Brownian motion on a submanifold
Consider an \(n\)-dimensional complete Riemannian manifold \(N\), and a closed \(m\)-dimensional submanifold \(M\). Using that the Laplacian on \(N\) is got by projection of the horizontal Laplacian on \(O(N): (\Delta^N\varphi)\circ\pi= \sum^n_{i=1} (H^N_i)^2(\varphi\circ\pi)\), one obtains classically the Brownian motion on \(N\) as \(\pi(p^N_t)\) where \(dp^N_t= \sum^n_{i=1} H^N_i(p^N_t)\circ dw^i_t\); moreover \(p^N_t\) performs the parallel transport along \(\pi(p^N_t)\). Comparing in some suitable way the canonical horizontal vector fields \(H^N_i\) with the orthogonal projection \(TN\to TM\), one gets horizontal vector fields \(H^M_i\), such that \((\Delta^M\varphi)\circ\pi= \sum^n_{i=1} (H^M_i)^2(\varphi\circ\pi)\), and then such that \(\pi(q^M_t)\) is a Brownian motion on \(M\), where \(dq^M_t= \sum^n_{i=1} H^n_i(q^M_t)\circ dw^i_t\). But \(q^M_t\) performs a parallel transport along \(\pi(q^M_t)\) in \(N\), and not in \(M\). To get the right parallel transport in \(M\), the author finds vector fields \(K^M_1,\dots, K^M_m\), which are obtained by a suited vertical modification of \(H^N_1,\dots, H^N_m\), such that \[ (\Delta^M\varphi)\circ\pi= \sum^m_{i=1} (K^M_i)^2(\varphi\circ\pi). \] Then \(dp^M_t= \sum^m_{i=1} K^M_i(p^M_t)\circ dw^i_t\) defines again a Brownian motion \(\pi(p^M_t)\) on \(M\), such that now \(p^M_t\) performs the stochastic parallel transport along \(\pi(p^M_t)\) in \(M\).
scientific article; zbMATH DE number 2133085