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Generalized span categories provide a framework for formalizing mathematical models of open systems in classical mechanics. We introduce categories LagSy and HamSy that, respectively, provide a categorical framework for the Lagrangian and Hamiltonian descriptions of open classical mechanical systems. The morphisms of LagSy and HamSy correspond to such open systems, and the composition of morphisms models the construction of systems from subsystems. The Legendre transformation gives rise to a functor from LagSy to HamSy that translates from the Lagrangian to the Hamiltonian perspective.

For each prime p, a diffusion constant together with a positive exponent specify a Vladimirov operator and an associated p-adic diffusion equation. The fundamental solution of this pseudo-differential equation gives rise to a measure on the Skorokhod space of p-adic valued paths that is concentrated on the paths originating at the origin. We calculate the first exit probabilities of paths from balls and estimate these probabilities for the brownian bridges.

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