Convergence of processes time-changed by Gaussian multiplicative chaos

Takumu Ooi

Published 2023-05-01Version 1

As represented by Liouville measure, Gaussian multiplicative chaos is a random measure constructed from Gaussian fields. We prove the convergence of processes time-changed by Gaussian multiplicative chaos under certain assumptions when Gaussian multiplicative chaos on bounded measurable sets is square integrable (the $L^2$-regime). As examples of the main result, we prove that, in half of the $L^2$-regime, the scaling limit of the ``Liouville simple random walk'' on $\mathbb{Z}^2$ is Liouville Brownian motion and, in the whole $L^2$-regime, ``Liouville $\alpha$-stable processes'' on $\mathbb{R}$ converge weakly to the ``Liouville Cauchy process'' as $\alpha \to 1$.