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Freidlin, Mark; Koralov, Leonid

Perturbations of parabolic equations and diffusion processes with degeneration: boundary problems, metastability, and homogenization. (English) Zbl 1531.35036

Ann. Probab. 51, No. 5, 1752-1784 (2023).
Summary: 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 nondegenerate 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.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B40 Asymptotic behavior of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K65 Degenerate parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
60F10 Large deviations
60J60 Diffusion processes

Keywords:

asymptotic problems for PDEs; equations with degeneration on the boundary; metastability; stabilization in parabolic equations
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References:

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