Some exact solutions to the forward Chapman-Kolmogorov equation are derived for processes driven by both Gaussian and compound Poisson (shot) noise. The combined action of these two forms of white noise is analyzed in transient and equilibrium conditions for different jump distributions and additive Gaussian noise. Steady-state distributions with power-law tails are obtained for exponentially distributed jumps and multiplicative linear Gaussian noise. Two applications are discussed: namely, the virtual waiting-time or Takàcs process including Gaussian oscillations and a simplified model of soil moisture dynamics, in which rainfall is modeled as a compound Poisson process and fluctuations in potential evapotranspiration are Gaussian.