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

We study the -th order variation seminorm of a general Ornstein--Uhlenbeck semigroup ( Hₜ)t>0 in Rⁿ, 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 > 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 Calder\'on--Zygmund techniques in the local region, and on the fact that the t derivative of the integral kernel of Hₜ in the global region has a bounded number of zeros in $(0,1]$. A counterexample is given for = 2; in fact, we prove that the second order variation seminorm of ( Hₜ)t>0, and therefore also the -th order variation seminorm for any ∈ [1,2), is not of strong nor weak type $(p,p)$ for any p ∈ [1,∞) with respect to the invariant measure.