Lyapunov exponents for stochastic differential equations on semi-simple Lie groups

Paulo R. C. Ruffino^{∗} Luiz A. B. San Martin^{†} Instituto de Matemática Universidade Estadual de Campinas Cx. Postal 6065 13.081-970 Campinas-SP, Brasil

February 12, 1998

Abstract

With an intrinsic approach on semi-simple Lie groups we find a Furstenberg– Khasminskii type formula for the limit of the diagonal component in the Iwasawa decomposition. It is an integral formula with respect to the invariant measure in the maximal flag manifold of the group (i.e. the Furstenberg boundary B = G/MAN). Its integrand involves the Takeuchi-Kobayashi Riemannian metric in the flag manifolds. When applied to linear stochastic systems which generate a semi-simple group the formula provides a diagonal matrix whose entries are the Lyapunov spectrum. Some Brownian motions on homogeneous spaces are discussed and an entropy of large deviations is computed.

Key words: Lyapunov exponents, stochastic differential equations, semi-simple Lie groups, flag manifolds, large deviations. AMS 1991 subject classification: 60H10 (58G32, 22E46).

∗Research partially supported by CNPq grant n^{∘} 300670/95-8 and FAPESP 96/3134-0.

†Research partially supported by CNPq grant n^{∘} 301060/94-0 and FAPESP 96/3134-0.

1 Introduction

In this article we consider right invariant stochastic differential equations in a semisimple Lie group G with the purpose of studying the asymptotic time average of the logarithm of the A-part of the Iwasawa decomposition of the trajectories. After constructing a convenient radial-spherical decomposition, we get an integral formula by applying a Furstenberg-Khasminskii type argument. Interesting algebraic and geometrical interpretations come out of this formula when we consider the Takeuchi-Kobayashi metric on the flag manifolds.

The motivation for having such a formula is that, for many well known interesting systems, that limit describes the stability of the system since it contains the Lyapunov spectrum. Among those systems, the linear ones have been quite well studied by several authors who developed formulae in different contexts. We mention, for instance, Khasminskii [17], Arnold, Kliemann and Oeljeklaus [1], Arnold, Oeljeklaus and Pardoux [2] for linear systems, and Carverhill [7], [8], Arnold and San Martin [24] for extensions to nonlinear systems.

Most of those formulae detect only the top Lyapunov exponent. A creative method to calculate the whole Lyapunov spectrum was established by Baxendale [5]. He used the same kind of argument for the calculation of the top exponent but applied to the induced system on the Grassmannian Gr_{k} (n). Another approach to find the whole spectrum is due to Arnold and Imkeller [3], who got formulae to calculate these numbers via anticipative calculus, where each exponent is given by a Khasminskii type formula plus a correction term which are expressed in terms of a Malliavin derivative of the orthogonal projectors on the Osseledets spaces.

Dealing with the Iwasawa decomposition of systems in Sl(n, ℝ), Liao [19], in a geometrical context analogous to [5], obtained the whole spectrum as an integral formula with respect to an invariant measure for the induced system on the special orthogonal group. Our approach in this article is close to [19]. Here however we work intrinsically in a general semi-simple Lie group. The link to linear systems is established either by taking linear representations of the group or by starting with a linear system and assuming that the Lie algebra generated by its coefficients is semisimple. The advantage of this intrinsic set up is that the assumptions regarding the non-degeneracy of the systems - which ensures ergodicity of the invariant measure - are less demanding, in the sense that it requires only that the Lie algebra generated by the system is semi-simple.

The intrinsic approach also allows applications to other systems, like the geodesic flow in symmetric spaces (see Malliavin and Malliavin [21] and Carverhill and Elworthy [10]) or geodesic systems and other kinds of Brownian motions.

This article is organized as follows: in section 2 we present some algebraic preliminaries on semi-simple Lie algebra, flag manifolds and the Takeuchi-Kobayashi metric. Section 3 shows that the homogeneous space G/MN is a trivial principal fibre bundle over the maximal flag manifold with A as the structural group such that there exists a spherical-radial decomposition of this space. In section 4 we study the asymptotic behavior of solutions of stochastic differential equations in these radial fibres. Then, we close the argument in section 5 where we show that for many interesting systems (cf. Guivarc’h and Raugi [14]) this asymptotic behavior, as limiting elements in 𝔞, provides the Lyapunov spectrum of the system. In section 6 we calculate some geometrically interesting examples, and finally in section 7 we prove a result on large deviations for systems generated by symmetric elements in the Lie algebra.

We mention that although we work in the semi-simple context, the results are easily extended to a reductive Lie algebra, that is, which decomposes as a sum of a semi-simple Lie algebra plus the center. At this regard recall that a Lie algebra of matrices which is irreducible in the sense that it does not have invariant proper subspaces, is reductive. This implies that this method applies also to linear systems which generate an irreducible Lie algebra of matrices.

2 Algebraic Preliminaries

The purpose of this section is to present some known algebraic and geometrical facts about semi-simple Lie groups, their algebras and associated flag manifolds. We refer to Helgason [15] or Warner [27] for unexplained concepts.

Before starting we set the following notation: if G is a Lie group, a homogeneous space of G is a coset space G/H with H a closed subgroup. By left translation, G acts transitively on G/H. Let 𝔤 be the Lie algebra of G and take X ∈ 𝔤. Then X induces the vector field X̃ on G/H given by

X̃ (x) = \frac{d}{dt} (exp tX) (x)_{|t=0}

whose flow is the action of exp (tX), t ∈ ℝ, on G/H. When it is necessary to emphasize the specific homogenous space G/H the induced vector field will be denoted by X̃|_{G/H}. Since the action is transitive the tangent space at x is

T_{x} (G/H) = {X̃ (x) : X ∈ 𝔤}.

2.1 Semi-simple Lie algebras

Let 𝔤 be a semi-simple Lie algebra. Given a Cartan decomposition 𝔤 = 𝔨 ⊕ 𝔰, let θ stand for the corresponding Cartan involution (θ = id in 𝔨 and θ = −id in 𝔰). Let Π ⊂ 𝔞^{∗} stand for the set of roots of the pair (𝔤, 𝔞); the eigenvalues of ad_{𝔤} (H), H ∈ 𝔞 are 0 and α (H), α ∈ Π. The root space

𝔤_{α} = {X ∈ 𝔤 : [H, X] = α (H) X for all H ∈ 𝔞}

is the common eigenspace for ad_{𝔤} (H) associated with the eigenvalue α (H), α ∈ Π. By fixing a lexicographic order in the dual 𝔞^{∗} of 𝔞 we have Π = Π^{+} ∪ Π^{−} where Π^{+} is the set of positive roots with respect to this order, and Π^{−} = −Π^{+}. The direct sum

𝔫^{+} = ∑_{α∈Π^{+}} 𝔤_{α}

is a nilpotent subalgebra of 𝔤. We denote by 𝔞^{+} the Weyl chamber associated with Π^{+}:

𝔞^{+} = {H ∈ 𝔞 : α (H) > 0, α ∈ Π^{+}}.

The choice of one among Π^{+}, 𝔫^{+} or 𝔞^{+} determines the others. From the decomposition of 𝔤 into ad (𝔞)-eigenspaces we have

𝔤 = 𝔫^{−} ⊕ 𝔪 ⊕ 𝔞 ⊕ 𝔫^{+}

where

𝔫^{−} = θ (𝔫^{+}) = ∑_{α∈Π^{−}} 𝔤_{α}

is the subalgebra opposed to 𝔫^{+} and 𝔪 = {X ∈ 𝔨 : [X, 𝔞] = 0} is the centralizer of 𝔞 in 𝔨. A Weyl chamber 𝔞^{+} determines the Iwasawa decomposition:

𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔫^{+}.

In the particular case of 𝔰𝔩 (n, ℝ) with the canonical Iwasawa decomposition, 𝔨 is the algebra of skew-symmetric matrices, 𝔞 is the abelian algebra of diagonal matrices and 𝔫^{+} the upper triangular matrices with zeros on the main diagonal.

Denote by ⟨·, ·⟩ the Cartan-Killing form of 𝔤. We recall the following facts (see e.g. [15]):

• θ (𝔤_{α}) = 𝔤_{−α}.

• ⟨𝔤_{α}, 𝔤_{β}⟩ = 0 unless β = −α.

• The bilinear form in 𝔤 defined by B_{θ} (X, Y) = −⟨X, θY ⟩ is an inner product. In particular, the restriction of ⟨·, ·⟩ to 𝔰 is an inner product and for every 0 ≠ X ∈ 𝔤_{α}, ⟨X, θX⟩ ≠ 0.

2.2 Flag Manifolds

Let G be a connected and noncompact semi-simple Lie group with Lie algebra 𝔤. An Iwasawa decomposition 𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔫^{+} extends to an Iwasawa decomposition G = KAN where the groups K, A and N are the exponentials of 𝔨, 𝔞 and 𝔫^{+} respectively.

Let M be the centralizer of A in K. The Lie algebra of M is the centralizer 𝔪 of 𝔞 in 𝔨. The product P = MAN is a closed subgroup of G with Lie algebra 𝔭 = 𝔪 ⊕ 𝔞 ⊕ 𝔫^{+}. The subgroup P is the normalizer of 𝔭 in G. It is a minimal parabolic subgroup and the quotient B = G/P is a compact homogeneous space of G known as the maximal flag manifold or the Furstenberg boundary of G. The subgroup K also acts transitively on B. Through the transitive action of K we have B = K/M. We remark that B is the same, regardless the specific G having Lie algebra 𝔤. This is due to the fact that M contains the center of G so that the action of G on B factors through the group of inner automorphisms of 𝔤 which is centerless.

