SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\frac {\partial \widehat {\Psi} _ {t}}{\partial t} = \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \widehat {\Psi} _ {t} - \nabla_ {\mathbf {x}} \widehat {\Psi} _ {t} ^ {\mathrm {T}} \mathbf {v} \tag {20}
$$
$$
s. t \quad \Psi (\mathbf {x}, \mathbf {v}, 0) \widehat {\Psi} (\mathbf {x}, \mathbf {v}, 0) = p (\mathbf {x}, \mathbf {v}, 0), \quad \Psi (\mathbf {x}, \mathbf {v}, T) \widehat {\Psi} (\mathbf {x}, \mathbf {v}, T) = p (\mathbf {x}, \mathbf {v}, T)
$$
Proof. By Lemma.B.1, we notice that the optimal control is:
By leveraging Hopf-Cole [43, 44] transformation, here we define
$$
\Psi = \exp {(\phi)},
$$
$$
\widehat {\Psi} = \mu \exp (- \phi).
$$
Then we can have the following expressions:
$$
\begin{array}{l} \nabla \Psi = \exp (\phi) \nabla \phi \\ \Delta \Psi = \nabla \cdot (\nabla \Psi) \\ = \sum_ {i} \frac {\partial}{\partial \mathbf {m} _ {i}} [ \exp (\phi) \nabla \phi ] \\ = \left[ \nabla \phi^ {\mathsf {T}} \left(\exp (\phi) \frac {\partial \phi_ {i}}{\partial \mathbf {m} _ {i}}\right) + \exp (\phi) \frac {\partial (\nabla \phi) _ {i}}{\partial \mathbf {m} _ {i}} \right] \\ = \exp (\phi) \left[ \| \nabla \phi \| _ {2} ^ {2} + \Delta \phi \right] \\ \end{array}
$$
$$
\begin{array}{l} \nabla \hat {\Psi} = \mu \exp (- \phi) (- \nabla \phi) + \exp (- \phi) \nabla \mu \\ = \exp (- \phi) (- \mu \nabla \phi + \nabla \mu) \\ \end{array}
$$
$$
\begin{array}{l} \Delta \widehat {\Psi} = \nabla \cdot \left(\nabla \widehat {\Psi}\right) \\ = \sum_ {i} \frac {\partial}{\partial \mathbf {m} _ {i}} [ \exp (- \phi) (- \mu \nabla \phi + \nabla \mu) ] \\ = \sum_ {i} \left[ (- \mu \nabla \phi + \nabla \mu) ^ {\top} \left(\exp (- \phi) \frac {- \partial [ \nabla \phi ] _ {i}}{\partial \mathbf {m} _ {i}}\right) \right. \\ \left. + \exp (- \phi) \left(\frac {\partial}{\partial \mathbf {m} _ {i}} [ \nabla \mu ] _ {i} - \mu \frac {\partial}{\partial \mathbf {m} _ {i}} [ \nabla \phi ] _ {i} - \nabla \phi^ {\top} \frac {\partial}{\partial \mathbf {m} _ {i}} [ \nabla \mu ] _ {i}\right) \right] \\ = \exp (- \phi) [ \mu \| \nabla \phi \| _ {2} ^ {2} - \nabla \mu^ {\mathsf {T}} \nabla \phi + \Delta \mu - \mu \Delta \phi - \nabla \phi^ {\mathsf {T}} \nabla \mu ] \\ = \exp (- \phi) [ \mu \| \nabla \phi \| _ {2} ^ {2} - 2 \nabla \mu^ {\mathsf {T}} \nabla \phi + \Delta \mu - \mu \Delta \phi ] \\ \end{array}
$$
Thus, we can have.
