Diffusion algebras

We define the notion of 'diffusion algebras'. They are quadratic Poincare-B irkhoff-Witt algebras which are useful in order to find exact expressions f or the probability distributions of stationary states appearing in one-dime nsional stochastic processes with exclusion. One considers processes in whi ch one has N species, the number of particles of each species being conserv ed. All diffusion algebras are obtained. The known examples already used in applications are special cases in our classification. To help the reader i nterested in physical problems, the cases N = 3 and 4 are listed separately .