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Some Analytic Results for Kimura Diffusion Operators

$icon-email$ Charles L. Epstein , Jon Wilkening

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 $C^{0} (P)$ -graph closure of $L$ acting on $C^{2} (P)$ always has a compact resolvent. In the $1 d$ -case, where $P = [0, 1]$ , we also establish a gradient estimate $‖ \partial_{x} f ‖_{C^{0} ([0, 1])} \leq C ‖ L f ‖_{C^{0} ([0, 1])}$ , provided that $L$ has strictly positive weights at $\partial [0, 1] = {0, 1}$ . This in turn leads to a precise characterization of the domain of the $C^{0}$ -graph closure in this case.

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Some Analytic Results for Kimura Diffusion Operators

Abstract