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

通过目前的研究项目，我们得到了以下结果：在Osada和Spohn的论文中，他们讨论了关于Ginzburg-Landau连续体模型的Gibbs测度的存在性和唯一性问题。他们对自我电位和相互作用电位的假设非常温和。其唯一性的充分条件与Papangelou的格情形相同。他们还给出了相位转换发生的例子。最近，Hariya将这一结果推广到有反射条件的情况。Hariya和Osada构造了一个与上面定义的吉布斯测度相对应的随机过程。他们考虑了时间演化符合Malliavin随机演算的情况。因为他们使用狄利克雷形式，所以对势的假设是非常温和的。Sugiura研究了上述构造的吉布斯态的混合性质，并导出了Dobrushin-Shlosman型混合性质。由于模型是连续的，近自旋相关的衰减也很重要。他的结果包含了这个可能是最好的属性。他还考虑了受圆柱形布朗运动扰动的随机过程，得到了该模型的log-Sobolev不等式的部分结果。相应结果的论文正在准备中。他还把这个问题简化为协方差的估计。这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.