# Behavior near walls in the mean-field approach to crowd dynamics

## TL;DR

This paper develops a mean-field stochastic model for pedestrian dynamics near walls, incorporating sticky boundaries and boundary diffusion, and demonstrates its mathematical properties, control possibilities, and empirical relevance.

## Contribution

It introduces a novel mean-field SDE model with boundary effects for pedestrian motion, including existence, uniqueness, control, and numerical validation.

## Key findings

- Model admits a unique weak solution.
- Pedestrian paths are semimartingales with smoothing boundary effects.
- Numerical simulations confirm empirical crowd behavior in corridors.

## Abstract

This paper introduces a system of stochastic differential equations (SDE) of mean-field type that models pedestrian motion. The system lets the pedestrians spend time at, and move along, walls, by means of sticky boundaries and boundary diffusion. As an alternative to Neumann-type boundary conditions, sticky boundaries and boundary diffusion have a 'smoothing' effect on pedestrian motion. When these effects are active, the pedestrian paths are semimartingales with first-variation part absolutely continuous with respect to the Lebesgue measure $dt$, rather than an increasing processes (which in general induces a measure singular with respect to $dt$) as is the case under Neumann boundary conditions. We show that the proposed mean-field model for pedestrian motion admits a unique weak solution and that it is possible to control the system in the weak sense, using a Pontryagin-type maximum principle. We also relate the mean-field type control problem to the social cost minimization in an interacting particle system. We study the novel model features numerically and we confirm empirical findings on pedestrian crowd motion in congested corridors.