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. 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 L¹, since we control the tail term only in L¹. This yields a linear L¹ to L^ bound. Consequentially, we prove polynomial upper and exponential lower bounds on the fundamental solution by adapting Aronson's method to non-local hypoelliptic equations.