$$
\begin{array}{l} \frac {\partial \Psi}{\partial t} = \exp (\phi) \frac {\partial \phi}{\partial t} \\ = \exp (\phi) \left(- \frac {1}{2} \| g \nabla_ {\mathbf {v}} \phi \| _ {2} ^ {2} - \mathbf {v} ^ {\top} \nabla_ {\mathbf {x}} \phi - \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \phi\right) \\ = - \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \Psi - \nabla_ {\mathbf {x}} \Psi^ {\mathsf {T}} \mathbf {v} \\ \end{array}
$$
$$
\begin{array}{l} \frac {\partial \widehat {\Psi}}{\partial t} = \exp (- \phi) \frac {\partial \mu}{\partial t} - \mu \exp (- \phi) \frac {\partial \phi}{\partial t} \\ = \exp (- \phi) \left(\frac {\partial \mu}{\partial t} - \mu \frac {\partial \phi}{\partial t}\right) \\ = \exp (- \phi) \left[ - \nabla_ {\mathbf {m}} \cdot \left\{\left[ (\boldsymbol {f} + g \mathbf {u}) \mathbf {I} _ {d} \right] \mu \right\} + \frac {1}{2} g ^ {2} \Delta_ {\mathbf {m}} \mu + \mu \left(\frac {1}{2} \| g \nabla_ {\mathbf {v}} \phi \| _ {2} ^ {2} + \mathbf {v} ^ {\mathsf {T}} \nabla_ {\mathbf {x}} \phi + \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \phi\right) \right] \\ = \exp (- \phi) \left[ - \nabla_ {\mathbf {v}} \cdot (g ^ {2} \mu \nabla_ {\mathbf {v}} \phi) - \mathbf {v} ^ {\mathsf {T}} \nabla_ {\mathbf {x}} \mu + \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \mu + \frac {\mu}{2} \| g \nabla_ {\mathbf {v}} \phi \| _ {2} ^ {2} + \mu \mathbf {v} ^ {\mathsf {T}} \nabla_ {\mathbf {x}} \phi + \mu \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \phi \right] \\ = \exp (- \phi) \left[ - g ^ {2} \nabla_ {\mathbf {v}} \mu^ {\mathsf {T}} \nabla_ {\mathbf {v}} \phi - g ^ {2} \mu \Delta_ {\mathbf {v}} \phi - \mathbf {v} ^ {\mathsf {T}} \nabla_ {\mathbf {x}} \mu + \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \mu + \frac {\mu}{2} \| g \nabla_ {\mathbf {v}} \phi \| _ {2} ^ {2} + \mu \mathbf {v} ^ {\mathsf {T}} \nabla_ {\mathbf {x}} \phi + \mu \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \phi \right] \\ = \exp (- \phi) \left[ - g ^ {2} \nabla_ {\mathbf {v}} \mu^ {\mathsf {T}} \nabla_ {\mathbf {v}} \phi - \mu \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \phi - \mathbf {v} ^ {\mathsf {T}} \nabla_ {\mathbf {x}} \mu + \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \mu + \frac {\mu}{2} \| g \nabla_ {\mathbf {v}} \phi \| _ {2} ^ {2} + \mu \mathbf {v} ^ {\mathsf {T}} \nabla_ {\mathbf {x}} \phi \right] \\ = \exp (- \phi) \left[ \frac {\mu}{2} \| g \nabla_ {\mathbf {v}} \phi \| _ {2} ^ {2} - g ^ {2} \nabla_ {\mathbf {v}} \mu^ {\mathsf {T}} \nabla_ {\mathbf {v}} \phi - \mu \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \phi + \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \mu - \mathbf {v} ^ {\mathsf {T}} \nabla_ {\mathbf {x}} \mu + \mu \mathbf {v} ^ {\mathsf {T}} \nabla_ {\mathbf {x}} \phi \right] \\ = \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \hat {\boldsymbol {\Psi}} - \nabla_ {\mathbf {x}} \hat {\boldsymbol {\Psi}} ^ {\mathsf {T}} \mathbf {v} \\ \end{array}
$$
Then we can represent the optimal control as:
$$
\begin{array}{l} \mathbf {u} _ {t} ^ {f *} := \mathbf {Z} _ {t} \equiv \left( \begin{array}{c} \mathbf {0} \\ \mathbf {z} _ {t} \end{array} \right) (21) \\ \equiv \binom {\mathbf {0}} {g \nabla_ {\mathbf {v}} \phi} \stackrel {\text {H o p f - C o l e}} {=} \binom {\mathbf {0}} {g \nabla_ {\mathbf {v}} \log \Psi_ {t}} (22) \\ \end{array}