For the construction of the non maximal flag manifolds we need the simple system of roots Σ associated to Π^{+}. This is a basis of 𝔞^{∗} such that every α ∈ Π^{+} is a linear combination of Σ with nonnegative integers as coefficients.

Given Θ ⊂ Σ let ⟨Θ⟩ be the subset of positive roots generated by Θ. Denote by 𝔫^{−} (Θ) the subalgebra spanned by the root spaces 𝔤_{−α}, α ∈ ⟨Θ⟩ and let 𝔭_{Θ} be the parabolic subalgebra defined by

𝔭_{Θ} = 𝔫^{−} (Θ) ⊕ 𝔭.

Its normalizer P_{Θ} in G is a parabolic subgroup whose Lie algebra is 𝔭_{Θ}. We put B_{Θ} = G/P_{Θ} for the corresponding flag manifold. If M_{Θ} = P_{Θ}∩K then B_{Θ} = K/M_{Θ}, that is, K is transitive in B_{Θ}. It turns out that M_{Θ} is the centralizer in K of any H in the “subchamber”

{H ∈ cl (𝔞^{+}) : α (H) = 0 if α ∈ ⟨Θ⟩ and α (H) > 0 if α ∈ Π^{+} − ⟨Θ⟩}.

By this transitivity of K, B_{Θ} identifies with the Ad (K)-orbit of H in 𝔰. In this case the notation B_{H} and M_{H} are also used instead of B_{Θ} and M_{Θ} respectively. The Lie algebra of M_{H} is the centralizer 𝔪_{H} of H in 𝔨:

𝔪_{H} = {X ∈ 𝔨 : [X, H] = 0}.

A relation between the Iwasawa and Cartan components is provided by the following simple algebraic lemma, which generalizes the symmetrization and skewsymmetrization of matrices.

Lemma 2.1 Let 𝔞^{+}, and hence 𝔫^{+} be given.

1. Define 𝔨_{𝔞^{+}} = {Y + θ (Y) : Y ∈ 𝔫^{+}}. Then 𝔨_{𝔞^{+}} ⊂ 𝔨 and 𝔨 = 𝔨_{𝔞^{+}} ⊕ 𝔪. Moreover, the skew-symmetrization map

χ : Y ∈ 𝔫^{+} ⟼ Y + θ (Y) ∈ 𝔨_{𝔞^{+}}

is an isomorphism of vector spaces.

2. Define 𝔰_{𝔞^{+}} = {Y − θ (Y) : Y ∈ 𝔫^{+}}. Then 𝔰_{𝔞^{+}} ⊂ 𝔰 and 𝔰 = 𝔰_{𝔞^{+}} ⊕ 𝔞. Moreover, the symmetrization map

σ : Y ∈ 𝔫^{+} ⟼ Y − θ (Y) ∈ 𝔰_{𝔞^{+}}

is an isomorphism of vector spaces.

Proof: To see item (1) note that 𝔨 is the subspace of points fixed by θ so that 𝔨_{𝔞^{+}} ⊂ 𝔨. We have 𝔨 ⊂ 𝔫^{−} ⊕ 𝔪 ⊕ 𝔫^{+}. Hence X ∈ 𝔨 is written uniquely as X = Z + A + Y, with Z ∈ 𝔫^{−}, A ∈ 𝔪 and Z ∈ 𝔫^{+}. Then

X = θ (X) = θ (Z) + A + θ (Y)

with θ (Z) ∈ 𝔫^{+}and θ (Y) ∈ 𝔫^{−}. Therefore θ (Y) = Z so that X = (Y + θ (Y))+A ∈ 𝔨_{𝔞^{+}}⊕𝔪. The isomorphism is a consequence of the fact that for Y ∈ 𝔫^{+}, Y +θ (Y) = 0 if and only if Y = 0.

Item (2) follows in the same way: 𝔰 ⊂ 𝔫^{−} ⊕𝔞⊕𝔫^{+} and consider Y −θ (Y) instead of Y + θ (Y). □

With the isomorphisms χ and σ of this lemma we construct the isomorphism

ζ = σ ∘ χ^{−1} : 𝔨_{𝔞^{+}} → 𝔰_{𝔞^{+}}.

For H ∈ 𝔞 the flag manifold B_{H} is the Ad (K)-orbit of H so that its tangent space at H is

T_{H}B_{H} = {X̃ (H) = [X, H] : X ∈ 𝔨} ⊂ 𝔰.

Clearly [𝔪_{H}, H] = 0. Hence T_{H}B_{H} is the subspace of tangent vectors X̃ (H) with X running through the subspace χ (𝔫_{H}) where

𝔫_{H} = ∑{𝔤_{α} : α ∈ Π^{+}, α (H) ≠ 0}.  (1)

An easy computation shows that T_{H}B_{H}, as a subspace of 𝔰, coincides with σ (𝔫_{H}) so that it is the orthogonal complement of ζ (𝔪_{H}).

Later on we will use the following facts relating the isomorphism σ : 𝔫^{+} → 𝔰_{𝔞^{+}} with the Cartan-Killing form: If Y ∈ 𝔤_{α}, α > 0 then σY = Y − θY, and since ⟨Y, Y ⟩ = 0,

⟨σY, σY ⟩ = ⟨Y − θY, Y − θY ⟩ = 2B_{θ} (Y, Y) .

Also, if Z ∈ 𝔤_{β}, with α ≠ β > 0 then ⟨Y, Z⟩ = ⟨Y, θZ⟩ = ⟨θY, θZ⟩ = 0 so that ⟨σY, σZ⟩ = 0. From this fact we can construct orthonormal bases of 𝔰 as follows: take a basis {Y_{1}, …, Y_{m}} of 𝔫^{+} which is the union of bases of the root spaces 𝔤_{α}, α > 0. Then {σY_{1}, …, σY_{m}} is a basis of 𝔰_{𝔞^{+}} which can be complemented with a basis of 𝔞 to get a basis of 𝔰. Any such basis will be called adapted to 𝔞^{+}. In particular, if {Y_{1}, …, Y_{m}} is orthonormal with respect to the inner product B_{θ} then

\{\frac{\sqrt{2}}{2}σY_{1}, …, \frac{\sqrt{2}}{2}σY_{m}\}  (2)

is orthonormal in 𝔰_{𝔞^{+}} which can be complemented to an orthonormal basis of 𝔰.

2.3 Takeuchi-Kobayashi metric

It is possible to endow a flag manifold with a special Riemannian metric which depends on its realization as an Ad (K)-orbit, namely the Takeuchi-Kobayashi (T-K) metric (see Duistermmat, Kolk and Varadarajan [11]). For the definition of the T-K metric take H ∈ cl (𝔞^{+}). Then at the tangent space T_{H}B_{H} the T-K metric is given by

(X̃ (H) , Ỹ (H))_{H} = ⟨H, [X, ζ (Y)]⟩  (3)

for X, Y ∈ 𝔨. This expression actually defines an inner product in T_{H}B_{H} which is invariant under M_{H} so that it extends to a K-invariant Riemannian metric in B_{H}. This metric will play an essential role in the sequel for the computation of the Lyapunov exponents.

A crucial fact about the T-K metric is that the vector fields induced by 𝔰 are gradient. More precisely, for X ∈ 𝔰 let X̃ be the vector field it induces in B_{H} through the G-action in this flag manifold. Since B_{H} is embedded in 𝔰, it makes sense to define the function f_{X} : B_{H} → ℝ by f_{X} (Y) = ⟨X, Y ⟩.

Lemma 2.2 For any X ∈ 𝔰, X̃ = gradf_{X} where the gradient is taken with respect to T-K, that is, d (f_{X}) = (X̃, ·).

Proof: See Proposition 3.3 in [11]. □

The right hand side of equation (3) is linear in H showing that the T-K metric changes linearly with H. The exact meaning of this linear dependence is as follows: fix t > 0 and put H_{1} = tH. The centralizers of H and H_{1} in K coincide so that B_{H_{1}} = B_{H}, that is, both orbits Ad (K) H_{1} and Ad (K) H identify with the same homogeneous space of K. Under these identifications H_{1} and H give the same base point. The vector field X̃ is defined by means of the K-action and it is independent of the specific realization. From (3) we see that the T-K metric defined by H_{1} is t times the metric defined by H.

Another aspect about the T-K metric which needs to be discussed concerns its values on the vectors of 𝔰 which are tangent to Ad (K) H at H. Any such vector is of the form σ (A) with A ∈ 𝔫_{H}, defined in (1). We have,

Lemma 2.3 Let α > 0 be a root such that α (H) ≠ 0. Let A ∈ 𝔤_{α} and view σ (A) ∈ 𝔰 as a tangent vector to Ad (K) H at H. Then

(σ (A) , σ (A))_{H} = \frac{1}{α (H)}⟨σ (A) , σ (A)⟩.

Moreover, if β ≠ α is another positive root and B ∈ 𝔤_{β} then (σ (B) , σ (A))_{H} = 0.

