Constructive Potential Theory: A Linear Logic Approach

In this paper we propose a constructive and logical foundation of stochastic analysis (more precise its axiomatic heart, axiomatic potential theory) using Abrusci's weak-commutative linear logic and Wiklicky's Hilbert Machine quantitative computational model. A general process algebra, named continuous process algebra or continuous information processing systems has been developed. An important application of this process algebra is that we can associate to each Hilbert machine a Dirichlet space, providing in this way a logical and computational model to each class of applications of Dirichlet spaces. We study the possible applications of this refined model to a large diversity of applied mathematics (including financial mathematics, stochastic processes and mathematical physics) via Dirichlet spaces.