Measure Differential Equations

  • Benedetto PiccoliEmail author


A new type of differential equations for probability measures on Euclidean spaces, called measure differential equations (briefly MDEs), is introduced. MDEs correspond to probability vector fields, which map measures on an Euclidean space to measures on its tangent bundle. Solutions are intended in weak sense and existence, uniqueness and continuous dependence results are proved under suitable conditions. The latter are expressed in terms of the Wasserstein metric on the base and fiber of the tangent bundle. MDEs represent a natural measure-theoretic generalization of ordinary differential equations via a monoid morphism mapping sums of vector fields to fiber convolution of the corresponding probability vector Fields. Various examples, including finite-speed diffusion and concentration, are shown, together with relationships to partial differential equations. Finally, MDEs are also natural mean-field limits of multi-particle systems, with convergence results extending the classical Dobrushin approach.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2008)zbMATHGoogle Scholar
  2. 2.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA (1997)Google Scholar
  3. 3.
    Bertozzi, A.L., Laurent, T., Rosado, J.: \(L^p\) theory for the multidimensional aggregation equation. Commun. Pure Appl. Math. 64(1), 45–83 (2011)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bonaschi, G.A., Carrillo, J.A., Di Francesco, M., Peletier, M.A.: Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D. ESAIM Control Optim. Calc. Var. 21(2), 414–441 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bressan, A.: Hyperbolic Systems of Conservation Laws, Volume 20 of Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2000)Google Scholar
  6. 6.
    Carrillo, J.A., Fornasier, M., Rosado, J., Toscani, G.: Asymptotic flocking dynamics for the kinetic Cucker-Smale model. SIAM J. Math. Anal. 42(1), 218–236 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cavagnari, G., Marigonda, A., Piccoli, B.: Optimal synchronization problem for a multi-agent system. Netw. Heterog. Media 12(2), 277–295 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cristiani, E., Piccoli, B., Tosin, A.: Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints. In: Naldi, G., Pareschi, L., Toscani, G. (eds.) Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, pp. 337–364. Birkhäuser, Boston (2010)Google Scholar
  9. 9.
    DiPerna, R.J., Lions, P.-L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. 130(2), 321–366 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dobrušin, R.L.: Vlasov equations. Funktsional. Anal. i Prilozhen. 13(2), 48–58 (1979)MathSciNetGoogle Scholar
  11. 11.
    Golse, F.: The mean-field limit for the dynamics of large particle systems. In: Journées ``Équations aux Dérivées Partielles'', pp. Exp. No. IX, 47. Univ. Nantes, Nantes (2003)Google Scholar
  12. 12.
    Martin Jr., R.H.: Nonlinear operators and differential equations in Banach spaces. Wiley-Interscience, New York (1976)Google Scholar
  13. 13.
    Øksendal, B.: Stochastic Differential Equations. Universitext. 6th edn., Springer, Berlin (2003)Google Scholar
  14. 14.
    Poupaud, F., Rascle, M.: Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients. Commun. Partial Differ. Equ. 22(1–2), 337–358 (1997)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Santambrogio, F.: Optimal Transport for Applied Mathematicians, Volume 87 of Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Cham (2015)Google Scholar
  16. 16.
    Schwartz, J.T.: Nonlinear functional analysis. Gordon and Breach Science Publishers, New York (1969). Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher, Notes on Mathematics and its ApplicationsGoogle Scholar
  17. 17.
    Villani, C.: Topics in Optimal Transportation, Volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2003)Google Scholar
  18. 18.
    Villani, C.: Optimal Transport. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Rutgers University CamdenCamdenUSA

Personalised recommendations