# A Bakry-Émery approach to Lipschitz transportation on manifolds

Pablo López-Rivera  Université Paris Cité & Sorbonne Université, CNRS,
Laboratoire Jacques-Louis Lions (LJLL), F-75013 Paris, France
<EMAIL_ADDRESS>
###### Abstract.

On weighted Riemannian manifolds we prove the existence of globally Lipschitz
transport maps between the weight (probability) measure and log-Lipschitz
perturbations of it, via Kim and Milman’s diffusion transport map, assuming
that the curvature-dimension condition $\mathrm{CD}(\rho_{1},\infty)$ holds,
as well as a second order version of it, namely
$\Gamma_{3}\geq\rho_{2}\Gamma_{2}$. We get new results as corollaries to this
result, as the preservation of Poincaré’s inequality for the exponential
measure $(0,+\infty)$ when perturbed by a log-Lipschitz potential and a new
growth estimate for the Monge map pushing forward the gamma distribution on
$(0,+\infty)$ (then getting as a particular case the exponential one), via
Laguerre’s generator.

##  1\. Introduction

In the study of functional inequalities, the $d$-dimensional Gaussian space,
$(\mathbb{R}^{d},\gamma_{d})$, where $\gamma_{d}$ denotes the standard
$d$-dimensional Gaussian measure, has been of central importance, as
$\gamma_{d}$ satisfies fundamental model inequalities, as Poincaré’s
inequality: for each $f\colon\mathbb{R}^{d}\to\mathbb{R}$ smooth and square-
integrable,

$\operatorname{Var}_{\gamma_{d}}(f)\leq\mathbb{E}_{\gamma_{d}}[\left|\nabla
f\right|^{2}].$

Given a second measure $\nu$ on $\mathbb{R}^{d}$ and a $C$-Lipschitz map
$T\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$ such that $T$ pushes forward
$\gamma_{d}$ towards $\nu$, we note that it is easy to transport as well
Gaussian Poincaré’s inequality to $\nu$: let
$g\colon\mathbb{R}^{d}\to\mathbb{R}$ be a smooth function and let us apply
Gaussian Poincaré’s inequality to $f=g\circ T$:

$\operatorname{Var}_{\nu}(g)=\operatorname{Var}_{\gamma_{d}}(g\circ
T)\leq\mathbb{E}_{\gamma_{d}}[\left|\nabla(g\circ
T)\right|^{2}]\leq\mathbb{E}_{\gamma_{d}}[\left|DT\right|^{2}\left|(\nabla
g)\circ T\right|^{2}]\leq C^{2}\mathbb{E}_{\nu}[\left|\nabla g\right|^{2}],$
(1)

where in the last inequality we stressed the fact that $T$ is a $C$-Lipschitz
map. We remark that the same argument applies to log-Sobolev or isoperimetric
inequalities.

In [Caf00] Caffarelli proved his famous contraction theorem, which states that
for a measure $\nu$ on $\mathbb{R}^{d}$ which is uniformly log-concave (i.e.,
there exists $\alpha>0$ such that $\nabla^{2}V(x)\succcurlyeq\alpha I_{d\times
d}$, for each $x\in\mathbb{R}^{d}$), then the Brenier map [Bre91] (i.e., the
transport map solving Monge’s problem; see [Vil09] for more context on Monge’s
problem), which pushes forward $\gamma_{d}$ towards $\nu$, is $C$-Lipschitz,
with a constant $C$ which does not depend on the dimension $d$. Thus
Caffarelli’s regularity theorem plays an important role when one wants to
obtain novel inequalities; not just functional but concentration and
isoperimetric ones, which are still independent of the dimension parameter $d$
(see for example [CE02] and [Har04].)

There are some extensions to Caffarelli’s theorem: for example in [CFJ17] it
was proven that the Brenier map between a log-concave measure and a compactly
supported perturbation of it is Lipschitz, again with a constant not depending
on the dimension. Unfortunately, we remark that there are no known analogs for
Caffarelli’s result for the Riemannian setting.

As it was seen in the transport of the Gaussian Poincaré’s inequality (1),
optimality of the transport map —in the sense of Monge’s problem— does not
play any role in the proof. In particular, in this note we prove the existence
of a globally Lipschitz transport map (not necessarily optimal in Monge’s
sense) between a measure on a manifold and a log-Lipschitz perturbation of it,
using a construction originally developed in [KM12] (and which traces its
origins back to [OV00]), revisited as well recently in [KP21], [Nee22], [MS22]
and [FMS23], which we will call the diffusion transport map, as it is based on
the heat flow interpolation between two measures induced by a diffusion
semigroup. Similar schemes, using Polchinski’s flow and multiscale Bakry-Émery
criteria, have been recently explored in [Ser22] and [She23], for example.

In our approach, we consider a diffusion operator $\mathrm{L}$ on a Riemannian
manifold $(M,g)$ which has an ergodic measure $\mu$. Fix a $K$-Lipschitz
potential $V\colon M\to\mathbb{R}$ and define the perturbation of $\mu$ by $V$
as the measure $d\nu=e^{-V}\,d\mu$. Using the aforementioned construction, we
are able to obtain a map $T\colon M\to M$ pushing forward the measure $\mu$
towards $\nu$.

Defining the carré du champ operator $\Gamma$, its iteration $\Gamma_{2}$ and
its second iteration $\Gamma_{3}$ (which will be precisely defined in Section
3), if we suppose that there exist positive constants $\rho_{1},\rho_{2}>0$
such that for each smooth and compactly supported function $f\colon
M\to\mathbb{R}$, $\Gamma_{2}(f)\geq\rho_{1}\Gamma(f)$ (i.e, classical Bakry-
Émery’s curvature-dimension condition $\mathrm{CD}(\rho,\infty)$) and
$\Gamma_{3}(f)\geq\rho_{2}\Gamma_{2}(f)$, then we are able to say that $T$ is
$C$-Lipschitz, where $C=C(\rho_{1},\rho_{2},K)$. The following theorem is the
precise statement of these ideas, corresponding to Theorem 4.5 in this note.

######  Theorem.

Let $(M,g)$ be a complete and connected $d$-dimensional Riemannian manifold.
Let $W\colon M\to\mathbb{R}$ be smooth and such that
$\int_{M}e^{-W}d\mathrm{Vol}=1$. Consider the diffusion operator
$\mathrm{L}=\Delta-\nabla W\cdot\nabla$. Let us assume that there exist
constants $\rho_{1},\rho_{2}>0$ such that

  1. (i)

$\forall
f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M),\Gamma_{2}(f)\geq\rho_{1}\Gamma(f)$;
and

  2. (ii)

$\forall
f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M),\Gamma_{3}(f)\geq\rho_{2}\Gamma_{2}(f)$.

Let $V\colon M\to\mathbb{R}$ smooth and $K$-Lipschitz and define
$d\mu=e^{-W}d\mathrm{Vol}$ and $d\nu=e^{-V}d\mu$. Then there exists a
Lipschitz mapping $T\colon M\to M$ pushing forward $\mu$ towards $\nu$ which
is
$\exp\left(\sqrt{\frac{2\pi}{\rho_{2}}}Ke^{\frac{K^{2}}{2\rho_{1}}}\right)$-Lipschitz.

The idea of working with $\Gamma_{3}$ is not new in the literature; it has
been explored previously in [Led93], [Bak94] and [LNP15]. In particular, in
[LNP15], the condition $\Gamma_{3}(f)\geq\rho_{2}\Gamma_{2}(f)$ was employed
to obtain regularity of the solutions of the PDE

$\Delta f-\nabla W\cdot\nabla f=V-\int V\,d\mu$

for Lipschitz data $V$, which corresponds to the linearization of Monge-
Ampère’s equation associated to the (quadratic) optimal transport problem for
$\mu$ and $\nu$, which coincides as well with the linearization of the
diffusion transport map.

###  1.1. Organization

In Section 2 we introduce all the necessary preliminary notions, starting from
diffusion operators on Riemannian manifolds, their probabilistic
interpretation and the continuity equation in the distributional sense. In
Section 3 we recall classical Bakry-Émery’s $\Gamma$-calculus and we study a
second order version of it, namely $\Gamma_{3}\geq\rho\Gamma_{2}$, and its
consequences. In Section 4 we detail the construction of the diffusion
transport map and we prove its Lipschitz regularity (i.e. Theorem 4.5.)
Finally, in Section 5 we give some applications of the main result.

###  1.2. Acknowledgments

I want to thank my advisor, Max Fathi, for his guidance, corrections and
continuous support with this project, and for introducing me to the subject. I
also thank Maxime Laborde for his help. The author acknowledges the support of
the French Agence Nationale de la Recherche (ANR) via the project CONVIVIALITY
( ANR-23-CE40-0003 ). This project has received funding from the European
Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska-Curie grant agreement No 945322.

##  2\. Preliminaries and notations

In this section we present all the main objects on which this note is based
and their most essential properties; on the other hand we fix the notations
for the rest of this note. We first introduce the language of Riemannian
geometry in order to talk about diffusion operators on weighted manifolds, to
then remind as well the probabilistic interpretation of these objects. Finally
we recall the continuity equation in the context of measure-valued flows and
the structure of its solutions.

###  2.1. Markov diffusion generators on a manifold

Let $(M,g)$ be a $d$-dimensional smooth Riemannian manifold which we assume to
be complete and connected, unless otherwise stated, and let us denote its
tangent bundle by $TM$. Usually the action of the metric $g$ onto two vector
fields $X,Y\colon M\to TM$ will be succinctly written by $X\cdot Y:=g(X,Y)$,
omitting the dependence on $x\in M$, unless necessary. The metric $g$ induces
a distance $d_{g}$ on $M$, which in turn generates the Riemannian volume
measure $d\mathrm{Vol}$ on $M$ (i.e., the $d$-dimensional Hausdorff measure
associated to the metric space $(M,d_{g})$.)

