Let X be an exponentially killed Lévy process on $T^n$ ,
the $n$ -dimensional torus, that satisfies a sector condition. (This includes
symmetric Lévy processes.) Let$\mathscr{F}_e$ denote the extended
Dirichlet space of X. Let $h \subset \mathscr{F}_e$ and let ${h_y, y \ subset
T^n}$ denote the set of translates of $h$. That is, $h_y(\dot) = h(\dot - y)$.
We consider the family of zero-energy continuous additive functions
${N_t^{[h_y]}, (y,t) \subset T^n \times R^+}$ defined by Fukushima. For a very
large class of random functions h we show that
$$J_\rho (T^n) = \int (\log N_\rho
(T^n,\varepsilon))^{1/2} d\varepsilon < \infty$$
is a necessary and sufficient condition for the
family ${N_t^{[h_y]}, (y,t) \subset T^n \times R^+}$ to have a continuous
version almost surely. Here $N_p(T^n, \varepsilon)$ is the minimum number of
balls of radius $\varepsilon$ in the metric p that covers $T^n$, where the
metric p is the energy metric. We argue that this condition is the natural
extension of the necessary and sufficient condition for continuity of local
times of Lévy processes of Barlow and Hawkes.
¶ Results on the bounded variation and p-variation (in t )of
$N_t^{[h_y]}$, for y fixed, are also obtained for a large class of random
functions h.