  An abrupt climatic transition could be triggered by a single extreme event,
an $\alpha$-stable non-Gaussian L\'evy noise is regarded as a type of noise to
generate such extreme events. In contrast with the classic Gaussian noise, a
comprehensive approach of the most probable transition path for systems under
$\alpha$-stable L\'evy noise is still lacking. We develop here a probabilistic
framework, based on the nonlocal Fokker-Planck equation, to investigate the
maximum likelihood climate change for an energy balance system under the
influence of greenhouse effect and L\'evy fluctuations. We find that a period
of the cold climate state can be interrupted by a sharp shift to the warmer one
due to larger noise jumps, and the climate change for warming $1.5\rm ^oC$
under an enhanced greenhouse effect generates a step-like growth process. These
results provide important insights into the underlying mechanisms of abrupt
climate transitions triggered by a L\'evy process.
