= Solution
With an external force $\mathbf F$, the required forward and backward noise histories become
$$
\mathbf f_F=E+\zeta\dot{\mathbf x}-\mathbf F,
\qquad
\mathbf f_B=E-\zeta\dot{\mathbf x}-\mathbf F.
$$
Repeating the difference of squares and using $2\zeta/\sigma^2=\beta$ gives
$$
\boxed{\frac{\mathbb P_F}{\mathbb P_B}
=\exp\left[-\beta\Delta H
+\beta\int_{t_1}^{t_2}
\mathbf F(\mathbf x)\mathbin\cdot\dot{\mathbf x}\,dt\right].}
$$
The second term is $\beta$ times the <work> performed by the external force.
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