On differential operators associated with Markov operators

In this paper we introduce and study a new class of elliptic second-order differential operators on a convex compact subset K   of RdRd, d⩾1d⩾1, which are associated with a Markov operator T   on C(K)C(K) and which degenerate on a suitable subset of K containing its extreme points. Among other things, we show that the closures of these operators generate Markov semigroups. Moreover, we prove that these semigroups can be approximated by means of iterates of suitable positive linear operators, which are referred to as the Bernstein–Schnabl operators associated with T. As a consequence we show that the semigroups preserve polynomials of a given degree as well as Hölder continuity, which gives rise to some spatial regularity properties of the solutions of the relevant evolution equations.