Duality approach to proving uniqueness

The martingale problem is a powerful method for the characterization and study of stochastic processes. The duality approach is one of the tools for proving uniqueness of solutions to martingale problems. In its original version it requires the construction of a dual process which describes any solution of the martingale problem. In recent years this method has proved to be extremely useful for proving uniqueness results for a variety of interacting measure-valued diffusions.

We will describe the duality approach, its modification (the so-called ``approximate duality" method), and give several examples of measure-valued diffusions and stochastic partial differential equations where the method has been successfully applied.