UC Berkeley

My thesis consists of three different projects.

\begin{itemize}

\item [1)] We consider a $2 \times 2$ matrix-valued process $(x_t)_{t\geq 0}$

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 $t\geq 0$. The QR factorization decomposes

$(x_t)_{t\geq 0}$ into a ``radial'' part $(T_t)_{t\geq 0}$

that is an autonomous diffusion on the set of upper triangular matrices

with positive determinant and

an ``angular'' process $(U_{R_t})_{t \ge 0}$, where $U$ is a Brownian motion

on the group $SO(2)$ of $2\times 2$ orthogonal matrices with

determinant one and the time-change $(R_t)_{t \ge 0}$

is adapted to the filtration generated by $(T_t)_{t\geq 0}$.

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 $(U_t)_{t\geq 0}$ on $SO(2)$

is not independent of the radial part $(T_t)_{t\geq 0}$.

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}.

\item [2)] In Chapter 2, which is based on \cite{HRW14}, we study the diffusion limit of a transport process that models the trajectory in $\bR^2$ of a particle under the influence of a conservative, spherically symmetric force field $\cU$. The particle travels along the trajectory determined by its initial conditions and $\cU$ 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.

\item [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 Carath {e}odory 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 Carath {e}odory 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 Carath {e}odory 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 Carath {e}odory 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}