Regularity properties of semigroups generated by some Fleming-Viot type operators

We study different qualitative properties of the semigroup generated by some degenerate differential elliptic operators on the standard simplex of $\R^d$. Some methods are new and are based on the representation formulas of the semigroup in terms of iterates of suitable positive operators. The main result is the ultracontractivity property which is obtained in the setting of weighted $L^p$-spaces. We describe the asymptotic behavior of the semigroup and obtain the compactness property in the same setting and also in spaces of continuous functions.