# Lipschitz Transport Maps via the Föllmer Flow

Yin Dai  ,  Yuan Gao  ,  Jian Huang  ,  Yuling Jiao  ,  Lican Kang  and  Jin
Liu  School of Mathematics and Statistics, Wuhan University, Wuhan, 430072,
China  Department of Applied Mathematics, The Hong Kong Polytechnic
University, Hong Kong, China  Department of Applied Mathematics, The Hong Kong
Polytechnic University, Hong Kong, China  School of Mathematics and
Statistics, Wuhan University, Wuhan, 430072, China<EMAIL_ADDRESS>School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China
School of Data Science, Chinese University of Hong Kong, Shenzhen, China

###### Abstract.

Inspired by the construction of the Föllmer process [42], we construct a unit-
time flow on the Euclidean space, termed the Föllmer flow, whose flow map at
time 1 pushes forward a standard Gaussian measure onto a general target
measure. We study the well-posedness of the Föllmer flow and establish the
Lipschitz property of the flow map at time 1. We apply the Lipschitz mapping
to several rich classes of probability measures on deriving dimension-free
functional inequalities and concentration inequalities for the empirical
measure.

###### Key words and phrases:

Lipschitz transport maps, functional inequalities, empirical measures,
Gaussian mixtures.

‡ Authors are listed in alphabetical order.

##  1\. Introduction

Functional inequalities, such as Poincaré-type and Sobolev-type inequalities,
are fundamental tools in studying sampling algorithms [24, 62, 58, 25, 63, 19,
5, 6], stochastic optimization [70, 81, 49, 61], and score-based generative
modeling [10, 55, 53, 80] in machine learning, statistics, and applied
probability. Establishing functional inequalities with dimension-free
constants has attracted widespread attention in various fields of mathematics
like probability, geometry, and analysis [78, 9, 29, 21, 19, 56, 22, 51, 46,
28, 50]. The seminal work [7] introduces the Bakry-Émery criterion to verify
dimension-free Poincaré and log-Sobolev inequalities for strongly log-concave
measures. Holley-Stroock perturbation principle [45] implies any bounded
perturbation of a strongly log-concave measure satisfies dimension-free
Poincaré and log-Sobolev inequalities. Up to now, it remains an attractive
topic to explore general classes of probability measures that satisfy
dimension-free functional inequalities.

One of the basic principles for proving functional inequalities regarding
probability measures on the Euclidean space is Lipschitz changes of variables
between a source and a target probability measures [17, 69, 11, 48, 27, 40,
52, 65, 66, 67, 23, 75]. For instance, provided that $\mu$ and $\nu$ are Borel
probability measures defined on $\mathbb{R}^{d}$, we would seek a Lipschitz
transport map $\varphi:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ such that
$\nu$ can be represented as a push-forward measure under $\varphi$, namely
$\nu=\mu\circ\varphi^{-1}$. $\mu$ is transported onto $\nu$ in the sense that
for every Borel set $B\subseteq\mathbb{R}^{d}$, $\nu(B)=\mu(\varphi^{-1}(B))$.
The Lipschitz nature of transport map $\varphi$ plays a profound impact on
transferring desirable analytic results from the source measure $\mu$ to the
target measure $\nu$. However, existence of such Lipschitz transport maps is
not guaranteed unless proper convexity restrictions are placed on the
measures. Our main goal is to extend quantitative Lipschitz regularity of
transport maps to measures that are not necessarily strongly log-concave.

Let us first recall the celebrated Caffarelli’s contraction theorem [17,
Theorem 2]. Let $\mu(\mathrm{d}x)=\exp(-U(x))\mathrm{d}x$ and
$\nu(\mathrm{d}x)=\exp(-W(x))\mathrm{d}x$ be two probability measures defined
on $\mathbb{R}^{d}$ with $U,W\in C^{2}({\mathbb{R}}^{d})$. Suppose that
$\nabla^{2}U(x)\preceq\beta\mathbf{I}_{d}$ and
$\nabla^{2}W(x)\succeq\alpha\mathbf{I}_{d}\succ 0$. Then the optimal transport
map $\varphi_{\mathrm{opt}}:=\nabla\psi$ from $\mu$ to $\nu$ is
$\sqrt{\beta/\alpha}$-Lipschitz, where
$\psi:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is the convex Brenier potential.
The optimal transport map
$\varphi_{\mathrm{opt}}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ pushes
forward $\mu$ onto $\nu$ in the sense that
$\nu=\mu\circ(\varphi_{\mathrm{opt}})^{-1}$. In particular, if $\gamma_{d}$ is
the standard Gaussian measure on $\mathbb{R}^{d}$ and probability measure
$\mu$ has a log-concave density with respect to $\gamma_{d}$, then there
exists a 1-Lipschitz map $\varphi_{\mathrm{opt}}$ such that
$\nu=\gamma_{d}\circ\left(\varphi_{\mathrm{opt}}\right)^{-1}$. The 1-Lipschitz
transport map $\varphi_{\mathrm{opt}}$ enables dimension-free functional
inequalities to be transferred from $\gamma_{d}$ to $\nu$. Recently, [65]
defines a Brownian transport map, based on the Föllmer process defined in
Definition 3.3, that transports the infinite-dimensional Wiener measure onto
probability measures on the Euclidean space. Lipschitz properties of the
Brownian transport map are investigated extensively while no analogous results
for optimal transport maps are known. [67] and [66] also utilize a Lipschitz
transport map along the reverse heat flow, which previously appears in [69]
and is further studied by [48], to establish functional inequalities, perform
eigenvalues comparisons, and study domination of distribution functions. On
the Caffarelli’s contraction theorem, [40] and [23] provide new proofs using
the entropic interpolation between the source and the target measures.

In this work, we construct a flow over the unit time interval on the Euclidean
space, named the Föllmer flow as in Definition 2.3 and Theorem 2.4. Our
construction is greatly enlightened by Föllmer’s derivation of the Föllmer
process. Then we define and analyze a new transport map, along the Föllmer
flow, which pushes forward the standard Gaussian measure to a general measure
satisfying mild regularity assumptions (see Assumptions 1, 2 and 3). The well-
posedness of the Föllmer flow and the Lipschitz property of its flow map at
time 1 are rigorously investigated under these regularity assumptions. By
virtue of the Lipschitz changes of variables principle, we prove dimension-
free $\Psi$-Sobolev inequalities, isoperimetric inequalities, $q$-Poincaré
inequalities and sharp non-asymptotic concentration bounds for the empirical
measure. Furthermore, we shall emphasize that both the Föllmer flow and its
flow map possess much computational flexibility in terms of the analytic
expression of its velocity field, which we believe may be of independent
interest, to develop sampling algorithms and generative models with
theoretical guarantees.

###  1.1. Related work

The work is notably relevant to the Brownian transport map built upon the
Föllmer process [65] and the transport map defined via the reverse heat flow
[69, 48, 67, 66]. The Brownian transport map, acquired from a strong solution
of the Föllmer process, pushes forward the Wiener measure onto probability
measures on the Euclidean space. The infinite-dimensional nature of the
Brownian transport map is quite different from that of the Föllmer flow which
is defined on the finite-dimensional Euclidean space. To produce the
randomness within the target measure, the Brownian transport map leverages the
randomness of the path while the Föllmer flow makes use of the randomness
delivered by the source measure. Meanwhile, [66] studies a transport map along
the reverse heat flow from the standard Gaussian measure to a target measure,
constructed by [69] and [48], as well as its Lipschitz property. The transport
map investigated in the work shares a similar Lipschitz property with the
transport map associated with the reverse heat flow. Nonetheless, the
transport map investigated by [48] and [66] is deduced via a limiting
argument, thus has no explicit expression. Under Assumptions 1, 2 and 3, our
considered transport map could be expressed as the flow map of the well-posed
Föllmer flow at time $t=1$ in a simple and explicit form. Towards connections
between the flows, the Föllmer flow over time interval $[0,1)$ (without time
$1$) is equivalent to the reverse heat flow through a deterministic change of
time, as revealed in Lemmas D.1 and D.2. Technically, the equivalence cannot
ensure well-posedness of the Föllmer flow at time $1$. We explicitly extend
the flow to time $1$ by deriving a uniform lower bound on the Jacobian matrix
of the velocity field via the Cramér-Rao inequality. Additionally, it is worth
mentioning that [2] and [1] introduce a unit-time normalizing flow, relevant
to the Föllmer flow, from the perspective of stochastic interpolation between
the Gaussian measure and a target measure. Nonetheless, the well-posedness of
this normalizing flow is not studied in the scope of their work.

###  1.2. Notations

For any integer $d\geq 1$, the Borel $\sigma$-algebra of $\mathbb{R}^{d}$ is
denoted by $\mathcal{B}(\mathbb{R}^{d})$. For $x,y\in\mathbb{R}^{d}$, define
$\left<x,y\right>:=\sum_{i=1}^{d}x_{i}y_{i}$ and the Euclidean norm
$|x|:=\left<x,x\right>^{1/2}$. Denote by
$\mathbb{S}^{d-1}:=\\{x\in\mathbb{R}^{d}:|x|=1\\}$. The operator norm of a
matrix $M\in\mathbb{R}^{m\times n}$ is denoted by
$\|M\|_{\mathrm{op}}:=\sup_{x\in\mathbb{R}^{n},|x|=1}|Mx|$ and $M^{\top}$ is
the transpose of $M$. Let $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ be a twice
continuously differentiable function. Denote by $\nabla f,\nabla^{2}f$ and
$\Delta f$ the gradient of $f$, the Hessian of $f$ and the Laplacian of $f$,
respectively. Let $\gamma_{d}$ denote the standard Gaussian measure on
$\mathbb{R}^{d}$, i.e.,
$\gamma_{d}(\mathrm{d}x):=(2\pi)^{-d/2}\exp(-|x|^{2}/2)\mathrm{d}x$. Let
$N(0,{\mathbf{I}}_{d})$ stand for a $d$-dimensional Gaussian random variable
with mean $0$ and covariance $\mathbf{I}_{d}$ being the $d\times d$ identity
matrix. Moreover, we use $\phi(x)$ to denote its probability density function
with respect to the Lebesgue measure.

The set of probability measures defined on a measurable space
$(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$ is denoted as
$\mathcal{P}(\mathbb{R}^{d})$. For any $\mathbb{R}^{d}$-valued random vector,
$\mathbb{E}[X]$ is used to denote its expectation. We say that $\Pi$ is a
transference plan of $\mu$ and $\nu$ if it is a probability measure on
$(\mathbb{R}^{d}\times\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d})\times\mathcal{B}(\mathbb{R}^{d}))$
such that for any Borel set $A$ of $\mathbb{R}^{d}$,
$\Pi(A\times\mathbb{R}^{d})=\mu(A)$ and $\Pi(\mathbb{R}^{d}\times A)=\nu(A)$.
We denote $\mathcal{C}(\mu,\nu)$ the set of transference plans of $\mu$ and
$\nu$. Furthermore, we say that a couple of $\mathbb{R}^{d}$-valued random
variables $(X,Y)$ is a coupling of $\mu$ and $\nu$ if there exists
$\Pi\in\mathcal{C}(\mu,\nu)$ such that $(X,Y)$ is distributed according to
$\Pi$. For two probability measures $\mu,\nu\in\mathcal{P}(\mathbb{R}^{d})$,
the Wasserstein distance of order $p\geq 1$ is defined as

$W_{p}(\mu,\nu):=\inf_{\Pi\in\mathcal{C}(\mu,\nu)}\left(\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^{p}\,\Pi(\mathrm{d}x,\mathrm{d}y)\right)^{1/p}.$

Let $\mu,\nu\in\mathcal{P}(\mathbb{R}^{d})$. The relative entropy of $\nu$
with respect to $\mu$ is defined by

$H(\nu\mid\mu)=\begin{cases}\int_{\mathbb{R}^{d}}\log\left(\frac{\mathrm{d}\nu}{\mathrm{d}\mu}\right)\nu(\mathrm{d}x),&\text{if
$\nu\ll\mu$, }\\\ +\infty,&\text{otherwise.}\end{cases}$

##  2\. Main results

We first present two definitions to characterize convexity properties of
probability measures and some useful notations.

######  Definition 2.1 ([18, 65]).

A probability measure $\mu(\mathrm{d}x)=\exp(-U(x))\mathrm{d}x$ is
$\kappa$-semi-log-concave for some $\kappa\in{\mathbb{R}}$ if its support
$\Omega\subseteq{\mathbb{R}}^{d}$ is convex and $U\in C^{2}(\Omega)$ satisfies

$\nabla^{2}U(x)\succeq\kappa\mathbf{I}_{d},\quad\forall x\in\Omega.$

######  Definition 2.2 ([38]).

A probability measure $\mu(\mathrm{d}x)=\exp(-U(x))\mathrm{d}x$ is
$\beta$-semi-log-convex for some $\beta>0$ if its support
$\Omega\subseteq{\mathbb{R}}^{d}$ is convex and $U\in C^{2}(\Omega)$ satisfies

$\nabla^{2}U(x)\preceq\beta\mathbf{I}_{d},\quad\forall x\in\Omega.$

Let $\nu(\mathrm{d}x)=p(x)\mathrm{d}x$ be a probability measure on
${\mathbb{R}}^{d}$ and define an operator $(\mathcal{Q}_{t})_{t\in[0,1]}$,
acting on function $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ by

$\displaystyle\mathcal{Q}_{1-t}f(x):=\int_{\mathbb{R}^{d}}\varphi^{tx,1-t^{2}}(y)f(y)\mathrm{d}y=\int_{\mathbb{R}^{d}}f\left(tx+\sqrt{1-t^{2}}z\right)\mathrm{d}\gamma_{d}(z)$

where $\varphi^{tx,1-t^{2}}(y)$ is the density of the $d$-dimensional Gaussian
measure with mean $tx$ and covariance $(1-t^{2})\mathbf{I}_{d}$.

Our first result is that we construct a flow over the unit time interval,
named the Föllmer flow, that pushes forward a standard Gaussian measure
$\gamma_{d}$ to a general target measure $\nu$ at time $t=1$. Before
rigorously defining the Föllmer flow, let us specify several regularity
assumptions that would ensure well-definedness and well-posedness of the
Föllmer flow.

######  Assumption 1.

The probability measure $\nu$ has a finite third moment and is absolutely
continuous with respect to the standard Gaussian measure $\gamma_{d}$.

######  Assumption 2.

The probability measure $\nu$ is $\beta$-semi-log-convex for some $\beta>0$.

