GENERALIZED SIMPLE EXCLUSION PROCESSES WITH POTENTIAL TRANSLATION INVARIANT PROBABILITY

Let S be a countable set. X=0, 1, …, m^s. P(=（p（x, y））_x,y∈s) a transition probability matrix, g （·） is a strictly monotonically increasing function with g （0） = 0. We say that （η_t, P^η） is a generalized simple exclusion process if it is uniquely determined by the generator \Omega f(\eta)=\sum_u \in G g(\eta(u)) \sum_v \in \mathbbB p(u, v)\leftf\left(\eta_u v\right)-f(\eta)\right, f \in \mathscrF(X), \eta \in \boldsymbolX. where \mathscrF(X) is the set of the all cylindrical functions on X; if η（u） =0 or η（v）=m or u=v then η（uv）=η, otherwise η_uv（u）=η（u）-1, η_uv（v）=η（v） +1, η_uv（w）=η（w）, w \bar\in\u, v\. When m=1, it is a simple exclusion process proposed and studied by Spitzer and Liggett. When m≥1 and P is positive recurrent and reversible, we obtained the ergodic theorem. In this paper we deal with the case of m≥1 and a potential random walk P. We obtain the description of all the translation invariant and invariant measures for the processes. A part of results of Liggett on simple exclusion processes is extended. Theorem. Let s=Z^d, P is a potential irreducible random walk on Z^d. Then the set of all extreme points of the translation invariant and invariant measures for the process coincides with V_p: 0≤ρ≤∞, where v₀ and v_∞ are the unit masses on 0（0∈X, 0（x）=0, \forall x \in S)） and M（M∈X,M（x）=m, \forall x \in S)） respectively; v₀ （0＜ρ＜∞）are product measures with marginal distributions \nu, \eta(x)=k)=\frac\rho^kg(k) g(k-1) \cdots g(1) / \sum_i=1^m \frac\rho^ig(i) g(i-1) \cdots g(1)+1 \quad 0 \leqslant k \leqslant m, x \in S.