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Existence, uniqueness and the strong Markov property of solutions to Kimura diffusions with singular drift


Author: Camelia A. Pop
Journal: Trans. Amer. Math. Soc. 369 (2017), 5543-5579
MSC (2010): Primary 60J60; Secondary 35J70
DOI: https://doi.org/10.1090/tran/6853
Published electronically: March 1, 2017
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Abstract: Motivated by applications to proving regularity of solutions to degenerate parabolic equations arising in population genetics, we study existence, uniqueness, and the strong Markov property of weak solutions to a class of degenerate stochastic differential equations. The stochastic differential equations considered in our article admit solutions supported in the set $ [0,\infty )^n\times \mathbb{R}^m$, and they are degenerate in the sense that the diffusion matrix is not strictly elliptic, as the smallest eigenvalue converges to zero at a rate proportional to the distance to the boundary of the domain, and the drift coefficients are allowed to have power-type singularities in a neighborhood of the boundary of the domain. Under suitable regularity assumptions on the coefficients, we establish existence of solutions that satisfy the strong Markov property, and uniqueness in law in the class of Markov processes.


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Additional Information

Camelia A. Pop
Affiliation: School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street SE, Minneapolis, Minnesota 55455
Email: capop@umn.edu

DOI: https://doi.org/10.1090/tran/6853
Keywords: Kimura diffusions, singular drift coefficient, degenerate diffusions, degenerate elliptic operators, strong Markov property, anisotropic H\"older spaces
Received by editor(s): July 20, 2014
Received by editor(s) in revised form: August 17, 2015, and September 2, 2015
Published electronically: March 1, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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