Eilenberg--Moore algebras for stochastic relations
A stochastic relation between two Polish spaces is a Kleisli morphism for the so-called Giry monad [\textit{M. Giry}, ``A categorical approach to probability theory'', Lect. Notes Math. 915, 68--85 (1982; Zbl 0486.60034)]. Hence, the Kleisli algebras of this monad could give interesting informations on stochastic relations. For categories of algebras, the Kleisli algebras describe only the free algebras whereas the category of Eilenberg-Moore algebras of a monad contains all algebras defined by this monad and gives therefore more information about the algebraic component of stochastic relations on Polish spaces. \textit{M. Giry} (loc. cit.) investigated the functor assigning to a Polish space \(X\) the space \({\mathbf P}(X)\) of probability measures on the Borel sets \({\mathcal B}(X)\) of \(X\). Together with the mapping \(\mu_X:{\mathbf P}({\mathbf P}(X))\to{\mathbf P}(X)\) assigning to a Borel measure \(M\in{\mathbf P}({\mathbf P}(X))\) and a \(B\in{\mathcal B}(X)\) the integral \[ \mu_X(M)(B)= \int_{{\mathbf P}(X)} \tau(B)\,M(d\tau), \] the probability functor \({\mathbf P}\) defines the Giry monad. The author considers a modification of the Giry monad namely the analogous monad induced by the subprobability functor \({\mathbf S}(X)\), \(X\) a Polish space. He succeeds to characterize the Eilenberg-Moore algebras of this modified Giry monad completely; they are the positively convex structures [called ``positively convex modules'' in \textit{D. Pumplün}, J. Convex Anal. 10, 109--127 (2003; Zbl 1056.52002)] on Polish spaces with the additional property that the positively convex operations are continuous mappings. He also proves the analogous results for the original Giry monad induced by \({\mathbf P}(X)\) and gets as algebras the convex structures on Polish spaces with continuous convex operations. Since the introduction of the Giry monad in 1981 this is the first complete and canonical characterization of these algebras. With this representation of the Eilenberg-Moore algebras one has a very well-manageable tool to investigate the theory of stochastic relations on Polish spaces algebraically.
scientific article; zbMATH DE number 5117647
scientific article