Duality and the well-posedness of a martingale problem.

For two Polish state spaces E and E, and an operator G, we obtain existence and uniqueness of a G-martingale problem provided there is a bounded continuous duality function H on E×E together with a dual process Y on E which is the unique solution of a G-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[H(X,y)]=E[H(x,Y)] holds for all (x,y)∈E×E and t≥0. While duality is well-known to imply uniqueness of the G-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.

The grapheme-valued Wright-Fisher diffusion with mutation.

In Athreya et al. (2021), models from population genetics were used to define stochastic dynamics in the space of graphons arising as continuum limits of dense graphs. In the present paper we exhibit an example of a simple neutral population genetics model for which this dynamics is a Markovian diffusion that can be characterized as the solution of a martingale problem.