [0,+\infty[. The generators of these semigroups are showed to be the differential operators A_{i}(u) = ½ p_{i}u’’ defined on suitable domains, where p_{1}(x) = x, p_{2}(x) = x(1+x) and p_{3}(x)=x^{2}. The three above mentioned approximation processes fall within a more general class of positive operators, referred to as exponential-type operators, which are generated by an analytic function p in C([0,+\infty[) which is strictly positive on ]0,+\infty[. So, we are naturally led to investigate whether, also in this more general situation, the differential operator Au = ½ p u’’, defined on a suitable domain, generates a C_{0}- semigroup of positive operators and whether the semigroup can be represented as a limit of iterates of the exponential-type operators corresponding to p. We prove that, under suitable assumptions on the growth at infinity of p and its derivatives, the above problem has a positive answer. In addition we show that the semigroup is the transition semigroup of a continuous Markov process on [0,+\infty]. In the particular case p(x) = x^2, starting from the stochastic differential equation associated with A, we also find an integral representation of the semigroup and we determine its asymptotic behaviour on bounded continuous functions.