The It\^o exponential on Lie Groups

TL;DR

This paper introduces the Itô exponential and logarithm on Lie groups, linking stochastic calculus with geometric structures, and explores their properties, including a Campbell-Hausdorff formula and applications to harmonic maps.

Contribution

It defines the Itô exponential and logarithm on Lie groups considering geometric connections, generalizing stochastic exponential concepts and deriving new formulas and subgroup structures.

Findings

It characterizes martingales in Lie groups via the Itô exponential and logarithm.

Establishes a Campbell-Hausdorff formula for these stochastic maps.

Shows that products of harmonic maps are harmonic, using the developed formulas.

Abstract

Let $G$ be a Lie Group with a left invariant connection $\nabla^{G}$. Denote by $\g$ the Lie algebra of $G$, which is equipped with a connection $\nabla^{\g}$. Our main is to introduce the concept of the It\^o exponential and the It\^o logarithm, which take in account the geometry of the Lie group $G$ and the Lie algebra $\g$. This definition characterize directly the martingales in $G$ with respect to the left invariant connection $\nabla^{G}$. Further, if any $\nabla^{\g}$ geodesic in $\g$ is send in a $\nabla^{G}$ geodesic we can show that the It\^o exponential and the It\^o logarithm are the same that the stochastic exponential and the stochastic logarithm due to M. Hakim-Dowek and D. L\'epingle in [10]. Consequently, we have a Campbell-Hausdorf formula. From this formula we show that the set of affine maps from $(M,\nabla^{G})$ into $(G,\nabla^{G})$ is a subgroup of the Loop group. As in general, the Lie algebra is considered as smooth manifold with a flat connection, we show a Campbell-Hausdorf formula for a flat connection on $\g$ and a bi-invariant connection on $G$. To this main we introduce the definition of the null quadratic variation property. To end, we use the Campbell-Hausdorff formula to show that a product of harmonic maps with value in $G$ is a harmonic map.