MARK KAC SEMINAR

We consider a discrete-time version of the parabolic Anderson model. This may be described as a model for a directed (1+d)-dimensional polymer interacting with a random potential, which is constant in the deterministic direction and i.i.d. in the d orthogonal directions. The potential at each site is a positive random variable with a polynomial tail at infinity. We show that, as the size of the system diverges, the polymer extremity is localized almost surely at one single point which grows ballistically. We give an explicit characterization of the localization point and of the typical paths of the model.

The main object of the discussion is a continuum system of interacting randomly moving particles. The motion of such a system can be treated as a flow on the phase space, which carries all the points of the space. The description of the motion involves several things such as noise, external forces, the law of interaction etc. Sometimes the structure of the noise or nature of the interaction lead to difficulties in constructing a mathematical model of the flow. In these cases instead of a flow of particles one should study a flow of probability kernels or a random measure on the space of trajectories adapted to the initial noise.

In the first part of the talk we discuss different descriptions of the joint particle motion such as stochastic flows, random measures and stochastic differential equations with interaction. The second part of the talk is devoted to the geometry of the obtained dynamical systems.