######  Assumption 3.

Let $D:=(1/\sqrt{2})\mathrm{diam}(\mathrm{supp}(\nu))$. The probability
measure $\nu$ satisfies one or more of the following assumptions:

  * (i)

$\nu$ is $\kappa$-semi-log-concave for some $\kappa>0$ with $D\in(0,\infty]$;

  * (ii)

$\nu$ is $\kappa$-semi-log-concave for some $\kappa\leq 0$ with
$D\in(0,\infty)$;

  * (iii)

$\nu=N(0,\sigma^{2}\mathbf{I}_{d})*\rho$ where $\rho$ is a probability measure
supported on a ball of radius $R$ on $\mathbb{R}^{d}$.

Let us move to a formal definition of the Föllmer flow and the exhibition of
its well-posedness. A complete exposition would be found in Section 3.

######  Definition 2.3.

Suppose that probability measure $\nu$ satisfies Assumption 1. If
$(X_{t})_{t\in[0,1]}$ solves the initial value problem (IVP)

$\frac{\mathrm{d}X_{t}}{\mathrm{d}t}=V(t,X_{t}),\quad
X_{0}\sim\gamma_{d},\quad t\in[0,1]$ (1)

where the velocity field $V$ is defined by

$V(t,x):=\frac{\nabla\log\mathcal{Q}_{1-t}r(x)}{t},\qquad\forall t\in(0,1]$
(2)

with
$V(0,x):=\mathbb{E}_{\nu}[X],r(x):=\frac{\mathrm{d}\nu}{\mathrm{d}\gamma_{d}}(x)$.
We call $(X_{t})_{t\in[0,1]}$ a Föllmer flow and $V(t,x)$ a Föllmer velocity
field associated to $\nu$.

######  Theorem 2.4 (Well-posedness).

Suppose that Assumptions 1, 2 and 3 hold. Then the Föllmer flow
$(X_{t})_{t\in[0,1]}$ associated to $\nu$ is a unique solution to the IVP (1).
Moreover, the push-forward measure $\gamma_{d}\circ(X_{1}^{-1})=\nu$.

The following results show that the Föllmer flow map at time $t=1$ is
Lipschitz when the target measure satisfies either the strong log-concavity
assumption or the bounded support assumption.

######  Theorem 2.5 (Lipschitz mapping).

Assume that Assumptions 1, 2, 3-(i) or 3-(ii) hold.

  * (i)

If $\kappa D^{2}\geq 1$, then $X_{1}(x)$ is a Lipschitz mapping with constant
$\tfrac{1}{\sqrt{\kappa}}$, i.e.,

$\|\nabla X_{1}(x)\|_{\mathrm{op}}\leq\frac{1}{\sqrt{\kappa}},\quad\forall
x\in{\mathbb{R}}^{d}.$

  * (ii)

If $\kappa D^{2}<1$, then $X_{1}(x)$ is a Lipschitz mapping with constant
$\exp\left(\frac{1-\kappa D^{2}}{2}\right)D$, i.e.,

$\|\nabla X_{1}(x)\|_{\mathrm{op}}\leq\exp\left(\frac{1-\kappa
D^{2}}{2}\right)D,\quad\forall x\in{\mathbb{R}}^{d}.$

######  Theorem 2.6 (Gaussian mixtures).

Assume that Assumptions 1, 2 and 3-(iii) hold. Then $X_{1}(x)$ is a Lipschitz
mapping with constant $\sigma\exp\left(\frac{R^{2}}{2\sigma^{2}}\right)$,
i.e.,

$\|\nabla
X_{1}(x)\|_{\mathrm{op}}\leq\sigma\exp\left(\frac{R^{2}}{2\sigma^{2}}\right),\quad\forall
x\in{\mathbb{R}}^{d}.$

######  Remark 2.7.

Combining $\mathrm{Lip}(X_{1}(x))\leq\|\nabla X_{1}(x)\|_{\mathrm{op}}$ and
Theorem 2.6, we get

$\mathrm{Lip}(X_{1}(x))\leq\sigma\exp\left(\frac{R^{2}}{2\sigma^{2}}\right),\quad\forall
x\in{\mathbb{R}}^{d}.$ (3)

For Gaussian mixtures, the Lipschitz constants of (3) are better than those
provided by the Brownian transport map [65, Theorem 1.4] and match those
presented in [66]. Meanwhile, the Lipschitz constants of $X_{1}$ lead to a
dimension-free logarithmic Sobolev constant and a dimension-free Poincaré
constant

$C_{\mathrm{LS}}(p)\leq
2\sigma^{2}\exp\left(\frac{R^{2}}{\sigma^{2}}\right),\quad
C_{\mathrm{P}}(p)\leq\sigma^{2}\exp\left(\frac{R^{2}}{\sigma^{2}}\right).$ (4)

On the one hand, (4) implies a Gaussian log-Sobolev constant $2\sigma^{2}$ and
a Gaussian Poincaré constant $\sigma^{2}$ as $R$ goes to zero. In fact,
Poincaré constant $\sigma^{2}$ and log-Sobolev constant $2\sigma^{2}$ are
optimal for Gaussian measure $N(0,\sigma^{2}\mathbf{I}_{d})$ on
$\mathbb{R}^{d}$. On the other hand, the Poincaré constant obtained by (4) is
obviously smaller than the result in [9, Theorem 1.2]. In fact, the upper
bound of Poincaré constant for distribution
$p=N(0,\sigma^{2}\mathbf{I}_{d})*\rho$ in [9, Theorem 1.2] is
$\sigma^{2}\exp\left(4R^{2}/\sigma^{2}\right)$. Similarly, the log-Sobolev
constant (4) we obtained is slightly better than that in [21, Corollary 1].
Indeed, the upper bound of log-Sobolev constant for distribution
$\nu=N(0,\sigma^{2}\mathbf{I}_{d})*\rho$ in [21, Corollary 1] is
$6(4R^{2}+\sigma^{2})\exp\left(4R^{2}/\sigma^{2}\right)$. Nonetheless, it is
worthwhile to remark that [21] considers a rich class of probability measures
with the convolutional structure, which leads to general results on dimension-
free log-Sobolev and Poincaré inequlities.

##  3\. The Föllmer flow and its well-posedness

Let us present our motivations to derive the Föllmer flow. We are largely
inspired by the construction of the Föllmer process [42, 57, 38, 39], which
provides a probabilistic solution to the Schrödinger problem [74, 60], though
our construction of the Föllmer flow is partially heuristic using a similar
time-reversal argument.

###  3.1. The Föllmer process

In Föllmer’s lecture notes at the École d’Été de Probabilités de Saint-Flour
in 1986 [42], the Föllmer process is constructed with time reversal of a
linear Itô SDE under a finite relative entropy condition, which rigorously
determines a Schrödinger bridge from a source Dirac measure $\delta_{0}$ to a
general target measure $\nu$. Let us briefly revisit Föllmer’s arguments to
derive such a process.

######  Definition 3.1 ([42]).

A diffusion process $\overline{P}:=\left(\overline{X}_{t}\right)_{t\in[0,1]}$
starting with marginal distribution $\nu$ at time $t=0$ and reaching $0$ at
time $t=1$ is defined by the following Itô SDE

$\mathrm{d}\overline{X}_{t}=-\frac{1}{1-t}\overline{X}_{t}\mathrm{d}t+\mathrm{d}\overline{W}_{t},\
\overline{X}_{0}\sim\nu,\ t\in[0,1)$ (5)

with an extended solution at time $t=1$, i.e.,
$\overline{X}_{1}\sim\delta_{0}$. The transition probability distribution of
(5) from $\overline{X}_{0}$ to $\overline{X}_{t}$ is given by
$\overline{X}_{t}|\overline{X}_{0}\sim
N((1-t)\overline{X}_{0},t(1-t)\mathbf{I}_{d})$ for every $0\leq t<1$.

######  Lemma 3.2 ([41]).

Suppose that the diffusion process $Q$ has finite relative entropy with
respect to a standard Wiener process $W_{t}$ over the unit time interval,
i.e., $t\in[0,1]$. Then for almost all $t\in[0,1]$, the logarithmic derivative
of marginal density $\rho_{t}$ of $Q$ satisfies the duality equation
$\nabla\log\rho_{t}(x)=b(x,t)+\overline{b}(x,1-t)$ for almost all
$x\in{\mathbb{R}}^{d}$, where $b(x,t)$ and $\overline{b}(x,t)$ are drifts of
diffusion process $Q$ and its time-reversed diffusion process $\overline{Q}$,
respectively.

######  Definition 3.3 ([42, 57]).

Föllmer process $P=(X_{t})_{t\in[0,1]}$ is defined by the Itô SDE

$\mathrm{d}X_{t}=\nabla\log\mathcal{P}_{1-t}r(X_{t})\mathrm{d}t+\mathrm{d}W_{t},\
X_{0}=0,\ t\in[0,1]$ (6)

where $W_{t}$ is a standard Wiener process and $\mathcal{P}_{t}$ is the heat
semigroup defined by $\mathcal{P}_{t}h(x):=\mathbb{E}\left[h(x+W_{t})\right]$.
Moreover, the drift $\nabla\log\mathcal{P}_{1-t}r(X_{t})$ is called the
Föllmer drift.

######  Remark 3.4.

According to Lemma 3.2, the Föllmer process $P$ can be obtained by taking the
time reversal of the diffusion process $\overline{P}$ over $t\in[0,1]$. It
implies that the Föllmer drift has an alternative representation, i.e., for
any $t\in(0,1]$, $\nabla\log\mathcal{P}_{1-t}r(X_{t})=X_{t}/t+\nabla\log
p_{t}(X_{t})$, where $p_{t}$ is the marginal density of the Föllmer process
$P$.

###  3.2. The Föllmer flow via time reversal

Since $\delta_{0}$ is a degenerate distribution in the sense that its mass is
concentrated at $0$, we consider constructing a diffusion process that starts
with a marginal distribution $\nu$ and would be able to keep the nonzero
variance of its marginal distribution at time $t=1$. Let us present the
constructed diffusion process first. For any $\varepsilon\in(0,1)$, we
consider a diffusion process
$\left(\overline{X}_{t}\right)_{t\in[0,1-\varepsilon]}$ defined by the
following Itô SDE

$\mathrm{d}\overline{X}_{t}=-\frac{1}{1-t}\overline{X}_{t}\mathrm{d}t+\sqrt{\frac{2}{1-t}}\mathrm{d}\overline{W}_{t},\quad\overline{X}_{0}\sim\nu$
(7)

for all $t\in[0,1-\varepsilon]$. By Theorem 2.1 in [72, Chapter IX], the
diffusion process $\overline{X}_{t}$ defined in (7) has a unique strong
solution on $[0,1-\varepsilon]$. Moreover, the transition probability
distribution of (7) from $\overline{X}_{0}$ to $\overline{X}_{t}$ is given by
$\overline{X}_{t}|\overline{X}_{0}=x_{0}\sim N((1-t)x_{0},\
t(2-t){\mathbf{I}}_{d})$ for every $t\in[0,1-\varepsilon]$. It is a
straightforward observation that the variance of
$\overline{X}_{1-\varepsilon}|\overline{X}_{0}$ for SDE (7) will approach the
identity matrix $\mathbf{I}_{d}$ when $\varepsilon$ is small enough. That is
why we could expect the marginal distribution of
$\overline{X}_{1-\varepsilon}$ would have a nonzero variance. In contrast, for
the time-reversed Föllmer process (5), the variance of
$\overline{X}_{1-\varepsilon}|\overline{X}_{0}$ will approach constant $0$ as
$\varepsilon\to 0$, which indicates the variance of its marginal distribution
vanishes at time $t=1$. However, SDE (7) is not well-defined at time $t=1$ due
to unbounded drift and diffusion coefficients. Then we leverage the fact that
the marginal distribution $\overline{\mu}_{t}$ of the diffusion process
$\overline{X}_{t}$ defined in (7) has been determined in the sense that
$\overline{X}_{t}\overset{d}{=}(1-t)X+\sqrt{t(2-t)}Y$ with
$X\sim\nu,Y\sim\gamma_{d}$, and concentrate on an ODEs system sharing the same
marginal distribution flow with SDE (7) in order to circumvent the singularity
of SDE (7) at time $t=1$.

Note that the marginal distribution flow
$(\overline{\mu}_{t})_{t\in[0,1-\varepsilon]}$ of the diffusion process (7)
satisfies the Fokker-Planck-Kolmogorov equation in an Eulerian framework [12]

$\partial_{t}\overline{\mu}_{t}=\nabla\cdot(\overline{\mu}_{t}V(1-t,x))\quad\textrm{on}\
[0,1-\varepsilon]\times{\mathbb{R}}^{d},\ \overline{\mu}_{0}=\nu$ (8)

in the sense that $\overline{\mu}_{t}$ is continuous in $t$ under the weak
topology, i.e.,

$\displaystyle\overline{\mu}_{t}(f):=\int_{\mathbb{R}^{d}}f(x)\mu_{t}(\mathrm{d}x)=\nu(f)-\int_{0}^{t}\overline{\mu}_{s}\left(\left<V(1-s,\cdot),\nabla
f\right>\right)\mathrm{d}s$

for all $f\in C^{\infty}_{0}(\mathbb{R}^{d})$ and the velocity field is given
by

$V(1-t,x):=\frac{1}{1-t}\left[x+S(1-t,x)\right],\quad t\in[0,1-\varepsilon]$
(9)

and

$S(t,x):=\nabla\log\int_{\mathbb{R}^{d}}(2\pi(1-t^{2}))^{-\frac{d}{2}}\exp\left(-\frac{|x-ty|^{2}}{2(1-t^{2})}\right)p(y)\mathrm{d}y$

for all $t\in[\varepsilon,1]$. Due to the classical Cauchy-Lipschitz theory
[4, Section 2] with a Lipschitz velocity field or the well-established
Ambrosio-DiPerna-Lions theory with lower Sobolev regularity assumptions on the
velocity field [35, 3], we shall define a flow
$(X^{*}_{t})_{t\in[0,1-\varepsilon]}$ in a Lagrangian formulation via the
following ODEs system

$\displaystyle\mathrm{d}X^{*}_{t}=-V\left(1-t,X^{*}_{t}\right)\mathrm{d}t,\quad
X^{*}_{0}\sim\nu,\quad t\in[0,1-\varepsilon].$ (10)

######  Proposition 3.5.

Assume the velocity field $V(t,x)$ satisfies $V\in
L^{1}([\varepsilon,1];W^{1,\infty}_{\mathrm{loc}}({\mathbb{R}}^{d};{\mathbb{R}}^{d}))$
and $|V|/(1+|x|)\in L^{1}([\varepsilon,1];L^{\infty}({\mathbb{R}}^{d}))$. Then
the push-forward measure associated with the flow map $X^{*}_{t}$ satisfies
$X^{*}_{t}\overset{d}{=}(1-t)X+\sqrt{t(2-t)}Y$ with
$X\sim\nu,Y\sim\gamma_{d}$. Moreover, the push-forward measure
$\nu\circ({X^{*}_{1-\varepsilon}})^{-1}$ converges to the Gaussian measure
$\gamma_{d}$ in the sense of Wasserstein-2 distance as $\varepsilon$ tends to
zero, i.e., $W_{2}(\nu\circ({X^{*}_{1-\varepsilon}})^{-1},\gamma_{d})\to 0$.

