In the talk I will discuss the construction of high-excited wave functions in spin-chains models, chaotic and pseudo-integrable quantum billiards. Special attention is given to the calculation of fractal properties of their wave functions.

The objective of this talk is the study of semimartingales and cylindrical semimartingales in infinite dimensional spaces. Our main motivation is the introduction of a theory of stochastic integration on duals of nuclear spaces, in particular on spaces of distributions. We start by considering conditions for a cylindrical semimartingale in the strong dual E' of a nuclear space E to have a E'-valued semimartingale version whose paths are right-continuous with left limits. Results of similar nature but for more specific classes of cylindrical semimartingales and examples will be provided. Later, we will show that under some general conditions every E'-valued semimartingale has a canonical decomposition. In the last part of our talk we explain how our aforementioned results on duals of nuclear spaces can be used to establish sufficient conditions for a cylindrical semimartingale in a Hilbert space H to have a H-valued semimartingale version whose paths are right-continuous with left limits. As an application of our result we will obtain the theorem of radonification by a single Hilbert-Schmidt operator for cylindrical semimartingales in H.