Null-controllability of the Kolmogorov equation in the whole phase space

We prove the null controllability, in arbitrary positive time, of the Kolmogorov equation ∂t+v⋅∇x−△v∂t+v⋅∇x−△v with (x,v)∈Rd×Rd(x,v)∈Rd×Rd, with a control region of the form ω=ωx×ωvω=ωx×ωv, where both ωxωx and ωvωv are open subsets of RdRd that are sufficiently spread out throughout the whole space RdRd. The proof is based on, on the one hand, a spectral inequality in RdRd with an observation on ωxωx, and, on the other hand, a Carleman-based observability inequality for a family of parabolic operators, ∂t−iv⋅ξ−△v∂t−iv⋅ξ−△v, coupled with a knowledge of the decay rate of the free solutions of the Kolmogorov equation.