$$
Then the solution of such mSB is characterized by the forward SDE:
$$
\mathrm {d} \mathbf {m} _ {t} = \left[ \boldsymbol {f} + g \mathbf {u} _ {t} ^ {f *} \right] \mathrm {d} t + g (t) \mathrm {d} \mathbf {w} _ {t}, \tag {23}
$$
Due to the structure of Hopf-Cole transform, one can have
$$
p _ {t} ^ {S B} = p _ {t} ^ {e q. (2 3)} = \Psi_ {t} \widehat {\Psi} _ {t} \tag {24}
$$
According to [27, 28], the reverse drift of such SDE (eq.23)  $\mathbf{u}_t^{b*}$  should admits,
$$
\mathbf {u} _ {t} ^ {f *} + \mathbf {u} _ {t} ^ {b *} = \mathbf {g} \nabla_ {\mathbf {v}} \log p _ {t} ^ {S B} \tag {25}
$$
$$
\binom {\mathbf {0}} {g \nabla_ {\mathbf {v}} \log \Psi_ {t}} + \mathbf {u} _ {t} ^ {b *} = \binom {\mathbf {0}} {g \nabla_ {\mathbf {v}} \log \Psi_ {t} + g \nabla_ {\mathbf {v}} \log \widehat {\Psi} _ {t}} \tag {26}
$$
$$
\mathbf {u} _ {t} ^ {b *} = \binom {\mathbf {0}} {g \nabla_ {\mathbf {v}} \log \hat {\Psi} _ {t}} \tag {27}
$$
which yields The backward optimal control
$$
\mathbf {u} _ {t} ^ {b *} := \hat {\mathbf {Z}} _ {t} \equiv \left( \begin{array}{c} \mathbf {0} \\ \hat {\mathbf {z}} _ {t} \end{array} \right) \equiv \left( \begin{array}{c} \mathbf {0} \\ g \nabla_ {\mathbf {v}} \log \hat {\Psi} _ {t} \end{array} \right) \tag {28}
$$
Thus, the optimal forward and backward process is
$$
\mathrm {d} \mathbf {m} _ {t} = \left[ \boldsymbol {f} + g \mathbf {u} _ {t} ^ {f *} \right] \mathrm {d} t + g (t) \mathrm {d} \mathbf {w} _ {t} \tag {29}
$$
$$
\mathrm {d} \mathbf {m} _ {s} = \left[ \boldsymbol {f} + g \mathbf {u} _ {s} ^ {b *} \right] \mathrm {d} t + g (t) \mathrm {d} \widehat {\mathbf {w}} _ {s} \tag {30}
$$
And  $\Psi$  and  $\hat{\Psi}$  satisfy following PDEs,
$$
\frac {\partial \hat {\boldsymbol {\Psi}} _ {t}}{\partial t} = \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \hat {\boldsymbol {\Psi}} _ {t} - \nabla_ {\mathbf {x}} \hat {\boldsymbol {\Psi}} _ {t} ^ {\mathsf {T}} \mathbf {v}
$$
Lemma B.3. By specifying  $\mathbf{f} = [\mathbf{v},\mathbf{0}]^{\top}$ , The PDE shown in 20 can be represented by following SDEs
$$
\left( \begin{array}{c} \mathrm {d} \mathbf {x} \\ \mathrm {d} \mathbf {v} \end{array} \right) = \left( \begin{array}{c} \mathbf {v} \\ - g ^ {2} \nabla_ {\mathbf {v}} \log \Psi \end{array} \right) \mathrm {d} t + \left( \begin{array}{c c} \mathbf {0} & \mathbf {0} \\ \mathbf {0} & g \end{array} \right) \mathrm {d} \mathbf {w} \tag {31}
$$
$$
\mathrm {d} \mathbf {y} = \frac {1}{2} \| \mathbf {z} \| ^ {2} \mathrm {d} t + \mathbf {z} ^ {\top} \mathrm {d} \mathbf {w} _ {t} \tag {32}
$$
$$
\mathrm {d} \widehat {\mathbf {y}} = \left[ \frac {1}{2} \| \widehat {\mathbf {z}} \| ^ {2} + \mathbf {z} ^ {\top} \widehat {\mathbf {z}} + \nabla_ {\mathbf {v}} \cdot g \widehat {\mathbf {z}} \right] \mathrm {d} t + \widehat {\mathbf {z}} ^ {\top} \mathrm {d} \mathbf {w} _ {t} \tag {33}
$$
$$
s. t: \exp \left(\mathbf {y} _ {0} + \widehat {\mathbf {y}} _ {0}\right) = p (\mathbf {x}, \mathbf {v}, 0), \quad \exp \left(\mathbf {y} _ {T} + \widehat {\mathbf {y}} _ {T}\right) = p (\mathbf {x}, \mathbf {v}, T) \tag {34}
$$
$$
