From Döblin to Aleksandrov
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…
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