We prove the irreducibility of a Dirichlet form on the based loop space on a compact Riemannian manifold. The Dirichlet form is defined by the gradient operator due to DRIVER and LÉANDRE. We also prove the uniqueness of the ground states of the Schrödinger operator for which the Dirichlet form satisfies the logarithmic Sobolev inequality. This is an extension of the corresponding results of GROSS (, ) to the case of general compact Riemannian manifolds.
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