-   The *scattering kernel* is a Borel function $\sigma\colon {{\mathcal V}} \times {{\mathcal V}}\to [0,+\infty)$ satisfying $\sigma(v,v')=\sigma(v',v)$, $(v,v') \in {{\mathcal V}}\times {{\mathcal V}}$.

-   The *scattering rate* $\lambda\colon {{\mathcal V}} \to
        [0,+\infty)$ is defined by $\lambda(v):=\int\!\pi({\mathop{}\!\mathrm{d}}v')\,
        \sigma(v,v')$. We require that it has all exponential moments with respect to $\pi$ namely, $\pi\big[ e^{\gamma \lambda}\big]
        <+\infty$ for any $\gamma\in {{\mathbb R}}_+$.

-   The *drift* is a Borel function $b\colon {{\mathcal V}} \to
        {{\mathbb R}}^d$. We require that it has all exponential moments with respect to $\pi$ namely, $\pi\big[ e^{\gamma |b| }\big]
        <+\infty$ for any $\gamma\in {{\mathbb R}}_+$, where $|b|$ is the Euclidean norm of $b$.

Given $T>0$ let $C\big([0,T]; {{\mathcal P}} ({{\mathbb T}}^d\times {{\mathcal V}})\big)$ be the set of continuous paths on ${{\mathcal P}}({{\mathbb T}}^d \times {{\mathcal V}})$ endowed with the topology of uniform convergence. Denote by ${{\mathcal M}}_\mathrm{a}\big([0,T]\times {{\mathbb T}}^d\times {{\mathcal V}} \times {{\mathcal V}}\big)$ the set of finite Radon measures on $[0,T]\times {{\mathbb T}}^d\times {{\mathcal V}}\times
{{\mathcal V}}$ antisymmetric with respect to the exchange of the last two variables endowed with the weak\* topology. Set ${{\mathcal S}}:=C\big([0,T];
{{\mathcal P}} ({{\mathbb T}}^d \times {{\mathcal V}})\big) \times {{\mathcal M}}_\mathrm{a}
\big([0,T]\times {{\mathbb T}}^d \times {{\mathcal V}}\times {{\mathcal V}} \big)$ endowed with the product 