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January  2019, 18(1): 323-340. doi: 10.3934/cpaa.2019017

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

Received  January 2018 Revised  April 2018 Published  August 2018

Fund Project: Work partially supported by the Italian INDAM-GNAMPA.

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.

Citation: Francesco Altomare, Mirella Cappelletti Montano, Vita Leonessa. On the positive semigroups generated by Fleming-Viot type differential operators. Communications on Pure & Applied Analysis, 2019, 18 (1) : 323-340. doi: 10.3934/cpaa.2019017
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