A family of functional inequalities: Lojasiewicz inequalities and displacement convex functions

Understand

For displacement convex functionals in the probability space equip\-ped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type \L oja\-sie\-wicz inequalities.

• \chg{We also discuss the more general case of $\lambda$-convex functions and we provide a general convergence theorem for the corresponding gradient dynamics.
• Specialising our results to the Boltzmann entropy, we recover Otto-Villani's theorem asserting the equivalence between logarithmic Sobolev and Talagrand's inequalities.
• The choice of power-type entropies shows a new equivalence between Gagliardo-Nirenberg inequality and a nonlinear Talagrand inequality.