# The strong data processing inequality under the heat flow

B. Klartag, O. Ordentlich Bo’az Klartag is with the Department of Mathematics,
Weizmann Institute of Science, Rehovot 76100, Israel
(boaz.klartag@weizmann.ac.il). Or Ordentlich is with the Hebrew University of
Jerusalem, Israel (or.ordentlich@mail.huji.ac.il). The work of BK was
supported by the Israel Science Foundation (ISF), grant No. 765/19. The work
of OO was supported by the Israel Science Foundation (ISF), grant No. 1641/21.

###### Abstract

Let $\nu$ and $\mu$ be probability distributions on $\mathbb{R}^{n}$, and
$\nu_{s},\mu_{s}$ be their evolution under the heat flow, that is, the
probability distributions resulting from convolving their density with the
density of an isotropic Gaussian random vector with variance $s$ in each
entry. This paper studies the rate of decay of $s\mapsto D(\nu_{s}\|\mu_{s})$
for various divergences, including the $\chi^{2}$ and Kullback-Leibler (KL)
divergences. We prove upper and lower bounds on the strong data-processing
inequality (SDPI) coefficients corresponding to the source $\mu$ and the
Gaussian channel. We also prove generalizations of de Brujin’s identity, and
Costa’s result on the concavity in $s$ of the differential entropy of
$\nu_{s}$. As a byproduct of our analysis, we obtain new lower bounds on the
mutual information between $X$ and $Y=X+\sqrt{s}Z$, where $Z$ is a standard
Gaussian vector in $\mathbb{R}^{n}$, independent of $X$, and on the minimum
mean-square error (MMSE) in estimating $X$ from $Y$, in terms of the Poincaré
constant of $X$.

##  1 Introduction

For two probability distributions $\nu$ and $\mu$ on $\mathbb{R}^{n}$, where
$\nu$ is absolutely continuous with respect to $\mu$, and a smooth, convex
function $\varphi:(0,\infty)\to\mathbb{R}$ with $\varphi(1)=0$, the
$\varphi$-divergence [17] is defined as

$\displaystyle
D_{\varphi}(\nu\|\mu)=\mathbb{E}_{\mu}\left[\varphi\left(\frac{d\nu}{d\mu}\right)\right],$
(1)

where $\varphi(0)=\lim_{t\rightarrow 0^{+}}\varphi(t)$. Let
$Z\sim\mathcal{N}(0,{\rm Id})$ be a standard Gaussian random variable in
$\mathbb{R}^{n}$. For $s\geq 0$, denote by $\nu_{s}$ the probability
distribution of the random variable $Y=X+\sqrt{s}Z$ when $X\sim\nu$ is
independent of $Z$. Similarly, write $\mu_{s}$ for the probability
distribution of $Y$ when $X\sim\mu$ is independent of $Z$. By the data-
processing inequality [17, 37] for $\varphi$-divergences, we have that

$s\mapsto D_{\varphi}(\nu_{s}\|\mu_{s})$

is non-increasing in $s\geq 0$. The goal of this paper is to provide estimates
on the rate of decay of $s\mapsto D_{\varphi}(\nu_{s}\|\mu_{s})$ as a function
of $\mu$ and uniformly over the measure $\nu$. We stress that here both
measures evolve according to the heat flow. This is in contrast to prior work
that analyzed the evolution of the entropy and similar functionals of
$\nu_{s}$ according to the heat flow. This corresponds to taking $\mu$ as the
Lebesgue measure whose heat flow evolution satisfies $\mu_{s}=\mu$. See more
details below.

To that end, we study the _strong data-processing inequality (SDPI)
constant/contraction coefficient_ for the probability distribution $\mu$ and
the Gaussian channel [1], [37, chapter 33], which is defined as

$\displaystyle\eta_{\varphi}(\mu,s)\triangleq\sup_{\nu:0<D_{\varphi}(\nu\|\mu)<\infty}\frac{D_{\varphi}(\nu_{s}\|\mu_{s})}{D_{\varphi}(\nu\|\mu)}.$
(2)

Recalling that $D_{\varphi}(\nu_{s}\|\mu_{s})\geq 0$ (with equality if and
only if $\nu_{s}=\mu_{s}$), by the data-processing inequality we have that
$0\leq\eta_{\varphi}(\mu,s)\leq 1$. We mostly analyze two choices of
$\varphi$. The first is $\varphi(x)=\varphi_{\text{KL}}(x)=x\log x$,111In this
paper, logarithms are always taken to the natural basis. which results in the
Kullback-Leibler (KL) divergence

$\displaystyle
D(\nu\|\mu)=D_{\text{KL}}(\nu\|\mu)=\int_{\mathbb{R}^{n}}d\nu\log\frac{d\nu}{d\mu},$
(3)

and the second is $\varphi(x)=\varphi_{\chi^{2}}(x)=(x-1)^{2}$, which results
in the $\chi^{2}$-divergence

$\chi^{2}(\nu\|\mu)=D_{\chi^{2}}(\nu\|\mu)=\int_{\mathbb{R}^{n}}\left(\frac{d\nu}{d\mu}-1\right)^{2}d\mu.$
(4)

It is common to rewrite the integrand in (4), by abuse of notation, as
$(d\nu-d\mu)^{2}/d\mu$, where $d\mu$ refers to the “density of $\mu$” with
respect to some ambient measure. We denote the corresponding contraction
coefficients by $\eta_{\text{KL}}(\mu,s)$ and $\eta_{\chi^{2}}(\mu,s)$. By
[36, Theorem 2] we have that

$\eta_{\chi^{2}}(\mu,s)\leq\eta_{\varphi}(\mu,s),$ (5)

for any $\varphi$ with $\varphi^{\prime\prime}(1)>0$. In particular,
$\eta_{\chi^{2}}(\mu,s)\leq\eta_{\text{KL}}(\mu,s)$. Intuitively, inequality
(5) follows by considering measures $\nu$ that are very close to $\mu$, and
using Taylor approximation.

Some of our results hold for further choices of $\varphi$ beyond
$\varphi_{\chi^{2}}$ and $\varphi_{\text{KL}}$. In particular, the crucial
requirement for our results is that $1/\varphi^{\prime\prime}$ is concave, as
in Chafaï [11]. This requirement is fulfilled for $\varphi_{\chi^{2}}$ and
$\varphi_{\text{KL}}$ (in fact for those functions, $1/\varphi^{\prime\prime}$
is linear). Another important class of functions for which
$1/\varphi^{\prime\prime}$ is concave is $\varphi_{\lambda}(x)=x^{\lambda}-1$,
for $\lambda\in(1,2)$. The corresponding divergence
$D_{\varphi_{\lambda}}(\nu\|\mu)$, defined by (1), is related to the Rényi
divergence of order $\lambda$, via the monotone transformation

$\displaystyle
D_{\lambda}(\nu\|\mu)=\frac{1}{\lambda-1}\log(1+D_{\varphi_{\lambda}}(\nu\|\mu)).$
(6)

Thus, although $D_{\lambda}(\nu\|\mu)$ is not a $\varphi$-divergence, it does
satisfy the data processing inequality, and bounds on the decay rate of
$s\mapsto D_{\varphi_{\lambda}}(\nu_{s}\|\mu_{s})$ immediately imply bounds on
the decay rate of the Rényi divergence $s\mapsto
D_{\lambda}(\nu_{s}\|\mu_{s})$. We note that the concavity requirement on
$1/\varphi^{\prime\prime}$ fails to hold for many other important
$\varphi$-divergences, such as the Jensen-Shannon divergence
($\varphi(x)=x\log\frac{2x}{x+1}+\log\frac{2}{x+1}$), the Le Cam divergence
($\varphi(x)=(1-x)/(2x+2)$), the total variation divergence
($\varphi(x)=|x-1|/2$) and the Hellinger divergence
($\varphi(x)=(1-\sqrt{x})^{2}$).

It is well-known, see. e.g., [37, Theorem 33.12] that

$\displaystyle\sqrt{\eta_{\chi^{2}}(\mu,s)}=S(X,X+\sqrt{s}Z)$ (7)

where $X\sim\mu$ is independent of the standard Gaussian $Z$. Here $S(X,Y)$ is
the Hirschfeld-Gebelein-Rényi maximal correlation [39] between the random
variables $X$ and $Y$ defined as

$\displaystyle S(X,Y)$
$\displaystyle=\sup_{f,g:\mathbb{R}^{n}\to\mathbb{R}}\frac{\mathbb{E}f(X)g(Y)-\mathbb{E}(f(X))\mathbb{E}(g(Y))}{\sqrt{\mathrm{Var}(f(X))\mathrm{Var}(g(Y))}}$
(8) $\displaystyle=\sup_{f}\sqrt{\mathrm{Var}(\mathbb{E}[f(X)|Y])}.$ (9)

The supremum in (9) runs over all measurable functions
$f:\mathbb{R}^{n}\to\mathbb{R}$ with $\mathbb{E}f(X)=0$ and
$\mathrm{Var}(f(X))=1$. In the passage from (8) to (9) we use the Cauchy-
Schwartz inequality.

The quantities $\eta_{\text{KL}}(\mu,s)$ and $\eta_{\chi^{2}}(\mu,s)$ can be
thought of as (normalized) measures of statistical dependence between
$X\sim\mu$ and $Y=X+\sqrt{s}Z$. Two other (un-normalized) measures of
dependence we will consider are the minimum mean-square error (MMSE)222For a
vector $z=(z_{1},\ldots,z_{n})\in\mathbb{R}^{n}$ we denote by
$|z|=\|z\|_{2}=\sqrt{\sum_{j}z_{j}^{2}}$ the $\ell_{2}$ norm of $z$.

$\displaystyle\mathrm{mmse}(\mu,s)=\mathbb{E}|X-\mathbb{E}[X|X+\sqrt{s}Z]|^{2},$
(10)

and the mutual information

$\displaystyle
I(\mu,s)=I(X;X+\sqrt{s}Z)=h(X+\sqrt{s}Z)-h(X+\sqrt{s}Z|X)=h(\mu_{s})-\frac{n}{2}\log(2\pi
es),$

where for a probability distribution $\mu$ with density $\rho$ in
$\mathbb{R}^{n}$, the differential entropy of $X\sim\mu$ is

$\displaystyle h(X)=h(\mu)=-\int_{\mathbb{R}^{n}}\rho(x)\log\rho(x)dx.$ (11)

Our results will be expressed in terms of the Poincaré constant
$C_{\mathrm{P}}(X)=C_{\mathrm{P}}(\mu)$, of $X\sim\mu$, and the log-Sobolev
constant $C_{\mathrm{LS}}(X)=C_{\mathrm{LS}}(\mu)$. The Poincaré constant of a
random vector $X$ in $\mathbb{R}^{n}$, denoted by $C_{P}(X)$, is the infimum
over all $C\geq 0$ such that for any locally-Lipschitz function
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ satisfying $\mathbb{E}|\nabla
f(X)|^{2}<\infty$,

$\mathrm{Var}(f(X))\leq C\cdot\mathbb{E}|\nabla f(X)|^{2}.$ (12)

The log-Sobolev constant of a random vector $X$ in $\mathbb{R}^{n}$, denoted
by $C_{\mathrm{LS}}(X)$, is the infimum over all $C\geq 0$ such that for any
locally-Lipschitz function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ satisfying
$\mathbb{E}f^{2}(X)=1$ and $\mathbb{E}|\nabla f(X)|^{2}<\infty$,

$\mathbb{E}f^{2}(X)\log f^{2}(X)\leq 2C\cdot\mathbb{E}|\nabla f(X)|^{2}.$ (13)

We list a few important properties of the Poincaré and log-Sobolev constants,
see [5] for more explanations, proofs and references.

  * •

The Poincaré constant and the log-Sobolev constant of the standard Gaussian
random vector $Z\in\mathbb{R}^{n}$ satisfy
$C_{\mathrm{P}}(Z)=C_{\mathrm{LS}}(Z)=1$. Furthermore, for any random variable
we have that

$\frac{C_{\mathrm{P}}(X)}{\frac{1}{n}\mathbb{E}|X-\mathbb{E}X|^{2}}\geq
1\qquad\text{and}\qquad\frac{C_{\mathrm{LS}}(X)}{\frac{1}{n}\mathbb{E}|X-\mathbb{E}X|^{2}}\geq
1.$

and in both cases the lower bound is attained if and only if $X$ is an
isotropic Gaussian random variable.

  * •

$C_{\mathrm{P}}(\mu)\leq C_{\mathrm{LS}}(\mu)$.

  * •

For $p\geq 1$, let
$\bar{\mathcal{B}}_{p}=\\{x\in\mathbb{R}^{n}~{}:~{}\|x\|_{p}<\alpha\\}$, where
$\alpha$ is chosen such that $\mathrm{vol}(\bar{\mathcal{B}}_{p})=1$, be the
unit-volume $\ell_{p}$-ball. For
$U_{p}\sim\mathrm{Uniform}(\bar{\mathcal{B}}_{p})$, we have that

$\displaystyle c<C_{\mathrm{P}}(U_{p})<C$ (14)

for certain explicit universal constants $c,C>0$. For $p\in[2,+\infty]$ we
furthermore have

$\displaystyle c<C_{\mathrm{LS}}(U_{p})<C$ (15)

for universal constants $c,C>0$. The log-Sobolev constant is not bounded by a
universal constant when $p<2$. These results may be extracted from the
literature in the following way: First, up to a universal constant, Poincaré
and log-Sobolev inequalities follows from a corresponding isoperimetric
inequality (see Ledoux [32] for the log-Sobolev case and Cheeger [12] for the
Poincaré case). Second, the corresponding isoperimetric inequalities were
proven in Sodin [42] ($1\leq p\leq 2$) and Latała and Wojtaszczyk [31].

  * •

$C_{\mathrm{P}}(X)$ is finite for any log-concave random vector $X$ in
$\mathbb{R}^{n}$, see Bobkov [9].

  * •

When $X$ is a log-concave random vector in $\mathbb{R}^{n}$, the Kanann-
Lovász-Simonovits (KLS) conjecture from [27] suggests that

$\|\mathrm{Cov}(X)\|_{op}\leq C_{\mathrm{P}}(X)\leq
C\cdot\|\mathrm{Cov}(X)\|_{op},$ (16)

where $C>0$ is a universal constant. Here,
$\mathrm{Cov}(X)\in\mathbb{R}^{n\times n}$ is the covariance matrix of $X$,
whose $i,j$ entry is $\mathbb{E}X_{i}X_{j}-\mathbb{E}X_{i}\mathbb{E}X_{j}$. We
write $\|\cdot\|_{op}$ for the operator norm, thus333We denote the standard
inner product of two vectors $x,y\in\mathbb{R}^{n}$ by $x\cdot y=\langle
x,y\rangle=x^{T}y$.

$\displaystyle\|\mathrm{Cov}(X)\|_{op}=\sup_{|\theta|=1}\mathrm{Var}(X\cdot\theta)$
(17)

where the supremum runs over all unit vectors $\theta\in\mathbb{R}^{n}$. The
left-hand side inequality in (16) follows from the definitions (12) and (17).
The current state of the art regarding the conjectural right-hand side
inequality in (16) is the bound from [28]

$\displaystyle C_{\mathrm{P}}(X)\leq C\log n\cdot\|\mathrm{Cov}(X)\|_{op},$
(18)

where $C>0$ is a universal constant and $X$ is an arbitrary log-concave random
variable in $\mathbb{R}^{n}$.

We recall that an absolutely continuous probability distribution $\mu$ on
$\mathbb{R}^{n}$, or a random variable $X\sim\mu$ is log-concave if it is
supported in an open, convex set $K\subseteq\mathbb{R}^{n}$ (which could be
$\mathbb{R}^{n}$ itself) and has a density of the form $e^{-H}$ in $K$ where
$H$ is a convex function. Thus a Gaussian measure is log-concave, as well as
the uniform measure on an arbitrary convex body.

###  1.1 Main results on $\chi^{2}$ divergence

For $\eta_{\chi^{2}}(\mu,s)$, we prove the following results.

