This book proves the existence, uniqueness and regularity results for a class of degenerate elliptic operators known as generalized Kimura diffusions, which act on functions defined on manifolds with corners. It presents a generalization of the Hopf boundary point maximum principle that demonstrates, in the general case, how regularity implies uniqueness. The book is divided in three parts. Part I deals with Wright–Fisher geometry and the maximum principle; Part II is devoted to an analysis of model problems, and includes degenerate Hölder spaces; and Part III discusses generalized Kimura diffusions. This introductory chapter provides an overview of generalized Kimura diffusions and their applications in probability theory, model problems, perturbation theory, main results, and alternate approaches to the study of similar degenerate elliptic and parabolic equations.
Keywords: generalized Kimura diffusion, manifold with corners, Hopf boundary point, regularity, uniqueness, Wright–Fisher geometry, Hölder space, probability theory, perturbation theory, degenerate elliptic operator
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