Variations of random fields and application to computation

研究期间（2000 - 2001学年）的计划是研究随机场和有关的计算问题。本文的主要研究成果都是建立在创新理论的基础上的，创新理论为研究提供了有力的工具。研究结果如下：1.高斯随机场。由等高线或闭曲面参数化的高斯随机场的新息可以通过其变分来获得。通常，这样的新息是一个白色噪声，其中给定的字段表示为一个随机积分。核表示了场的概率性质，类似于依赖的方式，包括参数有序时的可逆性.一般的随机场，对于一个表示为齐次混沌积分的场，我们可以显式地构造新息，从而得到最佳的预报子。此外，一般可加过程的随机线性函数允许我们以非线性方式形成新息，并且我们解决了通信理论中的可计算性问题，如跳跃发现.白色噪声理论的应用：对加性过程的从属性进行了深入的研究，并将其应用于候选黑洞X射线数据的处理。在谱外我们计算了它的特征泛函，从而有了一个很好的数学模型。

The plan during the research period (2000 - 2001 academic year) is to study random fields and related problems of computation. The main results have been obtained are all based on the Innovation Theory, which provides useful tools of study. The results are as follows.1. Gaussian random fields.The innovation of a Gaussian random field parameterized by a contour or a closed surface can be obtained by its variation. Usually, such an innovation is a white noise, by which the given field is expressed as a stochastic integral. The kernel represents the probabilistic properties of the field like way of dependence including reversibility when the parameters are ordered.2. General random fields.For a field expressed as an integral of homogeneous chaos we can form innovations explicity, so that the best predictor is obtained. Also, random linear functions of a general additive process allow us to form innovation in a nonlinear manner, and we solved the computability problems, like jump finding, m the theory of communications.3. Application of white noise theory.Having had profound study on the subordination for additive processes, we applied to the processing of the X-ray data from the black-hole candidate. Beyond the spectrum we have computed its characteristic functional to have a good mathematical model.