######  Remark 3.6.

Suppose that the target measure $\nu$ has a finite third moment. By Lemma A.1,
we can supplement the definition of velocity field $V(1-t,x)$ at time $t=1$,
i.e.,

$V(0,x):=\lim_{t\downarrow 0}V(t,x)=\lim_{t\downarrow
0}\frac{x+S(t,x)}{t}=\mathbb{E}_{\nu}[X].$

Then we extend the flow $(X^{*}_{t})_{t\in[0,1)}$ to time $t=1$ such that
$X^{*}_{1}\sim\gamma_{d}$, which solves the IVP

$\displaystyle\mathrm{d}X^{*}_{t}=-V\left(1-t,X^{*}_{t}\right)\mathrm{d}t,\quad
X^{*}_{0}\sim\nu,\quad t\in[0,1],$ (11)

where the velocity field

$V(1-t,x)=\frac{1}{1-t}\left[x+S(1-t,x)\right],\quad\forall t\in[0,1)$

and $V(0,x)=\mathbb{E}_{\nu}[X]$.

In order to exploit a time-reversal argument inspired by Föllmer, it remains
crucial to establish the well-posedness of a flow $(X^{*}_{t})_{t\in[0,1]}$
that solves the IVP (11). We proceed to study regularity properties of the
velocity field $V$ on $[0,1]\times{\mathbb{R}}^{d}$ by imposing structural
assumptions on the target measure $\nu$. By Theorem B.3, we know that there
exists $0\leq\theta^{\star}_{t}<\infty$ such that

$\|-\nabla V(t,x)\|_{\mathrm{op}}=\|\nabla
V(t,x)\|_{\mathrm{op}}\leq\theta^{\star}_{t}$ (12)

for any $t\in[0,1]$. Furthermore, the velocity field $-V(1-t,x)$ is smooth and
with the bounded derivative for any $t\in[0,1]$ and $x\in\mathbb{R}^{d}$.
Therefore, the IVP (11) has a unique solution and the flow map $x\mapsto
X^{*}_{t}(x)$ is a diffeomorphism from $\mathbb{R}^{d}$ onto $\mathbb{R}^{d}$
at any time $t\in[0,1]$. A standard time-reversal argument of ODE would yield
a formal definition of the Föllmer flow.

######  Definition 3.7.

Suppose that probability measure $\nu$ satisfies Assumption 1. If
$(X_{t})_{t\in[0,1]}$ solves the IVP

$\frac{\mathrm{d}X_{t}}{\mathrm{d}t}=V(t,X_{t}),\quad
X_{0}\sim\gamma_{d},\quad t\in[0,1]$ (13)

where the velocity field

$V(t,x)=\frac{1}{t}\left[x+S(t,x)\right],\quad\forall t\in(0,1],\quad
V(0,x)=\mathbb{E}_{\nu}[X],$

we call $(X_{t})_{t\in[0,1]}$ a Föllmer flow and $V(t,x)$ a Föllmer velocity
field associated to $\nu$.

######  Remark 3.8.

Notice that

$\displaystyle\mathcal{Q}_{1-t}r(x)=(2\pi)^{d/2}\exp\left(\frac{|x|^{2}}{2}\right)\frac{1}{(2\pi(1-t^{2}))^{d/2}}\int_{\mathbb{R}^{d}}p(y)\exp\left(-\frac{|x-ty|^{2}}{2(1-t^{2})}\right)\mathrm{d}y$

where $\mathcal{Q}_{1-t}r(x)$ is defined in (22). We further obtain
$\nabla\log\mathcal{Q}_{1-t}r(x)=x+S(t,x),\ \forall t\in[0,1]$. Therefore, we
have that (1) and (13) are equivalent, which satisfy $X_{0}\sim\gamma_{d}$ and
$X_{1}\sim\nu$.

Finally, let us conclude with the well-posedness properties of the Föllmer
flow, which is presented in Theorem 2.4 and summarized below.

######  Theorem 3.9 (Well-posedness).

Suppose that Assumptions 1, 2 and 3 hold. Then the Föllmer flow
$(X_{t})_{t\in[0,1]}$ associated to $\nu$ is a unique solution to the IVP
(13). Moreover, the push-forward measure $\gamma_{d}\circ(X_{1}^{-1})=\nu$.

##  4\. Applications

Owing to the Lipschitz transport properties proved in Theorems 2.5 and 2.6, we
are motivated to establish a variety of functional inequalities and
concentration inequalities for several classes of probability measures on
Euclidean space.

###  4.1. Dimension-free inequalities

In this subsection, we provide dimension-free results for the $\Psi$-Sobolev
inequalities. For completeness, we incorporate classical results for strongly-
log-concave measures ($\kappa>0$), which have been studied with the optimal
transport maps [17]. Compared with [65], we obtain that the upper bound
constants of $\Psi$-Sobolev inequalities, Isoperimetric inequalities and
$q$-Poincaré inequalities are the same for $\kappa D^{2}\geq 1$. When $\kappa
D^{2}<1$, our upper bound constants to $\Psi$-Sobolev inequalities and
Isoperimetric inequalities are in the same order with the results of Lemmas
5.3-5.5 in [65]. For the Gaussian mixtures case, we obtain that the constants
of these inequalities are slightly better than the result of Lemmas 5.3-5.5 in
[65].

######  Definition 4.1.

Let $\mathcal{I}$ be a closed interval (not necessarily bounded) and let
$\Psi:\mathcal{I}\rightarrow\mathbb{R}$ be a twice differentiable function. We
say that $\Psi$ is a divergence if each of the functions
$\Psi,\Psi^{\prime\prime}$ and $-1/\Psi^{\prime\prime}$ is a convex function.
Given a probability measure $\nu(\mathrm{d}x)=p(x)\mathrm{d}x$ on
$\mathbb{R}^{d}$ and a function $\zeta:\mathbb{R}^{d}\rightarrow\mathcal{I}$
such that $\int_{\mathbb{R}^{d}}\zeta(x)p(x)\mathrm{d}x\in\mathcal{I}$, we
define

$\mathrm{Ent}^{\Psi}_{p}(\zeta):=\int_{\mathbb{R}^{d}}\Psi(\zeta(x))p(x)\mathrm{d}x-\Psi\left(\int_{\mathbb{R}^{d}}\zeta(x)p(x)\mathrm{d}x\right).$

Some examples of the divergences are $\Psi:\mathbb{R}\rightarrow\mathbb{R}$
with $\Psi(x)=x^{2}$ (Poincaré inequality) and
$\Psi:\mathbb{R}_{+}\rightarrow\mathbb{R}$ with $\Psi(x)=x\log x$ (log-Sobolev
inequality).

######  Theorem 4.2 ($\Psi$-Sobolev inequalities).

Let Assumptions 1, 2 and 3 hold.

  * (1)

Let $\zeta:\mathbb{R}^{d}\rightarrow\mathcal{I}$ be any continuously
differentiable function such that
$\int_{\mathbb{R}^{d}}\zeta^{2}(x)p(x)\mathrm{d}x\in\mathcal{I}$.

    * (i)

If $\kappa D^{2}\geq 1$, then

$\mathrm{Ent}^{\Psi}_{p}(\zeta)\leq\frac{1}{2\kappa}\int_{\mathbb{R}^{d}}\Psi^{\prime\prime}(\zeta(x))|\nabla\zeta(x)|^{2}p(x)\mathrm{d}x.$

    * (ii)

If $\kappa D^{2}<1$, then

$\displaystyle\mathrm{Ent}^{\Psi}_{p}(\zeta)\leq\frac{\exp(1-\kappa
D^{2})}{2}D^{2}\int_{\mathbb{R}^{d}}\Psi^{\prime\prime}(\zeta(x))|\nabla\zeta(x)|^{2}p(x)\mathrm{d}x.$

  * (2)

Fix a probability measure $\rho$ on $\mathbb{R}^{d}$ supported on a ball of
radius $R$ and let $p:=N(a,\Sigma)*\rho$ and denote
$\lambda_{\min}:=\lambda_{\min}(\Sigma)$ and
$\lambda_{\max}=\lambda_{\max}(\Sigma)$. Then for any continuously
differentiable function $\zeta:\mathbb{R}^{d}\rightarrow\mathcal{I}$ such that
$\int_{\mathbb{R}^{d}}\zeta^{2}(x)p(x)\mathrm{d}x\in\mathcal{I}$, we have

$\displaystyle\mathrm{Ent}^{\Psi}_{p}(\zeta)\leq\frac{1}{2}\lambda_{\max}\exp\left(\frac{R^{2}}{\lambda_{\min}}\right)\int_{\mathbb{R}^{d}}\Psi^{\prime\prime}(\zeta(x))|\nabla\zeta(x)|^{2}p(x)\mathrm{d}x.$

######  Theorem 4.3 (Isoperimetric inequalities).

Assume that Assumptions 1, 2 and 3 hold. Let $\Phi$ be the cumulative
distribution function of $\gamma_{1}$ on $\mathbb{R}$, that is,

$\Phi(x)=\gamma_{1}(-\infty,x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}\exp\left(-\frac{y^{2}}{2}\right)\mathrm{d}y,\quad-\infty<\forall
x<+\infty$

and $B_{2}^{d}:=\\{x\in\mathbb{R}^{d}:|x|\leq 1\\}$ be the unit ball in
$\mathbb{R}^{d}$.

  * (1)

Let $A_{t}:=A+tB_{2}^{d}$ for any Borel set $A\subseteq\mathbb{R}^{d}$ and
$t\geq 0$, then

$p\left(A_{t}\right)\geq\Phi\left(p(A)+\frac{t}{C}\right),\quad
C:=\begin{cases}1/\sqrt{\kappa},\ &\text{if $\kappa D^{2}\geq 1$, }\\\
\exp\left(\frac{1-\kappa D^{2}}{2}\right)D,\ &\text{if $\kappa
D^{2}<1$.}\end{cases}$

  * (2)

Let $p:=N(a,\Sigma)*\rho$ where $\rho$ is a probability measure on
$\mathbb{R}^{d}$ and is supported on a ball of radius $R$. Set
$\lambda_{\min}:=\lambda_{\min}(\Sigma),\lambda_{\max}:=\lambda_{\max}(\Sigma)$
and

$C:=(\lambda_{\min}\lambda_{\max})^{1/2}\exp\left(\frac{R^{2}}{2\lambda_{\min}}\right).$

Then

$p(A_{t})\geq\Phi\left(p(A)+\frac{t}{C}\right),\quad\quad
A_{t}:=A+tB_{2}^{d}.$

Finally, let $\eta:\mathbb{R}^{d}\rightarrow\mathbb{R}$ be any continuously
differentiable function such that
$\int_{\mathbb{R}^{d}}\eta(x)p(x)\,\mathrm{d}x=0.$

######  Theorem 4.4 ($q$-Poincaré inequalities).

Suppose that Assumptions 1, 2 and 3 hold.

  * (1)

Let $q\geq 2$ be an even integer and $\eta\in L^{q}(\gamma_{d})$, then it
holds that

$\int_{\mathbb{R}^{d}}\eta^{q}(x)p(x)\,\mathrm{d}x\leq\left(\int_{\mathbb{R}^{d}}|\nabla\eta(x)|^{q}p(x)\,\mathrm{d}x\right)\begin{cases}C^{\star}_{1}&\text{if
$\kappa D^{2}\geq 1$},\\\ C^{\star}_{2}&\text{if $\kappa
D^{2}<1$.}\end{cases}$

where

$C^{\star}_{1}:=\left(\frac{q-1}{\kappa}\right)^{q/2},\quad
C^{\star}_{2}:=D^{q}\exp\left(\frac{q(1-\kappa D^{2})}{2}\right).$

  * (2)

Fix a probability measure $\rho$ on $\mathbb{R}^{d}$ supported on a ball of
radius $R$, and let $p:=N(a,\Sigma)*\rho$ and denote
$\lambda_{\min}:=\lambda_{\min}(\Sigma)$ and
$\lambda_{\max}:=\lambda_{\max}(\Sigma)$. Then for any $\eta\in
L^{q}(\gamma_{d})$ with even integer $q\geq 2$, it holds that

$\int_{\mathbb{R}^{d}}\eta^{q}(x)p(x)\,\mathrm{d}x\leq(q-1)^{\frac{q}{2}}(\lambda_{\min}\lambda_{\max})^{\frac{q}{2}}\exp\left(\frac{qR^{2}}{2\lambda_{\min}}\right)\int_{\mathbb{R}^{d}}|\nabla\eta(x)|^{q}p(x)\,\mathrm{d}x.$

###  4.2. Non-asymptotic bounds for empirical measures

Let $\mu$ be a probability distribution on $\mathbb{R}^{d}$ and

$\mu_{n}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}},$ (14)

be the empirical measure, where $(X_{i})_{i=1}^{n}$ are i.i.d. samples drawn
from $\mu$. Deriving the non-asymptotic convergence rate under the Wasserstein
distance of the empirical measure $\mu_{n}$ and the probability measure $\mu$
on Polish space is one of the most important topics in statistics,
probability, and machine learning. In recent years, significant progress has
been made on this topic. When $p=1$, the Kantorovich-Rubinstein duality [47]
implies that $W_{1}(\mu_{n},\mu)$ is equivalent to the supremum of the
empirical process indexed by Lipschitz functions. As a consequence, [37]
provides sharp lower and upper bounds of
$\mathbb{E}\left[W_{1}(\mu_{n},\mu)\right]$ for $\mu$ supported on a bounded
finite dimensional set. Subsequently, [76] studies the case when $\mu$ is the
uniform distribution on a $d$-dimensional unit cube. For general
distributions, [13, 34, 43] establish sharp upper bounds of
$\mathbb{E}\left[W_{p}(\mu_{n},\mu)\right]$ in finite dimensional Euclidean
spaces. Recently, by extending finite dimensional spaces to infinite
dimensional functional spaces, [59] establishes similar results for general
distributions.

