It has been known that stochastic differential equations with non-degenerate diffusion admit a unique solution for sub-critical drifts. In this talk, we extend this in two different directions: the critical drift case and the degenerate diffusion case. I will briefly introduce a hypoelliptic theory, and then explain how to obtain probabilistic results on stochastic differential equations.

In this presentation, we consider the random-cluster model which is a generalization of the standard edge percolation model. For the random-cluster model on lattice, we prove that the Glauber dynamics exhibits a phenomenon known as the cut-off, especially for the very subcritical regime for all dimensions. This is a joint work with Shirshendu Ganguly.