Elliptic differential operators and positive semigroups associated with generalized Kantorovich operators

Abstract

Deepening the study of a new approximation sequence of positive linear operators we introduced and studied in [12], in this paper we disclose its relationship with the Markov semigroup (pre)generation problem for a class of degenerate second-order elliptic differential operators which naturally arise through an asymptotic formula, as well as with the approximation of the relevant Markov semigroups in terms of the approximating operators themselves.The analysis is carried out in the context of the space C(K) of all continuous functions defined on an arbitrary convex compact subset K of Rd, d≥1, having non-empty interior and a not necessarily smooth boundary, as well as, in some particular cases, in Lp(K) spaces, 1≤p<+∞. The approximation formula also allows to infer some preservation properties of the semigroup such as the preservation of the Lipschitz-continuity as well as of the convexity. We finally apply the main results to some noteworthy particular settings such as balls and ellipsoids, the unit interval and multidimensional hypercubes and simplices. In these settings the relevant differential operators fall into the class of Fleming–Viot operators.