\mathbf {y} \equiv \mathbf {y} (\mathbf {x} _ {t}, \mathbf {v}, t) = \log \Psi (\mathbf {x} _ {t}, \mathbf {v} _ {t}, t), \quad \mathbf {z} \equiv \mathbf {z} (\mathbf {x} _ {t}, \mathbf {v} _ {t}, t) = g \nabla_ {\mathbf {v}} \log \Psi (\mathbf {x} _ {t}, \mathbf {v} _ {t}, t)
$$
$$
\widehat {\mathbf {y}} \equiv \widehat {\mathbf {y}} (\mathbf {x} _ {t}, \mathbf {v} _ {t}, t) = \log \widehat {\Psi} (\mathbf {x} _ {t}, \mathbf {v} _ {t}, t), \quad \widehat {\mathbf {z}} \equiv \widehat {\mathbf {z}} (\mathbf {x} _ {t}, \mathbf {v} _ {t}, t) = g \nabla_ {\mathbf {v}} \log \widehat {\Psi} (\mathbf {x} _ {t}, \mathbf {v} _ {t}, t)
$$
Proof. One can write
$$
\begin{array}{l} \frac {\partial \log \Psi}{\partial t} = \frac {1}{\Psi} \left(- \nabla_ {\mathbf {x}} \Psi^ {\mathsf {T}} \mathbf {v} - \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \Psi\right) \\ = - \nabla_ {\mathbf {x}} \log \Psi^ {\mathsf {T}} \mathbf {v} - \frac {1}{2} g ^ {2} \frac {\Delta_ {\mathbf {v}} \Psi}{\Psi} \\ = - \nabla_ {\mathbf {x}} \log \boldsymbol {\Psi} ^ {\mathsf {T}} \mathbf {v} - \frac {1}{2} g ^ {2} \operatorname {T r} \left[ \frac {1}{\boldsymbol {\Psi}} \nabla_ {\mathbf {v}} ^ {2} \boldsymbol {\Psi} \right] \\ \end{array}
$$
$$
\begin{array}{l} \frac {\partial \log \widehat {\Psi}}{\partial t} = \frac {1}{\widehat {\Psi}} \left(- \nabla_ {\mathbf {x}} \widehat {\Psi} ^ {\mathsf {T}} \mathbf {v} + \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \widehat {\Psi}\right) \\ = - \nabla_ {\mathbf {x}} \log \widehat {\boldsymbol {\Psi}} ^ {\mathsf {T}} \mathbf {v} - \frac {1}{2} g ^ {2} \operatorname {T r} \left[ \frac {1}{\widehat {\boldsymbol {\Psi}}} \nabla_ {\mathbf {v}} ^ {2} \widehat {\boldsymbol {\Psi}} \right] \\ \end{array}
$$
By applying Itô's lemma,
$$
\begin{array}{l} \mathrm {d} \log \Psi = \frac {\partial \log \Psi}{\partial t} \mathrm {d} t + \left[ \nabla_ {\mathbf {x}} \log \Psi^ {\mathsf {T}} \mathbf {v} + g ^ {2} \| \nabla_ {\mathbf {v}} \log \Psi \| _ {2} ^ {2} + \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \log \Psi \right] \mathrm {d} t + \left[ \nabla_ {\mathbf {m}} \log \Psi^ {\mathsf {T}} \right] g \mathrm {d} \mathbf {w} _ {t} \\ = \left[ - \nabla_ {\mathbf {x}} \log \Psi^ {\mathsf {T}} \mathbf {v} - \frac {1}{2} g ^ {2} \operatorname {T r} \left[ \frac {1}{\Psi} \nabla_ {\mathbf {v}} ^ {2} \Psi \right] \right] \mathrm {d} t \\ + \left[ \nabla_ {\mathbf {x}} \log \Psi^ {\mathsf {T}} \mathbf {v} + g ^ {2} \| \nabla_ {\mathbf {v}} \log \Psi \| _ {2} ^ {2} + \frac {1}{2} g ^ {2} \operatorname {T r} \left[ \frac {1}{\Psi} \nabla_ {\mathbf {v}} ^ {2} \Psi - \frac {1}{\Psi^ {2}} \nabla_ {\mathbf {v}} \Psi \nabla_ {\mathbf {v}} \Psi^ {\mathsf {T}} \right] \right] \mathrm {d} t + g \left[ \nabla_ {\mathbf {v}} \log \Psi^ {\mathsf {T}} \right] \mathrm {d} \mathbf {w} _ {t} \\ = \left[ g ^ {2} \| \nabla_ {\mathbf {v}} \log \Psi \| _ {2} ^ {2} - \frac {1}{2} g ^ {2} \operatorname {T r} \left[ \frac {1}{\Psi^ {2}} \nabla_ {\mathbf {v}} \Psi \nabla_ {\mathbf {v}} \Psi^ {\mathsf {T}} \right] \right] \mathrm {d} t + g \left[ \nabla_ {\mathbf {v}} \log \Psi^ {\mathsf {T}} \right] \mathrm {d} \mathbf {w} _ {t} \\ = \left[ \frac {1}{2} g ^ {2} \| \nabla_ {\mathbf {v}} \log \Psi \| _ {2} ^ {2} \right] \mathrm {d} t + g \left[ \nabla_ {\mathbf {v}} \log \Psi^ {\mathsf {T}} \right] \mathrm {d} \mathbf {w} _ {t} \\ \end{array}
