Short-time parametrix for the Fokker--Planck semigroup and applications

We construct a short-time parametrix for the Fokker–Planck semigroup in Euclidean space. Among possible applications, we obtain smoothing and localization properties of the semigroup, the derivation of an approximate short-time polar decomposition for the semigroup, the construction of a parametrix of the resolvent, pseudospectral estimates, and estimates of the asymptotics of the number of eigenvalues of the Fokker–Planck operator. As a step in the proofs, we introduce a class of operators, which we call subsectorial operators, to which the Fokker–Planck operator belongs, and describe some of their functional analytic and spectral properties.