Authors:

Thomas Martinod Saldarriaga

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About Paper:

It is known that classical solutions to the Hamilton-Jacobi equation (HJE) do not exist in general settings, as is the case for the HJE arising from overdamped Langevin dynamics. To address this, in this work we present a rigorous and explicit variational construction of weak KAM solutions for the HJE associated with overdamped Langevin dynamics on the one-dimensional torus. We begin by deriving the HJE corresponding to this specific stochastic system. Then, under the assumption that the external drift field has at least one root, we prove that the Mañé critical value is zero and compute the Aubry set. We proceed to explicitly compute the Peierls barrier and describe the global energy landscape in the one-dimensional conservative case, while discussing extensions to higher dimensions. Finally, we show that the dynamic solution to the HJE is given by the global energy landscape, which remains invariant under the Lax-Oleinik semigroup and thus provides a viscosity solution to the PDE. Keywords: Weak KAM Theory; Overdamped Langevin Dynamics; Global Energy Landscape

Source:

Purdue University / 2025

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Co-authors:

Thomas Martinod Saldarriaga