  In this paper we prove the boundedness of the Gaussian Riesz potentials
$I_{\beta}$, for $\beta\geq 1$ on $L^{p(\cdot)}(\gamma_d)$, the Gaussian
variable Lebesgue spaces under a certain additional condition of regularity on
$p(\cdot)$ following \cite{DalSco}. Additionally, this result trivially gives
us an alternative proof of the boundedness of Gaussian Riesz potentials
$I_\beta$ on Gaussian Lebesgue spaces $L^p(\gamma_d)$.
