by $\tilde
f({\lambda})$ its Laplace transform $$\tilde
f({\lambda})=\int_0^{+\infty}e^{-{\lambda}t}f(t)dt,\quad {\rm Re}\,{\lambda}>M.$$

Some function spaces
--------------------

For a given $ G\in {\cal S}({{\mathbb R}}\times{{\mathbb T}})$ - the class of Schwartz functions on ${{\mathbb R}}\times{{\mathbb T}}$ - we let $$\widehat G(\eta,k)=\int_{{{\mathbb R}}}e_y^\star(\eta)G(y,k)dy$$ be its Fourier transform in the first variable. Let $$\label{AC}
{\cal
  A}_c:=[G:\,\hat G\in C_c^\infty({{\mathbb R}}\times{{\mathbb T}})].$$ Let ${\cal A}$ be the Banach space obtained by the completion of ${\cal A}_c$ in the norm $$\label{060805-19}
\|G\|_{\cal A}:=\int_{{{\mathbb R}}}\sup_{k\in{{\mathbb T}}}|\widehat G(\eta,k)|d\eta,\quad G\in{\cal
  A}_c.$$ Space ${\cal A}'$ - the dual to ${\cal A}$ - consists of all distributions $G\in {\cal
  S}'({{\mathbb R}}\times{{\mathbb T}})$ of the form $$\langle G, F\rangle=\int_{{{\mathbb R}}\times {{\mathbb T}}} \widehat G^\star(\eta,k)
\widehat F(\eta,k)d\eta dk,\quad F\in {\cal A}$$ for some measurable function $\widehat G:{{\mathbb R}}\times{{\mathbb T}}\to\mathbb C$, equipped with the norm $$\label{060805-19a}
\|G\|_{{\cal A}'}=\sup_{\eta\in{{\mathbb R}}}\int_{{{\mathbb T}}}|\widehat G(\eta,k)|d k<+\infty.$$

We shall also consider the spaces ${\cal
  L}_{2,{\epsilon}}:=\ell_{2,{\epsilon}}\otimes L^2({{\mathbb T}})$. The respective norms of $G:{{\mathbb Z}}_{\epsilon}\times{{\mathbb T}}\to\mathbb C$ and $\widehat
G:{{\mathbb T}}_{\epsilon}\times{{\mathbb T}}\to\mathbb C$ are given by $$\label{011505-19}
\|G\|_{{\cal L}_{2,{\epsilon}}}:=\left\{\frac{{\epsilon}}{2}\sum_{y\in{{\mathbb Z}}_{\epsilon}}\|G_y\|_{L^2({{\mathbb T}})}^2\right\}^{1/2}
=\|\widehat
G\|_{L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})}.$$

Infinite system of interacting harmonic oscillators
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### Thermostatted Hamiltonian dynamics with momentum and energy conserving noise {#sec2.2.1}

We consider a stochastically perturbed chain of harmonic oscillators thermostatted 