Ergodicity and metastability for the stochastic quantisation equation

In this thesis we study ergodicity and metastability of solutions to the stochastic quantisation equation of the P(')2Euclidean Quantum Field theory. The main difficulty arises from the fact that solutions of this equation can be interpreted only in a renormalised sense and classical methods from SPDE Theory do not apply in this case. I. Ergodicity: In this part we study the long time behaviour of the law of the solutions. We first prove three main results: A strong dissipative bound for the solutions uniformly in the initial condition, the strong Feller property (and in particular local H¨older continuity of the associated Markov semigroup) and a support theorem. As a corollary, we prove exponential mixing of the law of the solutions with respect to the total variation distance. II. Metastability: In this part we restrict ourselves to the special case of the 2dimensional Allen–Cahn equation perturbed by small noise and study the long time behaviour of solutions pathwisely. We prove that solutions that start close to the minimisers of the potential of the deterministic system contract exponentially fast with overwhelming probability. The exponential rate is explicit in the parameters of the equation. As an application, we prove an Eyring–Kramers law for the transition times of the solutions between the minimisers of the potential of the deterministic system