On Measure-Valued Processes Generated by Differential Equations

Abstract

We study the problem of representation of a homogeneous semigroup {Θ _t } _{t ≥ 0} of transformations of probability measures on $\mathbb{R}^d$ in the form $\Theta _t (\mu) = \mu \circ u_{\mu}^{-1} (\cdot ,t),$ where $u_{\mu} :\mathbb{R}^d \times [0, T] \to \mathbb{R}^d$ satisfies a differential equation of a special form dependent on the measure μ. We give necessary and sufficient conditions for this representation.