Variational inequalities for the Ornstein–Uhlenbeck semigroup: the higher–dimensional case

We study the ϱ-th order variation seminorm of a general Ornstein–Uhlenbeck semigroup (ℋ_t)_t>0 in ℝ^n, taken with respect to t. We prove that this seminorm defines an operator of weak type (1,1) with respect to the invariant measure when ϱ> 2. For large t, one has an enhanced version of the standard weak-type (1,1) bound. For small t, the proof hinges on vector-valued Calderón–Zygmund techniques in the local region, and on the fact that the t derivative of the integral kernel of ℋ_t in the global region has a bounded number of zeros in (0,1]. A counterexample is given for ϱ= 2; in fact, we prove that the second order variation seminorm of (ℋ_t)_t>0, and therefore also the ϱ-th order variation seminorm for any ϱ∈ [1,2), is not of strong nor weak type (p,p) for any p ∈ [1,∞) with respect to the invariant measure.