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

通过本课题的研究，我们取得了以下成果。在Osada和Spohn的论文中，他们讨论了关于Ginzburg-Landau连续介质模型的Gibbs测度的存在唯一性问题。他们对自势和相互作用势的假设非常温和。其唯一性的充分条件与Papangelou的格子情形相同。他们还给出了一个相移发生的例子。最近，Hariya将这一结果推广到有反射条件的情况。Hariya和Osada构造了一个对应于上面定义的Gibbs测度的随机过程。他们考虑了时间演化对应于Malliavin随机演算的情况。由于他们使用狄利克雷形式，势的假设是非常温和的。Sugiura研究了上述构造在O中的Gibbs态的混合性质。他导出了Dobrushin-Shlosman型混合性质。由于该模型是连续的，近自旋关联的衰变也很重要。他的结果包括这一财产，这是可能的最好的。他还考虑了受柱面布朗运动扰动的随机过程，得到了关于该模型的log-Soblev不等式的部分结果。论文的相应成果正在筹备中。他还把这个问题归结为协方差的一些估计。做这件事是我们的下一个研究项目。

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.