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We introduce a new formulation for differential equation describing dynamicsof measures on an Euclidean space, that we call Measure Differential Equationswith sources. They mix two different phenomena: on one side, a transport-typeterm, in which a vector field is replaced by a Probability Vector Field, thatis a probability distribution on the tangent bundle; on the other side, asource term. Such new formulation allows to write in a unified way bothclassical transport and diffusion with finite speed, together with creation ofmass.  The main result of this article shows that, by introducing a suitableWasserstein-like functional, one can ensure existence of solutions to MeasureDifferential Equations with sources under Lipschitz conditions. We also prove auniqueness result under the following additional hypothesis: the measuredynamics needs to be compatible with dynamics of measures that are sums ofDirac masses.

Measure dynamics with Probability Vector Fields and sources