Let $\nabla$ be the Riemannian gradient, which acts on smooth functions
$f\colon M\to\mathbb{R}$ as a vector field by $\nabla
f=(\nabla^{i}f)_{i=1}^{d}$, where $\nabla^{i}f=g^{ij}\partial_{j}f$ (using
Einstein’s summation convention to handle operations with tensors) and let
$\nabla^{2}$ be the Hessian operator, which acts on smooth functions $f\colon
M\to\mathbb{R}$ via $\nabla^{2}f=(\nabla^{i}\nabla^{j}f)_{i,j=1}^{d}$.
Laplace-Beltrami’s operator, denoted by $\Delta$, acts on smooth functions and
this action can be characterized, for a smooth function $f\colon
M\to\mathbb{R}$, by

$\forall g\in\mathcal{C}^{\infty}_{\mathrm{c}}(M),\quad\int_{M}g\Delta
f\,d\mathrm{Vol}=-\int_{M}\nabla f\cdot\nabla g\,d\mathrm{Vol},$

where $\mathcal{C}^{\infty}_{\mathrm{c}}(M)$ is the set of real-valued smooth
functions with compact support on $M$. We recall Bochner’s formula: for each
$f\colon M\to\mathbb{R}$ smooth,

$\frac{1}{2}\Delta(\left|\nabla
f\right|^{2})=\left|\nabla^{2}f\right|_{\mathrm{HS}}^{2}+\nabla
f\cdot\nabla(\Delta f)+\operatorname{Ric}(\nabla f,\nabla f),$ (2)

where $\left|\cdot\right|_{\mathrm{HS}}$ denotes the Hilbert-Schmidt norm of a
tensor and $\operatorname{Ric}$ denotes the Ricci curvature tensor associated
to $(M,g)$.

Now let $W\colon M\to\mathbb{R}$ be a smooth function such that
$\int_{M}e^{-W}\,d\mathrm{Vol}=1$ to then define the (probability) measure
$d\mu(x):=e^{-W}\,d\mathrm{Vol}$. We say that $(M,g,\mu)$ is a weighted
manifold (that is, a Riemannian manifold equipped with a measure which has a
smooth strictly positive density with respect to the respective volume
measure.)

We define the diffusion operator $\mathrm{L}$, which acts on smooth functions
$f\colon M\to\mathbb{R}$ as

$\mathrm{L}f:=\Delta f-\nabla W\cdot\nabla f,$ (3)

which obviously has elliptic regularity. We note that the measure $\mu$ is
invariant for $\mathrm{L}$: for each
$f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$,

$\int_{M}\mathrm{L}f\,d\mu=0.$ (4)

Moreover, it can be easily seen that it holds the following integration by
parts formula: for all $f,h\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$,

$\int_{M}f\mathrm{L}h\,d\mu=-\int_{M}\nabla f\cdot\nabla h\,d\mu,$ (5)

which in turn implies that $\mathrm{L}$ is symmetric or reversible with
respect to the measure $\mu$: that is, for each
$f,h\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$,

$\int_{M}f\mathrm{L}h\,d\mu=\int_{M}h\mathrm{L}f\,d\mu.$ (6)

######  Remark 2.1.

Suppose now that we have a diffusion operator $\mathrm{L}$ defined on $M$, a
connected $d$-dimensional smooth manifold (note that we are not endowing $M$
with any Riemannian metric), which acts on smooth functions $f\colon
M\to\mathbb{R}$ (in local coordinates) via

$\mathrm{L}f:=a^{ij}\partial_{ij}^{2}f+b^{i}\partial_{i}f,$ (7)

such that the coefficients $x\mapsto a(x):=(a^{ij}(x))_{i,j=1}^{d}$, $x\mapsto
b(x):=(b^{i}(x))_{i=1}^{d}$ are smooth functions on $M$. Let us assume that
$\mathrm{L}$ is elliptic: that is, for each $x\in M$, the matrix $a(x)$ is
symmetric and positive-definite; then the matrix
$a^{-1}=(a_{ij}(x))_{i,j=1}^{d}$ can be regarded as a Riemannian metric on $M$
which we will denote by $g$. If we write $Z:=b^{i}\partial_{i}\colon M\to TM$,
then we can express $\mathrm{L}$ as

$\mathrm{L}f=\Delta_{g}f+Zf,$

where $\Delta_{g}$ denotes the Laplace-Beltrami operator on the (now)
Riemannian manifold $(M,g)$. Moreover, let us suppose that $\mathrm{L}$ is
invariant with respect to a measure $\mu$ of the form
$d\mu=e^{-W}\,d\mathrm{Vol}_{g}$, where $\mathrm{Vol}_{g}$ denotes the
Riemannian volume measure on $(M,g)$ and $W\colon M\to\mathbb{R}$ is a smooth
function; then $Zf=-\nabla_{g}W\cdot\nabla_{g}f$, so finally we get

$\mathrm{L}f=\Delta_{g}f-\nabla_{g}W\cdot\nabla_{g}f.$

That is, if we start with a linear diffusion operator in the form (7), we can
always express it in the form (3), for a well-chosen Riemannian structure,
which is natural for the operator $\mathrm{L}$. In other words, we can assume
without loss of generality that $\mathrm{L}=\Delta-\nabla W\cdot\nabla$.

From the perspective of operator theory, $\mathrm{L}$ could be regarded as
well as an unbounded linear operator with domain
$\mathcal{D}(\mathrm{L})\subseteq L^{2}(\mu)$ such that
$\mathcal{C}^{\infty}_{\mathrm{c}}(M)\subseteq\mathcal{D}(\mathrm{L})$.
Moreover, as we assumed that $(M,g)$ is complete, as $\mathrm{L}$ is an
elliptic diffusion operator on a smooth, connected and complete Riemannian
manifold, which is symmetric with respect to $\mu$, then Proposition 3.2.1 in
[BGL14] entails the essential self-adjointness of $\mathrm{L}$ on
$\mathcal{C}^{\infty}_{\mathrm{c}}(M)$, or equivalently, the density of
$\mathcal{C}^{\infty}_{\mathrm{c}}(M)$ into $\mathcal{D}(\mathrm{L})$.

######  Remark 2.2.

Following the discussion about the essential self-adjointness of $\mathrm{L}$,
we point out two possible issues that could happen in the practice.

  * •

First, sometimes we will not have the hypothesis of completeness of the
manifold $(M,g)$ (this is the case when $M$ is an open ball or more generally
an open connected domain $\mathcal{O}\neq\mathbb{R}^{d}$, both regarded as
submanifolds of the Euclidean space $\mathbb{R}^{d}$.) Fortunately, essential
self-adjointness is not an exclusive property of diffusion operators on
complete manifolds.

  * •

On the other hand, it could happen that the operator $\mathrm{L}$ is
essentially self-adjoint, but with respect to another class of functions, not
$\mathcal{C}^{\infty}_{\mathrm{c}}(M)$.

Following Hille-Yosida theory [Yos80], if we suppose that $\mathrm{L}$ is
essentially self-adjoint with respect to
$\mathcal{C}^{\infty}_{\mathrm{c}}(M)$ (or another class of smooth functions
dense in $L^{2}(\mu)$), then there exists a unique characteristic semigroup
$(\mathrm{P}_{t})_{t\geq 0}$ on $L^{2}(\mu)$ associated to $\mathrm{L}$
solving the heat equation for $\mathrm{L}$: for each $f\in L^{2}(\mu)$, then
$u(t,x):=\mathrm{P}_{t}f(x)$ is the unique solution of

$\left\\{\begin{aligned} \partial_{t}u(t,x)&=\mathrm{L}u(t,x)\\\
u(0,x)&=f(x);\end{aligned}\right.$ (8)

in other words, $\mathrm{L}$ corresponds to the infinitesimal generator of the
semigroup $(\mathrm{P}_{t})_{t\geq 0}$.

We defined our framework around the infinitesimal generator
$\mathrm{L}=\Delta-\nabla W\cdot\nabla$, but it is possible to translate these
properties, in an equivalent way, into the language of the semigroup
$(\mathrm{P}_{t})_{t\geq 0}$. The invariance (4) reads as having for each
$f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$, and any $t\geq 0$,

$\int_{M}\mathrm{P}_{t}f\,d\mu=\int_{M}f\,d\mu;$

and the symmetry (6) is the same as stating that for each
$f,h\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$ and any $t\geq 0$,

$\int_{M}f\mathrm{P}_{t}h\,d\mu=\int_{M}h\mathrm{P}_{t}f\,d\mu.$

Finally, we can see that $\mathrm{L}=\Delta-\nabla W\cdot\nabla$ is ergodic;
i.e., if $f\in\mathcal{C}^{\infty}_{\mathrm{c}}(E)$ and $\mathrm{L}f=0$, then
$f$ has to be identically constant. Indeed, if we suppose that
$\mathrm{L}f=0$, then by (5) we deduce that $\left|\nabla f\right|^{2}=0$ in
$M$, which in turn yields that $f$ is constant, as $M$ was assumed to be
connected. The following proposition translates ergodicity into a property for
the semigroup $(\mathrm{P}_{t})_{t\geq 0}$ (see for example [BGL14,
Proposition 3.1.13].)

######  Proposition 2.1.

Suppose that $\mathrm{L}$ is ergodic and that the corresponding semigroup
$(\mathrm{P}_{t})_{t\geq 0}$ is conservative, that is, for each $t\geq 0$,
$\mathrm{P}_{t}\mathds{1}=\mathds{1}$, where $\mathds{1}$ denotes the constant
function equal to 1 on $M$. Then for each $f\in L^{2}(\mu)$, the following
limit holds in the $L^{2}(\mu)$-sense:

$\lim_{t\to\infty}\mathrm{P}_{t}f=\int_{E}f\,d\mu.$

###  2.2. The probabilistic counterpart of $\mathrm{L}$

As we were able (via (8)) to construct the characteristic semigroup associated
to $\mathrm{L}$, then we can study $\mathrm{L}$ using probabilistic
techniques. First of all, for any $t\geq 0$, we denote by $\mathrm{P}_{t}^{*}$
the dual semigroup, which acts on a measure $\nu\in\mathcal{P}$ by

$\forall
f\in\mathcal{C}_{\mathrm{b}}(M),\quad\int_{M}f\,d(\mathrm{P}_{t}^{*}\nu)=\int_{M}\mathrm{P}_{t}f\,d\nu,$

where $\mathcal{C}_{\mathrm{b}}(M)$ denotes the space of bounded and
continuous real-valued functions on $(M,g)$. Now let $(X_{t})_{t\geq 0}$ be a
Markov process (defined on a probability space
$(\Omega,\mathcal{F},\mathbb{P})$) such that $X_{0}\sim\nu$ and that for each
$t\geq 0$,

$\forall
f\in\mathcal{C}_{\mathrm{b}}(M),\quad\mathbb{E}\left[f(X_{t})\right]=\int_{M}f\,d(\mathrm{P}_{t}^{*}\nu).$
(9)

In particular, if $x\in M$ and $\nu=\delta_{x}$ is the Dirac measure on $x$,
then $\mathbb{E}\left[f(X_{t})\right]=\mathrm{P}_{t}f(x)$.

