# Functional inequalities for a family of infinite-dimensional diffusions with
degenerate noise

Fabrice Baudoin∗ Department of Mathematics
University of Connecticut
Storrs, CT 06269, USA<EMAIL_ADDRESS>,  Maria Gordina∗∗ Department
of Mathematics
University of Connecticut
Storrs, CT 06269, USA<EMAIL_ADDRESS>,  David P. Herzog‡ Department
of Mathematics
Iowa State University
Ames, IA 50311, U.S.A<EMAIL_ADDRESS>,  Jina Kim‡ Department of
Mathematics
Trinity University
San Antonio, TX 78212, U.S.A<EMAIL_ADDRESS>and  Tai Melcher† Department
of Mathematics
University of Virginia
Charlottesville, VA 22903, USA<EMAIL_ADDRESS>
###### Abstract.

For a family of infinite-dimensional diffusions with degenerate noise, we
develop a modified $\Gamma$ calculus on finite-dimensional projections of the
equation in order to produce explicit functional inequalities that can be
scaled to infinite dimensions. The choice of our $\Gamma$ operator appears
canonical in our context, as the estimates depend only on the induced control
distance. We apply the general analysis to a number of examples, exploring
implications for quasi-invariance and uniqueness of stationary distributions.

###### Key words and phrases:

quasi-invariance, hypoellipticity, Kolmogorov diffusion, Wang-Harnack
inequality

###### 1991 Mathematics Subject Classification:

Primary 60J60, 28C20; Secondary 35H10

11footnotemark: 1$\ast$ This research was supported in part by NSF grant DMS-
DMS-2247117.

22footnotemark: 2$\ast\ast$ Research was supported in part by NSF grant
DMS-2246549. The author acknowledges the support by the Hausdorff Center of
Mathematics (Bonn, Germany) and the IHES (France), where parts of the work
were completed.

33footnotemark: 3${\ddagger}$ Research was supported in part by NSF grants
DMS-1612898 and DMS-1855504

44footnotemark: 4${\ddagger}$ Research was supported in part by NSF grants
DMS-1612898 and DMS-1855504

55footnotemark: 5$\dagger$ Research was supported in part by NSF grant
DMS-1255574.

###### Contents

  1. 1 Introduction
  2. 2 The finite-dimensional setting
    1. 2.1 The main equation and hypoellipticity
    2. 2.2 Elements of Bakry-Émery calculus
  3. 3 Functional inequalities by modified gradients
    1. 3.1 Picking $G$ based on 3.2
    2. 3.2 Reverse log-Sobolev and Wang-Harnack inequalities
    3. 3.3 Transportation cost inequalities
    4. 3.4 The iterate $G_{2}$ of $G$
  4. 4 Estimating the distance $\rho_{t}$ in examples
    1. 4.1 Kronecker product form of the examples
    2. 4.2 Examples
      1. 4.2.1 Linear kinetic Fokker-Planck equation
      2. 4.2.2 Coupled oscillators
      3. 4.2.3 Oscillators with some damping
  5. 5 From finite to infinite dimensions
    1. 5.1 The infinite-dimensional setting
    2. 5.2 Infinite-dimensional Wang-Harnack inequality
    3. 5.3 Mutual absolute continuity at time infinity
      1. 5.3.1 Linear kinetic Fokker-Planck
      2. 5.3.2 Oscillators with some damping

##  1\. Introduction

Understanding when measures on infinite-dimensional spaces possess smoothness
properties reminiscent of those on $\mathbf{R}^{n}$ is a fundamental problem
in the theory of diffusion processes. In the finite-dimensional context of
$\mathbf{R}^{n}$, Hörmander’s bracket generating condition [35] provides the
criteria to determine when the law of the diffusion with smooth $(C^{\infty})$
coefficients has a smooth density with respect to Lebesgue measure. At the
level of the associated diffusion, Hörmander’s condition translates to how the
_external_ randomness present in the equation propagates _internally_ to
produce a distribution with $C^{\infty}$ density. Thus, provided there is
sufficient randomness in the equation to ensure the needed propagation, the
existence and smoothness of the density follows from Hörmander’s result.

While the mantra of _sufficient noise implies smoothness of the law_ applies
readily in the finite-dimensional setting via Hörmander’s classical
hypoellipticity result, the story in infinite dimensions is more nuanced. For
example, even in relatively simple settings when noise acts on every basis
direction in an infinite-dimensional Hilbert space, which in particular acts
as a _natural_ phase space for the stochastic solution, the laws of two
solutions started from close initial conditions may be mutually singular at
any fixed time $t>0$. This, in turn, implies that the associated Markov
semigroup is not strong Feller [32, Example 3.15]. Furthermore, while
Hörmander’s condition in finite dimensions allows for the development of
regularity theory for the corresponding PDEs even in the absence of
ellipticity, such PDE techniques are not available in infinite dimensions. For
example, even in the context of an infinite-dimensional Brownian motion, the
classical Harnack inequality enjoyed by similar, non-degenerate finite-
dimensional processes fails to hold [4].

Because of the prevalence of degenerate noises in applications and these
nuances, understanding what hypoellipticity means for infinite-dimensional
diffusions continues to be an active area of research. On the one hand,
significant progress has been made in this direction in the context of fluid
mechanics. Indeed, building off of the pioneering works on unique ergodicity
of stochastically-forced PDEs [14, 16, 24, 37] as well as the known behavior
of the finite-dimensional Galerkin approximations [23, 46], Hairer and
Mattingly introduced the notion of _asymptotic strong Feller property_ in
order to prove unique ergodicity of the two-dimensional Navier–Stokes
equations on the period box under highly degenerate stochastic forcing [32].
This property was further developed and investigated in the works [33, 31,
41]. In this context, there is just enough smoothing _at time infinity_ , as
defined by the asymptotic strong Feller property, to conclude uniqueness of
steady states. We refer also to the work [25] which studies the Boussinesq
equations under degenerate stochastic forcing, validating the asymptotic
strong Feller property in that context.

On the other hand, smoothness of measures in infinite dimensions can be
understood as quasi-invariance under transformations such as translations.
This allows for the definition of smoothness even in the absence of a natural
reference measure such as Lebesgue measure. This point of view was pioneered
by Malliavin in [40] leading to development of Malliavin calculus. More
relevant to the current paper is the connection between hypoellipticity in
infinite dimensions to quasi-invariance properties and their relation to
classical functional inequalities, especially in the context of heat kernel
measures on infinite-dimensional Heisenberg-like groups [6, 5, 4, 20, 21, 28].
In these settings, the noise structure in the equations is different than in
the fluid models above, as the driving external randomness is infinite-
dimensional as opposed to acting on a few low frequencies in Fourier space.
Furthermore, techniques from Dirichlet forms, classical Cameron-Martin-
Girsanov theorem and $\Gamma$ calculus are often employed in place of Harris’
theorem and Malliavin calculus in the fluids setting. We refer also to [17,
26] for related work.

The goal of this paper is to make progress on understanding the meaning of
hypoellipticity in infinite-dimensions by developing a modified $\Gamma$
calculus. We build off of understanding from previous work in [9, 12, 5, 7].
Specifically, we design a modified $\Gamma$ operator in order to obtain
classical functional inequalities, e.g. Wang–Harnack and reverse log–Sobolev,
for a class of infinite-dimensional diffusions arising as solutions to certain
stochastic differential equations. These are generalizations of the Kolmogorov
diffusion studied in [9, 12, 5, 7]. An important contribution of this work is
that the choice of our modified $\Gamma$ operator appears canonical, as the
constants in the functional inequality bounds are independent of both the
noise and spatial dimension for finite-dimensional projections of the
equation. However, the dependence on these parameters is intrinsic in the
induced distance, which we show can be estimated in a variety of examples. The
existence of a natural notion of distance here is notable, as previously there
has been no geometry in which to work with this class of distributions. This
is in contrast to other hypoelliptic models (like the Heisenberg group) where
there is a natural geometric framework coming from the sub-Riemannian
distance. For this class of diffusions, we also study large-time properties
when the structure allows for it, developing a criteria for mutual absolute
continuity of stationary distributions for the associated Markov semigroup.
This is done by using the deduced Wang-Harnack type inequality in finite
dimensions, and scaling it appropriately to infinite dimensions.

The organization of this paper is as follows. In Section 2, we introduce the
finite-dimensional setting which will later be scaled to infinite dimensions
in Section 5. In Section 3, we develop our modified $\Gamma$ calculus in the
context of the finite-dimensional setting. In particular, we derive our choice
of $\Gamma$ operator and deduce a number of functional inequalities based on
this choice. In Section 4, we estimate the control distance associated to our
choice of $\Gamma$ operator in several concrete examples. Based on the
derivations of these functional inequalities, the control distance is the only
term one has left to estimate to produce fully explicit estimates. Finally in
Section 5, we scale the functional inequalities and the finite-dimensional
setting to infinite dimensions, obtaining criteria for quasi-invariance as
well as mutual absolute continuity of invariant probability measures. At the
end of Section 5, we revisit some of the examples discussed in Section 4,
applying the results obtained in this section.

##  2\. The finite-dimensional setting

###  2.1. The main equation and hypoellipticity

Let $\mathcal{B}$ denote the Borel sigma field of subsets of $\mathbf{R}^{n}$.
In the finite-dimensional setting, we consider the following stochastic
differential equation (SDE) on $\mathbf{R}^{n}$

(2.1) $\displaystyle dx_{t}=Ax_{t}\,dt+\sigma\,dB_{t},$

where $A$ and $\sigma$ are $n\times n$ real matrices, and $B_{t}$ is a
standard, $n$-dimensional Brownian motion defined on a probability space
$(\Omega,\mathcal{F},\mathbf{P},\mathbf{E})$. For all initial conditions
$x\in\mathbf{R}^{n}$, equation (2.1) has a unique pathwise solution
$x_{t}(x)$, defined for all times $t\geqslant 0$, which can be explicitly
written as

(2.2) $\displaystyle x_{t}(x)=e^{tA}x+\int_{0}^{t}e^{(t-s)A}\sigma dB_{s}.$

Unless we must emphasize the initial condition, we will write $x_{t}$ as
shorthand notation for a generic solution of (2.1).

Solutions of (2.1) are Markovian and we let $\\{P_{t}\\}_{t\geqslant 0}$
denote the corresponding Markov semigroup. We recall that
$\\{P_{t}\\}_{t\geqslant 0}$ acts on bounded, $\mathcal{B}$-measurable
functions $f:\mathbf{R}^{n}\rightarrow\mathbf{R}$ by

$\displaystyle P_{t}f(x):=\mathbf{E}f(x_{t}(x)),\,\,t\geqslant 0,$

and acts dually on a probability measure $\nu$ on
$\left(\mathbf{R}^{n},\mathcal{B}\right)$ via

$\displaystyle\nu
P_{t}(A):=\int_{\mathbf{R}^{n}}\nu(dx)P_{t}\mathbf{1}_{A}(x),\,\,\,\,t\geqslant
0,\,\,A\in\mathcal{B}.$

A probability measure $\nu$ on $\left(\mathbf{R}^{n},\mathcal{B}\right)$ is
called a _stationary distribution_ if $\nu P_{t}=\nu$ for all $t\geqslant 0$.
For $x\in\mathbf{R}^{n}$, $t\geqslant 0$ and $A\in\mathcal{B}$, we let

$\displaystyle P_{t}(x,A):=P_{t}\mathbf{1}_{A}(x)=\mathbf{P}\\{x_{t}(x)\in
A\\}$

denote the _Markov transition probability_ associated to
$\\{P_{t}\\}_{t\geqslant 0}$. Throughout, $L$ will denote the following
second-order operator

(2.3) $\displaystyle L=\sum_{i=1}^{n}(Ax)_{i}\frac{\partial}{\partial
x_{i}}+\frac{1}{2}\sum_{i,j=1}^{n}(\sigma\sigma^{\ast})_{ij}\frac{\partial^{2}}{\partial
x_{i}\partial x_{j}}.$

Note that $L$ corresponds to the action of the infinitesimal generator of
$\\{P_{t}\\}_{t\geqslant 0}$ on a domain of sufficiently smooth functions,
e.g. $C^{2}$ functions $f:\mathbf{R}^{n}\rightarrow\mathbf{R}$ with compact
support. We offer the slight abuse of terminology and refer to $L$ as the
generator of the Markov process $x_{t}$ throughout.

We are interested in the case when the noise in equation (2.1) is _degenerate_
, i.e. $\text{rank}(\sigma)<n$, but the process $x_{t}$ has a transition
probability density function $p_{t}(x,y)$ respect to Lebesgue measure which is
$C^{\infty}$ for all
$(t,x,y)\in(0,\infty)\times\mathbf{R}^{n}\times\mathbf{R}^{n}$. When
$\text{rank}(\sigma)<n$, the existence and smoothness of the transition
density, which will be referred to throughout as _hypoellipticity_ , is not
immediate precisely because the noise is degenerate. However, in this context,
hypoellipticity can be established under further conditions on the interaction
between $A$ and $\sigma$ in essentially two ways. The first and perhaps most
utilized way is to apply Hörmander’s hypoellipticity theorem [35] (see also
[48]). However, Hörmander’s result is more powerful than needed in the context
(2.1). For our purposes, a more direct way is to verify the _Kalman rank
condition_ [36] (see 1 below) and show that this condition implies
hypoellipticity. The Kalman rank condition is usually employed to ensure
controllability of the resulting ordinary differential equation (ODE) when the
independent Brownian motions in (2.1) are replaced by deterministic controls.

######  Assumption 1.

The _Kalman rank condition_ is satisfied; that is, if we define the matrix

$\displaystyle
A_{\sigma}:=\begin{bmatrix}\sigma&A\sigma&A^{2}\sigma&\ldots&A^{n-1}\sigma\end{bmatrix},$

then

$\displaystyle\text{rank}(A_{\sigma})=n.$

Consider the mean $m_{t}(x)$ and covariance $\Sigma_{t}$ of the process (2.2)
given by

(2.4) $\displaystyle m_{t}(x)=e^{tA}x\qquad\text{ and
}\qquad\Sigma_{t}=\int_{0}^{t}e^{(t-s)A}\sigma\sigma^{\ast}e^{(t-s)A^{\ast}}\,ds.$

A short argument (see 2.1 below) shows that 1 implies that $\Sigma_{t}$ as in
(2.4) is invertible for all $t>0$. This in turn implies that the process
$x_{t}$ is multivariate Gaussian with an explicit transition density given by

(2.5) $\displaystyle
p_{t}(x,y)=\frac{1}{\det(2\pi\Sigma_{t})^{1/2}}\exp\big{(}-\tfrac{1}{2}\langle\Sigma_{t}^{-1}(y-m_{t}(x)),y-m_{t}(x)\rangle\big{)},\,\,\,x,y\in\mathbf{R}^{n},\,t>0,$

where $\langle\cdot,\cdot\rangle$ denotes the standard inner product on
$\mathbf{R}^{n}$. Smoothness of $p_{t}(x,y)$ for
$(t,x,y)\in(0,\infty)\times\mathbf{R}^{n}\times\mathbf{R}^{n}$ can then be
readily verified from (2.4) and (2.5). It should be noted that the proof of
2.1 is Malliavin’s probabilistic proof of Hörmander’s theorem [38, 39, 44] in
the simplified context (2.1).

######  Lemma 2.1.

Suppose that 1 is satisfied. Then for every $t>0$, $\Sigma_{t}$ in (2.4) is
invertible.

###### Proof.

Fix $t>0$ and suppose that there exists $x\in\mathbf{R}^{n}$ such that
$\langle\Sigma_{t}x,x\rangle=0$. This implies

$\displaystyle
0=\langle\Sigma_{t}x,x\rangle=\int_{0}^{t}|\sigma^{\ast}e^{sA^{\ast}}x|^{2}\,ds.$

Hence, by continuity, $\sigma^{\ast}e^{sA^{\ast}}x=0$ for all $s\in[0,t]$.
Differentiating $\sigma^{\ast}e^{sA^{\ast}}x$ repeatedly with respect to $s$
and evaluating at $s=0$ implies that

$\displaystyle\sigma^{\ast}(A^{\ast})^{m}x=0\text{ for all }m=0,1,2,\ldots$

Note by 1 this implies $x=0$ since the rank of the matrix and its transpose
are the same. ∎

###  2.2. Elements of Bakry-Émery calculus

In Section 3, we provide a framework for establishing classical functional
inequalities in the finite-dimensional hypoelliptic setting (2.1), where the
constants in the bounds are independent of the spatial dimension $n$.
Importantly, we employ a modification of the Bakry-Émery calculus [1] by
proposing a _natural_ gradient associated to the system (2.1) which plays the
role analogous to the usual carré du champ in the uniformly elliptic setting,
e.g when $\text{rank}(\sigma)=n$ in equation (2.1) [1]. Here, we recall some
of the basic elements of the Bakry-Émery calculus.

