We consider the dynamical evolution of a simple climate system that describes the
average temperature of the Earth’s atmosphere owing to radiative forcing and coupling
to a positive feedback variable such as the concentration of greenhouse gases in the
presence of fluctuations. Analysing the resulting stochastic dynamical system shows that,
if the temperature relaxes rapidly relative to the concentration, the time-dependent and
stationary probability density functions (pdfs) for the temperature rise possess a fat tail.
In contrast, if the feedback variable relaxes rapidly relative to the temperature, the pdf
has no fat tail, and, instead, the system shows critical slowing down as the singular
limit of positive feedback is approached. However, if there is uncertainty in the feedback
variable itself, a fat tail can reappear. Our analysis may be generalized to more complex
models with similar qualitative results. Our results have policy implications: although fat
tails imply that the expectation of plausible damage functions is infinite, the pdfs permit
an examination of the trade-off between reducing emissions and reducing the positive
feedback gain.