$$
Similarly, one can have,
$$
\begin{array}{l} \mathrm {d} \log \widehat {\Psi} = \frac {\partial \log \widehat {\Psi}}{\partial t} \mathrm {d} t + \left[ \nabla_ {\mathbf {x}} \log \widehat {\Psi} ^ {\mathsf {T}} \mathbf {v} + g ^ {2} \nabla_ {\mathbf {v}} \log \Psi^ {\mathsf {T}} \nabla_ {\mathbf {v}} \log \widehat {\Psi} + \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \log \widehat {\Psi} \right] \mathrm {d} t + \left[ \nabla_ {\mathbf {m}} \log \widehat {\Psi} ^ {\mathsf {T}} \right] g \mathrm {d} \mathbf {w} _ {t} \\ = \left[ - \nabla_ {\mathbf {x}} \log \widehat {\Psi} ^ {\top} \mathbf {v} + \frac {1}{2} g ^ {2} \operatorname {T r} \left[ \frac {1}{\widehat {\Psi}} \nabla_ {\mathbf {v}} ^ {2} \widehat {\Psi} \right] \right] d t \\ + \left[ \nabla_ {\mathbf {x}} \log \widehat {\Psi} ^ {\mathsf {T}} \mathbf {v} + g ^ {2} \nabla_ {\mathbf {v}} \log \Psi^ {\mathsf {T}} \nabla_ {\mathbf {v}} \log \widehat {\Psi} + \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \log \widehat {\Psi} \right] d t + g \left[ \nabla_ {\mathbf {v}} \log \widehat {\Psi} ^ {\mathsf {T}} \right] d \mathbf {w} _ {t} \\ \end{array}
$$
Noticing:
$$
\begin{array}{l} \frac {1}{2} \left[ \frac {1}{\widehat {\Psi}} \nabla_ {\mathbf {v}} ^ {2} \widehat {\Psi} + \nabla_ {\mathbf {v}} ^ {2} \log \widehat {\Psi} \right] = \operatorname {T r} \left[ \frac {1}{\Psi} \nabla_ {\mathbf {v}} ^ {2} \widehat {\Psi} - \frac {1}{2} \| \nabla_ {\mathbf {v}} \log \widehat {\Psi} \| ^ {2} \right] \\ = \frac {1}{2} \| \nabla_ {\mathbf {v}} \log \widehat {\Psi} \| ^ {2} + \Delta_ {\mathbf {v}} \log \widehat {\Psi} \\ \end{array}
$$
Following the above derivation, one can have,
$$
\begin{array}{l} \mathrm {d} \log \widehat {\Psi} = \left[ - \nabla_ {\mathbf {x}} \log \widehat {\Psi} ^ {\mathsf {T}} \mathbf {v} + \frac {1}{2} g ^ {2} \operatorname {T r} \left[ \frac {1}{\widehat {\Psi}} \nabla_ {\mathbf {v}} ^ {2} \widehat {\Psi} \right] \right] \mathrm {d} t \\ + \left[ \nabla_ {\mathbf {x}} \log \hat {\Psi} ^ {\mathsf {T}} \mathbf {v} + g ^ {2} \nabla_ {\mathbf {v}} \log \Psi^ {\mathsf {T}} \nabla_ {\mathbf {v}} \log \hat {\Psi} + \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \log \hat {\Psi} \right] d t + g \left[ \nabla_ {\mathbf {v}} \log \hat {\Psi} ^ {\mathsf {T}} \right] d \mathbf {w} _ {t} \\ = \left[ g ^ {2} \nabla_ {\mathbf {v}} \log \Psi^ {\mathsf {T}} \nabla_ {\mathbf {v}} \log \widehat {\Psi} + \frac {1}{2} g ^ {2} \| \nabla_ {\mathbf {v}} \log \widehat {\Psi} \| ^ {2} + 2 \frac {1}{2} g ^ {2} \Delta_ {\mathbf {v}} \log \widehat {\Psi} \right] d t + g \left[ \nabla_ {\mathbf {v}} \log \widehat {\Psi} ^ {\mathsf {T}} \right] d \mathbf {w} _ {t} \\ \end{array}
$$
By defining
$$
\widehat {\mathbf {y}} \equiv \widehat {\mathbf {y}} (\mathbf {x} _ {t}, \mathbf {v} _ {t}, t) = \log \widehat {\boldsymbol {\Psi}} (\mathbf {x} _ {t}, \mathbf {v} _ {t}, t), \quad \widehat {\mathbf {z}} \equiv \widehat {\mathbf {z}} (\mathbf {x} _ {t}, \mathbf {v} _ {t}, t) = g \nabla_ {\mathbf {v}} \log \widehat {\boldsymbol {\Psi}} (\mathbf {x} _ {t}, \mathbf {v} _ {t}, t)
$$
One can conclude the results.