Proof: Let χ be the isomorphism of Lemma 2.1 and put X = −\frac{1}{α(H)}χ (A). Direct computations show that

(σ (A) , σ (A))_{H} = (X̃, X̃)_{H} = ⟨H, [X, ζX]⟩
= ⟨[H, X], ζX⟩ = ⟨−σ (A) , ζX⟩
= \frac{1}{α (H)}⟨σ (A) , σ (A)⟩

The orthogonality between σ (A) and σ (B) follows if we perform the computations with ζY instead of ζX where Y = −\frac{1}{β(H)}χ (B). □

This lemma has the following interesting consequence: suppose that H is such that α (H) = 1 for every positive root α such that α (H) ≠ 0. Then the T-K metric in B_{H} is just the metric induced by its immersion in 𝔰. When this happens we say that B_{H} is an immersed flag manifold.

For later reference we include here the computation of the T-K metric in the vector fields induced by the elements in 𝔰.

Lemma 2.4 Take H ∈ cl (𝔞^{+}) and denote by x_{0} the origin of B_{H}. Let α be a positive root. For X ∈ 𝔤_{α} put S = σ (X). Then

|S̃ (x_{0})|^{2} = α (H) ⟨σ (X) , σ (X)⟩.

Moreover, if β ≠ α is another positive root and Y ∈ 𝔤_{β} then (\begin{matrix} \\ σ & (X) \end{matrix}, \begin{matrix} \\ σ & (Y) \end{matrix})_{H} = 0.

Proof: At H the vector field \begin{matrix} \\ σ & (X) \end{matrix} is equal to (pr_{𝔨}σ (X))^{∼}. We have

σ (X) = X − θ (X) = (−X − θ (X)) + 2X.

The right hand side of this equality is the Iwasawa decomposition of σ (X) because −X − θ (X) ∈ 𝔨 and 2X ∈ 𝔫^{+}. Hence pr_{𝔨}σ (X) = − (X + θ (X)). Since ζ (σ (X)) = X − θ (X), a similar formula for Y yields

(\begin{matrix} \\ σ & (X) \end{matrix}, \begin{matrix} \\ σ & (Y) \end{matrix})_{H} = ⟨H, [X + θ (X) , Y − θ (Y)]⟩.

Now the Cartan-Killing form is invariant under the adjoint representation so that

(\begin{matrix} \\ σ & (Y) \end{matrix}, \begin{matrix} \\ σ & (Y) \end{matrix})_{H} = ⟨[H, X + θ (X)], Y − θ (Y)⟩.

But [H, X] = α (H) X and [H, θ (X)] = −α (H) X because X ∈ 𝔤_{α} and θ (X) ∈ 𝔤_{−α}. Since 𝔤_{α} + 𝔤_{−α} is orthogonal to 𝔤_{β} + 𝔤_{−β} if β ≠ α, this implies the second statement. On the other hand,

|\begin{matrix} \\ σ & (X) \end{matrix} (x_{0})|^{2} = α (H) ⟨X − θ (X) , X − θ (X)⟩

as claimed. □

3 Decomposition of G/MN

In this section we show that the homogeneous space G/MN is a trivial principal fiber bundle over the maximal flag manifold B such that the structural group is the A part of the Iwasawa decomposition G = KAN, i.e., there exists a kind of sphericalradial decomposition of this space. In the next subsections we study the asymptotic behavior of the trajectories in the fibers, so we find the Lyapunov spectrum in the entries of limiting elements in A.

Fix an Iwasawa decomposition of the Lie algebra 𝔤 = 𝔨⊕𝔞⊕𝔫 and let G = KAN be the corresponding decomposition on the group. We shall denote by pr_{𝔦} the projection of 𝔤 onto the Iwasawa component 𝔦 = 𝔨, 𝔞 or 𝔫.

The product L = MN ⊂ P is a closed subgroup which is normal in P because A normalizes N and M. Consider the canonical fibration

π : gL ∈ G/L ⟼ gP ∈ G/P

whose fiber is A = P/L. Since L is normal in P it turns out that G/L is a principal bundle over G/P with A as structural group. The right action of A on G/L is given by R_{h} (gL) = gLh = ghL, h ∈ A. It is clear that for any g ∈ G, gR_{h} = R_{h}g. Also, π is equivariant with respect to the actions of G on G/L and G/P in the sense that πg = gπ. The next proposition shows that the above principal bundle is trivial:

Proposition 3.1 The map

ϕ : (uM, h) ∈ (K/M) × A ⟼ uhL ∈ G/L.  (4)

is a diffeomorphism between G/L and B × A. Its inverse ϕ^{−1} maps gL ∈ G/L in (uM, h) ∈ K/M × A where g = uhn is the Iwasawa decomposition of g. Moreover, ϕ is a bundle map in the sense that ϕ (b, R_{a_{1}} (a)) = R_{a_{1}}ϕ (b, a) for all (b, a) ∈ B × A and a_{1} ∈ A.

Proof: Note that, by (4), ϕ does not depend on the representative u ∈ K because if m ∈ M and h ∈ A then umhL = uhmL = uhL. Let ψ be the map uhL ↦ (uM, h); we claim that ψ is the inverse of ϕ. Firstly we check that ψ is also well defined: let g = uhn and suppose that g_{1} = u_{1}h_{1}n_{1} is in the coset gL, i.e.

n^{−1}h^{−1}u^{−1}u_{1}h_{1}n_{1} ∈ L.

Since N ⊂ L it follows that h^{−1}u^{−1}u_{1}h_{1} ∈ L ⊂ P. Hence u^{−1}u_{1} ∈ P ∩ K = M. So that u^{−1}u_{1} commutes with h^{−1} and hence u^{−1}u_{1}h^{−1}h_{1} ∈ L which implies that h^{−1}h_{1} ∈ L ∩ A = {1}. Therefore h_{1} = h and uM = u_{1}M showing that ψ is well defined. The composition ϕψ is the identity because if g = uhn then gL = uhL. On the other hand, ψϕ (uM, h) = (uM, h) because uh is already written in Iwasawa decomposition. Since ϕ and ψ are differentiable we conclude that ϕ is a diffeomorphism between G/L and B × A. It is a bundle map because A normalizes L. □

This decomposition has an evident meaning as a polar decomposition with B playing the role of the spherical component while A is the radial component. Also, by identifying G/L with B × A through ϕ, the A-component of gL becomes the A-component of the Iwasawa decomposition of g.

3.1 Vector Fields

We look now at the behavior of vector fields in G/L under the above decomposition. Recall that the g-translation of X̃ is given by the adjoint in 𝔤:

g_{∗}X̃ = (Ad (g) X)^{∼} .  (5)

The vector field induced by X on G/L is right invariant under h ∈ A, i.e., R_{h∗}X = X because the action of G on G/L commutes with the right action of A. Also, taking the decomposition G/L = B × A and considering the trivial connection on this bundle, X decomposes as

X (b, h) = X_{H} (b, h) + X_{V} (b, h)

where X_{H} is the horizontal component (in the direction of B) while X_{V} stands for the vertical component (in the direction of A). We shall find explicit expressions for these components.

The horizontal component is just the vector field induced by X on B. In fact, the projection

π : G/L ⟶ G/P

is equivariant which implies that π_{∗} X̃|_{G/L} = X̃|_{G/P}. Since π_{∗}X_{V} = 0 it follows

that π_{∗} X̃|_{G/L} = X_{H}. Therefore X_{H} (b, h) = X_{H} (b) is independent of h ∈ A and

coincides with the vector field induced by X on B.

In order to get the vertical component we denote by H^{∗} the vertical vector field induced in B × A by H ∈ 𝔞 as an element of the Lie algebra of the structural group, hence, now the action is on the right. For every vertical vector v at (b, h) there exists H ∈ 𝔞 such that v = H^{∗} (b, h). Hence with a given X ∈ 𝔤 we have defined a map b ∈ B ↦ H_{X,b} ∈ 𝔞 such that

X (b, 1) = X_{H} (b) + H_{X,b}^{∗} (b, 1) .

From this equality we can obtain the vertical component. In fact, X is right invariant, i.e., X (b, h) = R_{h∗}X (b, 1). Now

\begin{aligned}R_{h∗} (X_{H} (b) + H_{X,b}^{∗} (b, 1)) & = & X_{H} (b) + R_{h∗} (H_{X,b}^{∗} (b, 1)) \\ & = & X_{H} (b) + (Ad (h^{−1}) H_{X,b})^{∗} (b, h) . \end{aligned}

Since A is abelian Ad (h^{−1}) H = H, hence

X (b, h) = X_{H} + H_{X,b}^{∗} (b, h)

i.e. the vertical component is determined by H_{X,b} ∈ 𝔞 which depends only on X and on b. We will find an explicit expression for this map. Consider the Iwasawa decomposition 𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔫 and let b_{0} = P be the origin of B. Since N is contained in the isotropy subgroup at (b_{0}, 1), the 𝔫-component of X becomes zero at this point. Under the diffeomorphism ϕ the horizontal component X̃|_{G/P} (b_{0}) is given

by (pr_{𝔨})^{}˜|_{G/P} and the vertical component is given by (pr_{𝔞})^{}˜|_{A}. Hence, since 𝔞 is

abelian, H_{X,b_{0}} = pr_{𝔞}X.

