  For a fixed right process $X$ we investigate those functions $u$ for which
$u(X)$ is a quasimartingale. We prove that $u(X)$ is a quasimartingale if and
only if $u$ is the dif- ference of two finite excessive functions. In
particular, we show that the quasimartingale nature of $u$ is preserved under
killing, time change, or Bochner subordination. The study relies on an analytic
reformulation of the quasimartingale property for $u(X)$ in terms of a certain
variation of $u$ with respect to the transition function of the process. We
provide sufficient conditions under which $u(X)$ is a quasimartingale, and
finally, we extend to the case of semi-Dirichlet forms a semimartingale
characterization of such functionals for symmetric Markov processes, due to
Fukushima.
