Smoothing properties of fractional Ornstein-Uhlenbeck semigroups and null-controllability

Abstract

We study fractional hypoelliptic Ornstein-Uhlenbeck operators acting on L 2 (R n) satisfying the Kalman rank condition. We prove that the semigroups generated by these operators enjoy Gevrey regularizing effects. Two byproducts are derived from this smoothing property. On the one hand, we prove the null-controllability in any positive time from thick control subsets of the associated parabolic equations posed on the whole space. On the other hand, by using interpolation theory, we get global L 2 subelliptic estimates for the these operators.

Exact Splitting Methods for Semigroups Generated by Inhomogeneous Quadratic Differential Operators

“…The setting of the exact splitting is the same as the setting of the classical ones. Nevertheless, we have to assume that exp(tL (1) ), . .…”

“…However, allowing the sub-time-steps of the splitting method to be nonlinear functions of δ t , the following splitting formula can be established (see subsection 7.4 of [2] for a proof) (2) e −δt(|x| 2 −∆) = e − 1 2 tanh(δt)|x| 2 e 1 2 sinh(2δt)∆ e − 1 2 tanh(δt)|x| 2 . Note that, contrary to the Strang splitting (1), this factorization is exact : there is no remainder term. Consequently, it is much more accurate than (1) and the time step can be taken quite large (the only possible restrictions coming from the spatial discretization).…”

“…Note that, contrary to the Strang splitting (1), this factorization is exact : there is no remainder term. Consequently, it is much more accurate than (1) and the time step can be taken quite large (the only possible restrictions coming from the spatial discretization). Furthermore, it is as cheap as (1) to compute.…”

“…A cheaper classical method consists in splitting the harmonic oscillator. For example, here, a natural splitting would be the following Strang splitting (1) e −δt(|x| 2 −∆) ≃ e −δt|x| 2 /2 e δt∆ e −δt|x| 2 /2 .…”

“…Consequently, it is much more accurate than (1) and the time step can be taken quite large (the only possible restrictions coming from the spatial discretization). Furthermore, it is as cheap as (1) to compute.…”