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Superposition principle and schemes for measure differential equations

  • * Corresponding author: Giulia Cavagnari

    * Corresponding author: Giulia Cavagnari 
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  • Measure Differential Equations (MDE) describe the evolution of probability measures driven by probability velocity fields, i.e. probability measures on the tangent bundle. They are, on one side, a measure-theoretic generalization of ordinary differential equations; on the other side, they allow to describe concentration and diffusion phenomena typical of kinetic equations. In this paper, we analyze some properties of this class of differential equations, especially highlighting their link with nonlocal continuity equations. We prove a representation result in the spirit of the Superposition Principle by Ambrosio-Gigli-Savaré, and we provide alternative schemes converging to a solution of the MDE, with a particular view to uniqueness/non-uniqueness phenomena.


    Erratum: The month information has been corrected from January to February. We apologize for any inconvenience this may cause.

    Mathematics Subject Classification: Primary:35S99, 28A50;Secondary:35F20, 35F25.


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  • Figure 1.  LAS and semi-discrete Lagrangian schemes on the left for $ N = 1 $. Mean-velocity scheme on the right

    Figure 2.  LAS and semi-discrete Lagrangian schemes on the left. Mean-velocity scheme on the right

    Figure 3.  LAS scheme: for N = 1 (left) and N=2 (right)

    Figure 4.  Semi-discrete Lagrangian scheme on the left. Mean-velocity scheme on the right

    Figure 5.  Peano's brush referred to Example 4

    Figure 6.  LAS scheme for N = 1, 2, 3

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