Given a symmetric positive definite matrix $\bf \Sigma \in \mathbb{R}^{n \times n}$, I want to find a matrix ${\bf \Gamma} \in \mathbb{R}^{n \times n}$ and a vector ${\bf m} \in \mathbb{R}^n$ such that if ${\bf X} \sim N(\bf{m}, \bf{\Gamma})$, then the random vector $\bf{Y} = (\exp(X_1), ..., \exp(X_n))$ has mean $\bf 0$ and covariance matrix $\bf \Sigma$. 
For $n=1$, this is possible, as shown e.g. in [this blog entry][1]

In higher dimensions $n > 1$, this is probably not possible for all matrices $\bf \Sigma$, so I would be interested in conditions for $\bf \Sigma$ under which such a matrix $\bf \Gamma$ exists, and how to find it. I have found [this previous question][2], but maybe after 3 years somebody has a better answer, and also I am only interested in the case where $\bf Y$ has zero mean. 


  [1]: https://www.johndcook.com/blog/2022/02/24/find-log-normal-parameters/
  [2]: https://stats.stackexchange.com/questions/439381/generate-multivariate-log-normal-variables-with-given-covariance-and-mean