Publ. Math. Debrecen 72/1-2 (2008), 199–225 Polynomial hypergroup structures and applications to probability theory By WALTER R. BLOOM (Murdoch) and HERBERT HEYER (T¨ ubingen) Abstract. A new trend in enlarging the repertoire of concrete hypergroups is the construction of polynomial hypergroup structures on higher-dimensional Euclidean spaces. It turns out that stochastic processes taking values in such structures and their duals reveal surprising phenomena. In the present exposition recent progress in the theory achieved by Koornwinder, Connett and Schwartz, and by the Tunisian School will be discussed and made accessible also to the non-specialized reader. 1. Introduction There are two significant aspects motivating an up-to-date exposition on progress in the theory of commutative hypergroups with base space a compact subset K of k-dimensional euclidean space and whose convolution in the set M (K) of bounded measures on K is defined via sequences of k-variable polynomials on K. Firstly, this survey shows how new hypergroup structures can be introduced for geometric configurations in R 2 such as the unit square, the unit disk, the parabolic bi-angle and the simplex. Secondly, it is of interest to study stochastic processes with independent increments in these configurations, their structure and their long-term behaviour. For general compact commutative hypergroups an elaborate harmonic analy- sis is available based on the generalized translation operation in M (K). Basic re- sults have been obtained in analogy to but remarkably distinct from the classical framework of a compact abelian group. It is worth recalling that in general there Mathematics Subject Classification: 43A62, 33C50, 60B99. Key words and phrases: polynomial hypergroups, generalized Laplacians, probability measures.