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arXiv · 2112.14224

Perturbations of Parabolic Equations and Diffusion Processes with Degeneration: Boundary Problems, Metastability, and Homogenization

Mark Freidlin·Leonid Koralov
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Abstract

We study diffusion processes that are stopped or reflected on the boundary of a domain. The generator of the process is assumed to contain two parts: the main part that degenerates on the boundary in a direction orthogonal to the boundary and a small non-degenerate perturbation. The behavior of such processes determines the stabilization of solutions to the corresponding parabolic equations with a small parameter. Metastability effects arise in this case: the asymptotics of solutions, as the size of the perturbation tends to zero, depends on the time scale. Initial-boundary value problems with both the Dirichlet and the Neumann boundary conditions are considered. We also consider periodic homogenization for operators with degeneration.

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Mark Freidlin, Leonid Koralov. 2021-12-28. Perturbations of Parabolic Equations and Diffusion Processes with Degeneration: Boundary Problems, Metastability, and Homogenization. https://arxiv.org/abs/2112.14224

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