Besides the above mentioned bounds in expectation, [79] obtains a high
probability bound on $W_{p}(\mu_{n},\mu)$ for measures $\mu$ with bounded
supports. By applying Sanov’s theorem to independent random variables, [14]
establishes concentration inequalities for empirical measures on non-compact
space. In this subsection, we will give a high probability bound on
$W_{2}(\mu_{n},\mu)$ by the Lipschitz transport properties proved in Theorems
2.5 and 2.6. To begin with, we will review the transportation inequality
defined in Definition 4.5, the non-asymptotic convergence rate of
$\mathbb{E}\left[W_{p}(\mu_{n},\mu)\right]$ as given in Theorem 4.7, and its
concentration inequality for $p=2$ as stated in Theorem 4.8. Then, we will
derive the non-asymptotic convergence rate for $W_{p}(\mu_{n},\mu)$, as stated
in Theorem 4.9 by combining the transportation inequality of the Gaussian
measure on $\mathbb{R}^{d}$ as established in [76] and the transportation
inequality of the push-forward measure of the Gaussian measure under Lipschitz
mapping, as shown in Lemma 4.6.

######  Definition 4.5 (Transportation inequality).

The probability measure $\mu$ satisfies the $L^{p}$-transportation inequality
on $\mathbb{R}^{d}$ if there is some constant $C>0$ such that for any
probability measure $\nu$, $W_{p}(\mu,\nu)\leq\sqrt{2CH(\nu\mid\mu)}$. To be
short, we write $\mu\in\mathrm{T_{p}}(C)$ for this relation.

######  Lemma 4.6 ([36]).

Assume that $\mu\in\mathrm{T_{p}}(C)$ on $\mathbb{R}^{d}$. If
$\Phi:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is Lipschitz continuous with
constant $\alpha>0$, then
$\nu=\mu\circ\Phi^{-1}\in\mathrm{T_{p}}(\alpha^{2}C)$ on $\mathbb{R}^{d}$.

######  Theorem 4.7 ([43]).

Let $p>0$, assume that for some $r>p$ and
$\int_{\mathbb{R}^{d}}|x|^{r}\,\mu(\mathrm{d}x)$ is finite. Then there exists
a constant $C>0$ depending only on $p,r,d$ such that for all $n\geq 1$,

$\displaystyle\mathbb{E}\left[W_{p}(\mu_{n},\mu)\right]\leq
C\left(\int_{\mathbb{R}^{d}}|x|^{r}\,\mu(\mathrm{d}x)\right)^{p/r}\begin{cases}n^{-\frac{1}{2}}+n^{-\frac{r-p}{r}},&\text{if
$p>d/2$ and $r\neq 2p$ }\\\
n^{-\frac{1}{2}}\log(1+n)+n^{-\frac{r-p}{r}},&\text{if $p=d/2$ and $r\neq 2p$
}\\\ n^{-\frac{p}{d}}+n^{-\frac{r-p}{r}},&\text{if $p<d/2$ and
$r\neq\frac{d}{d-p}$ }\end{cases}$

where the expectation is taken on the samples $X_{1},\cdots,X_{n}$.

The next result states that a $\mathrm{T_{2}}(C)$ inequality on $\mu$ implies
Gaussian concentration inequality for $W_{2}(\mu_{n},\mu)$.

######  Theorem 4.8 ([44]).

Let a probability measure $\mu$ on $\mathbb{R}^{d}$ satisfy the transportation
inequality $\mathrm{T_{2}}(C)$. The following holds:

$\mathbb{P}\left(W_{2}(\mu_{n},\mu)\geq\mathbb{E}\left[W_{2}(\mu_{n},\mu)\right]+t\right)\leq\exp\left(-\frac{nt^{2}}{C}\right).$

For any probability measure $\nu$ on $\mathbb{R}^{d}$ with a finite fifth
moment, let us define

$\displaystyle\mathsf{M}(\nu,d,n):=c_{d}\left(\int_{\mathbb{R}^{d}}|x|^{5}\,\nu(\mathrm{d}x)\right)^{2/5}\begin{cases}n^{-1/2}&\text{if
$d<4$}\\\ n^{-1/2}\log(1+n)&\text{if $d=4$ }\\\ n^{-2/d}&\text{if $d>4$
}\end{cases}$ (15)

where the constant $c_{d}$ depends only on $d$. On the other hand, for the
$L^{2}$-transportation inequality $\mathrm{T_{2}}(C)$, recall that Talagrand
[77] proved that the standard Gaussian measure $\gamma_{1}=N(0,1)$ satisfies
$\mathrm{T_{2}}(C)$ on $\mathbb{R}$ w.r.t. the Euclidean distance with the
sharp constant $C=1$ and found that $\mathrm{T_{2}}(C)$ is stable for product
(or independent) tensorization. Therefore, combining Lemma 4.6, Theorems 2.5,
2.6, 4.7 and 4.8, we obtain the following results.

######  Theorem 4.9 (Concentration for empirical measures).

Suppose that Assumptions 1, 2 and 3 hold, and let probability measure $\nu$
has a finite fifth moment.

  * (1)

If $\kappa D^{2}\geq 1$, then $\nu\in\mathrm{T_{2}}(1/\kappa)$. Moreover, for
any $\varepsilon\in(0,1)$, it holds that

$\displaystyle
W_{2}(\nu_{n},\nu)\leq\left(\frac{\log\varepsilon^{-1}}{n\kappa}\right)^{1/2}+\mathsf{M}(\nu,d,n)$

with probability at least $1-\varepsilon$ and constant $\mathsf{M}(\nu,d,n)$
given in (15).

  * (2)

If $\kappa D^{2}<1$, then $\nu\in\mathrm{T_{2}}\left(D^{2}\exp(1-\kappa
D^{2})\right)$. Moreover, for any $\varepsilon\in(0,1)$, it holds that

$\displaystyle
W_{2}(\nu_{n},\nu)\leq\left\\{\frac{\log\varepsilon^{-1}}{nD^{2}\exp(1-\kappa
D^{2})}\right\\}^{1/2}+\mathsf{M}(\nu,d,n)$

with probability at least $1-\varepsilon$ and constant $\mathsf{M}(\nu,d,n)$
given in (15).

  * (3)

If $\nu=N(0,\sigma^{2}\mathbf{I}_{d})*\rho$ where $\rho$ is a probability
measure supported on a ball of radius $R$ on $\mathbb{R}^{d}$, then
$\nu\in\mathrm{T_{2}}\left(\sigma^{2}\exp(R^{2}/\sigma^{2})\right)$. Moreover,
for any $\varepsilon\in(0,1)$, it holds that

$\displaystyle
W_{2}(\nu_{n},\nu)\leq\left\\{\frac{\log\varepsilon^{-1}}{n\sigma^{2}\exp(R^{2}/\sigma^{2})}\right\\}^{1/2}+\mathsf{M}(\nu,d,n)$

with probability at least $1-\varepsilon$ and constant $\mathsf{M}(\nu,d,n)$
given in (15).

##  5\. Conclusion

We have constructed the Föllmer flow originating from a standard Gaussian
measure and hitting a general target measure. By studying the well-posedness
of the Föllmer flow, we have established the Lipschitz property of its flow
map at time $t=1$. Such a Lipschitz transport map enables get functional
inequalities with dimension-free constants and derive concentration
inequalities for the empirical measure for rich classes of probability
measures. It is worthwhile to notice that the Föllmer velocity field has an
analytic expression that is compatible with Monte Carlo approximations.
Therefore, a possible direction of future research would be to design general-
purpose sampling algorithms and score-based generative models using the
Föllmer flow. Besides, being limited to scenarios covered in Assumptions 2 and
3, the work could be extended to explore weaker and even minimal regularity
assumptions on the target measure. For example, replacing semi-log-concavity
with “convexity at infinity” in [15, 18] is a potential step.

##  Appendix A Proof of Theorem 2.4 and Proposition 3.5

###  A.1. Well-definedness of the Föllmer flow

Recall that the velocity field $V(t,x)$ defined in (2) yields

$V(t,x):=\frac{\nabla\log\mathcal{Q}_{1-t}r(x)}{t},\quad
r(x):=\frac{p(x)}{\phi(x)}$

where $\nu(\mathrm{d}x)=p(x)\mathrm{d}x$. For any $t\in(0,1]$, then one
obtains

$\displaystyle\mathcal{Q}_{1-t}r(x)=\int_{\mathbb{R}^{d}}\varphi^{tx,1-t^{2}}(y)r(y)\mathrm{d}y=\int_{\mathbb{R}^{d}}\phi(z)r(tx+\sqrt{1-t^{2}}z)\mathrm{d}z,$

where $\varphi^{tx,1-t^{2}}(y)$ is the density of the $d$-dimensional Gaussian
measure with mean $tx$ and covariance $(1-t^{2})\mathbf{I}_{d}$. For the
convenience of subsequent calculation, we introduce the following symbols:

$S(t,x):=\nabla\log q_{t}(x),\quad
q_{t}(x):=\int_{\mathbb{R}^{d}}q(t,x|1,y)p(y)\mathrm{d}y$

where
$q(t,x|1,y):=(2\pi(1-t^{2}))^{-\frac{d}{2}}\exp\left(-\frac{|x-ty|^{2}}{2(1-t^{2})}\right)$
for any $t\in[0,1]$. Notice that

$\displaystyle\mathcal{Q}_{1-t}r(x)=(2\pi)^{d/2}\exp\left(\frac{|x|^{2}}{2}\right)\times\frac{1}{(2\pi(1-t^{2}))^{d/2}}\int_{\mathbb{R}^{d}}p(y)\exp\left(-\frac{|x-ty|^{2}}{2(1-t^{2})}\right)\mathrm{d}y.$

Then we have $\nabla\log\mathcal{Q}_{1-t}r(x)=x+S(t,x)$, for any $t\in[0,1]$.

Suppose that the target distribution $p$ satisfies the third moment condition,
we can supplement the definition of velocity field $V$ at time $t=0$, so that
$V$ is well-defined on the interval $[0,1]$. Then we have the following
result:

######  Lemma A.1.

Suppose that $\mathbb{E}_{p}[|X|^{3}]<\infty$, then

$\lim_{t\downarrow 0}V(t,x)=\lim_{t\downarrow
0}\frac{x+S(t,x)}{t}=\mathbb{E}_{p}[X].$

###### Proof.

Let $t\to 0$, then it yields

$\displaystyle\lim_{t\downarrow 0}V(t,x)=\lim_{t\downarrow
0}\partial_{t}S(t,x)=\lim_{t\downarrow
0}\left\\{\frac{\nabla[\partial_{t}q_{t}(x)]}{q_{t}(x)}-\frac{\partial_{t}q_{t}(x)}{q_{t}(x)}S(t,x)\right\\}.$

On the one hand, by simple calculation, it holds that

$\displaystyle\partial_{t}q_{t}(x)$
$\displaystyle=\partial_{t}\int_{\mathbb{R}^{d}}q(t,x|1,y)p(y)\mathrm{d}y=\partial_{t}\int_{\mathbb{R}^{d}}\left[2\pi(1-t^{2})\right]^{-\frac{d}{2}}\exp\left(-\frac{|x-ty|^{2}}{2(1-t^{2})}\right)p(y)\mathrm{d}y$
$\displaystyle=\frac{td}{1-t^{2}}q_{t}(x)-\frac{t}{(1-t^{2})^{2}}|x|^{2}q_{t}(x)+\frac{1+t^{2}}{(1-t^{2})^{2}}\int_{\mathbb{R}^{d}}x^{\top}yq(t,x|1,y)p(y)\mathrm{d}y$
$\displaystyle~{}~{}~{}-\frac{t}{(1-t^{2})^{2}}\int_{\mathbb{R}^{d}}|y|^{2}q(t,x|1,y)p(y)\mathrm{d}y.$

Furthermore, we also obtain

$\displaystyle\frac{\partial_{t}q_{t}(x)}{q_{t}(x)}=\frac{td}{1-t^{2}}-\frac{t}{(1-t^{2})^{2}}|x|^{2}+\frac{1+t^{2}}{(1-t^{2})^{2}}\int_{\mathbb{R}^{d}}x^{\top}yq(1,y|t,x)\mathrm{d}y-\frac{t}{(1-t^{2})^{2}}\int_{\mathbb{R}^{d}}|y|^{2}q(1,y|t,x)\mathrm{d}y.$

On the other hand, by straightforward calculation, it yields

$\displaystyle\nabla[\partial_{t}q_{t}(x)]$
$\displaystyle=-\frac{td}{(1-t^{2})^{2}}xq_{t}(x)+\frac{t^{2}d}{(1-t^{2})^{2}}\int_{\mathbb{R}^{d}}yq(t,x|1,y)p(y)\mathrm{d}y$
$\displaystyle\quad-\frac{2t}{(1-t^{2})^{2}}xq_{t}(x)+\frac{t}{(1-t^{2})^{3}}|x|^{2}xq_{t}(x)-\frac{t^{2}|x|^{2}}{(1-t^{2})^{3}}\int_{\mathbb{R}^{d}}yq(t,x|1,y)p(y)\mathrm{d}y$
$\displaystyle\qquad+\frac{1+t^{2}}{(1-t^{2})^{2}}\int_{\mathbb{R}^{d}}yq(t,x|1,y)p(y)\mathrm{d}y$
$\displaystyle\quad-\frac{1+t^{2}}{(1-t^{2})^{3}}\int_{\mathbb{R}^{d}}(x^{\top}y)xq(t,x|1,y)p(y)\mathrm{d}y+\frac{t(1+t^{2})}{(1-t^{2})^{3}}\int_{\mathbb{R}^{d}}(x^{\top}y)yq(t,x|1,y)p(y)\mathrm{d}y$
$\displaystyle\quad+\frac{t}{(1-t^{2})^{3}}\int_{\mathbb{R}^{d}}x|y|^{2}q(t,x|1,y)p(y)\mathrm{d}y-\frac{t^{2}}{(1-t^{2})^{3}}\int_{\mathbb{R}^{d}}y|y|^{2}q(t,x|1,y)p(y)\mathrm{d}y.$

Moreover, we also obtain

$\displaystyle\frac{\nabla[\partial_{t}q_{t}(x)]}{q_{t}(x)}$
$\displaystyle=-\frac{td}{(1-t^{2})^{2}}x+\frac{t^{2}d}{(1-t^{2})^{2}}\int_{\mathbb{R}^{d}}yq(1,y|t,x)\mathrm{d}y-\frac{2tx}{(1-t^{2})^{2}}+\frac{t|x|^{2}x}{(1-t^{2})^{3}}$
$\displaystyle\quad-\frac{t^{2}}{(1-t^{2})^{3}}|x|^{2}\int_{\mathbb{R}^{d}}yq(1,y|t,x)\mathrm{d}y+\frac{1+t^{2}}{(1-t^{2})^{2}}\int_{\mathbb{R}^{d}}yq(1,y|t,x)\mathrm{d}y$
$\displaystyle\quad-\frac{1+t^{2}}{(1-t^{2})^{3}}\int_{\mathbb{R}^{d}}(x^{\top}y)xq(1,y|t,x)\mathrm{d}y+\frac{t(1+t^{2})}{(1-t^{2})^{3}}\int_{\mathbb{R}^{d}}(x^{\top}y)yq(1,y|t,x)\mathrm{d}y$
$\displaystyle\quad+\frac{t}{(1-t^{2})^{3}}x\int_{\mathbb{R}^{d}}|y|^{2}q(1,y|t,x)\mathrm{d}y-\frac{t^{2}}{(1-t^{2})^{3}}\int_{\mathbb{R}^{d}}y|y|^{2}q(1,y|t,x)\mathrm{d}y.$

