Well-posedness of the martingale problem for some degenerate diffusion processes occurring in dynamics of populations

Abstract

In this paper we establish the well-posedness in C([0,);[0,1]d), for each starting point x [0,1]d, of the martingale problem associated with a class of degenerate elliptic operators which arise from the dynamics of populations as a generalization of the Fleming¿Viot operator. In particular, we prove that such degenerate elliptic operators are closable in the space of continuous functions on [0,1]d and their closure is the generator of a strongly continuous semigroup of contractions.

Author Keywords: Degenerate elliptic equations in nonregular domains; Generation of semigroup; Stochastic invariance; Fleming¿Viot operator; Martingale problem

35B45; 60J70; 35B65; 35J25