Semigroups and Moment Lyapunov Exponents

Author(s)

San Martin, Luiz A. B. ^[1]

Affiliation

^[1] Univ Estadual Campinas, Inst Math, Campinas, SP - Brazil

• Document type: Journal article
• Source: JOURNAL OF LIE THEORY; v. 30, n. 2, p. 587-616, 2020.
• Web of Science Citations: 0

Abstract

Let G be a noncompact semi-simple Lie group with finite center and mu a probability measure on G. We consider (i) the semigroup S-mu, generated by the support of mu (with the assumption that intS(mu) not equal empty set); (ii) The spectral radii r(lambda) of the operators U-lambda (mu) where U-lambda is a (nonunitary) representation of G induced by a real character and (iii) the moment Lyapunov exponents gamma (lambda, x) of the i.i.d. random product on G defined by mu. The equality r(lambda) = gamma (lambda, x) holds in many cases. We give a necessary and sufficient condition to have S-mu = G in terms of the analyticity of the map lambda bar right arrow r(lambda). The condition is applied to measures obtained by solutions of invariant stochastic differential equations on G yielding a necessary and sufficient condition for the controllability of invariant control systems on G in terms of the largest eigenvalues of second order differential operators. (AU)