Abstract. We study a class of elliptic operators A with unbounded coefficients defined in I × R d for some unbounded interval I ⊂ R. We prove that, for any s ∈ I, the Cauchy problem u(s,admits a unique bounded classical solution u. This allows to associate an evolution family {G(t, s)} with A, in a natural way. We study the main properties of this evolution family and prove gradient estimates for the function G(t, s)f . Under suitable assumptions, we show that there exists an evolution system of measures for {G(t, s)} and we study the first properties of the extension of G(t, s) to the L p -spaces with respect to such measures.

We prove convergence of the solutions Xn of semilinear stochastic evolution equations on a Banach space B, driven by a cylindrical Brownian motion in a Hilbert space H, dXn(t) = (AnX(t) + Fn(t, Xn(t))) dt + Gn(t, Xn(t)) dW H (t),τ (r) := inf{t > 0 : X (r) (t) > r} Date: August 14, 2018. 1991 Mathematics Subject Classification. 60H15 (primary); 35K37, 35A35, 35R60 (secondary).

Abstract. Motivated by applications to transition semigroups, we introduce the notion of a norming dual pair and study a Pettis-type integral on such pairs. In particular, we establish a sufficient condition for integrability. We also introduce and study a class of semigroups on such dual pairs which are an abstract version of transition semigroups. Using our results, we give conditions ensuring that a semigroup consisting of kernel operators has a Laplace transform which also consists of kernel operators. We also provide conditions under which a semigroup is uniquely determined by its Laplace transform.

We study continuity and equicontinuity of semigroups on norming dual pairs with respect to topologies defined in terms of the duality. In particular, we address the question whether continuity of a semigroup already implies (local/quasi) equicontinuity. We apply our results to transition semigroups and show that, under suitable hypothesis on E, every transition semigroup on C b (E) which is continuous with respect to the strict topology β 0 is automatically quasi-equicontinuous with respect to that topology. We also give several characterizations of β 0 -continuous semigroups on C b (E) and provide a convenient condition for the transition semigroup of a Banach space valued Markov process to be β 0 -continuous.