On a class of elliptic-parabolic equations on unbounded intervals

We study a class of degenerate elliptic second order differential operators acting on some polynomial weighted function spaces on [0,+infinity[. We show that these operators are the generators of C-0-semigroups of positive operators which, in turn, are the transition semigroups associated with right-continuous normal Markov processes with state space [0,+infinity]. Approximation and qualitative properties of both the semigroups and the Markov processes are investigated as well. Most of the results of the paper depend on a representation of the semigroups we give in terms of powers of particular positive operators of discrete type we introduced and studied in a previous paper.