A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle
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题目:A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures
报告人：Jian-Guo Liu（Duke University）
时间：-1 16:00 — 17:00
摘要：Many ideas in weak KAM theory are rooted in Freidlin-Wentzell's variational construction of the rate function of the large deviation principle for invariant measures. In this seminar, we reinterpret Freidlin-Wentzell's theory from a weak KAM perspective.
We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets, and the variational formulas for both self-consistent boundary data at each local attractors and the rate function. The weak KAM representation of Freidlin-Wentzell's variational construction of the rate function is proved based on the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers.
This rate function gives the maximal Lipschitz continuous viscosity solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set. The rate function is the selected unique weak KAM solution and serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.