Brownian Processes for Monte Carlo Integration on Compact Lie Groups

This article proposes a Monte Carlo approach for the evaluation of integrals of smooth functions defined on compact Lie groups. The approach is based on the ergodic property of Brownian processes in compact Lie groups. The article provides an elementary proof of this property and obtains the following results. It gives the rate of almost sure convergence of time averages along with a “large deviations” type upper bound and a central limit theorem. It derives probability of error bounds for uniform approximation of the paths of Brownian processes using two numerical schemes. Finally, it describes generalization to compact Riemannian manifolds.

[1]  Mtw,et al.  Stochastic flows and stochastic differential equations , 1990 .

[2]  M. Émery.  Stochastic Calculus in Manifolds , 1989 .

[3]  G. Pagès,et al.  RECURSIVE COMPUTATION OF THE INVARIANT DISTRIBUTION OF A DIFFUSION: THE CASE OF A WEAKLY MEAN REVERTING DRIFT , 2003 .

[4]  A. Shiryaev,et al.  Probability (2nd ed.) , 1995, Technometrics.

[5]  M. Piccioni,et al.  An Iterative Monte Carlo Scheme for Generating Lie Group-Valued Random Variables , 1994 .

[6]  S. Rosenberg.  The Laplacian on a Riemannian Manifold: The Laplacian on a Riemannian Manifold , 1997 .

[7]  B. Collins,et al.  Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group , 2004, math-ph/0402073.

[8]  P. Diaconis,et al.  The Subgroup Algorithm for Generating Uniform Random Variables , 1987, Probability in the Engineering and Informational Sciences.

[9]  Tammo tom Dieck,et al.  Representations of Compact Lie Groups , 1985 .

[10]  S. Varadhan,et al.  Large deviations , 2019, Graduate Studies in Mathematics.

[11]  S. Helgason.  Differential Geometry and Symmetric Spaces , 1964 .

[12]  K. Elworthy.  Stochastic Differential Equations on Manifolds , 1982 .

[13]  M. Pontier,et al.  Convergence des approximations de mcshane d'une diffusion sur une variete compacte , 1987 .

[14]  国田 寛.  Stochastic flows and stochastic differential equations , 1990 .

[15]  P. Kloeden,et al.  Numerical Solutions of Stochastic Differential Equations , 1995 .

[16]  P. Diaconis.  Group representations in probability and statistics , 1988 .

[17]  U. Porod.  The cut-off phenomenon for random reflections , 1996 .

[18]  R. Oliveira.  On the convergence to equilibrium of Kac’s random walk on matrices , 2007, 0705.2253.

[19]  R. Karandikar.  Multiplicative Decomposition of Non-Singular Matrix Valued Continuous Semimartingales , 1982 .

[20]  Nicola Bruti-Liberati.  Numerical Solution of Stochastic Differential Equations with Jumps in Finance , 2010 .

[21]  D. Applebaum.  Lévy Processes in Stochastic Differential Geometry , 2001 .

[22]  M. Liao.  Lévy Processes in Lie Groups , 2004 .

[23]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .