On differential Harnack bounds for a fractional heat equation

Consider the linear heat equation. The celebrated Li-Yau inequality states that for positive solutions we have $\bigtriangleup \log u \geq - \frac{n}{2t}$. By integrating this inequality along a straight space-time interval between two points, we may deduce the sharp Harnack estimate. In recent years there has been a considerable interest in extending the Li-Yau technique to the context of non-local diffusion, and in particular to the fractional heat equation. In the literature there are several Harnack inequalities concerning non-local diffusion, but finding the fractional Li-Yau estimate still poses a challenge. Previously, I showed the derivation of an optimal Harnack inequality in the basic fractional setting, which I will recall. Our goal is to find an optimal Li-Yau inequality, in the sense that it yields an identity when applied to a fundamental solution. The presentation shows the progress made on this topic, and is based on the joint work with Mikołaj Sierżęga.