For the values of H_{X,b} at other points of B, take u ∈ K and put b = ub_{0}. Then, by equation (5) and the fact that u_{∗}H^{∗} = H^{∗} we have:

X̃ (b, 1) = u_{∗} ((Ad (u^{−1}) X)^{∼} (b_{0}, 1))
= u_{∗} ((Ad (u^{−1}) X)^{∼}|_{G/P} (b_{0})) + u_{∗}H_{Ad(u^{−1})X,b_{0}}^{∗} (b_{0}, 1)
= X_{H} (b_{0}) + H_{Ad(u^{−1})X,b_{0}}^{∗} (b, 1) .

So, X_{V} (b, 1) = H_{Ad(u^{−1})X,b_{0}}^{∗} (b, 1), and we get for b = ub_{0}, u ∈ K the desired expression of H_{X,b}:

H_{X,b} = H_{Ad(u^{−1})X,b_{0}} = pr_{𝔞} (Ad (u^{−1}) X) .

The group A is diffeomorphic to its Lie algebra 𝔞 through the exponential map. Hence G/L is also diffeomorphic to B × 𝔞 and there is a decomposition of the vector fields at this level too. We have then:

Proposition 3.2 The differential equation induced in G/L = B × 𝔞 by X ∈ 𝔤 decomposes into the equations

\frac{db}{dt} = X̃|_{G/P} (b) , with b ∈ B and

\frac{dH}{dt} = pr_{𝔞} (Ad (u^{−1}) X) where b = ub_{0} and H ∈ 𝔞

Proof: We only remark that the second equation means that if a_{t} = exp H(t) ∈ A then ȧ_{t} = pr_{𝔞} (Ad (u^{−1}) X) a_{t}. □

In the following sections it will be convenient to use the notation

q_{X} (b) = pr_{𝔞} (Ad (u^{−1}) X) ∈ 𝔞  (6)

with X ∈ 𝔤, b = ub_{0} where u ∈ K. Note that (6) does not depend on the representative u ∈ K which satisfies ub_{0} = b. In fact, if u_{1}b_{0} = b then u_{1} = um for some m ∈ M, and pr_{𝔞} ∘Ad (m) = pr_{𝔞} because m centralizes 𝔞.

4 Stochastic Differential Equations

Consider the stochastic differential equation on the semi-simple Lie group G:

dg = X (g) + ∑_{i=1}^{m} Y^{i} (g) ∘ dW_{i}.  (7)

We shall assume the accessibility property of this system which means that X and Y^{1}, …, Y^{m} generate the Lie algebra 𝔤 of G. As in the case of vector fields this equation induces stochastic equations in the homogeneous spaces of G, in particular in G/L. By the preceding section, there is a decomposition of the process in G/L into radial and spherical components. In fact, using Itô’s formula the induced equation in B × 𝔞 has the components

db = X_{H} (b) + ∑_{i=1}^{m} Y_{H}^{i} (b) ∘ dW_{i}  (8)

in the direction of B and

dH = q_{X} (b) + ∑_{i=1}^{m} q_{Y^{i}} (b) ∘ dW_{i}  (9)

in the direction of 𝔞. Let g_{t} be the solution of (7) starting at the identity 1 ∈ G. Write g_{t} = u_{t}h_{t}n_{t} for its Iwasawa decomposition and put H_{t} = log h_{t}. Then H_{t} is driven by equation (9). In order to describe the asymptotic behavior of H_{t} it will be convenient to convert the Stratonovich equation (9) in Itô form:

dH = q_{X} (b) dt + \frac{1}{2} ∑_{i=1}^{m} r_{Y^{i}} (b) dt + ∑_{i=1}^{m} q_{Y^{i}} (b) dW_{i}  (10)

where r_{Z} for Z ∈ 𝔤 stands for the directional derivative:

r_{Z} (b) = (Z_{H} · q_{Z}) (b) .

We will find an expression for r_{Z} by reducing the computation of the derivative at the origin b_{0} ∈ B. Given b ∈ B let u ∈ K be such that b = ub_{0}. Then

\begin{aligned}r_{Z} (b) & = & d (q_{Z})_{b} (Z_{H} (b)) \\ & = & d (q_{Z})_{b} ∘ du_{b_{0}} ∘ du_{b}^{−1} (Z_{H} (b)) \end{aligned}

with u viewed as a diffeomorphism u : B → B. Hence:

r_{Z} (b) = d (q_{Z} ∘ u)_{b_{0}} (du_{b}^{−1} (Z_{H} (b))) .

Now, from (5), du_{b}^{−1} (Z_{H} (b)) is the vector field induced on B by Ad (u^{−1}) Z at b_{0}, i.e.,

r_{Z} (b) = d (q_{Z} ∘ u)_{b_{0}} ((Ad (u^{−1}) Z)^{∼} (b_{0})) .  (11)

We recall from the previous section that at the origin b_{0} of B, given X ∈ 𝔤, X̃ (b_{0}) = (pr_{𝔨}X)^{}˜ (b_{0}) (because 𝔞 + 𝔫 is contained in the isotropy subalgebra at b_{0}). Hence, if we denote

W (u) = pr_{𝔨} Ad (u^{−1}) Z,

then

r_{Z} (b) = \frac{d}{dt} (q_{Z} ∘ u) (e^{tW (u)}b_{0})_{|t=0} ,

which by the definition of q_{Z} becomes:

r_{Z} (b) = \frac{d}{dt} pr_{𝔞} (Ad (e^{−tW (u)}u^{−1}) Z)_{|t=0} .

A direct calculation shows that

r_{Z} (b) = pr_{𝔞}[Ad (u^{−1}) Z, W (u)].

Summarizing, we have the following formula:

Proposition 4.1 If Z ∈ 𝔤 and b = ub_{0} ∈ B with u ∈ K then

r_{Z} (b) = pr_{𝔞}[Ad (u^{−1}) Z, pr_{𝔨} Ad (u^{−1}) Z]

4.1 The Integral Formula

The assumption that {X, Y^{1}, …, Y^{m}} generates 𝔤 guarantees that in each compact homogeneous space of G there exists a unique (ergodic) invariant probability measure for the diffusion process which is the solution of the induced stochastic differential equation. In particular there exists a unique invariant probability measure ν on the maximal flag manifold for the process in this space. Applying the ergodic theorem to the skew-symmetric flow (see e.g. Arnold, Kliemann and Oeljeklaus [1] or Carverhill [7]) we have the following well known special case of the Law of Large Numbers:

lim_{t→∞} \frac{1}{t} log a_{t} = lim_{t→∞} \frac{1}{t} ∫_{0}^{t} Q(b_{s}) ds = ∫_{B} Q(b)ν (db) for ν ⊗ ℙ-almost every (b, ω)

where the function Q : B → 𝔞 is given by:

Q (b) = q_{X} + ∑_{i=1}^{m} r_{Y^{i}} (b)  (12)

with q_{X} (b) = pr_{𝔞} (Ad (u^{−1}) X) and r_{Y^{i}} (b) = pr_{𝔞}[Ad (u^{−1}) Y^{i}, pr_{𝔨}Ad (u^{−1}) Y^{i}] where b = ub_{0}.

4.2 The Integrand

We shall find an expression for the quadratic part r_{Z} (b), Z ∈ 𝔤, of the integrand in terms of the T-K metric.

The restriction of the Cartan-Killing form ⟨·, ·⟩ to 𝔞 is an inner product so that we determine r_{Z} (b) if we compute ⟨H, r_{Z} (b)⟩ for every H in a basis of 𝔞. In other words, we must calculate

⟨H, r_{Z} (b)⟩ = ⟨H, pr_{𝔞}[Ad (u^{−1}) Z, pr_{𝔨}Ad (u^{−1}) Z]⟩

for generic H ∈ 𝔞. Under the Cartan-Killing form 𝔨 and 𝔫^{+} are orthogonal to 𝔞 so that

⟨H, r_{Z} (b)⟩ = ⟨H, [Ad (u^{−1}) Z, pr_{𝔨}Ad (u^{−1}) Z]⟩.  (13)

Note first that if Z ∈ 𝔨 then Ad (u^{−1}) Z ∈ 𝔨. Hence pr_{𝔨}Ad (u^{−1}) Z coincides with Ad (u^{−1}) Z so that (13) vanishes trivially.

On the other hand for Z ∈ 𝔰 we can relate ⟨H, r_{Z} (b)⟩ with the T-K metric (·, ·) in B_{H}.

Lemma 4.2 If Z ∈ 𝔰 then

⟨H, [Z, pr_{𝔨}Z]⟩ = ((pr_{𝔨}Z)^{∼} , (pr_{𝔨}Z)^{∼})_{H}  (14)

where (pr_{𝔨}Z)^{∼} means the vector field on B_{H} induced by pr_{𝔨}Z.

Proof: By Lemma 2.1 there are Y ∈ 𝔫^{+} and H′ ∈ 𝔞 such that

Z = Y − θ (Y) + H′ = (−Y − θ (Y)) + H′ + 2Y.

