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      Tube estimates for diffusion processes under a weak H\"ormander condition

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          Abstract

          We consider a diffusion process under a local weak H\"{o}rmander condition on the coefficients. We find Gaussian estimates for the density in short time and exponential lower and upper bounds for the probability that the diffusion remains in a small tube around a deterministic trajectory (skeleton path), explicitly depending on the radius of the tube and on the energy of the skeleton path. We use a norm which reflects the anisotropic structure of the problem, meaning that the diffusion propagates in \(\R^2\) with different speeds in the directions \(\s\) and \([\s,b]\). We establish a connection between this norm and the standard control distance.

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          Journal
          2014-12-16
          2016-01-19
          Article
          1412.4917
          ce4e7859-1eb4-4ad4-945a-bebe807eb9db

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          math.PR

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