Metastability in stochastic dynamical systems plays a central role in understanding noise-induced transitions in climate models. In this talk, I present a set of theoretical tools for quantifying transition mechanisms beyond classical critical slowing down.
First, I revisit the concept of the Stochastic Basin of Attraction (SBA) for systems perturbed by Gaussian and L\'evy noise, where stability is characterized via escape probabilities and generator-based boundary value problems.
I then introduce a complementary geometric framework based on the committor function and stochastic separatrix structure, recently developed in a temperature--phytoplankton bistable model of Arctic under-ice blooms. A new geometric early warning indicator is defined via the arc-length averaged width of the probabilistic transition layer. In the weak-noise regime, we establish an explicit asymptotic coupling between this geometric quantity and the logarithmic mean first passage time.
Applications include the Amazonian vegetation and the Arctic under-ice blooms models, illustrating how probabilistic and geometric structures jointly govern metastable transitions in eco--climate tipping systems.
