Existence and upper bounds of fundamental solutions to kinetic equations with rough non-local diffusion

We study the fundamental solution operators associated to Kolmogorov equations with symmetric non-local diffusion with rough kernels. We prove that these operators are induced by measurable kernels and prove sharp upper bounds in terms of a kinetic profile governed by the interplay of transport and non-local diffusion. Our proof combines the small-jump with loss decomposition of P.-A. Meyer with the ideas of Nash. The proof is purely analytic.