Is there a way to simulate an N-dimensional Geometric Brownian Motion i.e. variable $$x_i, i \ \epsilon \ [1, N] $$ is diffusing in log-space such that $$log\ (x_i)$$ follows a Brownian motion with a given mean and variance and sum of all x_i is constrained to be 1. I don't know much about this topic so any references would be really useful.

Additionally can mean and variance of each variable be selected independently of each other.