$dX_N^\dagger = dX_N$ holds, because $dx_n = dx_n^\ast$.

Nonequilibrium Equalities
-------------------------

We now discuss nonequilibrium equalities. Let $Q_i[X_N, \lambda_N]$ be the heat that is absorbed by the system from the $i$th heat bath satisfying $Q_i[X_N, \Lambda_N] = -Q_i[X^\dagger_N, \Lambda^\dagger_N]$. We write $Q_i[X_N, \lambda_N]$ simply as $Q_i[X_N]$ for simplicity. It has been established that the following equality is satisfied for stochastic thermodynamic systems [@Crooks1; @Crooks2; @Jarzynski2; @Seifert2]: $$\frac{P^\dagger [X^\dagger_N | x^\ast_0]}{P[X_N | x_0]} = \exp \left( \sum_i \beta_i Q_i [X_N] \right),
\label{fluctuation1}$$ which is referred to as the detailed fluctuation theorem (or the transient fluctuation theorem). This is the starting point of our research. We can rewrite Eq. (\[fluctuation1\]) as $$\frac{P^\dagger [X^\dagger_N]}{P[X_N ]} = e^{ -\sigma [X_N] },
\label{fluctuation2}$$ where $$\sigma [X_N] := - \ln P^\dagger_0 [x^\dagger_0] + \ln P_0 [x_0] -  \sum_i \beta_i Q_i [X_N],
\label{entropy1}$$ which is called the entropy production along trajectory $X_N$.

Various proofs of the detailed fluctuation theorem \[Eqs. (\[fluctuation1\]) and (\[fluctuation2\])\] for stochastic systems have been presented, for example, in Refs. [@Crooks1; @Crooks2; @Seifert2] for the Markovian stochastic dynamics and in Ref. [@Ohkuma] for non-Markovian Langevin systems. A proof of Eqs. (\[fluctuation1\]) and (\[fluctuation2\]) has also been given in Ref. [@Jarzynski2] for the situations in 