A classification of the second order degenerate elliptic operators and its probabilistic characterization

1974 ◽  
Vol 30 (3) ◽  
pp. 235-254 ◽  
Author(s):  
Kanji Ichihara ◽  
Hiroshi Kunita
Keyword(s):  
Elliptic Operators ◽  
Second Order ◽  
Degenerate Elliptic Operators ◽  
Probabilistic Characterization ◽  
Degenerate Elliptic ◽  
Order Degenerate

scholarly journals On a class of degenerate elliptic equations

1974 ◽  
Vol 55 ◽  
pp. 181-204 ◽  
Author(s):  
Yoshiaki Hashimoto ◽  
Tadato Matsuzawa
Keyword(s):  
Elliptic Equations ◽  
Higher Order ◽  
Elliptic Operators ◽  
Second Order ◽  
General Boundary ◽  
Degenerate Elliptic Equations ◽  
Boundary Problems ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic

We shall prove in Chapter I the hypoellipticity for a class of degenerate elliptic operators of higher order. Chapter II will be devoted to the consideration of the regularity at the boundary for the solutions of general boundary problems for the equations considered in Chapter I being restricted to the second order.

Degenerate Diffusion Operators Arising in Population Biology (AM-185)

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Adjoint Operator ◽  
Integral Kernel ◽  
Mathematical Finance ◽  
Elliptic Operators ◽  
Complete Analysis ◽  
Time Behavior ◽  
Degenerate Elliptic Operators ◽  
Regularity Properties ◽  
Degenerate Elliptic ◽  
Mathematical Foundations

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the martingale problem and therefore the existence of the associated Markov process. The book uses an “integral kernel method” to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. The book establishes the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. It shows that the semigroups defined by these operators have holomorphic extensions to the right half plane. The book also demonstrates precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

scholarly journals Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators

2015 ◽  
Vol 103 (5) ◽  
pp. 1276-1293 ◽  
Author(s):  
Henri Berestycki ◽  
Italo Capuzzo Dolcetta ◽  
Alessio Porretta ◽  
Luca Rossi
Keyword(s):  
Maximum Principle ◽  
Principal Eigenvalue ◽  
Elliptic Operators ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic
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