To the generator $L$ as in (2.3), we associate the _carré du champ_ $\Gamma$
and its _iterate_ $\Gamma_{2}$, which for $f,g\in
C^{\infty}(\mathbf{R}^{n};\mathbf{R})$ are defined by

$\displaystyle\Gamma(f,g):=\tfrac{1}{2}L(fg)-$
$\displaystyle\tfrac{1}{2}gLf-\tfrac{1}{2}fLg,\qquad\Gamma(f):=\Gamma(f,f),$
$\displaystyle\Gamma_{2}(f)$
$\displaystyle:=\tfrac{1}{2}L\Gamma(f)-\Gamma(f,Lf).$

Importantly, $\Gamma$ and $\Gamma_{2}$ arise naturally in relation to the
Markov semigroup $\\{P_{t}\\}_{t\geqslant 0}$. If, for example, the Markov
process $x_{t}$ has a stationary distribution $\nu$ and
$f:\mathbf{R}^{n}\rightarrow\mathbf{R}$ is bounded measurable, then a formal
calculation using stationarity of $\nu$ gives

(2.6) $\displaystyle\frac{1}{2}\frac{d}{dt}\|P_{t}f\|_{L^{2}(\nu)}^{2}$
$\displaystyle=\int_{\mathbf{R}^{n}}(P_{t}f)LP_{t}f\,d\nu=-\int_{\mathbf{R}^{n}}\Gamma(P_{t}f)\,d\nu,$
$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\mathbf{R}^{n}}\Gamma(P_{t}f)\,d\nu$
$\displaystyle=\int_{\mathbf{R}^{n}}\Gamma(LP_{t}f,P_{t}f)\,d\nu=-\int_{\mathbf{R}^{n}}\Gamma_{2}(P_{t}f)\,d\nu.$

In particular, the forms $\Gamma,\Gamma_{2}$ are the resulting objects that
arise from iterating time with respect to the norm $\|\cdot\|_{L^{2}(\nu)}$.
This calculation has been exploited in several situations in order to study
convergence to equilibrium for SDEs with an explicit stationary distribution
$\nu$ [9, 1, 12, 19, 34, 30, 49]. Such a function was also used to prove a
logarithmic Sobolev inequality in [2], and this approach has been employed to
prove different functional inequalities as described in the monograph [3].

Crucially in what follows, we employ a generalization of $\Gamma$ and
$\Gamma_{2}$. Recalling that $\langle\cdot,\cdot\rangle$ denotes the standard
inner product on $\mathbf{R}^{n}$, for any $n\times n$ real matrix $G$ and
$f,g\in C^{\infty}(\mathbf{R}^{n};\mathbf{R})$, we introduce the notation

(2.7) $\displaystyle\Gamma^{G}(f,g)=\langle G\nabla f,\nabla
g\rangle,\qquad\Gamma^{G}(f)=\Gamma^{G}(f,f),$ (2.8)
$\displaystyle\Gamma_{2}^{G}(f)=\frac{1}{2}L\Gamma^{G}(f)-\Gamma^{G}(f,Lf).$

We see that $\Gamma^{G}$ is a generalization of $\Gamma$ since

$\displaystyle\Gamma(f,g)=\frac{1}{2}\sum_{\ell,m=1}^{n}(\sigma\sigma^{\ast})_{\ell
m}\partial_{\ell}f\partial_{m}g=\Gamma^{\tfrac{\sigma\sigma^{\ast}}{2}}(f,g).$

Similar functionals with a time-independent $G$ were considered in [8, 43].
Below, we find a convenient way to define a time-dependent, _natural_ $G$
associated to $x_{t}$ from which the desired functional inequalities can be
obtained and such that the constants in the bounds of these inequalities are
independent of the dimension $n$. Although we will not always need to assume
it, $G$ should be thought of as symmetric, positive-definite so that
$\Gamma^{G}(f)$ is equivalent to $|\nabla f|^{2}$, and so can be interpreted
as changing the underlying metric.

##  3\. Functional inequalities by modified gradients

The goal of this section is to construct an appropriate, time-dependent matrix
$G$ so that, by using $\Gamma^{G}$ and $\Gamma_{2}^{G}$ defined in (2.7) and
(2.8), we arrive at various functional inequalities where the constants in the
inequalities are independent of the spatial dimension $n$. We will see that by
using the framework outlined below, we arrive at a natural choice for $G$.
Note that a similar strategy was employed using a diagonal matrix in the
setting of the Kolmogorov diffusion in [5, 13]. Our construction holds in the
more general setting (2.1) for a convenient symmetric positive-definite matrix
$G$. We also remark that a similar construction was used in the paper [42] in
the context of chain of oscillators, but the matrix used there is not time-
dependent. We finally point out [27] where dimension dependent Hardy-
Littlewood-Sobolev inequalities for similar types of operators are obtained
using heat kernel estimates.

We begin by computing $\Gamma_{2}^{G}$ for a general, symmetric matrix $G$.
Here and in what follows, all matrices below are assumed to be spatially
constant.

######  Proposition 3.1.

Suppose that $G$ is an $n\times n$ symmetric matrix. Then for all $f\in
C^{\infty}(\mathbf{R}^{n};\mathbf{R})$, we have the formula

(3.1)
$\displaystyle\Gamma_{2}^{G}(f)=-\Gamma^{AG}(f)+\frac{1}{2}\sum_{\ell}\Gamma^{G}((\sigma^{\ast}\nabla
f)_{\ell}).$

If we furthermore assume that $G$ is nonnegative-definite, then

(3.2) $\displaystyle\Gamma_{2}^{G}(f)\geqslant-\Gamma^{AG}(f)$

for all $f\in C^{\infty}(\mathbf{R}^{n};\mathbf{R})$.

###### Proof.

Let $f\in C^{\infty}(\mathbf{R}^{n};\mathbf{R})$. Standard calculations using
symmetry of $G$ give

$\displaystyle L\Gamma^{G}(f)$
$\displaystyle=2\Gamma^{G}(f,Lf)-2\Gamma^{AG}(f)+\sum_{\ell,m}(\sigma\sigma^{\ast})_{\ell
m}\langle G\nabla(\partial_{\ell}f),\nabla(\partial_{m}f)\rangle.$

Relation (3.1) then follows using matrix arithmetic on the last term on the
right-hand side above. Under the additional assumption that $G$ is
nonnegative-definite, the inequality (3.2) follows immediately from (3.1). ∎

To obtain a gradient estimate using a time-dependent matrix $G$ in
$\Gamma^{G}$, we fix a horizon time $t>0$, and $f\in
L^{\infty}(\mathbf{R}^{n};\mathbf{R})$ and consider the functional

(3.3) $\displaystyle\phi(s):=P_{s}\Gamma^{G(s)}(P_{t-s}f),\,\,\,s\in[0,t),$

for a differentiable matrix $s\mapsto G(s)$ on $[0,t]$ which is $n\times n$,
symmetric. Below, we see that the expression (3.3) plays a role similar to the
$L^{2}(\nu)$ norm in (2.6). The following lemma yields the expression which
will, in turn, lead to the definition of a convenient $G$.

######  Lemma 3.2.

Fix $t>0$. Suppose that 1 is satisfied and that $s\mapsto G(s)\in
C^{1}([0,t];M_{n\times n})$ is symmetric and nonegative-definite on $[0,t]$.
For any $f\in L^{\infty}(\mathbf{R}^{n};\mathbf{R})$, let $\phi$ be as in
(3.3). Then $\phi$ is defined and continuously differentiable on $(0,t)$ and,
moreover, for $s\in(0,t)$ we have

(3.4)
$\displaystyle\phi^{\prime}(s)=P_{s}2\Gamma_{2}^{G(s)}(P_{t-s}f)+P_{s}\Gamma^{G^{\prime}(s)}(P_{t-s}f)\geqslant
P_{s}\Gamma^{G^{\prime}(s)-2AG(s)}(P_{t-s}f).$

######  Remark 3.3.

The fact that $\phi$ as in (3.3) makes sense, is continuously differentiable
on $(0,t)$ and has _nice_ properties permitting the calculations leading to
the proof of 3.2 follows from 1 and the explicit representation

(3.5) $\displaystyle P_{t}f(x)$
$\displaystyle=\frac{1}{\det(2\pi\Sigma_{t})^{1/2}}\int_{\mathbf{R}^{n}}f(y)\exp\big{(}-\tfrac{1}{2}\langle\Sigma_{t}^{-1}(y-m_{t}(x)),y-m_{t}(x)\rangle\big{)}\,dy$
(3.6) $\displaystyle=\mathbf{E}f(\Sigma_{t}^{1/2}Z+m_{t}(x)),$

where $t>0$, $m_{t}(x)$ and $\Sigma_{t}$ are as in (2.4), and $Z$ is the
standard normal distribution on $\mathbf{R}^{n}$. Recalling that the
covariance matrix $\Sigma_{t}$ is invertible for $t>0$ under 1 by 2.1, it
follows from (3.5) that the semigroup $\\{P_{t}\\}_{t\geq 0}$ is strong Feller
and Mehler. Furthermore, one can use formula (3.5) to explicitly find
derivatives of $P_{t}f(x)$, provided $t>0$, to fully justify the regularity of
$\phi$ on $(0,t)$ and the remaining derivative calculations in the proof of
3.2. Later we will also use formula (3.6) to do similar calculations under
further regularity hypotheses on the test function $f$.

###### Proof of 3.2.

Following 3.3, we observe that for $s\in(0,t)$ and $f\in
L^{\infty}(\mathbf{R}^{n};\mathbf{R})$, symmetry of $G(s)$ implies

$\displaystyle\phi^{\prime}(s)$
$\displaystyle=P_{s}L\Gamma^{G(s)}(P_{t-s}f)-2P_{s}\Gamma^{G(s)}(P_{t-s}f,LP_{t-s}f)+P_{s}\Gamma^{G^{\prime}(s)}(P_{t-s}f)$
$\displaystyle=P_{s}2\Gamma_{2}^{G(s)}(P_{t-s}f)+P_{s}\Gamma^{G^{\prime}(s)}(P_{t-s}f).$

The bound in the result follows after applying 3.1. ∎

###  3.1. Picking $G$ based on 3.2

There are many workable choices for the matrix $G(s)$ corresponding to the
dynamics (2.1). Given the bound in relation (3.4), we argue that the choice
given in the remainder of this section is canonical. In particular, fixing a
horizon time $t>0$, we define the matrix-valued function $s\mapsto G(s,t)$ on
$[0,t]$ by

(3.7) $\displaystyle
G(s,t)=\int_{s}^{t}e^{(s-v)A}\sigma\sigma^{\ast}e^{(s-v)A^{\ast}}\,dv.$

Note that this matrix satisfies the backward matrix-valued ODE

$\displaystyle\begin{cases}\partial_{s}G(s,t)=-\sigma\sigma^{\ast}+AG(s,t)+G(s,t)A^{\ast},\,s\in[0,t]&\\\
G(t,t)=0.&\end{cases}$

By symmetry of $G(s,t)$, for any $x\in\mathbf{R}^{n}$ we have that

$\displaystyle\langle\partial_{s}G(s,t)x,x\rangle=-\langle\sigma\sigma^{\ast}x,x\rangle+\langle
2AG(s,t)x,x\rangle.$

We observe that picking $G$ in this way allows us to replace the right-hand
side of the bound in (3.4) with

$\displaystyle-
P_{s}\Gamma^{\sigma\sigma^{\ast}}(P_{t-s}f)=-2P_{s}\Gamma(P_{t-s}f)=-\frac{d}{ds}P_{s}(P_{t-s}f)^{2}.$

As a consequence, we produce a reverse Poincaré-type inequality/gradient bound
for the semigroup associated to the modified operator $\Gamma^{G}$.

######  Proposition 3.4 (Reverse Poincaré-type inequality).

Let $t>0$ and $s\mapsto G(s,t)$ be as in (3.7) and suppose that 1 is
satisfied. Then for all $f\in L^{\infty}(\mathbf{R}^{n};\mathbf{R})$ we have
the explicit bound

(3.8) $\displaystyle\Gamma^{G(0,t)}(P_{t}f)\leqslant P_{t}f^{2}-(P_{t}f)^{2}.$

######  Remark 3.5.

By a nearly identical argument to the one used in 2.1, under 1, the matrix
$G(0,t)$ is invertible for $t>0$. Thus the modified gradient on the lefthand
side of (3.8) is comparable to the usual gradient.

###### Proof of Proposition 3.4.

By 1 and a density argument, it suffices to prove estimate (3.8) for $f\in
C^{\infty}_{0}(\mathbf{R}^{n};\mathbf{R})$ where
$C_{0}^{\infty}(\mathbf{R}^{n};\mathbf{R})$ denotes the space of smooth real-
valued functions with compact support in $\mathbf{R}^{n}$. If we let $\phi(s)$
be as in (3.3) with $G(s):=G(s,t)$, then 3.2 implies

(3.9) $\displaystyle\phi^{\prime}(s)$
$\displaystyle=2P_{s}\Gamma_{2}^{G(s)}(P_{t-s}f)+P_{s}\Gamma^{G^{\prime}(s)}(P_{t-s}f)\geqslant-2P_{s}\Gamma(P_{t-s}f)$

for $s\in(0,t)$. Next, notice that for $s\in(0,t)$

$\displaystyle\frac{d}{ds}P_{s}(P_{t-s}f)^{2}=2P_{s}\Gamma(P_{t-s}f).$

Thus for $\epsilon>0$ sufficiently small, integrating both sides of (3.9) from
$\epsilon$ to $t-\epsilon$ we arrive at the bound

$\displaystyle
P_{\epsilon}\Gamma^{G(\epsilon,t)}(P_{t-\epsilon}f)-P_{t-\epsilon}\Gamma^{G(t-\epsilon,t)}(P_{\epsilon}f)\leqslant
P_{t-\epsilon}(P_{\epsilon}f)^{2}-P_{\epsilon}(P_{t-\epsilon}f)^{2}.$

In order to pass to the limit as $\epsilon\rightarrow 0$ in the above to
arrive at (3.8) for $f\in C^{\infty}_{0}(\mathbf{R}^{n};\mathbf{R})$, it
remains to show that

$\displaystyle
P_{t-\epsilon}\Gamma^{G(t-\epsilon,t)}(P_{\epsilon}f)\rightarrow 0\text{ as
}\epsilon\rightarrow 0.$

Using (3.6) for $t=\epsilon$, we find that

$\displaystyle\Gamma^{G(t-\epsilon,t)}(P_{\epsilon}f)=\langle
G(t-\epsilon,t)\mathbf{E}e^{\epsilon A^{\ast}}\nabla
f(\Sigma_{\epsilon}^{1/2}Z+m_{\epsilon}(x)),\mathbf{E}e^{\epsilon
A^{\ast}}\nabla f(\Sigma_{\epsilon}^{1/2}Z+m_{\epsilon}(x))\rangle$

is both uniformly bounded in $\epsilon$ and converges to $0$ as
$\epsilon\rightarrow 0$. This now establishes the bound (3.8) for $f\in
C^{\infty}_{0}(\mathbf{R}^{n};\mathbf{R})$, finishing the proof. ∎

Because of the significance of the matrix $G(0,t)$, we introduce the following
notation.

######  Notation 3.1.

Throughout the remainder of the paper, we set

(3.10) $\displaystyle G_{t}:=G(0,t)$

where $G(s,t)$ is as in (3.7).

###  3.2. Reverse log-Sobolev and Wang-Harnack inequalities

We now see that a similar calculation to the one used to prove the reverse
Poincaré inequality in Section 3.1 can also be used to establish a reverse log
Sobolev inequality and the Wang-Harnack inequality.

To prove the log-Sobolev inequality, for computational ease, we introduce the
spaces $C^{\infty}_{\epsilon}(\mathbf{R}^{n};[0,\infty))$, $\epsilon>0$,
defined by

$\displaystyle
C_{\epsilon}^{\infty}(\mathbf{R}^{n};[0,\infty)):=\\{f\,:\,f=\varphi+\epsilon,\varphi\in
C_{0}^{\infty}(\mathbf{R}^{n};[0,\infty))\\}.$

Here we recall again that $C_{0}^{\infty}(\mathbf{R}^{n};[0,\infty))$ denotes
the space of smooth functions $\varphi:\mathbf{R}^{n}\rightarrow[0,\infty)$
with compact support in $\mathbf{R}^{n}$. Similar to 3.2, we need the
following computation.

######  Proposition 3.6.

Fix $t,\epsilon>0$ and suppose that $s\mapsto G(s)\in C^{1}([0,t];M_{n\times
n})$ is symmetric for any $s\in[0,t]$ and that 1 is satisfied. For $f\in
C_{\epsilon}^{\infty}(\mathbf{R}^{n};[0,\infty))$ and $s\in(0,t)$, define

(3.11) $\displaystyle\psi(s):=P_{s}[(P_{t-s}f)\Gamma^{G(s)}(\log P_{t-s}f)].$

Then

$\displaystyle\psi^{\prime}(s)$
$\displaystyle=2P_{s}[(P_{t-s}f)\Gamma_{2}^{G(s)}(\log
P_{t-s}f)]+P_{s}[(P_{t-s}f)\Gamma^{G^{\prime}(s)}(\log P_{t-s}f)].$

###### Proof.

Let $t,\epsilon>0$ and $f\in
C_{\epsilon}^{\infty}(\mathbf{R}^{n};[0,\infty))$. Observe that for
$s\in(0,t)$

$\displaystyle\psi^{\prime}(s)$
$\displaystyle=P_{s}L[(P_{t-s}f)\Gamma^{G(s)}(\log
P_{t-s}f)]-P_{s}[L(P_{t-s}f)\Gamma^{G(s)}(\log P_{t-s}f)]$ (3.12)
$\displaystyle\qquad+P_{s}[(P_{t-s}f)\frac{d}{ds}(\Gamma^{G(s)}(\log
P_{t-s}f))].$

Next, notice that

$\displaystyle\frac{d}{ds}(\Gamma^{G(s)}(\log P_{t-s}f))$
$\displaystyle=\Gamma^{G^{\prime}(s)}(\log
P_{t-s}f)-2\Gamma^{G(s)}(LP_{t-s}f/P_{t-s}f,\log P_{t-s}f)$ (3.13)
$\displaystyle=\Gamma^{G^{\prime}(s)}(\log P_{t-s}f)-2\Gamma^{G(s)}(L(\log
P_{t-s}f),\log P_{t-s}f)$ $\displaystyle\qquad-2\Gamma^{G(s)}(\Gamma(\log
P_{t-s}f),\log P_{t-s}f).$

Combining (3.12) with (3.13) gives

$\displaystyle\psi^{\prime}(s)$
$\displaystyle=P_{s}L[(P_{t-s}f)\Gamma^{G(s)}(\log
P_{t-s}f)]-P_{s}[L(P_{t-s}f)\Gamma^{G(s)}(\log P_{t-s}f)]$
$\displaystyle\qquad+P_{s}[(P_{t-s}f)\Gamma^{G^{\prime}(s)}(\log P_{t-s}f)]$
$\displaystyle\qquad-2P_{s}[(P_{t-s}f)\Gamma^{G(s)}(L(\log P_{t-s}f),\log
P_{t-s}f)]$ $\displaystyle\qquad-2P_{s}[(P_{t-s}f)\Gamma^{G(s)}(\Gamma(\log
P_{t-s}f),\log P_{t-s}f)]$ (3.14)
$\displaystyle=2P_{s}[(P_{t-s}f)\Gamma^{G(s)}_{2}(\log
P_{t-s}f)]+2P_{s}[\Gamma(P_{t-s}f,\Gamma^{G(s)}(\log P_{t-s}f))]$
$\displaystyle\qquad+P_{s}[(P_{t-s}f)\Gamma^{G^{\prime}(s)}(\log
P_{t-s}f)]-2P_{s}[(P_{t-s}f)\Gamma^{G(s)}(\Gamma(\log P_{t-s}f),\log
P_{t-s}f)].$

Now observe that if $B=\sigma\sigma^{\ast}/2$, we have

$\displaystyle\Gamma(P_{t-s}f,\Gamma^{G(s)}(\log P_{t-s}f))$
$\displaystyle=\sum_{\ell,m}B_{m\ell}\partial_{\ell}(P_{t-s}f)\partial_{m}(G\nabla(\log
P_{t-s}f)\cdot\nabla(\log P_{t-s}f))$
$\displaystyle=2\sum_{\ell,m,j,k}B_{m\ell}\partial_{\ell}(P_{t-s}f)G_{jk}\partial_{km}^{2}(\log
P_{t-s}f)\partial_{j}(\log P_{t-s}f)$

and

$\displaystyle(P_{t-s}f)\Gamma^{G(s)}(\Gamma(\log P_{t-s}f),\log P_{t-s}f)$
$\displaystyle=(P_{t-s}f)\sum_{j,k}G_{jk}\partial_{k}(\Gamma(\log
P_{t-s}f))\partial_{j}(\log P_{t-s}f)$
$\displaystyle=2(P_{t-s}f)\sum_{\ell,m,j,k}G_{jk}B_{m\ell}\partial_{km}^{2}(\log
P_{t-s}f)\partial_{\ell}(\log P_{t-s}f))\partial_{j}(\log P_{t-s}f)$
$\displaystyle=2\sum_{\ell,m,j,k}B_{m\ell}\partial_{\ell}(P_{t-s}f)G_{jk}\partial_{km}^{2}(\log
P_{t-s}f)\partial_{j}(\log P_{t-s}f).$

The result now follows after considering (3.14) since we just proved that

$\displaystyle 2P_{s}[\Gamma(P_{t-s}f,\Gamma^{G(s)}(\log
P_{t-s}f))]=2P_{s}[(P_{t-s}f)\Gamma^{G(s)}(\Gamma(\log P_{t-s}f),\log
P_{t-s}f)].$

∎

As a corollary of the previous computation, we obtain the following result.
Below, we extend $x\mapsto x\log x$ to $[0,\infty)$ by defining it to be zero
at $x=0$.

######  Corollary 3.7 (Reverse log Sobolev inequality).

Fix $t>0$, suppose 1 is satisfied and that $s\mapsto G(s,t)$ is as in (3.7) .
Then we have the following reverse log Sobolev inequality

(3.15) $\displaystyle(P_{t}f)\Gamma^{G_{t}}(\log P_{t}f)\leqslant 2P_{t}(f\log
f)-2(P_{t}f)\log P_{t}f,$

which is satisfied for all $f\in L^{\infty}(\mathbf{R}^{n};[0,\infty))$ which
are not identically zero.

###### Proof.

Fix $t,\epsilon>0$. We first prove the bound for $f\in
C_{\epsilon}^{\infty}(\mathbf{R}^{n};[0,\infty))$. First note that if $\psi$
is given by (3.11) with $G(s):=G(s,t)$ where $G(s,t)$ is as in (3.7), by 3.6
we have for $s\in(0,t)$

$\displaystyle\psi^{\prime}(s)$
$\displaystyle=2P_{s}[(P_{t-s}f)\Gamma_{2}^{G(s,t)}(\log
P_{t-s}f)]+P_{s}[(P_{t-s}f)\Gamma^{\partial_{s}G(s,t)}(\log P_{t-s}f)]$
$\displaystyle\geqslant-2P_{s}[(P_{t-s}f)\Gamma(\log
P_{t-s}f)]=-\frac{d}{ds}2P_{s}[(P_{t-s}f)\log P_{t-s}f].$

For $\delta>0$ small, integrating the previous inequality from $\delta$ to
$t-\delta$ produces

$\displaystyle P_{\delta}[(P_{t-\delta}f)\Gamma^{G(\delta,t)}(\log
P_{t-\delta}f)]-\ P_{t-\delta}[(P_{\delta}f)\Gamma^{G(t-\delta,t)}(\log
P_{\delta}f)]$ $\displaystyle\leqslant 2P_{t-\delta}[(P_{\delta}f)\log
P_{\delta}f]-2P_{\delta}[(P_{t-\delta}f)\log P_{t-\delta}f].$

Applying a line of reasoning similar to the one used in the proof of 3.6,
taking $\delta\rightarrow 0$ gives the desired estimate (3.15) in the case
when $f\in C_{\epsilon}^{\infty}(\mathbf{R}^{n};[0,\infty))$. Taking
$\epsilon\rightarrow 0$, we obtain the bound for $f\in
C_{0}^{\infty}(\mathbf{R}^{n};[0,\infty))$ which are not identically zero.
Density arguments give the claimed bound for all $f\in
L^{\infty}(\mathbf{R}^{n};[0,\infty))$ which are not identically zero. ∎

Next, we turn our attention to establishing a Wang-Harnack inequality. Later,
this will be used to prove a quasi-invariance result when we allow the spatial
dimension $n\rightarrow\infty$. To setup the statement of the result, suppose
that $K$ is $n\times n$ symmetric, positive-definite. We call a curve
$\gamma\in C^{1}([0,T];\mathbf{R}^{n})$ _subunit_ for $K$ if for all $f\in
C^{1}(\mathbf{R}^{n};\mathbf{R})$

$\displaystyle\bigg{|}\frac{d}{ds}f(\gamma(s))\bigg{|}\leqslant\sqrt{(\Gamma^{K}f)(\gamma(s))}.$

For $x,y\in\mathbf{R}^{n}$ and $t>0$, let

$\displaystyle S_{T}^{K}(x,y):=\\{\gamma:[0,T]\rightarrow\mathbf{R}^{n}\text{
subunit for }K\text{ and }\,\gamma(0)=x,\gamma(T)=y\\}$

and define the _control distance_ between $x,y\in\mathbf{R}^{n}$ associated to
$K$:

$\displaystyle\rho^{K}(x,y):=\inf\\{T>0\,:\,S_{T}^{K}(x,y)\text{
nonempty}\\}.$

Note that, in our simplified setting of $\mathbf{R}^{n}$, a curve $\gamma\in
C^{1}([0,T];\mathbf{R}^{n})$ is subunit for $K$ if and only if for every
$t\in[0,T]$, $\|K^{-1/2}\gamma^{\prime}(t)\|\leq 1$ and we therefore have

(3.16) $\displaystyle\rho^{K}(x,y)=\|K^{-1/2}(x-y)\|=\sqrt{\langle
x-y,K^{-1}(x-y)\rangle}.$

For simplicity, when 1 is satisfied, we use the notation

(3.17) $\displaystyle\rho_{t}(x,y):=\rho^{G_{t}}(x,y)$

to denote the control distance associated to the matrix $G_{t}$ as in (3.10).

######  Theorem 3.8 (Wang-Harnack inequality).

Fix $t>0$, $\alpha>1$ and suppose that 1 is satisfied. Then we have the
estimate

(3.18) $\displaystyle(P_{t}f(x))^{\alpha}\leqslant
P_{t}f^{\alpha}(y)\exp\bigg{(}\frac{\alpha\rho_{t}^{2}(x,y)}{2(\alpha-1)}\bigg{)}.$

for all $x,y\in\mathbf{R}^{n}$ and all $f\in
L^{\infty}(\mathbf{R}^{n};[0,\infty))$.

###### Proof.

We use a similar proof as in [10, Proposition 3.4]. By standard approximation
arguments, we may suppose without loss of generality that $f\in
C_{\epsilon}^{\infty}(\mathbf{R}^{n};[0,\infty))$ for some $\epsilon>0$. Let
$x,y\in\mathbf{R}^{n}$. Since $G_{t}$ is strictly positive-definite by 2.1,
there exists $T>0$ and a curve $\gamma\in S_{T}^{G_{t}}(x,y)$. Let
$\ell:[0,T]\rightarrow\mathbf{R}$ be given by
$\ell(s)=1+(\alpha-1)\tfrac{s}{T}$ and set

$\displaystyle\psi(s)=\frac{\alpha}{\ell(s)}\log P_{t}f^{\ell(s)}(\gamma(s)).$

Then applying 3.7 and using subuniticity of $\gamma$, we find that

$\displaystyle\psi^{\prime}(s)$
$\displaystyle=\frac{\alpha(\alpha-1)}{T\ell^{2}(s)}\frac{P_{t}f^{\ell(s)}\log
f^{\ell(s)}(\gamma(s))-\log
P_{t}f^{\ell(s)}(\gamma(s))P_{t}f^{\ell(s)}(\gamma(s))}{P_{t}f^{\ell(s)}(\gamma(s))}$
$\displaystyle\qquad+\frac{\alpha}{\ell(s)}\frac{\nabla(P_{t}f^{\ell(s)})(\gamma(s))\cdot\gamma^{\prime}(s)}{P_{t}f^{\ell(s)}(\gamma(s))}$
$\displaystyle\geqslant\frac{\alpha(\alpha-1)}{2T\ell^{2}(s)}\Gamma^{G_{t}}(\log
P_{t}f^{\ell(s)}(\gamma(s)))-\frac{\alpha}{\ell(s)}\sqrt{\Gamma^{G_{t}}(\log
P_{t}f^{\ell(s)}(\gamma(s)))}.$

Now for every $\lambda>0$,

$-\sqrt{\Gamma^{G_{t}}(\log
P_{t}f^{\ell(s)}(\gamma(s)))}\geqslant-\frac{1}{2\lambda^{2}}\Gamma^{G_{t}}(\log
P_{t}f^{\ell(s)}(\gamma(s)))-\frac{\lambda^{2}}{2}.$

By choosing $\lambda^{2}=\frac{T\ell(s)}{(\alpha-1)}$ we thus obtain

$\displaystyle\psi^{\prime}(s)\geqslant-\frac{\alpha}{2(\alpha-1)}T$

Integrating the inequality above from $0$ to $T$ we find that

$\displaystyle\alpha\log P_{t}f(x)-\log
P_{t}f^{\alpha}(y)\leqslant\frac{\alpha T^{2}}{2(\alpha-1)}.$

Exponentiating both sides of the inequality above and optimizing over all such
subunit curves $\gamma$ produces the claimed estimate when $f\in
C_{\epsilon}^{\infty}(\mathbf{R}^{n};[0,\infty))$, finishing the proof. ∎

Using similar computations to those for Corollary 3.7 and Theorem 3.8, we can
also obtain a bound on the total variation distance between the measures
$P_{t}(x,\,\cdot\,)$ and $P_{t}(y,\,\cdot\,)$ for $x,y\in\mathbf{R}^{n}$ using
the control distance $\rho_{t}$.

######  Corollary 3.9.

Let $t>0$ and suppose that 1 is satisfied. Then we have

$\displaystyle\|P_{t}(x,\,\cdot\,)-P_{t}(y,\,\cdot\,)\|_{TV}\leqslant\rho_{t}(x,y).$

for all $x,y\in\mathbf{R}^{n}$.

###### Proof.

Let $f\in L^{\infty}(\mathbf{R}^{n})$ with $\|f\|_{L^{\infty}}\leqslant 1$ and
observe that by 3.4 we have, for some $\xi\in\mathbf{R}^{n}$,

$\displaystyle|P_{t}f(x)-P_{t}f(y)|=|\nabla(P_{t}f)(\xi)\cdot(x-y)|$
$\displaystyle=|G_{t}^{1/2}\nabla(P_{t}f)(\xi)\cdot G_{t}^{-1/2}(x-y)|$
$\displaystyle\leqslant\sqrt{\Gamma^{G_{t}}(P_{t}f)(\xi)}\rho_{t}(x,y)\leqslant\rho_{t}(x,y),$

where in the final inequality we used (3.16) and 3.4. ∎

###  3.3. Transportation cost inequalities

Corollary 3.9 can be improved by using the Wasserstein(-Kantorovich-
Rubinstein) distance in addition to the total variation distance. Indeed,
functional inequalities like in Proposition 3.4 and 3.7 imply transportation
cost inequalities as in [11].

First recall the definitions of the Hellinger and $2$-Wasserstein distances.
Denote by $\mathcal{P}\left(\mathbf{R}^{n}\right)$ the set of Borel
probability measures on $\mathbf{R}^{n}$ and by
$\mathcal{P}^{2}\left(\mathbf{R}^{n}\right)\subset\mathcal{P}\left(\mathbf{R}^{n}\right)$
the space of probability measures having a finite second moment.

For $\mu,\nu\in\mathcal{P}^{2}\left(\mathbf{R}^{n}\right)$ the _$2$
-Wasserstein distance_ $W_{2}$ is defined by

$W_{2}\left(\mu,\nu\right)^{2}:=\inf\int_{\mathbf{R}^{n}\times\mathbf{R}^{n}}|x-y|^{2}\pi\left(dx,dy\right),$

where the infimum is taken over all couplings
$\pi\in\mathcal{P}\left(\mathbf{R}^{n}\times\mathbf{R}^{n}\right)$ with
marginals $\mu,\nu$.

We will also consider the _$2$ -Wasserstein distance_ $W^{\rho_{t}}_{2}$ for
the control distance associated to the matrix $G_{t}$ as in (3.10), which is
defined by

$W^{\rho_{t}}_{2}\left(\mu,\nu\right)^{2}:=\inf\int_{\mathbf{R}^{n}\times\mathbf{R}^{n}}\rho_{t}(x,y)^{2}\pi\left(dx,dy\right),$

where the infimum is taken over all couplings
$\pi\in\mathcal{P}\left(\mathbf{R}^{n}\times\mathbf{R}^{n}\right)$ with
marginals $\mu,\nu$. Letting

(3.19) $\displaystyle\lambda(t):=\inf_{|x|=1}\langle G_{t}x,x\rangle,$

we observe that 1 implies $\lambda(t)>0$ for all $t>0$. Furthermore, Note that

$W^{\rho_{t}}_{2}\left(\mu,\nu\right)^{2}\leq\frac{1}{\lambda(t)}W_{2}(\mu_{0},\mu_{1})^{2},\qquad\mu_{0},\mu_{1}\in\mathcal{P}(\mathbf{R}^{n}).$

The _$2$ -Hellinger distance_ $\mathsf{He}_{2}$ is defined by

$\mathsf{He}_{2}^{2}\left(\mu,\nu\right):=\int_{\mathbf{R}^{n}}\left(\sqrt{\frac{d\mu}{dm}}-\sqrt{\frac{d\nu}{dm}}\right)^{2}\,dm$

where $m$ is any measure that $\mu,\nu$ are both absolutely continuous with
respect to (for example, $m=\mu+\nu$). The definition of $\mathsf{He}_{2}$ is
independent of $m$. Convergence in the Hellinger distance is equivalent to
convergence in total variation, and
$\mathsf{He}_{2}^{2}\left(\mu,\nu\right)\leqslant 2$ for all
$\mu,\nu\in\mathcal{P}\left(\mathbf{R}^{n}\right)$, with
$\mathsf{He}_{2}^{2}\left(\mu,\nu\right)=2$ if and only if $\mu$ and $\nu$ are
mutually singular.

######  Proposition 3.10.

Let $t>0$ and suppose that 1 is satisfied. Then, we have

$\mathsf{He}_{2}(\mu_{0}P_{t},\mu_{1}P_{t})^{2}\leqslant\frac{1}{4}W^{\rho_{t}}_{2}(\mu_{0},\mu_{1})^{2},\qquad\mu_{0},\mu_{1}\in\mathcal{P}(\mathbf{R}^{n}).$

###### Proof.

This follows from Proposition 3.4 and [11, Theorem 3.7]. ∎

Let $\mathrm{Lip}_{b}(\mathbf{R}^{n})$ denote the space of all bounded
Lipschitz functions on $\mathbf{R}^{n}$, and for $a,b\geqslant 0$, let
$\mathcal{E}_{a,b}$ denote the class of all _positive_ functions $\varphi\in
C^{1}([0,1],\mathrm{Lip}_{b}(\mathbf{R}^{n}))$, bounded and bounded away from
$0$, satisfying the differential inequality

$\partial_{s}\varphi_{s}+a\varphi_{s}\Gamma^{G(t)}(\ln\varphi_{s})+b\varphi_{s}\ln\varphi_{s}\leqslant
0.$

For probability measures $\mu_{0},\mu_{1}\in\mathcal{P}(\mathbf{R}^{n})$, we
define

$T_{a,b}(\mu_{0},\mu_{1}):=\sup_{\varphi\in\mathcal{E}_{a,b}}\left[\int_{\mathbf{R}^{n}}\varphi_{1}\,d\mu_{1}-\int_{\mathbf{R}^{n}}\varphi_{0}\,d\mu_{0}\right].$

Provided $b>0$, note by [11, Proposition 5.11] we have

$T_{0,b}(\mu_{0},\mu_{1})=\begin{cases}C_{b}\int_{\mathbf{R}^{n}}\left(\frac{d\mu_{1}}{d\mu_{0}}\right)^{\frac{e^{b}}{e^{b}-1}}\,d\mu_{0},&\mu_{1}\ll\mu_{0}\\\
\infty,&\mu_{1}\not\ll\mu_{0}\end{cases}$

where $C_{b}=\frac{e^{b}-1}{e^{be^{b}}}$.

######  Theorem 3.11.

Let $t>0$ and suppose that 1 is satisfied. Then

(3.20) $T_{0,\kappa/\lambda(t)}(\mu_{0}P_{t},\mu_{1}P_{t})\leqslant
T_{\kappa,\kappa/\lambda(t)}(\mu_{0},\mu_{1}),\qquad\mu_{0},\mu_{1}\in\mathcal{P}(\mathbf{R}^{n}),\quad\kappa>0.$

In particular, for every $x,y\in\mathbf{R}^{n}$ and $t>0$

$\int_{\mathbf{R}^{n}}\left(\frac{p_{t}(x,z)}{p_{t}(y,z)}\right)^{1/(p-1)}p_{t}(x,z)\,dz\leqslant\exp\left(\frac{p}{(p-1)^{2}}\frac{\rho_{t}(x,y)^{2}}{2}\right),\qquad
p>1.$

###### Proof.

The entropic transportation inequality (3.20) follows from Corollary 3.7 and
[11, Theorem 5.15] since we have Wang-Harnack inequality (3.18) for any $p>1$

$(P_{t}f(x))^{p}\leqslant C^{p}P_{t}f^{p}(y),$

with

$C:=\exp\left(\frac{\rho_{t}^{2}(x,y)}{2(p-1)}\right).$

As was observed in different settings in [22, Lemma D1], [6, Lemma 2.11] and
[29, Proposition 4.1], this is equivalent to the integrated Harnack inequality
with $p^{\prime}=\frac{p}{p-1}$:

$\displaystyle\int_{\mathbf{R}^{n}}\left(\frac{p_{t}(x,z)}{p_{t}(y,z)}\right)^{p^{\prime}}p_{t}(x,z)\,dz\leqslant
C^{p^{\prime}}$
$\displaystyle=\exp\left(\frac{p^{\prime}\rho_{t}^{2}(x,y)}{2(p-1)}\right)=\exp\left(\frac{p}{(p-1)^{2}}\frac{\rho_{t}^{2}(x,y)}{2}\right).$

∎

######  Remark 3.12.

We now observe that 3.11 implies a quasi invariance result (see also [29]).
Relying on [11, Lemma 5.10] which says that if for $b>0$ we have
$T_{0,b}(\mu_{0},\mu_{1})<\infty$, then $\mu_{0}$ is absolutely continuous
with respect to $\mu_{1}$. Taking in (3.20) with $\mu_{0}=\delta_{x}$ and
$\mu_{1}=\delta_{y}$, then
$T_{0,\kappa/\lambda(t)}(\delta_{x}P_{t},\delta_{y}P_{t})<\infty$ and
therefore $\delta_{x}P_{t}$ is absolutely continuous with respect to
$\delta_{y}P_{t}$ and by symmetry $\delta_{y}P_{t}$ is absolutely continuous
with respect to $\delta_{x}P_{t}$.

###  3.4. The iterate $G_{2}$ of $G$

Before proceeding onto concrete applications of the matrix $G(s,t)$ as in
(3.7), in this section, we discuss the _iterate_ $G_{2}$ of this matrix. From
a practical standpoint, the iterate $G_{2}$ allows further stochastic mixing
while retaining some structure similar to $G_{t}$. In particular in some
applications, it will allows us to more easily control the smallest positive
eigenvalue $\lambda(t)$ of the matrix $G_{t}$ introduced in (3.19).

Fixing $t>0$, we define the _iterate_ $s\mapsto G_{2}(s,t)$ of the matrix
(3.7) on the interval $[0,t]$ by the formula

$\displaystyle G_{2}(s,t):=\int_{s}^{t}e^{(s-v)A}G(v,t)e^{(s-v)A^{\ast}}\,dv.$

Observe that $s\mapsto G_{2}(s,t)$ is the unique solution of the backwards ODE

$\displaystyle\begin{cases}\partial_{s}G_{2}(s,t)=-G(s,t)+AG_{2}(s,t)+G_{2}(s,t)A^{\ast}\\\
G_{2}(t,t)=0.\end{cases}$

Although we could re-do the Bakry-Émery calculus used in this section with
$G_{2}(s,t)$ in place of $G(s,t)$ to produce similar functional inequalities,
we find it more expedient to compare their respective spectra. In particular,
our main result in this section is 3.13 below.

To this end, for $t>0$, let $\lambda_{2}(t):=\inf_{|x|=1}\langle
G_{2}(0,t)x,x\rangle$ be the smallest eigenvalue of the matrix
$G_{2,t}:=G_{2}(0,t)$. We have the following:

######  Proposition 3.13.

For $t>s>0$, we have the bound

(3.21)
$\displaystyle\lambda(t)\geqslant\frac{\lambda_{2}(t)}{t}\geqslant\frac{\lambda(t-s)}{t}\inf_{|x|=1}\int_{0}^{s}|e^{-vA^{\ast}}x|^{2}\,dv.$

###### Proof.

Fix $t>0$ and $x\in\mathbf{R}^{n}$. We first prove the inequality
$\lambda(t)\geqslant\lambda_{2}(t)/t$. Observe that

$\displaystyle\langle G_{2,t}x,x\rangle=\int_{0}^{t}\langle
G(v,t)e^{-vA^{\ast}}x,e^{-vA^{\ast}}x\rangle\,dv$
$\displaystyle=\int_{0}^{t}\int_{v}^{t}|\sigma^{\ast}e^{(v-w)A^{\ast}}e^{-vA^{\ast}}x|^{2}\,dwdv$
$\displaystyle\leqslant\int_{0}^{t}\int_{0}^{t}|\sigma^{\ast}e^{-wA^{\ast}}x|^{2}\,dw\,dv=t\langle
G_{t}x,x\rangle.$

Taking the infimum over $x\in\mathbf{R}^{n}$ with $|x|=1$ implies
$\lambda(t)\geqslant\lambda_{2}(t)/t$.

For the remaining bound, fix $s>0$ with $s<t$. For $x\in\mathbf{R}^{n}$, since
$G(v,t)$ is nonnegative, we have that

$\displaystyle\langle G_{2,t}x,x\rangle$ $\displaystyle=\int_{0}^{t}\langle
G(v,t)e^{-vA^{\ast}}x,e^{-vA^{\ast}}x\rangle\,dv$ (3.22)
$\displaystyle\geqslant\int_{0}^{s}\langle
G(v,t)e^{-vA^{\ast}}x,e^{-vA^{\ast}}x\rangle\,dv.$

Now, for $v\in[0,s]$ we have for any $x\in\mathbf{R}^{n}$

(3.23) $\displaystyle\langle
G(v,t)x,x\rangle=\int_{v}^{t}|\sigma^{\ast}e^{(v-w)A^{\ast}}x|^{2}\,dw=\int_{0}^{t-v}|\sigma^{\ast}e^{-wA^{\ast}}x|^{2}\,dw\geqslant\lambda(t-s)|x|^{2}.$

Combining (3.22) with (3.23) produces

$\displaystyle\langle
G_{2,t}x,x\rangle\geqslant\lambda(t-s)\int_{0}^{s}|e^{-vA^{\ast}}x|^{2}\,dv,$

which implies the remaining inequality. ∎

######  Remark 3.14.

The bound (3.21) allows to utilize the underlying dynamics driven by the ODE
$\dot{x}=-A^{\ast}x$ to lower bound $\lambda(t)$. In practice, dissipative
dynamics for the system $\dot{x}=A^{\ast}x$ leads to explosive dynamics for
the time-reversed system $\dot{x}=-A^{\ast}x$. In such cases, it is often easy
to show explicitly that

$\displaystyle\int_{0}^{s}|e^{-vA^{\ast}}x|^{2}\,dv$

grows exponentially fast as $s\rightarrow\infty$. We refer the reader to
Section 4.2.3 for a concrete example.

##  4\. Estimating the distance $\rho_{t}$ in examples

The goal of this section is to bound the control distance $\rho_{t}$ as in
(3.17) from above in a number of conrete examples in the form (2.1) where the
noise is degenerate but 1 is satisfied. Of particular importance will be to
produce explicit estimates depending on key parameters in the specific system.
Looking ahead to the following section, how the estimates depend on the
spatial dimension parameter will be critical as we consider an infinite-
dimensional class of examples related to the examples treated in this section.

First, in Section 4.1, we outline the general form of each example and use it
to deduce a basic uppers bounds for $\rho_{t}$ in terms of the _underlying
matrices_ defined below. Importantly, the matrices $A$ and $\sigma$ will be
written in tensored form from which some simplifications can be deduced. Then,
in Section 4.2, we consider four, specific examples where we estimate
$\rho_{t}$ using the underlying matrices via 4.1 below.

###  4.1. Kronecker product form of the examples

Letting $j,k\in\mathbf{N}$, $I=I_{k\times k}$ denote the $k\times k$ identity
matrix and $Q$ denote a $k\times k$, symmetric strictly positive-definite
matrix, throughout this section $A$ and $\sigma$ are $n\times n$ matrices,
$n=jk$, of the following form

(4.1) $\displaystyle A=\begin{bmatrix}a_{11}I&a_{12}I&\ldots&a_{1j}I\\\
a_{21}I&a_{22}I&\ldots&a_{2j}I\\\ \vdots&\vdots&\ldots&\vdots\\\
a_{j1}I&a_{j2}I&\ldots&a_{jj}I\end{bmatrix}\,\,\,\,\,\text{ and
}\,\,\,\,\,\sigma=\begin{bmatrix}\sigma_{11}Q^{1/2}&\sigma_{12}Q^{1/2}&\ldots&\sigma_{1j}Q^{1/2}\\\
\sigma_{21}Q^{1/2}&\sigma_{22}Q^{1/2}&\ldots&\sigma_{2j}Q^{1/2}\\\
\vdots&\vdots&\ldots&\vdots\\\
\sigma_{j1}Q^{1/2}&\sigma_{j2}Q^{1/2}&\ldots&\sigma_{jj}Q^{1/2}\end{bmatrix}.$

To $A$ and $\sigma$, we associate $j\times j$ real matrices $\underline{A}$
and $\underline{\sigma}$, called the _underlying matrices_ corresponding to
$A$ and $\sigma$, given by

$\displaystyle\underline{A}=\begin{bmatrix}a_{11}&a_{12}&\ldots&a_{1j}\\\
a_{21}&a_{22}&\ldots&a_{2j}\\\ \vdots&\vdots&\ldots&\vdots\\\
a_{j1}&a_{j2}&\ldots&a_{jj}\end{bmatrix}\qquad\text{and}\qquad\underline{\sigma}=\begin{bmatrix}\sigma_{11}&\sigma_{12}&\ldots&\sigma_{1j}\\\
\sigma_{21}&\sigma_{22}&\ldots&\sigma_{2j}\\\ \vdots&\vdots&\ldots&\vdots\\\
\sigma_{j1}&\sigma_{j2}&\ldots&\sigma_{jj}\end{bmatrix}.$

The parameter $j$ will be called the _underlying dimension_. Observe that we
can write $A$ and $\sigma$ in terms of the underlying matrices $\underline{A}$
and $\underline{\sigma}$ using the Kronecker product as

$\displaystyle A=\underline{A}\otimes I\qquad\text{ and
}\qquad\sigma=\underline{\sigma}\otimes Q^{1/2}.$

Consequently, using symmetry of $Q$, we can write the corresponding matrix
$G_{t}$ as in (3.10) as

$\displaystyle G_{t}=\int_{0}^{t}e^{-vA}\sigma\sigma^{\ast}e^{-vA^{\ast}}\,dv$
$\displaystyle=\int_{0}^{t}(e^{-v\underline{A}}\otimes
I)(\underline{\sigma}\otimes Q^{1/2})(\underline{\sigma}\otimes
Q^{1/2})^{\ast}(e^{-v\underline{A}^{\ast}}\otimes I)\,dv$
$\displaystyle=\int_{0}^{t}e^{-v\underline{A}}\underline{\sigma}\,\underline{\sigma}^{\ast}e^{-v\underline{A}^{\ast}}\,dv\otimes
Q$ (4.2) $\displaystyle=:\underline{G}_{t}\otimes Q.$

Notationally, throughout this section, for any $x\in\mathbf{R}^{n}$ and any
$k\times k$ matrix $B$ we write

(4.3) $\displaystyle x=\begin{bmatrix}x_{1}\\\ x_{2}\\\ \vdots\\\
x_{j}\end{bmatrix}\in(\mathbf{R}^{k})^{j}\qquad\text{ and
}\qquad\hat{B}x=\begin{bmatrix}Bx_{1}\\\ Bx_{2}\\\ \vdots\\\
Bx_{j}\end{bmatrix}\in(\mathbf{R}^{k})^{j}$

where the $x_{\ell}$’s above are $k$-dimensional column vectors. Setting

(4.4) $\displaystyle\underline{A}_{\underline{\sigma}}$
$\displaystyle:=\begin{bmatrix}\underline{\sigma}&\underline{A}\,\underline{\sigma}&\ldots&\underline{A}^{j-1}\underline{\sigma}\end{bmatrix},$
(4.5) $\displaystyle\underline{\lambda}(j,t)$
$\displaystyle:=\inf_{|x|_{\mathbf{R}^{j}}=1}\langle\underline{G}_{t}x,x\rangle_{\mathbf{R}^{j}},$

we obtain the following result.

######  Proposition 4.1.

Suppose that $\underline{A}_{\underline{\sigma}}$ has full rank,
$x,y\in\mathbf{R}^{n}$ and $t>0$. Then $\underline{G}_{t}$ is invertible,
$\underline{\lambda}(j,t)>0$ and

(4.6) $\displaystyle\rho_{t}(x,y)^{2}$
$\displaystyle=\rho^{\underline{G}_{t}\otimes
I}(\widehat{Q^{-1/2}}x,\widehat{Q^{-1/2}}y)^{2}\leq\sum_{\ell=1}^{j}\frac{\langle
Q^{-1}(x_{\ell}-y_{\ell}),x_{\ell}-y_{\ell}\rangle_{\mathbf{R}^{k}}}{\underline{\lambda}(j,t)}.$

###### Proof.

The equality in (4.6) follows from (3.16) and properties of the Kronecker
product. The estimate in (4.6) follows by definition of
$\underline{\lambda}(j,t)$ and the fact that if
$v=(v^{1},\ldots,v^{j})\in\mathbf{R}^{j}$ is an eigenvector of a $j\times j$
matrix $C$ with eigenvalue $\lambda$, then for every $i=1,2,\ldots,k$ the
vector

$\displaystyle\begin{bmatrix}v^{1}e_{i}\\\ v^{2}e_{i}\\\ \vdots\\\
v^{j}e_{i}\end{bmatrix},$

with $e_{i}$ denoting the standard orthonormal basis element of
$\mathbf{R}^{k}$, is an eigenvector with eigenvalue $\lambda$ for the matrix
$C\otimes I$, $I=I_{k\times k}$. ∎

###  4.2. Examples

We next consider a number of examples with $A$ and $\sigma$ in the form (4.1)
in which we estimate the control distance $\rho_{t}$ as in (3.17) using 4.1
and the structure afforded in the specific dynamics.

######  Example 4.1 (Kolmogorov diffusion).

Consider the Kolmogorov diffusion, whose SDE is of the form (2.1) with $n=2k$
for some $k\in\mathbf{N}$, and

$\displaystyle A=\begin{bmatrix}0&0\\\ I&0\end{bmatrix}\qquad\text{ and
}\qquad\sigma=\begin{bmatrix}Q^{1/2}&0\\\ 0&0\end{bmatrix}$

where $I=I_{k\times k}$ and $Q$ is $k\times k$, symmetric and strictly
positive definite. In this case, the underlying dimension is $j=2$ and the
associated underlying matrices are given by

$\displaystyle\underline{A}=\begin{bmatrix}0&0\\\ 1&0\end{bmatrix}\qquad\text{
and }\qquad\underline{\sigma}=\begin{bmatrix}1&0\\\ 0&0\end{bmatrix}.$

Observe that

$\displaystyle\underline{A}_{\underline{\sigma}}=\begin{bmatrix}\underline{\sigma}&\underline{A}\underline{\sigma}\end{bmatrix}=\begin{bmatrix}1&0&0&0\\\
0&0&1&0\end{bmatrix}$

has full rank. Furthermore, $\underline{A}$ is nilpotent with
$\underline{A}^{2}=0$. Hence

$\displaystyle e^{-v\underline{A}}\sigma=(I_{2\times
2}-v\underline{A})\sigma=\begin{bmatrix}1&0\\\ -v&0\end{bmatrix},$

so that

$\displaystyle\underline{G}_{t}=\int_{0}^{t}e^{-v\underline{A}}\underline{\sigma}\,\underline{\sigma}^{\ast}e^{-v\underline{A}^{\ast}}\,dv=\int_{0}^{t}\begin{bmatrix}1&-v\\\
-v&v^{2}\end{bmatrix}dv=\begin{bmatrix}t&-\frac{t^{2}}{2}\\\\[3.0pt]
-\frac{t^{2}}{2}&\frac{t^{3}}{3}\end{bmatrix}.$

We observe that the matrix $\underline{G}_{t}$ has inverse given by

$\displaystyle\underline{G}_{t}^{-1}=\begin{bmatrix}\frac{4}{t}&\frac{6}{t^{2}}\\\
\frac{6}{t^{2}}&\frac{12}{t^{3}}\end{bmatrix},$

so that by 4.1, if $\hat{x}=\widehat{Q^{-1/2}}x$ and
$\hat{y}=\widehat{Q^{-1/2}}x$, we have

$\displaystyle\rho_{t}(x,y)^{2}=\rho^{\underline{G}_{t}\otimes
I}(\hat{x},\hat{y})^{2}$ $\displaystyle=\langle(\underline{G}_{t}^{-1}\otimes
I)(\hat{x}-\hat{y}),\hat{x}-\hat{y}\rangle$
$\displaystyle=\frac{4}{t}\|Q^{-1/2}(x_{1}-y_{1})\|^{2}_{\mathbf{R}^{k}}+\frac{12}{t^{2}}\langle
Q^{-1}(x_{1}-y_{1}),x_{2}-y_{2}\rangle_{\mathbf{R}^{k}}$
$\displaystyle\qquad+\frac{12}{t^{3}}\|Q^{-1/2}(x_{2}-y_{2})\|^{2}_{\mathbf{R}^{k}}.$

Furthermore, we can calculate the smallest positive eigenvalue
$\underline{\lambda}(2,t)>0$ of $\underline{G}_{t}$ to see that

$\displaystyle\underline{\lambda}(2,t)=\frac{t+\tfrac{t^{3}}{3}-\sqrt{(t+t^{3}/3)^{2}-t^{4}/3}}{2}$
$\displaystyle=\frac{t^{4}}{6(t+\tfrac{t^{3}}{3}+\sqrt{(t+t^{3}/3)^{2}-t^{4}/3})}$
$\displaystyle\simeq\begin{cases}\frac{t}{4}&\text{ as }t\rightarrow\infty\\\
\frac{t^{3}}{12}&\text{ as }t\rightarrow 0^{+}.\end{cases}$

Observe that the expression above for the distance $\rho_{t}$ also reveals a
scale invariance for this particular example. That is, if we define for each
$a>0$ a dilation $\delta_{a}:\mathbf{R}^{2k}\to\mathbf{R}^{2k}$ by

$\displaystyle\delta_{a}x=\delta_{a}\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}=\begin{bmatrix}ax_{1}\\\ a^{3}x_{2}\end{bmatrix},$

then we see that

$\rho_{t}(x,y)=\rho_{1}(\delta_{t^{-1/2}}x,\delta_{t^{-1/2}}y).$

This is appropriate for this example as it coincides with the natural scale
invariance that the solution $x_{t}$ inherits from the standard scaling
relation for Brownian motion

$\displaystyle\sqrt{a}B_{t}^{x}\overset{d}{=}B_{at}^{\sqrt{a}x},$

where $B_{t}^{x}:=x+B_{t}$. To see this explicitly, note that for the given
$A$ and $\sigma$, we may express the solution of (2.2) as

$x_{t}(x)=\begin{bmatrix}x_{1}+Q^{-1/2}B_{t}^{1}\\\
x_{2}+tx_{1}+\int_{0}^{t}Q^{-1/2}B_{s}^{1}\,ds\end{bmatrix},$

where $\\{B_{t}^{1}\\}_{t\geq 0}$ is a standard $k$-dimensional Brownian
motion. From this expression it follows that
$\\{\delta_{a}x_{t}(x)\\}\overset{d}{=}\\{x_{a^{2}t}(\delta_{a}(x))\\}$.

Note that control distances in more general settings are not so easily
computable. In particular, the scale invariance observed in Example 4.1 will
not hold for general $A$ and $\sigma$. However, there are other examples where
such relations are possible.

######  Example 4.2 (Iterated Kolmogorov diffusion).

Consider next the SDE (2.1) with $n=jk$ and $n\times n$ matrices $A$ and
$\sigma$ given by

$\displaystyle A=\begin{bmatrix}0&0&\cdots&0&0\\\ I&0&\cdots&0&0\\\
0&I&\cdots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&\cdots&I&0\\\
\end{bmatrix}\ \ \ \ \mbox{ and }\ \ \ \sigma=\begin{bmatrix}I&0&\cdots&0\\\
0&0&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&0\end{bmatrix},$

with $I=I_{k\times k}$. We observe that, in this case, equation (2.1) has
explicit solution given by

$x_{t}(x)=x+\left(B_{t},\int_{0}^{t}B_{s}\,ds,\ldots,\int_{\Delta_{j-1}(t)}B_{s_{j-1}}\,ds_{j-1}\cdots
ds_{1}\right)$

with $\\{B_{t}\\}_{t\geq 0}$ a standard $k$-dimensional Brownian motion and
$\Delta_{\ell}(t)=\\{0<s_{\ell}<s_{\ell-1}<\cdots<s_{1}<t\\}$. As with the
Kolmogorov diffusion, it is also possible to explicitly compute $G_{t}$, but
one may also see the scale invariance just from the above expression for
$x_{t}$. If we define $\delta_{a}:\mathbf{R}^{jk}\to\mathbf{R}^{jk}$ by

$\delta_{a}x=\delta_{a}(x_{1},\ldots,x_{j}):=\left(ax_{1},\ldots,a^{2j-1}x_{j}\right)$

then the standard scaling relation for $\\{B_{t}\\}$ implies that

$\\{\delta_{a}x_{t}(0)\\}\overset{d}{=}\\{x_{a^{2}t}(0)\\}$

and, similarly,

$\\{\delta_{a}(x_{t}(x))\\}\overset{d}{=}\\{x_{a^{2}t}(\delta_{a}(x))\\}.$

This can be expressed in semigroup form as

$P_{t}(f\circ\delta_{a})=(P_{a^{2}t}f)\circ\delta_{a}.$

In particular, taking $f=f\circ\delta_{a}$ in the Wang-Harnack inequality, cf.
3.8, yields

$\displaystyle((P_{t}(f\circ\delta_{a}))(x))^{\alpha}$
$\displaystyle=(P_{a^{2}t}f)^{\alpha}(\delta_{a}(x))$
$\displaystyle\leqslant\exp\left(\frac{\alpha}{2(\alpha-1)}\rho_{a^{2}t}(\delta_{a}x,\delta_{a}y)\right)(P_{a^{2}t}(f^{\alpha}))(\delta_{a}(y))$
$\displaystyle=\exp\left(\frac{\alpha}{2(\alpha-1)}\rho_{a^{2}t}(\delta_{a}x,\delta_{a}y)\right)(P_{t}((f\circ\delta_{a})^{\alpha}))(y).$

We may now take $f=f\circ\delta_{a^{-1}}$ which gives

$((P_{t}f)(x))^{\alpha}\leqslant\exp\left(\frac{\alpha}{2(\alpha-1)}\rho_{a^{2}t}(\delta_{a}x,\delta_{a}y)\right)(P_{t}(f^{\alpha}))(y).$

Note that the choice of $a=t^{-1/2}$ gives

$((P_{t}f)(x))^{\alpha}\leqslant\exp\left(\frac{\alpha}{2(\alpha-1)}\rho_{1}(\delta_{t^{-1/2}}x,\delta_{t^{-1/2}}y)\right)(P_{t}(f^{\alpha}))(y).$

####  4.2.1. Linear kinetic Fokker-Planck equation

Recalling that $I=I_{k\times k}$ is the identity and $Q$ is $k\times k$ and
positive-definite, consider the case when $A$ and $\sigma$ are $2k\times 2k$
of the following forms

$\displaystyle A=\begin{bmatrix}0&I\\\ -I&-\gamma I\end{bmatrix}\qquad\text{
and }\qquad\sigma=\begin{bmatrix}0&0\\\ 0&\sqrt{\gamma}Q^{1/2}\end{bmatrix}$

where $\gamma>0$ is a constant, called the _friction_ parameter, and the
matrix $Q$ is independent of $\gamma$. In this case, the underlying matrices
$\underline{A}$ and $\underline{\sigma}$ are $2\times 2$ and satisfy

$\displaystyle\underline{A}=\begin{bmatrix}0&1\\\
-1&-\gamma\end{bmatrix}\qquad\text{ and
}\qquad\underline{\sigma}=\begin{bmatrix}0&0\\\ 0&\sqrt{\gamma}\end{bmatrix}.$

Note that

(4.7)
$\displaystyle\underline{A}_{\underline{\sigma}}=\begin{bmatrix}\underline{\sigma}&\underline{A}\underline{\sigma}\end{bmatrix}=\begin{bmatrix}0&0&0&\sqrt{\gamma}\\\
0&\sqrt{\gamma}&0&-\gamma^{3/2}\end{bmatrix}$

has full rank so that $\underline{\lambda}(2,t)>0$ for all $t>0$. Our goal
will be to study $\underline{\lambda}(2,t)$ in the regimes where
$\gamma\approx 0$ and $\gamma\gg 1$. This analysis will be done on the
appropriate time scale depending on the regime. That is, when $\gamma\approx
0$, we set $t=t_{*}/\gamma$ and when $\gamma\gg 1$, we let $t=t_{*}\gamma$
where $t_{*}\geqslant 1$. These timescales correspond to the correct scaling
of the mixing rate of the Markovian dynamics with respect to $\gamma$ [15].
Below, we assume that $\gamma\neq 2$.

We find it convenient to diagonalize $\underline{A}$. First, observe that
$\underline{A}$ has distinct eigenvalues
$\lambda_{1},\lambda_{2}\in\mathbf{C}$ given by

(4.8)
$\displaystyle\lambda_{1}=\frac{-\gamma+\sqrt{\gamma^{2}-4}}{2}\qquad\text{
and }\qquad\lambda_{2}=\frac{-\gamma-\sqrt{\gamma^{2}-4}}{2}$

where $\sqrt{-1}=i$. Furthermore, we can write

$\displaystyle\underline{A}=\begin{bmatrix}1&1\\\
\lambda_{1}&\lambda_{2}\end{bmatrix}\begin{bmatrix}\lambda_{1}&0\\\
0&\lambda_{2}\end{bmatrix}\begin{bmatrix}1&1\\\
\lambda_{1}&\lambda_{2}\end{bmatrix}^{-1}.$

Hence,

$\displaystyle
e^{-\underline{A}t}\underline{\sigma}=-\frac{\sqrt{\gamma}}{\sqrt{\gamma^{2}-4}}\begin{bmatrix}0&e^{-\lambda_{2}t}-e^{-\lambda_{1}t}\\\
0&\lambda_{2}e^{-\lambda_{2}t}-\lambda_{1}e^{-\lambda_{1}t}\end{bmatrix}.$

Thus, if $c_{\gamma}=\gamma/(\gamma^{2}-4)$, we have

$\displaystyle\underline{G}_{t}=\int_{0}^{t}e^{-\underline{A}s}\underline{\sigma}\underline{\sigma}^{\ast}e^{-\underline{A}^{\ast}s}\,ds$
$\displaystyle=c_{\gamma}\int_{0}^{t}\begin{bmatrix}e^{-2\lambda_{1}s}-2e^{-(\lambda_{1}+\lambda_{2})s}+e^{-2\lambda_{2}s}&\lambda_{1}e^{-2\lambda_{1}s}-(\lambda_{1}+\lambda_{2})e^{-(\lambda_{1}+\lambda_{2})s}+\lambda_{2}e^{-2\lambda_{2}s}\\\
\
\lambda_{1}e^{-2\lambda_{1}s}-(\lambda_{1}+\lambda_{2})e^{-(\lambda_{1}+\lambda_{2})s}+\lambda_{2}e^{-2\lambda_{2}s}&\lambda_{1}^{2}e^{-2\lambda_{1}s}-2\lambda_{1}\lambda_{2}e^{-(\lambda_{1}+\lambda_{2})s}+\lambda_{2}^{2}e^{-2\lambda_{2}s}\end{bmatrix}\,ds$
$\displaystyle=:c_{\gamma}\begin{bmatrix}G_{11}&G_{12}\\\
G_{12}&G_{22}\end{bmatrix}.$

Observe that for all $\gamma\neq 2$, by positive-definiteness of
$\underline{G}_{t}$ via (4.7), the discriminant

$\displaystyle D:=G_{11}G_{22}-G_{12}^{2}$

is strictly positive for all $t>0$.

_Case 1_ ($\gamma\gg 1$). In this case, $\lambda_{1}\neq\lambda_{2}$ are both
real and we set $t=\gamma t_{*}$ where $t_{*}\geqslant 1$ is independent of
$\gamma$. Moreover, since $c_{\gamma}>0$,

$\displaystyle\underline{\lambda}(j,t)$
$\displaystyle=c_{\gamma}\frac{G_{11}+G_{22}-\sqrt{(G_{11}+G_{22})^{2}-4D}}{2}=c_{\gamma}\frac{2D}{G_{11}+G_{22}+\sqrt{(G_{11}+G_{22})^{2}-4D}}.$

Using the explicit expressions above along with the values of
$\lambda_{1},\lambda_{2}$ in (4.8), we obtain

(4.9) $\displaystyle G_{11}$
$\displaystyle=\frac{2}{\gamma}-\frac{\gamma}{2}-\frac{2}{\gamma}e^{\gamma
t}-\frac{\lambda_{2}}{2}e^{-2\lambda_{1}t}-\frac{\lambda_{1}}{2}e^{-2\lambda_{2}t},$
$\displaystyle G_{22}$
$\displaystyle=\frac{2}{\gamma}-\frac{\gamma}{2}-\frac{2}{\gamma}e^{\gamma
t}-\frac{\lambda_{1}}{2}e^{-2\lambda_{1}t}-\frac{\lambda_{2}}{2}e^{-2\lambda_{2}t},$
$\displaystyle G_{12}$ $\displaystyle=e^{\gamma
t}-\frac{e^{-2\lambda_{1}t}}{2}-\frac{e^{-2\lambda_{2}t}}{2}.$

Furthermore, note that as $\gamma\rightarrow\infty$ we have the following
asymptotic formulas

(4.10) $\displaystyle\lambda_{1}$
$\displaystyle=-\frac{1}{\gamma}+O(\gamma^{-2})\qquad\text{ and
}\qquad\lambda_{2}=-\gamma+\frac{1}{\gamma}+O(\gamma^{-2}).$

Substituting in $t=t_{*}\gamma$ into the expressions following (4.9) and using
(4.10) produces the following bound

$\displaystyle D$ $\displaystyle=G_{11}G_{22}-G_{12}^{2}$
$\displaystyle=\bigg{\\{}\frac{e^{-4\lambda_{2}\gamma
t_{*}}}{4}-e^{(-2\lambda_{2}+\gamma)\gamma
t_{*}}+\frac{\lambda_{2}^{2}}{4}e^{2\gamma^{2}t_{*}}+\frac{\lambda_{2}\gamma}{4}e^{-2\lambda_{2}\gamma
t_{*}}+R_{1}\bigg{\\}}$
$\displaystyle\qquad-\bigg{\\{}\frac{e^{-4\lambda_{2}\gamma
t_{*}}}{4}-e^{(-2\lambda_{2}+\gamma)\gamma
t_{*}}+\frac{3}{2}e^{2\gamma^{2}t_{*}}+R_{2}\bigg{\\}}$
$\displaystyle\geqslant\frac{\lambda_{2}^{2}}{4}e^{2\gamma^{2}t_{*}}\bigg{\\{}1-e^{-t_{*}}-C\gamma^{-2}\bigg{\\}}$

which is satisfied for all $t_{*}\geqslant 1$ and
$\gamma\geqslant\gamma_{\ell}\gg 1$ for some constant $C>0$ independent of
$t_{*},\gamma$. In a similar fashion, by increasing $\gamma_{\ell}>0$ and
$C>0$ if needed, we obtain

$\displaystyle G_{11}+G_{22}+\sqrt{(G_{11}+G_{22})^{2}-4D}$
$\displaystyle\leqslant-\frac{3\lambda_{2}}{2}e^{-2\lambda_{2}\gamma
t_{*}}\bigg{\\{}1+C/\gamma\bigg{\\}}$

for all $t_{*}\geqslant 1,\gamma\geqslant\gamma_{\ell}$. Consequently, by
increasing $\gamma_{\ell}$ again if needed, we see that for $t_{*}\geqslant 1$
and $\gamma\geqslant\gamma_{L}$

$\displaystyle\lambda(2,t_{*}\gamma)\geqslant\frac{-c_{\gamma}\lambda_{2}}{6}e^{t_{*}}\frac{1-e^{-t_{*}}-C\gamma^{-2}}{1+C\gamma^{-1}}\geqslant\frac{e^{t_{*}}}{16}.$

_Case II_ ($\gamma\approx 0$). In this case, we set $t=t_{*}/\gamma$ where
$t_{*}\geqslant 1$. In this case, since $c_{\gamma}<0$,

$\displaystyle\underline{\lambda}(2,t)=c_{\gamma}\frac{(G_{11}+G_{22})+\sqrt{(G_{11}+G_{22})^{2}-4D}}{2}=|c_{\gamma}|\frac{2D}{-(G_{11}+G_{22})+\sqrt{(G_{11}+G_{22})^{2}-4D}}.$

Again, since $\gamma\approx 0$, we find that if
$\beta=|\sqrt{\gamma^{2}-4}|/2$, then

$\displaystyle G_{11}$
$\displaystyle=-\frac{\gamma}{2}+\frac{2}{\gamma}[1-e^{\gamma
t}]+\frac{\gamma}{2}e^{\gamma t}\cos(2\beta t)+\beta e^{\gamma t}\sin(2\beta
t),$ $\displaystyle G_{22}$
$\displaystyle=-\frac{\gamma}{2}+\frac{2}{\gamma}[1-e^{\gamma
t}]+\frac{\gamma}{2}e^{\gamma t}\cos(2\beta t)-\beta e^{\gamma t}\sin(2\beta
t)$ $\displaystyle G_{12}$ $\displaystyle=e^{\gamma t}(1-\cos(2\beta t)).$

Setting $t=t_{*}/\gamma$ where $t_{*}\geqslant 1$ we find that there exists a
constant $C>0$ so that

(4.11) $\displaystyle
D\geqslant\frac{4}{\gamma^{2}}(1-e^{t_{*}})^{2}\\{1-C\gamma\\}$

for all $t_{*}\geqslant 1$, $0<\gamma\leqslant\gamma_{s}$, where
$\gamma_{s}>0$ is sufficiently small and $C>0$ is independent of $\gamma$. In
a similar fashion, by decreasing $\gamma_{s}>0$ and $C>0$ if needed, we obtain
the following bound

(4.12)
$\displaystyle-(G_{11}+G_{22})+\sqrt{(G_{11}+G_{22})^{2}-4D}\leqslant\frac{8}{\gamma}(e^{t_{*}}-1)\\{1+C\gamma\\}$

satisfied for all $t_{*}\geqslant 1$ and $0<\gamma\leqslant\gamma_{s}$.
Combining (4.11) with (4.12) and adjusting $\gamma_{s}$ smaller if needed we
obtain

$\displaystyle\underline{\lambda}(2,t_{*}/\gamma)\geqslant\frac{|c_{\gamma}|}{\gamma}(e^{t_{*}}-1)\frac{1-C\gamma}{1+C\gamma}\geqslant\frac{e^{t_{*}}-1}{8}\geqslant\frac{e^{t_{*}}}{16}$

for all $t_{*}\geqslant 1$ and $\gamma\leqslant\gamma_{s}$.

####  4.2.2. Coupled oscillators

For $a,d,b\in\mathbf{R}$, let $\text{Tri}_{j}(a,d,b)$ denote the $j\times j$
tridiagonal matrix satisfying

$\displaystyle(\text{Tri}_{j}(a,d,c))_{\ell m}=\begin{cases}a&\text{ if
}\,m=\ell-1\\\ d&\text{ if }\,m=\ell\\\ c&\text{ if }\,m=\ell+1\\\ 0&\text{
otherwise}\end{cases}$

and let $E_{\ell m}$ be the $j\times j$ matrix with $(\ell,m)$th entry equal
to $1$ and all other entries equal to $0$. Let the underlying matrices
$\underline{A}$ and $\underline{\sigma}$ be given by

$\displaystyle\underline{A}=\text{Tri}_{j}(1,0,-1)\qquad\text{ and
}\qquad\underline{\sigma}=E_{11}.$

A short calculation shows that for $m=0,1,\ldots,j-1$ and $\ell=1,2,\ldots,j$

$\displaystyle(\underline{A}^{m}\underline{\sigma})_{m+1,\ell}=E_{m+1,\ell}\qquad\text{
and }\qquad(\underline{A}^{m}\underline{\sigma})_{z,\ell}=0$

for $m+2\leq z\leq j-1$. Consequently, the matrix
$\underline{A}_{\underline{\sigma}}$ has full rank and so
$\underline{\lambda}(j,t)>0$ for all $t>0$. The goal of this calculation will
be to estimate $\underline{\lambda}(j,t)$ for $t\gg 1$. We find it again
convenient to diagonalize $\underline{A}$.

One can show that $\underline{A}$ has distinct eigenvalues $\lambda_{\ell}$,
$\ell=1,2,\ldots,j$, given by

(4.13) $\displaystyle\lambda_{\ell}=2i\cos\Big{(}\frac{\ell\pi}{j+1}\Big{)}$

with corresponding (right) eigenvectors $v_{\ell}$, $\ell=1,2,\ldots,j$,
defined by

$\displaystyle(v_{\ell})_{m}=i^{m}\sin\Big{(}\frac{\ell
m\pi}{j+1}\Big{)},\qquad m=1,2,\ldots,j.$

See, for example, [45, 47]. Moreover, using basic properties of trigonometric
functions, it can be checked that the set of eigenvectors
$\\{v_{1},\ldots,v_{j}\\}$ forms an orthogonal family with identical lengths
(see [50])

$\displaystyle|v_{\ell}|^{2}=\frac{j+1}{2},\qquad\ell=1,2,\ldots,j.$

Therefore, we define an orthonormal family of (column) eigenvectors
$\\{w_{1},\ldots,w_{j}\\}$ by $w_{\ell}=v_{\ell}/|v_{\ell}|$,
$\ell=1,2,\ldots,j$. Setting $P=[w_{1}\,\,w_{2}\,\,\ldots\,\,w_{j}]$ and
letting $P^{\ast}$ denote its Hermitian transpose, for any
$x\in\mathbf{R}^{j}_{\neq 0}$ we have by symmetry of $\sigma$ and antisymmetry
of $\underline{A}$

$\displaystyle\langle\underline{G}_{t}x,x\rangle=\int_{0}^{t}|\underline{\sigma}^{\ast}e^{-v\underline{A}^{\ast}}x|^{2}\,dv=\int_{0}^{t}|\underline{\sigma}e^{v\underline{A}}x|^{2}\,dv=\int_{0}^{t}|E_{11}P\text{diag}(e^{v\lambda_{1}},e^{v\lambda_{2}},\ldots,e^{v\lambda_{j}})P^{\ast}x|^{2}.$

Let $\xi=P^{\ast}x\in\mathbf{C}^{j}$ and notice that

$\displaystyle\langle\underline{G}_{t}x,x\rangle=\int_{0}^{t}\Big{|}\sum_{\ell=1}^{j}(w_{\ell})_{1}\xi_{\ell}e^{\lambda_{\ell}v}\Big{|}^{2}\,dv$
$\displaystyle=\sum_{\ell,m=1}^{j}(w_{\ell})_{1}\overline{(w_{m})_{1}}\xi_{\ell}\overline{\xi_{m}}\int_{0}^{t}e^{v\lambda_{\ell}+v\overline{\lambda_{m}}}\,dv$
$\displaystyle=t\sum_{\ell}|(w_{\ell})_{1}|^{2}|\xi_{\ell}|^{2}+\sum_{\ell\neq
m}(w_{\ell})_{1}\overline{(w_{m})_{1}}\xi_{\ell}\overline{\xi_{m}}\int_{0}^{t}e^{v\lambda_{\ell}+v\overline{\lambda_{m}}}\,dv.$

By definition of the eigenvalues $\lambda_{m}$ in (4.13), for $\ell\neq m$,
$\ell,m\in\\{1,2,\ldots,j\\}$, we obtain

$\displaystyle\bigg{|}\int_{0}^{t}e^{v\lambda_{\ell}+v\overline{\lambda_{m}}}\,dv\bigg{|}=\bigg{|}\frac{e^{2ti\cos(\frac{\ell\pi}{j+1})}-e^{2it\cos(\frac{m\pi}{j+1})}}{2i\cos\big{(}\frac{\ell\pi}{j+1}\big{)}-2i\cos\big{(}\frac{m\pi}{j+1}\big{)}}\bigg{|}\leqslant\frac{1}{\big{|}\cos\big{(}\frac{\ell\pi}{j+1}\big{)}-\cos\big{(}\frac{m\pi}{j+1}\big{)}\big{|}}.$

Using the mean value theorem, we find that for $\ell\neq m$,
$\ell,m\in\\{1,2,\ldots,j\\}$, there exists $p=p_{\ell m}$ strictly between
$\ell\pi/(j+1)$ and $m\pi/(j+1)$ such that

$\displaystyle\frac{1}{\big{|}\cos\big{(}\frac{\ell\pi}{j+1}\big{)}-\cos\big{(}\frac{m\pi}{j+1}\big{)}\big{|}}=\frac{1}{\big{|}\frac{(\ell-m)\pi}{j+1}\sin(p)\big{|}}\leqslant\frac{(j+1)}{\pi}\frac{1}{\sin(\tfrac{\pi}{j+1})}=:c_{j}.$

Hence, applying the bound above and using Cauchy-Schwarz we obtain

$\displaystyle\bigg{|}\sum_{\ell\neq
m}(w_{\ell})_{1}\overline{(w_{m})_{1}}\xi_{\ell}\overline{\xi_{m}}\int_{0}^{t}e^{v\lambda_{\ell}+v\overline{\lambda_{m}}}\,dv\bigg{|}\leqslant
c_{j}\bigg{(}\sum_{\ell}|(w_{\ell})_{1}\xi_{\ell}|\bigg{)}^{2}$
$\displaystyle\leqslant
c_{j}\sum_{\ell}|(w_{\ell})_{1}|^{2}\sum_{\ell}|\xi_{\ell}|^{2}$
$\displaystyle=c_{j}|x|^{2}.$

Thus,

$\displaystyle\langle G_{t}x,x\rangle\geqslant
t\sum_{\ell}|(w_{\ell})_{1}|^{2}|\xi_{\ell}|^{2}-c_{j}|x|^{2}\geqslant\big{(}t\sin^{2}\big{(}\tfrac{\pi}{j+1}\big{)}-c_{j}\big{)}|x|^{2}$

and so we obtain

$\displaystyle\underline{\lambda}(j,t)\geqslant
t\sin^{2}\big{(}\tfrac{\pi}{j+1}\big{)}-c_{j}$

for all $t>0$.

