Some Analytic Results for Kimura Diffusion Operators

Abstract

In this note, we prove several analytical results about generalized Kimura diffusion operators, L, defined on compact manifolds with corners, P. It is shown that the $\mathcal C^{0}(P)$-graph closure of L acting on $\mathcal C^{2}(P)$ always has a compact resolvent. In the 1d-case, where P = [0,1], we also establish a gradient estimate $\|\partial _{x} f\|_{\mathcal C^{0}([0,1])}\leq C\| L f\|_{\mathcal C^{0}([0,1])},$ provided that L has strictly positive weights at ∂[0,1] = {0,1}. This in turn leads to a precise characterization of the domain of the $\mathcal C^{0}$-graph closure in this case.