On the derivative of a semigroup
David A. Freedman · Bulletin of the American Mathematical Society · 1968
Let / be a countably infinite set, and P = {P(t, i, j)} a standard semigroup on J: that is, P(t) is a stochastic matrix, P(t+s) =*P(t)P(s) f and lim P(t, i, i) = P(0, i, i) « 1 for all i G /.As is well known, Q=P'(0) exists, although q(i)=Q(i, i) may be infinite for some or all i.When q(i) < oo, the numbers q(i) and Q(i, j)/q(i) have interesting known probabilistic interpretations, although the meaning of Q(i, j) itself is a little obscure.The object of this note is to "explain" Q(i, j) in a way which does not depend on q(i), finite or infinite.To state the explanation, give I the discrete topology, and let I^J{} be the one-point compactification.On a suitable probability triple, say (0, ïï, P*), construct an JU {$}-valued process X, which is Markov with stationary transitions P, starts from feGJ, and has smooth sample functions.More formally, for 0 = / 0 </i< • • •