Abordamos o problema do transporte otimo de Monge-Kantorovich no caso em que o custo e dado pelo quadrado da distância. Tal custo tem uma estrutura que permite a obtenção de resultados mais ricos do que o caso geral. Nosso objetivo e determinar se h a soluções para tal problema e caracteriza-las. Al em disso, tratamos informalmente do problema de transporte otimo para um custo geral. / We analyze the Monge-Kantorovich optimal transportation problem in the case where the cost function is given by the square of the Euclidean norm. Such cost has a structure which allow us to get more interesting results than the general case. Our main purpose is to determine if there are solutions to such problem and characterize them. We also give an informal treatment to the optimal transportation problem in the general case.

Abordamos o problema do transporte otimo de Monge-Kantorovich no caso em que o custo e dado pelo quadrado da distância. Tal custo tem uma estrutura que permite a obtenção de resultados mais ricos do que o caso geral. Nosso objetivo e determinar se h a soluções para tal problema e caracteriza-las. Al em disso, tratamos informalmente do problema de transporte otimo para um custo geral. / We analyze the Monge-Kantorovich optimal transportation problem in the case where the cost function is given by the square of the Euclidean norm. Such cost has a structure which allow us to get more interesting results than the general case. Our main purpose is to determine if there are solutions to such problem and characterize them. We also give an informal treatment to the optimal transportation problem in the general case.

Abordamos o problema do transporte otimo de Monge-Kantorovich no caso em que o custo e dado pelo quadrado da distância. Tal custo tem uma estrutura que permite a obtenção de resultados mais ricos do que o caso geral. Nosso objetivo e determinar se h a soluções para tal problema e caracteriza-las. Al em disso, tratamos informalmente do problema de transporte otimo para um custo geral. / We analyze the Monge-Kantorovich optimal transportation problem in the case where the cost function is given by the square of the Euclidean norm. Such cost has a structure which allow us to get more interesting results than the general case. Our main purpose is to determine if there are solutions to such problem and characterize them. We also give an informal treatment to the optimal transportation problem in the general case.

In this work we introduce some category-theoretical concepts and techniques to study probability distributions on metric spaces and ordered metric spaces. In Chapter 1 we give an overview of the concept of a probability monad, first defined by Giry. Probability monads can be interpreted as a categorical tool to talk about random elements of a space X. We can consider these random elements as formal convex combinations, or mixtures, of elements of X. Spaces where the convex combinations can be actually evaluated are called algebras of the probability monad. In Chapter 2 we define a probability monad on the category of complete metric spaces and 1-Lipschitz maps called the Kantorovich monad, extending a previous construction due to van Breugel. This monad assigns to each complete metric space X its Wasserstein space PX. It is well-known that finitely supported probability measures with rational coefficients, or empirical distributions of finite sequences, are dense in the Wasserstein space. This density property can be translated into categorical language as a colimit of a diagram involving certain powers of X. The monad structure of P, and in particular the integration map, is uniquely determined by this universal property. We prove that the algebras of the Kantorovich monad are exactly the closed convex subsets of Banach spaces. In Chapter 3 we extend the Kantorovich monad of Chapter 2 to metric spaces equipped with a partial order. The order is inherited by the Wasserstein space, and is called the stochastic order. Differently from most approaches in the literature, we define a compatibility condition of the order with the metric itself, rather then with the topology it induces. We call the spaces with this property L-ordered spaces. On L-ordered spaces, the stochastic order induced on the Wasserstein spaces satisfies itself a form of Kantorovich duality. The Kantorovich monad can be extended to the category of L-ordered metric spaces. We prove that its algebras are the closed convex subsets of ordered Banach spaces, i.e. Banach spaces equipped with a closed cone. The category of L-ordered metric spaces can be considered a 2-category, in which we can describe concave and convex maps categorically as the lax and oplax morphisms of algebras. In Chapter 4 we develop a new categorical formalism to describe operations evaluated partially. We prove that partial evaluations for the Kantorovich monad, or partial expectations, define a closed partial order on the Wasserstein space PA over every algebra A, and that the resulting ordered space is itself an algebra. We prove that, for the Kantorovich monad, these partial expectations correspond to conditional expectations in distribution. Finally, we study the relation between these partial evaluation orders and convex functions. We prove a general duality theorem extending the well-known duality between convex functions and conditional expectations to general ordered Banach spaces.