Appendix

Here we list side remarks as they appear in the text

• For some $f$ which is a function of $(t, x(t))$, we have $\frac{d f}{d t} = \frac{\partial f}{\partial t} + \nabla_x f \cdot \frac{d x}{d t}$

• For some $f: \mathbb R^d \rightarrow \mathbb R^d$ and some scalar $u: \mathbb R^d \rightarrow \mathbb R$, we have $\nabla \cdot (f u) = \nabla \cdot f u + \nabla u \cdot f$