at a fixed temperature $T\ge 0$ at $x=0$. Its dynamics is described by the system of Itô stochastic differential equations $$\begin{aligned}
&&d{\frak q}_{x}(t)={\frak p}_x(t)dt \nonumber
\\
&& d{\frak p}_x(t)=\left[-(\alpha\star{\frak
   q}(t))_x-\frac{{\epsilon}{\gamma}_0}{2}(\theta\star{\frak
   p}(t))_x\right]dt+\sqrt{{\epsilon}{\gamma}_0}\sum_{k=-1,0,1}(Y_{x+k}{\frak
   p}_x(t))dw_{x+k}(t)\label{eq:bas1} \\
&&
+\left(-{\gamma}_1{\frak p}_0(t)dt+\sqrt{2{\gamma}_1 T}dw(t)\right)\delta_{0,x},\quad x\in{{\mathbb Z}}.\nonumber\end{aligned}$$ Here $$\label{011210}
Y_x:=({\frak p}_x-{\frak p}_{x+1})\partial_{{\frak p}_{x-1}}+({\frak p}_{x+1}-{\frak p}_{x-1})\partial_{{\frak p}_{x}}+({\frak p}_{x-1}-{\frak p}_{x})\partial_{{\frak p}_{x+1}}$$ and $\left(w_x(t)\right)_{t\ge0}$, $x\in{{\mathbb Z}}$ with $\left(w(t)\right)_{t\ge0}$, are i.i.d. one dimensional, real valued, non-anticipative standard Brownian motions, over some filtered probability space $(\Sigma,{\cal F},\left({\cal F}_t\right)_{t\ge0},{{\mathbb P}})$. In addition, $$\theta_x=\Delta\theta^{(0)}_x:=\theta^{(0)}_{x+1}+\theta^{(0)}_{x-1}-2\theta^{(0)}_x$$ with $$\theta^{(0)}_x=\left\{
 \begin{array}{rl}
 -4,&x=0\\
 -1,&x=\pm 1\\
 0, &\mbox{ if otherwise.}
 \end{array}
 \right.$$ A simple calculation shows that $$\label{beta}
\hat \theta(k)=8{\frak s}^2(k)\left(1+2{\frak c}^2( k)\right),\quad k\in{{\mathbb T}}.$$ Parameters ${\epsilon}\gamma_0>0$, ${\gamma}_1$ describe the strength of the inter-particle and thermostat noises, respectively. In what follows we shall assume that ${\epsilon}>0$ is small, that corresponds to the low density hypothesis that results in atoms suffering finitely many ”collisions” in a macroscopic unit of time (Boltzmann-Grad limit). Although the noise considered here is continuous we believe that the results of the present paper extend to other type of noises, such as e.g. Poisson shots.

Since the vector field $Y_x$ is orthogonal both to a sphere ${\frak
  p}_{x-1}^2+{\frak p}_x^2+{\frak p}_{x+1}^2\equiv {\rm const}$ and plane ${\frak p}_{x-1}+{\frak p}_x+{\frak p}_{x+1}\equiv  