Brownian motion on a manifold as limit of stepwise conditioned standard Brownian motions (Q2702422)

Let \(M\) be a closed compact Riemannian \(m\)-dimensional manifold embedded in \(\mathbb{R}^n\), \(\Delta_M\) be the Laplace-Beltrami operator and \(q\) be a partition \(0=t_0<t_1<\dots<t_{n-1}<t_n=T\). NEWLINENEWLINENEWLINEThe authors study a continuous process \(\xi_q(t)\) in \(\mathbb{R}^n,\quad \xi_q(0)=x\in M\) that coincides with the standard Brownian motion \(w(t)\in\mathbb{R}^n\) conditioned to \(w(t_i)\in M\) for each \(i=1,\dots n\). NEWLINENEWLINENEWLINEThey prove that for each \(x\in M\) the probability measure \(W^x_{M,q}\) of the process \(\xi_q(t)\) converges in law to the distribution \(W_M^x\) of a diffusion process \(\xi(t)\in M, \quad \xi(0)=x\) as the mesh of the partition \(q\) tends to zero. In addition a generator of \(\xi(t)\) is proved to be equal to \({ 1\over 2} \Delta_M\).NEWLINENEWLINEFor the entire collection see [Zbl 0953.00049].