Semi-local behaviour of non-local hypoelliptic equations: divergence form

We derive the Strong Harnack inequality for a class of hypoelliptic integro-differential equations in divergence form. Our result is semi-local, in the sense that we require the equation to hold globally in velocity. In fact, it is known that the Strong Harnack inequality fails in the kinetic case if the equation is merely satisfied in a bounded velocity domain [arXiv:2405.05223]. This is in stark contrast to the case of parabolic equations, for which a local Strong Harnack inequality holds. In a first step, we derive a local bound on the non-local tail on upper level sets by exploiting the coercivity of the cross terms. In a second step, we perform a De Giorgi argument in $L^1$, since we control the tail term only in $L^1$. This yields a linear $L^1$ to $L^\infty$ bound. Consequently, we prove polynomial upper bounds for $s\in(0,1)$ and exponential lower bounds for $s\in(\frac12,1)$ on the fundamental solution by adapting Aronson's method to non-local hypoelliptic equations.