$$
\begin{array}{l} I _ {4} \leq \| X (t, \boldsymbol {x} _ {1}, \nu_ {1}) - X (t, \boldsymbol {x} _ {2}, \nu_ {2}) \| ^ {p - 1} \cdot \| X (t, \boldsymbol {x} _ {2}, \nu_ {2}) \| \cdot \\ \int_ {\mathbb {R} ^ {2 d}} \sup  _ {\theta \in [ 0, 1 ]} \left| \left| \nabla^ {2} V \left(\theta X \left(t, \boldsymbol {x} _ {1} ^ {\prime}, \nu_ {1}\right) + (1 - \theta) X \left(t, \boldsymbol {x} _ {2} ^ {\prime}, \nu_ {2}\right)\right) \right| \right|. \\ \left\| X \left(t, \boldsymbol {x} _ {1} ^ {\prime}, \nu_ {1}\right) - X \left(t, \boldsymbol {x} _ {2} ^ {\prime}, \nu_ {2}\right) \right\| \cdot \left\| X \left(t, \boldsymbol {x} _ {2} ^ {\prime}, \nu_ {2}\right) \right\| \pi^ {0} \left(\mathrm {d} \boldsymbol {x} _ {1} ^ {\prime} \mathrm {d} \boldsymbol {x} _ {2} ^ {\prime}\right) \\ \leq \left\| X (t, \boldsymbol {x} _ {1}, \nu_ {1}) - X (t, \boldsymbol {x} _ {2}, \nu_ {2}) \right\| ^ {p - 1} \left\| X (t, \boldsymbol {x} _ {2}, \nu_ {2}) \right\| \cdot \\ \left(\int_ {\mathbb {R} ^ {2 d}} \sup  _ {\theta \in [ 0, 1 ]} \left\| \nabla^ {2} V \left(\theta X \left(t, \boldsymbol {x} _ {1} ^ {\prime}, \nu_ {1}\right) + (1 - \theta) X \left(t, \boldsymbol {x} _ {2} ^ {\prime}, \nu_ {2}\right)\right) \right\| ^ {q} \| X \left(t, \boldsymbol {x} _ {2} ^ {\prime}, \nu_ {2}\right) \| ^ {q} \pi^ {0} \left(\mathrm {d} \boldsymbol {x} _ {1} ^ {\prime} \mathrm {d} \boldsymbol {x} _ {2} ^ {\prime}\right)\right) ^ {1 / q}. \\ \left(\int_ {\mathbb {R} ^ {2 d}} \left\| X \left(t, \boldsymbol {x} _ {1} ^ {\prime}, \nu_ {1}\right) - X \left(t, \boldsymbol {x} _ {2} ^ {\prime}, \nu_ {2}\right) \right\| ^ {p} \pi^ {0} \left(\mathrm {d} \boldsymbol {x} _ {1} ^ {\prime} \mathrm {d} \boldsymbol {x} _ {2} ^ {\prime}\right)\right) ^ {1 / p} \\ \stackrel {a} {\leq} C _ {V} ^ {1 / q} \left\| X (t, \boldsymbol {x} _ {1}, \nu_ {1}) - X (t, \boldsymbol {x} _ {2}, \nu_ {2}) \right\| ^ {p - 1} \left\| X (t, \boldsymbol {x} _ {2}, \nu_ {2}) \right\|. \\ \left(\int_ {\mathbb {R} ^ {2 d}} \left(\| V (X (t, \boldsymbol {x} _ {1} ^ {\prime}, \nu_ {1})) \| + \| V (X (t, \boldsymbol {x} _ {2} ^ {\prime}, \nu_ {2})) \| + \| X (t, \boldsymbol {x} _ {2} ^ {\prime}, \nu_ {2}) \| ^ {q}\right) \pi^ {0} (\mathrm {d} \boldsymbol {x} _ {1} ^ {\prime} \mathrm {d} \boldsymbol {x} _ {2} ^ {\prime})\right) ^ {1 / q}. \\ \left(\int_ {\mathbb {R} ^ {2 d}} \left\| X \left(t, \boldsymbol {x} _ {1} ^ {\prime}, \nu_ {1}\right) - X \left(t, \boldsymbol {x} _ {2} ^ {\prime}, \nu_ {2}\right) \right\| ^ {p} \pi^ {0} \left(\mathrm {d} \boldsymbol {x} _ {1} ^ {\prime} \mathrm {d} \boldsymbol {x} _ {2} ^ {\prime}\right)\right) ^ {1 / p} \\ \leq C _ {V} ^ {1 / q} \left(\| \mu_ {1, t} \| _ {\mathcal {P} _ {V}} ^ {1 / q} + \| \mu_ {2, t} \| _ {\mathcal {P} _ {V}} ^ {1 / q} + \| \mu_ {2, t} \| _ {\mathcal {P} ^ {q}}\right) \| X (t, \boldsymbol {x} _ {1}, \nu_ {1}) - X (t, \boldsymbol {x} _ {2}, \nu_ {2}) \| ^ {p - 1} \| X (t, \boldsymbol {x} _ {2}, \nu_ {2}) \| \cdot \\ \left(\int_ {\mathbb {R} ^ {2 d}} \left\| X \left(t, \boldsymbol {x} _ {1} ^ {\prime}, \nu_ {1}\right) - X \left(t, \boldsymbol {x} _ {2} ^ {\prime}, \nu_ {2}\right) \right\| ^ {p} \pi^ {0} \left(\mathrm {d} \boldsymbol {x} _ {1} ^ {\prime} \mathrm {d} \boldsymbol {x} _ {2} ^ {\prime}\right)\right) ^ {1 / p} \\ \overset {b} {\leq} C _ {V} ^ {1 / q} \left(e ^ {C _ {1} t / q} \| \nu_ {1} \| _ {\mathcal {P} _ {V}} ^ {1 / q} + e ^ {C _ {1} t / q} \| \nu_ {2} \| _ {\mathcal {P} _ {V}} ^ {1 / q} + e ^ {C _ {2} t} \| \nu_ {2} \| _ {\mathcal {P} ^ {p}}\right) \| X (t, \boldsymbol {x} _ {1}, \nu_ {1}) - X (t, \boldsymbol {x} _ {2}, \nu_ {2}) \| ^ {p - 1}. \\ \| X (t, \boldsymbol {x} _ {2}, \nu_ {2}) \| \left(\int_ {\mathbb {R} ^ {2 d}} \| X (t, \boldsymbol {x} _ {1} ^ {\prime}, \nu_ {1}) - X (t, \boldsymbol {x} _ {2} ^ {\prime}, \nu_ {2}) \| ^ {p} \pi^ {0} (\mathrm {d} \boldsymbol {x} _ {1} ^ {\prime} \mathrm {d} \boldsymbol {x} _ {2} ^ {\prime})\right) ^ {1 / p} \tag {74} \\ \end{array}
$$