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Lagrangian for pinned diffusion process. (English) Zbl 0867.58073

Ikeda, N. (ed.) et al., Itô’s stochastic calculus and probability theory. Tribute dedicated to Kiyosi Itô on the occasion of his 80th birthday. Tokyo: Springer. 117-128 (1996).
Let \(x(t)\) be a nondegenerate conservative diffusion process on a Riemannian manifold \(M\) such that \(x(t)\) is associated with the drifted heat equation constructed with the Laplace-Beltrami operator on \(M\) and a certain smooth vector field (drift) on \(M\). For a smooth spacetime \(\gamma(t)\), \(t\in [0,T]\), the authors investigate the asymptotics that \(x(t)\) pinned at time \(T\) sojourns in a small tubular neighbourhood of \(\gamma\) up to time \(T\). A formula for the asymptotics is established which in particular involves the action functional with respect to a certain Lagrangian depending on the scalar curvature of \(M\), divergence of the drift and some other terms.
For the entire collection see [Zbl 0852.00016].

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
58J37 Perturbations of PDEs on manifolds; asymptotics
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