On the equivalence between some jumping SDEs with rough coefficients and some non-local PDEs

We study some jumping SDE and the corresponding Fokker-Planck (or Kolmogorov forward) equation, which is a non-local PDE. We assume only some measurability and growth conditions on the coefficients. We prove that for any weak solution $(f_t{)}_{t\in [0,T]}$ of the PDE, there exists a weak solution to the SDE of which the time marginals are given by $(f_t{)}_{t\in [0,T]}$. As a corollary, we deduce that for any given initial condition, existence for the PDE is equivalent to weak existence for the SDE and uniqueness in law for the SDE implies uniqueness for the PDE. This extends some ideas of Figalli [5] concerning continuous SDEs and local PDEs.