The Markov process $(X_{t})_{t\geq 0}$ defined above is a diffusion process
(with random initial condition $\nu\in\mathcal{P}(M)$) on $M$ satisfying the
following stochastic differential equation (SDE):

$\left\\{\begin{aligned} dX_{t}&=\sqrt{2}\,dB_{t}-\nabla W(X_{t})\,dt\\\
X_{0}&\sim\nu,\end{aligned}\right.$ (10)

where $(B_{t})_{t\geq 0}$ is the Brownian motion on $(M,g)$ (see for example
[É89] for a succinct introduction to the language of stochastic calculus on
manifolds.)

If we write, for each $t\geq 0$, $\rho_{t}:=\mathrm{Law}(X_{t})$ (in
particular, $\rho_{0}=\nu$), then $(\rho_{t})_{t\geq 0}$ satisfies the Fokker-
Planck equation (which is adjoint of the heat equation (8))

$\partial_{t}\rho_{t}=\mathrm{L}^{*}\rho_{t}=\Delta\rho_{t}+\nabla
W\cdot\nabla\rho_{t}$ (11)

in the distributional sense; i.e., for all $t>0$ and each smooth function with
compact support $\varphi\colon M\to\mathbb{R}$,

$\frac{d}{dt}\int_{M}\varphi\,d\rho_{t}=\int_{M}\mathrm{L}\varphi\,d\rho_{t}=\int_{M}\left(\Delta\varphi-\nabla
W\cdot\nabla\varphi\right)\,d\rho_{t}.$

###  2.3. About the continuity equation

Fix $A_{\bullet}\colon[0,+\infty)\times M\to TM$ a time-depending vector field
on $M$. Given a fixed probability measure $\nu$ on $M$, we are interested in
solving the continuity equation on $M$, with velocity $A_{\bullet}$ and
initial condition $\nu$,

$\left\\{\begin{aligned}
\partial_{t}\rho_{t}+\nabla\cdot(\rho_{t}A_{t})&=0,\quad t>0\\\
\rho_{0}&=\nu,\end{aligned}\right.$ (12)

where $\nabla\cdot$ denotes the divergence operator on $(M,g)$, which acts on
a vector field $Z\colon M\to TM$ by

$\forall\varphi\in\mathcal{C}^{\infty}_{\mathrm{c}}(M),\quad\int_{M}\varphi\nabla\cdot
Z\,d\mathrm{Vol}=-\int_{M}\nabla\varphi\cdot Z\,d\mathrm{Vol}.$

We will be interested in measure-valued solutions for (12). To be precise, we
say that $(\rho_{t})_{t\geq 0}$, a sequence of probability measures on $M$
with $\rho_{0}=\nu$, satisfies the continuity equation (12) in the
distributional sense, if for any $t>0$ and each smooth function with compact
support $\varphi\colon M\to\mathbb{R}$,

$\frac{d}{dt}\int_{M}\varphi\,d\rho_{t}=\int_{M}\nabla\varphi\cdot
A_{t}\,d\rho_{t}.$

Now we introduce the flow of diffeomorphisms $(S_{t})_{t\geq 0}$ induced by
the vector field $A_{\bullet}$, which is defined by

$\left\\{\begin{aligned} \frac{d}{dt}S_{t}(x)&=A_{t}(S_{t}(x)),\quad t>0\\\
S_{0}(x)&=x,\end{aligned}\right.$ (13)

which is well-defined if $A_{\bullet}\colon$ is at least locally Lipschitz.
The following classic result (see for example [Vil09, Chapter 1])
characterizes weak solutions of the system (12) in terms of the associated
flow $(S_{t})_{t\geq 0}$.

######  Theorem 2.2.

Let $(M,g)$ be a complete and connected Riemannian manifold and fix
$A_{\bullet}\colon[0,+\infty)\times M\to TM$ a locally Lipschitz time-
depending vector field on $M$. Let $\nu$ be a probability measure on $M$ and
let $(\rho_{t})_{t\geq 0}$ be a sequence of probability measures on $M$ such
that $\rho_{0}=\nu$, which is continuous on $t\geq 0$ (for the weak topology)
and such that

$\int_{0}^{+\infty}\int_{M}\left|A_{t}(x)\right|\,d\rho_{t}\,dt<+\infty.$ (14)

Then $(\rho_{t})_{t\geq 0}$ is a distributional solution to the continuity
equation

$\left\\{\begin{aligned}
\partial_{t}\rho_{t}+\nabla\cdot(\rho_{t}A_{t})&=0,\quad t>0\\\
\rho_{0}&=\nu\end{aligned}\right.$

if and only if $\rho_{t}={S_{t}}_{\\#}\nu$, where $(S_{t})_{t\geq 0}$ is the
flow of diffeomorphisms associated to $A_{\bullet}$, in the sense of (13).

##  3\. Revisiting Bakry-Émery’s $\Gamma$-calculus

In this section we revisit Bakry-Émery’s $\Gamma$-calculus, firstly recalling
the basic definitions of $\Gamma$, its iteration $\Gamma_{2}$ and the
curvature-dimension condition $\mathrm{CD}(\rho,\infty)$ together with its
most essential consequences for our work. Secondly, we study further
iterations of this theory, which give birth to the operators $\Gamma_{n}$. In
particular, we will be interested in higher analogs
$\mathrm{CD}(\rho,\infty)$, in the sense of
$\Gamma_{n+1}\geq\rho_{n}\Gamma_{n}$, and their consequences for the
associated semigroup $(\mathrm{P}_{t})_{t\geq 0}$.

We assume, unless we express the contrary, that $(M,g)$ is a complete and
connected Riemannian manifold with weight $d\mu=e^{-W}\,d\mathrm{Vol}$ and we
consider $\mathrm{L}=\Delta-\nabla W\cdot\nabla$.

###  3.1. Classical $\Gamma$-calculus

For each $f,h\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$, we define the usual
carré du champ operator $\Gamma$ as

$\Gamma(f,h):=\frac{1}{2}\left(\mathrm{L}(fh)-f\,\mathrm{L}h-h\,\mathrm{L}f\right).$

$\Gamma$ is a bilinear symmetric form which is moreover positive: for all
$f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$, $\Gamma(f,f)\geq 0$. We mostly
work with evaluations of the form $\Gamma(f,f)$, so we define, doing an abuse
of notation, $\Gamma(f):=\Gamma(f,f)$. In the case of a Markov diffusion
operator $\mathrm{L}$ as (3), then $\Gamma$ has an explicit and natural
representation in terms of Riemannian gradient:

$\Gamma(f)=\left|\nabla f\right|^{2}.$ (15)

Using (15), we also have the Riemannian distance $d_{g}$ can be written in
terms of the operator $\Gamma$. Indeed, for each $x,y\in M$,

$d_{g}(x,y)=\sup_{\begin{subarray}{c}f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M);\\\
\Gamma(f)\leq 1\end{subarray}}\left(f(x)-f(y)\right).$

This directly leads to the following characterization of Lipschitz functions,
which plays an important role in this note.

######  Proposition 3.1.

Let $f\colon M\to\mathbb{R}$ be a smooth function. Then $f$ is $K$-Lipschitz
for the distance $d_{g}$ if and only if $\sqrt{\Gamma(f)}\leq K$ uniformly on
$M$.

In their seminal work [BE85], Bakry and Émery defined the iterated carré du
champ operator $\Gamma_{2}$: for $f,g\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$,

$\Gamma_{2}(f,h):=\frac{1}{2}\left(\mathrm{L}\Gamma(f,h)-\Gamma(f,\mathrm{L}h)-\Gamma(h,\mathrm{L}f)\right).$

As we did so with $\Gamma$, we set $\Gamma_{2}(f):=\Gamma_{2}(f,f)$. Now if
$\mathrm{L}=\Delta-\nabla W\cdot\nabla$, then using Bochner’s formula (2) we
can deduce an explicit formula for $\Gamma_{2}$:

$\Gamma_{2}(f)=\left|\nabla^{2}f\right|_{\mathrm{HS}}^{2}+\operatorname{Ric}(\nabla
f,\nabla f)+\nabla^{2}W(\nabla f,\nabla f).$ (16)

With both definitions of $\Gamma$ and $\Gamma_{2}$, we recall the classical
$\mathrm{CD}(\rho,\infty)$ curvature-dimension condition, primevally defined
by Bakry and Émery.

######  Definition 3.2 (Curvature-dimension condition
$\mathrm{CD}(\rho,\infty)$).

We say that the generator $\mathrm{L}$ satisfies the curvature-dimension
condition $\mathrm{CD}(\rho,\infty)$ if there exists $\rho\in\mathbb{R}$ such
that for each $f\in\mathcal{C}^{\infty}_{\mathrm{c}}(E)$,

$\Gamma_{2}(f)\geq\rho\Gamma(f).$

######  Remark 3.1.

For the weighted manifold $(M,g,e^{-W}d\mathrm{Vol})$ we define the Bakry-
Émery-Ricci tensor $\operatorname{Ric}_{W}:=\operatorname{Ric}+\nabla^{2}W$.
Then we see that $\mathrm{CD}(\rho,\infty)$ holds if and only if
$\operatorname{Ric}_{W}\succeq\rho g$. More generally, the Bakry-Émery-Ricci
tensor is an object of interest in itself in the context of weighted manifolds
as it encodes the structure of the weight measure $e^{-W}d\mathrm{Vol}$ into a
tensor. Assuming bounds on $\operatorname{Ric}_{W}$ implies as well
topological and geometrical properties (see for example [Lot03].)

The curvature-dimension condition $\mathrm{CD}(\rho,\infty)$ entails lots of
properties for $\mathrm{L}$ and its associated semigroup
$(\mathrm{P}_{t})_{t\geq 0}$. First of all, if we suppose that $(M,g)$ is
complete and $\mathrm{CD}(\rho,\infty)$ holds, then its associated semigroup
is conservative or non-explosive: for each $t\geq 0$,
$\mathrm{P}_{t}\mathds{1}=\mathds{1}$, where $\mathds{1}$ denotes the constant
function equal to 1 on $M$ (see for example Theorem 3.2.6 in [BGL14].)
Equivalently, if $\zeta$ denotes the explosion time of the process $(X_{t})$
generated by $(\mathrm{P}_{t})_{t\geq 0}$, then $\mathbb{P}(\zeta=+\infty)=1$.

