On the positive semigroups generated by Fleming-Viot type differential operators

In this paper we study a class of degenerate second-order elliptic differential operators, often referred to as Fleming-Viot type operators, in the framework of function spaces defined on the d-dimensional hypercube Qd of Rd, d ≥ 1. By making mainly use of techniques arising from approximation theory, we show that their closures generate positive semigroups both in the space of all continuous functions and in weighted Lp-spaces. In addition, we show that the semigroups are approximated by iterates of certain polynomial type positive linear operators, which we introduce and study in this paper and which generalize the Bernstein-Durrmeyer operators with Jacobi weights on [0, 1]. As a consequence, after determining the unique invariant measure for the approximating operators and for the semigroups, we establish some of their regularity properties along with their asymptotic behaviours.