Analysis and Probability seminar

Abstract: We study the variation seminorm of order $\varrho$ for a general Ornstein--Uhlenbeck semigroup $\left(H_t\right)_{t > 0}$, in any finite dimension. This is a way of measuring the speed of the convergence $H_t\,f \to f$ as $t\to 0$. For $\varrho>2$ this variation is known to define a bounded operator on $\L^p$ with respect to the invariant measure, and we prove the corresponding weak type (1,1). But when  $\varrho \le 2$ no strong nor weak $\L^p$ bounds hold.