Dynamical aspects of generalized Schr{\"o}dinger problem via Otto calculus -- A heuristic point of view

The defining equation $(\ast ):\dot{\omega }\mathrm{\_}t=-F^{\mathrm{\prime }}(\omega \mathrm{\_}t),$ of a gradient flow is kinetic in essence. This article explores some dynamical (rather than kinetic) features of gradient flows (i) by embedding equation $(\ast )$ into the family of slowed down gradient flow equations: $\dot \omega ^{ \varepsilon}_t=- \varepsilon F’( \omega ^{ \varepsilon}_t),$where$\varepsilon>0$,and(ii)byconsideringthe\mathit{\text{accelerations}}$\ddot \omega ^{ \varepsilon}_t$. We shall focus on Wasserstein gradient flows. Our approach is mainly heuristic. It relies on Otto calculus.A special formulation of the Schr{"o}dinger problem consists in minimizing some action on the Wasserstein space of probability measures on a Riemannian manifold subject to fixed initial and final data. We extend this action minimization problem by replacing the usual entropy, underlying Schr{"o}dinger problem, with a general function of the Wasserstein space. The corresponding minimal cost approaches the squared Wasserstein distance when some fluctuation parameter tends to zero. We show heuristically that the solutions satisfy a Newton equation, extending a recent result of Conforti. The connection with Wasserstein gradient flows is established and various inequalities, including evolutional variational inequalities and contraction inequality under curvature-dimension condition, are derived with a heuristic point of view. As a rigorous result we prove a new and general contraction inequality for the Schr{"o}dinger problem under a Ricci lower bound on a smooth and compact Riemannian manifold.