In applications such as molecular dynamics it is of interest to fit Smoluchowski
and Langevin equations to data. Practitioners often achieve this by a variety of seemingly ad hoc
procedures such as fitting to the empirical measure generated by the data, and fitting to properties of
auto-correlation functions. Statisticians, on the other hand, often use estimation procedures which fit
diffusion processes to data by applying the maximum likelihood principle to the path-space density
of the desired model equations, and through knowledge of the properties of quadratic variation. In
this note we show that these procedures used by practitioners and statisticians to fit drift functions
are, in fact, closely related and can be thought of as two alternative ways to regularize the (singular)
likelihood function for the drift. We also present the results of numerical experiments which probe
the relative efficacy of the two approaches to model identification and compare them with other
methods such as the minimum distance estimator
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