I would like to calculate
$$F^{-1}(\exp\{[F(g(x))-1]t\}),$$ where $F(g(x)) = \int_{\infty}^\infty dx e^{isx}g(x)$ is the Fourier transform, and $F^{-1}(\phi(s)) = \frac{1}{2\pi}\int_{-\infty}^\infty ds e^{-i sx} \phi(s)$ is the inverse Fourier transform.
Are there any special properties I might leverage to compute this transform? This arises in a stochastic Langevin equation with a Poisson shot noise driving force. $g(x)$ is the distribution of noise strengths.
Thanks for any guidance!