On tail probability of the covariance matrix in Eldan's stochastic localization

The Eldan's stochastic localization is a new kind of stochastic evolution in the space of probability measures which provides a novel way to study high dimensional convex body. A central object in the study of the stochastic localization is the stochastic process of its covariance matrix. The main result of this paper is some exponential-type tail probability estimate of the covariance process for the general time. This estimate implies a weaker version of a $p$-moment conjecture by Klartag and Lehec. The stochastic localization considered here is a simplified version by Lee and Vemplala.