  In finite-dimensions, minimal surfaces that fill in the space delineated by
closed curves and have minimal area arose naturally in classical physics in
several contexts. No such concept seems readily available in infinite
dimensions. The present work is motivated by the need for precisely such a
concept that would allow natural coordinates for a surface with a boundary of a
closed curve in the Wasserstein space of probability distributions (space of
distributions with finite second moments). The need for such a concept arose
recently in stochastic thermodynamics, where the Wasserstein length in the
space of thermodynamic states quantifies dissipation while area integrals
(albeit, presented in a special finite-dimensional parameter setting) relate to
useful work being extracted. Our goal in this work is to introduce the concept
of a minimal surface and explore options for a suitable construction. To this
end, we introduce the basic mathematical problem and develop two alternative
formulations. Specifically, we cast the problem as a two-parameter
Benamou-Breiner type minimization problem and derive a minimal surface equation
in the form of a set of two-parameter partial differential equations. Several
explicit solutions for minimal surface equations in Wasserstein spaces are
presented. These are cast in terms of covariance matrices in Gaussian
distributions.
