A random Trotter product formula

Let X(t) be a pure jump process with state space S and let , , ,2, be the succession of states visited by X(t), A0A. the sojourn times in each state, N(t) the number of transitions before t and A=t= t-N(t)-0 Sk. For each x1 S let Tz(t) be an operator semigroup on a Banach space L. Define TA(t, w)= T$0((l1)Ao)T41((11/)A1).**. T... t)((1I)At). Conditions are given under which rA(t, w) will converge almost surely (or in probability) to a semigroup of operators as A-oo. With S={1, 2} and X(t) = 1, 2n _ t < 2n + 1, = 2, 2n + 1 t < 2n + 2, n=O, 1, 2, * the result is just the "Trotter product formula".