Duality and the well-posedness of a martingale problem

Abstract

For two Polish state spaces E X and E Y , and an operator G X , we obtain existence and uniqueness of a G X -martingale problem provided there is a dual process Y on E Y solving a G Y -martingale problem. Duality here means the existence of a rich function H and transition kernels (µ t ) t≥0 on E X such that E y [H(x, Y t )] = µ t (x, dx ′ )H(x ′ , y) for all (x, y) ∈ E X ×E Y and t ≥ 0. While duality is well-known to imply uniqueness of the G X -martingale problem, we give here a set of conditions under which duality also implies existence without using approximation techniques. As examples, we treat branching and resampling models, such as Feller's branching diffusion and the Fleming-Viot superprocess.