Under $\mathrm{CD}(\rho,\infty)$, the invariant measure $\mu$ satisfies at the
same time the Poincaré and logarithmic Sobolev inequalities (see for example
Propositions 4.8.1 and 5.7.1 in [BGL14].)

######  Theorem 3.3.

Suppose $\mathrm{CD}(\rho,\infty)$ holds for $\rho>0$. Then:

  1. (a)

$(M,g,\mu)$ satisfies a Poincaré inequality with constant $\frac{1}{\rho}$:
that is, for each $f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$,

$\operatorname{Var}_{\mu}(f)\leq\frac{1}{\rho}\int_{M}\Gamma(f)\,d\mu.$

  2. (b)

$(M,g,\mu)$ satisfies a log-Sobolev inequality with constant $\frac{1}{\rho}$:
that is, for each $f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$,

$\operatorname{Ent}_{\mu}(f^{2})\leq\frac{2}{\rho}\int_{M}\Gamma(f)\,d\mu,$

where $\operatorname{Ent}_{\mu}(h):=\int_{M}h\log
h\,d\mu-\left(\int_{M}h\,d\mu\right)\log\left(\int_{M}h\,d\mu\right).$

Another remarkable consequence of $\mathrm{CD}(\rho,\infty)$ is reflected
analytically on the semigroup, via the local Poincaré and logarithmic Sobolev
inequalities (for a proof, see for example Theorems 4.7.2 and 5.5.2 in
[BGL14].)

######  Theorem 3.4.

The following assertions are equivalent.

  1. (i)

$\mathrm{CD}(\rho,\infty)$ holds.

  2. (ii)

For each $f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$ and every $t\geq 0$,

$\Gamma(\mathrm{P}_{t}f)\leq e^{-2\rho t}\mathrm{P}_{t}(\Gamma(f)).$

  3. (iii)

For each $f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$ and every $t\geq 0$,

$\sqrt{\Gamma(\mathrm{P}_{t}f)}\leq e^{-\rho
t}\mathrm{P}_{t}(\sqrt{\Gamma(f)}).$

  4. (iv)

For each $f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$ and every $t\geq 0$, if
$\rho\neq 0$,

$\mathrm{P}_{t}(f^{2})-(\mathrm{P}_{t}f)^{2}\leq\frac{1-e^{-2\rho
t}}{\rho}\mathrm{P}_{t}(\Gamma(f));$

if $\rho=0$, then

$\mathrm{P}_{t}(f^{2})-(\mathrm{P}_{t}f)^{2}\leq 2t\mathrm{P}_{t}(\Gamma(f)).$

  5. (v)

For each $f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$ positive and every $t\geq
0$, if $\rho\neq 0$,

$\mathrm{P}_{t}(f^{2}\log
f^{2})-\mathrm{P}_{t}(f^{2})\log\mathrm{P}_{t}(f^{2})\leq 2\frac{1-e^{-2\rho
t}}{\rho}\mathrm{P}_{t}\left(\Gamma(f)\right);$

if $\rho=0$, then

$\mathrm{P}_{t}(f^{2}\log
f^{2})-\mathrm{P}_{t}(f^{2})\log\mathrm{P}_{t}(f^{2})\leq
4t\mathrm{P}_{t}\left(\Gamma(f)\right).$

######  Remark 3.2.

In this note we aim to develop and use properties arising from
$\Gamma$-calculus theory, which is capable to give insights on the analytical
or probabilistic object represented by the generator $\mathrm{L}$ in terms of
more manipulable objects (algebraically speaking) such as the $\Gamma_{n}$
operators (to be defined in the next subsection.) To do so, a class of
functions $\mathcal{A}_{0}$ has to be fixed, in order to define those
functionals and work properly with them. In fact, this was already done
(implicitly) in the very first paragraphs of this subsection, by taking
$\mathcal{A}_{0}=\mathcal{C}^{\infty}_{\mathrm{c}}(M)$ as $\mathrm{L}$ is
essentially self-adjoint on this class of functions (c.f. [BGL14, Chapter 3].)

###  3.2. Further iterations of $\Gamma$-calculus

As it was seen before, the construction of $\Gamma_{2}$ was iterated from
$\Gamma$, so it is possible to go further and construct $\Gamma_{n}$, for
$n\geq 3$; this idea was already explored before in [Led93], [Bak94] and
[LNP15]. If we set $\Gamma_{0}(f,h):=fh$, then for $n\geq 0$ and
$f,h\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$ we define

$\Gamma_{n+1}(f,h):=\frac{1}{2}\left(\mathrm{L}\Gamma_{n}(f,h)-\Gamma_{n}(f,\mathrm{L}h)-\Gamma_{n}(h,\mathrm{L}f)\right),$
(17)

and again we set $\Gamma_{n}(f):=\Gamma_{n}(f,f)$.

######  Remark 3.3.

Iterations in the sense of (17) are consistent with classical
$\Gamma$-calculus, as $\Gamma_{1}=\Gamma$. For the sequel, we will always
assume that the iterations start from $\Gamma_{0}$: that is, $\Gamma_{1}$ is
the first iteration, $\Gamma_{2}$ the second and so on. For the purposes of
this note we are concerned only up to the third iteration, that is,
$\Gamma_{3}$, but the results in this section are stated and proved for
general iterations.

The goal of this section is to obtain analytical bounds for the semigroup
$(\mathrm{P}_{t})_{t\geq 0}$ assuming that
$\Gamma_{n+1}(f)\geq\rho_{n}\Gamma_{n}(f)$, in the spirit of Theorem 3.4. Now
we provide some illustrative examples of diffusion operators satisfying these
criteria.

######  Example 3.1 (Laplace-Beltrami operator on $\mathbb{S}^{d}$).

Let $d\geq 2$ and consider the $d$-dimensional sphere
$\mathbb{S}^{d}:=\\{x\in\mathbb{R}^{d+1}:\left|x\right|=1\\}$ together with
its standard Riemannian metric $g$ (induced by its embedding on
$\mathbb{R}^{d+1}$), which has a constant Ricci tensor:
$\operatorname{Ric}=(d-1)g$. Let us fix $\mu=\mathrm{Vol}$, the uniform
measure on $\mathbb{S}^{d}$, and let $\mathrm{L}=\Delta$, the Laplace-Beltrami
operator on $\mathbb{S}^{d}$, which is essentially self-adjoint on
$\mathcal{C}^{\infty}_{\mathrm{c}}(M)=\mathcal{C}^{\infty}(M)$ as
$\mathbb{S}^{d}$ is complete.

The stochastic process generated by $\mathrm{L}$ is the (rescaled) Brownian
motion on $\mathbb{S}^{d}$: $X_{t}=\sqrt{2}B_{t}$, for $t\geq 0$, where
$(B_{t})_{t\geq 0}$ the Brownian motion on the sphere. Following the previous
discusion, we have immediately that for $f\colon\mathbb{S}^{d}\to\mathbb{R}$
smooth,

$\Gamma(f)=\left|\nabla f\right|^{2}.$

For $\Gamma_{2}$, we can stress the fact that Ricci tensor is constant, so by
(16), we get

$\displaystyle\Gamma_{2}(f)$
$\displaystyle=\left|\nabla^{2}f\right|_{\mathrm{HS}}^{2}+(d-1)\left|\nabla
f\right|^{2}$

and by polarization we get

$\displaystyle\Gamma_{2}(f,h)$
$\displaystyle=\langle\nabla^{2}f,\nabla^{2}h\rangle_{\mathrm{HS}}+(d-1)\nabla
f\cdot\nabla h.$

We can observe that $\Gamma_{2}(f)\geq(d-1)\Gamma(f)$, so $\mathrm{L}$
satisfies $\mathrm{CD}(d-1,\infty)$. In particular, we have that the process
$(X_{t})_{t\geq 0}$ is non-explosive and thus neither the Brownian motion on
the sphere.

Now let us compute $\Gamma_{3}$:

$\displaystyle\Gamma_{3}(f)$
$\displaystyle=\frac{1}{2}\Delta\Gamma_{2}(f)-\Gamma_{2}(f,\Delta f)$
$\displaystyle=\left(\frac{1}{2}\Delta\left|\nabla^{2}f\right|_{\mathrm{HS}}^{2}+\frac{1}{2}(d-1)\Delta\left|\nabla
f\right|^{2}\right)-\left(\langle\nabla^{2}f,\nabla^{2}(\Delta
f)\rangle_{\mathrm{HS}}+(d-1)\nabla f\cdot\nabla(\Delta f)\right).$

Using Bochner’s formula (2) and again the constant Ricci curvature, we obtain

$\frac{1}{2}(d-1)\Delta\left|\nabla
f\right|^{2}=(d-1)\left|\nabla^{2}f\right|_{\mathrm{HS}}^{2}+(d-1)\nabla
f\cdot\nabla(\Delta f)+(d-1)^{2}\left|\nabla f\right|^{2},$

so

$\displaystyle\Gamma_{3}(f)$
$\displaystyle=\frac{1}{2}\Delta\left|\nabla^{2}f\right|_{\mathrm{HS}}^{2}-\langle\nabla^{2}f,\nabla^{2}(\Delta
f)\rangle_{\mathrm{HS}}+(d-1)\left|\nabla^{2}f\right|_{\mathrm{HS}}^{2}+(d-1)^{2}\left|\nabla
f\right|^{2}$
$\displaystyle=\frac{1}{2}\Delta\left|\nabla^{2}f\right|_{\mathrm{HS}}^{2}-\langle\nabla^{2}f,\nabla^{2}(\Delta
f)\rangle_{\mathrm{HS}}+(d-1)\Gamma_{2}(f).$

Now if we choose a local system of coordinates, then we are able to use again
Bochner’s formula on each coordinate of $\nabla f$:

$\displaystyle\frac{1}{2}\Delta\left|\nabla\nabla^{i}f\right|^{2}$
$\displaystyle=\left|\nabla^{2}\nabla^{i}f\right|_{\mathrm{HS}}^{2}+\nabla\nabla^{i}f\cdot\nabla(\Delta(\nabla^{i}f))+\operatorname{Ric}(\nabla\nabla^{i}f,\nabla\nabla^{i}f)$
$\displaystyle=\left|\nabla^{2}\nabla^{i}f\right|_{\mathrm{HS}}^{2}+\nabla\nabla^{i}f\cdot\nabla\nabla^{i}(\Delta
f)+\operatorname{Ric}(\nabla\nabla^{i}f,\nabla\nabla^{i}f)$
$\displaystyle=\left|\nabla^{2}\nabla^{i}f\right|_{\mathrm{HS}}^{2}+\nabla\nabla^{i}f\cdot\nabla\nabla^{i}(\Delta
f)+(d-1)g(\nabla\nabla^{i}f,\nabla\nabla^{i}f)$
$\displaystyle\geq\left|\nabla^{2}\nabla^{i}f\right|_{\mathrm{HS}}^{2}+\nabla\nabla^{i}f\cdot\nabla\nabla^{i}(\Delta
f),$

so if we sum in $i$, we get

$\frac{1}{2}\Delta\left|\nabla^{2}f\right|_{\mathrm{HS}}^{2}\geq\left|\nabla^{3}f\right|_{\mathrm{HS}}^{2}+\langle\nabla^{2}f,\nabla^{2}(\Delta
f)\rangle,$

so

$\Gamma_{3}(f)\geq\left|\nabla^{3}f\right|_{\mathrm{HS}}^{2}+(d-1)\Gamma_{2}(f);$

thus we conclude that $\Gamma_{3}(f)\geq(d-1)\Gamma_{2}(f)$.

