Asymptotics for scaled Kramers-Smoluchowski equations (Q2817450)

The Kramers-Smoluchowski equation for density \(\rho\) of diffusing material is a linear parabolic equation, which takes the form of NEWLINE\[NEWLINE \rho_t = (\rho_{\xi} + \epsilon^{-2}\Phi'\rho)_{\xi} NEWLINE\]NEWLINE in the one-dimensional space, and is similarly generalized to the multidimensional settings. Here, \(\Phi\) is a dual-well one-dimensional potential or its multidimensional generalization, and \(\xi\) is the spatial coordinate. The objective is to develop a simple proof of the known asymptotic form of the solutions (which yields the corresponding ``asymptotic capacity'') of this equation in the limit of a very deep dual-well potential, which corresponds to \(\epsilon \to 0\). The proof is developed for the one-dimensional equation, and is then generalized for its multidimensional counterpart.