# Well-posedness of McKean-Vlasov SDEs with density-dependent drift

Anh-Dung Le

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Anh-Dung Le. Well-posedness of McKean-Vlasov SDEs with density-dependent drift. 2024. hal-04561088

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# Well-posedness of McKean-Vlasov SDEs with density-dependent drift

Anh-Dung Le

► To cite this version:

Anh-Dung Le. Well-posedness of McKean-Vlasov SDEs with density-dependent drift. 2024. hal-04561088

HAL Id: hal-04561088

https://hal.science/hal-04561088v1

Preprint submitted on 26 Apr 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Well-posedness of McKean-Vlasov SDEs with density-dependent drift*

Anh-Dung Le

Toulouse School of Economics, 1 Esplanade de l'Université, 31000 Toulouse, France
leanhdung1994@gmail.com

April 26, 2024

## Abstract

In this paper, we study the well-posedness of McKean-Vlasov stochastic differential equations (SDE) whose drift depends pointwisely on marginal density and satisfies a condition about local integrability in time-space variables. The drift is assumed to be Lipschitz continuous in distribution variable with respect to Wasserstein metric $W_p$. Our approach is by approximation with mollified SDEs. We establish a new estimate about Hölder continuity in time of marginal density. Then we deduce that the marginal distributions (resp. marginal densities) of the mollified SDEs converge in $W_p$ (resp. topology of compact convergence) to the solution of the Fokker-Planck equation associated with the density-dependent SDE. We prove strong existence of a solution. Weak and strong uniqueness are obtained when $p = 1$, the drift coefficient is bounded, and the diffusion coefficient is distribution free.

**AMS subject classification:** 60B10, 60H10.

**Keywords:** McKean-Vlasov SDEs, density-dependent SDEs, well-posedness, local integrability.

## 1 Introduction

The study of distribution-dependent SDEs started with McKean's seminal work [1] about Vlasov equation of plasma which had been proposed in [2]. Classical results about the solvability of McKean-Vlasov equations include [3, 4, 5]. Since then, the literature on the well-posedness of McKean-Vlasov SDEs has been extended significantly. For recent results, we refer to [6, 7, 8, 9, 10, 11], the survey [12], and references therein. For applications on mean-field games, see the two-volume monograph [13].

In this paper, we consider the density-dependent McKean-Vlasov SDEs studied in [14]. More precisely, let $p \in [1, \infty)$ and $P_p(\mathbb{R}^d)$ be the space of Borel probability measures on $\mathbb{R}^d$ with finite $p$-th moment. We endow $P_p(\mathbb{R}^d)$ with the Wasserstein metric $W_p$. Let $T > 0$ and $T$ be the interval $[0, T]$. We consider measurable functions

$$b: \mathbb{T} \times \mathbb{R}^d \times \mathbb{R}_+ \times P_p(\mathbb{R}^d) \to \mathbb{R}^d,$$

$$\sigma: \mathbb{T} \times \mathbb{R}^d \times P_p(\mathbb{R}^d) \to \mathbb{R}^d \otimes \mathbb{R}^m.$$

<footnote>*This work was supported by ANR MaSDOL (No. ANR-19-CE23-0017); Air Force Office of Scientific Research, Air Force Material Command, USAF (No. FA9550-19-7026); and ANR Chess (No. ANR-17-EURE-0010).</footnote>
# Well-posedness of McKean-Vlasov SDEs with density-dependent drift*

Anh-Dung Le

Toulouse School of Economics, 1 Esplanade de l'Université, 31000 Toulouse, France
leanhdung1994@gmail.com

April 26, 2024

## Abstract

In this paper, we study the well-posedness of McKean-Vlasov stochastic differential equations (SDE) whose drift depends pointwisely on marginal density and satisfies a condition about local integrability in time-space variables. The drift is assumed to be Lipschitz continuous in distribution variable with respect to Wasserstein metric $W_p$. Our approach is by approximation with mollified SDEs. We establish a new estimate about Hölder continuity in time of marginal density. Then we deduce that the marginal distributions (resp. marginal densities) of the mollified SDEs converge in $W_p$ (resp. topology of compact convergence) to the solution of the Fokker-Planck equation associated with the density-dependent SDE. We prove strong existence of a solution. Weak and strong uniqueness are obtained when $p = 1$, the drift coefficient is bounded, and the diffusion coefficient is distribution free.

**AMS subject classification:** 60B10, 60H10.

**Keywords:** McKean-Vlasov SDEs, density-dependent SDEs, well-posedness, local integrability.

## 1 Introduction

The study of distribution-dependent SDEs started with McKean's seminal work [1] about Vlasov equation of plasma which had been proposed in [2]. Classical results about the solvability of McKean-Vlasov equations include [3, 4, 5]. Since then, the literature on the well-posedness of McKean-Vlasov SDEs has been extended significantly. For recent results, we refer to [6, 7, 8, 9, 10, 11], the survey [12], and references therein. For applications on mean-field games, see the two-volume monograph [13].

