Duality and the well-posedness of a martingale problem

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 bounded continuous duality function H on E_X×E_Y together with a dual process Y on E_Y which is the unique solution of a G_Y-martingale problem. For the corresponding solutions [Formula: see text] and [Formula: see text] , duality with respect to a function H in its simplest form means that the relation E_x[H(X_t,y)]=E_y[H(x,Y_t)] holds 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 approximating sequences of processes of a different kind (e.g. jump processes to approximate diffusions) which is a widespread strategy for proving existence of solutions of martingale problems. Given the process [Formula: see text] and a duality function H, to prove existence of [Formula: see text] one has to show that the r.h.s. of the duality relation defines for each y a measure on E_X, i.e. there are transition kernels [Formula: see text] from E_X to 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 all t≥0. As examples, we treat resampling and branching models, such as the Fleming-Viot measure-valued diffusion and its spatial counterparts (with both, discrete and continuum space), as well as branching systems, such as Feller's branching diffusion. While our main result as well as all examples come with (locally) compact state spaces, we discuss the strategy to lift our results to genealogy-valued processes or historical processes, leading to non-compact (discrete and continuum) state spaces. Such applications will be tackled in forthcoming work based on the present article.