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

Understand

The defining equation $(\ast):\ \dot \omega\_t=-F'(\omega\_t),$ 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) by considering the \emph{accelerations} $\ddot \omega ^{ \varepsilon}\_t$.
• We shall focus on Wasserstein gradient flows.
• Our approach is mainly heuristic.