######  Example 3.2 (Laguerre generator on $(0,+\infty)^{d}$).

Fix $p\in(0,+\infty)^{d}$ and define on $(0,+\infty)^{d}$ the diffusion
operator

$\mathrm{L}_{p}f:=\sum_{i=1}^{d}x_{i}\frac{\partial^{2}f}{\partial
x_{i}^{2}}+\sum_{i=1}^{d}(p_{i}-x_{i})\frac{\partial f}{\partial x_{i}},$

acting on smooth functions $f\colon(0,+\infty)^{d}\to\mathbb{R}$ and let $\mu$
be the multivariate gamma law on $(0,+\infty)^{d}$; i.e.,
$d\mu=\prod_{i=1}^{d}\frac{1}{\gamma(p_{i})}x_{i}^{p_{i}-1}e^{-x_{i}}\,dx_{i}$
(here $\gamma$ denotes the usual gamma function). Following Remark 2.1, here
the function $x\mapsto(\frac{1}{x_{i}}\delta_{ij})$ defines a Riemannian
metric $g$ on $(0,+\infty)^{d}$.

Its associated diffusion process $(X_{t})_{t\geq 0}$ is non-explosive as soon
as for each $1\leq i\leq d$, $p_{i}\geq 1$ [GJY03] and on the other hand, it
is known (see for example [Kra82]) that if for each $1\leq i\leq d$,
$p_{i}\geq\frac{3}{2}$, then $\mathrm{L}_{p}$ is essentially self-adjoint on
$\mathcal{C}^{\infty}_{\mathrm{c}}((0,+\infty)^{d})$, so we will restrict our
attention towards $p_{i}\geq\frac{3}{2}$.

$\Gamma_{n}$ operators have an explicit representation on terms of the
derivatives of $f$ (see for example [LNP15]):

$\displaystyle\Gamma(f)$
$\displaystyle=\sum_{i=1}^{d}x_{i}\left(\frac{\partial f}{\partial
x_{i}}\right)^{2}$ $\displaystyle\Gamma_{2}(f)$
$\displaystyle=\sum_{i,j=1}^{d}x_{i}x_{j}\left(\frac{\partial^{2}f}{\partial
x_{i}\partial x_{j}}\right)^{2}+\sum_{i=1}^{d}x_{i}\,\frac{\partial
f}{\partial x_{i}}\,\frac{\partial^{2}f}{\partial
x_{i}^{2}}+\frac{1}{2}\sum_{i=1}^{d}(p_{i}+x_{i})\left(\frac{\partial
f}{\partial x_{i}}\right)^{2}$ $\displaystyle\Gamma_{3}(f)$
$\displaystyle=\sum_{i,j,k=1}^{d}x_{i}x_{j}x_{k}\left(\frac{\partial^{3}f}{\partial
x_{i}\partial x_{j}\partial
x_{k}}\right)^{2}+3\sum_{i,j=1}^{d}x_{i}x_{j}\,\frac{\partial^{2}f}{\partial
x_{i}\partial x_{j}}\,\frac{\partial^{3}f}{\partial x_{i}^{2}\partial x_{j}}$
$\displaystyle\,\,\,\,\,\,\,\,\,+\frac{3}{2}\sum_{i,j=1}^{d}(p_{i}+x_{i})x_{j}\left(\frac{\partial^{2}f}{\partial
x_{i}\partial
x_{j}}\right)^{2}+\frac{3}{2}\sum_{i=1}^{d}x_{i}\left(\frac{\partial^{2}f}{\partial
x_{i}^{2}}\right)^{2}$
$\displaystyle\,\,\,\,\,\,\,\,\,+\frac{3}{2}\sum_{i=1}^{d}x_{i}\,\frac{\partial
f}{\partial x_{i}}\,\frac{\partial^{2}f}{\partial
x_{i}^{2}}+\frac{1}{4}\sum_{i=1}^{d}(3p_{i}+x_{i})\left(\frac{\partial
f}{\partial x_{i}}\right)^{2}.$

As $p_{i}\geq\frac{3}{2}$, we can easily observe that
$\Gamma_{2}(f)\geq\frac{1}{2}\Gamma(f)$ and that
$\Gamma_{3}(f)\geq\frac{1}{2}\Gamma_{2}(f)$.

######  Remark 3.4.

In the special case when $d=1$ and $p=1$, then $\mu$ corresponds to the
exponential measure on $(0,+\infty)$. $\mathrm{L}_{p}$ is not essentially
self-adjoint on $\mathcal{C}^{\infty}_{\mathrm{c}}((0,+\infty))$ but it is on
the space $\mathcal{C}^{\infty}_{\mathrm{c,\,Neu}}((0,+\infty))$, defined by

$\mathcal{C}^{\infty}_{\mathrm{c,\,Neu}}((0,+\infty)):=\left\\{f\in\mathcal{C}^{\infty}_{\mathrm{c}}((0,+\infty)):\lim_{x\to
0}x^{p}e^{-x}f^{\prime}(x)=0\right\\}\subseteq\mathcal{C}^{\infty}_{\mathrm{c}}((0,+\infty)),$

which could be interpreted as imposing a Neumann boundary condition at 0 on
$(0,+\infty)$. On the other hand, it can be seen that in this case, it holds
that $\Gamma_{2}(f)\geq\frac{1}{2}\Gamma(f)$ and that
$\Gamma_{3}(f)\geq\frac{1}{2}\Gamma_{2}(f)$

Now we show a technical lemma which will be the basis of our calculations. It
corresponds to a classical result when $n=1$ [BGL14], which is used to obtain
Theorem 3.4, but for the sake of completeness we provide a full proof.

######  Lemma 3.5.

Let $n\geq 0$, $f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$, $t\geq 0$ and $x\in
M$. We define $\Lambda_{n}\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ as

$\Lambda_{n}(s):=\mathrm{P}_{s}(\Gamma_{n}(\mathrm{P}_{t-s}f))(x).$

Then for each $t\geq s$,

$\frac{d}{ds}\Lambda_{n}(s)=2\mathrm{P}_{s}(\Gamma_{n+1}(\mathrm{P}_{t-s}f))=2\Lambda_{n+1}(s).$

###### Proof.

Let us define

$\displaystyle\varphi_{1}$
$\displaystyle\colon[0,t]\to\mathcal{C}^{\infty}(M)\cap L^{\infty}(\mu),$
$\displaystyle s\mapsto\varphi_{1}(s):=\mathrm{P}_{t-s}f;$
$\displaystyle\varphi_{2}$ $\displaystyle\colon\mathcal{C}^{\infty}(M)\cap
L^{\infty}(\mu)\to\mathcal{C}^{\infty}(M)\cap L^{\infty}(\mu),$ $\displaystyle
f\mapsto\varphi_{2}(u):=\Gamma_{n}(u);$ $\displaystyle\varphi_{3}$
$\displaystyle\colon\mathbb{R}_{+}\times\mathcal{C}^{\infty}(M)\cap
L^{\infty}(\mu)\to\mathcal{C}^{\infty}(M)\cap L^{\infty}(\mu),$ $\displaystyle
f\mapsto\varphi_{3}(s,u):=\mathrm{P}_{s}u.$

We can note that $\Lambda_{n}(s)=\varphi_{3}(s,\varphi_{2}(\varphi_{1}(s)))$.
Using the chain rule we obtain

$\displaystyle\frac{d}{ds}\Lambda_{n}(s)=D_{s}\varphi_{3}(s,\Gamma_{n}(\mathrm{P}_{t-s}f))+D_{u}\varphi_{3}(s,\Gamma_{n}(\mathrm{P}_{t-s}f))D\varphi_{2}(\mathrm{P}_{t-s}f)(\varphi_{1}^{\prime}(s)).$
(18)

On the other hand, we observe that

$\displaystyle D_{s}\varphi_{3}(s,u)$ $\displaystyle=\lim_{\varepsilon\to
0}\frac{\varphi_{3}(s+\varepsilon,u)-\varphi_{3}(s,u)}{\varepsilon}=\lim_{\varepsilon\to
0}\frac{\mathrm{P}_{s+\varepsilon}u-\mathrm{P}_{s}u}{\varepsilon}=\lim_{\varepsilon\to
0}\mathrm{P}_{s}\left(\frac{\mathrm{P}_{\varepsilon}u-\mathrm{P}_{0}u}{\varepsilon}\right)$
$\displaystyle=\mathrm{P}_{s}\left(\lim_{\varepsilon\to
0}\frac{\mathrm{P}_{\varepsilon}u-\mathrm{P}_{0}u}{\varepsilon}\right)$
$\displaystyle=\mathrm{P}_{s}\mathrm{L}u.$

For the second variable, we observe that $\varphi_{3}$ is linear in its second
variable, so

$D_{u}\varphi_{3}(s,u)(h)=\varphi_{3}(s,h).$

We know that $\Gamma_{n}(f)=\Gamma_{n}(f,f)$, with $\Gamma_{n}(\cdot,\cdot)$ a
symmetric bilinear form, so

$D\varphi_{2}(u)(h)=2\Gamma_{n}(u,h).$

For $\frac{d}{ds}\varphi_{1}$, we note that

$\displaystyle\frac{d}{ds}\varphi_{1}(s)=\frac{d}{ds}\left(\varphi_{3}(t-s,f)\right)=-\partial_{s}\varphi_{3}(t-s,f)=-\mathrm{P}_{t-s}\mathrm{L}f.$

Finally, mixing all this into (18), we get

$\displaystyle\frac{d}{ds}\Lambda_{n}(s)$
$\displaystyle=\mathrm{P}_{s}(\mathrm{L}(\Gamma_{n}(\mathrm{P}_{t-s}f)))-2\mathrm{P}_{s}(\Gamma_{n}(\mathrm{P}_{t-s}f,\mathrm{P}_{t-s}\mathrm{L}f))=\mathrm{P}_{s}(\Gamma_{n+1}(\mathrm{P}_{t-s}f)).\
$

∎

Now the following proposition is just a generalization of one of the
consequences of the curvature condition $\mathrm{CD}(\rho_{1},\infty)$,
reflected in Theorem 3.4, in the context of higher iterations of
$\Gamma$-calculus.

