Large deviations for long range interacting particle systems with jumps (Q1347268)

The author proves large deviation principles for the empirical measure and the empirical process of Markovian particle systems with a long-range interaction as the number \(n\) of particles tends to infinity. Every particle \(i \in \{1, \dots, n\}\) sits at a site \(s^n_i \in {\mathcal S} \subset \mathbb{R}^k\), and its random value \(X_i^n(t) \in {\mathcal Z} \subset \mathbb{R}^d\), which for \({\mathcal Z} = \{ - 1, + 1\}\) would denote the spin, evolves with time \(t \in [0,T]\). The interaction at time \(t\) is given via the empirical measure \(\overline X^n (t) = {1 \over n} \sum^n_{i = 1} \delta_{(X^n_i (t), s_i^n)}\); the rate for \(X_i^n (t)\) to change to \(X^n_i (t) + \Delta\) is determined by a Lévy kernel \({\mathcal L} (X_i^n (t)\), \(s^n_i, \overline X^n (t)\); \(d \Delta)\) on the set \(E\) of possible changes. The kernel is assumed to have a suitably bounded continuous density with respect to a reference measure on \(E\). The large deviations for the càdlàg empirical process \(\{\overline X^n (t)\}_{t \in [0,T]}\) and the empirical measure \(\widehat X^n = {1 \over n} \sum^n_{i = 1} \delta_{(X^n_i, s^n_i)}\) are proved using a modification of Sanov's theorem, the Laplace-Varadhan principle and the Dawson-Gärtner approach.