Degenerate elliptic operators, Feller semigroups and modified Bernstein‐Schnabl operators

Abstract

In this paper we study a class of elliptic second-order differential operators on finite dimensional convex compact\ud sets whose principal part degenerates on a subset of the boundary of the domain. We show that the closures\ud of these operators generate Feller semigroups.\ud Moreover, we approximate these semigroups by iterates of suitable positive linear operators which we also introduce\ud and study in this paper for the first time, and which we refer to as modified Bernstein-Schnabl operators.\ud As a consequence of this approximation we investigate some regularity properties preserved by the semigroup.\ud Finally, we consider the special case of the finite dimensional simplex and the well-known Wright-Fisher diffusion\ud model of gene frequency used in population genetics

Feller semigroups and degenerate elliptic operators III

“…They prove that the closure A of the operator V generates a Feller semigroup and further that the Feller semigroup can be approximated by iterates of modified Bernstein–Schnabl operators ([4]). It should be emphasized that Theorem 1.2 coincides with [2, Theorem 4.1], [3, Theorem 4.3] and [5, Theorem 3.1] with if the boundary is smooth, as in Example 1.1. 2°Theorem 1.2 is proved by Bony–Courrège–Priouret [8] in the elliptic case (see [8, Théorème XVI]) and then by Cancelier [9] in the non‐characteristic case: (cf. [9, Théorème 7.2]).…”

“…This paper is inspired by the work of Altomare et al. [2, 3] and [5] (see Remark 1.3), and it is a continuation of the previous papers Taira [23] through [31] and Taira–Favini–Romanelli [32].…”

“…Remark Some remarks are in order: 1°Altomare et al. [2] through [5] consider a convex compact domain K with not necessarily smooth boundary and a second‐order differential operator V which degenerates on a subset of the boundary containing the extreme points of K . They prove that the closure A of the operator V generates a Feller semigroup and further that the Feller semigroup can be approximated by iterates of modified Bernstein–Schnabl operators ([4]).…”

On differential operators associated with Markov operators

“…Bernstein-Schnabl operators, introduced by Schnabl in [25] and successively by Grossmann [20] and Nishishiraho [22,23] in different contexts, were intensively studied by Altomare (see [2], [6, Section 6.1] and the references therein) in association with a positive projection on C (K) and through the last twenty years they have been the subject of several researches and generalizations. For more details, we refer the interested reader to the survey [12] and its numerous references, but also to [9][10][11]24], among many others.…”