######  Proposition 3.6.

Let $n\geq 0$ and suppose that there exists a constant $\rho_{n}\in\mathbb{R}$
such that for all $f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$,
$\Gamma_{n+1}(f)\geq\rho_{n}\Gamma_{n}(f)$. Then for each $t\geq 0$,

$\Gamma_{n}(\mathrm{P}_{t}f)\leq
e^{-2\rho_{n}t}\mathrm{P}_{t}(\Gamma_{n}(f)).$

###### Proof.

Applying Lemma 3.5 and imposing $\Gamma_{n+1}\geq\rho_{n}\Gamma_{n}$, we get

$\displaystyle\frac{d}{ds}\Lambda_{n}(s)=2\mathrm{P}_{s}(\Gamma_{n+1}(\mathrm{P}_{t-s}f))\geq
2\rho_{n}\mathrm{P}_{s}(\Gamma_{n}(\mathrm{P}_{t-s}f))=\Lambda_{n}(s).$

Then by Grönwall’s inequality, we obtain

$\Lambda_{n}(0)\leq e^{-2\rho_{n}t}\Gamma_{n}(t),$

so

$\Gamma_{n}(\mathrm{P}_{t}f)\leq
e^{-2\rho_{n}t}\mathrm{P}_{t}(\Gamma_{n}(f)).$

∎

Now we present another bound.

######  Proposition 3.7.

Let $n\geq 1$. Then for each $f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$ and
for all $t>0$, we have that

$\Gamma_{n}(\mathrm{P}_{t}f)\leq\frac{1}{2t}\mathrm{P}_{t}(\Gamma_{n-1}(f)).$

###### Proof.

Using Lemma 3.5, we get

$\Lambda_{n}^{\prime}(s)=2\mathrm{P}_{s}(\Gamma_{n+1}(\mathrm{P}_{t+s}f))\geq
0,$

so $\Lambda_{n}$ is non-decreasing. Then, for each $t\geq s\geq 0$, we infer
that

$\Gamma_{n}(\mathrm{P}_{t}f)=\mathrm{P}_{0}(\Gamma_{n}(\mathrm{P}_{t}f))=\Lambda_{n}(0)\leq\Lambda_{n}(s)=\mathrm{P}_{s}(\Gamma_{n}(\mathrm{P}_{t-s}f)).$

Thus integrating and applying again Lemma 3.5, we see that

$\displaystyle t\Gamma_{n}(\mathrm{P}_{t}f)$
$\displaystyle=\int_{0}^{t}\Gamma_{n}(\mathrm{P}_{t}f)\,ds\leq\int_{0}^{t}\mathrm{P}_{s}(\Gamma_{n}(\mathrm{P}_{t-s})f)\,ds=\frac{1}{2}\int_{0}^{t}\Lambda_{n-1}^{\prime}(s)\,ds$
$\displaystyle=\frac{1}{2}\left(\Lambda_{n-1}(t)-\Lambda_{n-1}(0)\right)$
$\displaystyle=\frac{1}{2}\left(\mathrm{P}_{t}(\Gamma_{n-1}(f))-\Gamma_{n-1}(\mathrm{P}_{t}f)\right),$

from where we get

$\Gamma_{n}(\mathrm{P}_{t}f)\leq\frac{1}{2t}\left(\mathrm{P}_{t}(\Gamma_{n-1}(f))-\Gamma_{n-1}(\mathrm{P}_{t}f)\right)\leq\frac{1}{2t}\mathrm{P}_{t}(\Gamma_{n-1}(f)).$

∎

By mixing the two previous propositions, we deduce the following inequality.

######  Proposition 3.8.

Let $n\geq 1$ and suppose that there exists a constant $\rho_{n}\in\mathbb{R}$
such that for each $f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$,
$\Gamma_{n+1}(f)\geq\rho_{n}\Gamma_{n}(f)$. Then for all $t>0$,

$\Gamma_{n}(\mathrm{P}_{t}f)\leq\frac{1}{t}e^{-\rho_{n}t}\mathrm{P}_{t}(\Gamma_{n-1}f).$

###### Proof.

Let $t>0$. By Proposition 3.6:

$\Gamma_{n}(\mathrm{P}_{t}f)=\Gamma_{n}(\mathrm{P}_{t/2}(\mathrm{P}_{t/2}f))\leq
e^{-\rho_{n}t}\mathrm{P}_{t/2}(\Gamma_{n}(\mathrm{P}_{t/2}f)).$ (19)

Then, applying Proposition 3.7, we know that

$\Gamma_{n}(\mathrm{P}_{t/2}f)\leq\frac{1}{t}\mathrm{P}_{t/2}(\Gamma_{n-1}(f)).$
(20)

Therefore, mixing up inequalities (19) and (20), we finally obtain

$\Gamma_{n}(\mathrm{P}_{t}f)\leq\frac{1}{t}e^{-\rho_{n}t}\mathrm{P}_{t}(\Gamma_{n-1}f).$

∎

##  4\. The diffusion transport map on smooth manifolds

In this section we prove the main result of this note, Theorem 4.5, which
employs Kim and Milman’s heat flow transport map [KM12] on a weighted manifold
setting to obtain a Lipschitz transport map between the prescripted measure on
the manifold and a log-Lipschitz perturbation of the former.

Firstly, the construction will be detailed, taking care with the proper
details that must be kept with attention when passing from the Euclidean
setting towards the manifold one. After that, using the machinery of
$\Gamma$-calculus exhibited in Section 3, we prove that the heat flow
transport map is Lipschitz, giving an explicit bound on its Lipschitz
constant.

Here the context will always be given by a complete and connected Riemannian
manifold $(M,g)$ with weight $d\mu=e^{-W}\,d\mathrm{Vol}$ and the natural
diffusion operator $\mathrm{L}$ given by $\mathrm{L}=\Delta-\nabla
W\cdot\nabla$, unless otherwise stated. On the other hand, we assume that
there exists constants $\rho_{1},\rho_{2}>0$ such that for any
$f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$,
$\Gamma_{2}(f)\geq\rho_{1}\Gamma(f)$ (i.e., $\mathrm{CD}(\rho_{1},\infty)$)
and $\Gamma_{3}(f)\geq\rho_{2}\Gamma_{2}(f)$.

We emphasize that under these assumptions, $\mathrm{L}$ is essentially self-
adjoint on $\mathcal{C}^{\infty}_{\mathrm{c}}(M)$ and its associated semigroup
is conservative (or equivalently, that the associated Markov process does not
explode.)

###  4.1. Construction of the transport map

Let $V\colon M\to\mathbb{R}$ be a smooth function which is $K$-Lipschitz for
the metric $d_{g}$; so, in the light of Proposition 3.1, $V$ is such that
$\sqrt{\Gamma_{1}(V)}\leq K$ uniformly on $M$. Let us define $f=e^{-V}$. The
following result justifies the integrability of $f$ with respect to $\mu$
under $\mathrm{CD}(\rho,\infty)$, with $\rho>0$; it corresponds to Herbst’s
argument111This unpublished result was developed by Ira Herbst in a letter
addressed to Leonard Gross: https://perso.math.univ-
toulouse.fr/ledoux/files/2018/10/Herbst.pdf (see for example Proposition 5.4.1
in [BGL14].)

######  Lemma 4.1.

Let $V\colon M\to\mathbb{R}$ be a smooth function such that
$\sqrt{\Gamma_{1}(V)}\leq K$. If there exists $\rho>0$ such that
$\mathrm{CD}(\rho,\infty)$ holds, then both $V$ and $e^{sV}$ are
$\mu$-integrable, for any $s\in\mathbb{R}$. In particular, $e^{-V}$ is in
$L^{p}(\mu)$, for each $p\geq 1$.

Without loss of generality we will assume
$\int_{M}f\,d\mu=\int_{M}e^{-V}\,d\mu=1$ so we can define the probability
measure $\nu$ on $M$ as $d\nu:=f\,d\mu$. For each $t\geq 0$, set
$d\rho_{t}=\mathrm{P}_{t}f\,d\mu$, which is a probability measure, as $\mu$ is
invariant for $(\mathrm{P}_{t})_{t\geq 0}$. In particular, we have that the
flow $(\rho_{t})_{t\geq 0}$ is such that $\rho_{0}=\nu$ and
$\rho_{\infty}:=\lim_{t\to+\infty}\rho_{t}=\mu$ in distribution. Indeed,
$f\in\mathrm{L}^{2}(\mu)$ (because of Lemma 4.1) and we have that $\mathrm{L}$
is ergodic and its semigroup is conservative, then $\mathrm{P}_{t}f\to 1$ in
$L^{2}(\mu)$, which yields the desired convergence.

Now let us define, for each $t\geq 0$ and $x\in M$,
$V_{t}(x):=-\log\mathrm{P}_{t}f(x)$, so that $\nabla V_{t}$ is a vector field
for each $t\geq 0$. The following proposition shows that actually that
$V_{\bullet}$ and $(\rho_{t})_{t\geq 0}$ are connected via the continuity
equation.

######  Proposition 4.2.

The flow $(\rho_{t})_{t\geq 0}$ satisfies the continuity equation

$\left\\{\begin{aligned} \partial_{t}\rho_{t}+\nabla\cdot(\rho_{t}\nabla
V_{t})&=0,\quad t>0\\\ \rho_{0}&=\nu\end{aligned}\right.$

in the distributional sense.

