_0\}$ we have $$\begin{aligned} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi(x,v) { f ^\varepsilon}(t,x,v)\,{ \mathrm{d}(x,v)}} & = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi ({ X ^\varepsilon }(t;0,x,v), { V ^\varepsilon }(t;0,x,v)){ f ^{\mathrm{in}}}(x,v)\,{ \mathrm{d}(x,v)}} \\ & = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi ({ X ^\varepsilon }(t;0,x,v), { V ^\varepsilon }(t;0,x,v)