The Ito exponential on Lie groups

Abstract

Let G be a Lie Group with a complete, left invariant connection ∇ G . Denote by g the Lie algebra of G which is equipped with a complete connection ∇ g . Our main goal is to introduce the concept of the Itô exponential and the Itô logarithm. As a result, we characterize the martingales in G with respect to the left invariant connection ∇ G . Also, assuming that connection function α : g × g → g associated to ∇ G satisfies α(M, M ) = 0 for all M ∈ g we obtain a stochastic Campbell-Hausdorff formula. Further, from this stochastic Campbell-Hausdorf formula we present a way to construct martingales in Lie group. In consequence, we show that a product of harmonic maps with value in G is a harmonic map. To end, we apply this study in some matrix Lie groups.

Harmonic Maps into Homogeneous Spaces According to a Darboux Homogeneous Derivative

“…In the proof we use freely the concepts and notations of Catuogno [3] and Emery [6]. (2) in [15]) we deduce that…”

Martingales in Reductive Homogeneous spaces

“…To do this, we first endow the Lie Groups G with a left invariant connection ∇ G and g with the flat connection ∇ g . In [10], the author defines the Itô's stochastic exponential with respect to ∇ G and ∇ g as the solution of the Itô's stochastic differential equation…”

“…Thus, the Itô's stochastic exponential, which the author defined in [10], is used to characterize the horizontal martingales in G. Consequently, the martingales in G/H are characterized.…”

A loop group method for affine harmonic maps into Lie groups

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We generalize the Uhlenbeck-Segal theory for harmonic maps into compact semi-simple Lie groups to general Lie groups equipped with torsion free bi-invariant connection.of harmonic maps into solvable Lie groups (and, as a special case, into nilpotent Lie groups), since, generally, these groups do not admit any bi-invariant metric.In this context we would like to mention that harmonic maps into Lie groups, especially nilpotent or solvable Lie groups, equipped with left-invariant affine connections have applications to probability theory, since it is known that harmonic maps have a probabilistic characterization: A smooth map between Riemannian manifolds is a harmonic map if and only if it sends Brownian motions to martingales [37,45]. Martingales and harmonic maps into Lie groups equipped with left-invariant affine connections have been studied in [3,53]. This paper is organized as follows: In Section 1, we will briefly give preliminary results on vector bundle valued differential forms.

…”

“…In this context we would like to mention that harmonic maps into Lie groups, especially nilpotent or solvable Lie groups, equipped with left-invariant affine connections have applications to probability theory, since it is known that harmonic maps have a probabilistic characterization: A smooth map between Riemannian manifolds is a harmonic map if and only if it sends Brownian motions to martingales [37,45]. Martingales and harmonic maps into Lie groups equipped with left-invariant affine connections have been studied in [3,53]. This paper is organized as follows: In Section 1, we will briefly give preliminary results on vector bundle valued differential forms.…”