Boundary-condition independence of the dynamic conjugate

Determine whether the dynamic conjugate \(\mathcal{Y}(x,t\mid x_0,0)\) for a force-biased Brownian particle is generally independent of the boundary conditions, beyond the observed agreement between free Brownian motion and diffusion on an interval with absorbing boundaries.

Background

The paper derives a dynamic conjugate by differentiating the logarithm of a force-dependent transition density with respect to the applied force. For free biased Brownian motion, this gives a quantity proportional to the displacement, $\mathcal{Y}=\mu(x-x_0)/(2D)$. In the example of a particle confined to an interval with absorbing boundaries, the corresponding non-proper transition density has an additional boundary-dependent Green's-function factor, but that factor is independent of the force; consequently, the same dynamic conjugate is obtained.

The authors explicitly note that the agreement between the free and absorbing-boundary cases does not establish whether this independence persists for other boundary conditions or more general settings. Resolving this question would clarify how universally the dynamic conjugate is determined by the force-induced drift rather than by the details of the boundary dynamics.