In this paper, we consider the density-dependent McKean-Vlasov SDEs studied in [14]. More precisely, let $p \in [1, \infty)$ and $P_p(\mathbb{R}^d)$ be the space of Borel probability measures on $\mathbb{R}^d$ with finite $p$-th moment. We endow $P_p(\mathbb{R}^d)$ with the Wasserstein metric $W_p$. Let $T > 0$ and $T$ be the interval $[0, T]$. We consider measurable functions

$$b: \mathbb{T} \times \mathbb{R}^d \times \mathbb{R}_+ \times P_p(\mathbb{R}^d) \to \mathbb{R}^d,$$

$$\sigma: \mathbb{T} \times \mathbb{R}^d \times P_p(\mathbb{R}^d) \to \mathbb{R}^d \otimes \mathbb{R}^m.$$

<footnote>*This work was supported by ANR MaSDOL (No. ANR-19-CE23-0017); Air Force Office of Scientific Research, Air Force Material Command, USAF (No. FA9550-19-7026); and ANR Chess (No. ANR-17-EURE-0010).</footnote>
Let $(B_t, t \ge 0)$ be a given $m$-dimensional Brownian motion and $\mathbb{F} := (\mathcal{F}_t, t \ge 0)$ a given admissible filtration on a given probability space $(\Omega, \mathcal{A}, \mathbb{P})$. We consider the SDE

$$dX_t = b(t, X_t, \ell_t(X_t), \mu_t) dt + \sigma(t, X_t, \mu_t) dB_t, \quad t \in \mathbb{T}, \quad (1.1)$$

where the distribution of $X_0$ is $\nu$, that of $X_t$ is $\mu_t$, and the probability density function (p.d.f.) of $\mu_t$ is $\ell_t$.

The papers [15, 16, 17] study (1.1) in case $\sigma$ is constant and $b$ belongs to a Besov space. On the other hand, [18, 19, 20, 21] study the generalized porous media equation where $b = 0$ and $\sigma$ is density-dependent but time-independent. The papers [22, 23, 24, 25, 14, 26, 27, 28, 29, 30, 31] study the case where $b, \sigma$ are density-dependent but time-independent. More generally, [32, 33] allow $b, \sigma$ to be time-dependent. However, $b$ and $\sigma$ do not depend on probability distribution in those previously mentioned papers. Our paper is closely related to [34, 35] in which $b$ is both distribution-dependent and density-dependent. In particular, [34] utilizes the total variation norm $\| \cdot \|_{\text{var}}$ on $\mathcal{P}(\mathbb{R}^d)$ while [35] utilizes the $L^\infty$-norm $\| \cdot \|_{\infty}$ on the space of bounded p.d.f.'s. For a recent survey on existence result of density-dependent SDEs, we refer to [36].

The current paper contains the following contributions:

(a) Our approach is based on a mollifying argument whereas [34, 35] employ a Picard iteration argument.

(b) We obtain Hölder continuity of marginal density in both space and time whereas [34, 35] obtain only Hölder continuity in space.

(c) To obtain uniqueness of a solution, [34, 35] estimate $L^\infty$-norm $\| \cdot \|_{\infty}$ between marginal densities of two weak solutions. In our case, the presence of Wasserstein metric $W_p$ renders this approach inapplicable. To overcome this difficulty, we use another metric between marginal densities.

We recall two different notions of a solution of (1.1):

**Definition 1.1.**

1. A strong solution to (1.1) is a continuous $\mathbb{R}^d$-valued $\mathbb{F}$-adapted process $(X_t, t \in \mathbb{T})$ on $(\Omega, \mathcal{A}, \mathbb{P})$ such that for each $t \in \mathbb{T}$:

$$X_t = X_0 + \int_0^t b(s, X_s, \ell_s(X_s), \mu_s) \mathrm{d}s + \int_0^t \sigma(s, X_s, \mu_s) \mathrm{d}B_s \quad \mathbb{P}\text{-a.s.} \quad (1.2)$$

and

$$\int_0^t \mathbb{E}[|b(s, X_s, \ell_s(X_s), \mu_s)| + |\sigma(s, X_s, \mu_s)|^2] \mathrm{d}s < \infty. \quad (1.3)$$

2. A weak solution to (1.1) is a continuous $\mathbb{R}^d$-valued process $(X_t, t \in \mathbb{T})$ on some probability space $(\Omega, \mathcal{A}, \mathbb{P})$ where there exist some $m$-dimensional Brownian motion $(B_t, t \ge 0)$ and some admissible filtration $\mathbb{F} := (\mathcal{F}_t, t \ge 0)$ such that $(X_t, t \in \mathbb{T})$ is $\mathbb{F}$-adapted and that (1.2) and (1.3) hold.

