Quasi-Linear Equations with a Small Diffusion Term and the Evolution of Hierarchies of Cycles
| Autor: | Lucas Tcheuko, Leonid Koralov |
|---|---|
| Rok vydání: | 2015 |
| Předmět: |
Statistics and Probability
General Mathematics 010102 general mathematics Mathematical analysis Order (ring theory) Term (logic) Lambda 01 natural sciences Parabolic partial differential equation 010104 statistics & probability Metastability Large deviations theory 0101 mathematics Statistics Probability and Uncertainty Diffusion (business) Linear equation Mathematics |
| Zdroj: | Journal of Theoretical Probability. 29:867-895 |
| ISSN: | 1572-9230 0894-9840 |
| DOI: | 10.1007/s10959-015-0601-4 |
| Popis: | We study the long-time behavior (at times of order \(\exp (\lambda /\varepsilon ^2\))) of solutions to quasi-linear parabolic equations with a small parameter \(\varepsilon ^2\) at the diffusion term. The solution to a PDE can be expressed in terms of diffusion processes, whose coefficients, in turn, depend on the unknown solution. The notion of a hierarchy of cycles for diffusion processes was introduced by Freidlin and Wentzell and applied to the study of the corresponding linear equations. In the quasi-linear case, it is not a single hierarchy that corresponds to an equation, but rather a family of hierarchies that depend on the timescale \(\lambda \). We describe the evolution of the hierarchies with respect to \(\lambda \) in order to gain information on the limiting behavior of the solution of the PDE. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|
Full text from SpringerLink