White Noise Analysis: Part I. Theory in Progress

Abstract

In the past quarter century white noise theory has made great progress. We are further given many interesting and important questions which should be discussed in line with white noise analysis. On the other hand, some of the new directions are before our eyes and they require new analytic methods to be investigated, by generalizing the method established so far. It is now the time to remind the original idea of our stochastic analysis and to think of further directions of this theory which are going to appear in front of us. While we are reviewing the results, we have recognized the foundation of the theory should be refined. This will be done also in Part I. The plan of Part II is as follows. From what have been established, there naturally arise several new topics to be discussed. This is, in a sense, a continuation of Part I. On the other hand, having suggested by many significant examples and problems in applications, we come to a framework of the new white noise theory which is going to be constructed. A significant development is to grow out of L2-theory and to establish an analysis on a wider class of suitable functionals of sample paths. This approach will be proposed.