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We consider the construction and classification of some new mathematicalobjects, called ergodic spatial stationary processes, on locally compactAbelian groups, which provide a natural and very general setting for studyingdiffraction and the famous inverse problems associated with it. In particularwe can construct complete families of solutions to the inverse problem from anygiven pure point measure that is chosen to be the diffraction. In this casethese processes can be classified by the dual of the group of relators based onthe set of Bragg peaks, and this gives a solution to the homometry problem forpure point diffraction.  An ergodic spatial stationary process consists of a measure theoreticaldynamical system and a mapping linking it with the ambient space in whichdiffracting density is supposed to exist. After introducing these processes westudy their general properties and link pure point diffraction to almostperiodicity.  Given a pure point measure we show how to construct from it and a given setof phases a corresponding ergodic spatial stationary process. In fact we dothis in two separate ways, each of which sheds its own light on the nature ofthe problem. The first construction can be seen as an elaboration of theHalmos--von Neumann theorem, lifted from the domain of dynamical systems tothat of stationary processes. The second is a Gelfand construction obtained bydefining a suitable Banach algebra out of the putative eigenfunctions of thedesired dynamics.

Stationary processes with pure point diffraction