Non-null-controllability of the fractional heat equation and of the Kolmogorov equation

We prove in this article that the Kolmogorov-type equation $(\partial_t -\partial_v^2 + v^2\partial_x)f(t,x,v) = \mathbf 1_\omega u(t,x,v)$ for $(t,x)\in \mathbb T\times \Omega_v$ with $\Omega_v = \mathbb R$ or $(-1,1)$ is not null-controllable in any time if $\omega$ is a vertical band $\omega_x\times \Omega_v$. The idea is to remark that, for some families of solutions, the Kolmogorov equation behaves like what we'll call the rotated fractional heat equation $(\partial_t + \sqrt i(-\Delta)^{1/4})g(t,x) = \mathbf 1_\omega u(t,x)$, $x\in \mathbb T$ and to disprove the observability inequality for rotated fractional equation by looking at how coherent states evolve