Laplace approximation for stochastic line integrals
Given a diffusion process \((z_t)_{t\in {\mathbb R}_+}\) on a Riemannian manifold \(M\), the author considers the current process \((Y_t)_{t\in {\mathbb R}_+}\) mapping a \(1\)-form \(\alpha\) to the martingale part \(Y_t ( \alpha )\) of the stochastic line integral \(\int_{z[0,t]} \alpha\), and the empirical measure \(l_t = \int_0^t \delta_{z_s} ds\). The main result of the paper is a precise Laplace asymptotics for the pair \(t^{-1}(Y_t , l_t )\), which can essentially be described as \[ \lim_{t\to \infty} e^{-tK_F} E_x \left[ \exp \left( t F \left( {1 \over t} Y_t , {1 \over t} l_t \right) \right) \right] = \sum_{(\omega , \mu ) } D_{F,\omega} h_\omega ( x) \int_M {1 \over h_\omega} d\mu, \] where \(F\) is a sufficiently regular functional, the sum is taken among the equilibrium states \((\omega , \mu )\) of the system, \(h_\omega \in C_+ (M)\), \(D_{F,\omega}>0\) is a constant, and \[ K_F = \lim_{t\to \infty} {1 \over t} E_x \left[ \exp \left( t F \left( {1 \over t} Y_t , {1 \over t} l_t \right) \right) \right] , \] under a non-degeneracy condition on the Hessian of \(F-I\). The Laplace approximation for stochastic line integrals of periodic diffusions follows as a particular case, and an application to the convergence of path measures is given.
scientific article; zbMATH DE number 5553450
scientific article