On Itô's formula for certain fields of geometric objects
type: Item
instance of: scholarly article
title: On Itô's formula for certain fields of geometric objects
zbMATH Open document ID: 0607.58045
author: Hiroshi Akiyama
published in: Journal of the Mathematical Society of Japan
publication date: 1987-01-01T00:00:00Z
review text: A stochastic formula is set up for cross sections of fiber bundles which implies an extension of Itô's formula (in stochastic analysis) to the case of geometric object fields. More precisely, let \(\pi_ E: E\to M\) be a \(C^{\infty}\) fiber bundle and let \(\{\eta_ t\); \(t\geq 0\}\) be the stochastic flow of diffeomorphisms generated by a stochastic differential equation on E such that \(\{\eta_ t\}\) induces a stochastic flow of diffeomorphisms \(\{\theta_ t\); \(t\geq 0\}\) of M by the projection \(\pi_ E\). Let \(\sigma\) be a \(C^{\infty}\) (local) cross section of E. This paper gives Itô's formula for the \(\pi_ E^{-1}\)-valued process \(\eta_ t^{-1}(\sigma (\theta_ t(x)))\), \(x\in Dom(\sigma)\subset M\). The result is applied to the case where E is a bundle of geometric objects associated with the principal fiber bundle of r-th order frames over M, (r: a positive integer). When M is endowed with a linear connection [resp. a Riemannian metric], the result is used to study stochastic flow of affine [resp. conformal] motions. The result is also used to describe the behavior of a stochastically deformed projective structure.
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 55R10
Mathematics Subject Classification ID: 53A55
zbMATH DE Number: 3982140
zbMATH Keywords: fiber bundles
zbMATH Keywords: Itô's formula
zbMATH Keywords: geometric object fields
zbMATH Keywords: stochastic flow of diffeomorphisms
zbMATH Keywords: stochastic differential equation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03910079
OpenAlex ID: W2041168954
DOI: 10.2969/JMSJ/03910079
description: scientific article; zbMATH DE number 3982140