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

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Work partially supported by the Italian INDAM-GNAMPA.
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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 $\mathbf{R}^d$, $d ≥1$.

    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.

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


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