average of this quantity is given by $$\langle A \rangle = \int dX_N P[X_N | \Lambda_N] A[X_N, \Lambda_N],
\label{ensemble_average}$$ where $dX_N := \prod_{n=0}^N dx_n$.

Backward Control
----------------

Before proceeding to the nonequilibrium equalities, we consider the stochastic dynamics with a backward control protocol. The backward control protocol means the time-reversal of protocol $\Lambda_N$, which is formulated as follows. Let $\lambda^\ast$ be the time-reversal of $\lambda$; for example, if $\lambda$ is a magnetic field, then $\lambda^\ast = - \lambda$. The time-reversed protocol of $\lambda (t)$ is then given by $\lambda^\dagger (t) := \lambda^\ast (\tau - t)$. The backward protocol can be discretized as $\Lambda^\dagger_n := (\lambda^\ast_{N-1}, \lambda^\ast_{N-2}, \cdots, \lambda^\ast_{N-n-1})$. We define $\lambda^\dagger_n := \lambda^\ast_{N-n-1}$, $\lambda^\dagger_{\rm int} := \lambda_{\rm fin}^\ast$, and $\lambda^\dagger_{\rm fin} := \lambda_{\rm int}^\ast$.

We consider the probability of realizing trajectory $x'(t)$ of the system with a backward control protocol. We define $x'_n := x'(n \Delta t)$ and $X'_n := (x'_0, x'_1, \cdots, x'_N)$. We denote as $P_0^\dagger [x'_0]$ the initial distribution of the backward processes. We stress that $P_0^\dagger [x'_0]$ is not necessarily equal to the final distribution of the forward experiments. In fact, we can prepare a new state for the system to perform the backward experiments after 