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

Paulo R. C. Ruffino; Luiz A. B. San Martin

Archivum Mathematicum (2001)

  • Volume: 037, Issue: 3, page 207-231
  • ISSN: 0044-8753

Abstract

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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 / M A N ). 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.

How to cite

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Ruffino, Paulo R. C., and San Martin, Luiz A. B.. "Lyapunov exponents for stochastic differential equations on semi-simple Lie groups." Archivum Mathematicum 037.3 (2001): 207-231. <http://eudml.org/doc/248742>.

@article{Ruffino2001,
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.},
author = {Ruffino, Paulo R. C., San Martin, Luiz A. B.},
journal = {Archivum Mathematicum},
keywords = {Lyapunov exponents; stochastic differential equations; semi-simple Lie groups; flag manifolds; Lyapunov exponents; flag manifolds; stochastic differential equations},
language = {eng},
number = {3},
pages = {207-231},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Lyapunov exponents for stochastic differential equations on semi-simple Lie groups},
url = {http://eudml.org/doc/248742},
volume = {037},
year = {2001},
}

TY - JOUR
AU - Ruffino, Paulo R. C.
AU - San Martin, Luiz A. B.
TI - Lyapunov exponents for stochastic differential equations on semi-simple Lie groups
JO - Archivum Mathematicum
PY - 2001
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 037
IS - 3
SP - 207
EP - 231
AB - 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.
LA - eng
KW - Lyapunov exponents; stochastic differential equations; semi-simple Lie groups; flag manifolds; Lyapunov exponents; flag manifolds; stochastic differential equations
UR - http://eudml.org/doc/248742
ER -

References

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