The right hand side of this equality is the Iwasawa decomposition of Z because −Y − θ (Y) ∈ 𝔨, H′ ∈ 𝔞 and Y ∈ 𝔫^{+}. Hence

pr_{𝔨}Z = −Y − θ (Y) ,

and

[Z, pr_{𝔨}Z] = [Y − θ (Y) , pr_{𝔨}Z] + [H′, pr_{𝔨}Z].  (15)

The term [H′, pr_{𝔨}Z] is orthogonal to H so that it does not contribute to (14). In fact, pr_{𝔨}Z = −Y − θ (Y) belongs to 𝔫^{−} + 𝔫^{+} and this subspace is orthogonal to 𝔞 and invariant under ad (𝔞). On the other hand, the first term in the right hand side of (15) is −[ζ (pr_{𝔨}Z) , pr_{𝔨}Z]. The Cartan-Killing product of this term with H is by definition ((pr_{𝔨}Z)^{∼} , (pr_{𝔨}Z)^{∼})_{H}, proving the lemma. □

The right hand side of (14) can be given an interpretation in terms of the Gaction on B_{H}: For X ∈ 𝔤 let as before X̃ stand for the vector field induced by X on B_{H}. If x_{0} ∈ B_{H} corresponds to H then 𝔞 + 𝔫^{+} is contained in the isotropy subalgebra at x_{0}. Hence for X ∈ 𝔤,

X̃ (x_{0}) = (pr_{𝔨}X)^{∼} (x_{0}) .

Therefore we have

Corollary 4.3 For Z ∈ 𝔰 it holds

⟨H, [Z, pr_{𝔨}Z]⟩ = (Z̃, Z̃)_{x_{0}} = |Z̃ (x_{0})|^{2} .

Using K-invariance we can transport this formula to every point of B_{H}.

Corollary 4.4 If Z ∈ 𝔰 then

⟨H, r_{Z} (b)⟩ = (Z̃, Z̃)_{ux_{0}}

where (·, ·) is the T-K metric in B_{H}. Here u and b are related by b = ux_{0} and x_{0} is the origin of B_{H}.

Proof: Put U = Ad (u^{−1}) Z. By definition of r_{Z} and the above lemma,

⟨H, r_{Z} (b)⟩ = |Ũ (x_{0})|^{2}

with the norm given by the T-K metric in B_{H}. However,

Ũ (x_{0}) = (Ad (u^{−1}) Z)^{∼} (x_{0}) = u_{∗}^{−1} (Z (ux_{0})) .

Since the metric is K-invariant it follows that

|Ũ (x_{0})|^{2} = |u_{∗}^{−1} (Z̃ (ux_{0}))|^{2} = |Z̃ (ux_{0})|^{2}

as claimed. □

In general, let Z = A + S with A ∈ 𝔨 and S ∈ 𝔰 and for u ∈ K put Z_{u} = Ad (u^{−1}) Z, A_{u} = Ad (u^{−1}) A and S_{u} = Ad (u^{−1}) S. Then Z_{u} = A_{u} + S_{u}. Plugging this into formula (13) and taking into account that [𝔨, 𝔨] ⊂ 𝔨 is orthogonal to H we get

⟨H, r_{Z} (b)⟩ = ⟨H, [S_{u}, A_{u}]⟩ + |S̃ (ux_{0})|^{2}

where b = ub_{0}. Now,

S_{u} = \frac{Z_{u} − θZ_{u}}{2} , A_{u} = \frac{Z_{u} + θZ_{u}}{2}

and Ad (u) commutes with θ. Hence there is the following expression for ⟨H, r_{Z} (b)⟩, which holds for arbitrary Z.

Proposition 4.5 If Z ∈ 𝔤 and H ∈ 𝔞 then

⟨H, r_{Z} (b)⟩ = \frac{1}{2}⟨H, [Ad (u) Z, Ad (u) (θZ)]⟩ + |S̃ (ux_{0})|^{2}

where b = ub_{0} and b_{0} and x_{0} are the origins in the flag manifolds B and B_{H} respectively.

5 Lyapunov Exponents

The right invariant stochastic differential equation (7) on G induces a stochastic differential

dx = X̃ (x) + ∑_{i=1}^{m} Ỹ^{i} (x) ∘ dW_{i}

on each space endowed with a G action. For many of the induced systems their Lyapunov exponents are described by the asymptotic of the A-part in the Iwasawa decomposition. We present below two classical situations covered by this construction, namely the linear systems induced by representations of the group and, secondly, Brownian motions in flag manifolds and symmetric spaces.

5.1 Linear Systems

Let ρ : G → Gl (d, ℝ) be a representation of G in ℝ^{d}. It induces a representation of 𝔤 (also denoted by ρ) so that the right invariant vector fields in G are mapped into linear vector fields in ℝ^{d}. Therefore, under the representation, the system given by equation (7) is mapped into the linear differential equation:

dx = ρ (X) xdt + ∑_{i=1}^{m} ρ (Y^{i}) x ∘ dW_{i} x ∈ ℝ^{d}.  (16)

The relation between the systems (7) and (16) is that if x_{t} is the solution of (16) starting at x_{0} then x_{t} = ρ (g_{t}) x_{0} where g_{t} is the solution of (7) starting at the identity. Clearly, ρ (g_{t}) is the solution of a right invariant differential equation in the Lie group ρ (G), image of (7) under ρ. Every data about (16) is contained in this image system and not in G itself. Since we are primarily interested in (16) we assume from now on that ρ is a faithful representation. This amounts to assume that G is a linear group and ρ is just the inclusion of G into the general linear group. Alternatively we may start with a linear system and make the assumption that the Lie algebra generated by the coefficients is a semi-simple subalgebra of matrices.

Our purpose here is to sketch a proof of the easily suspected fact that the Apart in the Iwasawa decomposition gives, through the representation, the Lyapunov exponents of (16)

lim_{t→+∞} \frac{1}{t} log ||x_{t}|| .  (17)

There are certainly different ways to prove this fact. All of them require some regularity property of the system. For our systems the regularity comes from the accessibility property, i.e., the assumption that the coefficients of the system generate 𝔤.

To find the Lyapunov exponents of equation (16) we use the theory of Guivarc’h and Raugi [14]. The first thing to do is to change our continuous-time system into a discrete-time one. This is easily achieved by taking the solution g_{t} of (7) starting at the identity at time 1. Let μ be the law of g_{1}. Then the law of g_{k} is the k-th convolution power μ^{⋆k}. Also, the flow property of g_{t} implies that for a random element

g_{k} (ω) = g_{1} (θ_{k−1} (ω)) ⋯ g_{1} (θ (ω)) · g_{1} (ω)  (18)

where θ is the shift in probability space. Therefore, in a convenient probability space, g_{k} can be regarded as a product of an i.i.d. sequence of random elements in G. On the other hand the limit in (17) can be discretized with the same results, that is, the Lyapunov exponents of the sequence ρ (g_{n}) of random matrices coincides with the Lyapunov exponents of the system (16) (see Carverhill [7] for further discussions about the continuous vs. discrete-time stochastic systems).

With this in mind we observe that the support suppμ of μ has nonempty interior in G. In fact, by the support theorem suppμ = cl (A (1)), where A (1) is the attainable set from the identity in G, at time 1, of the right invariant control system in G obtained from (7):

ġ = X (g) + ∑_{i=1}^{m} u_{i} (t) Y^{i} (g)

with u_{i} (t) piecewise constant controls (see e.g. Ikeda and Watanabe [16]). A general result of Sussmann and Jurdjevic [25] says that the attainable set of an analytic control system at a fixed time has nonvoid interior inside the leaf of a certain integrable distribution of codimension zero or one in the state space. An application of this result to a right invariant control system on a Lie group proves that A (1) has nonvoid interior in a coset of a connected normal subgroup H ⊂ G with codimension zero or one. Since we are working with a semi-simple Lie group, there are no normal subgroup of codimension one. Hence H = G and suppμ has nonvoid interior in G.

This fact ensures that the probability measure μ and the corresponding random product are under the basic assumptions of [14], namely that the semigroup T_{μ} generated by suppμ is contracting and strongly irreducible.

Consider now the Iwasawa decomposition of the product (18)

g_{k} (ω) = u_{n} (ω) h_{k} (ω) n_{k} (ω) ∈ KAN.

Write also the polar decomposition

g_{k} (ω) = x_{k} (ω) a_{k} (ω) y_{k} (ω) ∈ KĀ^{+}K,  (19)

where Ā^{+} stands for the exponential of the closure of a Weyl chamber in 𝔞. By [14, Cor. 2.8] h_{k} (ω) a_{k}^{−1} (ω) converges almost surely so that

lim \frac{1}{k} log h_{k} = lim \frac{1}{k} log a_{k}.

At this point we need the following well known fact about Cartan decompositions of Lie algebras and subalgebras (see e.g. [15], [27]):

Lemma 5.1 Let 𝔤 ⊂ 𝔤̃ be noncompact semi-simple Lie algebras and consider a Cartan decomposition 𝔤 = 𝔨 ⊕ 𝔰 of 𝔤. Then there exists a Cartan decomposition 𝔤̃ = 𝔨̃ ⊕ 𝔰̃ such that 𝔨 ⊂ 𝔨̃ and 𝔰 ⊂ 𝔰̃. Also, if 𝔞 ⊂ 𝔰 is maximal abelian then there exists a maximal abelian 𝔞̃ ⊂ 𝔰̃ such that 𝔞 ⊂ 𝔞̃.