$$
\left( \begin{array}{c} \mathrm {d} \mathbf {x} \\ \mathrm {d} \mathbf {v} \end{array} \right) = \left( \begin{array}{c} \mathbf {v} \\ - g ^ {2} \nabla_ {\mathbf {v}} \log \Psi \end{array} \right) \mathrm {d} t + \left( \begin{array}{c c} \mathbf {0} & \mathbf {0} \\ \mathbf {0} & g \end{array} \right) \mathrm {d} \mathbf {w}
$$
$$
\mathrm {d} \mathbf {y} = \frac {1}{2} \| \mathbf {z} \| ^ {2} \mathrm {d} t + \mathbf {z} ^ {\top} \mathrm {d} \mathbf {w} _ {t}
$$
$$
\mathrm {d} \widehat {\mathbf {y}} = \left[ \frac {1}{2} \| \widehat {\mathbf {z}} \| ^ {2} + \mathbf {z} ^ {\top} \widehat {\mathbf {z}} + \nabla_ {\mathbf {v}} \cdot g \widehat {\mathbf {z}} \right] \mathrm {d} t + \widehat {\mathbf {z}} ^ {\top} \mathrm {d} \mathbf {w} _ {t}
$$
$$
\mathbf {s}. \mathbf {t}: \exp (\mathbf {y} _ {0} + \widehat {\mathbf {y}} _ {0}) = p (\mathbf {x}, \mathbf {v}, 0), \quad \exp (\mathbf {y} _ {T} + \widehat {\mathbf {y}} _ {T}) = p (\mathbf {x}, \mathbf {v}, T)
$$
Proposition B.4. The log-likelihood at data point  $\mathbf{m}_0$  can be expressed as
$$
\begin{array}{l} \log p (\mathbf {m} _ {0}, 0) = \mathbb {E} _ {\mathbf {m} _ {t} \sim (1 7)} [ \log p (\mathbf {m} _ {T}, T) ] - \int_ {0} ^ {T} \mathbb {E} _ {\mathbf {m} _ {t} \sim (1 7)} \left[ \frac {1}{2} \| \mathbf {z} _ {t} \| ^ {2} d t + \frac {1}{2} \| \widehat {\mathbf {z}} _ {t} \| ^ {2} + \mathbf {z} _ {t} ^ {\top} \widehat {\mathbf {z}} _ {t} + \nabla_ {\mathbf {v}} \cdot g \widehat {\mathbf {z}} _ {t} \right] d t \\ = \mathbb {E} _ {\mathbf {m} _ {t} \sim (1 7)} [ \log p (\mathbf {m} _ {T}, T) ] - \\ \int_ {0} ^ {T} \mathbb {E} _ {\mathbf {m} _ {t} \sim (1 7)} \left[ \frac {1}{2} \| \mathbf {z} _ {t} \| ^ {2} + \underbrace {\frac {1}{2} \| \widehat {\mathbf {z}} _ {t} - g \nabla_ {\mathbf {v}} \log p ^ {(1 7)} + \mathbf {z} _ {t} \| ^ {2}} _ {\text {m e a n m a t c h i n g o b j e c t i v e}} - \frac {1}{2} \| g \nabla_ {\mathbf {v}} \log p ^ {(1 7)} - \mathbf {z} _ {t} \| ^ {2} \right] \mathrm {d} t \\ \propto \int_ {0} ^ {T} \mathbb {E} _ {\mathbf {m} _ {t} \sim (1 7)} \left[ \underbrace {\frac {1}{2} \left\| \widehat {\mathbf {z}} _ {t} - g \nabla_ {\mathbf {v}} \log p ^ {(1 7)} + \mathbf {z} _ {t} \right\| ^ {2}} _ {\text {m e a n m a t c h i n g o b j e c t i v e}} \right] d t \\ \end{array}
$$
$$
\begin{array}{l} \log p (\mathbf {m} _ {T}, T) = \mathbb {E} _ {\mathbf {m} _ {t} \sim (1 6)} \left[ \log p (\mathbf {m} _ {0}, 0) \right] - \int_ {0} ^ {T} \mathbb {E} _ {\mathbf {m} _ {t} \sim (1 6)} \left[ \frac {1}{2} \| \mathbf {z} _ {t} \| ^ {2} \mathrm {d} t + \frac {1}{2} \| \widehat {\mathbf {z}} _ {t} \| ^ {2} + \mathbf {z} _ {t} ^ {\mathsf {T}} \widehat {\mathbf {z}} _ {t} + \nabla_ {\mathbf {v}} \cdot g \mathbf {z} _ {t} \right] \mathrm {d} t \\ = \mathbb {E} _ {\mathbf {m} _ {t} \sim (1 6)} [ \log p (\mathbf {m} _ {0}, 0) ] - \\ \int_ {0} ^ {T} \mathbb {E} _ {\mathbf {m} _ {t} \sim (1 6)} \left[ \frac {1}{2} \| \widehat {\mathbf {z}} _ {t} \| ^ {2} + \underbrace {\frac {1}{2} \| \widehat {\mathbf {z}} _ {t} - g \nabla_ {\mathbf {v}} \log p ^ {(1 6)} + \mathbf {z} _ {t} \| ^ {2}} _ {\text {m e a n m a t c h i n g o b j e c t i v e}} - \frac {1}{2} \| g \nabla_ {\mathbf {v}} \log p ^ {(1 6)} - \widehat {\mathbf {z}} _ {t} \| ^ {2} \right] \mathrm {d} t \\ \propto \mathbb {E} _ {\mathbf {m} _ {t} \sim (1 6)} \left[ \underbrace {\frac {1}{2} \| \widehat {\mathbf {z}} _ {t} - g \nabla_ {\mathbf {v}} \log p ^ {(1 6)} + \mathbf {z} _ {t} \| ^ {2}} _ {\text {m e a n m a t c h i n g o b j e c t i v e}} \right] \mathrm {d} t \\ \end{array}
$$
By maximizing the log-likelihood at time  $t = 0$  then  $t = T$  iteratively,  $(\mathbf{z}_t, \widehat{\mathbf{z}}_t)$  will converge to the solution of phase space  $SB$ .
