A number of continuous interacting particle systems can be described as collections of Brownian particles on the real line whose collision dynamics are mediated by the local times associated with the gaps between adjacent particles.  Examples of systems in this class include the ordered particle dynamics of rank-based diffusions and certain eigenvalue processes arising in random matrix theory.  We are interested in systems of this sort in which the number of particles is countably infinite.  In this talk, we will present new results on the strong existence and pathwise uniqueness of solutions to the SDEs describing the evolution of such systems.  The main challenge in this setting is establishing uniqueness, which requires controlling effects “coming from infinity.”  Our approach uses a novel technique based on some concentration bounds related to last-passage percolation.  We will also discuss new results on the structure of the set of stationary measures for infinite systems of competing Brownian particles.  This talk is based on joint work with Sayan Banerjee and Amarjit Budhiraja.

Speaker’s website: https://sites.google.com/view/peter-rudzis/home