The compatible Cartan decompositions extend to the group level: Let G ⊂ G̃ be semi-simple Lie groups with Lie algebras 𝔤 ⊂ 𝔤̃. A Cartan decomposition G = KS comes from a Cartan decomposition of 𝔤, through the exponential mapping. Hence there exists a Cartan decomposition G̃ = K̃S̃ such that K ⊂ K̃ and S ⊂ S̃. We can apply this fact to our linear group G. Since G is semi-simple it is contained in Sl (d, ℝ) so that a Cartan decomposition G = KS extends to a Cartan decomposition of Sl (d, ℝ). This means that there is an inner product of ℝ^{d} such that with respect to it the elements of K are orthogonal matrices and those of S are symmetric and positive definite. The same way for a polar decomposition G = KĀ^{+}K there is a group Ã of diagonal matrices in Sl (d, ℝ) containing A. We remark that it is not true in general that Ā^{+} is contained in a unique Weyl chamber of Ã. Given the decomposition (19) of g_{k} (ω), the eigenvalues of

λ = lim \frac{1}{2k} log (g_{k} (ω) g_{k} (ω)^{∗})

are exactly the eigenvalues of lim \frac{1}{k} log a_{k} (ω), which coincides with our previously defined Lyapunov exponent matrix. By the approach in Ruelle [23], the eigenvalues of λ are the Lyapunov exponents of our system (16).

We state now these facts using the language of representation theory. Let ρ : 𝔤 → 𝔤𝔩 (V) be a representation of the semi-simple Lie algebra in the real vector space V. If 𝔞 ⊂ 𝔤 has the same meaning as before, a linear functional λ : 𝔞 → ℝ is said to be a weight of the representation if the weight space

V_{λ} = {v ∈ V : ρ (H) v = λ (H) v for all H ∈ 𝔞}

is not zero. If H ∈ 𝔞 then ρ (H) is diagonalizable and its eigenvalues are λ (H) with λ running through the set of weights. With this terminology we have the formula:

Theorem 5.2 Assume that the right invariant system (7) in G satisfies the accessibility property. Consider the linear differential equation (16) induced by the representation ρ. Then the Lyapunov exponents of (16) are the entries of

ρ (∫ Q (b) ν (db)) ,

which are λ (∫ Q (b) ν (db)) with λ running through the weights of the representation. Here Q is given by (12) and ν is the unique invariant probability measure in the maximal flag manifold of G.

5.2 Systems with Symmetric Vector Fields

Consider a system

dg = ∑_{i=1}^{m} Y^{i} (g) ∘ dW_{i}  (20)

without drift such that {Y^{1}, …, Y^{m}} is an orthonormal basis of 𝔰. Under the inverse mapping of G the right invariant vector field Y^{i} is mapped into the left invariant vector field whose value at the identity is −Y^{i}. It was proved by Malliavin and Malliavin [21] that the left invariant system thus obtained is the horizontal diffusion in the symmetric space G/K (see also Liao [20] and Taylor [26]). In our computations below we shall recover a result of [21] on the limit behavior of the A-part of the Iwasawa decomposition of the horizontal diffusion.

For the system (20) the integrand in the formula (12) becomes

Q (b) = \frac{1}{2} ∑_{i=1}^{m} r_{Y^{i}} (b) .

By Corollary 4.4 if H ∈ 𝔞 then

⟨H, Q (b)⟩ = ∑_{i=1}^{m} |Ỹ^{i} (ux_{0})|^{2}  (21)

with b = ub_{0} and the norm is with respect to T-K metric in B_{H}. We shall compute this expression explicitly. Firstly note that the right hand side of (21) is independent of the orthonormal basis, in fact:

Lemma 5.3 Let (Y^{i})_{i=1,...,m} and (Z^{i})_{i=1,...,m} be orthonormal bases of 𝔰. Then

∑_{i=1}^{m} |Ỹ^{i} (ux_{0})|^{2} = ∑_{i=1}^{m} |Z̃^{i} (ux_{0})|^{2}

for all u ∈ K.

Proof: Let a_{j}^{i} ∈ ℝ, i, j = 1, …, m, be such that

Z^{i} = ∑ a_{j}^{i}Y^{j}.

Since the restriction to 𝔰 of the Cartan-Killing form is an inner product, the m × m matrix (a_{j}^{i})_{i,j} is orthogonal. Hence

∑_{i=1}^{m} |Z̃^{i} (ux_{0})|^{2} = ∑_{i,j=1}^{m} (a_{j}^{i})^{2} |Ỹ^{j} (ux_{0})|^{2} + ∑_{i}^{m} ∑_{j≠k}^{m} a_{j}^{i}a_{k}^{i} (Ỹ^{j} (ux_{0}) , Ỹ^{k} (ux_{0}))

shows the lemma. □

Corollary 5.4 If (Y^{i})_{i=1,...,m} is an orthonormal basis of 𝔰 then ∑_{i=1}^{m} |Ỹ^{i} (ux_{0})|^{2} is independent of u ∈ K.

Proof: The translation formula for vector fields in a homogeneous spaces yields

Ỹ (ux_{0}) = u_{∗} (Ad (u^{−1}) Y^{i})^{∼} (x_{0}) .

Hence

∑_{i=1}^{m} |Ỹ^{i} (ux_{0})|^{2} = ∑_{i=1}^{m} |(Ad (u^{−1}) Y^{i})^{∼} (x_{0})|^{2} .

By the previous lemma the right hand side is equal to ∑_{i=1}^{m} |Ỹ^{i} (x_{0})|^{2} because Ad (u^{−1}) Y^{i}, i = 1, …, m form an orthonormal basis. □

In the light of this corollary we write

c_{H} = ∑_{i=1}^{m} |Ỹ^{i} (ux_{0})|^{2} .

This is a constant independent of u ∈ K and the orthonormal basis (Y^{i})_{i=1,...,m} of 𝔰. In order to compute this constant we put u = 1 and choose an orthonormal adapted basis, which complements

\{S_{1}, …, S_{m}\} = \{\frac{\sqrt{2}}{2}σY_{1}, …, \frac{\sqrt{2}}{2}σY_{m}\}

where {Y_{1}, …, Y_{m}} is a basis of 𝔫^{+}, as the construction in (2). By Lemma 2.4 {S̃_{1} (x_{0}) , …, S̃_{m} (x_{0})} is an orthogonal basis of T_{x_{0}}B_{H} with respect to the T-K metric. Moreover, for j = 1, …, m,

⟨S_{j}, S_{j}⟩ = \frac{1}{2}⟨Y_{j} − θY_{j}, Y_{j} − θY_{j}⟩ = B_{θ} (Y_{j}, Y_{j}) = 1

so that

|S̃_{j} (x_{0})|^{2} = α (H)

if Y_{j} ∈ 𝔤_{α}. Therefore, summing up over α > 0 gives the value of c_{H}:

Proposition 5.5 c_{H} = ∑_{α>0} d_{α} α (H) where d_{α} = dim 𝔤_{α}.

From this proposition we get easily an expression for the Lyapunov exponent matrix for the symmetric system. For a root α define H_{α} by α (·) = ⟨H_{α}, ·⟩. Let Q ∈ 𝔞 be the constant function Q (b). Then Q is the only element of 𝔞 satisfying

⟨Q, H⟩ = \frac{c_{H}}{2} = \frac{1}{2} ∑_{α>0} d_{α}α (H) .

for every H ∈ 𝔞. Hence

Q = \frac{1}{2} ∑_{α>0} d_{α}H_{α}.

After integrating this constant with respect to the invariant probability measure on B we get Theorem 5.6 For the symmetric system (20) the Lyapunov exponent matrix is

Λ = \frac{1}{2} ∑_{α>0} d_{α}H_{α}  (22)

This result agrees with Theorem 8.2 in Malliavin and Malliavin [21] (see also Taylor [26, Cor. 5.2]). The factor 1/2 appearing in (22) is due to our normalization of (20) which is given by an orthonormal basis of 𝔰. Also, in [21] appears a minus sign which is due to the fact that the horizontal diffusion is given by g_{t}^{−1} where g_{t} is the solution of (20) and the Iwasawa decomposition considered in [21] is NAK so that the two A-parts are inverse to each other.

The symmetric stochastic differential equation (20) also induces Brownian motions in the immersed flag manifolds. Indeed, by Lemma 2.3 and the remarks afterwards, if H ∈ 𝔞^{+} is such that α (H) = 1 for every positive root α such that α (H) ≠ 0 then the T-K metric in B_{H} is induced by its immersion in 𝔰. On the other hand Lemma 2.2 ensures that if Z ∈ 𝔰 then the vector field Z̃ induced on B_{H} is the gradient of the height function X ↦ ⟨Z, X⟩. Since (20) is made up from an orthonormal basis of 𝔰, it follows that the diffusion generated by the induced differential equation on B_{H} is a gradient Brownian system with respect to the T-K metric. More generally, Liao [20, Thm. 1] shows that in each flag manifold there is a canonical K-invariant Riemannian metric for which our symmetric system induces a Brownian motion.

Like in the case of linear systems, here, the A-part of the Iwasawa decomposition gives the Lyapunov spectrum of the induced system. Hence, if Λ stands for the Lyapunov exponent matrix then the eigenvalues of ad (Λ) in the tangent space at the origin of the flag manifold are the Lyapunov exponents, given by:

{−α (Λ) : α ∈ Π^{+}, α (H) ≠ 0}

(see [20, Section 5] for a detailed discussion of these Lyapunov exponents).

6 Examples

We will describe here some of the semi-simple Lie groups together with their flag manifolds, emphasizing the T-K metric which by Proposition 4.5 enters in the formula for the Lyapunov exponent matrix.

