Lagrangian for pinned diffusion process
In the 1960s Professor K. Itô tried to understand the Feynman path integral probabilistically and constructed its integral representation over time dependent Hilbert spaces (, also ). Near the end of the 1970s D.Fujiwara succeeded in proving the existence of the limit of finite dimensional path integrals for Schrödinger equations in a very strong sense , and later in showing “Itô’s version” . Inspired by their works and looking at the discussions on the effect of curvature among physicists, one of the authors started studying the problem of most probable path or the Lagrangian or the Onsager-Machlup function or the probability functional for diffusion process on manifolds (cf., e.g., ,), and which is completely determined by a collaboration with S. Watanabe  (see Theorem 1.2 below).
KeywordsStochastic Differential Equation Ergodic Theorem Radial Motion Tubular Neighborhood Radial Part
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