Since $\mathbb{E}_{p}[|X|^{3}]<\infty$, it yields

$\lim_{t\downarrow
0}\int_{\mathbb{R}^{d}}|y|^{3}q(1,y|t,x)\mathrm{d}y=\int_{\mathbb{R}^{d}}|y|^{3}\lim_{t\downarrow
0}q(1,y|t,x)\mathrm{d}y=\mathbb{E}_{p}\left[|X|^{3}\right]<+\infty.$

Furthermore, we have

$\displaystyle\lim_{t\downarrow
0}\frac{\partial_{t}q_{t}(x)}{q_{t}(x)}S(t,x)=-xx^{\top}\mathbb{E}_{p}[X],\quad\lim_{t\downarrow
0}\frac{\nabla[\partial_{t}q_{t}(x)]}{q_{t}(x)}$
$\displaystyle=\mathbb{E}_{p}[X]-xx^{\top}\mathbb{E}_{p}[X].$

Therefore, it yields $\lim_{t\downarrow 0}V(t,x)=\mathbb{E}_{p}[X]$, which
completes the proof. ∎

###  A.2. Cramér-Rao inequality

In order to obtain a lower bound of the Jacobian matrix of velocity field $V$
defined in (2), we apply the classical Cramér-Rao bound [71, 31] in
statistical parameter estimation to a special case for location parameter
estimation. This particular application is far from being new in information
theory and convex geometry (for example, see [33, 64, 26, 73]). We include it
here for the sake of completeness. A lower bound on the covariance matrix of a
prescribed probability measure directly follows the Cramér-Rao bound [30,
Theorem 11.10.1].

######  Lemma A.2 (Cramér-Rao bound).

Let $\mu_{\theta}(\mathrm{d}x)=f_{\theta}(x)\mathrm{d}x$ be a probability
measure on ${\mathbb{R}}^{d}$ such that the density $f_{\theta}(x)$ is of
class $C^{2}$ with respect to an unknown parameter $\theta\in\Theta$. Assume
that a few mild regularity assumptions hold. Then provided that
$\\{X_{i}\\}_{i=1}^{n}$ are i.i.d. samples from $\mu_{\theta}$ with size $n$,
the mean-squared error of any unbiased estimator $g(X_{1},X_{2},\cdots,X_{n})$
for the parameter $\theta$ is lower bounded by the inverse of the Fisher
information matrix:

$\displaystyle\mathbb{E}_{\mu_{\theta}}\left[(g(X_{1},X_{2},\cdots,X_{n})-\theta)^{\otimes
2}\right]\succeq\left(-n\mathbb{E}_{\mu_{\theta}}\left[\frac{\partial^{2}}{\partial\theta^{2}}\log
f_{\theta}(X_{1})\right]\right)^{-1}.$

We consider the example of location parameter estimation. Suppose $\theta$ is
the location parameter and let $f_{\theta}(x)=f(x-\theta)$ and $g(x)=x$.
Specifically, it yields a lower bound on the covariance matrix of the
probability measure $\mu_{\theta}$ in the case that
$\theta=\mathbb{E}_{\mu_{\theta}}[X]$, i.e., a random sample
$X\sim\mu_{\theta}$ is an unbiased estimator of the mean $\theta$. Apart from
this implication, an alternative proof of the same lower bound on the
covariance matrix is presented in [23]. It is worth noting that a compactly
supported probability measure $\mu$ would suffice to ensure the Cramér-Rao
inequality holds.

######  Lemma A.3.

Let $\mu(\mathrm{d}x)=\exp(-U(x))\mathrm{d}x$ be a probability measure on
${\mathbb{R}}^{d}$ such that $U$ of class $C^{2}$ on the interior of its
domain. Suppose $X$ is a random sample from $\mu$. Then the covariance matrix
is lower bounded as
$\mathrm{Cov}_{\mu}(X)\succeq\left(\mathbb{E}_{\mu}\left[\nabla^{2}U(X)\right]\right)^{-1}$.

###  A.3. Proof of Propositions 3.5

###### Proof.

By Itô SDE defined in (7), we have the distribution $\overline{X}_{t}$, which
is given by

$\overline{X}_{t}|\overline{X}_{0}=x_{0}\sim
N((1-t)x_{0},t(2-t)\mathbf{I}_{d}).$ (16)

Due to the Cauchy-Lipschitz theory [4, Section 2], the push-forward map
$X^{*}_{t}$ and process $\overline{X}_{t}$ have the same distribution for any
$t\in[0,1-\varepsilon]$. Then by (16), we obtain

$X^{*}_{t}\overset{d}{=}\overline{X}_{t}\overset{d}{=}(1-t)X+\sqrt{t(2-t)}Y$

with $X\sim\nu,Y\sim\gamma_{d}$. Recall that $X^{*}_{1-\varepsilon}$ and
$\varepsilon X+\sqrt{1-\varepsilon^{2}}Y$ have the same distribution.
Therefore, by the definition of $W_{2}$ and Cauchy-Schwarz’s inequality, it
yields

$\displaystyle W^{2}_{2}(\nu\circ({X^{*}_{1-\varepsilon}})^{-1},\gamma_{d})$
$\displaystyle\leq\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varepsilon
x+(\sqrt{1-\varepsilon^{2}}-1)y|^{2}p(x)\phi(y)\mathrm{d}x\mathrm{d}y$
$\displaystyle\leq
2\varepsilon^{2}\int_{\mathbb{R}^{d}}|x|^{2}p(x)\mathrm{d}x+2\left(\sqrt{1-\varepsilon^{2}}-1\right)^{2}\int_{\mathbb{R}^{d}}|y|^{2}\phi(y)\mathrm{d}y$
$\displaystyle=2\varepsilon^{2}\,\mathbb{E}_{p}[|X|^{2}]+2d\left(\sqrt{1-\varepsilon^{2}}-1\right)^{2}.$

Let $\varepsilon\rightarrow 0$, and it yields $\lim_{\varepsilon\to
0}W_{2}(\nu\circ({X^{*}_{1-\varepsilon}})^{-1},\gamma_{d})=0$, which completes
the proof. ∎

##  Appendix B Proof of Theorem 2.5 and Theorem 2.6

###  B.1. Bound on the Lipschitz constant of the flow map

We take a close look at Lipschitz properties of the Föllmer flow (1). In order
to derive functional inequalities, we deploy the approach of Lipschitz changes
of variables from the Gaussian measure $\gamma_{d}$ to the target measure
$\nu$.

One key argument is to bound the maximum eigenvalue of the Jacobian matrix of
velocity field denoted as $\lambda_{\max}(\nabla V(t,x))$. By integrating both
sides of (1) w.r.t. time $s\in[0,t]$, we have

$X_{t}(x)-X_{0}(x)=\int_{0}^{t}V(s,X_{s}(x))\mathrm{d}s,\quad X_{0}(x)=x.$
(17)

Taking the first-order derivative w.r.t $x$ on both sides of (17), we get

$\nabla X_{t}(x)-\nabla X_{0}(x)=\int_{0}^{t}\nabla V(s,X_{s}(x))\nabla
X_{s}(x)\mathrm{d}s.$ (18)

Taking the first-order derivative w.r.t $t$ on both sides of (18), we get

$\frac{\partial}{\partial t}\nabla X_{t}(x)=\nabla V(t,X_{t}(x))\nabla
X_{t}(x).$ (19)

Let $a_{t}=|\nabla X_{t}(x)r|^{2}$ with $|r|=1$. Assume $\lambda_{\max}(\nabla
V(t,x))\leq\theta_{t}$. By (19) we get

$\displaystyle\frac{\partial}{\partial t}a_{t}=2\left<(\nabla
X_{t})r,\frac{\partial}{\partial t}(\nabla X_{t})r\right>=2\left<(\nabla
X_{t})r,\left(\nabla V(t,X_{t})\nabla X_{t}\right)r\right>\leq
2\theta_{t}a_{t}.$

The above display and Grönwall’s inequality imply

$\displaystyle\|\nabla
X_{t}(x)\|_{\mathrm{op}}=\sup_{\|r\|_{2}=1}\sqrt{a_{t}}\leq\sup_{\|r\|_{2}=1}\sqrt{a_{0}}\exp\left(\int_{0}^{t}\theta_{s}\mathrm{d}s\right)=\exp\left(\int_{0}^{t}\theta_{s}\mathrm{d}s\right).$

Let $t=1$, then we get

$\mathrm{Lip}(X_{1}(x))\leq\|\nabla
X_{1}(x)\|_{\mathrm{op}}\leq\exp\left(\int_{0}^{1}\theta_{s}\mathrm{d}s\right).$
(20)

###  B.2. Lipschitz properties of transport maps

In this subsection, we show that the considered flow map is Lipschitz in
various settings. The following is the main result of this subsection and it
covers the Lipschitz statements of Theorem 2.5 and Theorem 2.6.

######  Theorem B.1.

  * (1)

Suppose that either $p$ is $\kappa$-semi-log-concave for some $\kappa>0$, or
$p$ is $\kappa$-semi-log-concave for some $\kappa\in\mathbb{R}$ and
$D<+\infty$. Then the Föllmer flow (1) has a unique solution for all
$t\in[0,1]$. Furthermore,

    * (a)

If $\kappa D^{2}\geq 1$, then $X_{1}(x)$ is a Lipschitz mapping with constant
$\tfrac{1}{\sqrt{\kappa}}$, or equivalently,

$\|\nabla X_{1}(x)\|^{2}_{\mathrm{op}}\leq\frac{1}{\kappa},\quad\forall
x\in\mathbb{R}^{d}.$

    * (b)

If $\kappa D^{2}<1$, then $X_{1}(x)$ is a Lipschitz mapping with constant
$\exp\left(\frac{1-\kappa D^{2}}{2}\right)D$, or equivalently,

$\|\nabla X_{1}(x)\|^{2}_{\mathrm{op}}\leq\exp\left(1-\kappa
D^{2}\right)D^{2},\quad\forall x\in\mathbb{R}^{d}.$

  * (2)

Fix a probability measure $\rho$ on $\mathbb{R}^{d}$ supported on a ball of
radius $R$ and let $p:=N(0,\sigma^{2}\mathbf{I}_{d})*\rho$. Then the Föllmer
flow (1) has a unique solution for all $t\in[0,1]$. Furthermore, $X_{1}(x)$ is
a Lipschitz mapping with constant
$\sigma\exp\left(\frac{R^{2}}{2\sigma^{2}}\right)$, or equivalently,

$\|\nabla
X_{1}(x)\|^{2}_{\mathrm{op}}\leq\sigma^{2}\exp\left(\frac{R^{2}}{\sigma^{2}}\right),\quad\forall
x\in\mathbb{R}^{d}.$

###### Proof.

Combine Theorem B.3-(4) and Corollaries B.4-B.5 and then complete the proof. ∎

In fact, the existence of a solution to the IVP (1) also relies on controlling
$\nabla V$. To this end, we represent $\nabla V$ as a covariance matrix. We
start by defining a measure $p^{tx,1-t^{2}}$ on ${\mathbb{R}}^{d}$, for fixed
$t\in[0,1)$ and $x\in{\mathbb{R}}^{d}$, by

$\displaystyle p^{tx,1-t^{2}}(y):=$
$\displaystyle\frac{\varphi^{tx,1-t^{2}}(y)r(y)}{\mathcal{Q}_{1-t}r(x)},\quad
r:=\frac{\mathrm{d}\nu}{\mathrm{d}\gamma_{d}}=\frac{p}{\phi},$ (21)

where $\varphi^{tx,1-t^{2}}(y)$ is the density of the $d$-dimensional Gaussian
measure with mean $tx$ and covariance $(1-t^{2})\mathbf{I}_{d}$ and

$\displaystyle\mathcal{Q}_{1-t}r(x)=\int_{\mathbb{R}^{d}}\varphi^{tx,1-t^{2}}(y)r(y)\mathrm{d}y=\int_{\mathbb{R}^{d}}\phi(z)r(tx+\sqrt{1-t^{2}}z)\mathrm{d}z.$
(22)

Notice that

$\displaystyle\mathcal{Q}_{1-t}r(x)=(2\pi)^{d/2}\exp\left(\frac{|x|^{2}}{2}\right)\frac{1}{(2\pi(1-t^{2}))^{d/2}}\int_{\mathbb{R}^{d}}p(y)\exp\left(-\frac{|x-ty|^{2}}{2(1-t^{2})}\right)\mathrm{d}y.$

Hence, we obtain

$V(t,x)=\frac{x+S(t,x)}{t}=\frac{\nabla\log\mathcal{Q}_{1-t}r(x)}{t},\quad
0<t\leq 1.$

######  Lemma B.2.

Suppose velocity field $V$ is defined in (2), then

$\nabla
V(t,x)=\frac{t}{(1-t^{2})^{2}}\mathrm{Cov}\left(p^{tx,1-t^{2}}\right)-\frac{t}{1-t^{2}}{\mathbf{I}}_{d},\quad\forall
t\in(0,1),\quad\nabla V(0,x)=0.$ (23)

###### Proof.

By taking the first-order and the second-order derivatives on both sides in
(22), we get

$\displaystyle\nabla\mathcal{Q}_{1-t}r(x)$
$\displaystyle=\frac{t}{1-t^{2}}\int_{\mathbb{R}^{d}}(y-tx)\varphi^{tx,1-t^{2}}(y)r(y)\mathrm{d}y,$
$\displaystyle\nabla^{2}\mathcal{Q}_{1-t}r(x)$
$\displaystyle=\frac{t^{2}}{(1-t^{2})^{2}}\int_{\mathbb{R}^{d}}(y-tx)^{\otimes
2}\varphi^{tx,1-t^{2}}(y)r(y)\mathrm{d}y-\left(\frac{t^{2}}{1-t^{2}}\int_{\mathbb{R}^{d}}\varphi^{tx,1-t^{2}}(y)r(y)\mathrm{d}y\right){\mathbf{I}}_{d}.$

Then we obtain

$\displaystyle\nabla^{2}\log\mathcal{Q}_{1-t}r(x)$ $\displaystyle=$
$\displaystyle\frac{\nabla^{2}\mathcal{Q}_{1-t}r(x)}{\mathcal{Q}_{1-t}r(x)}-\left(\frac{\nabla\mathcal{Q}_{1-t}r(x)}{\mathcal{Q}_{1-t}r(x)}\right)^{\otimes
2}$ $\displaystyle=$
$\displaystyle\frac{t^{2}}{(1-t^{2})^{2}}\left[\int_{\mathbb{R}^{d}}(y-tx)^{\otimes
2}p^{tx,1-t^{2}}(y)\mathrm{d}y-\left(\int_{\mathbb{R}^{d}}(y-tx)p^{tx,1-t^{2}}(y)\mathrm{d}y\right)^{\otimes
2}\right]-\frac{t^{2}}{1-t^{2}}{\mathbf{I}}_{d}$ $\displaystyle=$
$\displaystyle\frac{t^{2}}{(1-t^{2})^{2}}\left[\int_{\mathbb{R}^{d}}y^{\otimes
2}p^{tx,1-t^{2}}(y)\mathrm{d}y-\left(\int_{\mathbb{R}^{d}}yp^{tx,1-t^{2}}(y)\mathrm{d}y\right)^{\otimes
2}\right]-\frac{t^{2}}{1-t^{2}}{\mathbf{I}}_{d}$ $\displaystyle=$
$\displaystyle\frac{t^{2}}{(1-t^{2})^{2}}\mathrm{Cov}(p^{tx,1-t^{2}})-\frac{t^{2}}{1-t^{2}}{\mathbf{I}}_{d}.$

Therefore, we get

$\nabla
V(t,x)=\frac{t}{(1-t^{2})^{2}}\mathrm{Cov}(p^{tx,1-t^{2}})-\frac{t}{1-t^{2}}{\mathbf{I}}_{d}.$
(24)

This completes the proof. ∎

Next, we use the representation of (23) to estimate the upper bound of $\nabla
V(t,x)$.

