#### Vol. 9, No. 3, 2016

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Ethics Statement Contacts Author Index To Appear ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Other MSP Journals

### Alpár Richárd Mészáros and Filippo Santambrogio

Vol. 9 (2016), No. 3, 615–644
##### Abstract

In the spirit of the macroscopic crowd motion models with hard congestion (i.e., a strong density constraint $\rho \le 1$) introduced by Maury et al. some years ago, we analyze a variant of the same models where diffusion of the agents is also taken into account. From the modeling point of view, this means that individuals try to follow a given spontaneous velocity, but are subject to a Brownian diffusion, and have to adapt to a density constraint which introduces a pressure term affecting the movement. From the point of view of PDEs, this corresponds to a modified Fokker–Planck equation, with an additional gradient of a pressure (only living in the saturated zone $\left\{\rho =1\right\}$) in the drift. We prove existence and some estimates, based on optimal transport techniques.

##### Keywords
diffusive crowd motion model, Fokker–Planck equation, density constraint, optimal transportation
##### Mathematical Subject Classification 2010
Primary: 35K61, 49J40, 49J45