Variational Inequalities for the Ornstein–Uhlenbeck Semigroup: The Higher-Dimensional Case

We study the rho-th order variation seminorm of a general Ornstein–Uhlenbeck semigroup 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 rho>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 in the global region has a bounded number of zeros in (0, 1]. A counterexample is given for rho=2; in fact, we prove that the second-order variation seminorm of (H_t), and therefore also the rho-th order variation seminorm for any rho\in [1,2), is not of strong nor weak type (p, p) for any with respect to the invariant measure.