Large symmetrised systems of interacting Brownian bridges and random interlacements and their scaling limits

The overall theme is interacting particle systems and their critical phenomena. The novelty is to combine large deviation analysis for interacting Brownian bridges with the recently developed theory of random interlacements. Another novelty is the probabilistic scattering length for interacting Brownian bridges and random interlacements. A third novelty is to investigate the diffusive scaling limits under symmetrisation. Here, the aim is to prove the conjecture that the limits are, in fact, examples of the so-called Schrödinger diffusion. The idea is to study various probabilistic approaches to interacting Boson systems and the onset of the Bose-Einstein condensation as a critical phenomenon. The phase transition manifests in the onset of so-called random interlacements or the loss of probability mass on finite time horizon random loops. The project is primarily probabilistic, using large deviation techniques, stochastic analysis of nonlinear diffusions, multi-scale analysis and interacting particle systems. This project analyses one of the most fascinating and challenging models: interacting Bosons at positive temperatures. Since the experiments on cold atoms in the late 1990s and two Nobel Prizes, mathematical research has started aiming to prove the so-called Bose-Einstein condensation - critical phenomena like, for example, the superfluidity of liquid Helium at low temperatures. The project uses probabilistic methods for the quantum interacting systems - the so-called Feynman-Kac formula rewrites the quantum problem as a classical problem in probability theory. Outline: Main techniques to be used are variants of large deviation analysis, stochastic analysis for nonlinear diffusions and concentration inequalities. In particular, the project studies path measures and local times along with novel space-time isomorphism theorems, and the project involves the following steps: The research activity splits into three different approaches. In the first one, the effect symmetrisation has on large systems of particles, respectively stochastic processes, is studied using a combination of the spatial dependences of the permutations and the cycle structures. The aim is to show that the unique minimiser of the large deviation rate functions and scaling limits are examples of the so-called Schrödinger diffusion. In the second approach, one studies the Gross-Pitaevskii scaling of the interaction; see (II) below. The challenge is to find a suitable probabilistic representation of the scattering length. Furthermore, this representation may allow proving the Gross-Pitaevskii variational formula. We study large symmetrised systems of Brownian motions and interlacements in a third approach. Here, we aim for a complete local time analysis using novel isomorphism theorems for space-time diffusions, see (III).(I) Examination of the large N limit (number of Brownian bridges) coupled with the large time horizon limit for empirical path measures. The aim is to prove that the unique minimiser is Schrödinger diffusions whose pair measure has a density given by the product of the Gross-Pitaevskii functions for dilute systems respectively by the bridge density function, also known as the Wasserstein diffusion transport term. (II) Examination of the large N limit (number of Brownian bridges) coupled with the large time limit for interacting Brownian motions in trap potential in the Gross-Pitaevskii scaling limit. The two open questions concern, on the one hand, the limit to the ground state description in the zero-temperature limit and, on the other hand, the role of the scattering length. The scattering length appears in analysis via a variational problem (PDE). The project will have to develop and employ a probabilistic version. (III) The symmetrisation procedure for Brownian motion systems is a two-random mechanism involving drawing random permutations and then sampling $N$ random particle positions. In the past, different groups have analysed these