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Local spectral gaps on loop spaces. (English) Zbl 1037.58022

The author is well known for his counterexample to the existence of a spectral gap on the loop space over a compact manifold. Here he proposes some results in the reverse direction: he exhibits indeed a local spectral gap by restricting the loop space to some convenient subspace, namely localizing the loops near some geodesics or constant loops.
More precisely, consider the subspace \(\Omega^{R,N}_{x,y}\) of pinned trajectories (defined on \([0,1]\)) linking \(x\) to \(y\), which have oscillation on each segment \([j/N,(j+1)/N]\) controlled by \(R\). Consider on \(\Omega^{R,N}_{x,y}\) the law \(P^T_{x,y}\) of the Brownian bridge with speed \(T> 0\). Denote by \(\lambda^{R,N}_{x,y}(T)\) the second lowest eigenvalue of the Ornstein-Uhlenbeck operator associated to \((\Omega^{R,N}_{x,y}, P^T_{x,y})\). The main results assert that \(\lambda^{R,N}_{x,y}(T)> 0\) for \(R< R_0\), and evaluate \(\liminf_{T\downarrow 0}\, T_x\log(\lambda^{R,N}_{x,y}(T))\), \(\limsup_{T\downarrow 0}\,T_x\log(\lambda^{R,N}_{x,y}(T))\).

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
47A75 Eigenvalue problems for linear operators
60H07 Stochastic calculus of variations and the Malliavin calculus
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