######  Theorem B.3.

Let $p$ be a probability measure on $\mathbb{R}^{d}$ with
$D:=(1/\sqrt{2})\mathrm{diam}(\mathrm{supp}(p))$.

  * (1)

For every $t\in[0,1)$,

$\frac{t}{1-t^{2}}\mathbf{I}_{d}\preceq\nabla
V(t,x)\preceq\left(\frac{tD^{2}}{(1-t^{2})^{2}}-\frac{t}{1-t^{2}}\right){\mathbf{I}}_{d}.$
(25)

  * (2)

Suppose that $p$ is $\beta$-semi-log-convex with $\beta\in(0,+\infty)$. Then
for any $t\in[0,1]$,

$\nabla V(t,x)\succeq\frac{t(1-\beta)}{\beta(1-t^{2})+t^{2}}{\mathbf{I}}_{d}.$
(26)

In particular, when $p\sim N\left(0,\frac{1}{\beta}\mathbf{I}_{d}\right)$,
then

$\nabla V(t,x)=\frac{t(1-\beta)}{\beta(1-t^{2})+t^{2}}{\mathbf{I}}_{d}.$

  * (3)

Let $\kappa\in{\mathbb{R}}$ and suppose that $p$ is $\kappa$-semi-log-concave.
Then for any
$t\in\left[\sqrt{\frac{\kappa}{\kappa-1}\mathds{1}_{\kappa<0}},1\right]$,

$\nabla
V(t,x)\preceq\frac{t(1-\kappa)}{\kappa(1-t^{2})+t^{2}}{\mathbf{I}}_{d}.$ (27)

  * (4)

Fix a probability measure $\rho$ on ${\mathbb{R}}^{d}$ supported on a ball of
radius $R$ and let $p:=N(0,\sigma^{2}\mathbf{I}_{d})*\rho$ with $\sigma>0$.
Then for any $t\in[0,1]$,

$\displaystyle\frac{(\sigma^{2}-1)t}{1+(\sigma^{2}-1)t^{2}}\mathbf{I}_{d}\preceq\nabla
V(t,x)\preceq
t\left\\{\frac{(\sigma^{2}-1)[1+(\sigma^{2}-1)t^{2}]+R^{2}}{[1+(\sigma^{2}-1)t^{2}]^{2}}\right\\}{\mathbf{I}}_{d}.$
(28)

###### Proof.

The proof idea of this theorem follows similar arguments as in [65, Lemma
3.3].

  * (1)

By [32, Theorem 2.6], there exists a closed ball with radius less than
$D:=(1/\sqrt{2})\mathrm{diam}(\mathrm{supp}(p))$ that contains
$\mathrm{supp}(p)$ in ${\mathbb{R}}^{d}$. Then the desired bounds are a direct
result of $0{\mathbf{I}}_{d}\preceq\mathrm{Cov}(p^{tx,1-t^{2}})\preceq
D^{2}{\mathbf{I}}_{d}$ and (23).

  * (2)

For any $t\in(0,1)$, recall that (23) reads

$\nabla
V(t,x)=\frac{t}{(1-t^{2})^{2}}\mathrm{Cov}\left(p^{tx,1-t^{2}}\right)-\frac{t}{1-t^{2}}{\mathbf{I}}_{d}.$
(29)

On the one hand, let $p$ be $\beta$-semi-log-convex for some $\beta>0$. Then
for any $t\in[0,1),p^{tx,1-t^{2}}$ is
$\left(\beta+\frac{t^{2}}{1-t^{2}}\right)$-semi-log-convex because

$\displaystyle-\nabla^{2}\log\left(p^{tx,1-t^{2}}(y)\right)=-\nabla^{2}\log\left(r(y)\phi(y)\right)-\nabla^{2}\log\left(\frac{\varphi^{tx,1-t^{2}}(y)}{\phi(y)}\right)\preceq\left(\beta+\frac{t^{2}}{1-t^{2}}\right)\mathbf{I}_{d}$

where we use that $p(y)=r(y)\phi(y)$. On the other hand, by Lemma A.3, we
obtain

$\mathrm{Cov}\left(p^{tx,1-t^{2}}\right)\succeq\left(\beta+\frac{t^{2}}{1-t^{2}}\right)^{-1}{\mathbf{I}}_{d}.$

Furthermore, by (29), we obtain

$\displaystyle\nabla
V(t,x)\succeq\left\\{\frac{t}{(1-t^{2})^{2}}\left(\beta+\frac{t^{2}}{1-t^{2}}\right)^{-1}-\frac{t}{1-t^{2}}\right\\}\mathbf{I}_{d}=\frac{t(1-\beta)}{\beta(1-t^{2})+t^{2}}\mathbf{I}_{d}.$

Recall that $(X_{t})_{t\in[0,1]}$ satisfies the IVP (1), then we have

$\nabla V(t,x)=\frac{\nabla^{2}\log\mathcal{Q}_{1-t}r(x)}{t},\quad
r(x):=\frac{p(x)}{\phi(x)}.$

Since $p\sim N\left(0,\frac{1}{\beta}\mathbf{I}_{d}\right)$, then it yields

$r(x)=\beta^{d/2}\exp\left(-\frac{\beta-1}{2}|x|^{2}\right)\propto\exp\left(-\frac{\beta-1}{2}|x|^{2}\right),$

where the symbol $\propto$ signifies equality up to a constant which does not
depend on $x$. Then by straightforward calculation for
$\mathcal{Q}_{1-t}r(x)$, we obtain

$\displaystyle\mathcal{Q}_{1-t}r(x)$
$\displaystyle\propto\int_{\mathbb{R}^{d}}\exp\left\\{-\frac{\beta-1}{2}\left|tx+\sqrt{1-t^{2}}y\right|^{2}-\frac{|y|^{2}}{2}\right\\}\mathrm{d}y$
$\displaystyle=\int_{\mathbb{R}^{d}}\exp\left\\{-\frac{(\beta-1)t^{2}}{2}|x|^{2}-(\beta-1)t\sqrt{1-t^{2}}\left<x,y\right>-\frac{\beta(1-t^{2})+t^{2}}{2}|y|^{2}\right\\}\mathrm{d}y$
$\displaystyle=\exp\left(-\frac{(\beta-1)t^{2}|x|^{2}}{2\beta_{t}}\right)\int_{\mathbb{R}^{d}}\exp\left\\{-\frac{\beta_{t}}{2}\left|y+\frac{(\beta-1)t\sqrt{1-t^{2}}}{\beta_{t}}x\right|^{2}\right\\}\mathrm{d}y,$

where we denote $\beta_{t}:=(1-t^{2})\beta+t^{2}$. Considering that the
integrand in the last line is proportional to the density of a Gaussian
measure, then the value of the integral does not depend on $x$, and

$\displaystyle\mathcal{Q}_{1-t}r(x)\propto\exp\left(-\frac{(\beta-1)t^{2}|x|^{2}}{2\beta_{t}}\right)=\exp\left(-\frac{|x|^{2}}{2}\cdot\frac{(\beta-1)t^{2}}{(1-t^{2})\beta+t^{2}}\right).$

So we have

$\nabla
V(t,x)=\frac{\nabla^{2}\mathcal{Q}_{1-t}r(x)}{t}=\frac{(1-\beta)t}{(1-t^{2})\beta+t^{2}}\mathbf{I}_{d}.$

  * (3)

Let $p$ be $\kappa$-semi-log-concave. Then for any $t\in[0,1)$,
$p^{tx,1-t^{2}}$ is $\left(\kappa+\frac{t^{2}}{1-t^{2}}\right)$-semi-log-
concave because

$\displaystyle-\nabla^{2}\log\left(p^{tx,1-t^{2}}(y)\right)=-\nabla^{2}\log\left(r(y)\phi(y)\right)-\nabla^{2}\log\left(\frac{\varphi^{tx,1-t^{2}}(y)}{\phi(y)}\right)\succeq\left(\kappa+\frac{t^{2}}{1-t^{2}}\right)\mathbf{I}_{d}$

where we use $p(y)=r(y)\phi(y)$. If
$t\in\left[\sqrt{\frac{\kappa}{\kappa-1}\mathds{1}_{\kappa<0}},1\right]$, then
$\kappa+\frac{t^{2}}{1-t^{2}}\geq 0$. By the well-known Brascamp-Lieb
inequality [8, 16], applied to functions of the form $\mathbb{R}^{d}\ni
x\mapsto f(x)=\left<x,v\right>$ for any $v\in\mathbb{S}^{d-1}$, we obtain

$\mathrm{Cov}\left(p^{tx,1-t^{2}}\right)\preceq\left(\kappa+\frac{t^{2}}{1-t^{2}}\right)^{-1}{\mathbf{I}}_{d}$

and the result follows by (23).

  * (4)

On the one hand, we have

$\displaystyle
p^{tx,1-t^{2}}(y)=\frac{(N(0,\sigma^{2}\mathbf{I}_{d})*\rho)(y)}{\varphi^{0,1}(y)}\cdot\frac{\varphi^{tx,1-t^{2}}(y)}{\mathcal{Q}_{1-t}\left(\frac{N(0,\sigma^{2}\mathbf{I}_{d})*\rho)}{\varphi^{0,1}}\right)(x)}=A_{x,t}\int_{\mathbb{R}^{d}}\varphi^{z,\sigma^{2}}(y)\varphi^{\frac{x}{t},\frac{1-t^{2}}{t^{2}}}(y)\rho(\mathrm{d}z),$

where the constant $A_{x,t}$ depends only on $x$ and $t$. Moreover, we obtain

$\
p^{tx,1-t^{2}}(y)=\int_{\mathbb{R}^{d}}\varphi^{\frac{(1-t^{2})z+\sigma^{2}tx}{1+(\sigma^{2}-1)t^{2}},\frac{\sigma^{2}(1-t^{2})}{1+(\sigma^{2}-1)t^{2}}}(y)\tilde{\rho}(\mathrm{d}z)$

where $\tilde{\rho}$ is a probability measure on $\mathbb{R}^{d}$ which is a
multiple of $\rho$ by a positive function. In particular, $\tilde{\rho}$ is
supported on the same ball as $\rho$. On the other hand, let $Z\sim\gamma_{d}$
and $Y\sim\tilde{\rho}$ be independent. Then

$\displaystyle\sqrt{\frac{\sigma^{2}(1-t^{2})}{1+(\sigma^{2}-1)t^{2}}}Z+\frac{(1-t^{2})}{1+(\sigma^{2}-1)t^{2}}Y+\frac{t\sigma^{2}}{1+(\sigma^{2}-1)t^{2}}x\sim
p^{tx,1-t^{2}}.$

Due to $0{\mathbf{I}}_{d}\preceq\mathrm{Cov}(Y)\preceq R^{2}{\mathbf{I}}_{d}$,
it holds that

$\displaystyle\frac{\sigma^{2}(1-t^{2})}{1+(\sigma^{2}-1)t^{2}}\mathbf{I}_{d}\preceq\mathrm{Cov}(p^{tx,1-t^{2}})\preceq\frac{\sigma^{2}(1-t^{2})[1+(\sigma^{2}-1)t^{2}]+(1-t^{2})^{2}R^{2}}{[1+(\sigma^{2}-1)t^{2}]^{2}}\mathbf{I}_{d}.$

By applying (23) again, it yields

$\displaystyle\frac{(\sigma^{2}-1)t}{1+(\sigma^{2}-1)t^{2}}\mathbf{I}_{d}\preceq\nabla
V(t,x)\preceq
t\left\\{\frac{(\sigma^{2}-1)[1+(\sigma^{2}-1)t^{2}]+R^{2}}{[1+(\sigma^{2}-1)t^{2}]^{2}}\right\\}{\mathbf{I}}_{d}.$

This completes the proof of Theorem B.3. ∎

Next, we present an upper bound on $\lambda_{\max}(\nabla V(t,x))$ and its
exponential estimation.

######  Corollary B.4.

Let $p$ be a probability measure on $\mathbb{R}^{d}$ with
$D:=(1/\sqrt{2})\mathrm{diam}(\mathrm{supp}(p))$ and suppose that $p$ is
$\kappa$-semi-log-concave with $\kappa\in[0,+\infty)$.

