C1

The observations of the first traces of cosmic structure in the

Cosmic Microwave Background are in excellent agreement with the

predictions of Inflation. However as we shall see, that account

is not fully satisfactory, as it does not address the transition

from an homogeneous and isotropic early stage to a latter one

lacking those symmetries. We will argue that new physics along the

lines of the dynamical quantum state reduction theories is needed

to account for such transition and, motivated by Penrose's ideas

suggest that quantum gravity might be the place from where

this new physics emerges. Moreover we will show that observations

can be used to constrain the various phenomenological proposals

made in this regard.

Relative entropy weighted optimization is convex optimization problem over the space of probability measures. Many convex optimization problems can be rephrased as such a problem. This is particularly useful since this problem type admits a quasi-explicit solution (i.e. as the expectation over a random variable), which immediately provides a Monte-Carlo method for numerically computing the solution of the optimization problem.

In this talk we discuss the background and application of this approach to stochastic optimal control problems, which may be considered as relative entropy weighted problems with Wiener space as probability space, and its connection with the theory of large deviations for Brownian functionals. As a particular application we discuss the minimization of the local time in a given point of Brownian motion with drift.