Integration formulas for Brownian motion on classical compact Lie groups

Combinatorial formulas for the moments of the Brownian motion on classical compact Lie groups are obtained. These expressions are deformations of formulas of B. Collins and P. 'Sniady for moments of the Haar measure and yield a proof of the First Fundamental Theorem of invariant theory and of classical Schur-Weyl dualities based on stochastic calculus.