Liouville Brownian motion and thick points of the Gaussian free field

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• Henry Jackson, Cambridge
• Tuesday 17 February 2015, 16:00-17:00
• MR12, CMS.

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We find a lower bound for the Hausdorff dimension that a Liouville Brownian motion spends in alpha-thick points of the Gaussian Free Field, where Alpha is not necessarily equal to the parameter used in the construction of the geometry. This completes a conjecture by N. Berestycki, where the corresponding upper bound was shown. In the course of the proof, we obtain estimates on the (Euclidean) diffusivity exponent, which depends strongly on the nature of the starting point. For a Liouville typical point, it is 1/(1-gamma^2/2). In particular, for gamma > sqrt(2). the path is Lebesgue-almost everywhere differentiable, almost surely. This provides a detailed description of the multifractal nature of Liouville Brownian motion.

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