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