English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This paper reinterprets Freidlin-Wentzell's variational construction of therate function in the large deviation principle for invariant measures from theweak KAM perspective. Through a one-dimensional irreversible diffusion processon a torus, we explicitly characterize essential concepts in the weak KAMtheory, such as the Peierls barrier and the projected Mather/Aubry/Ma\~n\'esets. The weak KAM representation of Freidlin-Wentzell's variationalconstruction of the rate function is discussed based on the global adjustmentfor the boundary data on the Aubry set and the local trimming from the liftedPeierls barriers. This rate function gives the maximal Lipschitz continuousviscosity solution to the corresponding stationary Hamilton-Jacobi equation(HJE), satisfying Freidlin-Wentzell's variational formula for the boundarydata. Choosing meaningful self-consistent boundary data at each local attractorare essential to select a unique weak KAM solution to stationary HJE. Thisselected viscosity solution also serves as the global energy landscape of theoriginal stochastic process. This selection for stationary HJEs can bedescribed by first taking the long time limit and then taking the zero noiselimit, which also provides a special construction of vanishing viscosityapproximation.

A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures