# Essential m-dissipativity and hypocoercivity of Langevin dynamics with
multiplicative noise

Alexander Bertram , Department of Mathematics, TU Kaiserslautern, PO box 3049,
67653 Kaiserslautern<EMAIL_ADDRESS>(corresponding
author) Martin Grothaus11footnotemark: 1<EMAIL_ADDRESS>
###### Abstract

We provide a complete elaboration of the $L^{2}$-Hilbert space hypocoercivity
theorem for the degenerate Langevin dynamics with multiplicative noise,
studying the longtime behavior of the strongly continuous contraction
semigroup solving the abstract Cauchy problem for the associated backward
Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant
diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in
the corresponding Fokker-Planck framework, and made rigorous in the Kolmogorov
backwards setting by Grothaus and Stilgenbauer. We extend these results to
weakly differentiable diffusion coefficient matrices, introducing
multiplicative noise for the corresponding stochastic differential equation.
The rate of convergence is explicitly computed depending on the choice of
these coefficients and the potential giving the outer force. In order to
obtain a solution to the abstract Cauchy problem, we first prove essential
self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert
spaces, using prior elliptic regularity results and techniques from Bogachev,
Krylov and Röckner. We apply operator perturbation theory to obtain essential
m-dissipativity of the Kolmogorov operator, extending the m-dissipativity
results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov
approach is natural, as the theory of generalized Dirichlet forms implies a
stochastic representation of the Langevin semigroup as the transition kernel
of a diffusion process which provides a martingale solution to the Langevin
equation with multiplicative noise. Moreover, we show that even a weak
solution is obtained this way.

Keywords: Langevin equation, multiplicative noise, hypocoercivity, essential
m-dissipativity, essential self-adjointness, Fokker-Planck equation
MSC (2020): 37A25, 47D07, 35Q84, 47B44, 47B25

### Acknowledgment

This version of the article has been accepted for publication, after peer
review but is not the Version of Record and does not reflect post-acceptance
improvements, or any corrections.

The Version of Record is available online at:
https://doi.org/10.1007/s00028-022-00773-y

##  1 Introduction

We study the exponential decay to equilibrium of Langevin dynamics with
multiplicative noise. The corresponding evolution equation is given by the
following stochastic differential equation on $\mathbb{R}^{2d}$,
$d\in\mathbb{N}$, as

$\displaystyle dX_{t}$ $\displaystyle=V_{t}\,\mathrm{d}t,$ (1.1)
$\displaystyle dV_{t}$
$\displaystyle=b(V_{t})\mathrm{d}t-\nabla\Phi(X_{t})\,\mathrm{d}t+\sqrt{2}\sigma(V_{t})\,\mathrm{d}B_{t},$

where $\Phi:\mathbb{R}^{d}\to\mathbb{R}$ is a suitable potential whose
properties are specified later, $B=(B_{t})_{t\geq 0}$ is a standard
$d$-dimensional Brownian motion, $\sigma:\mathbb{R}^{d}\to\mathbb{R}^{d\times
d}$ a variable diffusion matrix with at least weakly differentiable
coefficients, and $b:\mathbb{R}^{d}\to\mathbb{R}^{d}$ given by

$b_{i}(v)=\sum_{j=1}^{d}\partial_{j}a_{ij}(v)-a_{ij}(v)v_{j},$

where $a_{ij}=\Sigma_{ij}$ with $\Sigma=\sigma\sigma^{T}$.

This equation describes the evolution of a particle described via its position
$(X_{t})_{t\geq 0}$ and velocity $(V_{t})_{t\geq 0}$ coordinates, which is
subject to friction, stochastic perturbation depending on its velocity, and
some outer force $\nabla\Phi$. To simplify notation, we split
$\mathbb{R}^{2d}$ into the two components $x,v\in\mathbb{R}^{d}$ corresponding
to position and velocity respectively. This extends to differential operators
$\nabla_{x},\nabla_{v}$, and the Hessian matrix $H_{v}$.

Using Itô’s formula, we obtain the associated Kolmogorov operator $L$ as

$L=\operatorname{tr}\left(\Sigma
H_{v}\right)+b(v)\cdot\nabla_{v}+v\cdot\nabla_{x}-\nabla\Phi(x)\cdot\nabla_{v}.$
(1.2)

Here $a\cdot b$ or alternatively $(a,b)_{\mathrm{euc}}$ denotes the standard
inner product of $a,b\in\mathbb{R}^{d}$. We introduce the measure
$\mu=\mu_{\Sigma,\Phi}$ on $(\mathbb{R}^{2d},\mathcal{B}(\mathbb{R}^{2d}))$ as

$\mu_{\Sigma,\Phi}=(2\pi)^{-\frac{d}{2}}\mathrm{e}^{-\Phi(x)-\frac{v^{2}}{2}}\,\mathrm{d}x\otimes\mathrm{d}v=\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}\mathrm{e}^{-\Phi(x)}\otimes\nu,$

i.e. $\nu$ is the normalized standard Gaussian measure on $\mathbb{R}^{d}$. We
consider the operator $L$ on the Hilbert space
$H\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=L^{2}(\mathbb{R}^{2d},\mu)$.

We note that the results below on exponential convergence to equilibrium can
also be translated to a corresponding Fokker-Planck setting, with the
differential operator $L^{\mathrm{FP}}$ given as the adjoint, restricted to
sufficiently smooth functions, of $L$ in
$L^{2}(\mathbb{R}^{2d},\mathrm{d}(x,v))$. The considered Hilbert space there
is
$\tilde{H}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=L^{2}(\mathbb{R}^{2d},\tilde{\mu})$,
where

$\tilde{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(2\pi)^{-\frac{d}{2}}\mathrm{e}^{\Phi(x)+\frac{v^{2}}{2}}\,\mathrm{d}x\otimes\mathrm{d}v.$

Indeed, this is the space in which hypocoercivity of the kinetic Fokker-Planck
equation associated with the classical Langevin dynamics was proven in [1].
The rigorous connection to the Kolmogorov backwards setting considered
throughout this paper and convergence behaviour of solutions to the abstract
Cauchy problem $\partial_{t}f(t)=L^{\mathrm{FP}}f(t)$ are discussed in Section
5.3.

The concept of hypocoercivity was first introduced in the memoirs of Cédric
Villani ([2]), which is recommended as further literature to the interested
reader. The approach we use here was introduced algebraically by Dolbeault,
Mouhot and Schmeiser (see [3] and [1]), and then made rigorous including
domain issues in [4] by Grothaus and Stilgenbauer, where it was applied to
show exponential convergence to equilibrium of a Fiber laydown process on the
unit sphere. This setting was further generalized by Wang and Grothaus in [5],
where the coercivity assumptions involving in part the classical Poincaré
inequality for Gaussian measures were replaced by weak Poincaré inequalities,
allowing for more general measures for both the spatial and the velocity
component. In this case, the authors still obtained explicit, but
subexponential rates of convergence. On the other hand, the stronger notion of
hypercontractivity was explored in [6] on general separable Hilbert spaces
without the necessity to explicitly state the invariant measure. The specific
case of hypocoercivity for Langevin dynamics on the position space
$\mathbb{R}^{d}$ has been further explored in [7] and serves as the basis for
our hypocoercivity result. However, all of these prior results assume the
diffusion matrix to be constant, while we allow for velocity-dependent
coefficients.

In contrast to [7], we do not know if our operator
$(L,C_{c}^{\infty}(\mathbb{R}^{2d}))$ is essentially m-dissipative, and are
therefore left to prove that first. This property of the Langevin operator has
been shown by Helffer and Nier in [8] for smooth potentials and generalized to
locally Lipschitz-continuous potentials by Conrad and Grothaus in [9,
Corollary 2.3]. However, a corresponding result for a non-constant second
order coefficient matrix $\Sigma$ is not known to the authors.

Moreover, the symmetric part $S$ of our operator $L$ does not commute with the
linear operator $B$ as in [7], hence the boundedness of the auxiliary operator
$BS$ needs to be shown in a different way, which we do in Proposition 3.10.

In Theorem 3.4, we show under fairly light assumptions on the coefficients and
the potential that the operator $(L,C_{c}^{\infty}(\mathbb{R}^{2d}))$ is
essentially m-dissipative and therefore generates a strongly continuous
contraction semigroup on $H$. The proof is given in Section 4 and follows the
main ideas as in the proof of [9, Theorem 2.1], where a corresponding result
for $\Sigma=I$ was obtained.

For that proof we rely on perturbation theory of m-dissipative operators,
starting with essential m-dissipativity of the symmetric part of $L$. To that
end, we state an essential self-adjointness result for a set of non-degenerate
elliptic Dirichlet differential operators $(S,C_{c}^{\infty}(\mathbb{R}^{d}))$
on $L^{2}$-spaces where the measure is absolutely continuous wrt. the Lebesgue
measure. This result is stated in Theorem 4.5 and combines regularity results
from [10] and [11] with the approach to show essential self-adjointness from
[12].

Finally, our main hypocoercivity result reads as follows:

######  Theorem 1.1.

Let $d\in\mathbb{N}$. Assume that $\Sigma:\mathbb{R}^{d}\to\mathbb{R}^{d\times
d}$ is a symmetric matrix of coefficients $a_{ij}:\mathbb{R}^{d}\to\mathbb{R}$
which is uniformly strictly elliptic with ellipticity constant $c_{\Sigma}$.
Moreover, let each $a_{ij}$ be bounded and locally Lipschitz-continuous, hence
$a_{ij}\in H_{\mathrm{loc}}^{1,p}(\mathbb{R}^{d},\nu)\cap
L^{\infty}(\mathbb{R}^{d})$ for each $p\geq 1$. Assume the growth behaviour of
$\partial_{k}a_{ij}$ for all $1\leq k\leq d$ to be bounded either by

$|\partial_{k}a_{ij}(v)|\leq M(1+|v|)^{\beta}$

for $\nu$-almost all $v\in\mathbb{R}^{d}$ and some $M<\infty$,
$\beta\in(-\infty,0]$ or by

$|\partial_{k}a_{ij}(v)|\leq M(\mathds{1}_{B_{1}(0)}(v)+|v|^{\beta})$

for $\nu$-almost all $v\in\mathbb{R}^{d}$ and some $M<\infty$,
$\beta\in(0,1)$. Define $N_{\Sigma}$ in the first case as
$N_{\Sigma}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sqrt{M_{\Sigma}^{2}+(B_{\Sigma}\vee
M)^{2}}$ and in the second case as
$N_{\Sigma}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sqrt{M_{\Sigma}^{2}+B_{\Sigma}^{2}+dM^{2}}$,
where

$\displaystyle M_{\Sigma}$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\max\\{\|a_{ij}\|_{\infty}\mid
1\leq i,j\leq d\\}\quad\text{ and }$ $\displaystyle B_{\Sigma}$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\max\left\\{|\partial_{j}a_{ij}(v)|:v\in\overline{B_{1}(0)},\
1\leq i,j\leq d\right\\}.$

Let further $\Phi:\mathbb{R}^{d}\to\mathbb{R}$ be bounded from below, satisfy
$\Phi\in C^{2}(\mathbb{R}^{d})$ and that $\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x$
is a probability measure on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$
which satisfies a Poincaré inequality of the form

$\|\nabla
f\|_{L^{2}(\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)}^{2}\geq\Lambda\left\|f-\int_{R^{d}}f\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x\right\|_{L^{2}(\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)}^{2}$

for some $\Lambda\in(0,\infty)$ and all $f\in C_{c}(\mathbb{R}^{d})$.
Furthermore assume the existence of a constant $c<\infty$ such that

$|H\Phi(x)|\leq c(1+|\nabla\Phi(x)|)\quad\text{ for all }x\in\mathbb{R}^{d},$

where $H$ denotes the Hessian matrix and $|H\Phi|$ the Euclidian matrix norm.
If $\beta>-1$, then also assume that there are constants $N<\infty$,
$\gamma<\frac{2}{1+\beta}$ such that

$|\nabla\Phi(x)|\leq N(1+|x|^{\gamma})\qquad\text{ for all
}x\in\mathbb{R}^{d}.$

Then the Langevin operator $(L,C_{c}^{\infty}(\mathbb{R}^{2d}))$ as defined in
(1.2) is closable on $H$ and its closure $(L,D(L))$ generates a strongly
continuous contraction semigroup $(T_{t})_{t\geq 0}$ on $H$. Further, it holds
that for each $\theta_{1}\in(1,\infty)$, there is some
$\theta_{2}\in(0,\infty)$ such that

$\left\|T_{t}g-(g,1)_{H}\right\|_{H}\leq\theta_{1}\mathrm{e}^{-\theta_{2}t}\left\|g-(g,1)_{H}\right\|_{H}$

for all $g\in H$ and all $t\geq 0$. In particular, $\theta_{2}$ can be
specified as

$\theta_{2}=\frac{\theta_{1}-1}{\theta_{1}}\frac{c_{\Sigma}}{n_{1}+n_{2}N_{\Sigma}+n_{3}N_{\Sigma}^{2}},$

and the coefficients $n_{i}\in(0,\infty)$ only depend on the choice of $\Phi$.

Finally, our main results may be summarized by the following list:

  * •

Essential m-dissipativity (equivalently essential self-adjointness) of non-
degenerate elliptic Dirichlet differential operators with domain
$C_{c}^{\infty}(\mathbb{R}^{d})$ on Hilbert spaces with measure absolutely
continuous wrt. the $d$-dimensional Lebesgue measure is proved, see Theorem
4.5.

  * •

Essential m-dissipativity of the backwards Kolmogorov operator
$(L,C_{c}^{\infty}(\mathbb{R}^{d}))$ associated with the Langevin equation
with multiplicative noise (1.1) on the Hilbert space $H$ under weak
assumptions on the coefficient matrix $\Sigma$ and the potential $\Phi$, in
particular not requiring smoothness, is shown, see Theorem 3.4.

  * •

Exponential convergence to a stationary state of the corresponding solutions
to the abstract Cauchy problem $\partial_{t}u(t)=Lu(t)$, see (5.1) on the
Hilbert space $H$ with explicitly computable rate of convergence, as stated in
Theorem 1.1, is proved.

  * •

Adaptation of this convergence result to the equivalent formulation as a
Fokker-Planck PDE on the appropriate Hilbert space
$\tilde{H}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=L^{2}(\mathbb{R}^{2d},\tilde{\mu})$
is provided. In particular, this yields exponential convergence of the
solutions to the abstract Fokker-Planck Cauchy problem
$\partial_{t}u(t)=L^{\mathrm{FP}}u(t)$, with $L^{\mathrm{FP}}$ given by (5.3),
to a stationary state, see Section 5.3.

  * •

A stochastic interpretation of the semigroup as a transition kernel for a
diffusion process is worked out. Moreover, we prove this diffusion process to
be a weak solution to the Langevin SDE (1.1) and derive for it strong mixing
properties with explicit rates of convergence, see Section 5.2.

##  2 The abstract hypocoercivity setting

We start by recalling some basic facts about closed unbounded operators on
Hilbert spaces:

######  Lemma 2.1.

Let $(T,D(T))$ be a densely defined linear operator on $H$ and let $L$ be a
bounded linear operator with domain $H$.

  1. (i)

The adjoint operator $(T^{*},D(T^{*}))$ exists and is closed. If $D(T^{*})$ is
dense in $H$, then $(T,D(T))$ is closable and for the closure
$(\overline{T},D(\overline{T}))$ it holds $\overline{T}=T^{**}$.

  2. (ii)

$L^{*}$ is bounded and $\|L^{*}\|=\|L\|$.

  3. (iii)

If $(T,D(T))$ is closed, then $D(T^{*})$ is automatically dense in $H$.
Consequently by (i), $T=T^{**}$.

  4. (iv)

Let $(T,D(T))$ be closed. Then the operator $TL$ with domain

$D(TL)=\\{f\in H\mid Lf\in D(T)\\}$

is also closed.

  5. (v)

$LT$ with domain $D(T)$ need not be closed, however

$(LT)^{*}=T^{*}L^{*}.$

Let us now briefly state the abstract setting for the hypocoercivity method as
in [4].

######  Data conditions (D).

We require the following conditions which are henceforth assumed without
further mention.

  1. (D1)

_The Hilbert space:_ Let $(E,\mathcal{F},\mu)$ be some probability space and
define $H$ to be $H=L^{2}(E,\mu)$ equipped with the standard inner product
$(\cdot,\cdot)_{H}$.

  2. (D2)

_The $C_{0}$-semigroup and its generator:_ $(L,D(L))$ is some linear operator
on $H$ generating a strongly continuous contraction semigroup $(T_{t})_{t\geq
0}$.

  3. (D3)

_Core property of $L$:_ Let $D\subset D(L)$ be a dense subspace of $H$ which
is a core for $(L,D(L))$.

  4. (D4)

_Decomposition of $L$:_ Let $(S,D(S)))$ be symmetric, $(A,D(A))$ be closed and
antisymmetric on $H$ such that $D\subset D(S)\cap D(A)$ as well as
$L|_{D}=S-A$.

  5. (D5)

_Orthogonal projections:_ Let $P:H\to H$ be an orthogonal projection
satisfying $P(H)\subset D(S),\,SP=0$ as well as $P(D)\subset
D(A),\,AP(D)\subset D(A)$. Moreover, let $P_{S}:H\to H$ be defined as

$P_{S}f\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=Pf+(f,1)_{H},\qquad
f\in H.$

  6. (D6)

_Invariant measure:_ Let $\mu$ be invariant for $(L,D)$ in the sense that

$(Lf,1)_{H}=\int_{E}Lf\,\mathrm{d}\mu=0\qquad\text{ for all }f\in D.$

  7. (D7)

_Conservativity:_ It holds that $1\in D(L)$ and $L1=0$.

Since $(A,D(A))$ is closed, $(AP,D(AP))$ is also closed and densely defined.
Hence by von Neumann’s theorem, the operator

$I+(AP)^{*}(AP):D((AP)^{*}AP)\to H,$

where $D((AP)^{*}AP)=\\{f\in D(AP)\mid APf\in D((AP)^{*})\\}$, is bijective
and admits a bounded inverse. We therefore define the operator
$(B,D((AP)^{*}))$ via

$B\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(I+(AP)^{*}AP)^{-1}(AP)*$

Then $B$ extends to a bounded operator on $H$.

As in the given source, we also require the following assumptions:

######  Assumption (H1).

_Algebraic relation:_ It holds that $PAP|_{D}=0$.

######  Assumption (H2).

_Microscopic coercivity:_ There exists some $\Lambda_{m}>0$ such that

$-(Sf,f)_{H}\geq\Lambda_{m}\|(I-P_{S})f\|^{2}\qquad\text{ for all }f\in D.$

######  Assumption (H3).

_Macroscopic coercivity:_ Define $(G,D)$ via $G=PA^{2}P$ on $D$. Assume that
$(G,D)$ is essentially self-adjoint on $H$. Moreover, assume that there is
some $\Lambda_{M}>0$ such that

$\|APf\|^{2}\geq\Lambda_{M}\|Pf\|^{2}\qquad\text{ for all }f\in D.$

######  Assumption (H4).

_Boundedness of auxiliary operators:_ The operators $(BS,D)$ and $(BA(I-P),D)$
are bounded and there exist constants $c_{1},c_{2}<\infty$ such that

$\|BSf\|\leq c_{1}\|(I-P)f\|\quad\text{ and }\quad\|BA(I-P)f\|\leq
c_{2}\|(I-P)f\|$

hold for all $f\in D$.

We now state the central abstract hypocoercivity theorem as in [4]:

######  Theorem 2.2.

Assume that (D) and (H1)-(H4) hold. Then there exist strictly positive
constants $\kappa_{1},\kappa_{2}<\infty$ which are explicitly computable in
terms of $\Lambda_{m},\Lambda_{M},c_{1}$ and $c_{2}$ such that for all $g\in
H$ we have

$\|T_{t}g-(g,1)_{H}\|\leq\kappa_{1}\mathrm{e}^{-\kappa_{2}t}\|g-(g,1)_{H}\|\quad\text{
for all }t\geq 0.$

More specifically, if there exist $\delta>0$, $\varepsilon\in(0,1)$ and
$0<\kappa<\infty$ such that for all $g\in D(L)$, $t\geq 0$, it holds

$\displaystyle\kappa\|f_{t}\|^{2}\leq\left(\Lambda_{m}-\varepsilon(1+c_{1}+c_{2})\left(1+\frac{1}{2\delta}\right)\right)$
$\displaystyle\|(I-P)f_{t}\|^{2}$ (2.1)
$\displaystyle+\varepsilon\left(\frac{\Lambda_{M}}{1+\Lambda_{M}}-(1+c_{1}+c_{2})\frac{\delta}{2}\right)$
$\displaystyle\|Pf_{t}\|^{2},$

where
$f_{t}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=T_{t}g-(g,1)_{H}$,
then the constants $\kappa_{1}$ and $\kappa_{2}$ are given by

$\kappa_{1}=\sqrt{\frac{1+\varepsilon}{1-\varepsilon}},\qquad\kappa_{2}=\frac{\kappa}{1+\varepsilon}.$

In order to prove (H4), we will make use of the following result:

######  Lemma 2.3.

Assume (H3). Let $(T,D(T))$ be a linear operator with $D\subset D(T)$ and
assume $AP(D)\subset D(T^{*})$. Then

$(I-G)(D)\subset D((BT)^{*})\quad\text{ with
}\quad(BT)^{*}(I-G)f=T^{*}APf,\quad f\in D.$

If there exists some $C<\infty$ such that

$\|(BT)^{*}g\|\leq C\|g\|\qquad\text{ for all }g=(I-G)f,\quad f\in D,$ (2.2)

then $(BT,D(T))$ is bounded and its closure $(\overline{BT})$ is a bounded
operator on $H$ with $\|\overline{BT}\|=\|(BT)^{*}\|$.

In particular, if $(S,D(S))$ and $(A,D(A))$ satisfy these assumptions, the
corresponding inequalities in (H4) are satisfied with $c_{1}=\|(BS)^{*}\|$ and
$c_{2}=\|(BA)^{*}\|$.

######  Proof:

Let $h\in D((AP)^{*})$ and $f\in D$. Set $g=(I-G)f$. By the representation of
$B$ on $D((AP)^{*})$ together with self-adjointness of $(I+(AP)^{*}AP)^{-1}$
and $D\subset D(AP)$, we get

$(h,B^{*}g)_{H}=(Bh,(I-G)f)_{H}=((AP)^{*}h,f)_{H}=(h,APf)_{H}.$

So $B^{*}g=APf\in D(T^{*})$. By Lemma 2.1 (v),
$((BT)^{*},D((BT)^{*}))=(T^{*}B^{*},D(T^{*}B^{*}))$, which implies
$(BT)^{*}g=T^{*}B^{*}g=T^{*}APf$.

By essential self-adjointness and hence essential m-dissipativity of $G$,
$(I-G)(D)$ is dense in $H$. Therefore by (2.2), the closed operator
$((BT)^{*},D((BT)^{*}))$ is a bounded operator on $H$. Since $(BT,D(T))$ is
densely defined, by Lemma 2.1 (i) and (ii), it is closable with
$\overline{BT}=(BT)^{**}$, which is a bounded operator on $H$ with the stated
norm.

The last part follows directly by $Sf=S(I-P)f$ for $f\in D$. $\square$

##  3 Hypocoercivity for Langevin dynamics with multiplicative noise

As stated in the introduction, the aim of this section is to prove exponential
convergence to equilibrium of the semigroup solving the abstract Kolmogorov
equation corresponding to the Langevin equation with multiplicative noise
(1.1).

We remark that most of the conditions are verified analogously to [7], the
main difference being the proof of essential m-dissipativity for the operator
$(L,C_{c}^{\infty}(\mathbb{R}^{2d}))$ as well as the first inequality in (H4).
Nevertheless, some care has to be taken whenever $S$ is involved, as it
doesn’t preserve regularity to the same extent as in the given reference.

###  3.1 The data conditions

We start by introducing the setting and verifying the data conditions (D). The
notations introduced in this part will be used for the remainder of the
section without further mention.

Let $d\in\mathbb{N}$ and set the state space as $E=\mathbb{R}^{2d}$,
$\mathcal{F}=\mathcal{B}(\mathbb{R}^{2d})$. In the following, the first $d$
components of $E$ will be written as $x$, the latter $d$ components as $v$.
Let $\nu$ be the normalised Gaussian measure on $\mathbb{R}^{d}$ with mean
zero and covariance matrix $I$, i.e.

$\nu(A)=\int_{A}(2\pi)^{-\frac{d}{2}}\
\mathrm{e}^{-\frac{x^{2}}{2}}\,\mathrm{d}x.$

######  Assumption (P).

The potential $\Phi:\mathbb{R}^{d}\to\mathbb{R}$ is assumed to depend only on
the position variable $x$ and to be locally Lipschitz-continuous. We further
assume $\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x$ to be a probability measure on
$(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$.

Note that the first part implies $\Phi\in
H_{\text{loc}}^{1,\infty}(\mathbb{R}^{d})$. Moreover, $\Phi$ is differentiable
$\mathrm{d}x$-a.e. on $\mathbb{R}^{d}$, such that the weak gradient and the
derivative of $\Phi$ coincide $\mathrm{d}x$-a.e. on $\mathbb{R}^{d}$. In the
following, we fix a version of $\nabla\Phi$.

The probability measure $\mu$ on $(E,\mathcal{F})$ is then given by
$\mu=\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x\otimes\nu$, and we set
$H\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=L^{2}(E,\mu)$, which
satisfies condition (D1). Next we assume the following about
$\Sigma=(a_{ij})_{1\leq i,j\leq d}$ with $a_{ij}:\mathbb{R}^{d}\to\mathbb{R}$:

######  Assumption ($\Sigma$1).

$\Sigma$ is symmetric and uniformly strictly elliptic, i.e. there is some
$c_{\Sigma}>0$ such that

$(y,\Sigma(v)y)\geq c_{\Sigma}\cdot|y|^{2}\quad\text{ for all
}y,v\in\mathbb{R}^{d}.$

######  Assumption ($\Sigma$2).

There is some $p>d$ such that for all $1\leq i,j\leq d$, it holds that
$a_{ij}\in H_{\text{loc}}^{1,p}(\mathbb{R}^{d},\nu)\cap
L^{\infty}(\mathbb{R}^{d})$. Additionally, $a_{ij}$ is locally Lipschitz-
continuous for all $1\leq i,j\leq d$.

Additionally, we will consider one of the following conditions on the growth
of the partial derivatives:

######  Assumption ($\Sigma$3).

There are constants $0\leq M<\infty$, $-\infty<\beta\leq 0$ such that for all
$1\leq i,j,k\leq d$

$|\partial_{k}a_{ij}(v)|\leq M(1+|v|)^{\beta}\quad\text{ for $\nu$-almost all
}v\in\mathbb{R}^{d}.$

######  Assumption ($\Sigma$3′).

There are constants $0\leq M<\infty$, $0<\beta<1$ such that for all $1\leq
i,j,k\leq d$

$|\partial_{k}a_{ij}(v)|\leq
M(\mathds{1}_{B_{1}(0)}(v)+|v|^{\beta})\quad\text{ for $\nu$-almost all
}v\in\mathbb{R}^{d}.$

We note that any of these imply $\partial_{j}a_{ij}\in
L^{2}(\mathbb{R}^{d},\nu)$ for all $1\leq i,j\leq d$.

######  Definition 3.1.

Let $\Sigma$ satisfy ($\Sigma$2). Then we set

$\displaystyle M_{\Sigma}$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\max\\{\|a_{ij}\|_{\infty}:1\leq
i,j\leq d\\}\qquad\text{ and }$ $\displaystyle B_{\Sigma}$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\max\\{|\partial_{j}a_{ij}(v)|:v\in\overline{B_{1}(0)},\
1\leq i,j\leq d\\}.$

If $\Sigma$ additionally satisfies ($\Sigma$3), then we define

$N_{\Sigma}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sqrt{M_{\Sigma}^{2}+(B_{\Sigma}\vee
M)^{2}}.$

If instead ($\Sigma$3′) is fulfilled, then we consider instead

$N_{\Sigma}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sqrt{M_{\Sigma}^{2}+B_{\Sigma}^{2}+dM^{2}}.$

######  Definition 3.2.

Let $D=C_{c}^{\infty}(E)$ be the space of compactly supported smooth functions
on $E$. We define the linear operators $S,A$ and $L$ on $D$ via

$\displaystyle Sf$
$\displaystyle=\sum_{i,j=1}^{d}a_{ij}\partial_{v_{j}}\partial_{v_{i}}f+\sum_{i=1}^{d}b_{i}\partial_{v_{i}}f,$
$\displaystyle\quad\text{ where
}b_{i}(v)=\sum_{j=1}^{d}(\partial_{j}a_{ij}(v)-a_{ij}(v)v_{j}),$
$\displaystyle Af$
$\displaystyle=\nabla\Phi(x)\cdot\nabla_{v}f-v\cdot\nabla_{x}f,$
$\displaystyle Lf$ $\displaystyle=(S-A)f,\qquad\text{ for }f\in D.$

Integration by parts shows that $(S,D)$ is symmetric and non-positive definite
on $H$, and $(A,D)$ is antisymmetric on $H$. Hence, all three operators with
domain $D$ are dissipative and therefore closable. We denote their closure
respectively by $(S,D(S)),(A,D(A))$ and $(L,D(L))$.

For $f\in D$ and $g\in H^{1,2}(E,\mu)$, integration by parts yields

$(Lf,g)_{H}=-\int_{E}\left(\nabla f,\begin{pmatrix}0&-I\\\
I&\Sigma\end{pmatrix}\nabla g\right)_{\mathrm{euc}}\,\mathrm{d}\mu.$

In particular, (D6) is obviously fulfilled. Next we provide an estimate which
will be needed later:

######  Proposition 3.3.

Let ($\Sigma$2) and either ($\Sigma$3) or ($\Sigma$3′) hold respectively and
recall Definition 3.1. Then for all $1\leq i,j\leq d$, it holds that

$\|\partial_{j}a_{ij}-a_{ij}v_{j}\|_{L^{2}(\nu)}\leq N_{\Sigma}.$

######  Proof:

Due to integration by parts, it holds that

$\int_{\mathbb{R}^{d}}a_{ij}^{2}v_{j}^{2}\,\mathrm{d}\nu=\int_{\mathbb{R}^{d}}a_{ij}^{2}+2a_{ij}v_{j}\partial_{j}a_{ij}\,\mathrm{d}\nu.$

Hence we obtain in the case ($\Sigma$3′)

$\displaystyle\int_{\mathbb{R}^{d}}(\partial_{j}a_{ij}-a_{ij}v_{j})^{2}\,\mathrm{d}\nu$
$\displaystyle=\int_{\mathbb{R}^{d}}(\partial_{j}a_{ij})^{2}+a_{ij}^{2}\,\mathrm{d}\nu$
$\displaystyle\leq\int_{B_{1}(0)}(\partial_{j}a_{ij})^{2}\,\mathrm{d}\nu+\int_{\mathbb{R}^{d}\setminus
B_{1}(0)}(\partial_{j}a_{ij})^{2}\,\mathrm{d}\nu+M_{\Sigma}^{2}$
$\displaystyle\leq B_{\Sigma}^{2}+\int_{\mathbb{R}^{d}\setminus
B_{1}(0)}(M|v|^{\beta})^{2}\,\mathrm{d}\nu+M_{\Sigma}^{2}$ $\displaystyle\leq
B_{\Sigma}^{2}+M_{\Sigma}^{2}+\sum_{k=1}^{d}M^{2}\int_{\mathbb{R}^{d}}v_{k}^{2}\,\mathrm{d}\nu=B_{\Sigma}^{2}+M_{\Sigma}^{2}+M^{2}d.$

The case ($\Sigma$3) follows from $(\partial_{j}a_{ij})^{2}\leq(B_{\Sigma}\vee
M)^{2}$. $\square$

We now state the essential m-dissipativity result, which will be proven in the
next section.

######  Theorem 3.4.

Let ($\Sigma$1), ($\Sigma$2) and either ($\Sigma$3) or ($\Sigma$3′) be
fulfilled, and let $\Phi$ be as in (P). Assume further that $\Phi$ is bounded
from below and that $|\nabla\Phi|\in
L^{2}(\mathbb{R}^{d},\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)$. If $\beta$ is
larger than $-1$, then assume additionally that there is some $N<\infty$ such
that

$|\nabla\Phi(x)|\leq N(1+|x|^{\gamma}),\quad\text{ where
}\gamma<\frac{2}{1+\beta}.$

Then the linear operator $(L,C_{c}^{\infty}(\mathbb{R}^{2d}))$ is essentially
m-dissipative and hence its closure $(L,D(L))$ generates a strongly continuous
contraction semigroup on $H$. In particular, the conditions (D2)-(D4) are
satisfied.

Let us now introduce the orthogonal projections $P_{S}$ and $P$:

######  Definition 3.5.

Define $P_{S}:H\to H$ as

$P_{S}f=\int_{\mathbb{R}^{d}}f\,\mathrm{d}\nu(v),\qquad f\in H,$

where integration is understood w.r.t the velocity variable $v$. By Fubini’s
theorem and the fact that $\nu$ is a probability measure on $(E,\mathcal{F})$,
it follows that $P_{S}$ is a well-defined orthogonal projection on $H$ with

$P_{S}f\in
L^{2}(\mathbb{R}^{d},\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x),\quad\|P_{S}f\|_{L^{2}(\mathbb{R}^{d},\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)}=\|P_{S}f\|_{H},\quad
f\in H,$

where $L^{2}(\mathbb{R}^{d},\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)$ is
interpreted as embedded in $H$.

Then define $P:H\to H$ via $Pf=P_{S}f-(f,1)_{H}$ for $f\in H$. Again, $P$ is
an orthogonal projection on $H$ with

$Pf\in
L^{2}(\mathbb{R}^{d},\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x),\quad\|Pf\|_{L^{2}(\mathbb{R}^{d},\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)}=\|Pf\|_{H},\quad
f\in H.$

Additionally, for each $f\in D$, $P_{S}f$ admits a unique representation in
$C_{c}^{\infty}(\mathbb{R}^{d})$, which we will denote by $f_{S}\in
C_{c}^{\infty}(\mathbb{R}^{d})$.

In order to show the last remaining conditions (D5) and (D7), we will make use
of a standard sequence of cutoff functions as specified below:

######  Definition 3.6.

Let $\varphi\in C_{c}^{\infty}(\mathbb{R}^{d})$ such that $0\leq\varphi\leq
1$, $\varphi=1$ on $B_{1}(0)$ and $\varphi=0$ outside of $B_{2}(0)$. Define
$\varphi_{n}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varphi(\frac{z}{n})$
for each $z\in\mathbb{R}^{d}$, $n\in\mathbb{N}$. Then there exists a constant
$C<\infty$ independent of $n\in\mathbb{N}$ such that

$|\partial_{i}\varphi_{n}(z)|\leq\frac{C}{n},\
|\partial_{ij}\varphi_{n}(z)|\leq\frac{C}{n^{2}}\quad\text{ for all
}z\in\mathbb{R}^{d},1\leq i,j\leq d.$

Moreover $0\leq\varphi_{n}\leq 1$ for all $n\in\mathbb{N}$ and $\varphi_{n}\to
1$ pointwisely on $\mathbb{R}^{d}$ as $n\to\infty$.

######  Lemma 3.7.

Let ($\Sigma$2) and either ($\Sigma$3) or ($\Sigma$3′) be fulfilled, and let
$\Phi$ be as in (P). Then the operator $L$ satisfies the following:

  1. ()

$P(H)\subset D(S)$ with $SPf=0$ for all $f\in H$,

  2. ()

$P(D)\subset D(A)$ and $APf=-v\cdot\nabla_{x}(P_{S}f)$,

  3. ()

$AP(D)\subset D(A)$ with $A^{2}Pf=\langle
v,\nabla_{x}^{2}(P_{S}f)v\rangle-\nabla\Phi\cdot\nabla_{x}(P_{S}f)$.

  4. ()

It holds $1\in D(L)$ and $L1=0$.

In particular, (D5) and (D7) are fulfilled.

######  Proof:

We only show (i), as the other parts can be shown exactly as in [7]. First,
let $f\in C_{c}^{\infty}(\mathbb{R}^{d})$ and define $f_{n}\in D$ via
$f_{n}(x,y)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=f(x)\varphi_{n}(v)$.
Then by Lebesgue’s dominated convergence theorem and the inequalities in the
previous definition,

$Sf_{n}=f\cdot\left(\sum_{i,j=1}^{d}a_{ij}\partial_{ij}\varphi_{n}+\sum_{i=1}^{d}b_{i}\partial_{i}\varphi_{n}\right)\to
0\quad\text{ in $H$ as }n\to\infty,$

since $a_{ij}\in L^{\infty}(\mathbb{R}^{d})\subset L^{2}(\mathbb{R}^{d},\nu)$,
$|v|\in L^{2}(\mathbb{R}^{d},\nu)$ and $\partial_{j}a_{ij}\in
L^{2}(\mathbb{R}^{d},\nu)$ for all $1\leq i,j\leq d$.

Since $f_{n}\to f$ in $H$ and by closedness of $(S,D(S))$, this implies $f\in
D(S)$ with $Sf=0$, where $f$ is interpreted as an element of $H$.

Now let $g\in P(H)$ and identify $g$ as an element of
$L^{2}(\mathbb{R}^{d},\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)$. Then there exist
$g_{n}\in C_{c}^{\infty}(\mathbb{R}^{d})$ with $g_{n}\to g$ in
$L^{2}(\mathbb{R}^{d},\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)$ as $n\to\infty$.
Identifying all $g_{n}$ and $g$ with elements in $H$ then yields $g_{n}\to g$
in $H$ as $n\to\infty$ and $g_{n}\in D(S)$, $Sg_{n}=0$ for all
$n\in\mathbb{N}$. Therefore, again by closedness of $(S,D(S))$, $g\in D(S)$
and $Sg=0$. $\square$

###  3.2 The hypocoercivity conditions

Now we verify the hypocoercivity conditions (H1)-(H4) for the operator $L$.
From here on, we will assume $\Sigma$ to satisfy ($\Sigma$1), ($\Sigma$2) and
either ($\Sigma$3) or ($\Sigma$3′), with $N_{\Sigma}$ referring to the
appropriate constant as in Definition 3.1. Analogously to [7] we introduce the
following conditions:

######  Hypocoercivity assumptions (C1)-(C3).

We require the following assumptions on $\Phi:\mathbb{R}^{d}\to\mathbb{R}$:

  1. (C1)

The potential $\Phi$ is bounded from below, is an element of
$C^{2}(\mathbb{R}^{d})$ and $\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x$ is a
probability measure on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$.

  2. (C2)

The probability measure $\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x$ satisfies a
Poincaré inequality of the form

$\|\nabla
f\|_{L^{2}(\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)}^{2}\geq\Lambda\|f-(f,1)_{L^{2}(\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)}\|_{L^{2}(\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)}^{2}$

for some $\Lambda\in(0,\infty)$ and all $f\in C_{c}^{\infty}(\mathbb{R}^{d})$.

  3. (C3)

There exists a constant $c<\infty$ such that

$|\nabla^{2}\Phi(x)|\leq c(1+|\nabla\Phi(x)|)\quad\text{ for all
}x\in\mathbb{R}^{d}.$

Note that in particular, (C1) implies (P). As shown in [2, Lemma A.24],
conditions (C3) and (C1) imply $\nabla\Phi\in
L^{2}(\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)$.

Since we only change the operator $(S,D(S))$ in comparison to the framework of
[7], the results stated there involving only $(A,D(A))$ and the projections
also hold here and are collected as follows:

######  Proposition 3.8.

Let $\Phi$ satisfy (P). Then the following hold:

  1. (i)

Assume additionally $\nabla\Phi\in L^{2}(\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)$.
Then (H1) is fulfilled.

  2. (ii)

Assume that $\Phi$ satisfies (C1) and that $\nabla\Phi\in
L^{2}(\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)$. Then the operator $(G,D)$ defined
by $G\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=PA^{2}P$ is
essentially self-adjoint, equivalently essentially m-dissipative. For $f\in
D$, it holds

$Gf=PAAPf=\Delta f_{S}-\nabla\Phi\cdot\nabla f_{S}.$

  3. (iii)

Assume that $\Phi$ satisfies (C1) and (C2) as well as $\nabla\Phi\in
L^{2}(\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)$. Then (H3) holds with
$\Lambda_{M}=\Lambda$.

  4. (iv)

Assume that $\Phi$ satisfies (C1)-(C3). Then the second estimate in (H4) is
satisfied, and the constant there is given as $c_{2}=c_{\Phi}\in[0,\infty)$,
which only depends on the choice of $\Phi$.

It remains to show (H2) and the first half of (H4):

######  Proposition 3.9.

Let $\Phi$ be as in (P). Then Condition (H2) is satisfied with
$\Lambda_{m}=c_{\Sigma}$.

######  Proof:

Let $g\in C_{c}^{\infty}(\mathbb{R}^{d})$. The Poincaré inequality for
Gaussian measures, see for example [13], states

$\|\nabla
g\|_{L^{2}(\nu)}^{2}\geq\left\|g-\int_{\mathbb{R}^{d}}g(v)\,\mathrm{d}\nu(v)\right\|_{L^{2}(\nu)}^{2}.$

Therefore, integration by parts yields for all $f\in D$:

$\displaystyle(-Sf,f)_{H}$
$\displaystyle=\int_{E}\langle\nabla_{v}f,\Sigma\nabla_{v}f\rangle\,\mathrm{d}\mu\geq\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}c_{\Sigma}|\nabla_{v}f(x,v)|^{2}\,\mathrm{d}\nu\,\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x$
$\displaystyle\geq
c_{\Sigma}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}(f-P_{S}f)^{2}\,\mathrm{d}\nu\,\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x=c_{\Sigma}\|(I-P_{S})f\|_{H}^{2}$

$\square$

Finally, we verify the first part of (H4):

######  Proposition 3.10.

Assume that $\Phi$ satisfies (C1) and (C2) as well as $\nabla\Phi\in
L^{2}(\mathrm{e}^{-\Phi}\,\mathrm{d}x)$. Then the first inequality of (H4) is
also satisfied with
$c_{1}=d_{\Sigma}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sqrt{2d^{3}}N_{\Sigma}$.

######  Proof:

For $f\in D$, define $Tf\in H$ by

$Tf\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{i=1}^{d}b_{i}\partial_{i}(f_{S})=\sum_{i,j=1}^{d}(\partial_{j}a_{ij}-a_{ij}v_{j})\partial_{x_{i}}(P_{S}f).$

We want to apply Lemma 2.3 to the operator $(S,D(S))$. Let $f\in D$, $h\in
D(S)$ and $h_{n}\in D$ such that $h_{n}\to h$ and $Sh_{n}\to Sh$ in $H$ as
$n\to\infty$. Then, by integration by parts,

$(Sh,APf)_{H}=\lim_{n\to\infty}(Sh_{n},-v\cdot\nabla_{x}(P_{S}f))_{H}=\lim_{n\to\infty}(h_{n},-Tf)_{H}=(h,-Tf)_{H}.$

This shows $APf\in D(S^{*})$ and by the first part of Lemma 2.3, $(I-G)f\in
D((BS)^{*})$ and $(BS)^{*}(I-G)f=S^{*}APf=-Tf$. Now set $g=(I-G)f$, then, via
Proposition 3.3,

$\displaystyle\|(BS)^{*}g\|_{H}^{2}$
$\displaystyle=\|Tf\|_{H}^{2}=\int_{E}\left(\sum_{i=1}^{d}b_{i}\partial_{i}f_{S}\right)^{2}\,\mathrm{d}\mu$
$\displaystyle\leq
d^{2}\sum_{i,j=1}^{d}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}(\partial_{j}a_{ij}(v)-a_{ij}(v)v_{j})^{2}\
\mathrm{d}\nu(v)\ (\partial_{x_{i}}(P_{S}f)(x))^{2}\,\mathrm{e}^{-\Phi(x)}\
\mathrm{d}x$ $\displaystyle\leq
d^{3}N_{\Sigma}^{2}\sum_{i=1}^{d}\int_{\mathbb{R}^{d}}\partial_{x_{i}}(Pf)\cdot\partial_{x_{i}}(P_{S}f)\,\mathrm{e}^{-\Phi(x)}\
\mathrm{d}x.$

A final integration by parts then yields

$\displaystyle\|(BS)^{*}g\|_{H}^{2}$
$\displaystyle\leq-d^{3}N_{\Sigma}^{2}\int_{\mathbb{R}^{d}}Pf\cdot(\Delta_{x}P_{S}f-\nabla\Phi\nabla_{x}(P_{S}f))\,\mathrm{e}^{-\Phi(x)}\
\mathrm{d}x$ $\displaystyle=-d^{3}N_{\Sigma}^{2}\int_{\mathbb{R}^{d}}Pf\cdot
Gf\,\mathrm{e}^{-\Phi(x)}\ \mathrm{d}x$ $\displaystyle\leq
d^{3}N_{\Sigma}^{2}\,\|Pf\|_{L^{2}(\text{e}^{-\Phi(x)}\,\mathrm{d}x)}\cdot\|Gf\|_{L^{2}(\text{e}^{-\Phi(x)}\,\mathrm{d}x)}$
$\displaystyle\leq d^{3}N_{\Sigma}^{2}\,\|Pf\|_{H}(\|(I-G)f\|_{H}+\|f\|_{H})$
$\displaystyle\leq 2d^{3}N_{\Sigma}^{2}\,\|g\|_{H}^{2},$

where the last inequality is due to dissipativity of $(G,D)$. $\square$

######  Proof (of Theorem 1.1):

Under the given assumptions, all conditions (C1)-(C3), ($\Sigma$1),
($\Sigma$2) and either ($\Sigma$3) or ($\Sigma$3′) are satisfied. Therefore
hypocoercivity follows by the previous propositions and Theorem 2.2. It
remains to show the stated convergence rate, which will be done as in [7] or
[14] using the determined values for $c_{1}$, $c_{2}$, $\Lambda_{M}$ and
$\Lambda_{m}$. Fix

$\delta\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{\Lambda}{1+\Lambda}\frac{1}{1+c_{\Phi}+d_{\Sigma}}.$

Then the coefficients on the right hand side of (2.1) can be written as
$c_{\Sigma}-\varepsilon r_{\Phi}(N_{\Sigma})$ and $\varepsilon s_{\Phi}$
respectively, where

$\displaystyle r_{\Phi}(N_{\Sigma})$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(1+c_{\Phi}+\sqrt{2d^{3}}N_{\Sigma})\left(1+\frac{1+\Lambda}{2\Lambda}(1+c_{\Phi}+\sqrt{2d^{3}}N_{\Sigma})\right)\quad\text{
and }$ $\displaystyle s_{\Phi}$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{2}\frac{\Lambda}{1+\Lambda}.$

and $\varepsilon=\varepsilon_{\Phi}(\Sigma)\in(0,1)$ still needs to be
determined. Write $r_{\Phi}(N_{\Sigma})+s_{\Phi}$ as the polynomial

$r_{\Phi}(N_{\Sigma})+s_{\Phi}=a_{1}+a_{2}N_{\Sigma}+a_{3}N_{\Sigma}^{2},$

where all $a_{i}\in(0,\infty)$, $i=1,\dots,3$ depend on $\Phi$. Then define

$\tilde{\varepsilon}_{\Phi}(N_{\Sigma})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{N_{\Sigma}}{r_{\Phi}(N_{\Sigma})+s_{\Phi}}=\frac{N_{\Sigma}}{a_{1}+a_{2}N_{\Sigma}+a_{3}N_{\Sigma}^{2}}.$

Some rough estimates show $\tilde{\varepsilon}_{\Phi}(N_{\Sigma})\in(0,1)$.
Now let $v>0$ be arbitrary and set

$\varepsilon\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{v}{1+v}\frac{c_{\Sigma}}{N_{\Sigma}}\tilde{\varepsilon}_{\Phi}(N_{\Sigma})\in(0,1).$

Then $\varepsilon r_{\Phi}(N_{\Sigma})+\varepsilon
s_{\Phi}=\frac{v}{1+v}c_{\Sigma}<c_{\Sigma}$, hence we get the estimate

$c_{\Sigma}-\varepsilon r_{\Phi}(N_{\Sigma})>\varepsilon
s_{\Phi}=\frac{v}{1+v}\frac{2c_{\Sigma}}{n_{1}+n_{2}N_{\Sigma}+n_{3}N_{\Sigma}^{2}}=\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}\kappa,$

where all $n_{i}\in(0,\infty)$ depend on $\Phi$ and are given by

$n_{i}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{2}{s_{\Phi}}a_{i},\qquad\text{
for each }i=1,\dots,3.$

Clearly, $\kappa$, $\varepsilon$ and $\delta$ now solve (2.1) and the
convergence rate coefficients are given via Theorem 2.2 by

$\displaystyle\kappa_{1}$
$\displaystyle=\sqrt{\frac{1+\varepsilon}{1-\varepsilon}}=\sqrt{\frac{1+v+\frac{c_{\Sigma}}{N_{\Sigma}}\tilde{\varepsilon}_{\Phi}(N_{\Sigma})v}{1+v-\frac{c_{\Sigma}}{N_{\Sigma}}\tilde{\varepsilon}_{\Phi}(N_{\Sigma})v}}\leq\sqrt{1+2v+v^{2}}=1+v\quad\text{
and }$ $\displaystyle\kappa_{2}$
$\displaystyle=\frac{\kappa}{1+\varepsilon}>\frac{1}{2}\kappa$

Hence, by choosing $\theta_{1}=1+v$ and
$\theta_{2}=\frac{1}{2}\kappa=\frac{\theta_{1}-1}{\theta_{1}}\frac{c_{\Sigma}}{n_{1}+n_{2}N_{\Sigma}+n_{3}N_{\Sigma}^{2}}$,
the rate of convergence claimed in the theorem is shown. $\square$

######  Remark 3.11.

We remark here that all previous considerations up to the explicit rate of
convergence can also be applied to the formal adjoint operator $(L^{*},D)$
with $L^{*}=S+A$, the closure of which generates the adjoint semigroup
$(T_{t}^{*})_{t\geq 0}$ on $H$. For example, the perturbation procedure to
prove essential m-dissipativity is exactly the same as for $L$, since the sign
of $A$ does not matter due to antisymmetry. We can use this to construct
solutions to the corresponding Fokker-Planck PDE associated with our Langevin
dynamics, see Section 5.3.

##  4 Essential m-dissipativity of the Langevin operator

The goal of this section is to prove Theorem 3.4. We start by giving some
basics on perturbation of semigroup generators.

###  4.1 Basics on generators and perturbation

######  Definition 4.1.

Let $(A,D(A))$ and $(B,D(B))$ be linear operators on $H$. Then $B$ is said to
be _$A$ -bounded_ if $D(A)\subset D(B)$ and there exist constants $a,b<\infty$
such that

$\|Bf\|_{H}\leq a\|Af\|_{H}+b\|f\|_{H}$ (4.1)

holds for all $f\in D(A)$. The number $\inf\\{a\in\mathbb{R}\mid\text{
\eqref{eq:a-bound} holds for some }b<\infty\\}$ is called the _$A$ -bound_ of
$B$.

######  Theorem 4.2.

Let $D\subset H$ be a dense linear subspace, $(A,D)$ be an essentially
m-dissipative linear operator on $H$ and let $(B,D)$ be dissipative and
$A$-bounded with $A$-bound strictly less than $1$. Then $(A+B,D)$ is
essentially m-dissipative and its closure is given by
$(\overline{A}+\overline{B},D(\overline{A}))$.

A useful criterion for verifying $A$-boundedness is given by:

######  Lemma 4.3.

Let $D\subset H$ be a dense linear subspace, $(A,D)$ be essentially
m-dissipative and $(B,D)$ be dissipative. Assume that there exist constants
$c,d<\infty$ such that

$\|Bf\|_{H}^{2}\leq c(Af,f)_{H}+d\|f\|_{H}^{2}$

holds for all $f\in D$. Then $B$ is $A$-bounded with $A$-bound $0$.

We also require the following generalization of the perturbation method:

######  Lemma 4.4.

Let $D\subset H$ be a dense linear subspace, $(A,D)$ be essentially
m-dissipative and $(B,D)$ be dissipative on $H$. Assume that there exists a
complete orthogonal family $(P_{n})_{n\in\mathbb{N}}$, i.e. each $P_{n}$ is an
orthogonal projection, $P_{n}P_{m}=0$ for all $n\neq m$ and
$\sum_{n\in\mathbb{N}}P_{n}=I$ strongly, such that

$P_{n}(D)\subset D,\qquad P_{n}A=AP_{n},\quad\text{ and }\quad P_{n}B=BP_{n}$

for all $n\in\mathbb{N}$. Set
$A_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=AP_{n}$,
$B_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=BP_{n}$, both
with domain
$D_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=P_{n}(D)$, as
operators on $P_{n}(H)$. Assume that each $B_{n}$ is $A_{n}$-bounded with
$A_{n}$-bound strictly less than $1$. Then $(A+B,D)$ is essentially
m-dissipative.

###  4.2 The symmetric part

We first prove essential self-adjointness, equivalently essential
m-dissipativity, for a certain class of symmetric differential operators on
specific Hilbert spaces. This is essentially a combination of two results by
Bogachev, Krylov, and Röckner, namely [10, Corollary 2.10] and [12, Theorem
7], however, the combined statement does not seem to be well known and might
hold interest as the basis for similar m-dissipativity proofs. We use the
slightly more general statement from [11, Theorem 5.1] in order to relax the
assumptions.

######  Theorem 4.5.

Let $d\geq 2$ and consider $H=L^{2}(\mathbb{R}^{d},\mu)$ where
$\mu=\rho\,\mathrm{d}x$, $\rho=\varphi^{2}$ for some $\varphi\in
H_{\mathrm{loc}}^{1,2}(\mathbb{R}^{d})$ such that $\frac{1}{\rho}\in
L_{\mathrm{loc}}^{\infty}(\mathbb{R}^{d})$. Let $A=(a_{ij})_{1\leq i,j\leq
d}:\mathbb{R}^{d}\to\mathbb{R}^{d\times d}$ be symmetric and locally strictly
elliptic with $a_{ij}\in L^{\infty}(\mathbb{R}^{d})$ for all $1\leq i,j\leq
d$. Assume there is some $p>d$ such that $a_{ij}\in
H_{\mathrm{loc}}^{1,p}(\mathbb{R}^{d})$ for all $1\leq i,j\leq d$ and that
$|\nabla\rho|\in L_{\mathrm{loc}}^{p}(\mathbb{R}^{d})$. Consider the bilinear
form $(B,D)$ given by $D=C_{c}^{\infty}(\mathbb{R}^{d})$ and

$B(f,g)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(\nabla
f,A\nabla g)_{H}=\int_{\mathbb{R}^{d}}(\nabla f(x),A(x)\nabla
g(x))_{\mathrm{euc}}\,\rho(x)\,\mathrm{d}x,\qquad f,g\in D.$

Define further the linear operator $(S,D)$ via

$Sf\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{i,j=1}^{d}a_{ij}\partial_{j}\partial_{i}f+\sum_{i=1}^{d}b_{i}\partial_{i}f,\qquad
f\in D,$

where
$b_{i}=\sum_{j=1}^{d}(\partial_{j}a_{ij}+a_{ij}\frac{\partial_{j}\rho}{\rho})\in
L_{\mathrm{loc}}^{p}(\mathbb{R}^{d})$, so that $B(f,g)=(-Sf,g)_{H}$. Then
$(S,D)$ is essentially self-adjoint on $H$.

######  Proof:

Analogously to the proof of [12, Theorem 7], it can be shown that $\rho$ is
continuous, hence locally bounded. Assume that there is some $g\in H$ such
that

$\int_{\mathbb{R}^{d}}(S-I)f(x)\cdot
g(x)\cdot\rho(x)\,\mathrm{d}x=0\quad\text{ for all }f\in D.$ (4.2)

Define the locally finite signed Borel measure $\nu$ via
$\nu=g\rho\,\mathrm{d}x$, which is then absolutely continuous with respect to
the Lebesgue measure. By definition it holds that

$\int_{\mathbb{R}^{d}}\left(\sum_{i,j=1}^{d}a_{ij}\partial_{j}\partial_{i}f+\sum_{i=1}^{d}b_{i}\partial_{i}f-f\right)\,\mathrm{d}\nu=0\quad\text{
for all }f\in D,$

so by [11, Theorem 5.1], the density $g\cdot\rho$ of $\nu$ is in
$H_{\mathrm{loc}}^{1,p}(\mathbb{R}^{d})$ and locally Hölder continuous, hence
locally bounded. This implies $g=g\rho\cdot\frac{1}{\rho}\in
L_{\mathrm{loc}}^{p}(\mathbb{R}^{d})\cap
L_{\mathrm{loc}}^{\infty}(\mathbb{R}^{d})$ and $\nabla
g=\nabla(g\rho)\cdot\frac{1}{\rho}-(g\rho)\frac{\nabla\rho}{\rho^{2}}\in
L_{\mathrm{loc}}^{p}(\mathbb{R}^{d})$. Hence $g\in
H_{\mathrm{loc}}^{1,p}(\mathbb{R}^{d})$, is locally bounded, and $g\cdot
b_{i}\in L_{\mathrm{loc}}^{p}(\mathbb{R}^{d})$ for all $1\leq i\leq d$.
Therefore, we can apply integration by parts to (4.2) and get for every $f\in
D$:

$\displaystyle 0$
$\displaystyle=-\sum_{i,j=1}^{d}(a_{ij}\partial_{i}f,\partial_{j}g)_{H}-\sum_{i=1}^{d}(\partial_{i}f,b_{i}g)_{H}+\sum_{i=1}^{d}(\partial_{i}f,b_{i}g)_{H}-(f,g)_{H}$
(4.3) $\displaystyle=-\int_{\mathbb{R}^{d}}(\nabla f,A\nabla
g)_{\mathrm{euc}}\,\mathrm{d}\mu-(f,g)_{H}.$

Note that this equation can then be extended to all $f\in
H^{1,2}(\mathbb{R}^{d})$ with compact support, since $p>2$ by definition. Now
let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and set $\eta=\psi g\in
H^{1,2}(\mathbb{R}^{d})$, which has compact support. The same then holds for
$f\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\psi\eta\in
H^{1,2}(\mathbb{R}^{d})$. Elementary application of the product rule yields

$(\nabla\eta,A\nabla(\psi g))_{\mathrm{euc}}=(\nabla f,A\nabla
g)_{\mathrm{euc}}-\eta(\nabla\psi,A\nabla
g)_{\mathrm{euc}}+g(\nabla\eta,A\nabla\psi)_{\mathrm{euc}}.$ (4.4)

From now on, for $a,b:\mathbb{R}^{d}\to\mathbb{R}^{d}$, let $(a,b)$ always
denote the evaluation of the Euclidean inner product $(a,b)_{\mathrm{euc}}$.
By using (4.4) and applying (4.3) to $f$, we get

$\displaystyle\int_{\mathbb{R}^{d}}$ $\displaystyle(\nabla(\psi
g),A\nabla(\psi g))\,\mathrm{d}\mu+\int_{\mathbb{R}^{d}}(\psi
g)^{2}\,\mathrm{d}\mu=\int_{\mathbb{R}^{d}}(\nabla\eta,A\nabla(\psi
g))\,\mathrm{d}\mu+\int_{\mathbb{R}^{d}}\eta\psi g\,\mathrm{d}\mu$
$\displaystyle=\int_{\mathbb{R}^{d}}(\nabla f,A\nabla
g)\,\mathrm{d}\mu-\int_{\mathbb{R}^{d}}\eta(\nabla\psi,A\nabla
g)\,\mathrm{d}\mu+\int_{\mathbb{R}^{d}}g(\nabla\eta,A\nabla\psi)\,\mathrm{d}\mu+\int_{\mathbb{R}^{d}}fg\,\mathrm{d}\mu$
$\displaystyle=-\int_{\mathbb{R}^{d}}\psi g(\nabla\psi,A\nabla
g)\,\mathrm{d}\mu+\int_{\mathbb{R}^{d}}g(\nabla(\psi
g),A\nabla\psi)\,\mathrm{d}\mu$
$\displaystyle=\int_{\mathbb{R}^{d}}g^{2}(\nabla\psi,A\nabla\psi)\,\mathrm{d}\mu,$

where the last step follows from the product rule and symmetry of $A$. Since
$A$ is locally strictly elliptic, there is some $c>0$ such that

$0\leq\int_{\mathbb{R}^{d}}c(\nabla(\psi g),\nabla(\psi
g))\,\mathrm{d}\mu\leq\int_{\mathbb{R}^{d}}(\nabla(\psi g),A\nabla(\psi
g))\,\mathrm{d}\mu$

and therefore it follows that

$\int_{\mathbb{R}^{d}}(\psi
g)^{2}\,\mathrm{d}\mu\leq\int_{\mathbb{R}^{d}}g^{2}(\nabla\psi,A\nabla\psi)\,\mathrm{d}\mu.$
(4.5)

Let $(\psi_{n})_{n\in\mathbb{N}}$ be as in Definition 3.6. Then (4.5) holds
for all $\psi=\psi_{n}$. By dominated convergence, the left part converges to
$\|g\|_{H}^{2}$ as $n\to\infty$. The integrand of the right hand side term is
dominated by $d^{2}C^{2}M\cdot g^{2}\in L^{1}(\mu)$, where $C$ is from Def.
3.6 and $M\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\max_{1\leq
i,j\leq d}\|a_{ij}\|_{\infty}$. By definition of the $\psi_{n}$, that
integrand converges pointwisely to zero as $n\to\infty$, so again by dominated
convergence it follows that $g=0$ in $H$.

This implies that $(S-I)(D)$ is dense in $H$ and therefore that $(S,D)$ is
essentially self-adjoint. $\square$

######  Remark 4.6.

The above theorem also holds for $d=1$, as long as $p\geq 2$. Indeed,
continuity of $\rho$ follows from similar regularity estimates, see [12,
Remark 2]. The proof of [11, Theorem 5.1] mirrors the proof of [10, Theorem
2.8], where $d\geq 2$ is used to apply [10, Theorem 2.7]. However, in the
cases where it is applied, this distinction is not necessary (since
$p^{\prime}<q$ always holds). Finally, the extension of (4.3) requires $p\geq
2$.

We use this result to prove essential m-dissipativity of the symmetric part
$(S,D)$ of our operator $L$:

######  Theorem 4.7.

Let $H,D$ and the operator $S$ be defined as in Section 3.1. Then $(S,D)$ is
essentially m-dissipative on $H$. Its closure $(S,D(S))$ generates a sub-
Markovian strongly continuous contraction semigroup on $H$.

######  Proof:

Define the operator $(\tilde{S},C_{c}^{\infty}(\mathbb{R}^{d}))$ on
$L^{2}(\mathbb{R}^{d},\nu)$ by

$\tilde{S}f\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{i,j=1}^{d}a_{ij}\partial_{j}\partial_{i}f+\sum_{i=1}^{d}b_{i}\partial_{i}f,\quad
f\in C_{c}^{\infty}(\mathbb{R}^{d}).$

The density $\rho$ of $\nu$ wrt. the Lebesgue measure is given by
$\rho(v)=\mathrm{e}^{-v^{2}/2}=(\mathrm{e}^{-v^{2}/4})^{2}$. Due to the
conditions ($\Sigma$1), ($\Sigma$2) and either ($\Sigma$3) or ($\Sigma$3′),
all assumptions from Theorem 4.5 are fulfilled and therefore,
$(\tilde{S},C_{c}^{\infty}(\mathbb{R}^{d}))$ is essentially m-dissipative in
$L^{2}(\nu)$. Let $g=g_{1}\otimes g_{2}\in
C_{c}^{\infty}(\mathbb{R}^{d})\otimes C_{c}^{\infty}(\mathbb{R}^{d})$ be a
pure tensor. Then there is a sequence $(\tilde{f}_{n})_{n\in\mathbb{N}}$ in
$C_{c}^{\infty}(\mathbb{R}^{d})$ such that $(I-\tilde{S})\tilde{f}_{n}\to
g_{2}$ in $L^{2}(\nu)$ as $n\to\infty$. Define $f_{n}\in D$ for each
$n\in\mathbb{N}$ by

$f_{n}(x,v)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=g_{1}(x)\tilde{f}_{n}(v).$

Then

$\|(I-S)f_{n}-g\|_{H}=\|g_{1}\otimes((I-\tilde{S})\tilde{f}_{n}-g_{2})\|_{H}=\|g_{1}\|_{L^{2}(\mathrm{e}^{-\Phi(x)}\,\mathrm{d}x)}\cdot\|(I-\tilde{S})\tilde{f}_{n}-g_{2}\|_{L^{2}(\nu)},$

which converges to zero as $n\to\infty$. By taking linear combinations, this
shows that $(I-S)(D)$ is dense in $C_{c}^{\infty}(\mathbb{R}^{d})\otimes
C_{c}^{\infty}(\mathbb{R}^{d})$ wrt. the $H$-norm. Since
$C_{c}^{\infty}(\mathbb{R}^{d})\otimes C_{c}^{\infty}(\mathbb{R}^{d})$ is
dense in $H$, $(S,D)$ is essentially m-dissipative and its closure $(S,D(S))$
generates a strongly continuous contraction semigroup.

It can easily be shown that $(Sf,f^{+})_{H}\leq 0$ for all $f\in D$.
Parallelly to the proof of (D7), it holds that $1\in D(S)$ and $S1=0$. This
together implies that $(S,D(S))$ is a Dirichlet operator and the generated
semigroup is sub-Markovian. $\square$

###  4.3 Perturbation of the symmetric part for nice coefficients

Now we extend the essential m-dissipativity stepwise to the non-symmetric
operator $L$ by perturbation. This follows and is mostly based on the method
seen in the proof of [15, Theorem 6.3.1], which proved that result for
$\Sigma=I$.

Since $S$ is dissipative on
$D_{1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=L_{0}^{2}(\mathrm{e}^{-\Phi}\,\mathrm{d}x)\otimes
C_{c}^{\infty}(\mathbb{R}^{d})\supset D$, the operator $(S,D_{1})$ is
essentially m-dissipative as well. The unitary transformation
$T:L^{2}(\mathbb{R}^{d},\mathrm{d}(x,v))\to H$ given by
$Tf(x,v)=\mathrm{e}^{\frac{v^{2}}{4}+\frac{\Phi(x)}{2}}f(x,v)$ leaves $D_{1}$
invariant. This implies that the operator $(S_{1},D_{1})$ on
$L^{2}(\mathbb{R}^{d},\mathrm{d}(x,v))$ , where $S_{1}=T^{-1}ST$, is again
essentially m-dissipative. Note that $S_{1}$ is explicitly given by

$S_{1}f=\sum_{i,j=1}^{d}a_{ij}\partial_{v_{j}}\partial_{v_{i}}f-\frac{1}{4}(v,\Sigma
v)f+\frac{1}{2}\operatorname{tr}(\Sigma)f+\sum_{i,j=1}^{d}\partial_{j}a_{ij}(\frac{v_{i}}{2}f+\partial_{v_{i}}f)$

Now consider the operator $(ivxI,D_{1})$, which is dissipative as
$\operatorname{Re}(ivxf,f)_{L^{2}(\mathbb{R}^{d},\mathrm{d}(x,v))}=0$ for
$f\in D_{1}$. We show the following perturbation result:

######  Proposition 4.8.

Let $\Sigma$ satisfy ($\Sigma$3) with $\beta\leq-1$. Then the operator
$(S_{1}+ivxI,D_{1})$ is essentially m-dissipative on
$L^{2}(\mathbb{R}^{d},\mathrm{d}(x,v))$.

######  Proof:

Define the orthogonal projections $P_{n}$ via
$P_{n}f(x,v)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\xi_{n}(x)f(x,v)$,
where $\xi_{n}$ is given by $\xi_{n}=\mathds{1}_{[n-1,n)}(|x|)$, which leave
$D_{1}$ invariant. Then the conditions for Lemma 4.4 are fulfilled, and we are
left to show the $A_{n}$-bounds. Note that due to the restriction on $\beta$,
there is some constant $C<\infty$ such that $\partial_{j}a_{ij}(v)v_{i}\leq C$
for all $1\leq i,j\leq d$, $v\in\mathbb{R}^{d}$. For each fixed
$n\in\mathbb{N}$ it holds for all $f\in P_{n}D_{1}$:

$\displaystyle\|ivxf\|_{L^{2}}^{2}$ $\displaystyle\leq
n^{2}\int_{\mathbb{R}^{2d}}|v|^{2}f^{2}\,\mathrm{d}(x,v)\leq
4c_{\Sigma}^{-1}n^{2}\int_{\mathbb{R}^{2d}}\frac{(v,\Sigma
v)}{4}f^{2}\,\mathrm{d}(x,v)$ $\displaystyle\leq
4c_{\Sigma}^{-1}n^{2}\int_{\mathbb{R}^{2d}}\frac{(v,\Sigma
v)}{4}f^{2}+(\nabla_{v}f,\Sigma\nabla_{v}f)\,\mathrm{d}(x,v)$
$\displaystyle=4c_{\Sigma}^{-1}n^{2}\int_{\mathbb{R}^{2d}}\left(-\sum_{i,j=1}^{d}a_{ij}\partial_{v_{j}}\partial_{v_{i}}f-\sum_{i,j=1}^{d}\partial_{j}a_{ij}\partial_{v_{i}}f+\frac{(v,\Sigma
v)}{4}f\right)f\,\mathrm{d}(x,v)$
$\displaystyle=4c_{\Sigma}^{-1}n^{2}\left((-P_{n}S_{1}f,f)+\int_{\mathbb{R}^{2d}}\frac{1}{2}\operatorname{tr}(\Sigma)f^{2}+\sum_{i,j=1}^{d}\partial_{j}a_{ij}\frac{v_{i}}{2}f^{2}\,\mathrm{d}(x,v)\right)$
$\displaystyle\leq
4c_{\Sigma}^{-1}n^{2}\left((-S_{1}f,f)+(d^{2}C+\frac{dM_{\Sigma}}{2})\|f\|_{L^{2}}^{2}\right).$

Hence by Lemma 4.3, $(ivxIP_{n},P_{n}D_{1})$ is $S_{1}P_{n}$-bounded with
Kato-bound zero. Application of Lemma 4.4 yields the statement. $\square$

Since $C_{c}^{\infty}(\mathbb{R}^{d})\otimes C_{c}^{\infty}(\mathbb{R}^{d})$
is dense in $D_{1}$ wrt. the graph norm of $S_{1}+ivxI$, we obtain essential
m-dissipativity of $(S_{1}+ivxI,C_{c}^{\infty}(\mathbb{R}^{d})\otimes
C_{c}^{\infty}(\mathbb{R}^{d}))$ and therefore also of its dissipative
extension $(S_{1}+ivxI,D_{2})$ with
$D_{2}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathcal{S}(\mathbb{R}^{d})\otimes
C_{c}^{\infty}(\mathbb{R}^{d}))$, where $\mathcal{S}(\mathbb{R}^{d})$ denotes
the set of smooth functions of rapid decrease on $\mathbb{R}^{d}$. Applying
Fourier transform in the $x$-component leaves $D_{2}$ invariant and shows that
$(L_{2},D_{2})$ is essentially m-dissipative, where $L_{2}=S_{1}+v\nabla_{x}$.
Now we add the part depending on the potential $\Phi$.

######  Proposition 4.9.

Let $\Sigma$ satisfy ($\Sigma$3) with $\beta\leq-1$ and $\Phi$ be Lipschitz-
continuous. Then the operator $(L^{\prime},D_{2})$ with
$L^{\prime}=L_{2}-\nabla\Phi\nabla_{v}$ is essentially m-dissipative on
$L^{2}(\mathbb{R}^{d},\mathrm{d}(x,v))$.

######  Proof:

It holds due to antisymmetry of $v\nabla_{x}$ that

$\displaystyle\|\nabla\Phi\nabla_{v}f\|_{L^{2}}^{2}$
$\displaystyle\leq\||\nabla\Phi|\|_{\infty}^{2}c_{\Sigma}^{-1}\left((\nabla_{v}f,\Sigma\nabla_{v}f)_{L^{2}}+\left(\frac{(v,\Sigma
v)}{4}f,f\right)_{L^{2}}-(v\nabla_{x}f,f)_{L^{2}}\right)$
$\displaystyle\leq\||\nabla\Phi|\|_{\infty}^{2}c_{\Sigma}^{-1}\left((-L_{2}f,f)_{L^{2}}+(d^{2}C+\frac{dM_{\Sigma}}{2})\|f\|_{L^{2}}^{2}\right),$

analogously to the proof of Proposition 4.8, which again implies that the
antisymmetric, hence dissipative operator $(\nabla\Phi\nabla_{v},D_{2})$ is
$L_{2}$-bounded with bound zero. This shows the claim. $\square$

Denote by $H_{c}^{1,\infty}(\mathbb{R}^{d})$ the space of functions in
$H^{1,\infty}(\mathbb{R}^{d})$ with compact support and set
$D^{\prime}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=H_{c}^{1,\infty}(\mathbb{R}^{d})\otimes
C_{c}^{\infty}(\mathbb{R}^{d})$. As $(L^{\prime},D^{\prime})$ is dissipative
and its closure extends $(L^{\prime},D_{2})$, it is itself essentially
m-dissipative. The unitary transformation $T$ from the beginning of this
section leaves $D^{\prime}$ invariant, and it holds that $TL^{\prime}T^{-1}=L$
on $D^{\prime}$. This brings us to the first m-dissipativity result for the
complete Langevin operator:

######  Theorem 4.10.

Let $\Sigma$ satisfy ($\Sigma$3) with $\beta\leq-1$ and $\Phi$ be Lipschitz-
continuous. Then $(L,D)$ with is essentially m-dissipative on $H$.

######  Proof:

By the previous considerations, $(L,D^{\prime})$ is essentially m-dissipative
on $H$. Let $f\in D^{\prime}$ with $f=g\otimes h$. It holds $g\in
H_{c}^{1,\infty}(\mathbb{R}^{d})\subset H^{1,2}(\mathbb{R}^{d})$. Choose a
sequence $(g_{n})_{n\in\mathbb{N}}$ with $g_{n}\in
C_{c}^{\infty}(\mathbb{R}^{d})$, such that $g_{n}\to g$ in
$H^{1,2}(\mathbb{R}^{d})$ as $n\to\infty$. Due to boundedness of
$\mathrm{e}^{-\Phi}$ and $v_{j}\mathrm{e}^{-v^{2}/2}$ for all $1\leq j\leq d$,
it follows immediately that $g_{n}\otimes h\to f$ and $L(g_{n}\otimes h)\to
Lf$ in $H$ as $n\to\infty$. This extends to arbitrary $f\in D^{\prime}$ via
linear combinations and therefore shows that
$C_{c}^{\infty}(\mathbb{R}^{d})\otimes C_{c}^{\infty}(\mathbb{R}^{d})$ and
hence also $D$, is a core for $(L,D(L))$. $\square$

###  4.4 Proof of Theorem 3.4

It is now left to relax the assumptions on $\Sigma$ and $\Phi$ by
approximation. Let the assumptions of Theorem 3.4 hold and wlog $\Phi\geq 0$.
For $n\in\mathbb{N}$ we define $\Sigma_{n}$ via

$\Sigma_{n}=(a_{ij,n})_{1\leq i,j\leq d},\quad
a_{ij,n}(v)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=a_{ij}\left(\left(\frac{n}{|v|}\wedge
1\right)v\right).$

Then each $\Sigma_{n}$ also satisfies ($\Sigma$1)-($\Sigma$3) with $\beta=-1$,
since $\partial_{k}a_{ij,n}=\partial_{k}a_{ij}$ on $B_{n}(0)$ and
$|\partial_{k}a_{ij,n}|\leq\frac{(1+\sqrt{d})nL_{\Sigma,n}}{|v|}$ outside of
$\overline{B_{n}(0)}$, where $L_{\Sigma,n}$ denotes the supremum of
$\max_{1\leq k\leq d}|\partial_{k}a_{ij}|$ on $\overline{B_{n}(0)}$. Let
further $\eta_{m}\in C_{c}^{\infty}(\mathbb{R}^{d})$ for each $m\in\mathbb{N}$
with $\eta=1$ on $B_{m}(0)$ and set $\Phi_{m}=\eta_{m}\Phi$, which is
Lipschitz-continuous. Define $H_{m}$ as
$L^{2}(\mathbb{R}^{2d},\mathrm{e}^{-\frac{v^{2}}{2}-\Phi_{m}(x)}\,\mathrm{d}(x,v))$
and $(L_{n,m},D)$ via

$L_{n,m}f=\sum_{i,j=1}^{d}a_{ij,n}\partial_{v_{j}}\partial_{v_{i}}f+\sum_{i=1}^{d}\sum_{j=1}^{d}(\partial_{j}a_{ij,n}(v)-a_{ij,n}(v)v_{j})\partial_{v_{i}}f+v\cdot\nabla_{x}f-\nabla\Phi_{m}\cdot\nabla_{v}f.$

Then Theorem 4.10 shows that for each $n,m\in\mathbb{N}$, $(L_{n,m},D)$ is
essentially m-dissipative on $H_{m}$, and it holds that $L_{n,m}f=Lf$ for all
$f\in D$ on $B_{m}(0)\times B_{n}(0)$. Note further that
$\|\cdot\|_{H}\leq\|\cdot\|_{H_{m}}$.

We need the following estimates:

######  Lemma 4.11.

Let $n,m\in\mathbb{N}$ and $\Sigma_{n}$, $\Phi_{m}$ as defined above. Then
there is a constant $D_{1}<\infty$ independent of $n,m$ such that for each
$1\leq j\leq d$, the following hold for all $f\in D$:

$\displaystyle\|v_{j}f\|_{H_{m}}$ $\displaystyle\leq
D_{1}n^{\frac{1+\beta}{2}}\|(I-L_{n,m})f\|_{H_{m}},$
$\displaystyle\|\partial_{v_{j}}f\|_{H_{m}}$ $\displaystyle\leq
D_{1}n^{\frac{1+\beta}{2}}\|(I-L_{n,m})f\|_{H_{m}}.$

######  Proof:

Recall the unitary transformations
$T_{m}:L^{2}(\mathbb{R}^{2d},\mathrm{d}(x,v))\to H_{m}$ defined by
$T_{m}f=\mathrm{e}^{\frac{v^{2}}{4}+\frac{\Phi_{m}(x)}{2}}f$, as well as the
operator $L_{n,m}^{\prime}=T_{m}^{-1}L_{n,m}T_{m}$, and let $f\in
T_{m}^{-1}D$. Then

$\displaystyle
L_{n,m}^{\prime}f=\sum_{i,j=1}^{d}a_{ij,n}\partial_{v_{j}}\partial_{v_{i}}f$
$\displaystyle-\frac{1}{4}(v,\Sigma_{n}v)f+\frac{1}{2}\operatorname{tr}(\Sigma_{n})f+\sum_{i,j=1}^{d}\partial_{j}a_{ij,n}(\frac{v_{i}}{2}f+\partial_{v_{i}}f)$
$\displaystyle-v\nabla_{x}f+\nabla\Phi_{m}\nabla_{v}f.$

Analogously to the proof of Proposition 4.8 and due to antisymmetry of
$v\nabla_{x}$ and $\nabla\Phi_{m}\nabla_{v}$ on $L^{2}(\mathrm{d}(x,v))$, it
holds that

$\displaystyle\|v_{j}T_{m}f\|_{H_{m}}^{2}$
$\displaystyle=\|v_{j}f\|_{L^{2}(\mathrm{d}(x,v))}^{2}\leq
4c_{\Sigma}^{-1}\int_{\mathbb{R}^{2d}}\frac{1}{4}(v,\Sigma_{n}v)f^{2}\,\mathrm{d}(x,v)$
$\displaystyle\leq
4c_{\Sigma}^{-1}\left((-L_{n,m}^{\prime}f,f)_{L^{2}(\mathrm{d}(x,v))}+\int_{\mathbb{R}^{2d}}\frac{f^{2}}{2}\left(\operatorname{tr}(\Sigma_{n})+\sum_{i,j=1}^{d}\partial_{j}a_{ij,n}v_{i}\right)\mathrm{d}(x,v)\right).$

Since $|\operatorname{tr}(\Sigma_{n})|\leq|\operatorname{tr}(\Sigma)|\leq
d\cdot M_{\Sigma}$ and

$|\partial_{j}a_{ij,n}(v)v_{i}|\leq|\partial_{j}a_{ij}(v)|\cdot|v_{i}|\leq\max\\{B_{\Sigma},M\cdot
n^{\beta+1}\\}\quad\text{ for all }v\in B_{n}(0),$

as well as

$|\partial_{j}a_{ij,n}(v)v_{i}|\leq(1+\sqrt{d})n\frac{|v_{i}|}{|v|}\max_{1\leq
k\leq d}\sup_{y\in B_{n}(0)}|\partial_{k}a_{ij}(y)|\leq
2\sqrt{d}Mn^{\beta+1}\quad\text{ for all }v\notin B_{n}(0),$

and wlog $B_{\Sigma}\leq M\cdot n^{\beta+1}$, it follows that

$\|v_{j}T_{m}f\|_{H_{m}}^{2}\leq
4c_{\Sigma}^{-1}(-L_{n,m}^{\prime}f,f)_{L^{2}(\mathrm{d}(x,v))}+2c_{\Sigma}^{-1}(dM_{\Sigma}+2d^{5/2}Mn^{\beta+1})\|f\|_{L^{2}(\mathrm{d}(x,v))}^{2}.$

Further, it clearly holds that

$\displaystyle(-L_{n,m}^{\prime}f,f)_{L^{2}(\mathrm{d}(x,v))}$
$\displaystyle\leq\frac{1}{4}\left(\|L_{n,m}^{\prime}f\|_{L^{2}(\mathrm{d}(x,v))}+\|f\|_{L^{2}(\mathrm{d}(x,v))}\right)^{2}\quad\text{
and }$ $\displaystyle\|f\|_{L^{2}(\mathrm{d}(x,v))}^{2}$
$\displaystyle\leq\left(\|L_{n,m}^{\prime}f\|_{L^{2}(\mathrm{d}(x,v))}+\|f\|_{L^{2}(\mathrm{d}(x,v))}\right)^{2}.$

Dissipativity of $(L_{n,m}^{\prime},T_{m}^{-1}D)$ on $L^{2}(\mathrm{d}(x,v))$
implies

$\|L_{n,m}^{\prime}f\|_{L^{2}(\mathrm{d}(x,v))}+\|f\|_{L^{2}(\mathrm{d}(x,v))}\leq\|(I-L_{n,m}^{\prime})f\|_{L^{2}(\mathrm{d}(x,v))}+2\|(I-L_{n,m}^{\prime})f\|_{L^{2}(\mathrm{d}(x,v))}.$

Overall, we get

$\displaystyle\|v_{j}T_{m}f\|_{H_{m}}^{2}$ $\displaystyle\leq
2c_{\Sigma}^{-1}(1+2(dM_{\Sigma}+2d^{5/2}Mn^{\beta+1}))\|(I-L_{n,m}^{\prime})f\|_{L^{2}(\mathrm{d}(x,v))}^{2}$
$\displaystyle\leq
18c_{\Sigma}^{-1}d^{3}n^{\beta+1}\max\\{M_{\Sigma},M\\}\|(I-L_{n,m}^{\prime})f\|_{L^{2}(\mathrm{d}(x,v))}^{2}.$

Since

$\|(I-L_{n,m}^{\prime})f\|_{L^{2}(\mathrm{d}(x,v))}^{2}=\|T_{m}^{-1}(I-L_{n,m})T_{m}f\|_{L^{2}(\mathrm{d}(x,v))}^{2}=\|(I-L_{n,m})T_{m}f\|_{H_{m}}^{2},$

this proves the first statement with
$D_{1}=\sqrt{18c_{\Sigma}^{-1}d^{3}\max\\{M_{\Sigma},M\\}}$.

