Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 344 / 4 / d / Solution

Conditioned on $f$, the additive-noise trajectory has the Onsager–Machlup functionalUnder time reversal, $\dot{x}$ changes sign while $x$ and the active force $f$ are even. Subtracting the forward and backward conditional actions givesThe corresponding ratio for an Ornstein–Uhlenbeck path conditioned on its initial endpoint isConsequentlywhereIf stationary endpoint densities are included in the path measures, their ratio cancels the Ornstein–Uhlenbeck boundary term; the displayed convention is the endpoint-conditioned path probability used in the calculation.