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

Journal article, 2025

We study the & rhov;-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 & rhov; > 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 Calderon-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 & rhov; = 2; in fact, we prove that the second-order variation seminorm of (H-t)(t>0), and therefore also the & rhov;-th order variation seminorm for any & rhov; is an element of [1,2), is not of strong nor weak type (p, p) for any p is an element of [1,infinity) with respect to the invariant measure.