Elliptic differential operators and positive semigroups associated with generalized Kantorovich operators

Altomare, Francesco; Montano, Mirella Cappelletti; Leonessa, Vita; Raşa, Ioan

doi:10.1016/j.jmaa.2017.08.034

Cited by 17 publications

(10 citation statements)

References 14 publications

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“…If f ∈ C [0, 1] is convex the inequality Inequalities of type (1) have important applications. They are useful when studying whether the Bernstein-Schnabl operators preserve convexity (see [2,3,4]).…”

Section: Introductionmentioning

confidence: 99%

A sharpening of a problem on Bernstein polynomials and convex functions

Abel¹,

Raşa²

2018

Mathematical Inequalities &Amp; Applications

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Section: Introductionmentioning

confidence: 99%

A sharpening of a problem on Bernstein polynomials and convex functions

Abel¹,

Raşa²

2018

Mathematical Inequalities &Amp; Applications

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“…They prove that the closure A of the operator V generates a Feller semigroup

{true{T_{t} true}}_{t \geq 0}

and further that the Feller semigroup

{true{T_{t} true}}_{t \geq 0}

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

K : = D

if the boundary

\partial K

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:

\partial D = {normalΣ}_{3}

(cf. [9, Théorème 7.2]).…”

Section: Introduction and Main Resultsmentioning

confidence: 89%

“…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].…”

Section: Introduction and Main Resultsmentioning

confidence: 95%

“…Remark Some remarks are in order: 1°Altomare et al. [2] through [5] consider a convex compact domain K with not necessarily smooth boundary

\partial K

and a second‐order differential operator V which degenerates on a subset of the boundary

\partial K

containing the extreme points of K . They prove that the closure A of the operator V generates a Feller semigroup

{true{T_{t} true}}_{t \geq 0}

and further that the Feller semigroup

{true{T_{t} true}}_{t \geq 0}

can be approximated by iterates of modified Bernstein–Schnabl operators ([4]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Feller semigroups and degenerate elliptic operators III

Taira

2020

Mathematische Nachrichten

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This paper is devoted to the functional analytic approach to the problem of construction of Feller semigroups in the characteristic case via the Fichera function. Probabilistically, our result may be stated as follows: We construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves continuously in the interior of the state space, without reaching the boundary. We make use of the Hille–Yosida–Ray theorem that is a Feller semigroup version of the classical Hille–Yosida theorem in terms of the positive maximum principle. Our proof is based on a method of elliptic regularizations essentially due to Oleĭnik and Radkevič. The weak convergence of approximate solutions follows from the local sequential weak compactness of Hilbert spaces and Mazur's theorem in normed linear spaces.

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“…Due to their intrinsic interest, more recently an increasing attention has been turned to them also in the setting of hypercubes (see, e.g., [3], [6,Section 5.8], [7,9,14,17] and the references therein).…”

mentioning

confidence: 99%

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

Altomare¹,

Montano²,

Leonessa³

2019

Communications on Pure &Amp; Applied Analysis

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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 Q d of R d , 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 L p-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.

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Elliptic differential operators and positive semigroups associated with generalized Kantorovich operators

Cited by 17 publications

References 14 publications

A sharpening of a problem on Bernstein polynomials and convex functions

A sharpening of a problem on Bernstein polynomials and convex functions

Feller semigroups and degenerate elliptic operators III

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

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