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.

I will present a criterion for the validity of the central limit theorem
for a class of dependent random variables and then I will discuss some applications of
it on random, boundary homogenization problems of nonlinear PDEs such nonlinear
parabolic ones and Navier walls.