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In this paper, we study the Onsager-Machlup function and its relationship tothe Freidlin-Wentzell function for measures equivalent to arbitrary infinitedimensional Gaussian measures. The Onsager-Machlup function can serve as adensity on infinite dimensional spaces, where a uniform measure does not exist,and has been seen as the Lagrangian for the ``most likely element". TheFreidlin-Wentzell rate function is the large deviations rate function forsmall-noise limits and has also been identified as a Lagrangian for the ``mostlikely element". This leads to a conundrum - what is the relationship betweenthese two functions?  We show both pointwise and $\Gamma$-convergence (which is essentially theconvergence of minimizers) of the Onsager-Machlup function under thesmall-noise limit to the Freidlin-Wentzell function - and give an expressionfor both. That is, we show that the small-noise limit of the most likelyelement is the most likely element in the small noise limit for infinitedimensional measures that are equivalent to a Gaussian. Examples of measuresinclude the law of solutions to path-dependent stochastic differentialequations and the law of an infinite system of random algebraic equations.

The Small-Noise Limit of the Most Likely Element is the Most Likely Element in the Small-Noise Limit