On optimal Harnack bounds for a fractional heat equation

Considering the linear heat equation, the celebrated Li-Yau inequality states that for positive solutions we have $\Delta \log u \geq - \frac{n}{2t}$. 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. Finding the fractional Li-Yau estimate still poses a challenge, but in the literature there are several Harnack inequalities concerning non-local diffusion. Our goal is to find an optimal Harnack 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.