Conference Agenda

We propose a stochastic description of the time dependent quantum Bose-Einstein condensate at zero temperature, within the context of Nelson stochastic mechanics. We describe an infinite particle limit of interacting diffusions which corresponds to the mean field limit in the related quantum system. We are able to extend the framework of Nelson stochastic mechanics to nonlinear systems in particular to the case of the nonlinear Schrödinger equation. We also propose how to extend to this nonlinear case the Guerra-Morato variational approach. Our work can also be seen in the context of a mean field limit of McKean–Vlasov processes in a general situation where the drift is a very singular function depending non-trivially on all the particles.

We consider the existence of weak solutions of McKean-Vlasov SDEs with common noise and the propagation of chaos for the associated weakly interacting finite particle systems. Our strategy consists of two main components allowing us to analyze settings with general nonlinear but uniformly elliptic coefficients possessing only low spatial regularity through a marriage of probabilistic and analytic techniques. First, we explore the emergence of regularity in limit points of McKean-Vlasov particle systems, leading to a priori regularity estimates for large-system limits of the empirical measure flows from finite particle systems. Second, we leverage this regularity to establish the existence of weak solutions for McKean-Vlasov SDEs and to identify more nuanced conditions under which chaos propagates, i.e. under which an asymptotic decoupling of the particles takes place and the dynamics in large systems become conditionally independent in law. Next to its applicability to low-regularity regimes, the approach we take to obtain weak solutions and the propagation of chaos may also be useful in future applications to mean-field games and controlled problems.

We introduce a novel class of particle processes governed by stochastic differential equations, featuring both local and mean-field interactions. The impact of the (countably infinite) particle population on an individual is felt through the state of designated neighbors and the average state across the entire population. Under a suitable homogeneity condition on the coefficients of the SDEs, we prove that the dynamics of the infinite system is well-defined, and that the average dynamics can be characterized as the unique solution to a non-linear Fokker-Planck-Kolmogorov equation. In view of the latter, the average dynamics can be decoupled from the local one, rendering the interaction within the population purely local. We further prove that the infinite system can be viewed as the limit of an increasing sequence of finite systems, in which both the individual and aggregate quantitates converge to their respective counterparts.

This is joint work with Ulrich Horst.

We condition a Brownian motion on spending an atypically small amount of time outside a compact interval and characterize the resulting process in terms of an SDE. In particular, we encounter situations where the process almost surely does not leave the interval at all, discovering a very rare extreme example of entropic repulsion. Moreover, we explicitly determine the exact asymptotic behavior of associated conditioning probabilities on $[0,T]$, as $T\to\infty$.