  We study the impact of applying stochastic forcing to the Ghil-Sellers energy
balance climate model in the form of a fluctuating solar irradiance. Through
numerical simulations, we explore the noise-induced transitions between the
competing warm and snowball climate states. We consider multiplicative
stochastic forcing driven by Gaussian and $\alpha$-stable L\'evy -
$\alpha\in(0,2)$ - noise laws, examine the statistics of transition times, and
estimate most probable transition paths. While the Gaussian noise case has been
carefully studied in a plethora of investigations on metastable systems, much
less is known about the L\'evy case, especially in the case of high- and
infinite-dimensional systems. In the weak noise limit, the expected residence
time in each metastable state scales in a fundamentally different way in the
Gaussian vs. L\'evy noise case with respect to the intensity of the noise. In
the former case, the classical Kramers-like exponential law is recovered. In
the latter case, power laws are found, with the exponent equal to $-\alpha$, in
apparent agreement with rigorous results obtained for additive noise in a
related - yet different - reaction-diffusion equation as well as in simpler
models. This can be better understood by treating the L\'evy noise as a
compound Poisson process. The transition paths are studied in a projection of
the state space and remarkable differences are observed between the two
different types of noise. The snowball-to-warm and the warm-to-snowball most
probable transition path cross at the single unstable edge state on the basin
boundary. In the case of L\'evy noise, the most probable transition paths in
the two directions are wholly separated, as transitions apparently take place
via the closest basin boundary region to the outgoing attractor. This property
can be better elucidated by considering singular perturbations to the solar
irradiance.
