Is there a way to simulate an $N$-dimensional geometric Brownian motion i.e. variable $$x_i, i \in [1, N] $$ 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.