Diffusions with Gaussian Drifts
We continue our investigation of the passive tracer model introduced in Chap. 11 under an additional assumption that the drift is Gaussian. In this case the generator of the environment process satisfies the graded sector condition introduced in Sect. 2.7.4. Using this approach we reprove first the central limit theorem for diffusions whose drift has a finite Péclet number. This result can be extended to some classes of time dependent flows whose Péclet number are infinite by taking into account time decorrelation properties of the drift. We apply the variational principles derived in Chap. 4 to prove that the sufficient conditions for the central limit theorem obtained in this way are in some sense optimal. We give examples of families of isotropic flows for which the motion of a tracer is superdiffusive.
KeywordsVariational Principle Central Limit Theorem Stochastic Differential Equation Spectral Measure Environment Process
- Tóth B, Valko B (2010) Superdiffusive bounds on self-repellent Brownian polymers and diffusion in the curl of the Gaussian free field in d=2. arXiv:1012.5698