######  Theorem 1.1.

For any probability distribution $\mu$ and $s>0$

$\displaystyle
1-\frac{\mathrm{mmse}(\mu,s)}{\mathbb{E}_{\mu}|X-\mathbb{E}X|^{2}}\leq\eta_{\chi^{2}}(\mu,s)=S^{2}(X,X+\sqrt{s}Z)\leq\frac{1}{1+\frac{s}{C_{\mathrm{P}}(\mu)}}.$
(19)

If $\mu$ is additionally log-concave, then,

$\eta_{\chi^{2}}(\mu,s)\geq e^{-s/C_{P}(\mu)}.$ (20)

######  Theorem 1.2.

For any log-concave probability distribution $\mu$ and function
$f:\mathbb{R}^{n}\to\mathbb{R}$, the function
$s\mapsto\log\mathrm{Var}(\mathbb{E}[f(X)|X+\sqrt{s}Z])$ is convex. In
particular, whenever $\mu$ is log-concave, $s\mapsto\log S(X,X+\sqrt{s}Z)$ is
convex, and consequently, $s\mapsto\eta_{\chi^{2}}(\mu,s)$ is also convex.

Inequality (20) as well as Theorem 1.2 are mostly an interpretation of the
results of [29]. Theorem 20 implies that for log-concave $\mu$ the rate at
which $s\mapsto\eta_{\chi^{2}}(\mu,s)$ decreases with $s$ is largely
determined by $C_{\mathrm{P}}(\mu)$. In particular, if
$s^{\chi^{2}}_{\mu}(\alpha)=\min\\{s~{}:~{}\eta_{\chi^{2}}(\mu,s)\leq\alpha\\}$
is the required time for $\eta_{\chi^{2}}(\mu,s)$ to drop below
$\alpha\in(0,1)$, then

$\displaystyle C_{\mathrm{P}}(\mu)\cdot\log\frac{1}{\alpha}\leq
s^{\chi^{2}}_{\mu}(\alpha)\leq
C_{\mathrm{P}}(\mu)\cdot\left(\frac{1}{\alpha}-1\right).$ (21)

For $\alpha$ close to $1$, the upper bound and lower bound nearly coincide. We
refer to the quantity $s^{\chi^{2}}_{\mu}(\alpha)$ for $\alpha=1/2$ as the
“$\chi^{2}$ half-blurring time” of the measure $\mu$. In the log-concave case,
the $\chi^{2}$ half-blurring time of $\mu$ has the order of magnitude of
$C_{P}(\mu)$, up to an explicit multiplicative universal constant. A Corollary
of Theorem 20 is the following:

######  Corollary 1.3.

$\displaystyle\mathrm{mmse}(\mu,s)\geq\frac{\mathbb{E}_{\mu}|X-\mathbb{E}X|^{2}}{1+\frac{C_{\mathrm{P}}(\mu)}{s}}=\frac{nP}{1+\frac{P}{s}\cdot\bar{C}_{\mathrm{P}}(\mu)}$
(22)

where

$\displaystyle P=\frac{1}{n}\mathbb{E}_{\mu}|X-\mathbb{E}X|^{2},$ (23)

and

$\displaystyle\bar{C}_{\mathrm{P}}(\mu)=\frac{C_{\mathrm{P}}(\mu)}{P}.$ (24)

Let us compare Corollary 1.3 with known bounds in Estimation Theory. Assume
that $\mu$ has smooth density $\rho$ and recall that the Bayesian Cramér-Rao
lower bound (CRLB)/multivariate van Trees inequality [22, eq.10] for the
additive isotropic Gaussian noise case, gives

$\displaystyle\mathrm{mmse}(\mu,s)\geq\frac{n^{2}s}{n+s\mathcal{J}(\mu)}=\frac{nP}{\bar{\mathcal{J}}(\mu)+\frac{P}{s}},$
(25)

where

$\displaystyle\mathcal{J}(\mu)=\int_{x\in\mathbb{R}^{n}}\frac{|\nabla\rho(x)|^{2}}{\rho(x)}dx.$
(26)

and

$\displaystyle\bar{\mathcal{J}}(\mu)=\frac{\mathcal{J}(\mu)\cdot P}{n}.$ (27)

We see that the lower bound from Corollary 1.3 is tighter than the Bayesian
CRLB whenever
$\frac{P}{s}(\bar{C}_{\mathrm{P}}(\mu)-1)<\bar{\mathcal{J}}(\mu)-1$. In
particular, whenever $\bar{C}_{\mathrm{P}}(\mu)$ is finite, the bound from
Corollary 1.3 is tighter than the Bayesian CRLB for low enough signal-to-noise
ratio (SNR) $\frac{P}{s}$. There are also cases where $\bar{\mathcal{J}}(\mu)$
is infinite, rendering the Bayesian CRLB useless for all $s>0$, whereas
$\bar{C}_{\mathrm{P}}(\mu)$ is finite. An example of such probability
distribution is the uniform distribution over a convex set. We remark that
there are also examples where $\mathcal{J}(\mu)$ is finite and
$C_{\mathrm{P}}(\mu)$ infinite, such as the density $t\mapsto
e^{-(t^{2}+1)^{\alpha}}$ on the real line for $0<\alpha<1/2$.

Using the I-MMSE identity of Guo, Shamai and Verdù [24, Theorem 2], we can
deduce the following.

######  Corollary 1.4.

$\displaystyle
I(\mu,s)=I(X;X+\sqrt{s}Z)\geq\frac{n}{2}\cdot\frac{1}{\bar{C}_{\mathrm{P}}(\mu)}\log\left(1+\frac{P}{s}\cdot\bar{C}_{\mathrm{P}}(\mu)\right),$
(28)

where $P$ and $\bar{C}_{\mathrm{P}}(\mu)$ are as defined in (23) and (24),
respectively.

It is easy to verify that the function
$t\mapsto\frac{1}{t}\log(1+\frac{P}{s}\cdot t)$ is decreasing. Thus, the lower
bound above is monotonically decreasing in $\bar{C}_{\mathrm{P}}(\mu)$, which
in turn is minimized for the isotropic Gaussian distribution, for which its
value is $1$. Thus, the lower bound is tight for the isotropic Gaussian
distribution, and controls the mutual information loss of the probability
distribution $\mu$ from the upper bound $\frac{n}{2}\log(1+\frac{P}{s})$ via
the distance from Gaussianity measure $\bar{C}_{\mathrm{P}}(\mu)\geq 1$.

It is insightful to compare the lower bound from Corollary 1.4 to the standard
entropy power inequality (EPI)-based [16, Theorem 17.7.3] lower bound

$\displaystyle
I(\mu,s)\geq\frac{n}{2}\log\left(1+\frac{P}{s}\cdot\bar{N}(\mu)\right).$ (29)

where

$\displaystyle\bar{N}(\mu)=\frac{e^{\frac{2}{n}h(\mu)}}{2\pi e\cdot P}$ (30)

is the normalized entropy power (recall that $\bar{N}(\mu)\leq 1$ with
equality iff $\mu$ is Gaussian isotropic). It is easy to see that whenever
$\bar{C}_{\mathrm{P}}(\mu)<\infty$, the new lower bound from Corollary 1.4 is
tighter than the EPI-based bound (29) for $s$ large enough (low SNR).

Note that using the Bayesian CRLB (25) and the I-MMSE identity, one easily
obtains the lower bound

$\displaystyle
I(\mu,s)\geq\frac{n}{2}\log\left(1+\frac{P}{s}\cdot\frac{1}{\bar{\mathcal{J}}(\mu)}\right).$
(31)

However, since for any probability distribution $\mu$ we have that
$\bar{\mathcal{J}}(\mu)\cdot\bar{N}(\mu)\geq 1$, with equality iff $\mu$ is
the isotropic Gaussian distribution [10], [19, Theorem 16], this bound is
subsumed by the EPI-based bound (29).

###  1.2 Main results on KL-divergence

We now move on to present our results concerning the KL divergence. Our first
result is a generalization of de Bruijn’s identity.

######  Proposition 1.5.

Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}^{n}$ with
$D(\nu\,\,||\,\,\mu)<\infty$, and assume $\mathbb{E}_{\mu}|X|^{4}<\infty$.
Then for any $s>0$,

$\displaystyle\frac{d}{ds}D(\nu_{s}\,\,||\,\,\mu_{s})=-\frac{1}{2}J(\nu_{s}\,\,||\,\,\mu_{s}),$
(32)

where the Fisher information of a probability distribution $\nu$ relative to
$\mu$ is, with $f=d\nu/d\mu$,

$\displaystyle J(\nu\,\,||\,\,\mu)=\int_{\mathbb{R}^{n}}\frac{|\nabla
f|^{2}}{f}d\mu.$ (33)

Proposition 1.5 easily implies de Bruijn’s identity.

Some of the proceeding results rely on the correctness of a certain
conjecture, namely Conjecture 4.2, stated in Section 4. Calling it a
conjecture is perhaps too modest: we provide an almost full of proof for
Conjecture 4.2. The only gap between our “proof” and a fully rigorous proof is
the justification of order exchange between integration and differentiation,
as well as justifying several integration by parts (that is, showing that the
involved functions vanish at infinity). While it is not trivial to rigorously
justify these steps, we are quite certain they are correct under some minimal
regularity assumptions. In fact, the justification of such steps is often
ignored in the literature. For example, to the best of our knowledge, the
first time a fully rigorous proof was given for de Bruijn’s identity was by
Barron in 1984 [6], while prior proofs, such as that of Blachman [8] did not
justify the exchange of differentiation and integration. Whenever a statement
relies on the correctness of Conjecture 4.2, we explicitly emphasize it.

We prove the following bounds for the KL contraction coefficient:

######  Theorem 1.6.

For any probability measure $\mu$ and $s>0$,

$\displaystyle
1-\frac{\mathrm{mmse}(\mu,s)}{\mathbb{E}_{\mu}|X-\mathbb{E}X|^{2}}\leq\eta_{\text{KL}}(\mu,s)\leq\frac{1}{1+\frac{s}{C_{\mathrm{LS}}(\mu)}}.$
(34)

If additionally $\mu$ is log-concave, then, assuming the validity of
Conjecture 4.2, we also have

$\displaystyle 1-\frac{s}{C_{\mathrm{LS}}(\mu)}\leq\eta_{\text{KL}}(\mu,s).$
(35)

The left-hand side inequality in (34) immediately follows from (5) and (19).

######  Theorem 1.7.

Assuming validity of Conjecture 4.2, for any probability distribution $\nu$
and any log-concave probability distribution $\mu$, the mapping $s\mapsto
D(\nu_{s}\|\mu_{s})$ is convex. In particular, assuming validity of Conjecture
4.2, whenever $\mu$ is log-concave, $s\mapsto\eta_{\text{KL}}(\mu,s)$ is
convex

Theorem 1.7 implies that $s\mapsto h(\nu_{s})$ is concave, which is [13,
Corollary 1]. Thus, Theorem 1.7 is stronger than Costa’a Corollary about
concavity of $s\mapsto h(\nu_{s})$. On the other hand, Theorem 1.7 does not
imply the concavity of $s\mapsto\exp\left(\frac{2}{n}h(\nu_{s})\right)$, which
is Costa’s main result in [13].

As in our analysis for $\eta_{\chi^{2}}(s,\mu)$, we see that Theorem 1.6
implies that for log-concave $\mu$ the rate at which
$s\mapsto\eta_{\text{KL}}(\mu,s)$ decreases with $s$ is largely determined by
$C_{\mathrm{LS}}(\mu)$. In particular, if
$s^{\text{KL}}_{\mu}(\alpha)=\min\\{s~{}:~{}\eta_{\text{KL}}(\mu,s)\leq\alpha\\}$
is the required time for $\eta_{\text{KL}}(\mu,s)$ to drop below
$\alpha\in(0,1)$, then

$\displaystyle(1-\alpha)\cdot C_{\mathrm{LS}}(\mu)\leq
s^{\text{KL}}_{\mu}(\alpha)\leq\left(\frac{1-\alpha}{\alpha}\right)C_{\mathrm{LS}}(\mu).$
(36)

For $\alpha$ close to $1$, the upper bound and lower bound nearly coincide. We
refer to the quantity $s^{\text{KL}}_{\mu}(\alpha)$ for $\alpha=1/2$ as the
“$KL$ half-blurring time” of the measure $\mu$. In the log-concave case, the
$KL$ half-blurring time of $\mu$ has the order of magnitude of
$C_{\mathrm{LS}}(\mu)$, up to an explicit multiplicative universal constant.

###  1.3 Main results on general divergences

Fix a smooth, convex function $\varphi:(0,\infty)\to\mathbb{R}$ with
$\varphi(1)=0$ and set $\varphi(0)=\lim_{t\rightarrow 0^{+}}\varphi(t)$. In
[11], Chafaï introduced the concepts of $\varphi$-Sobolev inequalities and
$\varphi$-entropy, which turn out to fit nicely with our study of the
evolution of the $\varphi$-divergence $D_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})$.

The $\varphi$-Sobolev constant of a random vector $X$ in $\mathbb{R}^{n}$,
denoted by $C_{\varphi}(X)$, is the infimum over all $C\geq 0$ such that for
any locally-Lipschitz function $f:\mathbb{R}^{n}\rightarrow(0,\infty)$ with
$\mathbb{E}\varphi(X)=1$,

$\mathbb{E}\varphi(f(X))\leq\frac{C}{2}\cdot\mathbb{E}\varphi^{\prime\prime}(f(X))|\nabla
f(X)|^{2}.$ (37)

This generalized the definition of the Poincaré constant and the log-Sobolev
constant. Indeed, in the case where $\varphi(x)=x\log x$ is the
$KL$-divergence, we have $C_{\varphi}(X)=C_{\mathrm{LS}}(X)$, and a simple
argument reveals that when $\varphi(x)=(x-1)^{2}$ we have
$C_{\varphi}(X)=C_{\mathrm{P}}(X)$. When $X\sim\mu$ we set
$C_{\varphi}(\mu)=C_{\varphi}(X)$. In the general case, our definition (37) is
slightly different from the one in Chafaï in that we impose the additional
requirement that $\mathbb{E}f(X)=1$, which implies that
$\varphi(\mathbb{E}f(X))=0$. In Chafaï the left-hand side of (37) is replaced
by $\mathbb{E}\varphi(f(X))-\varphi(\mathbb{E}f(X))$. In the cases of interest
just described, the two definitions coincide by homogeneity.

Given two probability distributions $\mu$ and $\nu$ on $\mathbb{R}^{n}$, with
$\nu$ absolutely continuous with a smooth density with respect to $\mu$, we
defined the $\varphi$-Fisher information via

$J_{\varphi}(\nu\,\,||\,\,\mu)=\int_{\mathbb{R}^{n}}\varphi^{\prime\prime}(f)|\nabla
f|^{2}d\mu,$

where $f=d\nu/d\mu$. Thus we have the defining inequality of the
$\varphi$-Sobolev constant,

$D_{\varphi}(\nu\,\,||\,\,\mu)\leq\frac{C_{\varphi}(\mu)}{2}\cdot
J_{\varphi}(\nu\,\,||\,\,\mu).$ (38)

Some of the properties of the Poincaré and log-Sobolev constants generalize to
the more general context of the $\varphi$-Sobolev constant. The
$\varphi$-Sobolev constant is $2$-homogeneous: For any fixed number
$\lambda>0$,

$C_{\varphi}(\lambda X)=\lambda^{2}C_{\varphi}(X).$

This follows from the definition (37), by substituting $g(x)=f(\lambda x)$.
The Poincaré constant is the smallest of all $\varphi$-Sobolev constants:

######  Proposition 1.8.

Assume that $\varphi^{\prime\prime}(1)>0$. Then for any random vector $X$ for
which $C_{\varphi}(X)<\infty$,

$C_{P}(X)\leq C_{\varphi}(X).$

Proposition 1.8 must be known, yet for lack of concise reference we provide
its proof below. The function $1/\varphi^{\prime\prime}$ plays an important
role in the sequel. When it is a concave function on $(0,\infty)$ – this is
equivalent to condition (H1) from Chafaï [11] – it is possible to compute the
$\varphi$-Sobolev constant of the Gaussian measure. The following proposition
follows from the results of Chafaï [11] and Proposition 1.8. For convenience
we provide a proof.

######  Proposition 1.9.

Let $\varphi:(0,\infty)\to\mathbb{R}$ be a smooth, convex function with
$\varphi(1)=0$ such that $1/\varphi^{\prime\prime}$ is concave. Let $X$ be a
standard Gaussian random vector in $\mathbb{R}^{n}$. Then,

$C_{\varphi}(X)=1.$

Proposition 1.5, the de Bruijn identity, may be generalized even further:

######  Proposition 1.10.

Let $\varphi(x)=\varphi_{\lambda}(x)=x^{\lambda}-1$, for $\lambda>1$. Let
$\mu$ and $\nu$ be probability distributions on $\mathbb{R}^{n}$ with
$D_{\varphi}(\nu\,\,||\,\,\mu)<\infty$, and assume
$\mathbb{E}_{\mu}|X|^{4}<\infty$. Then for any $s>0$,

$\displaystyle\frac{d}{ds}D_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})=-\frac{1}{2}J_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s}).$
(39)

This proposition formally implies Proposition 1.5 since
$D_{\text{KL}}(\nu\|\mu)=\lim_{\lambda\to
1}\frac{1}{\lambda-1}\log(1+D_{\varphi_{\lambda}}(\nu\,\,||\,\,\mu))$. While
we have only proved Proposition 1.10 for $\varphi(x)=\varphi_{\lambda}(x)$,
$\lambda>1$, we believe it should be valid for any smooth, convex function
$\varphi:(0,\infty)\to\mathbb{R}$ with $\varphi(1)=0$. We have only used the
assumption that $\varphi(x)=\varphi_{\lambda}(x)$, $\lambda>1$, for justifying
integration under the integral sign, and for justifying integration by parts.
The following is a generalization of Theorem 1.6.

######  Theorem 1.11.

Let $\varphi(x)=\varphi_{\lambda}(x)=x^{\lambda}-1$, for $1<\lambda\leq 2$,
such that $1/\varphi^{\prime\prime}$ is concave. Then, for any probability
measure $\mu$ and $s>0$,

$\displaystyle
1-\frac{\mathrm{mmse}(\mu,s)}{\mathbb{E}_{\mu}|X-\mathbb{E}X|^{2}}\leq\eta_{\varphi}(\mu,s)\leq\frac{1}{1+\frac{s}{C_{\varphi}(\mu)}}.$
(40)

If additionally $\mu$ is log-concave, then, assuming the validity of
Conjecture 4.2, we also have that

$\displaystyle 1-\frac{s}{C_{\varphi}(\mu)}\leq\eta_{\varphi}(\mu,s).$ (41)

The left-hand side inequality in (40) immediately follows from (5) and (19).
We proceed with a generalization of Theorem 1.7.

######  Theorem 1.12.

Let $\varphi(x)=\varphi_{\lambda}(x)=x^{\lambda}-1$, for $1<\lambda\leq 2$,
such that $1/\varphi^{\prime\prime}$ is concave, such that
$1/\varphi^{\prime\prime}$ is concave. Then, assuming the validity of
Conjecture 4.2, for any probability measure $\nu$ and any log-concave
probability measure $\mu$, the mapping $s\mapsto
D_{\varphi}(\nu_{s}\|\mu_{s})$ is convex. In particular, whenever $\mu$ is
log-concave, $s\mapsto\eta_{\varphi}(\mu,s)$ is convex.

For the geometric meaning of the $\varphi$-Sobolev constant, and for the
relations to concentration of measure, we refer the reader to [11, Section
3.4].

###  1.4 Related Work

Contraction coefficients for the Gaussian channel were studied in [35] (see
also [21]), and it was shown that there exist probability distributions $\mu$
with bounded second moment $\mathbb{E}_{\mu}|X-\mathbb{E}X|^{2}$ for which
$\eta_{\chi^{2}}(\mu,s)=\eta_{\text{KL}}(\mu,s)=1$. Our results show that
while a bounded second moment does not imply non-trivial
$\eta_{\chi^{2}}(\mu,s)$ or $\eta_{\text{KL}}(\mu,s)$, bounded Poincarè
constant $C_{\mathrm{P}}(\mu)$ or log-Sobolev constant $C_{\mathrm{LS}}(\mu)$,
respectively, do imply non-trivial contraction coefficient. Moreover, in the
log-concave case the Poincarè constant or log-Sobolev constant are essentially
equivalent to the corresponding contraction coefficient.

For discrete channels, Raginsky [38] has estimated the SDPI constants
$\eta_{\chi^{2}}(\mu,K_{Y|X})$ and $\eta_{\text{KL}}(\mu,K_{Y|X})$, as well as
for other choices of $\varphi$, of a source $\mu$ and a general discrete
channel $K_{Y|X}$ as a function of a Poincarè constant
$C_{\mathrm{P}}(\mu,K_{Y|X})$ or a log-Sobolev constant
$C_{\mathrm{LS}}(\mu,K_{Y|X})$, respectively, corresponding to _both_ the
source $\mu$ and _the channel $K_{Y|X}$_. See [38] for the precise definitions
of those constants and the relations to the contraction coefficients. We
stress that the Poincarè constant $C_{\mathrm{P}}(\mu)$ and log-Sobolev
constant $C_{\mathrm{LS}}(\mu)$ used in our bounds depend only on the source
$\mu$. Furthermore, our bounds only address the Gaussian channels, while the
estimates in [38] hold for any discrete channel (though this class does not
include the Gaussian channel).

The general problem of developing lower bounds, amenable to evaluation, on the
MMSE in estimating a random variable $X$ from a dependent random variable $Y$
is a classic topic in information theory, signal processing, and statistics,
and so is the special case of estimation in Gaussian noise. Classic references
include [46, 44, 48, 50]. See also [37, Chapter 30]. Our lower bound from
Corollary 1.3 varies smoothly with the noise level $s$, similarly to the
Bayesian CRLB, and therefore, cannot capture threshold phenomena/phase
transitions of the MMSE in $s$. However, to the best of our knowledge, no
previous bounds have estimated $\mathrm{mmse}(\mu,s)$ in terms of the Poincarè
constant $C_{\mathrm{P}}(\mu)$. A recent work by Zieder, Dytso and Cardone
[49] develops a lower bound on the MMSE for a class of additive channels from
$X$ to $Y$, including the Gaussian one. Their bound is expressed in terms of
the random variable $\kappa(X)$, where
$\kappa(x)=(C_{\mathrm{P}}(W_{Y|X=x}))^{-1}$ is the inverse of the Poincarè
constant corresponding to the conditional distributions of the channel’s
output given the input $x$, as well as the variance of the conditional
information density between $X$ and $Y$. Our bound from Corollary 1.3 (which
holds only for the Gaussian channel), on the other hand, depends _only_ on
Poincarè constant of the _source_ $C_{\mathrm{P}}(\mu)$.

The function $s\mapsto h(X+\sqrt{s}Z)$ for isotropic Gaussian $Z$
statistically independent of $X\sim\mu$, has been studied intensively, from
the first days of information theory. de Bruijn’s identity [43, 13] (see also
[16, Theorem 17.7.2]) shows that the derivative of this function with respect
to $s$ is $\frac{1}{2}\mathcal{J}(\mu_{s})$. As explained here, our
Proposition 1.5 is a stronger and more general identity. de Bruijn’s identity
was instrumental for proving the entropy power inequality [43, 8]. Decades
later, Costa used de Bruijn’s identity for showing that $s\mapsto
e^{\frac{2}{n}h(X+\sqrt{s}Z)}$ is concave, which immediately also implies the
concavity of $s\mapsto h(X+\sqrt{s}Z)$. Several alternative proofs for Costa’s
EPI were given in [20, 47, 25, 40, 14]. An important point of view on the
evolution of curvature under the heat flow, involving curvature and dimension
was developed by Bakry, Émery and the Toulouse school, see the book [5].

In Theorem 1.7 we establish the convexity of $s\mapsto D(\nu_{s}\|\mu_{s})$
for log-concave $\mu$, which generalizes the latter, and weaker, statement of
Costa. Unfortunately, we were not able to prove convexity of $s\mapsto\log
D(\nu_{s}\|\mu_{s})$. Further improvements of the entropy power inequality for
the random variable $X+\sqrt{s}Z$ were established by Courtade [14]. The
relation between $I(\mu,s)$ and $\mathrm{mmse}(\mu,s)$, called the I-MMSE
relation, which is intimately related to de Bruijn’s identity, was discovered
by Guo, Shamai and Verdú [24]. To the best of our knowledge, our Corollary 1.4
is the first lower bound on $I(\mu,s)$ in terms of $C_{\mathrm{P}}(\mu)$. An
upper bound in a somewhat similar spirit was derived in [3]. Inequalities
interpolating between the Poincaré and the log-Sobolev inequality appear in
Latała and Oleszkiewicz [30]. This is a family of inequalities parameterized
by a parameter $1\leq p\leq 2$.

In the remainder of this paper, we prove the theorems stated above.

##  2 Minimum-squared error and mutual information

###  2.1 Proof of the lower bound in Theorem 20

We show that for any pair of random variables $X,Y$ in
$\mathbb{R}^{n}\times\mathcal{A}$, where $\mathcal{A}$ is some abstract
alphabet, it holds that

$\displaystyle S^{2}(X,Y)\geq
1-\frac{\mathrm{mmse}(X|Y)}{\mathbb{E}|X-\mathbb{E}X|^{2}},$ (42)

where

$\displaystyle\mathrm{mmse}(X|Y)=\mathbb{E}|X-\mathbb{E}[X|Y]|^{2}.$ (43)

The idea is to use linear test functions in the definition (9) of the maximal
correlation $S(X,Y)$. For the case $n=1$, the lower bound was already observed
by Reǹyi [39], see also [4, eq. 17]. Here we give a proof for the general
case. Let

$\displaystyle
i^{*}=\operatorname*{\arg\\!\max}_{i\in[n]}\frac{\mathrm{Var}(\mathbb{E}[X_{i}|Y])}{\mathrm{Var}(X_{i})},$
(44)

and set $\tilde{f}:\mathbb{R}^{n}\to\mathbb{R}$ as

$\tilde{f}(X)=\frac{X_{i^{*}}-\mathbb{E}X_{i^{*}}}{\sqrt{\mathrm{Var}(X_{i^{*}})}},$

such that $\mathbb{E}\tilde{f}(X)=0$ and $\mathrm{Var}(\tilde{f}(X))=1$. Since
for non-negative numbers $a_{1},\ldots,a_{n},b_{1},\ldots,b_{n}$ it holds that
$\frac{\sum_{i=1}^{n}a_{i}}{\sum_{i=1}^{n}b_{i}}\leq\max_{i\in[n]}\frac{a_{i}}{b_{i}}$,
we have that

$\displaystyle\mathrm{Var}(\mathbb{E}[\tilde{f}(X)|Y])$
$\displaystyle=\frac{\mathrm{Var}(\mathbb{E}[X_{i^{*}}|Y])}{\mathrm{Var}(X_{i^{*}})}$
$\displaystyle\geq\frac{\sum_{i=1}^{n}\mathrm{Var}(\mathbb{E}[X_{i}|Y]}{\sum_{i=1}^{n}\mathrm{Var}(X_{i})}$
(45)
$\displaystyle=\frac{\sum_{i=1}^{n}\mathrm{Var}(X_{i})-\mathbb{E}(\mathrm{Var}[X_{i}|Y)]}{\sum_{i=1}^{n}\mathrm{Var}(X_{i})}$
(46)
$\displaystyle=1-\frac{\mathrm{mmse}(X|Y)}{\mathbb{E}_{\mu}|X-\mathbb{E}[X]|^{2}}.$
(47)

The statement follows by invoking (9)

$\displaystyle
S^{2}(X;Y)=\sup_{f}\mathrm{Var}(\mathbb{E}[f(X)|Y])\geq\mathrm{Var}(\mathbb{E}[\tilde{f}(X)|Y]).$
(48)

###  2.2 Proof of Corollary 1.4

Let

$\displaystyle\tilde{I}(\mu,\rho)$ $\displaystyle=I(X;\sqrt{\rho}X+Z),$ (49)
$\displaystyle\widetilde{\mathrm{mmse}}(\mu,\rho)$
$\displaystyle=\mathbb{E}|X-\mathbb{E}[X|\sqrt{\rho}X+Z]|^{2}.$ (50)

Clearly, $I(\mu,s)=\tilde{I}(\mu,\frac{1}{s})$ and
$\mathrm{mmse}(\mu,s)=\widetilde{\mathrm{mmse}}(\mu,\frac{1}{s})$. By
Corollary 1.3, we therefore have that

$\displaystyle\widetilde{\mathrm{mmse}}(\mu,\rho)=\mathrm{mmse}\left(\mu,\frac{1}{\rho}\right)\geq\frac{nP}{1+\rho\bar{C}_{\mathrm{P}}(\mu)P}.$
(51)

By the I-MMSE identity [24, Theorem 2],

$\displaystyle\frac{d}{d\rho}\tilde{I}(\mu,\rho)=\frac{1}{2}\widetilde{\mathrm{mmse}}(\mu,\rho).$
(52)

Thus,

$\displaystyle\tilde{I}(\mu,\rho)=\frac{1}{2}\int_{0}^{\rho}\widetilde{\mathrm{mmse}}(\mu,t)dt\geq\frac{1}{2}\int_{0}^{\rho}\frac{nP}{1+t\bar{C}_{\mathrm{P}}(\mu)P}dt=\frac{n}{2}\frac{1}{\bar{C}_{\mathrm{P}}(\mu)}\log\left(1+\bar{C}_{\mathrm{P}}(\mu)P\cdot\rho\right).$
(53)

where in the last equality we have used the identity
$\int_{0}^{\rho}\frac{a}{1+bt}dt=\frac{a}{b}\log(1+b\rho)$ for $a,b>0$. The
claimed result follows by recalling that
$I(\mu,s)=\tilde{I}(\mu,\frac{1}{s})$.

##  3 $\chi^{2}$-divergence and proofs for blurring time bounds

In this section we complete the proof of Theorem 20. Let $X$ be a random
vector attaining values in $\mathbb{R}^{n}$. When proving Theorem 20 we may
assume that $C_{P}(X)<\infty$ as otherwise the conclusion is vacuous. Recall
from above that a log-concave random vector has a finite Poincaré constant,
and that for any additional random vector $Y$,

$S^{2}(X,Y)=\sup_{f}\mathrm{Var}(\mathbb{E}[f(X)|Y])$ (54)

where the supreumum runs over all measurable functions
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ with $\mathbb{E}f(X)=0$ and
$\mathrm{Var}(f(X))=1$. The requirement that $\mathbb{E}f(X)=0$ is actually
not necessary. Let $Z$ be a standard Gaussian random vector in
$\mathbb{R}^{n}$, independent of $X$. Recall that we are interested in the
$\chi^{2}$ contraction coefficient

$\eta_{\chi^{2}}(\mu,s)=S^{2}(X,X+\sqrt{s}Z).$

Following Klartag and Putterman [29], for a function
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ with $\mathbb{E}|f(X)|<\infty$ and for
$s>0$ we write

$Q_{s}f(X+\sqrt{s}Z)=\mathbb{E}[f(X)|X+\sqrt{s}Z].$

Let $\mu$ be the probability distribution on $\mathbb{R}^{n}$ that is the
distribution law of the random vector $X$, and let $\mu_{s}$ be the
distribution law of $X+\sqrt{s}Z$. The operator

$Q_{s}:L^{2}(\mu)\rightarrow L^{2}(\mu_{s})$

is of norm at most one, since

$\int_{\mathbb{R}^{n}}|Q_{s}f|^{2}d\mu_{s}=\mathbb{E}\left|Q_{s}f(X+\sqrt{s}Z)\right|^{2}=\mathbb{E}\left|\mathbb{E}[f(X)|X+\sqrt{s}Z]\right|^{2}\leq\mathbb{E}|f(X)|^{2}.$

######  Lemma 3.1.

For any $s>0$, the quantity $S^{2}(X,X+\sqrt{s}Z)$ is the square of the
operator norm of $Q_{s}:L^{2}(\mu)\rightarrow L^{2}(\mu_{s})$ restricted to
the subspace of functions of $\mu$-average zero. In other words,
$S^{2}(X,X+\sqrt{s}Z)$ is the minimal number $M\geq 0$ such that for any $f\in
L^{2}(\mu)$ with $\int fd\mu=0$,

$\|Q_{s}f\|_{L^{2}(\mu_{s})}^{2}\leq M\cdot\|f\|_{L^{2}(\mu)}^{2}.$ (55)

###### Proof.

By (54), for any $s>0$,

$S^{2}(X,X+\sqrt{s}Z)=\sup_{f}\mathrm{Var}(\mathbb{E}[f(X)|X+\sqrt{s}Z])=\sup_{f}\mathrm{Var}(Q_{s}f(X+\sqrt{s}Z)),$

where the supremum runs over all $f$ with $\mathbb{E}f(X)=0$ and
$\mathrm{Var}f(X)=1$. Therefore,

$S^{2}(X,X+\sqrt{s}Z)=\sup_{f}\int_{\mathbb{R}^{n}}|Q_{s}f|^{2}d\mu_{s}=\sup_{f}\|Q_{s}f\|_{L^{2}(\mu_{s})}^{2},$

where the supremum runs over all $f$ with $\int fd\mu=0$ and $\int
f^{2}d\mu=1$.

The operator $Q_{s}$ may be expressed as an integral operator. Write

$\gamma_{s}(x)=(2\pi s)^{-n/2}\exp(-|x|^{2}/(2s))$ (56)

for the density in $\mathbb{R}^{n}$ of a Gaussian random vector of mean zero
and covariance $s\cdot{\rm Id}$. By considering the joint distribution of
$(X,X+\sqrt{s}Z)$ and writing conditional expectation as an integral, we see
that

$Q_{s}f(y)=\frac{\int_{\mathbb{R}^{n}}\gamma_{s}(x-y)f(x)d\mu(x)}{(\rho*\gamma_{s})(y)}.$
(57)

The proof of Theorem 20 requires two differentiations of the expression on the
left-hand side of (55) with respect to $s$. It will be convenient to consider
a subclass of well-behaved functions of $L^{2}(\mu)$ in order to justify the
differentiations under the integral sign. Write $\rho$ for the density of
$\mu$.

As in Klartag and Putterman [29], we say that a function
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ has subexponential decay relative to
$\rho$ if there exist $C,a>0$ such that

$|f(x)|\leq\frac{C}{\sqrt{\rho(x)}}e^{-a|x|}\qquad\qquad\qquad(x\in\mathbb{R}^{n}).$
(58)

If $\rho$ decays exponentially at infinity – for instance, if $\rho$ is log-
concave – then all polynomials have subexponential decay relative to $\rho$.
We say that a function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is
$\mu$-tempered if it is smooth and if all of its partial derivatives of all
orders have subexponential decay relative to $\rho$. Write $\mathcal{F}_{\mu}$
for the collection of all $\mu$-tempered functions on $\mathbb{R}^{n}$. Since
$\mathcal{F}_{\mu}$ contains all compactly-supported smooth functions, it is a
dense subspace of $L^{2}(\mu)$.

######  Lemma 3.2.

If $f$ is $\mu$-tempered, then $Q_{s}f$ is $\mu_{s}$-tempered for any $s>0$.

Lemma 3.2 is proven in [29, Lemma 2.2] under the additional assumption that
$\mu$ is log-concave. The log-concavity assumption is only used in the proof
of Lemma 2.2 in [29] in order to show that for any $p>0$,

$\int_{\mathbb{R}^{n}}|x|^{p}d\mu(x)<\infty.$ (59)

However, a measure $\mu$ with $C_{P}(\mu)<\infty$ clearly satisfies (59). In
fact, since $f(x)=|x|$ is a $1$-Lipschitz function, we even know that
$\int_{\mathbb{R}^{n}}e^{\alpha f}d\mu<\infty$ for some positive $\alpha>0$,
this goes back to Gromov and Milman [23]. Hence Lemma 3.2 applies for any
probability measure $\mu$ with a finite Poincaré constant.

We write $\nabla^{2}f(x)\in\mathbb{R}^{n\times n}$ for the Hessian matrix of
$f$ at the point $x$. We abbreviate $X_{s}=X+\sqrt{s}Z$. The following lemma
is proven in [29] as well:

######  Lemma 3.3.

For any $s>0$ and $f\in\mathcal{F}_{\mu}$, setting $f_{s}=Q_{s}f$,

$\frac{\partial}{\partial s}\mathrm{Var}(f_{s}(X_{s}))=-\mathbb{E}|\nabla
f_{s}(X_{s})|^{2},$ (60)

and

$\frac{\partial}{\partial s}\mathbb{E}|\nabla
f_{s}(X_{s})|^{2}=-\mathbb{E}\|\nabla^{2}f_{s}(X_{s})\|_{HS}^{2}-2\mathbb{E}(\nabla^{2}\psi_{s})(X_{s})\nabla
f_{s}(X_{s})\cdot\nabla f_{s}(X_{s}),$ (61)

where $\rho_{s}=e^{-\psi_{s}}$ is the density of $X_{s}$ and $\|A\|_{HS}$ is
the Hilbert-Schmidt norm (also known as the Frobenius norm) of the matrix
$A\in\mathbb{R}^{n\times n}$.

Recall that the density of the random vector $X=X_{0}$ is the function
$\rho=e^{-\psi}$. The Laplace operator associated with $\mu$ is defined for
$u\in\mathcal{F}_{\mu}$ via

$Lu=L_{\mu}u=\Delta u-\nabla\psi\cdot\nabla u.$ (62)

For any $u,v\in\mathcal{F}_{\mu}$ we have the integration by parts formula

$\int_{\mathbb{R}^{n}}(Lu)vd\mu=-\int_{\mathbb{R}^{n}}\langle\nabla u,\nabla
v\rangle d\mu$ (63)

and the (integrated) Bochner formula

$\int_{\mathbb{R}^{n}}(Lu)^{2}d\mu=\int_{\mathbb{R}^{n}}\|\nabla^{2}u\|_{HS}^{2}d\mu+\int_{\mathbb{R}^{n}}\langle(\nabla^{2}\psi)\nabla
u,\nabla u\rangle d\mu,$ (64)

where $\|\nabla^{2}u\|_{HS}$ is the Hilbert-Schmidt norm of the Hessian matrix
$\nabla^{2}u$. Formulae (63) and (64) are proven by intergation by parts, see
e.g., Ledoux [33, Section 2.3]. The $\mu$-temperedness of $u,v$ are used in
order to discard the boundary terms. The operator $-L$ is symmetric and
positive semi-definite in $\mathcal{F}_{\mu}\subseteq L^{2}(\mu)$. Recall the
well-known subadditivity property of the Poincaré constant (e.g. [15] or
Theorem 5.1),

$C_{\mathrm{P}}(X_{s})=C_{\mathrm{P}}(X+\sqrt{s}Z)\leq
C_{\mathrm{P}}(X)+C_{\mathrm{P}}(\sqrt{s}Z)=C_{\mathrm{P}}(X)+s.$

###### Proof of Theorem 20.

We begin with the proof of the right-hand side inequality in (19), as the
left-hand side inequality was already proven above. If $C_{P}(X)=+\infty$ then
this inequality is vacuously true, hence we may assume that $C_{P}(X)<\infty$.
Recall that $X_{s}=X+\sqrt{s}Z$. By Lemma 55, we need to show that for any
function $f$ with $\mathbb{E}|f(X)|^{2}<\infty$,

$\mathrm{Var}\left(\mathbb{E}\left[f(X)|X_{s}\right]\right)=\mathrm{Var}(Q_{s}f(X_{s}))\leq\frac{1}{1+s/C_{P}(\mu)}\cdot\mathrm{Var}f(X).$
(65)

Recall that $\mu$ is the distribution law of $X$. Since
$Q_{s}:L^{2}(\mu)\rightarrow L^{2}(\mu_{s})$ is a bounded operator and
$\mathcal{F}_{\mu}$ is dense in $L^{2}(\mu)$, it suffices to prove (65) under
the additional assumption that $f\in\mathcal{F}_{\mu}$. Abbreviate
$f_{s}=Q_{s}(f)$. By using (60) and the Poincaré inequality, we get

$\frac{\partial}{\partial
s}\mathrm{Var}(f_{s}(X_{s}))\leq-\frac{1}{C_{\mathrm{P}}(X_{s})}\mathrm{Var}(f_{s}(X_{s}))\leq-\frac{1}{s+C_{\mathrm{P}}(X)}\mathrm{Var}(f_{s}(X_{s})),$

where we used subadditivity in the last passage. Therefore

$\frac{\partial}{\partial
s}\log\mathrm{Var}(f_{s}(X_{s}))\leq-\frac{1}{s+C_{\mathrm{P}}(X)}.$

By integrating from $0$ to $s$ we conclude that

$\log\frac{\mathrm{Var}(f_{s}(X_{s}))}{\mathrm{Var}f(X)}\leq-\int_{0}^{s}\frac{dx}{x+C_{\mathrm{P}}(X)}=\log\frac{1}{1+s/C_{\mathrm{P}}(X)},$

proving (65). This proves the right-hand side inequality in (19). Let us now
assume that $X$ is log-concave and prove (20). This is in fact proven in [29],
but let us briefly repeat the argument here for convenience. Let
$0<\varepsilon<C_{\mathrm{P}}(\mu)$. By the definition of
$C_{\mathrm{P}}(\mu)$, there exists a non-constant function $f$ such that

$\mathrm{Var}f(X)\geq(C_{\mathrm{P}}(\mu)-\varepsilon)\cdot\mathbb{E}|\nabla
f(X)|^{2}.$ (66)

Thanks to the appendix of [7], we may assume that $f$ is smooth and compactly-
supported in $\mathbb{R}^{n}$, and in particular $f\in\mathcal{F}_{\mu}$.
According to (60) and (61) we may differentiate the Rayleigh quotient

$\frac{\partial}{\partial s}\frac{\mathbb{E}|\nabla
f_{s}(X_{s})|^{2}}{\mathrm{Var}(f_{s}(X_{s}))}=-\mathbb{E}|\nabla^{2}\tilde{f}_{s}(X_{s})|^{2}-2\mathbb{E}(\nabla^{2}\psi_{s})(X_{s})\nabla\tilde{f}_{s}(X_{s})\cdot\nabla\tilde{f}_{s}(X_{s})-\mathbb{E}|\nabla\tilde{f_{s}}(X_{s})|^{4}$

where we normalize $\tilde{f}_{s}=f_{s}/\sqrt{\mathrm{Var}(f_{s}(X_{s})}$. As
explained in the proof of [29, Theorem 2.4], it follows from the spectral
theorem that

$\langle
L_{\mu_{s}}^{2}\tilde{f}_{s},\tilde{f}_{s}\rangle_{L^{2}(\mu_{s})}\geq\langle
L\tilde{f}_{s},\tilde{f}_{s}\rangle_{L^{2}(\mu_{s})}^{2}=\left(\int_{\mathbb{R}^{n}}|\nabla\tilde{f}_{s}|^{2}d\mu_{s}\right)^{2}.$

Therefore, from the Bochner formula (64),

$\frac{\partial}{\partial s}\frac{\mathbb{E}|\nabla
f_{s}(X_{s})|^{2}}{\mathrm{Var}(f_{s}(X_{s}))}\leq-\mathbb{E}(\nabla^{2}\psi_{s})(X_{s})\nabla\tilde{f}_{s}(X_{s})\cdot\nabla\tilde{f}_{s}(X_{s})\leq
0.$ (67)

where the last inequality is the only place where we use log-concavity;
indeed, since $X$ is log-concave, so is the random vector $X+\sqrt{s}Z$ by the
Prékopa-Leindler inequality (see e.g. the first pages of Pisier [34], or
Davidovič, Korenbljum and Hacet [18]). Thus $e^{-\psi_{s}}$ is a log-concave
density, which amounts to the fact that the symmetric matrix
$\nabla^{2}\psi_{s}$ is positive semi-definite. Hence the term involving the
Hessian $\nabla^{2}\psi_{s}$ in (67) is non-positive. Consequently,
$\mathbb{E}|\nabla f_{s}(X_{s})|^{2}/\mathrm{Var}(f_{s}(X_{s}))$ is non-
increasing in $s$ and

$\log\frac{\mathrm{Var}(f_{s}(X_{s}))}{\mathrm{Var}f(X)}=-\int_{0}^{s}\frac{\mathbb{E}|\nabla
f_{t}(X_{t})|^{2}}{\mathrm{Var}(f_{t}(X_{t}))}dt\geq-s\cdot\frac{\mathbb{E}|\nabla
f(X)|^{2}}{\mathrm{Var}(f(X))}\geq-\frac{s}{C_{\mathrm{P}}(\mu)-\varepsilon}.$

Hence for any $\varepsilon>0$ we found $f$ such that

$\mathrm{Var}\left(\mathbb{E}\left[f(X)|X_{s}\right]\right)=\mathrm{Var}(f_{s}(X_{s}))\geq\exp\left(-\frac{s}{C_{\mathrm{P}}(\mu)-\varepsilon}\right)\mathrm{Var}f(X).$

Since $\varepsilon>0$ is arbitrary, and in view of Lemma 55 this proves that
$S^{2}(X,X+\sqrt{s}Z)\geq\exp(-s/C_{\mathrm{P}}(\mu))$. Inequality (20) is
thus proven.

######  Remark 3.4.

Thomas Courtade and Joseph Lehec communicated to us an alternative proof of
the right-hand side inequality in (19). Their proof does not require
differentiation with respect to the parameter $s$.

##  4 Identities for KL-divergence and general divergences

In this section we prove Propositions 1.5 and 1.10, the generalized de Bruijn
identity. We keep the notation from the previous section. Let
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a $\mu$-integrable function.

As in the previous section, some care is needed when differentiating under the
integral sign and when integrating by parts and neglecting the boundary terms.
As opposed to the previous section analyzing the case of the
$\chi^{2}$-divergence, here we will be rather brief in the justification. As
in [29] and in Section 3, the basic idea is to introduce a suitable class
$\mathcal{F}_{\mu,\varphi}$ of smooth functions, which allow for integrations
by parts without boundary terms, and is preserved by the $Q_{s}$-dynamics.
Then one approximates the density of an arbitrary measure $\nu$ with
$J_{\varphi}(\nu\,\,||\,\,\mu)<\infty$ with a function from this class.

From Klartag and Putterman [29], we know that for any $f\in\mathcal{F}_{\mu}$
and $s>0$,

$\frac{d}{ds}Q_{s}f=\Box_{s}Q_{s}f$ (68)

where

$\Box_{s}=L_{s}-\frac{\Delta}{2}$

and $L_{s}=L_{\mu_{s}}$ is the Laplace operator associated with the measure
$\mu_{s}$. We write $\rho_{s}$ for the density of $\mu_{s}$.

