For one-dimensional drift-diffusion equations with small noise, the long-time limit of the solution at each critical time scale equals a Markov chain on a hierarchical set of metastable wells with explicit jump rates.
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From one-dimensional diffusion processes metastable behaviour to parabolic equations asymptotics
For one-dimensional drift-diffusion equations with small noise, the long-time limit of the solution at each critical time scale equals a Markov chain on a hierarchical set of metastable wells with explicit jump rates.