The classical heat equation is a beautiful, simple and immensely useful model of diffusion which - with due adjustments - may be found in a wide variety of scientific theories ranging from physics, through biology to finance. The Li-Yau inequality, an instance of a so-called differential Harnack estimate, is a fundamental lower bound on the rate of decay of positive solutions. In line with the canonical nature of the heat equation, the Li-Yau bound is a mathematical gem with powerful applications. It is only natural to look for counterparts of this bound in other models of diffusion.

However, take a step in a non-linear direction or introduce some non-locality and you find yourself in trouble. Not only the complexity of the problem increases dramatically, it is not even clear what to calculate.

The Li-Yau inequality sits in the intersection of PDEs, geometry, probability, thermodynamics and optimal transport and each of these disciplines offers a perspective on possible generalisations. Despite a concerted effort on various fronts, we are still far from having a convincing unified approach to the problem. A central issue lies in identifying a suitable interpretation of the Li-Yau inequality that carries through to generalised settings.

In this talk I will discuss some conceptual and technical challenges on the path to non-linear and non-local generalisations of the Li-Yau inequality and describe where such results may be put to good use.