Weak transport identity for characteristic mean-field solutions

For ${\mu }_0$-almost every label $x$, the path $X_s(x)$ is absolutely continuous. For a fixed label, its velocity is continuous in time: the integrand converges pointwise for almost every second label and is dominated by the integrable path supremum in the definition, so [F5] applies. Thus its path is $C^1$, and the ordinary chain rule [F2] and defining ODE [F1] give $\phi (X_t(x))-\phi (x)={\int }_0^t\mathrm{\nabla }\phi (X_s(x))\cdot \int K(X_s(x),X_s(y))d{\mu }_0(y)ds\mathrm{.}$ The integrand is absolutely integrable over time and both labels: $\mathrm{\nabla }\phi$ is bounded, and [F1] bounds $\mathrm{\mid }K(X_s(x),X_s(y))\mathrm{\mid }$ by a constant plus $L(\mathrm{\mid }{X}_s(x)\mathrm{\mid }+\mathrm{\mid }{X}_s(y)\mathrm{\mid })$, whose label integrals are finite uniformly in $s$. This last bound and [F4] justify the exchanges; [F5] supplies the stated velocity continuity.

Integrate step 1.1 in $x$ against ${\mu }_0$. The left side becomes the two displayed measure integrals by the pushforward formula [F1]. By [F3] and the absolute-integrability check in step 1.1, exchange the time and label integrals. Replacing $X_s(x),X_s(y)$ by their pushforward variables gives the right side of the Statement.