Summary
We study properties of Brownian bridges on a complete Riemannian manifoldM. LetQ t x,y be the law of Brownian bridge fromx toy with lifetimet. Q t x,y is a probability measure on the space Ω x,y of continuous paths ω with ω(0)=x and ω(1)=y. We prove thatQ t x,y possesses the large deviation property with the rate function
We show that ifM and its metric are analytic then forany x, y onM there exists a probability measure μ x,y which is supported by a subset of the space of minimizing geodesics joiningx andy such that Ω x,y t →μ x,y weakly in Ω x,y ast→0. We also give a complete characterization of the exact support of μ x,y .
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Research supported in part by the grant NSF-DMS-86-00233. Current address: Department of Math. Northwestern University
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Hsu, P. Brownian bridges on Riemannian manifolds. Probab. Th. Rel. Fields 84, 103–118 (1990). https://doi.org/10.1007/BF01288561
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DOI: https://doi.org/10.1007/BF01288561