Study on semi-Levy processes and semi-selfsimilar processes

今年是三年项目的最后一年，我们花了大部分时间研究无穷可分分布的几个子类的随机积分表征，这与整个项目的目的有关。结果，我们成功地对G型分布的类及其子类的递减序列进行了随机积分表征，这是2005学年开始时的目标之一。此外，在无限可分分布特征函数的Levy- khintchine表示中，我们发现了具有非常简单的Levy测度形式的G型自分解分布的一个新子类。2005年11月，Maejima在德国慕尼黑工业大学的研讨会上就这一结果以及其他相关结果做了一次演讲，这次演讲非常受欢迎。虽然我们在半列维过程上没有取得很大的成功，但Maejima在研讨会上提出了用半列维过程代替随机积分中的列维过程的想法，这个想法得到了听众的认可，Maejima感到了一股新的国际联合研究的浪潮。这些作品主要是由前岛和佐藤，前岛，他的学生和田纳西大学的Jan Rosinski完成的。本学年开始时提到的另一个话题是一个新话题，即广义Ornstein-Uhlenbeck过程。研究了广义Ornstein-Uhlenbeck过程的极限分布的无限可分性，得到了广义Ornstein-Uhlenbeck过程在温和条件下不仅是无限可分的，而且是自可分的。还构造了几个有趣的例子。这些结果是由前岛在2006年3月墨西哥举行的一次国际会议上提出的。从三年的研究成果中，我们可以看到半列维过程的研究路线，以及对无限可分分布类的更精确分类。除此之外，Maejima和墨西哥的Victor Perez-Abrue一起发现了一些随机矩阵的例子，它们的行列式是无限可分的。这些例子很有趣，因为我们知道原始的随机矩阵不是无限可分的。关于无穷可除性还有很多有趣的未解问题。少

This year was the last year of the three year project, and we spent most time for the study of stochastic integral characterizations of several subclasses of infinitely divisible disitirutions in connection with the purpose of the whole project. As a result, we succeeded in stochastic integral characterization of the class of type G distributions and its decreasing sequence of subclasses, which was one of the aims at the beginning of the academic year of 2005. Furthermore, we found a new subclass of type G and seldecomposable distributions which has very simple form of Levy measures in the Levy-Khintchine representation of the characteristic function of infinitely divisible distribution. Maejima gave a talk on this result together with more other related results at the seminar of Munich Technical University, Germany, November 2005, and the talk was very much appreciated. Although we were not able to get a big success on semi-Levy processes, Maejima suggested in the seminar the idea of … More replacing Levy process in the stochastic integral by semi-Levy process, and this idea was well accepted by the audience, and Maejima felt a wave of a new international joint research on this topic. These works were mainly done by Maejima and Sato, Maejima, his student and Jan Rosinski at University of Tenessee.Another topic which was mentioned at the beginning of this academic year was a new topic, which is generalized Ornstein-Uhlenbeck processes. We investigated the infinite divisibility of the limiting distributions of generalized Ornstein-Uhlenbeck processes and we found that under a mild condition they are not only infinitely divisible but also seldecomposable. Also several interesting examples were constructed. These results were presented by Maejima at an International Conference in Mexico, March 2006.Based on the results obtained during the three year project, we can see the route of the study of Semi-Levy processes and more precise classification of the class of infinitely divisible distributions. Besides those, Maejima found, together with Victor Perez-Abrue at Maxico, examples of random matrices whose determinants are infinitely divisible. These examples are interesting because it is known that the original random matrices are not infinitely divisible. A lot of interesting unsolved problems around the infinitely divisibility remain. Less