Proof. from Lemma.B.3, one can have:
$$
\begin{array}{l} \log p (\mathbf {m} _ {0}, 0) = \mathbb {E} [ \mathbf {y} _ {0} + \widehat {\mathbf {y}} _ {0} ] \\ = \mathbb {E} \left[ \mathbf {y} _ {T} + \widehat {\mathbf {y}} _ {T} \right] - \int_ {0} ^ {T} \mathbb {E} \left[ \frac {1}{2} \| \mathbf {z} _ {t} \| ^ {2} d t + \frac {1}{2} \| \widehat {\mathbf {z}} _ {t} \| ^ {2} + \mathbf {z} _ {t} ^ {\top} \widehat {\mathbf {z}} _ {t} + \nabla_ {\mathbf {v}} \cdot g \widehat {\mathbf {z}} _ {t} \right] d t \\ = \mathbb {E} \left[ \log p (\mathbf {m} _ {T}, T) \right] - \int_ {0} ^ {T} \mathbb {E} \left[ \frac {1}{2} \| \mathbf {z} _ {t} \| ^ {2} + \frac {1}{2} \| \widehat {\mathbf {z}} _ {t} \| ^ {2} + \mathbf {z} _ {t} ^ {\top} \widehat {\mathbf {z}} _ {t} + \nabla_ {\mathbf {v}} \cdot g \widehat {\mathbf {z}} _ {t} \right] d t \\ = \mathbb {E} \left[ \log p (\mathbf {m} _ {T}, T) \right] - \int_ {0} ^ {T} \mathbb {E} \left[ \frac {1}{2} \| \mathbf {z} _ {t} \| ^ {2} + \frac {1}{2} \| \widehat {\mathbf {z}} _ {t} \| ^ {2} - \widehat {\mathbf {z}} _ {t} ^ {\top} (g \nabla_ {\mathbf {v}} \log p ^ {S B}) + \mathbf {z} _ {t} ^ {\top} \widehat {\mathbf {z}} _ {t} \right] d t \\ = \mathbb {E} [ \log p (\mathbf {m} _ {T}, T) ] \\ - \int_ {0} ^ {T} \mathbb {E} \left[ \frac {1}{2} \| \mathbf {z} _ {t} \| ^ {2} + \frac {1}{2} \| \widehat {\mathbf {z}} _ {t} - g \nabla_ {\mathbf {v}} \log p ^ {S B} + \mathbf {z} _ {t} \| ^ {2} - \frac {1}{2} \| g \nabla_ {\mathbf {v}} \log p ^ {S B} - \mathbf {z} _ {t} \| ^ {2} \right] d t \\ \end{array}
$$
A similar result can be obtained for  $\log p(\mathbf{m}_T,T)$
One can notice that the likelihood objective is a continuous time analog of the mean matching objective proposed in [15], and iterative optimization between  $\log p(\mathbf{m}_0, 0)$  and  $\log p(\mathbf{m}_T, T)$  are the continuous analog of IPF. Hence, the convergence proof will keep valid (see Proposition 4 in [15]).
The equivalence of KL divergence optimization in IPF and likelihood optimization is widely analyzed in [14, 15, 18]. The objective function will eventually boil down to the mean matching objective shown in the above proposition.B.4.
