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Asymptotic spectral analysis of a small diffusion operator and the life times of the corresponding diffusion processстатья
Статья опубликована в журнале из списка RSCI Web of Science
Статья опубликована в журнале из списка Web of Science и/или ScopusДата последнего поиска статьи во внешних источниках: 27 октября 2021 г.
- Авторы: Kolokol’tsov V.N., Makarov K.A.
- Журнал: Russian Journal of Mathematical Physics
- Том: 4
- Номер: 3
- Год издания: 1996
- Издательство: Maik Nauka/Interperiodica Publishing
- Местоположение издательства: Russian Federation
- Первая страница: 341
- Последняя страница: 360
- Аннотация: The diffusion operator D = -∈Δ + (∇φ, ∇) on a Riemannian manifold M is considered, where φ is a Morse function on M and ∈ > 0 is a small parameter. Let Ωi denote the domains of attraction of the dynamical system ẋ = -∇φ(x). It is proved that the set of eigenvalues of D exponentially small in ∈ asymptotically coincides with the spectrum of the matrix G = Gij, where Gij is the mean life time in the domain Ωi of the corresponding diffusion process starting from Ωj. This result is used to derive the asymptotics of the exponentially small eigenvalues of D (including the pre-exponential terms) for the one-dimensional case. To this end, we also develop a generalization of the Wentzell graphical technique for computing the spectra of matrices with exponentially small entries. This method reduces the calculation of the low-lying spectrum of the diffusion operator to a problem of combinatorial analysis that can be easily solved by computer. © 1997 by John Wiley & Sons, Inc.
- Добавил в систему: Колокольцов Василий Никитич