Ergodicity for the Stochastic Quantization Problems on the 2D-Torus

In this paper we study the stochastic quantization problem on the two dimensional torus and establish ergodicity for the solutions. Furthermore, we prove a characterization of the Φ24 quantum field on the torus in terms of its density under translation. We also deduce that the Φ24 quantum field on the torus is an extreme point in the set of all L-symmetrizing measures, where L is the corresponding generator.