  Motivated by entropic optimal transport, time reversal of Markov jump
processes in $\mathbb{R}^n$ is investigated. Relying on an abstract integration
by parts formula for the carr\'e du champ of a Markov process recently obtained
by Cattiaux, Gentil and the authors, and using an entropic improvement strategy
discovered by F\"ollmer in the eighties, we compute the semimartingale
characteristics of the time reversed process for a wide class of jump processes
with possibly unbounded variation sample paths and singular intensities of
jump.
