We consider three monads on $\mathsf{Top}$ , the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second one is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads $V \to H$ . In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. We show that V can be restricted to a submonad of $\tau$ -smooth probability measures on $\mathsf{Top}$ . By composing these morphisms of monads, we obtain that taking the supports of $\tau$ -smooth probability measures is also a morphism of monads.

Let $(X,T)$ be a topological dynamical system consisting of a compact metric space X and a continuous surjective map $T : X \to X$ . By using local entropy theory, we prove that $(X,T)$ has uniformly positive entropy if and only if so does the induced system $({\mathcal {M}}(X),\widetilde {T})$ on the space of Borel probability measures endowed with the weak* topology. This result can be seen as a version for the notion of uniformly positive entropy of the corresponding result for topological entropy due to Glasner and Weiss.

We consider the stochastic difference equation ${{\eta }_{k}}\,=\,{{\xi }_{k}}\phi \left( {{\eta }_{k-1}} \right),\,\,k\,\in \,\mathbb{Z}$ on a locally compact group $G$, where $\phi $ is an automorphism of $G$, ${{\xi }_{K}}$ are given $G$-valued random variables, and ${{\eta }_{k}}$ are unknown $G$-valued random variables. This equation was considered by Tsirelson and Yor on a one-dimensional torus. We consider the case when ${{\xi }_{K}}$ have a common law $\mu $ and prove that if $G$ is a distal group and $\phi $ is a distal automorphism of $G$ and if the equation has a solution, then extremal solutions of the equation are in one-to-one correspondence with points on the coset space $K\backslash G$ for some compact subgroup $K$ of $G$ such that $\mu $ is supported on $Kz\,=\,z\phi \left( K \right)$ for any $z$ in the support of $\mu $. We also provide a necessary and sufficient condition for the existence of solutions to the equation.

A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume classical models with a finite number of alleles, as well as models with a continuum of possible alleles as used in quantitative genetics. The dynamics of the corresponding probability distributions is governed by an integro-differential equation in the Banach space of Borel measures on a locally compact space. Existence and uniqueness of the solutions of the initial value problem is proved using basic semigroup theory. A complete characterization of the structure of stationary distributions is presented. Then, existence and uniqueness of stationary distributions is proved under mild conditions by applying operator theoretic generalizations of Perron–Frobenius theory. For an extension of Kingman's original house-of-cards model, a classification of possible stationary distributions is obtained.