On Itô's formula for certain fields of geometric objects
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.
scientific article; zbMATH DE number 3982140