MARK KAC SEMINAR

Continuous-time Markov chains (on discrete spaces) are Markov processes such that when they visit a state of the space, they remain there during an exponentially distributed random time before next jump. The parameter of the exponential distribution depends on the current state. Therefore, continuous-time Markov chains can be seen as discrete time Markov chains equipped with an internal clock regulating the pace with which they evolve.

Firstly, we shall describe precisely the process and present the semi-martingale method for determining its type (i.e. its being recurrent or transient). Secondly, the occurrence of the phenomenon of explosion (i.e. the departing of the process towards infinity in a finite time) will be also established by semi-martingale methods. Thirdly, a new phenomenon we have termed implosion will be presented and studied.

These general results will be applied to various types of random walks on the d-dimensional lattice or in quadrants. In particular, we shall show that, depending on the pace of internal clock, the random walk on the 2-dimensional lattice (that is normally null-recurrent) can become positive recurrent.

The Allen-Cahn equation is a partial differential equation describing phase separation. It can be obtained from the continuum limit of a particle system which is similar to the Ising model, but with real-valued spins. The stochastic version of this equation shows a metastable behaviour similar to the Ising model at low temperature: the time needed for a transition between the "plus" phase and the "minus" phase is exponentially long in the inverse temperature. I will discuss some recent results on Eyring-Kramers type asymptotics for the prefactor of this transition time, for the Allen-Cahn equation on the torus in dimensions 1 and 2. Based on joint works with Giacomo Di Gesù, Bastien Fernandez, Barbara Gentz and Hendrik Weber.

The slides of this lecture are available via this link.