######  Lemma 4.1.

Let $\varphi(x)=\varphi_{\lambda}(x)=x^{\lambda}-1$, for $\lambda>1$. Let
$\mu$ and $\nu$ be probability distributions on $\mathbb{R}^{n}$ with
$D_{\varphi}(\nu\,\,||\,\,\mu)<\infty$, and assume
$\mathbb{E}_{\mu}|X|^{4}<\infty$. Denote $f=d\nu/d\mu$ such that
$f_{s}=Q_{s}f=d\mu_{s}/d\nu_{s}$, for $s>0$. we have

$\displaystyle\frac{d}{ds}\mathbb{E}\varphi(f_{s}(X_{s}))=-\frac{1}{2}\cdot\mathbb{E}\varphi^{\prime\prime}(f_{s}(X_{s}))|\nabla
f_{s}(X_{s})|^{2}=-\frac{1}{2}\int_{\mathbb{R}^{n}}\varphi^{\prime\prime}(f_{s})|\nabla
f_{s}|^{2}d\mu_{s}.$ (69)

In particular, if $D_{\text{KL}}(\nu\,\,||\,\,\mu)<\infty$

$\frac{d}{ds}\mathbb{E}f_{s}(X_{s})\log
f_{s}(X_{s})=-\frac{1}{2}\cdot\mathbb{E}\frac{|\nabla
f(X_{s})|^{2}}{f_{s}(X_{s})}$ (70)

###### Proof.

We prove (69) below for any smooth, convex function
$\varphi:(0,\infty)\to\mathbb{R}$ with $\varphi(1)=0$, without justifying
differentiation under the integral sign, and without justifying integration by
parts. The justifications for these steps for
$\varphi(x)=\varphi_{\lambda}(x)=x^{\lambda}-1$, for $\lambda>1$, under the
assumptions that $D_{\varphi}(\nu\,\,||\,\,\mu)<\infty$, and
$\mathbb{E}_{\mu}|X|^{4}<\infty$, are given in the appendix. We also prove in
the appendix that (69) implies (70).

Recall that $\partial\rho_{s}/\partial s=\Delta\rho_{s}/2$ by the heat
equation, as $\mu_{s}=\mu*\gamma_{s}$ and $\rho_{s}$ is the density of
$\mu_{s}$. This and (68) are the main places where we use the heat equation in
our arguments. Let us compute:

$\displaystyle\frac{d}{ds}\int_{\mathbb{R}^{n}}\varphi(f_{s})\rho_{s}=\int_{\mathbb{R}^{n}}\varphi^{\prime}(f_{s})\Box_{s}f_{s}\cdot\rho_{s}+\int_{\mathbb{R}^{n}}\varphi(f_{s})\frac{\Delta\rho_{s}}{2}.$

Integrating by parts we continue with

$\displaystyle=\int_{\mathbb{R}^{n}}\left[\varphi^{\prime}(f_{s})\left(L_{s}f_{s}-\frac{\Delta
f_{s}}{2}\right)+\frac{\Delta(\varphi(f_{s}))}{2}\right]\rho_{s}$
$\displaystyle=\int_{\mathbb{R}^{n}}\left[\varphi^{\prime}(f_{s})L_{s}f_{s}-\varphi^{\prime}(f_{s})\frac{\Delta
f_{s}}{2}+\frac{\varphi^{\prime}(f_{s})\Delta
f_{s}+\varphi^{\prime\prime}(f_{s})|\nabla f_{s}|^{2}}{2})\right]\rho_{s}$
$\displaystyle=\int_{\mathbb{R}^{n}}\left[-\varphi^{\prime\prime}(f_{s})|\nabla
f_{s}|^{2}+\frac{\varphi^{\prime\prime}(f_{s})|\nabla
f_{s}|^{2})}{2}\right]\rho_{s},$

where we used the integration by parts (63) in the last passage.

###### Proof of Proposition 1.5 and Proposition 1.10.

Follow directly from Lemma 4.1, and the definition of $D_{\varphi}$ and
$J_{\varphi}$.

We move on to computing the second derivative of
$\mathbb{E}\varphi(f(X_{s}))$.

######  Conjecture 4.2.

Denoting

$\kappa=-\frac{1}{2\varphi^{\prime\prime}}\left(\frac{1}{\varphi^{\prime\prime}}\right)^{\prime\prime},$

and

$g_{s}=\varphi^{\prime}(f_{s})$

we have, under mild regularity assumptions, the “Bochner-type” formula

$\displaystyle\frac{d}{ds}\int_{\mathbb{R}^{n}}\varphi^{\prime\prime}(f_{s})|\nabla
f_{s}|^{2}d\mu_{s}=-\int_{\mathbb{R}^{n}}\left[2(\nabla^{2}\psi_{s})\nabla
g_{s}\cdot\nabla g_{s}+\left|\nabla^{2}g_{s}\right|^{2}+\kappa(f_{s})|\nabla
g_{s}|^{4}\right]\frac{d\mu_{s}}{\varphi^{\prime\prime}(f_{s})},$

where we recall that $\rho_{s}=e^{-\psi_{s}}$.

###### Sketch of proof.

We assume below that differentiation under the integral sign is valid, and
that all functions involved in integration in parts vanish. Justifying these
assumptions is the missing step in proving this conjecture. We now prove the
“Bochner-typ” formula above, under the assumption that those steps are indeed
valid (i.e., ignoring all regularity issues).

Our approach follows the $M$-function of Ivanishvili and Volberg [26] and the
dynamic approach to the $\Gamma$-calculus is presented in [29]. We consider a
function $M(x,y)$ of two real variables of the form

$M(x,y)=y\varphi^{\prime\prime}(x).$ (71)

Thus $M(x,y)=y/x$ in the case where $\varphi(x)=x\log x$ and $M(x,y)=2y$ if
$\varphi(x)=(x-1)^{2}$. In view of Lemma 4.1, we would need to compute

$\frac{d}{ds}\mathbb{E}M(f_{s}(X_{s}),|\nabla f_{s}(X_{s})|^{2}).$

By the chain rule, for any functions $f,g$,

$\Delta M(f,g)=M_{x}\Delta f+M_{y}\Delta g+M_{xx}|\nabla f|^{2}+2M_{xy}\nabla
f\cdot\nabla g+M_{yy}|\nabla g|^{2},$ (72)

where we abbreviate $M_{x}=\partial M/\partial x,M_{y}=\partial M/\partial y$
etc. Let us also introduce the dynamic $\Gamma$-calculus notation from [29].
We set

$\Gamma_{1}(u,v)=\nabla u\cdot\nabla v$

and

$\Gamma_{2}(u,v):=\Box_{s}\Gamma_{1}(u,v)-\Gamma_{1}(\Box_{s}u,v)-\Gamma_{1}(u,\Box_{s}v)$

which satisfies

$\Gamma_{2}(u,u)=\|\nabla^{2}u\|_{HS}^{2}-2\nabla^{2}(\log\rho_{s})\nabla
u\cdot\nabla u.$ (73)

We abbreviate $\Gamma_{1}(u)=\Gamma_{1}(u,u)$ and
$\Gamma_{2}(u)=\Gamma_{2}(u,u)$.

We first prove that for a general function $M$ of two variables,

$\displaystyle\frac{d}{ds}$ $\displaystyle\mathbb{E}M(f_{s}(X_{s}),|\nabla
f_{s}(X_{s})|^{2})$ (74)
$\displaystyle=-\int_{\mathbb{R}^{n}}\left[M_{y}\Gamma_{2}(f_{s})+\frac{M_{xx}}{2}\Gamma_{1}(f_{s})+M_{xy}\Gamma_{1}(f_{s},\Gamma_{1}(f_{s}))+\frac{M_{yy}}{2}\Gamma_{1}(\Gamma_{1}(f_{s}))\right]d\mu_{s}.$

To this end, abbreviate $M=M(f_{s},|\nabla
f_{s}|^{2}),M_{x}=M_{x}(f_{s},|\nabla f_{s}|^{2})$ etc. Then,

$\displaystyle\frac{d}{ds}\int_{\mathbb{R}^{n}}M(f_{s},|\nabla
f_{s}|^{2})\rho_{s}$ $\displaystyle=\int_{\mathbb{R}^{n}}\left[\frac{\Delta
M}{2}+M_{x}\Box_{s}f_{s}+2M_{y}\Gamma_{1}(\Box_{s}f_{s},f_{s})\right]\rho_{s}$
(75) $\displaystyle=\int_{\mathbb{R}^{n}}\left[\frac{\Delta
M}{2}+M_{x}\Box_{s}f_{s}+M_{y}\left(\Box_{s}\Gamma_{1}(f_{s},f_{s})-\Gamma_{2}(f_{s},f_{s})\right)\right]\rho_{s}$
$\displaystyle=\int_{\mathbb{R}^{n}}\left[\frac{\Delta
M}{2}+M_{x}\left(L_{s}-\frac{\Delta}{2}\right)f_{s}+M_{y}\left(\left(L_{s}-\frac{\Delta}{2}\right)\Gamma_{1}(f_{s})-\Gamma_{2}(f_{s})\right)\right]\rho_{s}.$

By (72) with $f=f_{s},g=\Gamma_{1}(f_{s})$,

$\Delta M=M_{x}\Delta
f_{s}+M_{y}\Delta\Gamma_{1}(f_{s})+M_{xx}\Gamma_{1}(f_{s})+2M_{xy}\Gamma_{1}(f_{s},\Gamma_{1}(f_{s}))+M_{yy}\Gamma_{1}(\Gamma_{1}(f_{s})).$

Therefore,

$\frac{1}{2}\left(\Delta M-M_{x}\Delta
f_{s}-M_{y}\Delta\Gamma_{1}(f_{s})\right)=\frac{M_{xx}}{2}\Gamma_{1}(f_{s})+M_{xy}\Gamma_{1}(f_{s},\Gamma_{1}(f_{s}))+\frac{M_{yy}}{2}\Gamma_{1}(\Gamma_{1}(f_{s})).$

Consequently, the integral in (75) equals

$\int_{\mathbb{R}^{n}}\left[M_{x}L_{s}f_{s}+M_{y}L_{s}\Gamma_{1}(f_{s})-M_{y}\Gamma_{2}(f_{s})+\frac{M_{xx}}{2}\Gamma_{1}(f_{s})+M_{xy}\Gamma_{1}(f_{s},\Gamma_{1}(f_{s}))+\frac{M_{yy}}{2}\Gamma_{1}(\Gamma_{1}(f_{s}))\right]\rho_{s}$

Integrate by parts the two terms involving $L_{s}$ using (63) to obtain

$\displaystyle\int_{\mathbb{R}^{n}}$
$\displaystyle\left[M_{x}L_{s}f_{s}+M_{y}L_{s}\Gamma_{1}(f_{s})\right]\rho_{s}=\int_{\mathbb{R}^{n}}\left[-\nabla
M_{x}\cdot\nabla f_{s}-\nabla
M_{y}\cdot\nabla\Gamma_{1}(f_{s})\right]\rho_{s}$
$\displaystyle=\int_{\mathbb{R}^{n}}\left[-(M_{xx}\nabla
f_{s}+M_{xy}\nabla\Gamma_{1}(f_{s}))\cdot\nabla f_{s}-(M_{xy}\nabla
f_{s}+M_{yy}\nabla\Gamma_{1}(f_{s}))\cdot\nabla\Gamma_{1}(f_{s})\right]\rho_{s}$
$\displaystyle=\int_{\mathbb{R}^{n}}\left[-M_{xx}\Gamma_{1}(f_{s})-2M_{xy}\Gamma_{1}(\Gamma_{1}(f_{s}),f_{s})-M_{yy}\Gamma_{1}(\Gamma_{1}(f_{s}))\right]\rho_{s}.$

Hence, the integral in (75) equals

$-\int_{\mathbb{R}^{n}}\left[M_{y}\Gamma_{2}(f_{s})+\frac{M_{xx}}{2}\Gamma_{1}(f_{s})+M_{xy}\Gamma_{1}(f_{s},\Gamma_{1}(f_{s}))+\frac{M_{yy}}{2}\Gamma_{1}(\Gamma_{1}(f_{s}))\right]\rho_{s}$

and (74) is proven. In the specific case where $M$ is the function given by
(71) we conclude from (74) that

$\displaystyle\frac{d}{ds}$
$\displaystyle\int_{\mathbb{R}^{n}}\varphi^{\prime\prime}(f_{s})|\nabla
f_{s}|^{2}d\mu_{s}$ (76)
$\displaystyle=-\int_{\mathbb{R}^{n}}\left[\varphi^{\prime\prime}(f_{s})\Gamma_{2}(f_{s})+\frac{\varphi^{(4)}(f_{s})}{2}\Gamma_{1}(f_{s})^{2}+\varphi^{(3)}(f_{s})\Gamma_{1}(f_{s},\Gamma_{1}(f_{s}))\right]d\mu_{s}$
$\displaystyle=-\int_{\mathbb{R}^{n}}\left[2\varphi^{\prime\prime}(\nabla^{2}\psi_{s})\nabla
f_{s}\cdot\nabla
f_{s}+\varphi^{\prime\prime}|\nabla^{2}f_{s}|^{2}+\frac{\varphi^{(4)}}{2}|\nabla
f_{s}|^{4}+2\varphi^{(3)}(\nabla^{2}f_{s})\nabla f_{s}\cdot\nabla
f_{s}\right]d\mu_{s}$
$\displaystyle=-\int_{\mathbb{R}^{n}}\left[2\varphi^{\prime\prime}(\nabla^{2}\psi_{s})\nabla
f_{s}\cdot\nabla
f_{s}+\varphi^{\prime\prime}\left|\nabla^{2}f_{s}+\frac{\varphi^{(3)}}{\varphi^{\prime\prime}}\nabla
f_{s}\otimes\nabla
f_{s}\right|^{2}+\left\\{\frac{\varphi^{(4)}}{2}-\frac{\left(\varphi^{(3)}\right)^{2}}{\varphi^{\prime\prime}}\right\\}|\nabla
f_{s}|^{4}\right]d\mu_{s}$

Note that

$\frac{\varphi^{(4)}}{2}-\frac{\left(\varphi^{(3)}\right)^{2}}{\varphi^{\prime\prime}}=-\frac{(\varphi^{\prime\prime})^{2}}{2}\left(\frac{1}{\varphi^{\prime\prime}}\right)^{\prime\prime}=(\varphi^{\prime\prime})^{3}\kappa.$

Consequently,

$\displaystyle\frac{d}{ds}$
$\displaystyle\int_{\mathbb{R}^{n}}\varphi^{\prime\prime}(f_{s})|\nabla
f_{s}|^{2}d\mu_{s}$
$\displaystyle=-\int_{\mathbb{R}^{n}}\left[2\varphi^{\prime\prime}(\nabla^{2}\psi_{s})\nabla
f_{s}\cdot\nabla
f_{s}+\varphi^{\prime\prime}\left|\nabla^{2}f_{s}+\frac{\varphi^{(3)}}{\varphi^{\prime\prime}}\nabla
f_{s}\otimes\nabla f_{s}\right|^{2}+\kappa(\varphi^{\prime})^{3}|\nabla
f_{s}|^{4}\right]d\mu_{s}.$

Since $g_{s}=\varphi^{\prime}(f_{s})$ we have $\nabla
g_{s}=\varphi^{\prime\prime}(f_{s})\nabla f_{s}$ and
$\nabla^{2}g_{s}=\varphi^{\prime\prime}(f_{s})\nabla^{2}f_{s}+\varphi^{(3)}(f_{s})\nabla
f_{s}\otimes\nabla f_{s}$, and the lemma is proven.

Specializing Conjecture 4.2 to the case where $\varphi(x)=x\log x$, we obtain
the following:

######  Proposition 4.3.

Assuming the validity of Conjecture 4.2, With $g_{s}=\log Q_{s}f$ we have

$\frac{d}{ds}\mathbb{E}\frac{|\nabla
f_{s}(X_{s})|^{2}}{f_{s}(X_{s})}=-\int_{\mathbb{R}^{n}}\left[|\nabla^{2}g_{s}|^{2}+2\langle(\nabla^{2}\psi_{s})\nabla
g_{s},\nabla g_{s}\rangle\right]f_{s}d\mu_{s}.$

###### Proof.

This follows from Conjecture 4.2 with $\varphi(x)=x\log x$, since in this case
$1/\varphi^{\prime\prime}(x)=x$ and $\kappa\equiv 0$.

##  5 Proofs of inequalities related to $\varphi$-Sobolev constants

We keep the notation and assumptions of the previous section. We need to use
the subadditivity property of the $\varphi$-Sobolev constants, proven by
Chafaï [11, Corollary 3.1] in greater generality. For the convenience of the
reader we include here a statement and a proof of the following proposition,
as well as of Proposition 1.9, which also goes back to Chafaï [11].

