A Skew-product decomposition of diffusions on a manifold equipped with a group action, A Lorentz model with variable density in a conservative force field, and Reconstruction of a manifold from the intrinsic metric of an associated Markov chain

My thesis consists of three different projects.;1) We consider a 2 x 2 matrix-valued process (xt) t≥0 that is obtained by taking a matrix-valued process with entries that are independent one-dimensional standard Brownian motions and time-changing it in a natural way so that the determinant is nonzero for all t ≥ 0. The QR factorization decomposes ( xt)t ≥ 0 into a "radial'' part (Tt)t ≥ 0 that is an autonomous diffusion on the set of upper triangular matrices with positive determinant and an "angular'' process (U Rt)t≥0, where U is a Brownian motion on the group SO(2) of 2 x 2 orthogonal matrices with determinant one and the time-change (R t)t≥0 is adapted to the filtration generated by (Tt) t≥0. In this project we show that, unlike classical skew-products such as the celebrated skew-product decomposition of planar Brownian motion into its radial and angular parts, the Brownian motion ( Ut)t≥0 on SO(2) is not independent of the radial part (Tt) t≥0. We observe that our process fits into the framework of a theorem from cite L09 on the existence of a skew-product decomposition of a general continuous Markov process on a smooth manifold whose distribution is equivariant under the action of a Lie group. Our result is a counterexample to the main result of cite L09, but the conclusion of that result holds after a slight strengthening of the hypotheses. These results appear in cite EHW14.;2) In Chapter 2, which is based on cite HRW14, we study the diffusion limit of a transport process that models the trajectory in R 2 of a particle under the influence of a conservative, spherically symmetric force field U. The particle travels along the trajectory determined by its initial conditions and U until, according to a Poisson process with variable intensity on this trajectory, it reflects in a uniform direction. We show that under a proper rescaling of time, energy and the density of obstacles, the trajectory converges to a diffusion whose generator can be found explicitly. This generalizes cite BR14, where the force field was taken to be constant, to a large class of force fields.;3) A Dirichlet form on a Hilbert space naturally induces a metric on its domain in terms of the energy measure of the form. This metric, which is known as the Caratheodory or intrinsic metric, is studied extensively in cite Dav93 where it is used to establish estimates for the heat kernel of a discrete Laplacian operator on a weighted graph. We study the Caratheodory metric associated with the generator of a continuous time Markov chain on a graph of points sampled independently from a distribution on an embedded manifold. Under a proper rescaling of the edge weights, the generator of the Markov chain converges to a weighted Laplacian on the manifold as the number of points goes to infinity. In this third project we conjecture that a rescaling of the Caratheodory distances between any two fixed points on the graph converges to the geodesic distance on the manifold as the number of points on the graph goes to infinity. We prove that the geodesic distances form a limiting lower bound for the Caratheodory distances, and provide some heuristic arguments to indicate why they may be limiting upper bounds as well. However, the upper bound limit remains an open question for future study. end itemize.