  Motivated by a problem from incompressible fluid mechanics of Brenier (JAMS
1989), and its recent entropic relaxation by Arnaudo, Cruizero, L\'eonard &
Zambrini (AIHP PS 2020), we study a problem of entropic minimization on the
path space when the reference measure is a generic Feller semimartingale. We
show that, under some regularity condition, our problem connects naturally with
a, possibly non-local, version of the Hamilton-Jacobi-Bellman equation.
Additionally, we study existence of minimizers when the reference measure in a
Ornstein-Uhlenbeck process.
