  In this paper, we establish $\mathcal B$-valued variational inequalities for
differential operators, ergodic averages and symmetric diffusion semigroups
under the condition that Banach space $\mathcal B$ has martingale cotype
property. These results generalize, on the one hand Pisier and Xu's result on
the variational inequalities for $\mathcal B$-valued martingales, on the other
hand many classical variational inequalities in harmonic analysis and ergodic
theory. Moreover, we show that Rademacher cotype $q$ is necessary for the
$\mathcal B$-valued $q$-variational inequalities. As applications of the
variational inequalities, we deduce the jump estimates and obtain quantitative
information on the rate of convergence. It turns out the rate of convergence
depends on the geometric property of the Banach space under consideration,
which considerably improve Cowling and Leinert's result where it is shown that
the convergence always holds for all Banach spaces.
