Laplace Asymptotic Expansions for Gaussian Functional Integrals

Abstract

We obtain a Laplace asymptotic expansion, in orders of $\lambda$, of $$ E^\rho_x \left\{ G(\lambda x) e^{-\lambda ^{-2} F(\lambda x)}\right\}$$ the expectation being with respect to a Gaussian process. We extend a result of Pincus and build upon the previous work of Davies and Truman. Our methods differ from those of Ellis and Rosen in that we use the supremum norm to simplify the application of the result.