On a class of degenerate elliptic operators arising from Fleming-Viot processes

Abstract

We are dealing with the solvability of an elliptic problem related to a class of degenerate second order operators which arise from the theory of Fleming-Viot processes in population genetics. In the one dimensional case the problem is solved in the space of continuous functions. In higher dimension we study the problem in \( L^2 \) spaces with respect to an explicit measure which, under suitable assumptions, can be taken invariant and symmetrizing for the operators. We prove the existence and uniqueness of weak solutions and we show that the closure of the operator in such \( L^2 \) spaces generates an analytic \( C_0 \)-semigroup.