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Hyperbolic Dynamics and Brownian MotionAn Introduction$
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Jacques Franchi and Yves Le Jan

Print publication date: 2012

Print ISBN-13: 9780199654109

Published to Oxford Scholarship Online: January 2013

DOI: 10.1093/acprof:oso/9780199654109.001.0001

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Brownian motions on groups of matrices

Brownian motions on groups of matrices

(p.146) Chapter Seven: Brownian motions on groups of matrices
Hyperbolic Dynamics and Brownian Motion

Jacques Franchi

Yves Le Jan

Oxford University Press

This chapter is devoted to (left and right) Brownian motions on groups of matrices, which the chapter constructs as solutions to linear stochastic differential equations. The chapter establishes in particular that the solution of such an equation lives in the subgroup associated with the Lie subalgebra generated by the coefficients of the equation. Reversed processes, Hilbert–Schmidt estimates, approximation by stochastic exponentials, Lyapunov exponents and diffusion processes are also considered. Then the chapter concentrates on important examples: the Heisenberg group, PSL(2), SO(d), PSO(1, d), the affine group A d and the Poincaré group P d +1. By means of a projection, we obtain the spherical and hyperbolic Brownian motions, and relativistic diffusion in Minkowski space.

Keywords:   stochastic differential equations, left Brownian motion, diffusion processes, hyperbolic Brownian motion, relativistic diffusion

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