Proposition B.5 (Optimality w.r.t.  $\kappa_{\mathrm{boundary}}$ ). Given the reference path measure  $\bar{\pi}$  driven by the policy  $\widehat{\mathbf{z}}_t$  from boundary  $\mu_{t_{i + 1}}$  in the reverse time direction, the optimal path measure in the forward time direction of the following problem
$$
\min  _ {\pi} \mathcal {J} (\pi) := \sum_ {i = 0} ^ {N - 1} K L \left(\pi_ {t _ {i}: t _ {i + 1}} | \bar {\pi} _ {t _ {i}: t _ {i + 1}}\right), \quad s. t \quad \pi \quad \in \left\{\int \pi_ {t _ {i}: t _ {i + 1}} d \mathbf {m} _ {t _ {i + 1}} = \mu_ {t _ {i}}, \int \mu_ {t _ {i}} d \mathbf {v} _ {t _ {i}} = \rho_ {t _ {i}} \right\}
$$
$$
i s: \qquad \pi_ {t _ {i}: t _ {i + 1}} ^ {*} = \frac {\rho_ {t _ {i}} \bar {\pi} _ {t _ {i} : t _ {i + 1}}}{\int \bar {\pi} _ {t _ {i} : t _ {i + 1}} \mathrm {d} \mathbf {m} _ {t _ {i + 1}} \mathrm {d} \mathbf {v} _ {t _ {i}}}.
$$
$$
\left\| \mathbf {z} _ {t} + \widehat {\mathbf {z}} _ {t} - g \nabla_ {\mathbf {v}} \log \hat {p} _ {t} \right\| _ {2} ^ {2} = 0, \tag {35a}
$$
$$
p _ {t _ {i}} \left(\mathbf {v} _ {t _ {i}} \mid \mathbf {x} _ {t _ {i}}\right) \equiv \hat {q} _ {t _ {i}} \left(\mathbf {v} _ {t _ {i}} \mid \mathbf {x} _ {t _ {i}}\right), \tag {35b}
$$
where  $\hat{p}_t$  and  $\hat{q}_t$  denote the marginal density and conditional velocity distribution of the reference path measure at time  $t$  and  $t_i$ , respectively.
Proof. Due to the similarity of optimization for  $\kappa_{\mathrm{boundary}}$ , the close form solution of the next path measure is (see §4 in [1] for detail):
$$
\pi_ {t _ {i}: t _ {i + 1}} ^ {*} = \frac {\rho_ {t _ {i}} \bar {\pi} _ {t _ {i} : t _ {i + 1}}}{\int \bar {\pi} _ {t _ {i}: t _ {i + 1}} \mathrm {d} \mathbf {m} _ {t _ {i + 1}} \mathrm {d} \mathbf {v} _ {t _ {i}}}.
$$
By denoting the transition kernel of parameterized SDE driven by backward policy  $\widehat{\mathbf{z}}_t$  as  $q(\cdot |\cdot)$ , and the time range between  $t_i$  and  $t_{i + 1}$  is discretized into  $S$  interval by EM discretization. Then one can
Doing eq.36 revursively
$$
\begin{array}{l} = p _ {t _ {i}} \left(\mathbf {x} _ {t _ {i}}\right) q \left(\mathbf {v} _ {t _ {i}} \mid \mathbf {x} _ {t _ {i}}\right) q \left(\mathbf {m} _ {t _ {i} + \delta_ {t}} \mid \mathbf {m} _ {t _ {i}}\right) \prod_ {s = 1} ^ {S - 1} q _ {s} \left(\mathbf {m} _ {t _ {i} + (s + 1) \cdot \delta_ {t}} \mid \mathbf {m} _ {t _ {i} + s \cdot \delta_ {t}}\right) \\ = p _ {t _ {i}} \left(\mathbf {x} _ {t _ {i}}\right) q \left(\mathbf {v} _ {t _ {i}} \mid \mathbf {x} _ {t _ {i}}\right) \prod_ {s = 0} ^ {S - 1} q _ {s} \left(\mathbf {m} _ {t _ {i} + (s + 1) \cdot \delta_ {t}} \mid \mathbf {m} _ {t _ {i} + s \cdot \delta_ {t}}\right) \\ \end{array}
$$
According to [15], given the policy  $\widehat{\mathbf{z}}_t$ , the transition kernel  $q_{s}(\mathbf{m}_{t_i + (s + 1)\cdot \delta_t}|\mathbf{m}_{t_i + s\cdot \delta_t})$  can be estimated by  $\widehat{\mathbf{z}}_t$  (see Proposition 3 in [15]) and it can be treated as the label for the forward policy  $\mathbf{z}_t$  for all  $s$ . Thus, if  $\pi_{t_it_{i + 1}}$  is aligned with  $\pi_{t_it_{i + 1}}^*$ , then one can construct following objective function for policy  $\mathbf{z}_t$ :