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

Francesco Altomare; Mirella Cappelletti Montano; Vita Leonessa

doi:10.3934/cpaa.2019017

Article Contents

2019, Volume 18, Issue 1: 323-340. Doi: 10.3934/cpaa.2019017

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

1.
Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Campus Universitario, Via Edoardo Orabona n. 4, 70125 Bari, Italy
2.
Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Campus di Macchia Romana, Viale Dell' Ateneo Lucano n. 10, 85100 Potenza, Italy

* Corresponding author
* Corresponding author

Received: December 31, 2017

Revised: March 31, 2018

Published: August 2018

Work partially supported by the Italian INDAM-GNAMPA.

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Abstract

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 $\mathbf{R}^d$, $d ≥1$.

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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Mathematics Subject Classification: Primary: 47D06, 47D07; Secondary: 41A36, 35K65.

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