######  Theorem 5.1 (“Subadditivity of the $\varphi$-Sobolev constants”,
Chafaï [11]).

Let $X$ and $Y$ be independent random vectors in $\mathbb{R}^{n}$. Let
$\varphi:(0,\infty)\to\mathbb{R}$ be a smooth, convex function with
$\varphi(1)=0$ such that $1/\varphi^{\prime\prime}$ is concave. Then,

$C_{\varphi}(X+Y)\leq C_{\varphi}(X)+C_{\varphi}(Y).$

###### Proof.

for any locally-Lipschitz function $f:\mathbb{R}^{n}\rightarrow[0,\infty)$
with $\mathbb{E}|f(X+Y)|<\infty$, denoting $g(x)=\mathbb{E}f(x+Y)$ we have

$\displaystyle\mathbb{E}$
$\displaystyle\varphi(f(X+Y))-\varphi(\mathbb{E}(f(X+Y)))$
$\displaystyle=\mathbb{E}_{X}\left[\mathbb{E}_{Y}\varphi(f(X+Y))-\varphi(\mathbb{E}_{Y}\varphi(f(X+Y)))\right]+\mathbb{E}_{X}\varphi(g(X))-\varphi(\mathbb{E}_{X}g(X))$
$\displaystyle\leq\frac{C_{\varphi}(Y)}{2}\mathbb{E}_{X}\mathbb{E}_{Y}\varphi^{\prime\prime}(f(X+Y))|\nabla
f(X+Y)|^{2}+\frac{C_{\varphi}(X)}{2}\mathbb{E}_{X}\varphi^{\prime\prime}(g(X))|\nabla
g(X)|^{2}.$

To conclude the proof it remains to show that for any fixed
$x\in\mathbb{R}^{n}$,

$\varphi^{\prime\prime}(g(x))|\nabla
g(x)|^{2}\leq\mathbb{E}_{Y}\varphi^{\prime\prime}(f(x+Y))|\nabla f(x+Y)|^{2}.$

Indeed, this would imply that

$\mathbb{E}\varphi(f(X+Y))-\varphi(\mathbb{E}(\varphi(X+Y)))\leq\frac{C_{\varphi}(Y)+C_{\varphi}(X)}{2}\mathbb{E}\varphi^{\prime\prime}(f(X+Y))|\nabla
f(X+Y)|^{2},$

and hence $C_{\varphi}(X+Y)\leq C_{\varphi}(Y)+C_{\varphi}(X)$. Since $\nabla
g(x)=\mathbb{E}\nabla f(x+Y)$, all that remains is to show that

$\varphi^{\prime\prime}(\mathbb{E}f(x+Y))|\mathbb{E}\nabla
f(x+Y)|^{2}\leq\mathbb{E}_{Y}\varphi^{\prime\prime}(f(x+Y))|\nabla
f(x+Y)|^{2}.$

By Jensen’s inequality, this would follow once we show that the function

$(x,y)\mapsto\varphi^{\prime\prime}(x)|y|^{2}\qquad\qquad(x>0,y\in\mathbb{R}^{n})$
(77)

is jointly convex in $x$ and $y$. Since
$\varphi^{\prime\prime}(x)|y|^{2}=\sum_{j}\varphi^{\prime\prime}(x)y_{j}^{2}$,
it suffices to show that the function

$(x,y)\mapsto\varphi^{\prime\prime}(x)y^{2}\qquad\qquad(x>0,y\in\mathbb{R})$
(78)

is jointly convex in $(x,y)$. The Hessian of this function is the matrix

$\left(\begin{matrix}\varphi^{(4)}(x)y^{2}&2\varphi^{(3)}(x)y\\\
2\varphi^{(3)}(x)y&2\varphi^{\prime\prime}(x)\end{matrix}\right)$

Since this symmetric $2\times 2$ matrix has a non-negative entry on the
diagonal, namely $2\varphi^{\prime\prime}(x)$, by the Sylvester criterion it
is positive semi-definite if and only if its determinant is non-negative. The
determinant equals

$2y^{2}\left[\varphi^{\prime\prime}\varphi^{(4)}-2(\varphi^{(3)})^{2}\right]=-2y^{2}(\varphi^{\prime\prime})^{3}\left(\frac{1}{\varphi^{\prime\prime}}\right)^{\prime\prime},$

where we used the formula
$(1/\varphi^{\prime\prime})^{\prime\prime}=-(\varphi^{\prime\prime}\varphi^{(4)}-2(\varphi^{(3)})^{2})/(\varphi^{\prime\prime})^{3}$.
Since $\varphi^{\prime\prime}\geq 0$ and $1/\varphi^{\prime\prime}$ is
concave, the determinant is non-negative, and the function in (78) is convex.
This completes the proof.

###### Proof of Proposition 1.8.

For any locally-Lipschitz function $f$,

$\mathbb{E}\varphi(f(X))-\varphi(\mathbb{E}f(X))\leq\frac{C_{\varphi}(X)}{2}\cdot\mathbb{E}\varphi^{\prime\prime}(f(X))|\nabla
f(X)|^{2}.$

Let $g:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a Lipschitz, bounded function.
Then there exists $\varepsilon_{0}>0$ such that for any
$0<\varepsilon<\varepsilon_{0}$, the function $g_{0}=1+\varepsilon g$ attains
values at $I$. In fact, since $\varphi$ is smooth, there exists $M>0$ and
$\varepsilon_{1}<\varepsilon_{0}$ such that for $\varepsilon<\varepsilon_{1}$,

$\left|\varphi(g_{0}(x))-\varepsilon\varphi^{\prime}(1)g(x)-\frac{\varepsilon^{2}}{2}\varphi^{\prime\prime}(1)g^{2}(x)\right|\leq\varepsilon^{3}M$

and

$\varphi^{\prime\prime}(g_{0}(x))\geq(1-\varepsilon
M)\cdot\varphi^{\prime\prime}(1).$

Consequently, for any $0<\varepsilon<\varepsilon_{1}$,

$\frac{\varepsilon^{2}}{2}\varphi^{\prime\prime}(1)Var(g(X))-2M\varepsilon^{3}\leq\mathbb{E}\varphi(f(X))-\varphi(\mathbb{E}f(X))$

and

$\mathbb{E}\varphi^{\prime\prime}(f(X))|\nabla f(X)|^{2}\geq(1-\varepsilon
M)\varepsilon^{2}\varphi^{\prime\prime}(1)\cdot\mathbb{E}|\nabla g(X)|^{2}.$

By using the $\varphi$-Sobolev inequality and dividing by
$\varepsilon^{2}\varphi^{\prime\prime}(1)>0$, and then letting $\varepsilon$
tends to zero we obtain

$\frac{1}{2}Var(g(X))\leq\frac{C_{\varphi}(X)}{2}\mathbb{E}|\nabla g(X)|^{2}.$
(79)

This inequality holds for a Lipschitz, bounded function $g$. A standard
truncation argument shows that (79) holds true for an arbitrary locally-
Lipschitz function $g$. Indeed, It suffices to replace $g(x)$ by
$g_{M}(x)=\theta(x/M)g(x)$ where $\theta:\mathbb{R}^{n}\rightarrow[0,1]$ is a
compactly supported smooth function that equals one in a neighborhood of the
origin. The function $g_{M}$ is a Lipschitz, bounded function, and a short
argument detailed e.g. in the proof of Proposition 27 in [7] allows to take
the limit as $M\rightarrow\infty$ in the Poincaré inequality. This shows that
$C_{P}(X)\leq C_{\varphi}(X)$.

###### Proof of Proposition 1.9.

Let $Z$ be a standard Gaussian random vector, independent of $X$. When $X$ is
a standard Gaussian, the operator $Q_{s}$ has a pleasant form given by the
Mehler formula. Recall that in general,

$Q_{s}f(X+\sqrt{s}Z)=\mathbb{E}\left[f(X)|X+\sqrt{s}Z\right]$

and hence

$\mathbb{E}Q_{s}f(X+\sqrt{s}Z)=\mathbb{E}f(X).$

In the case where $X$ is Gaussian, the density of $X$ conditioned on
$X+\sqrt{s}Z=y$ is

$\frac{\gamma_{1}(x)\gamma_{s}(y-x)}{\gamma_{1+s}(y)}=\gamma_{\frac{s}{s+1}}\left(x-\frac{y}{s+1}\right)$

where $\gamma_{s}$ is the Gaussian density, as in (56). In other words, the
distribution of $X$ conditioned on $X+\sqrt{s}Z$ is that of a Gaussian random
vector of mean $\frac{X+\sqrt{s}Z}{s+1}$ and covariance $\frac{s}{s+1}{\rm
Id}$. Therefore, for any $y\in\mathbb{R}^{n}$,

$Q_{s}f(y)=\mathbb{E}f\left(\frac{y}{s+1}+\sqrt{\frac{s}{s+1}}Z\right)$ (80)

and

$\nabla Q_{s}f(y)=\frac{1}{s+1}\mathbb{E}\nabla
f\left(\frac{y}{s+1}+\sqrt{\frac{s}{s+1}}Z\right).$

Consequently, in all of $\mathbb{R}^{n}$ we have the identity,

$\varphi^{\prime\prime}(Q_{s}f)|\nabla
Q_{s}f|^{2}=\frac{1}{(s+1)^{2}}\cdot\varphi^{\prime\prime}(Q_{s}f)|Q_{s}(\nabla
f)|^{2}$ (81)

where $Q_{s}$ is acting on a vector field entry by entry, i.e., $Q_{s}(\nabla
f)=(Q_{s}(\partial_{1}f),\ldots Q_{s}(\partial_{n}f))$. Since the function in
(77) is convex in $x$ and $y$, and since $Q_{s}$ is an averaging operator as
we see from (80), we may use Jensen’s inequality. We obtain from (81) that

$\varphi^{\prime\prime}(Q_{s}f)|\nabla Q_{s}f|^{2}\leq\frac{1}{(s+1)^{2}}\cdot
Q_{s}\left[\varphi^{\prime\prime}(f)|\nabla f|^{2}\right]$ (82)

Abbreviate $f_{s}=Q_{s}f$ and $X_{s}=X+\sqrt{s}Z$, and observe that

$\mathbb{E}\varphi(f(X))-\varphi(\mathbb{E}f(X))=-\int_{0}^{\infty}\frac{\partial}{\partial
s}\mathbb{E}\varphi(\mathbb{E}\left[f(X)|X+\sqrt{s}Z\right])ds=-\int_{0}^{\infty}\frac{\partial}{\partial
s}\mathbb{E}\varphi(f_{s}(X_{s}))ds.$

From the generalized de Bruijn identity in the form of Lemma 4.1 above, and
from (82),

$\displaystyle\mathbb{E}\varphi(f(X))-\varphi(\mathbb{E}f(X))$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\mathbb{E}\varphi^{\prime\prime}(f_{s}(X_{s}))|\nabla
f_{s}(X_{s})|^{2}ds$
$\displaystyle\leq\int_{0}^{\infty}\frac{1}{2(s+1)^{2}}\mathbb{E}Q_{s}\varphi^{\prime\prime}(f(X_{s}))|\nabla
f(X_{s})|^{2}ds$
$\displaystyle=\int_{0}^{\infty}\frac{1}{2(s+1)^{2}}\mathbb{E}\left[\varphi^{\prime\prime}(f)|\nabla
f|^{2}\right](X))ds$
$\displaystyle=\frac{1}{2}\mathbb{E}\left[\varphi^{\prime\prime}(f)|\nabla
f|^{2}\right](X).$

This shows that $C_{\varphi}(X)\leq 1$. The inequality $C_{\varphi}(X)\geq
C_{P}(X)=1$ follows from Proposition 1.8.

######  Question 5.2.

Is there an extremality property of the log-Sobolev constant
$C_{\mathrm{LS}}(X)$ among all $\varphi$-Sobolev constants $C_{\varphi}(X)$
where $\varphi$ ranges over the class of convex function with $\varphi(1)=0$
such that $1/\varphi^{\prime\prime}$ is concave?

######  Remark 5.3.

In the proof of Proposition 1.9 we used the fact that when $X$ is a standard
Gaussian random vector, for any $f$,

$|\nabla Q_{s}f|^{2}\leq\frac{1}{(s+1)^{2}}Q_{s}(|\nabla f|^{2}).$ (83)

Since $\varphi^{\prime\prime}\geq 0$, inequality (83) and the convexity of the
function $(x,y)\mapsto\varphi^{\prime\prime}(x)y^{2}$, suffices for concluding
(82). Inequality (83) is the only property of a Gaussian random vector that
was used in the proof of Proposition 1.9. Arguing as in the proof of [29,
Lemma 2.5], one can show that (83) holds true in the case where the law $\mu$
of $X$ is more log-concave than the Gaussian measure, i.e. when its density
$e^{-\psi}$ satisfies $\nabla^{2}\psi\geq{\rm Id}$ everywhere in
$\mathbb{R}^{n}$. We omit the details. This yields another proof of a result
from Chafaï [11]: If $1/\varphi^{\prime\prime}$ is concave and $X$ is more
log-concave than the Gaussian, then,

$C_{\varphi}(X)\leq 1.$

###### Proof of Theorem 1.11 and Theorem 1.12.

We mimick the proof of Theorem 20. Begin with the proof of the right-hand side
inequality in (40), as the left-hand side inequality was already proven above.
If $C_{\varphi}(X)=+\infty$ then this inequality is vacuously true, hence we
may assume that $C_{\varphi}(X)<\infty$. Assume that
$D_{\varphi}(\nu\,\,||\,\,\mu)<\infty$, and write $f=d\nu/d\mu$, so that
$f\geq 0$ satisfies $\int fd\mu=1$. Clearly $f_{s}=Q_{s}f$ is a probability
density with respect to $\mu_{s}$, and in fact $d\nu_{s}/d\mu_{s}=Q_{s}f$. We
need to show that

$D_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})=\int_{\mathbb{R}^{n}}\varphi(Q_{s}f)d\mu_{s}\leq\frac{1}{1+s/C_{\varphi}(\mu)}\cdot\int_{\mathbb{R}^{n}}\varphi(f)d\mu=\frac{1}{1+s/C_{\varphi}(\mu)}\cdot
D_{\varphi}(\nu\,\,||\,\,\mu).$ (84)

The two equalities in (84) follow from the definition of $D_{\varphi}$, and it
remains to prove the inequality. Let $X\sim\mu$ and recall that
$X_{s}=X+\sqrt{s}Z$. By the generalized de Bruijn identity, i.e. Proposition
1.10, and the definition of the $\varphi$-Sobolev constant,

$\frac{\partial}{\partial
s}D_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})=-\frac{1}{2}J_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})\leq-\frac{1}{C_{\varphi}(X_{s})}D_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})\leq-\frac{1}{C_{\varphi}(X)+C_{\varphi}(\sqrt{s}Z)}D_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s}),$

where in the last passage we used subadditivity, i.e., Theorem 5.1. By
homogeneity and Proposition 1.9,

$\frac{\partial}{\partial
s}D_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})\leq-\frac{1}{C_{\varphi}(X)+s}D_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s}).$

Therefore

$\frac{\partial}{\partial s}\log
D_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})\leq-\frac{1}{s+C_{\varphi}(X)}.$

By integrating from $0$ to $s$ we conclude that

$\log\frac{D_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})}{D_{\varphi}(\nu\,\,||\,\,\mu)}\leq-\int_{0}^{s}\frac{dx}{x+C_{\varphi}(X)}=\log\frac{1}{1+s/C_{\varphi}(X)}.$

This proves the right-hand side inequality in (40).

