Invariance Principle for Lifts of Geodesic Random Walks

We consider a certain class of Riemannian submersions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>π</mml:mi> <mml:mo>:</mml:mo> <mml:mi>N</mml:mi> <mml:mo>→</mml:mo> <mml:mi>M</mml:mi></mml:mrow> </mml:math> and study lifted geodesic random walks from the base manifold <i>M</i> to the total manifold <i>N</i>. Under appropriate conditions on the distribution of the speed of the geodesic random walks, we prove an invariance principle, i.e., convergence to horizontal Brownian motion for the lifted walks. This gives us a natural probabilistic proof of the geometric identity relating the horizontal Laplacian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Δ</mml:mi> <mml:mi>H</mml:mi></mml:msub> </mml:math> on <i>N</i> and the Laplace-Beltrami operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Δ</mml:mi> <mml:mi>M</mml:mi></mml:msub> </mml:math> on <i>M</i>. In the setting where <i>N</i> is the orthonormal frame bundle <i>O</i>(<i>M</i>), this identity is central in the Malliavin-Eells-Elworthy construction of Riemannian Brownian motion.