Differential calculus on path and loop spaces. II: Irreducibility of Dirichlet forms on loop spaces
[For part I see the author and \textit{D. Elworthy}, C. R. Acad. Sci., Paris, Sér. I 321, No. 1, 97-102 (1995; Zbl 0837.60053)]. Let \((M,g)\) be a compact Riemannian manifold and \(\xi(t)\in M\) be a diffusion manifold with the generator \({1\over 2}\Delta\) where \(\Delta\) is the Laplace operator defined by the metric \(g\). The conditional diffusion on \(M\) defines the pinned Brownian motion on the based loop space \(L_x(M)\), \(x\in M\), consisting of continuous paths. Let \(\nabla\) denote a gradient on \(L_x(M)\) defined by a Riemannian connection and \(E\) be a Dirichlet form on it with the domain \(D({ E})\). The author proves the irreducibility of the symmetric diffusive process on \(L_x(M)\). In other words he proves that if a Riemannian connection is the Lévy-Civita connection and \(F\in D( E)\) possesses the property \(\nabla F(\gamma)=0\) for \(\nu_x\)-almost all \(\gamma\) on the probability space \((L_x(M),\nu_x)\), then \(F\) is constant on homotopy equivalent classes of \(L_x(M)\). As a corollary the uniqueness of the ground states for the Schrödinger operator on each homotopy class is proved. The methods and results of the article are the extension and combination of the methods and results recently developed in papers by Gross, Kusuoka and the author.
scientific article; zbMATH DE number 1239982
scientific article