Random Fourier series and continuous additive functionals of Lévy processes on the torus

Abstract

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.