A NOISE WITH SPACE PARAMETER AND INVARIANCE OF THE CLASS OF INTENSITIES

Abstract

We are going to discuss a new noise depending on the space variable u. When we discuss random phenomena, we can see significant different characters between the phenomena depending on time and those on space. We can see this fact clearly when we take noises, which we mean systems involving independent idealized random variables, like white noise (B) over dot(t) or many other examples. We shall take noises depending specifically on space to discuss different properties from what are found in time-dependent noises. We shall also discuss phenomena where Poisson type variables are involved, however we emphasize that their significant properties are described in terms of the intensity, and we shall therefore consider the phenomena in question with special emphasis on intensity which is often parametrized by the space variables. Interesting results will be seen in the two-dimensional space parameter case, where the group SL(2, R) plays an important role in the representation of the class of intensities.