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. The proof is based on a priori estimates, and as such extends the first non-stochastic approach of the non-local parabolic Strong Harnack inequality by Kassmann-Weidner [arXiv:2303.05975] to hypoelliptic equations; however, in contrast to the parabolic case, we only obtain a semi-local result, in the sense that we require the equation to hold globally in velocity, which in particular does not contradict the counterexample constructed in [arXiv:2405.05223]. 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. Consequentially, we prove polynomial upper and exponential lower bounds on the fundamental solution by adapting Aronson's method to non-local hypoelliptic equations.