Convergence of Processes Time-Changed by Gaussian Multiplicative Chaos

Gaussian multiplicative chaos is a random measure constructed from a Gaussian field. An example of this is the Liouville measure, which is constructed from a Gaussian free 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 L2-regime). As examples of the main result, we prove that, in the whole L2-regime, the scaling limit of the Liouville simple random walk on Z2 is Liouville Brownian motion and, as α→1, Liouville α-stable processes on R converge weakly to the Liouville Cauchy process.