For the second part, note that
$\partial_{v_{j}}T_{m}f=T_{m}\partial_{v_{j}}f+\frac{v_{j}}{2}T_{m}f$ and that

$\displaystyle\|T_{m}\partial_{v_{j}}f\|_{H_{m}}^{2}$
$\displaystyle=(\partial_{v_{j}}f,\partial_{v_{j}}f)_{L^{2}(\mathrm{d}(x,v))}^{2}\leq
c_{\Sigma}^{-1}\int_{\mathbb{R}^{2d}}(\nabla_{v}f,\Sigma_{n}\nabla_{v}f)_{\mathrm{euc}}\,\mathrm{d}(x,v)$
$\displaystyle\leq
c_{\Sigma}^{-1}\left((-L_{n,m}^{\prime}f,f)_{L^{2}}+\int_{\mathbb{R}^{2d}}\frac{1}{2}\operatorname{tr}(\Sigma_{n})f^{2}+\sum_{i,j=1}^{d}\partial_{j}a_{ij,n}\frac{v_{i}}{2}f^{2}\,\mathrm{d}(x,v)\right).$

Repeating all calculations of the first part yields

$\|\partial_{v_{j}}T_{m}f\|_{H_{m}}\leq\left(\frac{D_{1}}{2}+\frac{D_{1}}{2}\right)n^{1+\beta}\|(I-L_{n,m})T_{m}f\|_{H_{m}}.$

$\square$

Fix some pure tensor $g\in C_{c}^{\infty}(\mathbb{R}^{d})\otimes
C_{c}^{\infty}(\mathbb{R}^{d})$. We prove that for every $\varepsilon>0$, we
can find some $f\in D$ such that $\|(I-L)f-g\|_{H}<\varepsilon$. This then
extends to arbitrary $g\in C_{c}^{\infty}(\mathbb{R}^{d})\otimes
C_{c}^{\infty}(\mathbb{R}^{d})$ via linear combinations and therefore implies
essential m-dissipativity of $(L,D)$ on $H$, since
$C_{c}^{\infty}(\mathbb{R}^{d})\otimes C_{c}^{\infty}(\mathbb{R}^{d})$ is
dense in $H$. If $\beta\leq-1$, then the proof is easier and follows
analogously to the proof of of [15, Theorem 6.3.1]. Therefore we will assume
$\beta>-1$. Recall that in this case, we have $|\nabla\Phi(x)|\leq
N(1+|x|^{\gamma})$ for all $x\in\mathbb{R}^{d}$, where
$\gamma<\frac{2}{1+\beta}$, see the assumptions of Theorem 3.4.

Denote the support of $g$ by $K_{x}\times K_{v}$, where $K_{x}$ and $K_{v}$
are compact sets in $\mathbb{R}^{d}$. By a standard construction, for each
$\delta_{x},\delta_{v}>0$, there are smooth cutoff functions
$0\leq\phi_{\delta_{x}},\psi_{\delta_{v}}\leq 1\in
C_{c}^{\infty}(\mathbb{R}^{d})$ with
$\operatorname{supp}(\phi_{\delta_{x}})\subset B_{\delta_{x}}(K_{x})$,
$\operatorname{supp}(\psi_{\delta_{v}})\subset B_{\delta_{v}}(K_{v})$,
$\phi_{\delta_{x}}=1$ on $K_{x}$, $\psi_{\delta_{v}}=1$ on $K_{v}$. Moreover,
there are constants $C_{\phi},C_{\psi}$ independent of $\delta_{x}$ and
$\delta_{v}$ such that

$\|\partial^{s}\phi_{\delta_{x}}\|_{\infty}\leq
C_{\phi}\delta_{x}^{-|s|}\quad\text{ and
}\quad\|\partial^{s}\psi_{\delta_{v}}\|_{\infty}\leq
C_{\psi}\delta_{v}^{-|s|}$

for all multi-indices $s\in\mathbb{N}^{d}$. Fix $\alpha$ such that
$\frac{1+\beta}{2}<\alpha<\frac{1}{\gamma}$. For any $\delta>0$, we set
$\delta_{x}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\delta^{\alpha}$
and
$\delta_{v}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\delta$,
and then define
$\chi_{\delta}(x,v)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\phi_{\delta_{x}}(x)\psi_{\delta_{v}}(v)=\phi_{\delta^{\alpha}}(x)\psi_{\delta}(v)$.

For $f\in D$, $\delta>0$, consider
$f_{\delta}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\chi_{\delta}f$,
which is an element of $D$, as $\chi_{\delta}\in D$. Without loss of
generality, we consider $\delta$ and hence $\delta^{\alpha}$ sufficiently
large such that $\operatorname{supp}(\phi_{\delta^{\alpha}})\subset
B_{2\delta^{\alpha}}(0)$, $\operatorname{supp}(\psi_{\delta})\subset
B_{2\delta}(0)$ and that there are $n,m\in\mathbb{N}$ that satisfy

$\operatorname{supp}(\phi_{\delta^{\alpha}})\times\operatorname{supp}(\psi_{\delta})\subset
B_{m}(0)\times B_{n}(0)\subset B_{2\delta^{\alpha}}(0)\times B_{2\delta}(0).$
(4.6)

The following then holds:

######  Lemma 4.12.

Let $g\in C_{c}^{\infty}(\mathbb{R}^{d})\otimes
C_{c}^{\infty}(\mathbb{R}^{d})$ and $\phi,\psi$ as above. Then there is a
constant $D_{2}<\infty$ and a function $\rho:\mathbb{R}\to\mathbb{R}$
satisfying $\rho(s)\to 0$ as $s\to\infty$, such that for any $\delta$, $n$ and
$m$ satisfying (4.6),

$\|(I-L)f_{\delta}-g\|_{H}\leq\|(I-L_{n,m})f-g\|_{H_{m}}+D_{2}\cdot\rho(\delta)\|(I-L_{n,m})f\|_{H_{m}}$

holds for all $f\in D$.

######  Proof:

By the product rule,

$\displaystyle\|(I-L)f_{\delta}-g\|_{H}$
$\displaystyle\leq\|\chi_{\delta}((I-L)f-g)\|_{H}+\sum_{i,j=1}^{d}\|a_{ij}\phi_{\delta^{\alpha}}(x)\partial_{j}\partial_{i}\psi_{\delta}(v)f\|_{H}$
$\displaystyle+2\sum_{i,j=1}^{d}\|a_{ij}\phi_{\delta^{\alpha}}(x)\partial_{i}\psi_{\delta}(v)\partial_{v_{j}}f\|_{H}+\sum_{i,j=1}^{d}\|\partial_{j}a_{ij}\phi_{\delta^{\alpha}}(x)\partial_{i}\psi_{\delta}(v)f\|_{H}$
$\displaystyle+\sum_{i,j=1}^{d}\|a_{ij}v_{j}\phi_{\delta^{\alpha}}(x)\partial_{i}\psi_{\delta}(v)f\|_{H}+\sum_{i=1}^{d}\|v_{i}\partial_{i}\phi_{\delta^{\alpha}}(x)\psi_{\delta}(v)f\|_{H}$
$\displaystyle+\sum_{i=1}^{d}\|\partial_{i}\Phi\phi_{\delta^{\alpha}}(x)\partial_{i}\psi_{\delta}(v)f\|_{H}.$

Due to the choice of $n$ and $m$, every $\|\cdot\|_{H}$ on the right hand side
can be replaced with $\|\cdot\|_{H_{m}}$, $a_{ij}$ by $a_{ij,n}$, and $\Phi$
by $\Phi_{m}$, hence $L$ by $L_{n,m}$.

We now give estimates for each summand of the right hand side, in their order
of appearance:

  1. (1)

$\|\chi_{\delta}((I-L)f-g)\|_{H}\leq\|(I-L_{n,m})f-g\|_{H_{m}}$,

  2. (2)

$\|a_{ij}\phi_{\delta^{\alpha}}(x)\partial_{j}\partial_{i}\psi_{\delta}(v)f\|_{H}\leq
M_{\Sigma}C_{\psi}\delta^{-2}\|f\|_{H_{m}}$,

  3. (3)

$\|a_{ij}\phi_{\delta^{\alpha}}(x)\partial_{i}\psi_{\delta}(v)\partial_{v_{j}}f\|_{H}\leq
M_{\Sigma}C_{\psi}\delta^{-1}\|\partial_{v_{j}}f\|_{H_{m}}$,

  4. (4)

$\|\partial_{j}a_{ij}\phi_{\delta^{\alpha}}(x)\partial_{i}\psi_{\delta}(v)f\|_{H}\leq\max\\{B_{\Sigma},M\cdot(2\delta)^{\beta\vee
0}\\}C_{\psi}\delta^{-1}\|f\|_{H_{m}}$,

  5. (5)

$\|a_{ij}v_{j}\phi_{\delta^{\alpha}}(x)\partial_{i}\psi_{\delta}(v)f\|_{H}\leq
M_{\Sigma}C_{\psi}\delta^{-1}\|v_{j}f\|_{H_{m}}$,

  6. (6)

$\|v_{i}\partial_{i}\phi_{\delta^{\alpha}}(x)\psi_{\delta}(v)f\|_{H}\leq
C_{\phi}\delta^{-\alpha}\|v_{i}f\|_{H_{m}}$,

  7. (7)

$\|\partial_{i}\Phi\phi_{\delta^{\alpha}}(x)\partial_{i}\psi_{\delta}(v)f\|_{H}\leq
N(1+(2\delta^{\alpha})^{\gamma})C_{\psi}\delta^{-1}\|f\|_{H_{m}}$,

where the last inequality is due to $|\partial_{i}\Phi(x)|\leq
N(1+|x|^{\gamma})$ for all $x\in\mathbb{R}^{d}$ and the support of the cutoff
as in (4.6). Application of Lemma 4.11 shows the existence of $D_{2}$
independent of $n,m$, such that

$\|(I-L)f_{\delta}-g\|_{H}\leq\|(I-L_{n,m})f-g\|_{H_{m}}+D_{2}\cdot\rho(\delta)\|(I-L_{n,m})f\|_{H_{m}}$

where

$\rho(\delta)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\delta^{-2}+2^{\frac{1+\beta}{2}}\delta^{\frac{1+\beta}{2}-1}+2^{\beta\vee
0}\delta^{(\beta\vee
0)-1}+2^{\frac{1+\beta}{2}}\delta^{\frac{1+\beta}{2}-\alpha}+\delta^{-1}+2^{\gamma}\delta^{\alpha\gamma-1}.$

Clearly $\rho(\delta)\to 0$ as $\delta\to\infty$ due to $\beta<1$ and the
definition of $\alpha$. $\square$

Now finally we show that for each $\varepsilon>0$, we can find some
$f_{\delta}\in D$ such that

$\|(I-L)f_{\delta}-g\|_{H}<\varepsilon.$

Choose $\delta>0$ large enough such that
$\rho(\delta)<\frac{\varepsilon}{4D_{2}\|g\|_{H}}$ (where $\rho$ ans $D_{2}$
are provided by Lemma 4.12) and that there exist $n,m$ satisfying (4.6).

Then choose $f\in D$ via Theorem 4.10 such that
$\|(I-L_{n,m})f-g\|_{H_{m}}<\min\\{\frac{\varepsilon}{2},\|g\|_{H}\\}$ and
define $f_{\delta}$ as before. Note that due to the choice of the cutoffs, it
holds $\|g\|_{H}=\|g\|_{H_{m}}$, therefore

$\|(I-L)f_{\delta}-g\|_{H}<\frac{\varepsilon}{2}+\frac{\varepsilon}{4\|g\|_{H_{m}}}(\|(I-L_{n,m})f-g\|_{H_{m}}+\|g\|_{H_{m}})<\varepsilon.$

As mentioned earlier, this shows essential m-dissipativity of the operator
$(L,D)$ on $H$ and therefore concludes the proof of Theorem 3.4.

##  5 Applications

###  5.1 The associated Cauchy problem

We consider the abstract Cauchy problem associated with the operator $L$.
Given the initial condition $u_{0}\in H$, $u:[0,\infty)\to H$ should satisfy

$\partial_{t}u(t)=\left(\operatorname{tr}\left(\Sigma
H_{v}\right)+b\cdot\nabla_{v}+v\cdot\nabla_{x}-\nabla\Phi\cdot\nabla_{v}\right)u(t)\quad\text{
and }\quad u(0)=u_{0}.$ (5.1)

If we set
$u(t)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=T_{t}u_{0}$,
where $(T_{t})_{t\geq 0}$ is the semigroup on $H$ generated by the closure
$(L,D(L))$ of $(L,D)$, then the map $t\mapsto u(t)$ is continuous in $H$. For
all $t\geq 0$, it holds that $\int_{0}^{t}u(s)\,\mathrm{d}s\in D(L)$ with
$L\int_{0}^{t}u(s)\,\mathrm{d}s=T_{t}u_{0}-u_{0}=u(t)-u_{0}$, hence $u$ is the
unique mild solution to the abstract Cauchy problem.

