math.AP — Analysis of PDEs

From Döblin to Aleksandrov

Contributed by Amélie Loher, Connor Mooney, Clément Mouhot

We prove Aleksandrov maximum principles for linear second-order kinetic equations in non-divergence form with measurable, uniformly elliptic diffusion coefficients. For coefficients $A(t,v)$ we obtain every source exponent $p>2d+1$. For autonomous coefficients $a(x,v)$ in one space dimension, we obtain every $p>p_*(\frac\Lambda\lambda)$, where $p_*(\frac\Lambda\lambda)<4$ depends only on the ellipticity ratio. The first step in both proofs is Döblin--Fourier cancellation: two positive parts of the kernel have nearly opposite phases, and a common lower bound for their velocity marginals allows us to cancel fixed portions of them. For the autonomous problem, this argument applies on velocity intervals away from zero. A preliminary Aleksandrov estimate gives Hölder continuity and then a Harnack comparison. A homogeneous Bellman barrier controls concentration in position and velocity; together with Harnack, it makes the local density estimates summable across zero velocity, including at $p=4$. We give the full proofs and the passage from the maximum principle to local Hölder continuity.