The following stochastic flow
\[
d\mathbf{r}=\mathbf{v}dt,\quad d\mathbf{v}=-(\mathbf{v/}\tau \mathbf{)}%
dt+d\mathbf w(t,\mathbf{r)},\quad \mathbf{r,v\in }R^{2}
\]
is considered which \ is used to describe tracer particles in turbulent
flow, drifters in the upper ocean, cloud formation, ultrasonic aggregation
of aerosols, mammal migration, iterating functions, and other phenomena. An
exact expression for the top Lyapunov exponent of the flow is given for
isotropic Brownian forcing $\mathbf w(t,\mathbf{r)}$ in terms of Airy functions.

In this talk, we will show that the a.c. spectrum of the multidimensional Dirac operator is being preserved under very weak perturbations. The conditions on the decay of potential are optimal in some sense. The case
of Schrodinger operator will be discussed too.

We study the (signed) flow of spectral multiplicity for a family of magnetic Schrodinger operators in R^2,
$$
H(\lambda a) = (-i \nabla -\lambda a)^2 +V(x),
\lambda \ge 0,
$$
in the large coupling limit, i.e., $\lambda \to 0$. Our main assumption is for the magnetic field
$ B curl \lambda a$ to have compact support consisting of a finite number of components. The total magnetic flux may be non-zero.