Suppose I have a $D$ dimensional Gaussian random variable, given by
$$Y\sim N(\mu, \Sigma).$$
I would like to study the distribution of
$$X=\Big(\frac{e^{Y_1}}{\sum_ie^{Y_i}},\ldots,\frac{e^{Y_D}}{\sum_ie^{Y_i}}\Big).$$
Specifically, I would like to determine the joint pdf $p(x_1,\ldots,x_D)$, defined over the $D-1$ simplex $\Delta^{D-1}$.
I'm aware of the constructions given here, however neither one quite seems to fit with what I'm looking for.
The former construction essentially sets $Y_D=1$, while the latter is actually the density for
$$X=\Big(\frac{e^{Y_1}}{1+e^{Y_1}},\ldots,\frac{e^{Y_D}}{1+e^{Y_D}}\Big),$$
which is not what I want.