Brownian motion on an embedded manifold as the limit of Brownian motions with reflection in its tubular neighborhoods. (English. Russian original) Zbl 1053.60090

Sidorova, N. A.

Math. Notes 73, No. 6, 895-899 (2003); translation from Mat. Zametki 73, No. 6, 947-950 (2003).

Introduction: A surface measure on the path space in a compact Riemannian manifold embedded in \(\mathbb{R}^n\) is studied. This measure is defined as the weak limit as \(\varepsilon\to 0\) of the family of measures \(\mathbb{W}_\varepsilon\) corresponding to the Brownian motion on \(\mathbb{R}^n\) starting at a point on the manifold with reflection on the boundary of the tubular \(\varepsilon\)-neighborhood of the manifold. We prove that this limit exists and the surface measure it defines coincides with the Wiener measure on the manifold.

MSC:

60J65

Brownian motion

Keywords:

Brownian motion with reflection; surface measure; path space; Fermi decomposition; Wiener measure