On differential operators associated with Markov operators

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

doi:10.1016/j.jfa.2014.01.001

Cited by 10 publications

(7 citation statements)

References 18 publications

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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: 88%

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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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“…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: 88%

“…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: 93%

Feller semigroups and degenerate elliptic operators III

Taira

2020

Mathematische Nachrichten

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“…The iterates converge to the linear interpolation operator that interpolates at the endpoints of [0,1]. In this example we demonstrated the underlying framework for finite-rank operators that reproduce constant and linear functions.…”

Section: An Introductory Examplementioning

confidence: 84%

“…Recently, Altomare [2] has shown a different approach using the concept of Choquet-boundaries and results from Korovkin-type approximation theory. Altomare et al [1] have shown an application where they discussed differential operators associated with Markov operators, where also the knowledge of limit of the iterates is significant. Another application has been shown in the field of approximation theory, where the iterates can be used to prove lower estimates for Markov operators with sufficient smooth range, see Nagler et al [17].It is worthwhile to mention that in most methods the limiting operator has to be known apriori.…”

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confidence: 99%

Iterates of Markov operators and their limits

Nagler

2018

Journal of Mathematical Analysis and Applications

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It is well known that iterates of quasi-compact operators converge towards a spectral projection, whereas the explicit construction of the limiting operator is in general hard to obtain. Here, we show a simple method to explicitly construct this projection operator, provided that the fixed points of the operator and its adjoint are known which is often the case for operators used in approximation theory.We use an approach related to Riesz-Schauder and Fredholm theory to analyze the iterates of operators on general Banach spaces, while our main result remains applicable without specific knowledge on the underlying framework. Applications for Markov operators on the space of continuous functions C(X) are provided, where X is a compact Hausdorff space.The behaviour of the iterates of Markov operators has been studied extensively in modern ergodic theory, while in general the limiting operator is not explicitly given. A comprehensive overview on limit theorems for quasi-compact Markov operators can be found in Hennion and Hervé [8]. In this article, we construct the limit of the iterates of quasi-compact operators that satisfy a spectral condition. It will be shown under which conditions the limit exists and how the limiting projection operator can be explicitly constructed using the inverse of a Gram matrix. The explicit knowledge of the limiting operator is of interest in many applications.This research is motivated by studying general Markov operators on the space of continuous functions C(X), where X is a compact Hausdorff space. Lotz [16] has already shown uniform ergodic theorems for Markov operators on C(X). For specific classes of operators, the limiting operator has been provided as shown for instance by Kelisky and Rivlin [12], Karlin and Ziegler [10] and Gavrea and Ivan [6,7]. Recently, Altomare [2] has shown a different approach using the concept of Choquet-boundaries and results from Korovkin-type approximation theory. Altomare et al. [1] have shown an application where they discussed differential operators associated with Markov operators, where also the knowledge of limit of the iterates is significant. Another application has been shown in the field of approximation theory, where the iterates can be used to prove lower estimates for Markov operators with sufficient smooth range, see Nagler et al. [17].It is worthwhile to mention that in most methods the limiting operator has to be known apriori. Here, we show an elegant extension to general Banach spaces for quasi-compact Markov operators. This extension provides a very general framework to explicitly construct the limiting operator with a simple method without prior knowledge of this operator.After an introductory example, we introduce briefly our notation and recall the most important results that are necessary to prove our results. All of these results are well-known and can be found, e. g., in the classical books of Ruston [20], Rudin [19], Heuser [9]. In the next section, we discuss how the complemented subspace for some finite-dimensional eigens...

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“…In the above mentioned works, though, it was assumed that the underlying operator T is a positive projection satisfying suitable assumptions. Francesco overcame such a limitation first in the context of the interval [0, 1], in the joint work [23] with the second author of this paper and Ioan Raşa, and then in the context of compact convex sets of R d (see [18,20]).…”

Section: Some Important Scientific Contributions Of Francesco Altomarementioning

confidence: 99%