We study the \(\varrho \) -th order variation seminorm of a general Ornstein–Uhlenbeck semigroup \(\left( \mathcal H_t\right) _{t>0}\) in \(\mathbb {R}^n\) , 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 \(\varrho > 2\) . 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 \({\mathcal {H}}_t\) in the global region has a bounded number of zeros in (0, 1]. A counterexample is given for \(\varrho = 2\) ; in fact, we prove that the second-order variation seminorm of \(\left( \mathcal H_t\right) _{t>0}\) , and therefore also the \(\varrho \) -th order variation seminorm for any \(\varrho \in [1,2)\) , is not of strong nor weak type (p, p) for any \(p \in [1,\infty )\) with respect to the invariant measure.
