Stochastic semigroups: their construction by perturbation and approximation H. R. Thieme 1 (Tempe) and J. Voigt (Dresden) Abstract. The main object of the paper is to present a criterion for the minimal semigroup associated with the Kolmogorov differential equations to be stochastic in ‘ 1 . Our criterion uses a weighted ‘ 1 - space. As an abstract preparation we present a perturbation theorem for substochastic semigroups which generalizes known results to the case of ordered Banach spaces which need not be AL-spaces We also consider extensions of Kolmogorov’s equations to spaces of measures. In an appendix we present a version of the Miyadera perturbation theorem for positive semigroups on ordered Banach spaces. Key words: semigroup, substochastic, stochastic, perturbation, approximation, Kolmogorov differential equations MSC 2000: 47D06, 47D07, 47B60, 47B65, 60J25, 60J35 1 Introduction Let X be an ordered real Banach space with a generating (reproducing) cone X + , X = X + - X + , and a norm which is additive on X + , kx + yk = kxk + kyk for all x, y ∈ X + . A C 0 -semigroup (S (t)); t > 0) (of bounded linear operators) on X is called substochastic (stochastic) if S (t) is positive and kS (t)xk 6 kxk (kS (t)xk = kxk) for all x ∈ X + , t > 0. The norm on X + can be uniquely extended to a positive bounded linear functional ϕ ∈ X * satisfying ϕx = kxk for all x ∈ X + . The generator A of a stochastic semigroup necessarily satisfies ϕAx =0 for all x ∈ D(A) ∩ X + . It is the aim of this note to indicate conditions for a C 0 -semigroup defined by a limiting procedure to be stochastic. Differently from [17], [24; Sec. 2], [3] we work with an auxiliary Banach space X 1 that is continuously and densely embedded into X . This space may be of its own interest: in case of the Kolmogorov differential equations (Section 4), it can be chosen as the first moment space ‘ 11 = {x =(x j ) ∞ j =0 ; kxk 1 := ∑ ∞ j =0 (1 + j )|x j | < ∞}. By perturbation (Section 2) we will construct substochastic C 0 -semigroups on X which leave the auxiliary Ba- nach space invariant and induce C 0 -semigroups thereon. These semigroups can also obtained by an approximation procedure (Section 3). As a second application we present a measure-valued generalization of Kolmogorov’s differential equations (Section 5) whose solutions are associ- ated with Markov transition functions. In an appendix we present a version of the Miyadera perturbation theorem for positive semigroups on ordered Banach spaces. This result is needed in Section 2. 1 partially supported by NSF grant DMS-0314529