Consider a crystal formed of two types of atoms placed at the nodes of the
integer lattice. The type of each atom is chosen at random, but the crystal
is statistically shift-invariant. Consider next an electron hopping from atom
to atom. This electron performs a random walk on the integer lattice with
randomly chosen transition probabilities (since the configuration seen by
the electron is different at each lattice site). This process is highly
non-Markovian, due to the interaction between the walk and the
environment.

We will present a martingale approach to proving the invariance principle
(i.e. Gaussian fluctuations from the mean) for (irreversible) Markov chains
and show how this can be transferred to a result for the above process
(called random walk in random environment).

This is joint work with Timo Sepp\"al\"ainen.

For a class of semilinear stochastic parabolic equations of Ito type, under suitable conditions, we shall prove the existence of positive local solutions and their Lp-moments will blow up in a finte time for any p greater or equal to one.

We consider a polymer measure based on random walks which are based on sums of iid stable random variables.
A Gibbs measure is defined which models an attraction to the origin for these walks. A phase transition occurs as the the strength of the attraction to the origin occurs.
We examine various "thermodynamic" quantities and show they are all related to each other in a simple way and exhibit universality.