Chapter

Brownian motions on groups of matrices

Jacques Franchi and Yves Le Jan

in Hyperbolic Dynamics and Brownian Motion

Published in print August 2012 | ISBN: 9780199654109
Published online January 2013 | e-ISBN: 9780191745676 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199654109.003.0007

Series: Oxford Mathematical Monographs

Brownian motions on groups of matrices

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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 Ad and the Poincaré group Pd+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

Chapter.  28419 words. 

Subjects: Mathematical and Statistical Physics

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