Properties and Construction of Mappings

The Pdf equation derived in the previous chapter can be solved with the aid of certain closure assumptions and the mapping method suggested by Chen, Chen, and Kraichnan in 1989. This method worked very well for the case of a single scalar, and this chapter is devoted to the extensions to multiple variables and two points in time and/or space. Elementary properties of mappings are introduced and discussed; injective, surjective, and bijective mappings are defined and bijective mappings for probability measures are considered with regard to relation to Gaussian measures. Bijective mappings for characteristic functionals are considered for the case of nuclear definition and image spaces and restrictions are established to insure that the image of a characteristic function is also a characteristic function. Methods for the construction of local and global mappings are covered and illustrated with examples.