In this talk we review some recent results, obtained in a joint work with Paolo Ciatti and Peter Sjögren, concerning variational and jump inequalities for a nonsymmetric Ornstein–Uhlenbeck semigroup $(\mathcal H_t)_{t> 0}$. First, we introduce variation seminorms and consider the variation operator of $(\mathcal H_t)_{t> 0}$, taken with respect to $t$, in $\mathbb R^n$.

We discuss boundedness properties of this operator in Lebesgue spaces with respect to the invariant measure, focusing, in particular, on the weak type $(1,1)$ of the variation operator of order $\varrho$ of $(\mathcal H_t)_{t> 0}$ for $\varrho>2$. For $1\le \varrho\le 2$ we present a negative result.

If time permits, we shall discuss jump inequalities (which provide an endpoint refinement when $\varrho=2$) in this setting.