Abstract
Every regular, local Dirichlet form on a locally compact, separable space X defines in an intrinsic way a pseudo metric ρ on the state space. Assuming that this is actually a complete metric (compatible with the original topology), we prove that (X, ρ) is a geodesic space. That is, any two points in X are joined by a minimal geodesic.
Also analogues of the Hopf-Rinow Theorem and of the Cartan-Hadamard Theorem are obtained. The latter requires the notion of curvature which is defined by means of the CAT-inequality.
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Sturm, KT. (1995). On the Geometry Defined by Dirichlet Forms. In: Bolthausen, E., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications. Progress in Probability, vol 36. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7026-9_17
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DOI: https://doi.org/10.1007/978-3-0348-7026-9_17
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