Convergence of processes time-changed by Gaussian multiplicative chaos

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