# Brownian motions on groups of matrices

# Brownian motions on groups of matrices

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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