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In this paper we deepen the study of a sequence of positive linear operators acting on L¹([0, 1]^N), N ≥ 1, that have been introduced in [3] and that generalize the multidimensional Kantorovich operators (see [15]). We show that particular iterates of these operators converge on C ([0, 1]^N) to a Markov semigroup and on L^p([0, 1]^N), 1 ≤ p < +∞, to a positive contractive C₀-semigroup (that is an extension of the previous one). The generators of these C₀-semigroups are the closures of some partial differential operators that belong to the class of Fleming-Viot operators arising in population genetics.

Mathematics Subject Classification (2010): 41A36, 47D06, 47F05.

Keywords: Multidimensional Kantorovich operator, positive approximation process, iterate of operators, positive C₀-semigroup, approximation of semigroups, Fleming-Viot operator