6.1 Real rank 1 groups

The real rank of 𝔤 (or G) is the dimension of the subalgebra 𝔞. For a rank one Lie group there is just one flag manifold, which is diffeomorphic to a sphere S^{n} with dimension n = dim 𝔰 − 1. There is just one simple root α and the positive roots are α and possibly 2α. Hence for a symmetric system the Lyapunov exponent matrix is

Λ = d_{α}H_{α} + d_{2α}H_{2α} = (d_{α} + 2d_{2α}) H_{α}

with d_{2α} = 0 if 2α is not a root. Since Ad (K) is transitive on the spheres of 𝔰, modulo constant multiples there is just one T-K metric. It is given by the isometric immersion of S^{n} into 𝔰 when d_{2α} = 0, hence in this case it is the canonical Riemannian metric in S^{n}.

The simple rank one Lie algebras are composed of three series of algebras and an exceptional one (see [15, Ch. X, Table V]). Below we list them with the corresponding dimensions of the root spaces and of 𝔰.

• 𝔰𝔬 (1, n); d_{α} = n − 1, d_{2α} = 0; dim 𝔰 = n. (This class includes 𝔰𝔩 (2, ℝ) ≈ 𝔰𝔭 (1, ℝ) ≈ 𝔰𝔬 (1, 2) and 𝔰𝔲^{∗} (4) ≈ 𝔰𝔬 (1, 5))

• 𝔰𝔲 (1, n); d_{α} = 2 (n − 1), d_{2α} = 1; dim 𝔰 = 2n. (This class includes 𝔰𝔬^{∗} (6) ≈ 𝔰𝔲 (1, 3))

• 𝔰𝔭 (1, n); d_{α} = 4 (n − 1), d_{2α} = 3; dim 𝔰 = 4n.

• A real form of the exceptional Lie algebra F_{4}; d_{α} = 8, d_{2α} = 7; dim 𝔰 = 16.

Symmetric systems in 𝔰𝔬 (1, n) were studied by Baxendale [4]. This is the Lie algebra of real matrices of the form

( \begin{matrix}0 & γ \\ γ^{t} & B \end{matrix} )  (23)

with B skew-symmetric and γ a 1 × n-matrix. For a Cartan decomposition we can take 𝔰 to be the subspace of matrices in (23) with B = 0 and

𝔞 = \{H (γ) = ( \begin{matrix}0 & γ \\ γ^{t} & 0 \end{matrix} ) : γ = (x, 0, …, 0) , x ∈ ℝ\}.  (24)

The computation of adjoints and the Cartan-Killing form gives, for the simple root α,

H_{α} = \frac{1}{2 (n − 1)}H (γ)

where γ = (1, 0, …, 0). Hence the matrix Lyapunov exponent is Λ = \frac{1}{4}H (γ). The eigenvalues of this matrix are the Lyapunov exponents computed in [4, Thm. 2.6], with λ = \sqrt{1/ (2 (n − 1))} (the notation is as in [4]). This normalization of the system of [4] comes from the fact that we consider systems made up of an orthonormal basis in 𝔰. In fact, with H (γ) and γ as in (23), ⟨H (γ) , H (γ)⟩ = 2 (n − 1) |γ|^{2} with |γ| given by the canonical inner product in ℝ^{n}.

The Lie algebra 𝔰𝔲 (1, n) is the algebra of the skew-Hermitian matrices with respect to a Hermitian form of signature (1, n). It is realized as the algebra of complex matrices of the form

( \begin{matrix}it & z \\ z^{∗} & B \end{matrix} )  (25)

where t ∈ ℝ, z ∈ ℂ^{n} is a 1 × n complex matrix and B + B^{∗} = 0. We denote the matrix in (25) by (t, z, B) ∈ ℝ × ℂ^{n} × 𝔲 (n). There are the following data:

𝔨 = {(t, 0, B) : trB = −it}, 𝔰 = {(0, z, 0) : z ∈ ℂ^{n}} ≈ ℂ^{n}

and 𝔞 = span_{ℝ}H_{0} with H_{0} = (0; 1, 0, …, 0; 0) is a maximal abelian subalgebra of 𝔰. The eigenvalues of ad (H_{0}) are 0, ±1, ±2, so that the roots are ±α, ±2α with α (H_{0}) = 1. Since dim 𝔤_{±α} = 2 (n − 1) and dim 𝔤_{±2α} = 1,

⟨H_{0}, H_{0}⟩ = 4 (n − 1) + 8 = 4 (n + 1) .

Hence the Cartan-Killing form is ⟨C, D⟩ = 2 (n + 1) tr (CD) for C, D ∈ 𝔰𝔲 (1, n).

If z ∈ ℂ^{n} is such that its first component is purely imaginary we denote by [z] the n × n skew-Hermitian matrix B = (b_{jk}) whose first row is z and b_{jk} = 0 if j, k ≥ 2. By computing the eigenspaces of ad (H_{0}), it follows that 𝔫^{+} = 𝔤_{α} + 𝔤_{2α} is the subalgebra whose elements are of the form (t; z; [z]) with z = (−it, z_{2}, …, z_{n}). If Y = (t; z; [z]) ∈ 𝔫^{+} then Y + θ (Y) = 2 (t; 0; [z]) because θ fixes 𝔨 and changes the sign in 𝔰. This implies that

ζ : ( \begin{matrix}it & 0 \\ 0 & [z] \end{matrix} ) ⟼ ( \begin{matrix}0 & z \\ z^{∗} & 0 \end{matrix} ) .

Now, let X = (t; 0; [z]) and Y = (s; 0; [w]) with z_{1} = it and w_{1} = is. Then

[H_{0}, X] = (0; 2it, z_{2}, …, z_{n}; 0) .

So that

(X̃, Ỹ )_{H_{0}} = 4 (n + 1) (2ts + Re (z_{2}w̄_{2} + ⋯ + z_{n}w̄_{n})) .

Through the identification of 𝔰 with ℂ^{n}, X̃ (H_{0}) becomes the tangent vector

− (2it, z_{2}, …, z_{n}) .

Since a similar expression holds for Ỹ, the T-K metric in the tangent space to S^{2n−1} at (1, 0, …, 0) is

((it, z_{2}, …, z_{n}) , (is, w_{2}, …, w_{n})) = (n + 1) (2ts + Re (z_{2}w̄_{2} + ⋯ + z_{n}w̄_{n})) .

Note that (it, 0, …, 0) is in the direction of the complex line spanned by (1, 0, …, 0). This inner product extends to the whole sphere through the action of SU (n) which is contained in K. Since SU (n) maps complex subspaces into complex subspaces, we have the following description of the T-K metric at z ∈ S^{2n−1}: it is 2 (n + 1) times the canonical inner product in the direction of the complex line spanned by z and (n + 1) times the canonical inner product in the subspace orthogonal to this complex line.

We refrain to write down the details for the algebra 𝔰𝔭 (1, n). The description is similar to 𝔰𝔲 (1, n) with a quaternionic space playing the role of ℂ^{n}.

6.2 Real Forms

Up to isomorphism each complex semi-simple Lie algebra 𝔤_{ℂ} has just one normal real form 𝔤 (see e.g. [15]). For such a real form a maximal abelian subalgebra 𝔞 ⊂ 𝔰 is a Cartan subalgebra of 𝔤, the root spaces 𝔤_{α} are one dimensional and the subalgebra 𝔪 ⊂ 𝔨 reduces to zero. We describe below the normal real forms of the simple complex classical Lie algebras, together with their flag manifolds. We use the notation E_{ij} for the matrix whose only nonzero entry is 1 in position i, j. Also,

λ_{i} : diag{a_{1}, …, a_{n}} ⟼ a_{i}

is a linear functional in the space of diagonal matrices.

A_{l} The Lie algebra 𝔰𝔩 (n, ℝ) is the normal real form of 𝔰𝔩 (n, ℂ). We have 𝔨 = 𝔰𝔬 (n, ℝ), 𝔰 the space of trace zero symmetric n × n matrices and 𝔞 the subalgebra of diagonal matrices in 𝔰𝔩 (n, ℝ). The Cartan-Killing form is

⟨X, Y ⟩ = 2n tr (XY) .

The set of (𝔤, 𝔞)-roots is Π = {α_{ij} = λ_{i} − λ_{j} : i ≠ j}. The root space g_{α_{ij}} is spanned by E_{ij}, i ≠ j. The co-root α_{ij} with respect to the Cartan-Killing is the matrix

H_{α_{ij}} = \frac{E_{ii} − E_{jj}}{2n} .

We can declare α_{ij} > 0 if i < j. Hence for a symmetric system normalized by the Cartan-Killing form its Lyapunov exponent matrix is

Λ = \frac{1}{2n}diag\{n − 1, n − 3 …, −n + 1\}.

