Discrete Approximation to Time-changed Brownian Motions

We develop a general discrete approximation scheme for time-changed Brownian motions on ${\mathbb{R}}^d$. Our approximation scheme works for any smooth measure with full quasi-support on ${\mathbb{R}}^d$ with suitable initial distributions. Under some mild conditions on the smooth measure, the discrete approximation scheme works for every starting point. Our results in particular give a discrete approximation scheme for Liouville Brownian motions.