####  4.2.3. Oscillators with some damping

We revisit the previous example, but this time we place damping on the first
coordinate. That is, we set

$\displaystyle\underline{A}=\text{Tri}_{j}(1,0,-1)-E_{11}\qquad\text{ and
}\qquad\underline{\sigma}=E_{11}.$

In this case, diagonalizing $\underline{A}$ as before seems challenging
because an explicit expression for eigenvalues and eigenvectors is not known.
In order to analyze $\underline{\lambda}(j,t)$, we appeal to 3.13.

Note that, in a similar fashion to the previous example, one can readily check
that $\underline{A}_{\underline{\sigma}}$ has full rank, so that
$\underline{\lambda}(j,t)>0$ for all $t>0$. We next seek to estimate

(4.14) $\displaystyle\int_{0}^{s}|e^{-v\underline{A}^{\ast}}x|^{2}\,dv.$

Observe that the integrand in (4.14) is precisely $|y(v)|^{2}$ where $y(v)$ is
the solution of the following ODE on $\mathbf{R}^{j}$ at time $v$:

$\displaystyle\begin{cases}\dot{y}=(E_{11}+\text{Tri}_{j}(1,0,-1))y\\\
y_{0}=x\in\mathbf{R}^{j}.\end{cases}$

We claim that

$\displaystyle|y(v)|^{2}\geqslant d_{j}|x|^{2}e^{r_{j}v}\text{ for all
}v\geqslant 0$

where $d_{j},r_{j}>0$ are constants. We will prove the claim using a
convenient Lyapunov function.

To define our Lyapunov function, let $a_{\ell}$, $\ell=0,1,2,\ldots,j-1$, be
positive constants to be determined and set $a_{j}=0$. Define and function
$V:\mathbf{R}^{j}\rightarrow\mathbf{R}$ by

$\displaystyle V(y)=\frac{a_{0}}{2}|y|^{2}-\sum_{i=1}^{j-1}a_{i}y_{i}y_{i+1}.$

We first pick a convenient form for the constants $a_{i}$; that is, we define

$\displaystyle a_{i-1}=a_{i}+b_{i},\,\,\,\,i=2,\ldots,j,$

for some positive constants $b_{i}$, $i=2,\ldots,j$, to be determined. Note
that, in particular, this means that

$\displaystyle a_{\ell}=\sum_{i=\ell+1}^{j}b_{i},\qquad\ell=1,\ldots,j,$

with the convention that the empty sum is zero.

First observe that we have the following explicit bound:

$\displaystyle\big{(}\tfrac{a_{0}}{2}+\textstyle{\sum_{i=2}^{j}}b_{i}\big{)}|y|^{2}$
$\displaystyle\geqslant(\tfrac{a_{0}}{2}+\tfrac{a_{1}}{2})y_{1}^{2}+\sum_{i=2}^{j}\big{(}\tfrac{a_{0}}{2}+\tfrac{a_{i-1}}{2}+\tfrac{a_{i}}{2}\big{)}y_{i}^{2}$
(4.15) $\displaystyle\geqslant V(y)$
$\displaystyle\geqslant\big{(}\tfrac{a_{0}}{2}-\tfrac{a_{1}}{2}\big{)}y_{1}^{2}+\sum_{i=2}^{j}\big{(}\tfrac{a_{0}}{2}-\tfrac{a_{i-1}}{2}-\tfrac{a_{i}}{2}\big{)}y_{i}^{2}$
$\displaystyle\geqslant\big{(}\tfrac{a_{0}}{2}-\textstyle{\sum}_{i=2}^{j}b_{i}\big{)}|y|^{2}.$

In particular, we need to choose

$\displaystyle a_{0}>2\sum_{i=2}^{j}b_{i}$

so that $V$ is nonnegative.

Next, observe that

$\displaystyle\frac{d}{dt}V(y(t))$
$\displaystyle=-a_{1}y_{1}(t)y_{2}(t)+\sum_{i=1}^{j}(a_{i-1}-a_{i})(y_{i}(t))^{2}+\sum_{i=1}^{j-2}(a_{i}-a_{i+1})y_{i}(t)y_{i+2}(t)$
$\displaystyle=-a_{1}y_{1}(t)y_{2}(t)+(a_{0}-a_{1})y_{1}^{2}+\sum_{i=2}^{j}b_{i}(y_{i}(t))^{2}+\sum_{i=1}^{j-2}b_{i+1}y_{i}(t)y_{i+2}(t)$
$\displaystyle\geqslant-
a_{1}y_{1}(t)y_{2}(t)+(a_{0}-a_{1}-\tfrac{b_{2}}{2})y_{1}^{2}+(b_{2}-\tfrac{b_{3}}{2})y_{2}^{2}+(b_{j}-\tfrac{b_{j-1}}{2})y_{j}^{2}$
$\displaystyle\qquad+\sum_{i=3}^{j-1}(b_{i}-\tfrac{b_{i+1}}{2}-\tfrac{b_{i-1}}{2})(y_{i}(t))^{2}.$

Pick $b_{i}=\log(i+1)$, $i=2,\ldots,j$, and notice by concavity we have

$\displaystyle b_{i}>\frac{b_{i+1}}{2}+\frac{b_{i-1}}{2},\,\,i=2,\ldots,j.$

On the other hand, for any $\beta>0$, we have

$\displaystyle\frac{d}{dt}V(y(t))$
$\displaystyle\geqslant(a_{0}-a_{1}-\tfrac{a_{1}^{2}}{2\beta}-b_{2})y_{1}^{2}+(b_{2}-\tfrac{b_{3}}{2}-\tfrac{\beta}{2})y_{2}^{2}+(b_{j}-\tfrac{b_{j-1}}{2})y_{j}^{2}$
$\displaystyle\qquad+\sum_{i=3}^{j-1}(b_{i}-\tfrac{b_{i+1}}{2}-\tfrac{b_{i-1}}{2})(y_{i}(t))^{2}$

Pick $\beta=\log(2)$, define $b_{j+1}=\log(j+2)$ and let

$\displaystyle
a_{0}=a_{1}+\tfrac{a_{1}^{2}}{2\beta}+b_{2}+b_{j}-\tfrac{b_{j+1}}{2}-\tfrac{b_{j-1}}{2}.$

Using the fact that the function $x\mapsto\log(x)-\log(x+1)/2-\log(x-1)/2$
defined on $[2,\infty)$ is strictly decreasing and using (4.15), we obtain

$\displaystyle\frac{d}{dt}V(y(t))\geqslant(b_{j}-\tfrac{b_{j+1}}{2}-\tfrac{b_{j-1}}{2})|y(t)|^{2}\geqslant\frac{(b_{j}-\tfrac{b_{j+1}}{2}-\tfrac{b_{j-1}}{2})}{a_{0}+2\sum_{i=2}^{j}b_{i}}V(y(t)).$

Hence, letting

(4.16) $\displaystyle
r_{j}=\frac{(b_{j}-\tfrac{b_{j+1}}{2}-\tfrac{b_{j-1}}{2})}{a_{0}+2\sum_{i=2}^{j}b_{i}}\qquad\text{
and }\qquad
c_{j}=\frac{\frac{a_{0}}{2}-\sum_{i=2}^{j}b_{i}}{\frac{a_{0}}{2}+\sum_{i=2}^{j}b_{i}},$

we have that

$\displaystyle|y(t)|^{2}\geqslant|x|^{2}c_{j}e^{r_{j}t}\,\,\,\text{ for all
}t\geqslant 0.$

Thus,

$\displaystyle\int_{0}^{t}|e^{-v\underline{A}^{\ast}}x|^{2}\,ds\geqslant|x|^{2}\frac{c_{j}}{r_{j}}(e^{r_{j}t}-1).$

Applying 3.13, we obtain the bound for $t\geq 2$

(4.17)
$\displaystyle\underline{\lambda}(j,t)\geqslant\underline{\lambda}(j,t/2)\frac{c_{j}}{tr_{j}}(e^{\frac{r_{j}t}{2}}-1)\geq\frac{\underline{\lambda}(j,1)}{tr_{j}}c_{j}(e^{\frac{r_{j}t}{2}}-1)$

##  5\. From finite to infinite dimensions

The goal of this section is to move from the finite-dimensional setting in
relation (2.1) to an infinite-dimensional version of the system where the
noise becomes infinite-dimensional. In the context of the examples of the
previous section, this means that if $n=jk$, where $j$ is the underlying
dimension $j$ and $k$ is the dimension of the noise, then we take
$k\rightarrow\infty$ leaving $j$ fixed. Building off of the general analysis
done with the modified gradients in Section 3, we will be able to extend the
the Wang-Harnack inequality in 3.8 to an infinite-dimensional version. We will
then use this infinite-dimensional version to conclude a quasi-invariance
result as well as a “time-infinity” Wang-Harnack inequality. We conclude the
section by revisiting some of the examples considered in Section 4 as they
relate to their associated infinite-dimensional versions.

###  5.1. The infinite-dimensional setting

Fix a separable, infinite-dimensional Hilbert space $H$ with inner product and
norm respectively denoted by

$\displaystyle\langle\cdot,\cdot\rangle_{H}\qquad\text{ and
}\qquad\|\cdot\|_{H}=\sqrt{\langle\cdot,\cdot\rangle_{H}}.$

Below, we make several slight abuses of notation to help connect with the
finite-dimensional setting (2.1) previously considered. In particular, we will
intentionally reuse the notations $Q,A,\sigma$, which were previously used to
denote operators on $\mathbf{R}^{k},\mathbf{R}^{n},\mathbf{R}^{n}$,
respectively. Here, in this section, they will be linear operators on the
respective spaces $H$, $H^{j}$, $H^{j}$. We let $W=H^{j}$ denote the separable
Banach space with norm given by

$\displaystyle\|X\|_{W}:=\sqrt{\|X^{1}\|_{H}^{2}+\cdots+\|X^{j}\|_{H}^{2}}$

where $X=(X^{1},X^{2},\ldots,X^{j})\in H^{j}$.

Throughout this section we will employ the following:

######  Assumption 2.

The mapping $Q:H\rightarrow H$ is a strictly positive symmetric bounded linear
operator of trace-class.

Note that under 2, we can diagonalize $Q$ in $H$; that is, there is an
orthonormal basis $\\{e_{\ell}\\}_{\ell\in\mathbf{N}}$ of $H$ and real numbers
$\alpha_{\ell}>0$ for which

(5.1) $\displaystyle Qe_{\ell}=\alpha_{\ell}e_{\ell}\,\,\text{ for all
}\,\,\ell\in\mathbf{N}.$

Note that $Q$ being trace class translates to the summability condition

(5.2) $\displaystyle\sum_{\ell=1}^{\infty}\alpha_{\ell}<\infty.$

Again, offering slight abuses of notation, we assume that

$\displaystyle B(t):=(B^{1}(t),B^{2}(t),\ldots,B^{j}(t))$

where $B^{i}(t)$, $i=1,2,\ldots,j$, are independent, $Q$-Brownian motions on
$H$. Note that this is the same as supposing that each $B^{i}$ can be written
as

$\displaystyle
B^{i}(t)=\sum_{\ell=1}^{\infty}\sqrt{\alpha_{\ell}}\beta_{\ell}^{i}(t)e_{\ell}$

where $\\{\beta_{k}^{i}\\}_{k=1,i=1}^{\infty,j}$ is a collection of mutually
independent standard, real-valued Brownian motions on
$(\Omega,\mathcal{F},\mathbf{P})$ [18]. The fact that each $B^{i}$ has
continuous paths in $\|\cdot\|_{H}$ with probability one follows by path
continuity of each $\beta_{k}^{i}$ and the summability condition (5.2).

Recall that associated to $Q$ is the Hilbert space $H_{Q}$ consisting of
elements $h$ such that

$\sum_{\ell=1}^{\infty}\frac{\langle
h,e_{\ell}\rangle_{H}^{2}}{\alpha_{\ell}}<\infty$

equipped with the inner product $\langle h,k\rangle_{Q}:=\langle
Q^{-1/2}h,Q^{-1/2}k\rangle_{H}$, where $Q^{-1/2}$ denotes the pseudo-inverse
of $Q^{1/2}$. The space $H_{Q}$ is called the _Cameron-Martin space_
associated to the Gaussian measure $\mu=\operatorname{Law}(B^{i}(1))$, and
enjoys various important analytic properties with respect to $\mu$. For
example, we recall the Cameron-Martin-Maruyama “quasi-invariance” theorem: The
measure $\mu^{h}$ defined by $\mu^{h}(\Gamma):=\mu(\Gamma+h)$ is mutually
absolutely continuous with respect to $\mu$ if and only if $h\in H_{Q}$. If
$h\notin H_{Q}$ then $\mu^{h}$ is singular with respect to $\mu$. Let
$\mathcal{H}_{Q}$ denote the Hilbert space $(H_{Q})^{j}$ equipped with inner
product coming from the product structure

$\langle X,Y\rangle_{\mathcal{H}_{Q}}:=\langle
X^{1},Y^{1}\rangle_{Q}+\cdots+\langle X^{j},Y^{j}\rangle_{Q}$

and induced norm

$\|X\|_{\mathcal{H}_{Q}}:=\sqrt{\|X^{1}\|_{Q}^{2}+\cdots+\|X^{j}\|_{Q}^{2}}=\|(I_{j\times
j}\otimes Q^{-1/2})X\|_{W}.$

Letting $I_{H}$ denote the identity operator on $H$, we set

(5.3) $\displaystyle A=(\underline{A}\otimes I_{H})\qquad\text{ and
}\qquad\sigma=(\underline{\sigma}\otimes I_{H})$

where $\underline{A}$ and $\underline{\sigma}$ are $j\times j$ real matrices.
In relation to (2.1), in this context we consider the integral equation on
$W:=H^{j}$

(5.4) $\displaystyle X(t;X_{0})$
$\displaystyle=X_{0}+\int_{0}^{t}AX(s)\,ds+\sigma B(t),\,\,\,\,X_{0}\in W.$

Using a standard iteration procedure, it is not hard to show that for every
$X_{0}\in W$, relation (5.4) has a unique solution which is a stochastic
process $X(t)$ on $W$ with continuous paths in the norm $\|\cdot\|_{W}$.
Furthermore, the process $X(t)$ is Markov and we let $\mathcal{P}_{t}$ denote
the corresponding Markov semigroup.

###  5.2. Infinite-dimensional Wang-Harnack inequality

Next, consider the projection operator $\pi_{k}:H\rightarrow H$ associated to
the orthonormal basis $\\{e_{k}\\}_{k\in\mathbf{N}}$ given by

$\displaystyle\pi_{k}(h):=\pi_{k}\bigg{(}\sum_{\ell=1}^{\infty}\langle
h,e_{\ell}\rangle e_{\ell}\bigg{)}=\sum_{\ell=1}^{k}\langle h,e_{\ell}\rangle
e_{\ell},$

and then define $\Pi_{k}:W\rightarrow W$ as

$\displaystyle\Pi_{k}(h_{1},h_{2},\ldots,h_{j}):=(\pi_{k}(h_{1}),\ldots,\pi_{k}(h_{j})).$

We observe that, by (5.3) and (5.4), the process $t\mapsto\Pi_{k}X(t)$
satisfies the finite-dimensional integral equation

$\displaystyle\Pi_{k}X(t;X_{0})=\Pi_{k}X_{0}+\int_{0}^{t}A\Pi_{k}X(s)\,ds+\sigma\Pi_{k}B_{t}.$

Let $\mathcal{P}_{t}^{k}$ denote the Markov semigroup corresponding to
$\Pi_{k}X(t)$.

######  Proposition 5.1.

Suppose that $\underline{A}_{\underline{\sigma}}$ as in (4.4) is of full rank.
Let $\alpha>1$. Then for all bounded, measurable
$\varphi:W\rightarrow\mathbf{R}$, $t>0$, $X_{0},Y_{0}\in W$, and
$k\in\mathbf{N}$ we have

(5.5)
$\displaystyle(\mathcal{P}_{t}^{k}\varphi(\Pi_{k}X_{0}))^{\alpha}\leqslant\mathcal{P}_{t}^{k}\varphi^{\alpha}(\Pi_{k}Y_{0})\exp\bigg{(}\frac{\alpha}{\alpha-1}\frac{\|(I_{j\times
j}\otimes
Q^{-1/2})\Pi_{k}(X_{0}-Y_{0})\|_{W}^{2}}{2\underline{\lambda}(j,t)}\bigg{)}.$

###### Proof.

This follows after combining 3.8 with 4.1. ∎

The following infinite-dimensional Wang-Harnack inequality follows immediately
from the previous result by allowing $k\to\infty$.

