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The Onsager-Machlup (OM) functional is well-known for characterizing the mostprobable transition path of a diffusion process with non-vanishing noise.However, it suffers from a notorious issue that the functional is unboundedbelow when the specified transition time $T$ goes to infinity. This hinders theinterpretation of the results obtained by minimizing the OM functional. Weprovide a new perspective on this issue. Under mild conditions, we show thatalthough the infimum of the OM functional becomes unbounded when $T$ goes toinfinity, the sequence of minimizers does contain convergent subsequences onthe space of curves. The graph limit of this minimizing subsequence is anextremal of the abbreviated action functional, which is related to the OMfunctional via the Maupertuis principle with an optimal energy. We furtherpropose an energy-climbing geometric minimization algorithm (EGMA) whichidentifies the optimal energy and the graph limit of the transition pathsimultaneously. This algorithm is successfully applied to several typicalexamples in rare event studies. Some interesting comparisons with theFreidlin-Wentzell action functional are also made.

The Graph Limit of The Minimizer of The Onsager-Machlup Functional and Its Computation