Numerical Analysis of Generalized Statistical Manifolds Associated with Stochastic Processes

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1.A generalized statistical manifold is constructed for (time-) discrete stochastic processes characterized by n parameters 6=(θ_1・・・θ_n). According to information geometry, a metric tensor and an α-connection are introduced on an n-parameter family S_N={p(X, θ, N) θ∈Ω} of probability density functions p(X, θ, N) for a random variable X(N) at a discrete time N. As the discrete time goesby, a time series of statistical manifolds・・・, S_1, S_2, ・・・, S_N・・・ is generated, andgathering them leads to a stratified statistical manifold, which is the direct product space S×Z of a statistical manifold S and a set Z of integers. On the generalized statistical manifold an extended connection connecting two layers is introduced after the Newton-Cartan theory of Newton gravity and a theory in elementary particle physics in which Higgs fields are formulated as extended gauge fields. And then extended curvature tensors are constructed from the extended connection and the α-connection. Applying this new geometrical method to a random walk model, we find that its generalized statistical manifold has a duality structure.2.The physical role of the Riemann scalar curvature R of thermodynamic equilibrium systems is studied. Janyszek and Mrugala (JM) interpreted the tfas a measure of the instability of the systems.3.To examine wider roles of curvatures in physics, we also construct new models of thermodynamic systems or stochastic processes. Three germs are obtained, but geometrical characteristics in these models are left as future works.