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

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*Paulo R. C. Ruffino and Luiz A. B. San Martin*

**Address.** Instituto de Matematica, Universidade Estadual de Campinas,
Cx. Postal 6065,

13.081-970 Campinas - SP, BRASIL
**E-mail:** smartin@ime.unicamp.br

ruffino@ime.unicamp.br

**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 Borel
type 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.

**AMSclassification.** 60H10 (58G32, 22E46)

**Keywords.** Lyapunov exponents, stochastic differential equations,
semi-simple Lie groups, flag manifolds.