Global characteristic flow for a globally Lipschitz mean-field kernel

On a time interval $[a,a+\delta ]$, use the space in [F2] over $({\mathbb{R}}^d,{\mu }_0)$ and fix the initial label map $X_a$. Define $\mathrm{\Phi }(Z{)}_t(x)=X_a(x)+{\int }_a^t\int K(Z_s(x),Z_s(y))d{\mu }_0(y)ds\mathrm{.}$ The growth bound in [F1], the integrability of $X_a$, and [F4] make the time integral finite in the weighted norm. For almost every label, the integrand is continuous in time; it is dominated in the second label by $\mathrm{\mid }K(0,0)\mathrm{\mid }+L({\sup }_s\mathrm{\mid }{Z}_s(x)\mathrm{\mid }+{\sup }_s\mathrm{\mid }{Z}_s(y)\mathrm{\mid })$, integrable by the path norm, so [F5] makes each output path continuous. The space is nonempty because the constant-in-time path $X_a$ belongs to it.

For any $Z,W$, applying [F1], integrating first over the label $x$ and then $y$, and using that ${\mu }_0$ has mass one gives $\mathrm{\parallel }\mathrm{\Phi }Z-\mathrm{\Phi }W\mathrm{\parallel }\le 2L\delta \mathrm{\parallel }Z-W\mathrm{\parallel }\mathrm{.}$ Choose $\delta >0$ with $2L\delta <1$ when $L>0$; if $L=0$ any finite $\delta$ works. The contraction theorem [F3] yields a unique fixed point on this interval.

The bound in step 2.1 applies on each successive interval with the attained endpoint path as initial data. By [F6], choose an integer $J>2LT$ and partition $[0,T]$ into $J$ equal intervals (or use one interval when $L=0$); each interval then has length below $1\mathrm{/}(2L)$ and successive fixed points construct the solution on all of $[0,T]$. Each fixed point belongs to the weighted path space, so its endpoint has finite first moment and is admissible for the next interval.

Any two solutions on the first interval are fixed points of the same contraction and hence agree by [F3]. Repeating on each successive interval gives equality for ${\mu }_0$-almost every label on all of $[0,T]$. The fixed point equation and the pushforward definition give exactly the characteristic solution in [F1] and Characteristic solution of a probability mean-field equation.