Abstract: In this talk we briefly discuss the martingale problem. Knowing the existence of a solution to this problem in closely related to knowing the existence of solutions to a closely related PDE or, in general, a pseudo-differential equation. In order to formalize this, we put a differentiable structure on the space of Borel probability measures and study the integral curves on this space. These integral curves help us solve the martingale problem for a large collection of operators. This talk will hopefully make the connection between probability theory and PDE’s more clear.