Stochastic analysis on loop space

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(1) We gave an estimate on the gap of spectrum of Schrodinger operators by using weak Poincare inequality. Also we gave an estimate on the distribution function of the ground state by the inequality.(2) Let H be the space of H^1-paths on a Euclidean space. Consider a Morse function on H which is a sum of the energy of the path and a smooth function on H which can be estended to a smooth function on the space of continuous paths. We defined a Witten Laplacian twisted by the Morse function on a Wiener space and proved that the first order behavior of the lowest eigenvalue under semiclassical limit is determined by the hessian of the Morse function.(3) Consider a continuous function F on the Cameron-Martin subspace of a classical Wiener space. Assume F can be extended to a continuous function F on the Wiener space. Then if the domain {F > 0} is a connected set, then weak Poincare inequalities hold on {F > 0}. We extend this result to the case where F is a continuous function of Brownian rough paths.(4) We proved very precise Gaussian estimates on heat kernels on Riemannian manifolds which possess poles under the assumptions that the curvature and the derivatives go to 0 sufficiently fast at infinity.