  * (1)

If $\kappa D^{2}\geq 1$, then

$\lambda_{\max}(\nabla
V(t,x))\leq\theta_{t}:=\frac{t(1-\kappa)}{t^{2}(1-\kappa)+\kappa}.$ (30)

and

$\exp\left(\int_{0}^{1}\theta_{s}\mathrm{d}s\right)=\frac{1}{\sqrt{\kappa}}.$
(31)

  * (2)

If $\kappa D^{2}<1$, then

$\lambda_{\max}(\nabla
V(t,x))\leq\theta_{t}:=\begin{cases}\frac{t(t^{2}+D^{2}-1)}{(1-t^{2})^{2}},\
&t\in[0,t_{0}],\\\ \frac{t(1-\kappa)}{t^{2}(1-\kappa)+\kappa},\
&t\in[t_{0},1],\end{cases}$ (32)

where $t_{0}=\sqrt{\frac{1-\kappa D^{2}}{(1-\kappa)D^{2}+1}}$ and

$\exp\left(\int_{0}^{1}\theta_{s}\mathrm{d}s\right)=\exp\left(\frac{1-\kappa
D^{2}}{2}\right)D.$ (33)

###### Proof.

By Theorem B.3, we obtain

$\displaystyle\lambda_{\max}(\nabla
V(t,x))\leq\frac{tD^{2}}{(1-t^{2})^{2}}-\frac{t}{1-t^{2}},\quad\lambda_{\max}(\nabla
V(t,x))\leq\frac{t(1-\kappa)}{\kappa(1-t^{2})+t^{2}},\quad\forall t\in[0,1].$

By simple algebra calculation, it yields

$\displaystyle\frac{t(D^{2}+t^{2}-1)}{(1-t^{2})^{2}}\leq\frac{t(1-\kappa)}{\kappa(1-t^{2})+t^{2}}\quad\text{if
and only if}\quad(1+D^{2}-\kappa D^{2})t^{2}\leq 1-\kappa D^{2}.$

We consider two cases.

  * (1)

$\kappa D^{2}\geq 1$: By considering $\kappa D^{2}=1$, we see that the bound
$(1+D^{2}-\kappa D^{2})t^{2}\leq 1-\kappa D^{2}$ cannot hold. So it would be
advantageous to use the bound

$\displaystyle\lambda_{\max}(\nabla
V(t,x))\leq\theta_{t}:=\frac{t(1-\kappa)}{\kappa(1-t^{2})+t^{2}}=\frac{t(1-\kappa)}{t^{2}(1-\kappa)+\kappa}.$

Next, we will compute $\exp\left(\int_{0}^{1}\theta_{t}\mathrm{d}t\right)$ and
we first check that the integral $\int_{0}^{1}\theta_{t}\mathrm{d}t$ is well-
defined. For this reason, we only need to consider whether the sign of the
denominator $(1-\kappa)t^{2}+\kappa$ is equal to 0.

The only case is $(1-\kappa)t^{2}+\kappa=0$ that happens when
$t^{2}_{0}:=\kappa/(\kappa-1)$. If $\kappa\in(0,1]$,
$(1-\kappa)t^{2}+\kappa\neq 0$. Thus, $\theta_{t}$ is integrable on $[0,1]$.
If $\kappa>1$, $t_{0}>1$. Then $\theta_{t}$ is integrable on $[0,1]$ as well.
The only case is $\kappa=0$ which results in $t_{0}=0$. However, in this case,
we cannot have $\kappa D^{2}\geq 1$ as $\kappa=0$. Then by simple calculation,

$\displaystyle\int_{0}^{1}\theta_{t}\mathrm{d}t=(1-\kappa)\int_{0}^{1}\frac{t\mathrm{d}t}{(1-\kappa)t^{2}+\kappa}=-\frac{1}{2}\log\kappa,\quad\exp\left(\int_{0}^{1}\theta_{t}\mathrm{d}t\right)\leq\frac{1}{\sqrt{\kappa}}.$

  * (2)

$\kappa D^{2}<1$: The condition $(1+D^{2}-\kappa D^{2})t^{2}\leq 1-\kappa
D^{2}$ is equivalent to

$t\leq\sqrt{\frac{1-\kappa D^{2}}{1+(1-\kappa)D^{2}}}$

since the denominator is nonnegative as $\kappa D^{2}<1$. Hence, we define

$\lambda_{\max}(\nabla
V(t,x))\leq\theta_{t}:=\begin{cases}\tfrac{t(t^{2}+D^{2}-1)}{(1-t^{2})^{2}},\
&0\leq t\leq t_{0},\\\ \tfrac{t(1-\kappa)}{t^{2}(1-\kappa)+\kappa},\
&t_{0}\leq t\leq 1,\end{cases}$

where $t_{0}:=\sqrt{\tfrac{1-\kappa D^{2}}{(1-\kappa)D^{2}+1}}$. In order to
compute integral $\int_{0}^{1}\theta_{t}\mathrm{d}t$, we note that, following
the discussion in the case $\kappa D^{2}\geq 1$, the denominators $1-t^{2}$
and $(1-\kappa)t^{2}+\kappa$ do not vanish in the intervals $[0,t_{0}]$ and
$[t_{0},1]$, respectively. For $t\in[0,t_{0}]$, using integral by parts, we
have

$\displaystyle\int_{0}^{t_{0}}\frac{t(t^{2}+D^{2}-1)}{(1-t^{2})^{2}}\mathrm{d}t=\frac{1}{2}\int_{0}^{t_{0}}(t^{2}+D^{2}-1)\mathrm{d}\left(\frac{1}{1-t^{2}}\right)$
$\displaystyle=\left.\frac{t^{2}+D^{2}-1}{2(1-t^{2})}\right|^{t=t_{0}}_{t=0}-\frac{1}{2}\int_{0}^{t_{0}}\frac{2t}{1-t^{2}}\mathrm{d}t=\frac{t^{2}_{0}}{2(1-t^{2}_{0})}D^{2}+\frac{1}{2}\log(1-t^{2}_{0})$
$\displaystyle=\frac{1-\kappa
D^{2}}{2}+\frac{1}{2}\log\left(\frac{D^{2}}{1+(1-\kappa)D^{2}}\right).$

For $t\in[t_{0},1]$, we have

$\displaystyle\int_{t_{0}}^{1}\frac{t(1-\kappa)}{\kappa+(1-\kappa)t^{2}}\mathrm{d}t=-\frac{1}{2}\log\left(t^{2}_{0}+(1-t^{2}_{0})\kappa\right)=-\frac{1}{2}\log\left(\frac{1}{1+(1-\kappa)D^{2}}\right).$

Hence, we obtain

$\int_{0}^{1}\theta_{t}\mathrm{d}t=\int_{0}^{t_{0}}\theta_{t}\mathrm{d}t+\int_{t_{0}}^{1}\theta_{t}\mathrm{d}t=\frac{1-\kappa
D^{2}}{2}+\log D.$

Then

$\exp\left(\int_{0}^{1}\theta_{t}\mathrm{d}t\right)=\exp\left(\frac{1-\kappa
D^{2}}{2}+\log D\right)=D\exp\left(\frac{1-\kappa D^{2}}{2}\right).$

This completes the proof of Corollary B.4. ∎

######  Corollary B.5.

Let $p$ be a probability measure on $\mathbb{R}^{d}$ with
$D:=(1/\sqrt{2})\mathrm{diam}(\mathrm{supp}(p))<\infty$ and suppose that $p$
is $\kappa$-semi-log-concave with $\kappa\in(-\infty,0)$. We have

$\lambda_{\max}(\nabla
V(t,x))\leq\theta_{t}:=\begin{cases}\tfrac{t(t^{2}+D^{2}-1)}{(1-t^{2})^{2}},\
&t\in[0,t_{0}]\\\ \tfrac{t(1-\kappa)}{t^{2}(1-\kappa)+\kappa},\
&t\in[t_{0},1]\end{cases}$ (34)

where $t_{0}=\sqrt{\tfrac{1-\kappa D^{2}}{(1-\kappa)D^{2}+1}}$ and

$\exp\left(\int_{0}^{1}\theta_{s}\mathrm{d}s\right)=\exp\left(\frac{1-\kappa
D^{2}}{2}\right)D.$ (35)

###### Proof.

By Theorem B.3, we obtain

$\displaystyle\lambda_{\max}(\nabla
V(t,x))\leq\frac{tD^{2}}{(1-t^{2})^{2}}-\frac{t}{1-t^{2}},\quad\forall
t\in[0,1),\quad\lambda_{\max}(\nabla
V(t,x))\leq\frac{t(1-\kappa)}{\kappa(1-t^{2})+t^{2}},\quad\forall
t\in\left[\sqrt{\frac{\kappa}{\kappa-1}},1\right].$

Then it yields

$\lambda_{\max}(\nabla
V(t,x))\leq\frac{t(t^{2}+D^{2}-1)}{(1-t^{2})^{2}},\quad\forall
t\in\left[0,\sqrt{\frac{\kappa}{\kappa-1}}\right).$

Next, since $0<\sqrt{\tfrac{\kappa}{\kappa-1}}<\sqrt{\tfrac{1-\kappa
D^{2}}{(1-\kappa)D^{2}+1}}\leq 1$ and $\kappa(1-t^{2})+t^{2}\geq 0$ for all
$t\geq\sqrt{\frac{\kappa}{\kappa-1}}$, then one obtains

$\frac{t(t^{2}+D^{2}-1)}{(1-t^{2})^{2}}\leq\frac{t(1-\kappa)}{\kappa(1-t^{2})+t^{2}}$

for all $t\in\left[\sqrt{\frac{\kappa}{\kappa-1}},\sqrt{\frac{-\kappa
D^{2}}{(1-\kappa)D^{2}+1}}\right]$. We define

$\lambda_{\max}(\nabla
V(t,x))\leq\theta_{t}:=\begin{cases}\tfrac{t(t^{2}+D^{2}-1)}{(1-t^{2})^{2}},\
&t\in[0,t_{0}]\\\ \tfrac{t(1-\kappa)}{t^{2}(1-\kappa)+\kappa},\
&t\in[t_{0},1]\end{cases}$

where $t_{0}:=\sqrt{\tfrac{1-\kappa D^{2}}{(1-\kappa)D^{2}+1}}$. As in the
proof of Corollary B.4, it holds that

$\displaystyle\int_{0}^{t_{0}}\theta_{t}\mathrm{d}t=\frac{1-\kappa
D^{2}}{2}+\frac{1}{2}\log\left(\frac{D^{2}}{1+(1-\kappa)D^{2}}\right),\quad\int_{t_{0}}^{1}\theta_{t}\mathrm{d}t=-\frac{1}{2}\log\left(\frac{1}{1+(1-\kappa)D^{2}}\right).$

Then we have

$\displaystyle\int_{0}^{1}\theta_{t}\mathrm{d}t=\frac{1-\kappa D^{2}}{2}+\log
D,\quad\exp\left(\int_{0}^{1}\theta_{t}\mathrm{d}t\right)=D\exp\left(\frac{1-\kappa
D^{2}}{2}\right).$

This completes the proof of Corollary B.5. ∎

##  Appendix C Proof of Theorems 4.2, 4.3 and 4.4

We start with a differential Lipschitz mapping
$T:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ associated with constant $C$. The
following result describes the Lipschitz properties of the derivatives of
composite mappings.

######  Lemma C.1.

Let $T:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ be a differential Lipschitz
mapping with constant $C$ and let $\zeta:\mathbb{R}^{d}\rightarrow\mathbb{R}$
be a continuously differentiable function. Then

$\nabla(\zeta\circ T)=[(\nabla\zeta)\circ T]\nabla T,$

where $(\nabla T)(x):\mathbb{R}^{d}\rightarrow\mathbb{R}^{d\times d}$ be a
Jacobian matrix for any $x\in\mathbb{R}^{d}$. Furthermore, we obtain

$\displaystyle|\nabla\zeta(T(x))|\leq\|\nabla
T(x)\|_{\mathrm{op}}\cdot|(\nabla\zeta)\circ(T(x))|\leq
C|(\nabla\zeta)\circ(T(x))|$

for all $x\in\mathbb{R}^{d}$.

Since the proof of this result is almost trivial by using the chain rule and
the Lipschitz mapping $T$, we omit it here. Through Lemma C.1, we can start
the proofs of the functional inequalities which follow from Theorems 2.5 and
2.6. We first begin with the $\Psi$-Sobolev inequalities defined in [20].

###  C.1. Proof of Theorem 4.2

###### Proof.

  * (1)

It can be seen from [20, Corollary 2.1] that for standard Gaussian measure
$\gamma_{d}$ on $\mathbb{R}^{d}$, we have the following $\Psi$-Sobolev
inequalities:

$\mathrm{Ent}^{\Psi}_{\gamma_{d}}(F)\leq\frac{1}{2}\int_{\mathbb{R}^{d}}\Psi^{\prime\prime}(F)|\nabla
F|^{2}\mathrm{d}\gamma_{d}$ (36)

for any smooth function $F:\mathbb{R}^{d}\rightarrow\mathcal{I}$. Let
$(X_{t})_{t\in[0,1]}$ be the solution of IVP (1) so that $X_{1}\sim p$ if
$X_{0}\sim N(0,\mathbf{I}_{d})$. Suppose that
$X_{1}(x):\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is a Lipschitz mapping with
constant $C$ and let $F:=\zeta\circ
X_{1}:\mathbb{R}^{d}\rightarrow\mathcal{I}$ with
$\zeta:\mathbb{R}^{d}\rightarrow\mathcal{I}$. Then combining Lemma C.1, (36)
and $p=\gamma_{d}\circ(X_{1})^{-1}$ we have

$\displaystyle\mathrm{Ent}^{\Psi}_{p}(\zeta)=\mathrm{Ent}^{\Psi}_{\gamma_{d}}(F)\leq\frac{1}{2}\int_{\mathbb{R}^{d}}\Psi^{\prime\prime}(F)|\nabla
F|^{2}\mathrm{d}\gamma_{d}$
$\displaystyle\leq\frac{C^{2}}{2}\int_{\mathbb{R}^{d}}\Psi^{\prime\prime}(\zeta\circ
X_{1})|\nabla\zeta\circ X_{1}|^{2}\mathrm{d}\gamma_{d}$
$\displaystyle=\frac{C^{2}}{2}\int_{\mathbb{R}^{d}}\Psi^{\prime\prime}(\zeta(x))|\nabla\zeta(x)|^{2}p(x)\mathrm{d}x.$

The proof is complete by Theorems 2.5 and 2.6.

