  In this paper, we introduce weak optimal entropy transport problems that
cover both optimal entropy transport problems and weak optimal transport
problems introduced by Liero, Mielke, and Savar\'{e} [27]; and Gozlan, Roberto,
Samson and Tetali [20], respectively. Under some mild assumptions of entropy
functionals, we establish a Kantorovich type duality for our weak optimal
entropy transport problem. We also introduce martingale optimal entropy
transport problems, and express them in terms of duality, homogeneous marginal
perspective functionals and homogeneous constraints.
