\, e^{-\beta W_d^F[\gamma^F]} \equiv
\int d\gamma^F\, {\cal Q}^F[\gamma^F] ,$$ where ${\cal Q}^F  = {\cal P}^Fe^{-\beta W_d^F}$. Let $\zeta_{\rm typ}^F$ and $\zeta_{\rm dom}^F$ denote the regions of trajectory space where ${\cal P}^F$ and ${\cal Q}^F$, respectively, are peaked, as illustrated in Fig.\[fig:pathSpace\]. Thus, while $\zeta_{\rm typ}^F$ contains the [*typical*]{} forward realizations, $\zeta_{\rm dom}^F$ contains the [*dominant*]{} ones, since the greatest contribution in Eq.\[eq:exp\_F.q\] comes from the peak region of ${\cal Q}^F$.

For the reverse process, we have $$1 = \Bigl\langle e^{-\beta W_d}\Bigr\rangle_R = 
\int d\gamma^R {\cal P}^R[\gamma^R] \, e^{-\beta W_d^R[\gamma^R]} \equiv
\int d\gamma^R\, {\cal Q}^R[\gamma^R],$$ and we define regions $\zeta_{\rm typ}^R$ and $\zeta_{\rm dom}^R$ where ${\cal P}^R$ and ${\cal Q}^R$ are peaked.

Combining Eq.\[eq:ratio\] with the definitions of ${\cal Q}^F$ and ${\cal Q}^R$, we get

\[eq:qFpR.qRpF\] $$\begin{aligned}
\label{eq:qFpR}
{\cal Q}^F[\gamma^F] &=& {\cal P}^R[\gamma^R] \\
\label{eq:qRpF}
{\cal Q}^R[\gamma^R] &=& {\cal P}^F[\gamma^F] .\end{aligned}$$

Eq. \[eq:qFpR\] states that the function ${\cal Q}^F$ is the conjugate image of the distribution ${\cal P}^R$; thus if we plot ${\cal P}^R$ in the lower box of Fig. \[fig:pathSpace\], then its mirror image in the upper box is ${\cal Q}^F$. It follows that the trajectories in $\zeta_{\rm dom}^F$ (the peak region of ${\cal Q}^F$) are the conjugate twins of 