Quenched Mass Transport of Particles Toward a Target

Bruno Bouchard, Boualem Djehiche, Idris Kharroubi

Research output: Contribution to journalArticlepeer-review


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. This kind of problems is motivated by limiting behavior of interacting particles systems with applications in, for example, agricultural crop management. 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 toward a given target, along the paths of a stochastic differential equation. Our results extend those of Soner and Touzi, Journal of the European Mathematical Society (2002) to our setting.

Original languageEnglish (US)
Pages (from-to)345-374
Number of pages30
JournalJournal of Optimization Theory and Applications
Issue number2
StatePublished - Aug 1 2020


  • Dynamic programming
  • Mass transportation
  • McKean–Vlasov SDEs
  • Stochastic target
  • Viscosity solutions

ASJC Scopus subject areas

  • Management Science and Operations Research
  • Control and Optimization
  • Applied Mathematics


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