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Browsing by Subject "Forman's theorem"

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  • Satukangas, Eetu Aatos Elmeri (2023)
    In this thesis we prove a short time asymptotic formula for a path integral solution to the Fokker Planck heat equation on a Riemannian manifold. The result is inspired by multiple developments regarding the theory of stochastic differential equations on a Riemannian manifold. Most notably the papers by Itô (1962) which describes the stochastic differential equation, Graham (1985) which describes the probabilistic time development of the stochastic differential equation via a path integral and Anderson and Driver (1999) which proves that Graham's path integral converges to the correct notion of probability. The starting point of the thesis is a paper by R. Graham (1985) where a path integral formula for the solution of the heat equation on a Riemannian manifold is given in terms of a stochastic differential equation in Itô sense. The path integral formula contains an integrand of the exponential of an action function. The action function is defined by the given stochastic differential equation and additional integration variables denoted as the momenta of the paths appearing in the integral. The path integral is defined as the time continuum limit of a product of integrals on a discrete time lattice. The result obtained in this thesis is proven by considering the saddle point approximation of the action appearing in the finite version of Graham's path integral formula. The saddle point approximation gives a power series approximation of the action up to the second order by taking the first and second variations of the action and setting the first variation as zero. We say that the saddle point approximation is evaluated along the critical path of the action which is defined by taking the first variation as zero. The second variation of the action is called the Hessian matrix. With the saddle point approximation of the action, we obtain an asymptotic formula of the path integral which contains the exponential of the action evaluated along the critical path and the determinant of the Hessian. The main part of the proof is the evaluation of the determinant of the Hessian in the continuum limit. To this end we prove a finite dimensional version of a theorem due to R. Forman (1987), called Forman's theorem, which allows us to calculate the ratio of determinants of the Hessian parametrized by two different boundary conditions as a ratio of finite dimensional determinants. We then show that in the continuum limit the ratio of determinants of the Hessian can be written in terms of the Jacobi ow. With the Forman's theorem we then get the short time asymptotic formula by evaluating the determinants on a short time interval.