Let us now assume that $X$ is log-concave and prove (41). Let
$0<\varepsilon<C_{\varphi}(\mu)$. There exists non-negative function $f$ with
$\mathbb{E}f(X)=1$ such that

$\mathbb{E}\varphi(f(X))\geq(C_{\varphi}(\mu)-\varepsilon)\cdot\mathbb{E}\varphi^{\prime\prime}(f(X))|\nabla
f(X)|^{2}.$ (85)

By an argument similar to the one from the appendix of [7], we may assume that
$f$ is smooth. Let $\nu$ satisfy $d\nu/d\mu=f$. By Conjecture 4.2,

$\displaystyle\frac{d}{ds}$ $\displaystyle
J_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})=\frac{d}{ds}\int_{\mathbb{R}^{n}}\varphi^{\prime\prime}(f_{s})|\nabla
f_{s}|^{2}d\mu_{s}$
$\displaystyle=-\int_{\mathbb{R}^{n}}\left[2(\nabla^{2}\psi_{s})\nabla
g_{s}\cdot\nabla g_{s}+\left|\nabla^{2}g_{s}\right|^{2}+\kappa(f_{s})|\nabla
g_{s}|^{4}\right]\frac{d\mu_{s}}{\varphi^{\prime\prime}(f_{s})}\leq 0,$ (86)

where $f_{s}=Q_{s}f$. Indeed, each of the three summands in the integral is
non-negative; the first is non-negative since $\nabla^{2}\psi_{s}\geq 0$ by
log-concavity, the second is always non-negative, and the third is non-
negative since $\kappa\geq 0$ as $1/\varphi^{\prime\prime}$ is concave. We
thus proved that $J_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})$ is decreasing with
$s$. Since $J_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})$ is the derivative of
$D_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})$, we conclude that
$D_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})$ is a concave function of $s$.
Moreover, by the generalized de Brujin identity,

$D_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})-D_{\varphi}(\nu\,\,||\,\,\mu)=-\int_{0}^{s}J_{\varphi}(\nu_{t}\,\,||\,\,\mu_{t})dt\geq-s\cdot
J_{\varphi}(\nu\,\,||\,\,\mu)\geq-\frac{s}{C_{\varphi}(\mu)-\varepsilon}\cdot
D_{\varphi}(\nu\,\,||\,\,\mu).$

and hence

$D_{\varphi}(\nu_{s}\,\,||\,\,\mu_{s})\geq\left(1-\frac{s}{C_{\varphi}(\mu)-\varepsilon}\right)D_{\varphi}(\nu\,\,||\,\,\mu).$

Since $\varepsilon>0$ is arbitrary, this proves that

$\eta_{\varphi}(\mu,s)\geq 1-\frac{s}{C_{\varphi}(\mu)}.$

###### Proof of Theorem 1.6 and Theorem 1.7.

These follow from Theorem 1.11 and Theorem 1.12, respectively.

##  Appendix A Technical justifications for Lemma 4.1

We assume $\varphi(x)=\varphi_{\lambda}(x)=x^{\lambda}-1$, for $\lambda>1$.
Let $\nu$ and $\mu$ be two probability distributions on $\mathbb{R}^{n}$. We
make the following regularity assumptions:

  1. 1.

$D_{\varphi}(\nu\|\mu)<\infty$

  2. 2.

$\mathbb{E}_{\mu}|X|^{4}<\infty$

Let $f=d\nu/d\mu$, such that $f_{s}=d\nu_{s}/d\mu_{s}$. The following
proposition will be useful in the derivation below.

######  Proposition A.1.

Let $\varphi(x)=\varphi_{\lambda}(x)=x^{\lambda}-1$, $\lambda>1$, and assume
$D_{\varphi}(\nu\|\mu)<\infty$. Then, for any $s>0$ we also have
$D_{\varphi^{2}}(\nu_{s}\|\mu_{s})<\infty$, and in particular,
$\chi^{2}(\nu\|\mu)<\infty$. Furthermore,
$D_{\varphi_{4\lambda}}(\nu_{s}\|\mu_{s})<\infty$.

###### Proof.

Recall that the Rènyi entropy of order $\lambda$ is defined as

$\displaystyle
D_{\lambda}(\nu_{s}\|\mu_{s})=\frac{1}{\lambda-1}\log\left(1+\mathbb{E}\left[\varphi_{\lambda}\left(\frac{d\nu_{s}}{d\mu_{s}}(X_{s})\right)\right]\right)=\frac{1}{\lambda-1}\log\left(1+\mathbb{E}\left[\varphi_{\lambda}\left(f_{s}(X_{s})\right)\right]\right),$
(87)

where $\varphi_{\lambda}(x)=x^{\lambda}-1$, and $f=d\nu/d\mu$, such that
$f_{s}=d\nu_{s}/d\mu_{s}$. It therefore follows that

$\displaystyle\varphi^{2}_{\lambda}(x)=(x^{\lambda}-1)^{2}=(x^{2\lambda}-1)-2(x^{\lambda}-1)=\varphi_{2\lambda}(x)-2\varphi_{\lambda}(x),$
(88)

so that

$\displaystyle D_{\varphi}(\nu_{s}\|\mu_{s})$
$\displaystyle=\mathbb{E}\varphi_{\lambda}(f_{s}(X_{s}))=e^{(\lambda-1)D_{\lambda}(\nu_{s}\|\mu_{s})}-1$
(89) $\displaystyle D_{\varphi^{2}}(\nu_{s}\|\mu_{s})$
$\displaystyle=\mathbb{E}\varphi_{2\lambda}(f_{s}(X_{s}))-2\mathbb{E}\varphi_{\lambda}(f_{s}(X_{s}))=e^{(2\lambda-1)D_{2\lambda}(\nu_{s}\|\mu_{s})}-2e^{(\lambda-1)D_{\lambda}(\nu_{s}\|\mu_{s})}+1.$
(90)

By the data-processing inequality, and the assumption that
$D_{\varphi}(\nu\|\mu)$ is finite, we have

$\displaystyle D_{\varphi}(\nu_{s}\|\mu_{s})\leq
D_{\varphi}(\nu\|\mu)<\infty.$ (91)

Therefore, by (87), it follows that $D_{\lambda}(\nu_{s}\|\mu_{s})$ is also
finite. From [2, Theorem 1] (see also [41] and [45, Theorem 30]) it follows
that finiteness of $D_{\lambda}(\nu_{s}\|\mu_{s})$ for $\lambda>1$ also
implies the finiteness of $D_{2\lambda}(\nu_{s}\|\mu_{s})$ (and also of
$D_{4\lambda}(\nu_{s}\|\mu_{s})$). To see this, note that for $\lambda>1$ [2,
Theorem 1] reads

$D_{\lambda}(\nu_{s}\|\mu_{s})=\sup_{\xi\ll\nu_{s}}\left[D_{\text{KL}}(\xi\|\mu_{s})-\frac{\lambda}{\lambda-1}D_{\text{KL}}(\xi\|\nu_{s})\right].$

Thus, finiteness of $D_{\lambda}(\nu_{s}\|\mu_{s})$ implies that for any
distribution $\xi\ll\nu_{s}$ we have $D_{\text{KL}}(\xi\|\mu_{s})<\infty$.
Using the same equation again, with $2\lambda$ (or $4\lambda)$, we see that
$D_{2\lambda}(\nu_{s}\|\mu_{s})$ must be finite as well. Thus

$\displaystyle D_{\varphi}(\nu_{s}\|\mu_{s})<\infty\Rightarrow
D_{\lambda}(\nu_{s}\|\mu_{s})<\infty\Rightarrow
D_{2\lambda}(\nu_{s}\|\mu_{s})<\infty\Rightarrow
D_{\varphi^{2}}(\nu_{s}\|\mu_{s})<\infty.$ (92)

To prove that $\chi^{2}(\nu\|\mu)$ is finite, we note that
$\chi^{2}(\nu\|\mu)=D_{\varphi_{\lambda^{\prime}}}(\nu\|\mu)$ with
$\lambda^{\prime}=2$. Since $\lambda>1$, we have that
$\lambda^{\prime}<2\lambda$. We have already shown that
$D_{2\lambda}(\nu\|\mu)$ is finite. This together with the fact that
$\lambda\mapsto D_{\lambda}(\nu\|\mu)$ is non-decreasing in $\lambda>1$, shows
that $D_{\lambda^{\prime}}(\nu\|\mu)$ is finite, which in turn implies that
$D_{\varphi_{\lambda^{\prime}}}(\nu\|\mu)$ is finite.

###  A.1 Justification of first integration in parts

We prove that

$\displaystyle\sum_{i=1}^{n}\int_{\mathbb{R}^{n}}|\varphi(f_{s})\frac{\partial}{\partial
x_{i}}\rho_{s}|<\infty.$ (93)

We have

$\displaystyle\varphi(f_{s})\nabla\rho_{s}=\varphi(f_{s})\sqrt{\rho_{s}}\cdot\frac{\nabla\rho_{s}}{\sqrt{\rho_{s}}}.$
(94)

Applying the Cauchy-Schwartz inequality, we therefore have that

$\displaystyle\sum_{i=1}^{n}\int_{\mathbb{R}^{n}}|\varphi(f_{s})\frac{\partial}{\partial
x_{i}}\rho_{s}|$
$\displaystyle=n\sum_{i=1}^{n}\frac{1}{n}\int_{\mathbb{R}^{n}}|\varphi(f_{s})\sqrt{\rho_{s}}|\cdot|\frac{\frac{\partial}{\partial
x_{i}}\rho_{s}}{\sqrt{\rho_{s}}}|$ (95) $\displaystyle\leq
n\sum_{i=1}^{n}\frac{1}{n}\sqrt{\int_{\mathbb{R}^{n}}\varphi^{2}(f_{s})\rho_{s}\cdot\int_{\mathbb{R}^{n}}\frac{\left(\frac{\partial}{\partial
x_{i}}\rho_{s}\right)^{2}}{\sqrt{\rho_{s}}}}$ (96)
$\displaystyle\leq\sqrt{n}\sqrt{\int_{\mathbb{R}^{n}}\varphi^{2}(f_{s})\rho_{s}\cdot\int_{\mathbb{R}^{n}}\frac{|\nabla\rho_{s}|^{2}}{\rho_{s}}}$
(97)
$\displaystyle=\sqrt{\mathbb{E}\varphi^{2}(f_{s}(X_{s}))}\sqrt{n\cdot\mathcal{J}(\mu_{s})}.$
(98)

We have that $\mathcal{J}(\mu_{s})<\infty$ for any $s>0$. Finiteness of
$\mathbb{E}\varphi^{2}(f_{s}(X_{s}))=D_{\varphi^{2}}(\nu_{s}\|\mu_{s})$
follows from Proposition A.1. We therefore conclude that (93) holds.

###  A.2 Justification of second integration in parts

We prove that

$\displaystyle\sum_{i=1}^{n}\int_{\mathbb{R}^{n}}|\rho_{s}\frac{\partial}{\partial
x_{i}}\varphi(f_{s})|<\infty.$ (99)

Recalling that $\varphi(x)=x^{\lambda}-1$ we have

$\displaystyle\varphi^{\prime}(x)$ $\displaystyle=\lambda
x^{\lambda-1},~{}~{}\varphi^{\prime\prime}(x)=\lambda(\lambda-1)x^{\lambda-2},$
(100)

and therefore

$\displaystyle\frac{\varphi^{\prime}(x)}{\sqrt{\varphi^{\prime\prime}(x)}}$
$\displaystyle=\left(\frac{\lambda}{\lambda-1}x^{\lambda}\right)^{1/2}=\left(\frac{\lambda}{\lambda-1}(\varphi(x)+1)\right)^{1/2}.$
(101)

Consequently, (recall that $\varphi^{\prime\prime}>0$)

$\displaystyle\rho_{s}\nabla\varphi(f_{s})$
$\displaystyle=\rho_{s}\varphi^{\prime}(f_{s})\nabla
f_{s}=\left(\sqrt{\varphi^{\prime\prime}(f_{s})\rho_{s}}\nabla
f_{s}\right)\left(\frac{\varphi^{\prime}(f_{s})}{\sqrt{\varphi^{\prime\prime}(f_{s})}}\sqrt{\rho_{s}}\right)$
(102)
$\displaystyle=\sqrt{\frac{\lambda}{\lambda-1}}\left(\sqrt{\varphi^{\prime\prime}(f_{s})\rho_{s}}\nabla
f_{s}\right)\left(\sqrt{(\varphi(f_{s})+1)\rho_{s}}\right).$ (103)

Applying the Cauchy-Schwartz inequality, we therefore have that

$\displaystyle\sum_{i=1}^{n}\int_{\mathbb{R}^{n}}|\rho_{s}\frac{\partial}{\partial
x_{i}}\varphi(f_{s})|$
$\displaystyle=\sqrt{\frac{\lambda}{\lambda-1}}n\sum_{i=1}^{n}\frac{1}{n}\int_{\mathbb{R}^{n}}\left(\sqrt{\varphi^{\prime\prime}(f_{s})\rho_{s}}\frac{\partial}{\partial
x_{i}}f_{s}\right)\left(\sqrt{(\varphi(f_{s})+1)\rho_{s}}\right)$ (104)
$\displaystyle\leq\sqrt{\frac{\lambda}{\lambda-1}}n\sum_{i=1}^{n}\frac{1}{n}\sqrt{\int_{\mathbb{R}^{n}}\varphi^{\prime\prime}(f_{s})\rho_{s}\left(\frac{\partial}{\partial
x_{i}}f_{s}\right)^{2}\int_{\mathbb{R}^{n}}(\varphi(f_{s})+1)\rho_{s}}$ (105)
$\displaystyle=\sqrt{\frac{\lambda}{\lambda-1}}\sqrt{n(1+\mathbb{E}\varphi(f_{s}(X_{s})))\cdot\int_{\mathbb{R}^{n}}\varphi^{\prime\prime}(f_{s})|\nabla
f_{s}|^{2}d\mu_{s}}.$ (106)

Consequently, the assumption that $D_{\varphi}(\nu_{s}\|\mu_{s})$ is finite
and that $\int_{\mathbb{R}^{n}}\varphi^{\prime\prime}(f_{s})|\nabla
f_{s}|^{2}d\mu_{s}$ is finite (otherwise the statement is void), implies that
(99) holds.

###  A.3 Justification of taking derivative under the integral sign

We show that
$\frac{d}{ds}\int_{\mathbb{R}^{n}}\varphi(f_{s})\rho_{s}=\int_{\mathbb{R}^{n}}\frac{d}{ds}\varphi(f_{s})\rho_{s}$.
To that end, we will show that for any $0<a<b$ close enough to each other, it
holds that $\sup_{s\in[a,b]}|\frac{d}{ds}\varphi(f_{s})\rho_{s}|$ is
integrable, and use the dominated convergence theorem. In particular, we may
assume $\frac{b}{a}\leq\frac{4\lambda-1}{4\lambda-2}$.

