Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 3 / a / Solution

Put $E(t)=-\delta A/\delta \mathbf{x}(t)$. For a specified forward trajectory, the equation of motion determineswhereas its time reverse requiresAssume the action is invariant under time-reversal symmetry, the position is even and velocity is odd under time reversal, and the trajectory-to-noise Jacobian is identical in the two directions. The Onsager--Machlup path probability is thenwith $\zeta \dot{\mathbf{x}}$ replaced by $-\zeta \dot{\mathbf{x}}$ for ${\mathbb{P}}_B$. Sincetheir ratio is