  * (2)

Let random vector $Y\sim\rho$, let $\tilde{\rho}$ be the law of
$\Sigma^{-1/2}Y$, and define $\tilde{p}:=\gamma_{d}*\tilde{\rho}$. Set
$\lambda_{\min}:=\lambda_{\min}(\Sigma)$ and
$\lambda_{\max}:=\lambda_{\max}(\Sigma)$. Then combining Lemma C.1, (36) and
$\tilde{p}=\gamma_{d}\circ(X_{1})^{-1}$, we have

$\mathrm{Ent}^{\Psi}_{\tilde{p}}(\zeta)\leq\frac{\exp(\lambda^{-1}_{\min}R^{2})}{2}\int_{\mathbb{R}^{d}}\Psi^{\prime\prime}(\zeta(x))|\nabla\zeta(x)|^{2}\tilde{p}(x)\mathrm{d}x.$

Let $p=N(a,\Sigma)*\rho$ and let $\tilde{X}\sim\tilde{p}$ such that

$\displaystyle\Sigma^{1/2}\tilde{X}+a=\Sigma^{1/2}\left(X+\Sigma^{-1/2}Y\right)+a=\left(\Sigma^{1/2}X+a\right)+Y\sim
p=N(a,\Sigma)*\rho,$

where $X\sim N(0,\mathbf{I}_{d})$. Given
$\zeta:\mathbb{R}^{d}\rightarrow\mathcal{I}$ and let
$\tilde{\zeta}(x):=\zeta(\Sigma^{1/2}x+a)$ so that

$\displaystyle\mathrm{Ent}^{\Psi}_{p}(\zeta)=\mathrm{Ent}^{\Psi}_{\tilde{p}}(\tilde{\zeta})\leq\frac{\exp(\lambda^{-1}_{\min}R^{2})}{2}\int_{\mathbb{R}^{d}}\Psi^{\prime\prime}(\tilde{\zeta}(x))|\nabla\tilde{\zeta}(x)|^{2}\tilde{p}(x)\mathrm{d}x.$

Since
$(\nabla\tilde{\zeta})(x)=\Sigma^{1/2}\left(\nabla\zeta(\Sigma^{1/2}x+a)\right)$,
we get

$|(\nabla\tilde{\zeta})(x)|^{2}\leq\lambda_{\max}\left|\left(\nabla\zeta(\Sigma^{1/2}x+a)\right)\right|^{2}.$

Furthermore, it yields that

$\displaystyle\mathrm{Ent}^{\Psi}_{p}(\zeta)\leq\frac{\lambda_{\max}\exp(\lambda^{-1}_{\min}R^{2})}{2}\int_{\mathbb{R}^{d}}\Psi^{\prime\prime}(\zeta(x))|\nabla\
\zeta(x)|^{2}p(x)\mathrm{d}x.$

This completes the proof. ∎

###  C.2. Proof of Theorem 4.3

###### Proof.

  * (1)

By using [54, Theorem 4.3], then the Gaussian measure $\gamma_{d}$ on
$\mathbb{R}^{d}$ satisfies the following Gaussian isoperimetric inequality:

$\gamma_{d}(K_{t})\geq\Phi\left(\gamma_{d}(K)+t\right),\quad t\geq 0$

for any Borel measurable set $K_{t}:=K+tB_{2}^{d}$ and
$K\subseteq\mathbb{R}^{d}$. Therefore, suppose $(X_{t})_{t\in[0,1]}$ be the
solution of IVP (1) so that $X_{1}\sim p$ if $X_{0}\sim N(0,\mathbf{I}_{d})$.
Moreover, suppose that $X_{1}(x):\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is a
Lipschitz mapping with constant $C$, then for any fixed $x\in\mathbb{R}^{d}$,

$|X_{1}(x+y)-X_{1}(x)|\leq C|y|,\quad\forall y\in\mathbb{R}^{d}.$

We first show the following result:

$X^{-1}_{1}(E)+\frac{t}{C}B_{2}^{d}\subseteq X^{-1}_{1}(E_{t}),\quad
E_{t}:=E+tB_{2}^{d}$ (37)

for any Borel measurable set $E\subseteq\mathbb{R}^{d}$ and $t\geq 0$. To
obtain (37), we only need to prove that

$X_{1}\left(X^{-1}_{1}(E)+\frac{r}{C}B_{2}^{d}\right)\subseteq E_{t},\quad
t\geq 0$

or, in other words, if $x\in X^{-1}_{1}(E)+\frac{t}{C}B_{2}^{d}$, then
$X_{1}(x)\in E_{t}$ for any Borel measurable set $K$. Furthermore, if we
assume

$x\in X^{-1}_{1}(E)+\frac{t}{C}B_{2}^{d}\quad\text{so that}\quad
x=\theta+\frac{t}{C}h$

for some $\theta\in X^{-1}_{1}(E)$ and $h\in B_{2}^{d}$, we have
$X_{1}\left(x-\frac{t}{C}h\right)\in E$. Then it yields that

$\left|X_{1}\left(x-\frac{t}{C}h\right)-X_{1}(x)\right|\leq t,\quad t\geq 0$

where $x-\frac{t}{C}h\in X^{-1}_{1}(E)$. Therefore, $X_{1}(x)\in E_{t}$ as
desired. Finally, combining the Gaussian isoperimetric inequality and (37), it
yields

$\displaystyle
p(E_{t})=\gamma_{d}\left(X^{-1}_{1}(E_{t})\right)\geq\gamma_{d}\left(X^{-1}_{1}(E)+\frac{t}{C}B_{2}^{d}\right)\geq\Phi\left(\gamma_{d}\left[X^{-1}_{1}(E)+\frac{t}{C}\right]\right)=\Phi\left(p(E)+\frac{t}{C}\right).$

This proof is completed by Theorems 2.5 and 2.6.

  * (2)

Let random vector $Y\sim\rho$, let $\tilde{\rho}$ be the law of
$\Sigma^{-1/2}Y$, and define measure $\tilde{p}:=\gamma_{d}*\tilde{\rho}$. Set
$\lambda_{\min}:=\lambda_{\min}(\Sigma)$ and
$\lambda_{\max}:=\lambda_{\max}(\Sigma)$. Similar to the argument of part (1),
for any Borel set $E\subset\mathbb{R}^{d}$ and $t\geq 0$, we obtain

$\tilde{p}\left(E_{t}\right)\geq\Phi\left(\tilde{p}(E)+\frac{t}{C}\right),C:=\left(\lambda_{\min}\right)^{1/2}\exp\left(\frac{R^{2}}{2\lambda_{\min}}\right).$

Let $p=N(a,\Sigma)*\rho$ and let $\tilde{X}\sim\tilde{p}$ so that

$\Sigma^{1/2}\tilde{X}+a=\left(\Sigma^{1/2}X+a\right)+Y\sim
p=N(a,\Sigma)*\rho$

and $X\sim N(0,\mathbf{I}_{d})$. Then for any Borel measurable set
$E\subset\mathbb{R}^{d}$ and $t\geq 0$, it yields that

$\displaystyle
p(E_{t})=\tilde{p}\left(\Sigma^{-1/2}(E-a)+\Sigma^{-1/2}tB_{2}^{d}\right)\geq\tilde{p}\left(\Sigma^{-1/2}(E-a)+t\lambda^{-1/2}_{\max}B_{2}^{d}\right).$

Hence, we obtain

$p(E_{t})\geq\Phi\left(\tilde{p}\left[\Sigma^{-1/2}(E-a)\right]+\frac{t\lambda^{-1/2}_{\max}}{C}\right).$

We obtain the desired result by applying
$\tilde{p}\left[\Sigma^{-1/2}(E-a)\right]=p(E)$.

This completes the proof. ∎

###  C.3. Proof of Theorem 4.4

###### Proof.

  * (1)

We will use the fact that [68, Proposition 3.1] the $q$-Poincaré inequality
holds for the standard Gaussian measure $\gamma_{d}$ on $\mathbb{R}^{d}$:

$\mathbb{E}_{\gamma_{d}}\left[F^{q}\right]\leq(q-1)^{q/2}\mathbb{E}_{\gamma_{d}}\left[|\nabla
F|^{q}\right],$ (38)

for any smooth function $F\in L^{q}(\gamma_{d})$ with
$\mathbb{E}_{\gamma_{d}}[F]=0$. Let $(X_{t})_{t\in[0,1]}$ be the solution of
IVP (1) so that $X_{1}\sim p$ if $X_{0}\sim N(0,\mathbf{I}_{d})$. Suppose that
$X_{1}(x):\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is a Lipschitz mapping with
constant $C$ and let $F:=\eta\circ X_{1}$. Then combining Lemma C.1, (38) and
$p=\gamma_{d}\circ(X_{1})^{-1}$ we have

$\displaystyle\mathbb{E}_{p}[\eta^{q}]=\mathbb{E}_{\gamma_{d}}[F^{q}]\leq(q-1)^{q/2}\mathbb{E}_{\gamma_{d}}[|\nabla
F|^{q}]\leq C^{q}(q-1)^{q/2}\mathbb{E}_{p}[|\nabla\eta|^{q}].$

The proof is complete by Theorems 2.5 and 2.6.

  * (2)

Let $Y\sim\rho$, let $\tilde{\rho}$ be the law of $\Sigma^{-1/2}Y$, and define
$\tilde{p}:=N(0,\mathbf{I}_{d})*\tilde{\rho}$. Set
$\lambda_{\min}:=\lambda_{\min}(\Sigma)$ and
$\lambda_{\max}:=\lambda_{\max}(\Sigma)$. The argument of part (1) gives,

$\mathbb{E}_{\tilde{p}}[\eta^{q}]\leq\exp\left(\frac{qR^{2}}{2\lambda_{\min}}\right)\lambda_{\min}^{q/2}(q-1)^{q/2}\mathbb{E}_{\tilde{p}}[|\nabla\eta|^{q}].$

Let $p=N(a,\Sigma)*\rho$ and let $\tilde{X}\sim\tilde{p}$ such that

$\Sigma^{1/2}\tilde{X}+a=\left(\Sigma^{1/2}X+a\right)+Y\sim
p=N(a,\Sigma)*\rho$

and $X\sim N(0,\mathbf{I}_{d})$. Let
$\tilde{\eta}(x):=\eta\left(\Sigma^{1/2}x+a\right)$ so that

$\displaystyle\mathbb{E}_{p}[\eta^{q}]=\mathbb{E}_{\tilde{p}}\left[(\tilde{\eta})^{q}\right]\leq\exp\left(\frac{qR^{2}}{2\lambda_{\min}}\right)\lambda_{\min}^{q/2}(q-1)^{q/2}\mathbb{E}_{\tilde{p}}[|\nabla\tilde{\eta}|^{q}].$

Since
$(\nabla\tilde{\eta})(x)=\Sigma^{1/2}\left(\nabla\eta\left(\Sigma^{1/2}x+a\right)\right)$
we have

$\left|(\nabla\tilde{\eta})(x)\right|^{q}\leq(\lambda_{\max})^{q/2}\left|\nabla\eta\left(\Sigma^{1/2}x+a\right)\right|^{q}.$

Further, we obtain

$\displaystyle\mathbb{E}_{p}[\eta^{q}]=\mathbb{E}_{\tilde{p}}\left[(\tilde{\eta})^{q}\right]\leq(\lambda_{\min}\lambda_{\max})^{q/2}\exp\left(\frac{qR^{2}}{2\lambda_{\min}}\right)(q-1)^{q/2}\mathbb{E}_{p}[|\nabla\eta|^{q}].$

This completes the proof. ∎

##  Appendix D Time changes

######  Lemma D.1.

Let $(\overline{X}_{t})_{t\in[0,1)}$ be a diffusion process defined by (7)
with $\varepsilon\to 0$ and let $(\overline{Y}_{s})_{s\geq 0}$ be an Ornstein-
Uhlenbeck process $(\overline{Y}_{s})_{s\geq 0}$ defined by

$\mathrm{d}\overline{Y}_{s}=-\overline{Y}_{s}\mathrm{d}s+\sqrt{2}\mathrm{d}\overline{W}_{s},\quad\overline{Y}_{0}\sim\nu,\quad
s\geq 0.$ (39)

Then $(\overline{X}_{t})_{t\in[0,1)}$ is equivalent to
$(\overline{Y}_{s})_{s\geq 0}$ through the change of time formula
$t=1-e^{-s}$.

###### Proof.

Let $s=-\log(1-t)$ for any $t\in[0,1)$. By applying (7), it yields

$\mathrm{d}\overline{X}_{1-e^{-s}}=-\overline{X}_{1-e^{-s}}\mathrm{d}s+\sqrt{2}\mathrm{d}\overline{W}_{s},\quad\overline{X}_{0}\sim\nu,\quad
s\geq 0.$

On the one hand, since (39) has a unique strong solution, it indicates
$\overline{Y}_{s}=\overline{X}_{1-e^{-s}}$ for all $s\geq 0$. On the other
hand, the infinitesimal generator of Markov process $(\overline{Y}_{s})_{s\geq
0}$ is given by

$L^{\overline{Y}}=\Delta-x\cdot\nabla.$ (40)

By using (7), the infinitesimal generator of $(\overline{X}_{t})_{t\in[0,1)}$
is given by

$L^{\overline{X}}_{t}=\frac{1}{1-t}(\Delta-x\cdot\nabla).$ (41)

Furthermore, combining the chain rule and straightforward calculation, we
obtain that processes $\overline{X}_{t}$ and $\overline{Y}_{s}$ have the same
infinitesimal generator, which implies $\overline{X}_{t}=\overline{Y}_{s}$ for
any $t\in[0,1),s=-\log(1-t)$. ∎

######  Lemma D.2.

Let $(X^{*}_{t})_{t\in[0,1)}$ be the time reversal of a Föllmer flow
associated to probability measure $\nu$ defined by (10) with $\varepsilon\to
0$ and let $(Y^{*}_{s})_{s\geq 0}$ be a heat flow from probability measure
$\nu$ to the standard Gaussian measure $\gamma_{d}$ defined by

$\displaystyle\mathrm{d}Y^{*}_{s}(x)=-\nabla\log\left\\{\int_{\mathbb{R}^{d}}r\left(e^{-s}Y^{*}_{s}(x)+\sqrt{1-e^{-2s}}z\right)\mathrm{d}\gamma_{d}(z)\right\\}\mathrm{d}s$
(42)

where $r(x):=(\mathrm{d}\nu/\mathrm{d}\gamma_{d})(x),Y^{*}_{0}\sim\nu$ for all
$s\geq 0$. Then $(X^{*}_{t})_{t\in[0,1)}$ is equivalent to $(Y^{*}_{s})_{s\geq
0}$ through the change of time formula $t=1-e^{-s}$.

###### Proof.

Let $s=-\log(1-t)$ for every $t\in[0,1)$. By (10), it yields

$\displaystyle\mathrm{d}X^{*}_{1-e^{-s}}(x)=-\nabla\log\left\\{\int_{\mathbb{R}^{d}}r\left(e^{-s}X^{*}_{1-e^{-s}}(x)+\sqrt{1-e^{-2s}}z\right)\mathrm{d}\gamma_{d}(z)\right\\}\mathrm{d}s$

where $X^{*}_{0}\sim\nu$ for all $s\geq 0$. The expression above indicates
that $Y^{*}_{s}:=X^{*}_{1-e^{-s}}$ satisfies (42). ∎

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