Convergence of moderately interacting particle systems to a diffusion-convection equation

We give a probabilistic interpretation of the solution of a diffusion-convection equation. To do so, we define a martingale problem in which the drift coefficient is nonlinear and unbounded for small times whereas the diffusion coefficient is constant. We check that the time marginals of any solution are given by the solution of the diffusion-convection equation. Then we prove existence and uniqueness for the martingale problem and obtain the solution as the propagation of chaos limit of a sequence of moderately interacting particle systems.