###### Proof.

Let $t>0$ and $\varphi\in\mathcal{C}^{\infty}_{\mathrm{c}}(M)$. We shall prove
that

$\frac{d}{dt}\int_{M}\varphi\,d\rho_{t}=\int_{M}\nabla\varphi\cdot\nabla
V_{t}\,d\rho_{t}.$

Indeed, as $(\rho_{t})_{t\geq 0}$ is solution of the SDE (10), then it
satisfies Fokker-Planck’s equation (11) in the distributional sense, hence we
can write

$\displaystyle\frac{d}{dt}\int_{M}\varphi\,d\rho_{t}$
$\displaystyle=\int_{M}\left(\Delta\varphi-\nabla
W\cdot\nabla\varphi\right)\,d\rho_{t}=\int_{M}\Delta\varphi
e^{-W}\mathrm{P}_{t}f\,d\mathrm{Vol}-\int_{M}\nabla
W\cdot\nabla\varphi\,d\rho_{t}$
$\displaystyle=-\int_{M}\nabla\varphi\cdot\nabla(e^{-W}\mathrm{P}_{t}f)\,d\mathrm{Vol}-\int_{M}\nabla
W\cdot\nabla\varphi\,d\rho_{t}$
$\displaystyle=-\int_{M}\nabla\varphi\cdot\nabla\mathrm{P}_{t}f\,d\mu=-\int_{M}\nabla\varphi\cdot\frac{\nabla\mathrm{P}_{t}f}{\mathrm{P}_{t}f}\,d\rho_{t}=\int_{M}\nabla\varphi\cdot\nabla
V_{t}\,d\rho_{t},$

which was the desired equality. ∎

Now let us consider the flow of diffeomorphisms $(S_{t})_{t\geq 0}$ induced by
the vector field $\nabla V_{t}$:

$\left\\{\begin{aligned} \frac{d}{dt}S_{t}(x)&=\nabla V_{t}(S_{t}(x)),\quad
t>0\\\ S_{0}(x)&=x.\end{aligned}\right.$ (21)

Our goal will be to characterize the flow $(\rho_{t})_{t\geq 0}$ in terms of
$(S_{t})_{t\geq 0}$ using Theorem 2.2. First of all, it is clear that $\nabla
V_{\bullet}$ is smooth (thus locally Lipschitz) as the semigroup
$(\mathrm{P}_{t})_{t\geq 0}$ preserves smoothness as $\mathrm{L}$ is an
elliptic diffusion operator. It is also clear that $t\mapsto\rho_{t}$ is
continuous for the weak topology. Finally, we just have to justify the
integrability condition (14), which will hold under the curvature-dimension
condition $\mathrm{CD}(\rho_{1},\infty)$.

######  Lemma 4.3.

Let $V_{\bullet}$ be defined as above. If there exists $\rho_{1}>0$ such that
$\mathrm{CD}(\rho_{1},\infty)$ holds, then

$\int_{0}^{+\infty}\int_{M}\left|\nabla
V_{t}(x)\right|\,d\rho_{t}\,dt<+\infty.$

###### Proof.

Under $\mathrm{CD}(\rho_{1},\infty)$, then using the local Poincaré inequality
(item (iii) in Theorem 3.4) we have that for each $t\geq 0$,

$\sqrt{\Gamma(\mathrm{P}_{t}f)}\leq
e^{-\rho_{1}t}\mathrm{P}_{t}(\sqrt{\Gamma(f)}).$

If we develop $\mathrm{P}_{t}(\sqrt{\Gamma(f)})$, we obtain

$\mathrm{P}_{t}(\sqrt{\Gamma(f)})=\mathrm{P}_{t}(\left|\nabla
f\right|)=\mathrm{P}_{t}(\left|\nabla V\right|f).$

As $V$ is a $K$-Lipschitz potential, then by definition we have that
$\left|\nabla V\right|\leq K$; thus we finally get

$\left|\nabla\mathrm{P}_{t}f\right|=\sqrt{\Gamma(\mathrm{P}_{t}f)}\leq
e^{-\rho_{1}t}K\mathrm{P}_{t}f,$

so it follows that $\left|\nabla V_{t}(x)\right|\leq Ke^{-\rho_{1}t}$. Then

$\int_{0}^{+\infty}\int_{M}\left|\nabla V_{t}(x)\right|\,d\rho_{t}\,dt\leq
K\int_{0}^{+\infty}e^{-\rho_{1}t}\,dt<+\infty,$

as $\rho_{1}>0$. ∎

As we reunited the hypotheses of Theorem 2.2, we therefore deduce that for
each $t\geq 0$, ${S_{t}}_{\\#}\nu=\rho_{t}$. For each $t\geq 0$, let
$T_{t}:=S_{t}^{-1}$, so ${T_{t}}_{\\#}\rho_{t}=\nu$.

If we suppose that for each $t\geq 0$ there exists a constant $K_{t}>0$ such
that $T_{t}$ is $K_{t}$-Lipschitz, and that
$K:=\limsup_{t\to+\infty}K_{t}<+\infty$, then Lemma 1 in [MS22] states that
(modulo subsequence) for each $x\in M$, the limit $\lim_{t\to+\infty}T_{t}(x)$
exists and moreover, it defines a $K$-Lipschitz mapping $T\colon M\to M$ such
that $T_{\\#}\mu=\nu$, as a consequence of Arzelà-Ascoli’s theorem. Now the
following lemma (Proposition 1 in [FMS23] and Lemma 2.1 in [Nee22]) exhibits a
condition on $V_{t}$ to ensure that the mapping $T$ is Lipschitz, giving an
estimate on its Lipschitz constant.

######  Lemma 4.4.

If there exists $\lambda\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ integrable such
that for all $t\geq 0$ and $x\in M$,

$-\nabla^{2}V_{t}(x)\preceq\lambda(t)g_{x},$

then $T$ is $\exp(\int_{0}^{+\infty}\lambda(t)\,dt)$-Lipschitz.

Now we are ready to state the main result in this section and in this note.

######  Theorem 4.5.

Let $(M,g)$ be a complete and connected $d$-dimensional Riemannian manifold.
Let $W\colon M\to\mathbb{R}$ be smooth and such that
$\int_{M}e^{-W}d\mathrm{Vol}=1$. Consider the diffusion operator
$\mathrm{L}=\Delta-\nabla W\cdot\nabla$. Let us assume that there exist
constants $\rho_{1},\rho_{2}>0$ such that

  1. (i)

$\forall
f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M),\Gamma_{2}(f)\geq\rho_{1}\Gamma(f)$;
and

  2. (ii)

$\forall
f\in\mathcal{C}^{\infty}_{\mathrm{c}}(M),\Gamma_{3}(f)\geq\rho_{2}\Gamma_{2}(f)$.

Let $V\colon M\to\mathbb{R}$ smooth and $K$-Lipschitz and define
$d\mu=e^{-W}d\mathrm{Vol}$ and $d\nu=e^{-V}d\mu$. Then there exists a
Lipschitz mapping $T\colon M\to M$ pushing forward $\mu$ towards $\nu$ which
is
$\exp\left(\sqrt{\frac{2\pi}{\rho_{2}}}Ke^{\frac{K^{2}}{2\rho_{1}}}\right)$-Lipschitz.

Before proceeding with the proof of the theorem, we will make a few remarks.

######  Remark 4.1.

  * •

First of all, we observe that the estimate for the Lipschitz constant given by
Theorem 4.5 does not depend on the dimension $d$ of the manifold $(M,g)$.

  * •

Theorem 5 in [FMS23] gives another estimate for the Lipschitz constant of $T$,
in the manifold context as well. The assumptions of that result are similar to
those of Theorem 4.5, but instead of supposing that
$\Gamma_{3}\geq\rho_{2}\Gamma_{2}$, it is assumed a uniform bound on $R$, the
Riemann tensor of curvature (which fully characterizes curvature in
manifolds.) The estimates are of the same type, in the sense that both are of
the form $O\left(\exp(\exp(K^{2}))\right)$, with $K$ the Lipschitz constant of
the potential $V$, but the assumptions are different.

Before starting with the proof of Theorem 4.5, we recall the following lemma,
which again is a consequence of Herbst’s argument (see [Mil23, p. 298].)

######  Lemma 4.6.

Let $(E,\delta,m)$ be a metric measure space and suppose that $m$ is a
probability measure. If $m$ satisfies a log-Sobolev inequality with constant
$\lambda_{\mathrm{LS}}>0$, then for each $g\colon E\to\mathbb{R}$
$C$-Lipschitz, and any $-\infty<q<p<+\infty$,

$\left\lVert
e^{g}\right\rVert_{L^{p}(m)}\leq\exp\left(C^{2}\frac{(p-q)}{2\lambda_{\mathrm{LS}}}\right)\left\lVert
e^{g}\right\rVert_{L^{q}(m)}.$

###### Proof (of Theorem 4.5).

In view of what has been discussed in this section, and more particularly, in
the light of Lemma 4.4, we only have to prove that we can find an integrable
bound $\lambda$ for $\nabla^{2}\log\mathrm{P}_{t}f$; that is,
$\nabla^{2}\log\mathrm{P}_{t}f(x)\preceq\lambda(t)g_{x}$. We note that for
each $t>0$, $x\in M$ and $Y\in T_{x}M$ with $\left|Y\right|=1$:

$\displaystyle\nabla^{2}\log\mathrm{P}_{t}f(x)(Y,Y)$
$\displaystyle=\frac{\nabla^{2}\mathrm{P}_{t}f(x)(Y,Y)}{\mathrm{P}_{t}f(x)}-\left|\nabla\log\mathrm{P}_{t}f(x)\cdot
Y\right|^{2}$
$\displaystyle\leq\frac{\nabla^{2}\mathrm{P}_{t}f(x)(Y,Y)}{\mathrm{P}_{t}f(x)}$
$\displaystyle\leq\frac{\left|\nabla^{2}\mathrm{P}_{t}f(x)\right|_{\mathrm{HS}}}{\mathrm{P}_{t}f(x)};$

that is,

$\nabla^{2}\log\mathrm{P}_{t}f(x)(Y,Y)\leq\frac{\left|\nabla^{2}\mathrm{P}_{t}f(x)\right|_{\mathrm{HS}}}{\mathrm{P}_{t}f(x)}.$
(22)

On the one hand, in the light of Bochner’s formula (or more precisely (16)),
we note that
$\left|\nabla^{2}\mathrm{P}_{t}f\right|_{\mathrm{HS}}\leq\sqrt{\Gamma_{2}(\nabla^{2}\mathrm{P}_{t}f)}$.
On the other hand, by Proposition 3.8, we have that

$\Gamma_{2}(\mathrm{P}_{t}f)\leq\frac{1}{t}e^{-\rho_{2}t}\mathrm{P}_{t}(\Gamma_{1}(f));$

thus

$\left|\nabla^{2}\mathrm{P}_{t}f\right|_{\mathrm{HS}}^{2}\leq\frac{1}{t}e^{-\rho_{2}t}\mathrm{P}_{t}(\Gamma_{1}(f)).$
(23)

