A homogeneous flow preserves spherical support separation

Since $R(m_{{\mu }_t})\le F({\mu }_0)$, $\mathrm{\parallel }d{R}_{m_{{\mu }_t}}\mathrm{\parallel }$ is bounded on $[0,T]$ by [F1]. Differentiating $\mathrm{\Phi }(ru)=r^2\mathrm{\Phi }(u)$ gives $d{\mathrm{\Phi }}_{ru}=rd{\mathrm{\Phi }}_u$. If $\zeta \in {\mathrm{\partial }}_{{\lambda }_V}V(u)$, substitute $y=rz$ into its defining inequality and use $V(rz)=r^{2}V(z)$; division by $r^2$ shows $r\zeta \in {\mathrm{\partial }}_{{\lambda }_V}V(ru)$, and applying the same argument with $1\mathrm{/}r$ proves equality of the scaled subdifferentials. The feature-energy bounds on the unit sphere therefore imply $\mathrm{\mid }{v}_t(u)\mathrm{\mid }\le C\mathrm{\mid }u\mathrm{\mid }$ uniformly on $[0,T]$. Along a characteristic, $r(t)=\mathrm{\mid }{X}_t(u_0)\mathrm{\mid }$ obeys $\mathrm{\mid }{r}^{\mathrm{\prime }}(t)\mathrm{\mid }\le Cr(t)$ whenever $r(t)>0$. Grönwall gives $e^{-Ct}\mathrm{\mid }{u}_0\mathrm{\mid }\le \mathrm{\mid }{X}_t(u_0)\mathrm{\mid }\le e^{Ct}\mathrm{\mid }{u}_0\mathrm{\mid }$; the lower bound prevents a nonzero trajectory from reaching zero, while two-homogeneity fixes the zero trajectory.

By [F3], the flow law has a path-space representation concentrated on AC trajectories solving its velocity equation, with initial law ${\mu }_0$. Each such path solves the nonautonomous characteristic inclusion for the moment path $m_{{\mu }_t}$; uniqueness in [F2] makes it $t\mapsto X_t(x)$ from its initial point $x$. Therefore ${\mu }_t=(X_t{)}_{\mathrm{\#}}{\mu }_0$. The compact set $K$ has compact image $X_t(K)$; [F4] gives $\mathrm{spt}{\mu }_t=\overline{X_t(K)}=X_t(K)$. Continuity on $[0,T]\times K$ makes $M_K={\sup }_{s\le T,u\in K}\mathrm{\mid }{X}_s(u)\mathrm{\mid }$ finite.

Let $C_{\mathrm{\infty }}$ be the unbounded component of ${\mathbb{R}}^d\setminus K$. It is unique because the complement of a sufficiently large ball is connected. It is an open connected subset of Euclidean space, hence path connected: small open balls show that each path component is open, and its complement, being a union of the other path components, is open as well. No point of $C_{\mathrm{\infty }}$ lies in $B(0,r_a)$: a path in $C_{\mathrm{\infty }}$ from such a point to a point outside $B(0,r_b)$ would contain a subpath from $r_a{\mathbb{S}}^{d-1}$ to $r_b{\mathbb{S}}^{d-1}$ lying in $C_{\mathrm{\infty }}$ and avoiding $K$, contrary to the initial separation. Choose $\alpha >\max (r_b,M_K)$ and then $\beta >e^{CT}\alpha$. Set $A=B(0,\beta )\cap C_{\mathrm{\infty }}$. Its boundary lies in $K\cup \beta {\mathbb{S}}^{d-1}$, and $A\cap B(0,r_a)=\mathrm{\partial }A\cap B(0,r_a)=\mathrm{\varnothing }$.

For $y\in \alpha {\mathbb{S}}^{d-1}$, $y\in A$ and $X_s(\mathrm{\partial }A)$ avoids $y$ for all $s\le T$: $X_s(K)\subset B(0,M_K)\subset B(0,\alpha )$, while the radial lower bound sends $\beta {\mathbb{S}}^{d-1}$ outside $B(0,\alpha )$. Homotopy invariance [F5] gives $D(X_t,A,y)=D(\mathrm{id},A,y)=1$. Choose $0<\eta <e^{-CT}{r}_a$. For every $y\in B(0,\eta )$ and $x\in \mathrm{\partial }A$, $\mathrm{\mid }{X}_s(x)\mathrm{\mid }\ge e^{-Cs}\mathrm{\mid }x\mathrm{\mid }\ge e^{-CT}{r}_a>\eta$, so the target avoids $X_s(\mathrm{\partial }A)$. Since $B(0,r_a)\cap A=\mathrm{\varnothing }$, the identity degree there is zero; hence $D(X_t,A,y)=0$.

Consider any path from $\eta {\mathbb{S}}^{d-1}$ to $\alpha {\mathbb{S}}^{d-1}$ and stop it the first time it reaches $\alpha {\mathbb{S}}^{d-1}$. Before that time it stays in $\overline{B(0,\alpha )}$, which is disjoint from $X_t(\beta {\mathbb{S}}^{d-1})$. If the stopped path avoided $X_t(K)$, then $\mathrm{\partial }A\subset K\cup \beta {\mathbb{S}}^{d-1}$ would imply that it avoided $X_t(\mathrm{\partial }A)$. The degree along that moving target path would be constant by [F5], contradicting its endpoint values $0$ and $1$ from step 4.1. Thus every such path meets $X_t(K)=\mathrm{spt}{\mu }_t$, proving separation.