Let $\beta$ be the density corresponding to $\nu$, and
$\beta_{s}=\beta*\gamma_{s}$ the density corresponding to $\nu_{s}$. Recalling
that $\varphi(x)=x^{\lambda}-1$, we have that
$\varphi(f_{s})=\left(\frac{\beta_{s}}{\rho_{s}}\right)^{\lambda}-1$, so that

$\displaystyle\frac{d}{ds}\varphi(f_{s})\rho_{s}$
$\displaystyle=\frac{d}{ds}\left(\beta_{s}^{\lambda}\rho_{s}^{1-\lambda}-\rho_{s}\right)$
(107)
$\displaystyle=\lambda\beta^{\prime}_{s}\left(\frac{\beta_{s}}{\rho_{s}}\right)^{\lambda-1}+(1-\lambda)\rho^{\prime}_{s}\left(\frac{\beta_{s}}{\rho_{s}}\right)^{\lambda}-\rho^{\prime}_{s}$
(108)
$\displaystyle=\rho_{s}\left(\frac{\beta_{s}}{\rho_{s}}\right)^{\lambda}\left(\lambda\cdot\frac{\beta^{\prime}_{s}}{\beta_{s}}+(1-\lambda)\cdot\frac{\rho^{\prime}_{s}}{\rho_{s}}\right)-\rho^{\prime}_{s}.$
(109)

Thus,

$\displaystyle\int_{\mathbb{R}^{n}}\sup_{s\in[a,b]}|\frac{d}{ds}\varphi(f_{s})\rho_{s}|$
(110)
$\displaystyle\leq\lambda\underbrace{\int_{\mathbb{R}^{n}}\sup_{s\in[a,b]}\left|\rho_{s}\left(\frac{\beta_{s}}{\rho_{s}}\right)^{\lambda}\frac{\beta^{\prime}_{s}}{\beta_{s}}\right|}_{I}+|1-\lambda|\underbrace{\int_{\mathbb{R}^{n}}\sup_{s\in[a,b]}\left|\rho_{s}\left(\frac{\beta_{s}}{\rho_{s}}\right)^{\lambda}\frac{\rho^{\prime}_{s}}{\rho_{s}}\right|}_{II}+\underbrace{\int_{\mathbb{R}^{n}}\sup_{s\in[a,b]}|\rho^{\prime}_{s}|}_{III}.$
(111)

Before bounding integrals $I$, $II$, and $III$, we develop explicit
expressions for $\rho_{s}$ ,$\beta_{s}$, $\rho^{\prime}_{s}$ and
$\beta^{\prime}_{s}$. Recall that

$\displaystyle\rho_{s}(y)$ $\displaystyle=\mathbb{E}_{\mu}[\gamma_{s}(y-X)],$
(112) $\displaystyle\frac{d}{ds}\gamma_{s}(y-X)$
$\displaystyle=\frac{1}{s}\gamma_{s}(y-X)\left[\frac{|y-X|^{2}}{2s}-\frac{n}{2}\right].$
(113)

Differentiation under the integral sign is valid, see e.g. [6], and therefore

$\displaystyle\rho^{\prime}_{s}(y)=\frac{d}{ds}\rho_{s}(y)$
$\displaystyle=\mathbb{E}_{\mu}\left[\frac{d}{ds}\gamma_{s}(y-X)\right]=\frac{1}{s}\left[\mathbb{E}_{\mu}\left[\frac{|y-X|^{2}}{2s}\gamma_{s}(y-X)\right]-\frac{n}{2}\rho_{s}(y)\right].$
(114)

Similarly,

$\displaystyle\beta_{s}(y)=\mathbb{E}_{\nu}[\gamma_{s}(y-X)],~{}~{}~{}\beta^{\prime}_{s}(y)=\frac{d}{ds}\beta_{s}(y)=\frac{1}{s}\left[\mathbb{E}_{\nu}\left[\frac{|y-X|^{2}}{2s}\gamma_{s}(y-X)\right]-\frac{n}{2}\beta_{s}(y)\right].$
(115)

We begin with showing that the integral $I$ is finite. Using the Cauchy-
Schwartz inequality, we have

$\displaystyle\int_{\mathbb{R}^{n}}\sup_{s\in[a,b]}\left|\rho_{s}\left(\frac{\beta_{s}}{\rho_{s}}\right)^{\lambda}\frac{\beta^{\prime}_{s}}{\beta_{s}}\right|\leq\left(\int_{\mathbb{R}^{n}}\sup_{s\in[a,b]}\left|\rho_{s}\left(\frac{\beta_{s}}{\rho_{s}}\right)^{2\lambda}\right|\cdot\int_{\mathbb{R}^{n}}\sup_{s\in[a,b]}\left|\rho_{s}\left(\frac{\beta^{\prime}_{s}}{\beta_{s}}\right)^{2}\right|\right)^{1/2}.$
(116)

We bound each of the two integrals in the product above. We have that

$\displaystyle\sup_{s\in[a,b]}\left|\rho_{s}\left(\frac{\beta_{s}}{\rho_{s}}\right)^{2\lambda}(y)\right|\leq\sup_{s\in[a,b]}(2\pi
s)^{-n/2}\sup_{s\in[a,b]}\bigg{(}\mathbb{E}_{\nu}\left[\exp\left(-\frac{1}{2s}|y-X|^{2}\bigg{)}\right]\right)^{2\lambda}$
(117)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\cdot\sup_{s\in[a,b]}\left(\mathbb{E}_{\mu}\left[\exp\left(-\frac{1}{2s}|y-X|^{2}\right)\right]\right)^{1-2\lambda}$
(118) $\displaystyle=(2\pi
a)^{-n/2}\bigg{(}\mathbb{E}_{\nu}\left[\exp\left(-\frac{1}{2b}|y-X|^{2}\bigg{)}\right]\right)^{2\lambda}\left(\mathbb{E}_{\mu}\left[\exp\left(-\frac{1}{2a}|y-X|^{2}\right)\right]\right)^{1-2\lambda}$
(119)
$\displaystyle=C\rho_{a}^{1-2\lambda}\beta_{b}^{2\lambda}=C\rho_{a}\left(\frac{\beta_{b}}{\rho_{a}}\right)^{2\lambda}=C\rho_{a}\left(\frac{\rho_{b}}{\rho_{a}}\frac{\beta_{b}}{\rho_{b}}\right)^{2\lambda}$
(120)
$\displaystyle=C\rho_{a}\left[\left(\frac{\rho_{b}}{\rho_{a}}\right)^{2\lambda-1/2}\right]\left[\left(\frac{\rho_{b}}{\rho_{a}}\right)^{1/2}\left(\frac{\beta_{b}}{\rho_{b}}\right)^{2\lambda}\right]$
(121)

where $C=(b/a)^{n\lambda}$. With this, we may apply the Cauchy-Schwartz
inequality and obtain

$\displaystyle\int_{\mathbb{R}^{n}}\sup_{s\in[a,b]}\left|\rho_{s}\left(\frac{\beta_{s}}{\rho_{s}}\right)^{2\lambda}\right|\leq
C\left(\mathbb{E}_{\mu_{a}}\left[\left(\frac{\rho_{b}}{\rho_{a}}\right)^{4\lambda-1}\right]\mathbb{E}_{\mu_{b}}\left[\left(\frac{\beta_{b}}{\rho_{b}}\right)^{4\lambda}\right]\right)^{1/2}.$
(122)

We have that
$\mathbb{E}_{\mu_{b}}\left[\left(\frac{\beta_{b}}{\rho_{b}}\right)^{4\lambda}\right]=1+D_{\varphi_{4\lambda}}(\nu_{b}\|\mu_{b})$
is finite due to Proposition A.1. Furthermore, by the data-processing
inequality

$\displaystyle\mathbb{E}_{\mu_{a}}\left[\left(\frac{\rho_{b}}{\rho_{a}}\right)^{4\lambda-1}\right]=1+D_{\varphi_{4\lambda-1}}(\mu_{b}\|\mu_{a})\leq
1+D_{\varphi_{4\lambda-1}}(\mathcal{N}(0,b\cdot{\rm
Id})\|\mathcal{N}(0,a\cdot{\rm Id})).$ (123)

It is easy to see that
$D_{\varphi_{\lambda}}(\mathcal{N}(0,\sigma^{2}_{0}\cdot{\rm
Id})\|\mathcal{N}(0,\sigma^{2}_{1}\cdot{\rm Id})<\infty$ provided that
$\frac{\sigma_{0}^{2}}{\sigma_{1}^{2}}<\frac{\lambda}{\lambda-1}$, see [45,
eq. (10)]. Thus,
$\mathbb{E}_{\mu_{a}}\left[\left(\frac{\rho_{b}}{\rho_{a}}\right)^{4\lambda-1}\right]<\infty$,
since we assumed $\frac{b}{a}\leq\frac{4\lambda-1}{4\lambda-2}$.

We move on to upper bounding the second integral in (116). Using (115) we have
that

$\displaystyle\left|\frac{\beta^{\prime}_{s}}{\beta_{s}}\right|=\frac{1}{s}\left|\frac{\mathbb{E}_{\nu}\left[\frac{|y-X|^{2}}{2s}\gamma_{s}(y-X)\right]-\frac{n}{2}\beta_{s}}{\beta_{s}}\right|\leq\frac{1}{s}\left[\frac{n}{2}+\frac{\mathbb{E}_{\nu}\left[\frac{|y-X|^{2}}{2s}\gamma_{s}(y-X)\right]}{\beta_{s}}\right]$
(124)

The function $|y-X|\mapsto\frac{|y-X|^{2}}{2s}$ is increasing, while the
function $|y-X|\mapsto\gamma_{s}(|y-X|)$ is decreasing. Consequently,

$\displaystyle\mathbb{E}_{\nu}\left[\frac{|y-X|^{2}}{2s}\gamma_{s}(y-X)\right]\leq\mathbb{E}_{\nu}\left[\frac{|y-X|^{2}}{2s}\right]\mathbb{E}_{\nu}\left[\gamma_{s}(y-X)\right]=\beta_{s}\cdot\mathbb{E}_{\nu}\left[\frac{|y-X|^{2}}{2s}\right].$
(125)

Using the Cauchy Schwartz inequality, we can further bound

$\displaystyle\mathbb{E}_{\nu}\left[\frac{|y-X|^{2}}{2s}\right]$
$\displaystyle=\mathbb{E}_{\mu}\left[\frac{d\nu}{d\mu}(X)\frac{|y-X|^{2}}{2s}\right]\leq\left(\mathbb{E}_{\mu}\left[\left(\frac{d\nu}{d\mu}\right)^{2}\right]\cdot\mathbb{E}_{\mu}\left[\frac{|y-X|^{4}}{4s^{2}}\right]\right)^{1/2}$
(126)
$\displaystyle=\left((\chi^{2}(\nu\|\mu)+1)\cdot\mathbb{E}_{\mu}\left[\frac{|y-X|^{4}}{4s^{2}}\right]\right)^{1/2}$
(127) $\displaystyle\leq
c\cdot\left(\mathbb{E}_{\mu}\left[\frac{|y-X|^{4}}{4s^{2}}\right]\right)^{1/2},$
(128)

where the fact that $\chi^{2}(\nu\|\mu)$ is bounded follows from Proposition
A.1. Thus,

$\displaystyle\left|\frac{\beta^{\prime}_{s}}{\beta_{s}}\right|\leq\frac{1}{s}\left[\frac{n}{2}+c\cdot\left(\mathbb{E}_{\mu}\left[\frac{|y-X|^{4}}{4s^{2}}\right]\right)^{1/2}\right].$
(129)

In particular, for $s\in[a,b]$, we have

$\displaystyle\left|\frac{\beta^{\prime}_{s}}{\beta_{s}}\right|^{2}$
$\displaystyle\leq\frac{1}{a^{2}}\left[\frac{n^{2}}{4}+cn\left(\mathbb{E}_{\mu}\left[\frac{|y-X|^{4}}{4a^{2}}\right]\right)^{1/2}+c^{2}\cdot\mathbb{E}_{\mu}\left[\frac{|y-X|^{4}}{4a^{2}}\right]\right]$
(130)
$\displaystyle=c_{1}+c_{2}\left(\mathbb{E}_{\mu}\left[\frac{|y-X|^{4}}{4b^{2}}\right]\right)^{1/2}+c_{3}\mathbb{E}_{\mu}\left[\frac{|y-X|^{4}}{4b^{2}}\right].$
(131)

Noting further that for all $s\in[a,b]$ we have that
$\rho_{s}\leq(b/a)^{n/2}\rho_{b}$, it holds that

$\displaystyle\int_{\mathbb{R}^{n}}\sup_{s\in[a,b]}$
$\displaystyle\left|\rho_{s}\left(\frac{\beta^{\prime}_{s}}{\beta_{s}}\right)^{2}\right|\leq\int_{\mathbb{R}^{n}}c^{\prime}_{1}\rho_{b}(y)+c^{\prime}_{2}\rho_{b}(y)\left(\mathbb{E}_{\mu}\left[\frac{|y-X|^{4}}{4b^{2}}\right]\right)^{1/2}+c^{\prime}_{3}\rho_{b}(y)\mathbb{E}_{\mu}\left[\frac{|y-X|^{4}}{4b^{2}}\right]dy$
(132) $\displaystyle\leq
c^{\prime}_{1}+c^{\prime}_{2}\left(\mathbb{E}_{Y\sim\rho_{b}}\mathbb{E}_{X\sim\mu}\left[\frac{|Y-X|^{4}}{4b^{2}}\right]\right)^{1/2}+c^{\prime}_{3}\mathbb{E}_{Y\sim\rho_{b}}\mathbb{E}_{X\sim\mu}\left[\frac{|Y-X|^{4}}{4b^{2}}\right],$
(133)

where we have used Jensen’s inequality above. The expression above is finite
by our assumption that $\mathbb{E}_{\mu}|X|^{4}<\infty$. Thus, integral $I$ is
finite.

The proof that integral $II$ is finite is nearly identical, where the only
difference is that we do not need the change of measure from $\nu$ to $\mu$
“trick” we have used in (126)-(128), and instead we have the trivial bound
$\mathbb{E}_{\mu}\left[\frac{|y-X|^{2}}{2s}\right]\leq\left(\mathbb{E}_{\mu}\left[\frac{|y-X|^{4}}{4s^{2}}\right]\right)^{1/2}$.

We are left with showing that integral $III$ converges. This follows since

$\displaystyle\int_{\mathbb{R}^{n}}\sup_{s\in[a,b]}|$
$\displaystyle\rho^{\prime}_{s}|=\int_{\mathbb{R}^{n}}\sup_{s\in[a,b]}\rho_{s}\left|\frac{\rho^{\prime}_{s}}{\rho_{s}}\right|\leq\left(\int_{\mathbb{R}^{n}}\sup_{s\in[a,b]}\rho_{s}\cdot\int_{\mathbb{R}^{n}}\sup_{s\in[a,b]}\rho_{s}\left|\frac{\rho^{\prime}_{s}}{\rho_{s}}\right|^{2}\right)^{1/2}$
(134)
$\displaystyle\leq(b/a)^{n/2}\left(\int_{\mathbb{R}^{n}}\sup_{s\in[a,b]}\rho_{s}\left|\frac{\rho^{\prime}_{s}}{\rho_{s}}\right|^{2}\right)^{1/2},$
(135)

and we have already shown that this integral converges.

###  A.4 Proof of that (69) implies (70)

Let $\lambda>1$ and $\varphi_{\lambda}(x)=x^{\lambda}-1$. By (69), we have
that for any $s>0$,

$\displaystyle\frac{d}{ds}D_{\varphi_{\lambda}}(\nu_{s}\,\,||\,\,\mu_{s})=-\frac{1}{2}J_{\varphi_{\lambda}}(\nu_{s}\,\,||\,\,\mu_{s}).$
(136)

where

$J_{\varphi_{\lambda}}(\nu\,\,||\,\,\mu)=\int_{\mathbb{R}^{n}}\varphi_{\lambda}^{\prime\prime}(f)|\nabla
f|^{2}d\mu=\lambda(\lambda-1)\int_{\mathbb{R}^{n}}f^{\lambda-1}\frac{|\nabla
f|^{2}}{f}d\mu,$

where $f=d\nu/d\mu$. Recalling the definition of Rényi divergence (87), we
have that for any $\lambda>1$

$\displaystyle\frac{d}{ds}D_{\lambda}(\nu_{s}\|\mu_{s})=-\frac{1}{2}\frac{\lambda\int_{\mathbb{R}^{n}}f_{s}^{\lambda-1}\frac{|\nabla
f_{s}|^{2}}{f_{s}}d\mu_{s}}{1+D_{\varphi_{\lambda}}(\nu_{s}\|\mu_{s})}\triangleq
G(\lambda,s).$ (137)

We have that $G(\lambda,s)$ is continuous in both $\lambda>1$ and $s>0$, and
that $\lim_{\lambda\to 1}G(\lambda,s)=-\frac{1}{2}J(\nu_{s}\|\mu_{s})$. Recall
that

$\displaystyle D_{\text{KL}}(\nu_{s}\|\mu_{s})=\lim_{\lambda\to
1}D_{\lambda}(\nu_{s}\|\mu_{s}).$ (138)

The theorem then follows by exchanging of limits in

$\displaystyle\frac{d}{ds}\lim_{\lambda\to
1}D_{\lambda}(\nu_{s}\|\mu_{s})=\lim_{\lambda\to
1}\frac{d}{ds}D_{\lambda}(\nu_{s}\|\mu_{s})=\lim_{\lambda\to
1}G(\lambda,s)=-\frac{1}{2}J(\nu_{s}\|\mu_{s}),$ (139)

which is valid because $G(\lambda,s)$ is continuous in
$(\lambda,s)\in[1,\infty)\times(0,\infty)$.

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