The flag manifolds of a group whose Lie algebra is 𝔰𝔩 (n, ℝ) are the standard flag manifolds: Given a sequence of positive integers r = (r_{1}, … r_{k}) with r_{1} + ⋯ + r_{k} = n let 𝔽 (r) stand for the manifold of all flags (V_{1} ⊂ ⋯ ⊂ V_{k}) where V_{i} is a subspace of ℝ^{n} with dim V_{i} = r_{1} + ⋯ + r_{i}. Let

H = diag{a_{1}, …, a_{n}} a_{1} + ⋯ + a_{n} = 0

be such that a_{1} = ⋯ = a_{r_{1}} > a_{r_{1}+1} = ⋯ = a_{r_{1}+r_{2}} > · · ·. Then 𝔽 (r) identifies with the orbit of H under conjugations by orthogonal matrices. By varying H we get different embeddings of 𝔽 (r) into 𝔰 with corresponding T-K metric.. In case H has only two eigenvalues with multiplicities d and n − d, 𝔽 (r) is the Grassmannian Gr_{d} (n) of d-dimensional subspaces of ℝ^{n}. In this case the T-K metric is just the metric induced by the embedding in 𝔰. It is a multiple of the canonical metric in the Gr_{d} (n) which turns it into a locally symmetric space (see Kobayashi and Nomizu [18]). The Grassmannians are the only immersed flag manifolds.

C_{l} The normal real form of the complex simple Lie algebra 𝔰𝔭 (n, ℂ) is the Lie algebra 𝔰𝔭 (n, ℝ) of real symplectic matrices. It is the Lie algebra of matrices which are skew with respect to the canonical symplectic form

ω (u, v) = u^{t} ( \begin{matrix}0 & −1_{n×n} \\ 1_{n×n} & 0 \end{matrix} ) v u, v ∈ ℝ^{2n}.

Hence it is given by the 2n × 2n real matrices of the form

(A, B, C) = ( \begin{matrix}A & B \\ C & −A^{t} \end{matrix} )

with B and C symmetric n × n matrices. A Cartan decomposition is given by the symmetric and skew symmetric matrices in 𝔰𝔭 (n, ℝ), that is,

𝔨 = {(A, B, −B^{t}) : A + A^{t} = 0} ≈ 𝔲 (n) 𝔰 = {(A, B, B^{t}) : A = A^{t}}.

We can choose

𝔞 = {(A, 0, 0) : A = diag (a_{1}, …, a_{n})}.

Then the roots are α_{ij} = λ_{i} − λ_{j}, i ≠ j and ±β_{ij} = ± (λ_{i} + λ_{j}) with Π^{+} = {α_{ij}, i < j; β_{ij}}. Since the Cartan-Killing form is ⟨C, D⟩ = 2ntr (CD), the co-roots are

H_{α_{ij}} = \frac{1}{2n} (E_{ii} − E_{jj}, 0, 0) H_{β_{ij}} = \frac{1}{2n} (E_{ii} + E_{jj}, 0, 0) .

Adding up the positive co-roots we get the Lyapunov exponent matrix for a symmetric system

Λ = \frac{1}{2 (n + 1)} (diag\{2n − 1, 2n − 3, …, 1\}, 0, 0) .

A subspace V ⊂ ℝ^{2n} is Lagrangian if the restriction of ω to V is identically zero. By force dim V ≤ n. Denote by L_{d} (n), d = 1, …, n the set of all d-dimensional Lagrangian subspaces of ℝ^{2n}. Similarly, denote by 𝔽L (r) the subset of 𝔽 (r) consisting of flags (V_{1} ⊂ ⋯ ⊂ V_{k}) which are made up of Lagrangian subspaces. Any flag manifold of 𝔰𝔭 (n, ℝ) is some 𝔽L (r). In particular, the minimal flag manifolds are L_{d} (n), d = 1, …, n. For a sequence r let

A_{r} = diag{a_{1}, …, a_{n}}

be such that a_{1} = ⋯ = a_{r_{1}} > a_{r_{1}+1} = ⋯ = a_{r_{1}+r_{2}} > ⋯, and put H_{r} = (A_{r}, 0, 0). Then the Ad (K)-orbit of H identifies with 𝔽L (r). Since H_{r} has just one positive eigenvalue if and only if r = (n), L_{n} (n) is the only flag manifold of 𝔰𝔭 (n, ℝ) which is immersed.

B-D_{l} The normal real form of 𝔰𝔬 (n, ℂ) is 𝔰𝔬 (l, n − l) the Lie algebra of the matrices which are skew-symmetric with respect to a quadratic form of signature (l, n − l). Here n = 2l or 2l + 1 according if it is even or odd. The description here parallels that of the symplectic Lie algebra, with the quadratic form instead of the symplectic one. We shall avoid it here, but record that, in the canonical realization of the algebras, the Lyapunov exponent matrix for a symmetric system is given by

Λ = \frac{1}{2 (l − 1)} ( \begin{matrix}A & 0 \\ 0 & −A \end{matrix} ) A = diag\{2l − 2, 2l − 4, …, 0\}

for n = 2l even and

Λ = \frac{1}{2l − 1} ( \begin{matrix}0 & 0 & 0 \\ 0 & A & 0 \\ 0 & 0 & −A \end{matrix} ) A = diag\{2l − 1, 2l − 3, …, 1\}

for n = 2l + 1 odd.

7 Large Deviations

In this section we go back to the set up of subsection 5.2 to calculate the entropy of large deviations for the Lyapunov exponent matrix of the stochastic equation (20), driven by the symmetric vector fields. According to Theorem 5.6 we have that

lim_{t→∞} \frac{1}{t} log A_{t} = Λ a.s.,

where

Λ = \frac{1}{2} ∑_{α>0} d_{α}H_{α}.

Let α be an element in the dual 𝔞^{∗}, we calculate the large deviation for the real random variable α (log A_{t} − tΛ). It is illustrative to think of the 𝔰𝔩(n, ℝ) case, so that we can see the large deviation for the convergence in each entry of the diagonal matrices. Essentially we check that these random variables satisfy the continuous version of a well known general result on large deviation, which for the convenience of the reader we reproduce here:

Theorem 7.1 Let (Z_{n})_{n≥1} be a sequence of real random variables such that for some η > 0 there exists the limit

lim_{n→∞} \frac{1}{n} log 𝔼 [exp sZ_{n}] = c (s)

for all s ∈ [−η, η]. Assume that c (s) is of C^{1}-class, strictly convex in [−η, η] and c′ (0) = 0. Then for ε ∈ (0, c (η) /η) ,

lim_{n→∞} \frac{1}{n} log ℙ [Z_{n} > nε] = ϕ (ε)

where ϕ is the Legendre transform

ϕ (ε) = − sup_{s∈(0,η)} (sε − c (s)) .

For a proof see e.g. Bougerol and Lacroix [6, Lemma 6.2, Chapter V] or Freidlin and Ventsel [13, Thm. 1.1 and 1.2, Chapter V]. Now, we state the following

Theorem 7.2 For the symmetric system (20), let α be an element of 𝔞^{∗}. Then

lim_{t→∞} \frac{1}{t} log ℙ [α(log A_{t} − tΛ) > tε] = −\frac{ε}{∥α∥^{2}}

where ∥α∥ is the norm with respect to the Cartan-Killing inner product on 𝔰.

Proof: We only have to show that the function

c (s) = lim_{t→∞} \frac{1}{t} log 𝔼 [exp sα (log A_{t} − tΛ)]

satisfies the conditions of Theorem 7.1. Since Λ is deterministic we write c (s) = λ_{a}(s) − sα (Λ), and compute the s-exponential moment

λ_{α} (s) = lim_{t→∞} \frac{1}{t} log 𝔼 [exp sα (log A_{t})] .

By the stochastic equation (9) which describes the process H_{t} = log A_{t} and the Itô’s formula we have that

e^{sα(log A_{t})} = 1 + s ∫_{0}^{t} e^{sα(H_{u})}α (q_{Y^{i}} (b_{u})) dW_{u}^{i}
+\frac{1}{2} ∫_{0}^{t} e^{sα(H_{u})} (sα (∑_{i=1}^{m} r_{Y^{i}} (b_{u})) + s^{2} ∑_{i=1}^{m} α (q_{Y^{i}} (b_{u}))^{2}) du.(26)

The first summation is clarified by the comments after Proposition 5.5: the function Q (b) = \frac{1}{2} ∑_{i=1}^{m} r_{Y^{i}} (b) is constant and equal to Λ for all b ∈ B. For the second summation, note that it is independent of the orthonormal basis {Y^{1}, …, Y^{m}}. In fact, if Z^{i} = ∑_{j} a_{ij}Y^{j} with (a_{ij})_{1≤i,j≤m} an orthogonal matrix then the linearity of Y ↦ q_{Y} implies that

∑_{i=1}^{m} α (q_{Z^{i}} (b))^{2} = ∑_{i=1}^{m} α (q_{Y^{i}} (b))^{2} .

By the definition of q_{X} in equation (6), for u ∈ K, q_{X} (ub) = q_{Ad(u^{−1})X} (b_{0}). Since Ad (u^{−1}) maps orthonormal bases into orthonormal bases, the above summation is also independent of b ∈ B. Moreover, by choosing a basis which is an extension of an orthonormal basis in 𝔞 one sees that this summation equals ∥α∥^{2}. Summing up these remarks, by equation (26) and Fubini theorem we conclude that the expectation r_{t} = 𝔼 [exp sα (log A_{t})] satisfies the linear ODE:

ṙ_{t} = (sα (Λ) + \frac{s^{2}}{2} ∥α∥^{2}) r_{t}.

Hence the s-moment exponent λ_{α} (s) = sα (Λ) + \frac{s^{2}}{2} ∥α∥^{2} and finally

c (s) = \frac{s^{2}}{2} ∥α∥^{2} .

By Theorem 7.1, the Legendre transform of c (s) provides the entropy of large deviations of the statement. □

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