Laplace approximation for stochastic line integrals
type: Item
instance of: scholarly article
title: Laplace approximation for stochastic line integrals
zbMATH Open document ID: 1181.60038
author: Kazumasa Kuwada
publication date: 2009-05-13T00:00:00Z
review text: 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.
reviewed by: Nicolas Privault
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60B10
zbMATH DE Number: 5553450
zbMATH Keywords: Laplace approximation
zbMATH Keywords: large deviations
zbMATH Keywords: stochastic line integrals
zbMATH Keywords: empirical measures
MaRDI profile type: Publication
full work available at URL: s00440-008-0140-3
OpenAlex ID: W2014839573
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DOI: 10.1007/S00440-008-0140-3
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 5553450