Research on several problems of stochastic processes concerning the Ginzburg-Landau continuum model

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Through the present research project, we obtain the following results.0. In Osada and Spohn's paper, they discuss the existence and uniqueness problems of the Gibbs measure concerning the Ginzburg-Landau continuum model. Their assumptions for the self-potential and interaction-potential are very mild. The sufficient condition of the uniqueness is the same as that for the lattice case derived by Papangelou. They also give an example in which the phase translation occurs.Recently, Hariya extended this results to the case with reflection condition.1. Hariya and Osada construct a stochastic process corresponding to the Gibbs measure defined above. They consider the case when the time evolution is corresponding to the Malliavin stochastic calculus. Since they use the Dirichlet form, the assumption for the potential is very mild.2. Sugiura investigates the mixing property for the Gibbs state constructed above in O.He derives the Dobrushin-Shlosman type mixing property. Since the model is continuum, the decay of the correlation of near spins is also important. His result includes this property which is the best possible one. He also considers the stochastic process perturbed by cylindrical Brownian motion and obtains a partial results about the log-Sobolev inequality for this model. The paper of the corresponding results is in preparation. He also reduces this problem to some estimates of covariance. Doing this is the next research project for us.