On a somewhat forgotten condition of Hasegawa and on Blackwell’s example

Abstract

Abstract. We show, by presenting two examples, that a somewhat forgotten condition of Hasegawa (Proc Jpn Acad 40:262-266, 1964) is useful in proving convergence of operator semigroups, and may be more handy than the standard range condition. Also, we present the semigroup related to Blackwell's example (Ann Math Statist 29:313-316, 1958) as an infinite product of commuting Markov semigroups. Intriguingly, it is hard to find a manageable description of the generator of this semigroup. As a result, it is much easier to prove the existence of the infinite product involved by direct argument than it is to do this using the Trotter-KatoSova-Kurtz-Hasegawa theory.Mathematics Subject Classification. 47D06, 47D09.

On a Certain Operator Related to Blackwell’s Markov Chain

“…Proof Theorem 2.1 of [2] specifies conditions under which an infinite product of commuting contraction semigroups exists. All those conditions are satisfied in our case (thanks to Theorem 1), especially the key one that D(A) is dense in l 1 (N).…”

“…α n < ∞. Now we follow the construction of the Blackwell semigroup as described in [2]. Firstly, let I be the set of functions i : N → {0, 1} admitting value 1 finitely many times.…”

“…Then for any x = i∈I δ i e i from l 1 (I) we have B n x = i∈I δ i B n e i . As explained in [2], B n is the generator of a Markov chain on I in which the nth coordinate of an i ∈ I jumps between 0 and 1. In terms of semigroups, B n generates a Markov semigroup {e t B n , t ≥ 0} which values on {e i } i∈I are given by…”

“…Theorem 4.2 of [2] says that semigroups {T n (t), t ≥ 0} converge, when n → ∞, to a strongly continuous semigroup {T (t), t ≥ 0} composed of Markov operators. In such a case, we say that the infinite product ∞ k=1 e t B k exists and denote ∞ k=1 e t B k := T (t).…”

“…It is also denoted by e t A B , where A B is the generator of {T (t), t ≥ 0}. Although a formula for T (t) is given in Lemma 4.1 of [2] it is not useful here, since it is expressed as an infinite sum of terms which are difficult to handle. The fact that semigroups T n (t) converge strongly is also a conclusion from Theorem 1, see Remark 1 in Sect.…”