Laplace asymptotic expansions for Gaussian functional integrals (Q1269718)

Let \(\mathbb{E}^\rho_x\) denote expectation with respect to the mean zero Gaussian process \(x= x(\tau)\), \(0\leq \tau\leq t\), with covariance function \(\rho(\sigma, \tau)\) and paths \(x\in \mathbb{C}[0,t]\). It is proved that under some conditions on the continuous functionals \(F(\cdot)\) and \(G (\cdot)\) on \(\mathbb{C}[0,t]\), \[ \mathbb{E}^\rho_x \biggl[G(\lambda x) \exp\bigl(-\lambda^{-2} F(\lambda x) \bigr)\biggr] =\exp(-b \lambda^{-2}) \left[\sum^{n-3}_{i=0} \lambda^i \Gamma_i \right]+ O (\lambda^{n-2}), \text{ as } \lambda\to 0. \]