Now, let us note that

$\displaystyle\mathrm{P}_{t}(\Gamma_{1}(f))$
$\displaystyle=\mathrm{P}_{t}(\left|\nabla
f\right|^{2})=\mathrm{P}_{t}(\left|f\nabla V\right|^{2})\leq
K^{2}\mathrm{P}_{t}(f^{2})\leq
K^{2}\exp\left(K^{2}\frac{1-e^{-2\rho_{1}t}}{\rho_{1}}\right)(\mathrm{P}_{t}f)^{2}$
$\displaystyle\leq K^{2}\exp\left(\frac{M^{2}}{\rho_{1}}\right)$

where we employed Lemma 4.6, for $p=2,q=1$ and the measure
$m=\mathrm{P}_{t}^{*}\delta_{x}$, as it satisfies a log-Sobolev inequality
with constant $\lambda_{\mathrm{LS}}=\frac{1-e^{-2\rho_{1}t}}{\rho_{1}}$
(condition (iv) in Theorem 3.4.); so if we mix this with both (22) and (23),
then we finally get

$\displaystyle\nabla^{2}\log\mathrm{P}_{t}f(x)(Y,Y)\leq
M\sqrt{\beta}e^{\frac{M^{2}}{2\rho_{1}}}\frac{1}{\sqrt{t}}e^{-\frac{1}{2}\rho_{2}t},$

hence the Lipschitz estimate follows from the fact that

$\displaystyle\int_{0}^{+\infty}\frac{1}{\sqrt{t}}e^{-\frac{1}{2}\rho_{2}t}\,dt=\sqrt{\frac{2}{\rho_{2}}}\int_{0}^{+\infty}\frac{1}{\sqrt{s}}e^{-s}\,ds=\sqrt{\frac{2\pi}{\rho_{2}}}.$

∎

##  5\. Applications

In this section we give different applications of Theorem 4.5.

###  5.1. Transfer of functional inequalities

As it is well known, having a Lipschitz map pushing forward a measure towards
a log-Lipschitz perturbation of it allows us to transport functional
inequalities from the source measure to the image one, as we did in the very
beginning of the article in equation (1), transporting Gaussian Poincaré’s
inequality. In particular, we recall Theorem 3.3 which says that under
$\mathrm{CD}(\rho,\infty)$ (for $\rho>0$), then the measure $\mu$ satisfies
both log-Sobolev and Poincaré’s inequalities (both with constant
$\frac{1}{\rho}$). More precisely, we have the following corollary.

######  Corollary 5.1.

Let $\mathrm{L}$ be a diffusion operator on $(M,g)$ with symmetric and ergodic
measure $\mu$ satisfying the hypotheses of Theorem 4.5. Then any $K$ log-
Lipschitz perturbation of the measure $\mu$ satisfies both log-Sobolev and
Poincaré’s inequalities with constant $\frac{C_{K}}{\rho}$, where
$C_{K}=\exp\left(2\sqrt{\frac{2\pi}{\rho_{2}}}Ke^{\frac{K^{2}}{2\rho_{1}}}\right)$.

######  Remark 5.1.

Transfer of functional inequalities for perturbations is a kind of result
which has been present in literature for a long time (see for example [AS94],
on the preservation of log-Sobolev’s inequality for perturbations.)
Nevertheless, our method allows us to transfer not just inequalities in the
form of Poincaré or log-Sobolev but isoperimetric and concentration
inequalities.

###  5.2. The sphere

In order to give a more concrete application, we use Theorem 4.5 in the
context of the sphere $\mathbb{S}^{d}$. If we consider $\mathbb{S}^{d}$ as a
weighted manifold with $\mu=\mathrm{Vol}$ and $\mathrm{L}=\Delta$, the
Laplace-Beltrami operator on $\mathbb{S}^{d}$, then following Example 3.1, we
have that $\Gamma_{2}\geq(d-1)\Gamma$ and $\Gamma_{3}\geq(d-1)\Gamma_{2}$, so
applying Theorem 4.5, we obtain that each $V\colon\mathbb{S}^{d}\to\mathbb{R}$
1-Lipschitz, then there exists a Lipschitz mapping $T$ pushing forward
$\mathrm{Vol}$ towards $e^{-V}\,\mathrm{Vol}$ with Lipschitz constant
$\exp\left(\sqrt{\frac{2\pi}{d-1}}e^{\frac{1}{2(d-1)}}\right)$.

Let us note that if we take $\mathbb{S}^{d}(\sqrt{d})$, the $d$-sphere of
radius $\sqrt{d}$, and we set $\mu_{d}$ as the uniform measure on it, then its
Ricci tensor is given by
$\operatorname{Ric}_{\mathbb{S}^{d}(\sqrt{d})}=\frac{d-1}{d}g_{\mathbb{S}^{d}(\sqrt{d})}$,
so it will satisfy $\Gamma_{2}\geq\frac{d-1}{d}\Gamma$ and
$\Gamma_{3}\geq\frac{d-1}{d}\Gamma_{2}$; thus we can apply Theorem 4.5 to get
a transport map with Lipschitz constant
$\exp\left(\sqrt{2\pi\frac{d-1}{d}}e^{\frac{1}{2\frac{d-1}{d}}}\right)$. Let
us remark that asymptotically the Lipschitz constant converges to
$\exp\left(\sqrt{2\pi}e^{\frac{1}{2}}\right)$ as $d\to+\infty$.

######  Remark 5.2.

It is clear that the Lipschitz constant obtained in this example is worser
than the specific one provided in Theorem 3 of [FMS23], which stresses the
constant curvature of $\mathbb{S}^{d}$. Nevertheless, we think that it
illustrates the applicability of Theorem 4.5; i.e., a diffusion generator
$\mathrm{L}$ on a manifold satisfying $\Gamma_{2}\geq\rho_{1}\Gamma$ and
$\Gamma_{3}\geq\rho_{2}\Gamma_{2}$ at the same time.

###  5.3. Laguerre generator

As we saw in Example 3.2, for any $p\in(0,+\infty)^{d}$ such that for all
$1\leq i\leq d$, $p_{i}\geq\frac{3}{2}$, then the Laguerre operator
$\mathrm{L}_{p}$ is essentially self-adjoint and its associated semigroup
conservative, so we may still apply Theorem 4.5. Thus we have a globally
Lipschitz map pushing forward the multivariate gamma distribution towards log-
Lipschitz perturbations of it.

For the particular case when $d=1$ and $p=1$, where the ergodic measure $\mu$
corresponds to the exponential distribution on $(0,+\infty)$, following Remark
3.4 we have that $\mathrm{L}_{p}$ is essentially self-adjoint with respect to
the class of functions $\mathcal{C}^{\infty}_{\mathrm{c,\,Neu}}((0,+\infty))$,
but this makes no difference in the arguments, so we may apply as well Theorem
4.5.

In the latter case, the operator $-\mathrm{L}_{1}$ has a discrete spectrum
equal to $\mathbb{N}$; thus it satisfies Poincaré’s inequality (with a better
constant than the one given by $\mathrm{CD}(\frac{1}{2},\infty)$): for each
$f\colon(0,+\infty)\to\mathbb{R}$ smooth,

$\operatorname{Var}_{\mu_{1}}(f)\leq\int_{\mathbb{R}_{+}}x(f^{\prime})^{2}\,d\mu_{1}.$

In particular, in the light of Corollary 5.1, it is possible to transfer this
inequality for any log-Lipschitz (for the metric generated by the Laguerre
generator $\mathrm{L}_{1}$) perturbation of the exponential measure.

######  Remark 5.3.

For the exponential measure there is a very rich theory in terms of the study
of functional inequalities (see for example [BL97] and [BK08]), but our
theorem cannot be linked with those results, as those employ the carré du
champ operator given by $f\mapsto(f^{\prime})^{2}$ instead of the natural one
for Laguerre’s generator given by $f\mapsto x(f^{\prime})^{2}$.

It is known that in dimension one, the diffusion transport map $T$ coincides
with the Monge map. The following result provides a new estimate on the growth
of Monge’s map for $d=1$ and $p\geq\frac{3}{2}$ or $p=\frac{1}{2}$ (i.e. when
the ergodic measure corresponds to the exponential measure.)

######  Proposition 5.2.

Let $\mu_{p}$ be the multivariate gamma distribution on $(0,+\infty)$ and let
$V\colon(0,+\infty)\to\mathbb{R}$ be a Lipschitz potential (for the metric
$x\mapsto\frac{1}{x}$) and let $T$ be the Monge map pushing forward $\mu_{p}$
towards $e^{-V}\mu_{p}$. Then there exists a constant $C>0$ such that for any
$x>0$,

$\left|T^{\prime}(x)\right|_{\mathbb{R}}\leq C\sqrt{x},$

where $\left|\cdot\right|_{\mathbb{R}}$ denotes the absolute value function on
$\mathbb{R}$.

###### Proof.

Let us denote by $g$ the metric $x\mapsto\frac{1}{x}$. Using Theorem 4.5, we
deduce the existence of a constant $C>0$ such that $\sup_{x>0}\left\lVert
T^{\prime}(x)\right\rVert_{\mathrm{op}}\leq C$. Therefore for each $x>0$,

$\displaystyle C\geq\left\lVert
T^{\prime}(x)\right\rVert_{\mathrm{op}}=\sup_{\left|v\right|_{g}=1}\left|T^{\prime}(x)\cdot
v\right|_{g}=\left|T^{\prime}(x)\right|_{g}=\sqrt{\frac{1}{x}(T^{\prime}(x))^{2}}=\frac{\left|T^{\prime}(x)\right|_{\mathbb{R}}}{\sqrt{x}},$

yielding the desired result. ∎

######  Remark 5.4.

Theorem 1.3 in [CF21] provides a growth estimate for the derivative of the
Monge map pushing forward the Gaussian measure $\gamma_{d}$ onto a log-concave
measure on $\mathbb{R}^{d}$ which moreover is log-Lipschitz (satisfying
certain technical bounds for the Hessian of its log-density): more precisely,
it states that $\left\lVert\nabla
T(x)\right\rVert_{\mathrm{op}}=O(1+\left|x\right|^{2})$ as
$\left|x\right|\to\infty$. We observe that in the exponential case, when we
pushforward the measure towards a log-Lipschitz perturbation, then the growth
of the derivative of Monge’s map is $O(\sqrt{x})$ as $x\to\infty$.

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