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I am trying to read and understand the article "Hypocoercivity for linear kinetic equations conserving mass." by Dolbeault, Mouhot, Schmeiser. doi: 10.1090/s0002-9947-2015-06012-7 (https://homepage.univie.ac.at/christian.schmeiser/DMS-I.pdf)

Already on first page we have the passage:

We consider linear kinetic equations which can be written as \begin{equation} \partial_t f + \mathrm{T}f = \mathrm{L}f, \end{equation} $ f=f(x,v,t) $ with $ (x,v,t) \in \mathbb{T}^d \times \mathbb{R}^d \times (0, \infty)$ and $d \in \mathbb{N}.$ The linear transport operator $$\mathrm{T} := v \cdot \nabla_x - \nabla_x V(x) \cdot \nabla_v $$ has characteristics given on the phase space $\mathbb{T}^d \times \mathbb{R}^d$ by the flow of the Hamiltonian $$ (x,v) \mapsto E(x,v) := \frac{1}{2} |v|^2 + V(x).$$
The external potential $V(x)$ is measurable function on $\mathbb{R}^d.$ I don't understand what is meant by "the characteristics of the transport operator are given by the flow the Hamiltonian"? And what is the connection of that Functional $E$ with the transport operator? I never heard the term "characteristics of operators", I just know this in context of the "method of characteristics" and Cauchy-Problems. Would be very grateful for further explanations and definitions.

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This is related to the standard formulation of the method of characteristics in the following way. Consider the time-independent homogeneous version of this equation, i.e., an equation just involving the linear operator $\mathrm{T}$: $$\mathrm{T}u = 0\iff v\cdot\nabla_x u - \left[\nabla_x V\right]\cdot\nabla_v u=0,$$ where $u$ is a function of $x$ and $v$ only. Once can apply the method of characteristics to this system and obtain a system of $2d+1$ ODEs, although one will simply state that $\dot{u}=0$ along characteristic lines. Here I will let $\dot{{}}$ denote differentiation with respect to some real parameter $s$. The corresponding equations for $x$ and $v$ are then $$\dot{x} = v,\quad \dot{v}=-\nabla_x V.$$

Switching gears, given a smooth function $H:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$, the corresponding Hamiltonian system on $\mathbb{R}^{2n}$ is defined to be $$\dot{q} = \nabla_p H, \quad\dot{p}=-\nabla_q H,$$ where $p,q\in\mathbb{R}^n$. Importantly, $H$ is constant along trajectories of this system.

If we now make the identification $(x,v) = (q,p)$ and define $H(q,p) = E(x,v)$, then Hamilton's equations for this function read $$ \begin{aligned} \dot{x} = \nabla_v E &= v \\ \dot{v} = -\nabla_x E &= -\nabla_x V, \end{aligned} $$ which are our original characteristic equations for the operator $\mathrm{T}$.

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