On the existence of a scalar pressure field in the Br\"odinger problem

This work deals with the entropic regularization of the Brenier problem for perfect incompressible fluids introduced by Arnaudon, Cruzeiro, L\'eonard and Zambrini. We show that as in the original setting, there exists a scalar pressure field which is interpreted as the Lagrange multiplier associated to the incompressibility constraint. The proof goes through a reformulation of the problem in PDE terms.