A variational approach to gradient flows in metric spaces,

In this note we report on a new variational principle for Gradient Flows in metric spaces.
This new variational formulation consists in a functional defined on entire trajectories
whose minimizers converge, in the case in which the energy is geodesically convex,
to curves of maximal slope.
The key point in the proof is a reformulation
of the problem in terms of a dynamic programming principle combined
with suitable a priori estimates on the minimizers.
The abstract result is applicable to a large class of evolution PDEs, including
Fokker Plack equation, drift diffusion and Heat flows in metric-measure spaces.