## Uniform positivity improving property, Sobolev inequalities, and spectral gaps.(English)Zbl 0914.47041

Markov semigroups are investigated by means of Dirichlet forms on Wiener spaces, Riemannian manifolds and loop spaces. Ergodicity conditions and Sobolev type inequalities are employed to estimate the spectral gap for Markov generator. That allows to prove e.g. whether a positive transition density can be associated with a diffusion process on a Riemannian manifold. Logarithmic Sobolev inequalities on connected complete Riemannian manifolds and the case of weighted Wiener measures are discussed as well.

### MSC:

 47D07 Markov semigroups and applications to diffusion processes 60E15 Inequalities; stochastic orderings 46G12 Measures and integration on abstract linear spaces
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### References:

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