We offer fairly simple and direct proofs of the asymptotics for the scaled Kramers-Smoluchowski equation in both one and higher dimensions. For the latter, we invoke the sharp asymptotic capacity asymptotics of Bovier-Eckhoff-Gayrard-Klein [B-E-G-K].

In this paper, we generalize the results of Evans and Tabrizian [3], by deriving asymptotics for the time-rescaled Kramers-Smoluchowski equations, in the case of a general non-symmetric potential function with multiple wells. The asymptotic limit is described by a system of reaction-diffusion equations whose coefficients are determined by the Kramers constants at the saddle points of the potential function and the Hessians of the potential function at global minima.2010 Mathematics Subject Classification. 35K15, 35K57, 35J20.

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