A note on kernel methods for multiscale systems with critical transitions

We study the maximum mean discrepancy (MMD) in the context of critical transitions modelled by fast-slow stochastic dynamical systems. We establish a new link between the dynamical theory of critical transitions with the statistical aspects of the MMD. In particular, we show that a formal approximation of the MMD near fast subsystem bifurcation points can be computed to leading order. This leading order approximation shows that the MMD depends intricately on the fast-slow systems parameters, which can influence the detection of potential early-warning signs before critical transitions. However, the MMD turns out to be an excellent binary classifier to detect the change-point location induced by the critical transition. We cross-validate our results by numerical simulations for a van der Pol-type model.