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A Special Problem of Brownian Motion, and a General Theory of Gaussian Random Functions
ECOLE POLYTECIHNIQUE Palaiseau France
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The subject of this paper is twofold a special problem and a general theory. The reader may wonder why the general theory is not stated in part 2 and then applied to the special problem. The answer is that the general theory appeared as a necessary generalization of theorems stated in part 2, after the two first parts had been written, and the author thought that he would not have enough time before this Symposium to reorganize the paper. Moreover, part 2 will be a good introduction to the general theory. In the introduction the author will begin with the general theory, and the reader who wishes to do so may begin with part 3. The problem considered in this theory is to find an explicit representation of Gaussian r.f.1 of a real variable t that may be considered as the canonical form of this function. By subtraction of a known function, it may be reduced to a Gaussian r.f. phit with identically zero expectation. Such a r.f. is generally defined by its covariance gammat1,ts, or by a stochastic differential equation with a Cauchy condition. None of these methods gives an explicit representation of phit.
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