A general non-convex large deviation result with applications to stochastic equations (Q1840521)

The author proves an abstract large deviation result for a sequence of random elements \((Y_n)\) in a topological vector space \(E\) which generalizes classical results of Gärtner and Ellis to a non-convex situation. Instead of considering the logarithmic moment generating function \(\theta_n(\xi)= \log{\mathbf E}\exp\langle Y_n\xi\rangle\), \(\xi\in E^*\), he stipulates the existence of functions \(\Phi_n: E\times E^*\to {\mathbf R}\) satisfying the exponential martingale condition \({\mathbf E}(\exp\langle Y_n\xi\rangle-\Phi_n(Y_n \xi))= 1\) and the existence of the limit \(\lim_{n\to\infty} n^{-1}\Phi_n(x,n\xi)\) for every \(\xi\in E^*\). The results are applied to large deviation properties of the stochastic differential equations.