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与整个项目的目的有关的几个子类的无限可分disirution的随机积分特征。结果，我们成功地在随机积分的性质的类的G型分布及其下降序列的子类，这是在2005学年开始的目标之一。在无穷可分分布的特征函数的Levy-Khintchine表示中，我们发现了一个新的G型自可分分布子类，它具有非常简单的Levy测度形式. 2005年11月，前岛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发表了演讲，受到了高度评价。虽然我们在半Levy工艺上未能取得巨大成功，但Maejima在研讨会上提出了以下想法： 关于我们 用半Levy过程代替随机积分中的Levy过程，这一想法得到了听众的广泛接受，Maejima也感受到了国际上对这一课题进行联合研究的新浪潮。这些工作主要是由Maejima和Sato，Maejima，他的学生和Tenessee大学的Jan Rosinski完成的。另一个在本学年开始时提到的话题是一个新的话题，这是广义Ornstein-Uhlenbeck过程。研究了广义Ornstein-Uhlenbeck过程极限分布的无穷可分性，发现在较弱的条件下，它们不仅是无穷可分的，而且是自分解的.还提出了几个有趣的例子。这些结果是Maejima在2006年3月于墨西哥召开的国际会议上发表的。基于三年来的研究成果，我们可以看到Semi-Levy过程的研究路线以及无穷可分分布类的更精确分类。除此之外，前岛发现，连同维克托佩雷斯，Abrue在Maxico，例子随机矩阵的决定因素是无限整除。这些例子很有趣，因为已知原始随机矩阵不是无限可分的。围绕无限可除性还有很多有趣的未解决的问题。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