######  Theorem 5.2.

Suppose that $\underline{A}_{\underline{\sigma}}$ as in (4.4) is of full rank
and that $X_{0},Y_{0}$ are such that $X_{0}-Y_{0}\in\mathcal{H}_{Q}$. Let
$\alpha>1$. Then for all bounded, measurable $\varphi:W\rightarrow\mathbf{R}$
and $t>0$ we have

$\displaystyle(\mathcal{P}_{t}\varphi(X_{0}))^{\alpha}\leqslant\mathcal{P}_{t}\varphi^{\alpha}(Y_{0})\exp\bigg{(}\frac{\alpha}{\alpha-1}\frac{\|X_{0}-Y_{0}\|_{\mathcal{H}_{Q}}^{2}}{{2}\underline{\lambda}(j,t)}\bigg{)}.$

###### Proof.

One need only show that for any $X_{0}\in W$,

$\Pi_{k}X(t)\xrightarrow{a.s.}X(t)$

as $k\rightarrow\infty$, and this follows immediately by uniqueness of
solutions and the definition of $\Pi_{k}$. ∎

The equivalence of Wang-Harnack inequalities with integrated Harnack
inequalities has been established in [22, 6, 29], along with how these
estimates imply quasi-invariance results of the following kind.

######  Theorem 5.3.

Fix $t>0$. The measure $\mu_{t}:=\operatorname{Law}(X_{t}(0))$ is quasi-
invariant under translations by elements of $\mathcal{H}_{Q}$, that is, for
$X_{0}\in\mathcal{H}_{Q}$ the measure
$\mu_{t}(X_{0}):=\operatorname{Law}(X_{t}(X_{0}))$ is mutually absolutely
continuous with respect to $\mu_{t}$. Moreover, for all $p>1$

$\left\|\frac{d\mu_{t}(X_{0})}{d\mu_{t}}\right\|_{L^{p}(W,\mu_{t})}\leqslant\exp\left(\frac{1+p}{{{2}\underline{\lambda}(j,t)}}\|X_{0}\|_{\mathcal{H}_{Q}}^{2}\right).$

######  Remark 5.4.

In each of the examples considered in Section 4 we saw that
$\underline{A}_{\underline{\sigma}}$ has full rank. Thus 5.3 applies to the
infinite-dimensional versions of these equations when the noise becomes
infinite dimensional ($k\rightarrow\infty$) as described above.

###  5.3. Mutual absolute continuity at time infinity

Under further conditions, we will use the results of Section 3 to deduce that
the laws of the infinite-dimensional Markov process $X(t)$ solving (5.4)
started from two initial conditions $X_{0}$ and $Y_{0}$ in $W$ become mutually
absolutely continuous at “time infinity”. This follows from a strengthening of
5.2, noting that that result only applies to sufficiently smooth initial
conditions in $W$ at a finite time $t>0$. Formally taking
$t,k\rightarrow\infty$ in the bound (5.5) suggests this should hold, but more
care needs to be taken when we pass to the limit.

To this end, for $k\in\mathbf{N}$ define

$\displaystyle\Delta_{k}:=(I_{W}-\Pi_{k}).$

######  Assumption 3.

The matrix $\underline{A}_{\underline{\sigma}}$ in (4.4) is of full rank and
the constant $\underline{\lambda}(j,t)>0$ as in (4.5) satisfies the following
conditions:

  * (i)

$\underline{\lambda}(j,t)\rightarrow\infty$ as $t\rightarrow\infty$.

  * (ii)

Define by (i) a sequence $\\{t_{k}\\}$ of positive times
$t_{k}\rightarrow\infty$ such that
$\underline{\lambda}(j,t_{k})=\alpha_{k}^{-1}$ for all $k$ where we recall
that the $\alpha_{k}$ are as in (5.1). Then there exists a $p>1$ such that for
any $X_{0}\in W$ we have that

$\displaystyle\mathbf{E}\|\Delta_{k}X(t_{k};X_{0})\|^{p}_{W}\rightarrow 0.$

Under 3, we can prove our main result at “time infinity”. However, we first
establish a natural condition which ensures 3(ii) is satisfied.

######  Proposition 5.5.

Suppose that the underlying matrix $\underline{A}$ as in (5.3) is such that
$\langle\underline{A}x,x\rangle_{\mathbf{R}^{j}}\leqslant 0$ for all
$x\in\mathbf{R}^{j}$. Suppose, furthermore, that 3(i) is satisfied and that
the sequence of times $\\{t_{k}\\}$ defined by
$\lambda(j,t_{k})=\alpha_{k}^{-1}$, $k\in\mathbf{N}$, satisfies the condition

(5.6) $\displaystyle t_{k}\sum_{\ell=k+1}^{\infty}\alpha_{\ell}\rightarrow
0\text{ as }k\rightarrow\infty.$

Then 3(ii) is satisfied with $p=2$.

######  Remark 5.6.

Suppose that $\underline{\lambda}(j,t)\geqslant c_{j}t^{3}-d_{j}$ for all
$t\geqslant 0$ for some constants $c_{j},d_{j}>0$ and that
$\alpha_{k}=1/k^{2}$. Then in this case,

$\displaystyle t_{k}\lesssim k^{2/3}$

so that

(5.7) $\displaystyle
t_{k}\sum_{\ell=k+1}^{\infty}\alpha_{\ell}\lesssim\frac{1}{k^{1/3}}\rightarrow
0\text{ as }k\rightarrow\infty.$

In short, the smallest positive eigenvalue $\underline{\lambda}(j,t)$ must
grow sufficiently fast as $t\rightarrow\infty$ with respect to the sequence
$\alpha_{k}$ so that (5.6) is satisfied.

###### Proof of 5.5.

Let $Y_{k}(t):=\Delta_{k}X(t;X_{0})$ and define

$\displaystyle\tau_{k,m}=\inf\\{t\geqslant 0\,:\,\|Y_{k}(t)\|_{W}\geqslant
m\\}.$

Set $\tau_{k,m}(t)=t\wedge\tau_{k,m}$ and observe that, by Itô’s formula and
the condition $\langle\underline{A}x,x\rangle_{\mathbf{R}^{j}}\leqslant 0$ we
have

$\displaystyle\mathbf{E}\|Y_{k}(\tau_{k,m}(t))\|_{W}^{2}$
$\displaystyle=\|Y_{k}(0)\|_{W}^{2}+2\mathbf{E}\int_{0}^{\tau_{k,m}(t)}\sum_{\ell=1}^{j}\langle(AY_{k}(s))_{\ell},(Y_{k}(s))_{\ell}\rangle_{H}\,ds$
$\displaystyle\qquad+\sum_{\ell_{1},\ell_{2}=1}^{j}\underline{\sigma}_{\ell_{1}\ell_{2}}^{2}\sum_{\ell=k+1}^{\infty}\alpha_{\ell}\mathbf{E}\tau_{k,m}(t)$
$\displaystyle\leqslant\|Y_{k}(0)\|_{W}^{2}+\sum_{\ell_{1},\ell_{2}=1}^{j}\underline{\sigma}_{\ell_{1}\ell_{2}}^{2}\sum_{\ell=k+1}^{\infty}\alpha_{\ell}t.$

Taking $m\rightarrow\infty$ and then plugging in $t=t_{k}$, we arrive at the
following estimate

$\displaystyle\mathbf{E}\|Y_{k}(t_{k})\|_{W}^{2}$
$\displaystyle\leqslant\|Y_{k}(0)\|_{W}^{2}+\sum_{\ell_{1},\ell_{2}=1}^{j}\underline{\sigma}_{\ell_{1}\ell_{2}}^{2}t_{k}\sum_{\ell=k+1}^{\infty}\alpha_{\ell}.$

Applying the hypothesis (5.7), we conclude the result. ∎

We are now prepared to state and prove our uniqueness result.

######  Corollary 5.7 (Uniqueness of stationary distributions via Wang-
Harnack).

Suppose that 3 is satisfied and $\mu$ and $\nu$ are stationary distributions
for the Markov process $X(t)$. Then for any $\alpha>1$ there exists a constant
$C=C(\mu,\nu,\alpha)>0$ such that

(5.8) $\displaystyle(\mu(\varphi))^{\alpha}\leqslant C\nu(\varphi^{\alpha})$

for all $\varphi:W\rightarrow[0,\infty)$ bounded, measurable. Consequently,
there is at most one stationary distribution corresponding to $X(t)$.

###### Proof.

Note that the final conclusion in the result follows immediately by symmetry
of the bound (5.8) and ergodic decomposition, for any two such stationary
distributions must be mutually absolutely continuous. Let $\alpha>1$. In order
to prove (5.8), it suffices to prove the bound for all
$\varphi:W\rightarrow[0,\infty)$ which are bounded, Lipschitz and such that
$\varphi^{\alpha}$ is also Lipschitz; that is,
$\|\varphi\|_{L^{\infty}}<\infty$,

$\displaystyle\|\varphi\|_{\text{Lip}}:=\sup_{\begin{subarray}{c}X,Y\in W\\\
X\neq
Y\end{subarray}}\frac{|\varphi(X)-\varphi(Y)|}{\|X-Y\|_{W}}<\infty\qquad\text{
and }\qquad\|\varphi^{\alpha}\|_{\text{Lip}}<\infty.$

Let $\epsilon\in(0,1)$ and pick $R=R(\mu,\nu,\alpha)>0$ large enough so that
$B_{R}:=\\{\|X\|_{W}<R\\}$ satisfies

$\displaystyle\mu(B_{R}^{c})<\tfrac{\epsilon}{2}\qquad\text{ and
}\qquad\nu(B_{R}^{c})<\tfrac{\epsilon}{2}.$

Then by invariance and Jensen’s inequality we have that

$\displaystyle\mu(\varphi)^{\alpha}=\bigg{[}\int\mu(dX)\mathcal{P}_{t_{k}}\varphi(X)\bigg{]}^{\alpha}$
$\displaystyle\leqslant\int\mu(dX)(\mathcal{P}_{t_{k}}\varphi(X))^{\alpha}\leqslant\int_{B_{R}}\mu(dX)(\mathcal{P}_{t_{k}}\varphi(X))^{\alpha}+\tfrac{\epsilon}{2}\|\varphi\|_{L^{\infty}}^{\alpha}.$

Multiplying the previous inequality by $\nu(W)=1$ and writing this as
$\nu(W)=\nu(B_{R})+\nu(B_{R}^{c})$ gives

$\displaystyle\mu(\varphi)^{\alpha}$
$\displaystyle\leqslant\nu(B_{R})\int_{B_{R}}\mu(dX)(\mathcal{P}_{t_{k}}\varphi(X))^{\alpha}+\epsilon\|\varphi\|_{L^{\infty}}^{\alpha}$
$\displaystyle=\int_{B_{R}}\int_{B_{R}}(\mathcal{P}_{t_{k}}\varphi(X))^{\alpha}\mu(dX)\nu(dY)+\epsilon\|\varphi\|_{L^{\infty}}^{\alpha}.$

In order to control the double integral above, observe that

$\displaystyle\mathcal{P}_{t_{k}}\varphi(X)=\mathcal{P}_{t_{k}}^{k}\varphi(\Pi_{k}X)+\mathbf{E}(\varphi(X(t_{k};X))-\varphi(\Pi_{k}X(t_{k};X)))$

and

$\displaystyle|\mathbf{E}(\varphi(X(t_{k};X))-\varphi(\Pi_{k}X(t_{k};X)))|\leqslant
2\|\varphi\|_{L^{\infty}}\wedge\big{\\{}\|\varphi\|_{\text{Lip}}(\mathbf{E}\|\Delta_{k}X(t_{k},X)\|^{p})^{1/p}\big{\\}}=:C_{k}(\varphi,X).$

By 3, we have that $C_{k}(\varphi,X)\rightarrow 0$ pointwise in $X$ as
$k\rightarrow\infty$. Hence applying 5.1 we have

$\displaystyle\int_{B_{R}}\int_{B_{R}}(\mathcal{P}_{t_{k}}\varphi(X))^{\alpha}\mu(dX)\nu(dY)$
$\displaystyle\leqslant
2^{\alpha}\int_{B_{R}}\int_{B_{R}}(\mathcal{P}_{t_{k}}^{k}\varphi(\Pi_{k}X))^{\alpha}\mu(dX)\nu(dY)+2^{\alpha}\int_{B_{R}}\mu(dX)C_{k}(\varphi,X)$
$\displaystyle\leqslant
2^{\alpha}C(\alpha,R)\int_{B_{R}}\int_{B_{R}}(\mathcal{P}_{t_{k}}^{k}\varphi^{\alpha}(\Pi_{k}Y))\mu(dX)\nu(dY)+2^{\alpha}\int_{B_{R}}\mu(dX)C_{k}(\varphi,X)$
$\displaystyle\leqslant
2^{\alpha}C(\alpha,R)\int_{B_{R}}\int_{B_{R}}(\mathcal{P}_{t_{k}}\varphi^{\alpha}(Y))\mu(dX)\nu(dY)+2^{\alpha}\int_{B_{R}}\mu(dX)C_{k}(\varphi,X)$
$\displaystyle\qquad+2^{2\alpha}C(\alpha,R)\int_{B_{R}}\nu(dY)C_{k}(\varphi^{\alpha},Y).$
$\displaystyle\leqslant
2^{\alpha}C(\alpha,R)\nu(\varphi^{\alpha})+2^{\alpha}\int_{B_{R}}\mu(dX)C_{k}(\varphi,X)+2^{2\alpha}C(\alpha,R)\int_{B_{R}}\nu(dY)C_{k}(\varphi^{\alpha},Y).$

Taking $k\rightarrow\infty$ using 3 and the bounded convergence theorem and
then letting $\epsilon\rightarrow 0$ we obtain the desired bound (5.8) for all
$\varphi:W\rightarrow[0,\infty)$ bounded, Lipschitz with $\varphi^{\alpha}$
Lipschitz. This finishes the proof. ∎

We now revisit the linear kinetic Fokker Planck equation and the degenerately
damped oscillators considered in Section 4.

####  5.3.1. Linear kinetic Fokker-Planck

In the setting of Section 4.2.1 in Case 1 ($\gamma\gg 1$), we recall $j=2$ and
that we obtained the following bound

(5.9) $\displaystyle\underline{\lambda}(2,t)\geqslant\frac{e^{t/\gamma}}{16}$

for all $t\geqslant\gamma\geqslant\gamma_{L}$. Fixing
$\gamma\geqslant\gamma_{L}$ and a summable sequence
$\\{\alpha_{k}\\}_{k\in\mathbf{N}}$, we observe by (5.9) that the sequence
$\\{t_{k}\\}$ in 3 satisfies, for all $k$ large enough,

$\displaystyle t_{k}\leqslant\gamma\log(\alpha_{k}^{-1})+\gamma\log(16).$

Furthermore, in this context it is clear that
$\langle\underline{A}x,x\rangle_{\mathbf{R}^{2}}\leqslant 0$. Thus by 5.5, for
3 to be satisfied, it suffices that $\\{\alpha_{k}\\}$ satisfy

(5.10)
$\displaystyle\log(\alpha_{k}^{-1})\sum_{\ell=k+1}^{\infty}\alpha_{\ell}\rightarrow
0\text{ as }k\rightarrow\infty.$

Clearly, condition (5.10) is satisfied if the noise decays fast enough at
large scales, e.g. $\alpha_{k}\sim 1/k^{p}$ or even $\alpha_{k}\sim
1/(k\log(k)^{p})$ for some $p>1$. In particular, under (5.10), uniqueness of
stationary distributions follows from 5.7. The same conclusion also holds in
the case when $\gamma\approx 0$.

####  5.3.2. Oscillators with some damping

In the setting of Section 4.2.3, we obtained the bound

$\displaystyle\underline{\lambda}(j,t)\geqslant\underline{\lambda}(j,1)\frac{c_{j}}{r_{j}}(e^{r_{j}(t-1)}-1)$

for all $t\geqslant 1$ (4.17) where $r_{j}$ and $c_{j}$ are positive constants
defined in (4.16). Because $\underline{\lambda}(j,1)>0$ and
$\langle\underline{A}x,x\rangle_{\mathbf{R}^{j}}\leqslant 0$, a similar
argument to the one used in Section 5.3.1 implies that, provided (5.10) is
satisfied, there is at most one stationary distribution for the corresponding
infinite-dimensional process (5.4) by 5.5 and 5.7.

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