  We explore a connection of the deterministically forced Burgers equation for
local velocity fields with probabilistic solutions of the Schr\"{o}dinger
boundary data problem. An issue of deducing the most likely interpolating
dynamics from the given initial and terminal probability density data is here
investigated to give account of the perturbation by external electromagnetic
fields. A suitable extension of the Hopf-Cole logarithmic transformation allows
to deal with nonconservative drift fields and implies the validity of
appropriately generalised heat equations, which completely determine the
dynamics.
