Abstract.-We present a class of Markov processes with multidimensional finite state space on which the infinitesimal transition movement is a shifting of one unit from one coordinate to some other to its right. We show how to represent the Kolmogorov forward equations as a triangular system. For positive integer m and for nonnegative integers N,rm,...),rjr0, we let SN,m denote the set of (m + 1)-tuples r = (rm,. . . ,r1,ro) with components summing to N. Let X(t) = (Xm(t), . ,Xj(t),Xo(t)) represent an (m + 1)-tuple random variable taking on values only from the set S'N,m For integers i and j satisfying m > i > i > 0 and for reS'N,m, we let Xij(r,t) be nonnegative continuous functions of t. Finally, when N = 1, we define e i to be the vector having rk = 6in k = 0, . . ,m. Thus em = (1,0,... ,0) and eo = (0,. . . ,0,1).Denote by p(rot rs) the probability that X(t) = ro, given that X(s) = r, 0 < s < t and define p(ro,tJ rs) = 0 whenever either ro or r is not an element of StNmn Our right-shift processes assume that for At > 0 p(r-ei+ejt+Atl't)= r X ,j(r-t) t+o(At), m > i>jWe denote by I a,} the initial distribution at time s and put p(ro,tIIar},s) = E p(ro,t| r,s)ar for roeS'N,in (2) the summation being over all r belonging to S'N,m. The Kolmogorov equations then read dp(rotII ar},s) -_p(ro,tIlar},s) E Xi,(ro,t) For nonnegative integers N and ri, we define recursively L(r1,ro;N,1) = 1 + r1,
