# American Institute of Mathematical Sciences

November  2019, 39(11): 6207-6230. doi: 10.3934/dcds.2019270

## Measure dynamics with Probability Vector Fields and sources

 1 Department of Mathematical Sciences, Rutgers University - Camden, Camden, NJ, USA 2 Dipartimento di Matematica "Tullio Levi–Civita", Università degli Studi di Padova, Padova, Italy

* Corresponding author: Francesco Rossi

Received  September 2018 Revised  April 2019 Published  August 2019

We introduce a new formulation for differential equation describing dynamics of measures on an Euclidean space, that we call Measure Differential Equations with sources. They mix two different phenomena: on one side, a transport-type term, in which a vector field is replaced by a Probability Vector Field, that is a probability distribution on the tangent bundle; on the other side, a source term. Such new formulation allows to write in a unified way both classical transport and diffusion with finite speed, together with creation of mass.

The main result of this article shows that, by introducing a suitable Wasserstein-like functional, one can ensure existence of solutions to Measure Differential Equations with sources under Lipschitz conditions. We also prove a uniqueness result under the following additional hypothesis: the measure dynamics needs to be compatible with dynamics of measures that are sums of Dirac masses.

Citation: Benedetto Piccoli, Francesco Rossi. Measure dynamics with Probability Vector Fields and sources. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6207-6230. doi: 10.3934/dcds.2019270
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