If $u_{0}\in D(L)$, then $u(t)\in D(L)$ for all $t\geq 0$, and
$\partial_{t}u(t)=LT_{t}u_{0}=Lu(t)$, so $u$ is even a classical solution to
the abstract Cauchy problem associated to $L$. In particular, this holds for
all $u_{0}\in C_{c}^{2}(\mathbb{R}^{d\times d})$, since $L$ is dissipative
there and it extends $D$, which implies $C_{c}^{2}(\mathbb{R}^{d\times
d})\subset D(L)$.

In this context, Theorem 1.1 shows exponential convergence of the unique
solution $u(t)$ to a constant as $t\to\infty$. More precisely, for each
$\theta_{1}>1$ we can calculate $\theta_{2}\in(0,\infty)$ depending on the
choice of $\Sigma$ and $\Phi$ such that for all $t\geq 0$,

$\left\|u(t)-\int_{E}u_{0}\,\mathrm{d}\mu\right\|_{H}\leq\theta_{1}\mathrm{e}^{-\theta_{2}t}\left\|u_{0}-\int_{E}u_{0}\,\mathrm{d}\mu\right\|_{H}.$

###  5.2 Connection to Langevin dynamics with multiplicative noise

So far, our considerations have been purely analytical, giving results about
the core property of $D$ for $L$ and rate of convergence for the generated
semigroup $(T_{t})_{t\geq 0}$ in $H$. However, this approach is still quite
natural in the context of the Langevin SDE (1.1), as the semigroup has a
meaningful stochastic representation. The connection is achieved via the
powerful theory of generalized Dirichlet forms as developed by Stannat in
[16], which gives the following:

Assume the context of Theorem 3.4. There exists a Hunt process

$\mathbf{M}=\left(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},(X_{t},V_{t}),(P_{(x,v)})_{(x,v)\in\mathbb{R}^{d}\times\mathbb{R}^{d}}\right)$

with state space $E=\mathbb{R}^{d}\times\mathbb{R}^{d}$, infinite lifetime and
continuous sample paths ($P_{(x,v)}$-a.s. for all $(x,v)\in E$), which is
properly associated in the resolvent sense with $(T_{t})_{t\geq 0}$. In
particular (see [15, Lemma 2.2.8]), this means that for each bounded
measurable $f$ which is also square-integrable with respect to the invariant
measure $\mu$ and all $t>0$, $T_{t}f$ is a $\mu$-version of $p_{t}f$, where
$(p_{t})_{t\geq 0}$ is the transition semigroup of $\mathbf{M}$ with

$p_{t}f:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R},\qquad(x,v)\mapsto\mathbb{E}_{(x,v)}\left[f(X_{t},V_{t})\right].$

This representation can be further extended to all $f\in H$, see for example
[17, Exercise IV.2.9]. Moreover, if $\mu$-versions of $\Sigma$ and $\Phi$ are
fixed, then $P_{(x,v)}$ solves the martingale problem for $L$ on
$C_{c}^{2}(E)$ for $L$-quasi all $(x,v)\in E$, i.e. for each $f\in
C_{c}^{2}(E)$, the stochastic process $(M_{t}^{[f]})_{t\geq 0}$ defined by

$M_{t}^{[f]}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=f(X_{t},V_{t})-f(X_{0},V_{0})-\int_{0}^{t}Lf(X_{s},V_{s})\,\mathrm{d}s,$

is a martingale with respect to $P_{(x,v)}$. If $h\in L^{2}(\mu)$ is a
probability density with respect to $\mu$, then the law
$P_{h}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{E}P_{(x,v)}h(x,v)\,\mathrm{d}\mu$
solves the martingale problem for $(L,D(L))$, without the need to fix specific
versions of $\Sigma$ and $\Phi$. In particular, this holds for $h=1$. As in
[15, Lemma 2.1.8], for $f\in D(L)$ with $f^{2}\in D(L)$ and $Lf\in
L^{4}(\mu)$, a martingale is also defined via

$N_{t}^{[f]}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(M_{t}^{[f]})^{2}-\int_{0}^{t}L(f^{2})(X_{s},V_{s})-(2fLf)(X_{s},V_{s})\,\mathrm{d}s,\qquad
t\geq 0,$

which may serve as a way to verify that $\mathbf{M}$ is already a weak
solution of (1.1), as it allows a representation of the quadratic variation
process. Indeed, if we set
$f_{n}^{i}(x,v)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varphi_{n}(x_{i})x_{i}$
for a suitable sequence $(\varphi_{n})_{n\in\mathbb{N}}$ of cutoff functions
as in Definition 3.6, evaluation of $N_{t}^{[f_{n}^{i}]}$ shows that the
quadratic variation $[M^{[f_{n}^{i}]}]_{t}$ of $M_{t}^{[f_{n}^{i}]}$ is
constantly zero, which implies the same for $M_{t}^{[f_{n}^{i}]}$. Hence, by
introducing appropriate stopping times, it follows that
$X_{t}^{i}-X_{0}^{i}=\int_{0}^{t}V_{s}^{i}\,\mathrm{d}s$, so the first line of
the SDE (1.1) is satisfied.

In an analogous procedure, using
$g_{n}^{i}(x,v)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varphi_{n}(v_{i})v_{i}$,
we can see that the quadratic covariation $[V^{i},V^{j}]_{t}$ is given by
$2\int_{0}^{t}a_{ij}(V_{s})\,\mathrm{d}s$. Since $\Sigma$ is strictly
elliptic, the diffusion matrix $\sigma$ is invertible and by Lévy’s
characterization, the process
$B_{t}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{0}^{t}\frac{1}{\sqrt{2}}\sigma^{-1}(V_{s})\,\mathrm{d}M_{s}$
is a standard $d$-dimensional Brownian motion, where
$M_{t}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(M_{t}^{[v_{1}]},\dots,(M_{t}^{[v_{d}]})$,
which is a local martingale. Moreover, it holds that

$\mathrm{d}V_{t}=\mathrm{d}M_{t}+b(V_{t})-\nabla\Phi(X_{t})\,\mathrm{d}t=\sqrt{2}\sigma(V_{t})\mathrm{d}B_{t}+b(V_{t})-\nabla\Phi(X_{t})\,\mathrm{d}t,$

so $(X_{t},V_{t})$ is a weak solution to the SDE (1.1) with initial
distribution $h\mu$ under $P_{h}$.

Finally, in this context, the statement on hypocoercivity (Theorem 1.1) shows
that for every $\theta_{1}>1$, there is an explicitly computable
$\theta_{2}\in(0,\infty)$ depending on the choice of $\Sigma$ and $\Phi$, such
that the transition semigroup $(p_{t})_{t\geq 0}$ satisfies

$\|p_{t}g-\int_{E}g\,\mathrm{d}\mu\|_{L^{2}(\mu)}\leq\theta_{1}\mathrm{e}^{-\theta_{2}t}\|g-\int_{E}g\,\mathrm{d}\mu\|_{L^{2}(\mu)}$
(5.2)

for all $g\in L^{2}(\mu)$ and $t\geq 0$. In particular, this implies that the
probability law $P_{\mu}$ on the space of continuous paths on $E$ with initial
distribution (and invariant measure) $\mu$ has the strong mixing property,
i.e. for any Borel sets $A_{1},A_{2}$ on the path space, it holds that

$P_{\mu}(\varphi_{t}A_{1}\cap A_{2})\to
P_{\mu}(A_{1})P_{\mu}(A_{2})\quad\text{ as }t\to\infty,$

where
$\varphi_{t}A_{1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{(Z_{s})_{s\geq
0}\in C([0,\infty),E)\mid(Z_{s+t})_{s\geq 0}\in A_{1}\\}$. This follows from
(5.2) and associatedness of the semigroups to the probability law $P_{\mu}$,
see for example [15, Remark 2.1.13].

###  5.3 Corresponding Fokker-Planck equation

In this part we give a reformulation of the convergence rate result detailed
in Section 5.1 for readers which are more familiar with the classical Fokker-
Planck formulation for probability densities. In the current literature,
Fokker-Planck equations are more often expressed as equations on measures,
rather than functions. For example, in the non-degenerate case, exponential
convergence in total variation to a stationary solution is studied in [18],
which includes further references to related works. Our goal here however is
simply to make the convergence result immediately applicable to less
specialized readers in the form of the estimate (5.4) for solutions to the
Cauchy problem associated with the operator defined in (5.3), hence we stick
to the expression via probability densities.

Given a Kolmogorov backwards equation of the form
$-\partial_{t}u(x,t)=L^{\mathrm{K}}u(x,t)$, the corresponding Fokker-Planck
equation is given by $\partial_{t}f(x,t)=L^{\mathrm{FP}}f(x,t)$, where
$L^{\mathrm{FP}}=(L^{\mathrm{K}})^{\prime}$ is the adjoint operator of
$L^{\mathrm{K}}$ in $L^{2}(\mathbb{R}^{d},\mathrm{d}x)$, restricted to smooth
functions. In our setting, $L^{\mathrm{K}}=L$ produces via integration by
parts for $f\in D$:

$L^{\mathrm{FP}}f=\sum_{i,j=1}^{d}\partial_{v_{i}}(a_{ij}\partial_{v_{j}}f+v_{j}a_{ij}f)-v\cdot\nabla_{x}f+\nabla\Phi\nabla_{v}f.$
(5.3)

Consider the Fokker-Planck Hilbert space
$\widetilde{H}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=L^{2}(E,\widetilde{\mu})$,
where

$\widetilde{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(2\pi)^{-\frac{d}{2}}\mathrm{e}^{\Phi(x)+\frac{v^{2}}{2}}\,\mathrm{d}x\otimes\mathrm{d}v.$

Then a unitary Hilbert space transformation between $H$ and $\widetilde{H}$ is
given by

$T:H\to\widetilde{H},\quad Tg=\rho g\quad\text{ with
}\quad\rho(x,v)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{e}^{-\Phi(x)-\frac{v^{2}}{2}}.$

Let $(T_{t})_{t\geq 0}$ be the semigroup on $H$ generated by $(L,D(L))$ and
denote by $(T_{t}^{*})_{t\geq 0}$ and $L^{*}$ the adjoint semigroup on $H$ and
its generator, respectively. It is evident that for $f\in D$, $L^{*}$ is given
as $L^{*}f=(S+A)f$, where $S$ and $A$ refer to the symmetric and antisymmetric
components of $L$ respectively, as defined in Definition 3.2. As mentioned in
3.11, we achieve the exact same results for the equation corresponding to
$L^{*}$ as for the one corresponding to $L$, which we considered in Section 3.
In particular, $(L^{*},D)$ is essentially m-dissipative and its closure
$(L^{*},D(L^{*}))$ generates $(T_{t}^{*})_{t\geq 0}$, which converges
exponentially to equilibrium with the same rate as $(T_{t})_{t\geq 0}$.

Let
$\widetilde{T}_{t}g\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=T(T_{t}^{*})T^{-1}g$
for $t\geq 0$, $g\in\widetilde{H}$. Then $(\widetilde{T}_{t})_{t\geq 0}$ is a
strongly continuous contraction semigroup on $\widetilde{H}$ with the
generator $(TL^{*}T^{-1},T(D(L^{*})))$. It is easy to see that
$L^{\mathrm{FP}}=TL^{*}T^{-1}$, so for each initial condition
$u_{0}\in\widetilde{H}$,
$u(t)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\widetilde{T}_{t}u_{0}$
is a mild solution to the Fokker-Planck Cauchy problem. Note that for $\Phi\in
C^{\infty}(\mathbb{R}^{d})$, the transformation $T$ leaves $D$ invariant,
which implies $D\subset T(D(L^{*}))$ and essential m-dissipativity of
$(L^{\mathrm{FP}},D)$ on $\widetilde{H}$.

If $u_{0}\in T(D(L^{*}))$, then

$\partial_{t}\widetilde{T}_{t}u_{0}=T(L^{*}T_{t}^{*})T^{-1}u_{0},$

and therefore

$\displaystyle\int_{E}\partial_{t}u(t)f\,\mathrm{d}(x,v)$
$\displaystyle=\int_{E}L^{*}T_{t}^{*}T^{-1}u_{0}f\,\mathrm{d}\mu=\int_{E}T_{t}^{*}T^{-1}u_{0}Lf\,\mathrm{d}\mu$
$\displaystyle=\int_{E}TT_{t}^{*}T^{-1}u_{0}Lf\,\mathrm{d}(x,v)=\int_{E}L^{\mathrm{FP}}u(t)f\,\mathrm{d}(x,v),$

so $u(t)$ is also a classical solution. Due to the invariance of $\mu$ for
$L$, a stationary solution is given by $\rho$ and by Theorem 1.1, for every
$\theta_{1}>1$ and the appropriate $\theta_{2}$ it holds that

$\displaystyle\left\|u(t)-\rho(u_{0},\rho)_{\widetilde{H}}\right\|_{\widetilde{H}}$
$\displaystyle=\left\|T_{t}^{*}T^{-1}u_{0}-(T^{-1}u_{0},1)_{H}\right\|_{H}$
(5.4)
$\displaystyle\leq\theta_{1}\mathrm{e}^{-\theta_{2}t}\left\|T^{-1}u_{0}-(T^{-1}u_{0},1)_{H}\right\|_{H}$
$\displaystyle=\theta_{1}\mathrm{e}^{-\theta_{2}t}\left\|u_{0}-\rho(u_{0},\rho)_{\widetilde{H}}\right\|_{\widetilde{H}}.$

This shows exponential convergence to a stationary state for solutions to the
Fokker-Planck equation.

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