Quenched mass transport of particles towards a target

TL;DR

This paper studies the problem of controlling the initial distribution of a mean-field stochastic system to ensure it reaches a target distribution at a fixed time, using geometric PDE methods.

Contribution

It extends stochastic target control theory to mean-field SDEs with a geometric dynamic programming principle and viscosity solutions.

Findings

Established a geometric dynamic programming principle for mean-field stochastic targets.

Proved the value function is a viscosity solution of a geometric PDE.

Characterized initial laws that can be transported to a target distribution almost surely.

Abstract

We consider the stochastic target problem of finding the collection of initial laws of a mean-field stochastic differential equation such that we can control its evolution to ensure that it reaches a prescribed set of terminal probability distributions, at a fixed time horizon. Here, laws are considered conditionally to the path of the Brownian motion that drives the system. We establish a version of the geometric dynamic programming principle for the associated reachability sets and prove that the corresponding value function is a viscosity solution of a geometric partial differential equation. This provides a characterization of the initial masses that can be almost-surely transported towards a given target, along the paths of a stochastic differential equation. Our results extend [16] to our setting.