Correspondingly, we have notions of uniqueness of (1.1):

**Definition 1.2.**
Let $(B_t, t \ge 0)$ be a given $m$-dimensional Brownian motion and $\mathbb{F} := (\mathcal{F}_t, t \ge 0)$ a given admissible filtration on a given probability space $(\Omega, \mathcal{A}, \mathbb{P})$. We consider the SDE

$$dX_t = b(t, X_t, \ell_t(X_t), \mu_t) dt + \sigma(t, X_t, \mu_t) dB_t, \quad t \in \mathbb{T}, \quad (1.1)$$

where the distribution of $X_0$ is $\nu$, that of $X_t$ is $\mu_t$, and the probability density function (p.d.f.) of $\mu_t$ is $\ell_t$.

The papers [15, 16, 17] study (1.1) in case $\sigma$ is constant and $b$ belongs to a Besov space. On the other hand, [18, 19, 20, 21] study the generalized porous media equation where $b = 0$ and $\sigma$ is density-dependent but time-independent. The papers [22, 23, 24, 25, 14, 26, 27, 28, 29, 30, 31] study the case where $b, \sigma$ are density-dependent but time-independent. More generally, [32, 33] allow $b, \sigma$ to be time-dependent. However, $b$ and $\sigma$ do not depend on probability distribution in those previously mentioned papers. Our paper is closely related to [34, 35] in which $b$ is both distribution-dependent and density-dependent. In particular, [34] utilizes the total variation norm $\| \cdot \|_{\text{var}}$ on $\mathcal{P}(\mathbb{R}^d)$ while [35] utilizes the $L^\infty$-norm $\| \cdot \|_{\infty}$ on the space of bounded p.d.f.'s. For a recent survey on existence result of density-dependent SDEs, we refer to [36].

The current paper contains the following contributions:

(a) Our approach is based on a mollifying argument whereas [34, 35] employ a Picard iteration argument.

(b) We obtain Hölder continuity of marginal density in both space and time whereas [34, 35] obtain only Hölder continuity in space.

(c) To obtain uniqueness of a solution, [34, 35] estimate $L^\infty$-norm $\| \cdot \|_{\infty}$ between marginal densities of two weak solutions. In our case, the presence of Wasserstein metric $W_p$ renders this approach inapplicable. To overcome this difficulty, we use another metric between marginal densities.

We recall two different notions of a solution of (1.1):

**Definition 1.1.**

1. A strong solution to (1.1) is a continuous $\mathbb{R}^d$-valued $\mathbb{F}$-adapted process $(X_t, t \in \mathbb{T})$ on $(\Omega, \mathcal{A}, \mathbb{P})$ such that for each $t \in \mathbb{T}$:

$$X_t = X_0 + \int_0^t b(s, X_s, \ell_s(X_s), \mu_s) \mathrm{d}s + \int_0^t \sigma(s, X_s, \mu_s) \mathrm{d}B_s \quad \mathbb{P}\text{-a.s.} \quad (1.2)$$

and

$$\int_0^t \mathbb{E}[|b(s, X_s, \ell_s(X_s), \mu_s)| + |\sigma(s, X_s, \mu_s)|^2] \mathrm{d}s < \infty. \quad (1.3)$$

2. A weak solution to (1.1) is a continuous $\mathbb{R}^d$-valued process $(X_t, t \in \mathbb{T})$ on some probability space $(\Omega, \mathcal{A}, \mathbb{P})$ where there exist some $m$-dimensional Brownian motion $(B_t, t \ge 0)$ and some admissible filtration $\mathbb{F} := (\mathcal{F}_t, t \ge 0)$ such that $(X_t, t \in \mathbb{T})$ is $\mathbb{F}$-adapted and that (1.2) and (1.3) hold.

Correspondingly, we have notions of uniqueness of (1.1):

**Definition 1.2.**
1. Equation (1.1) has strong uniqueness if any two strong solutions on a given probability space $(\Omega, \mathcal{A}, \mathbb{P})$ for a given Brownian motion and a given admissible filtration coincide $\mathbb{P}$-a.s. on the path space $C(\mathbb{T}; \mathbb{R}^d)$. Equation (1.1) has weak uniqueness if any weak solution induces the same distribution on $C(\mathbb{T}; \mathbb{R}^d)$.

2. Equation (1.1) is strongly well-posed if it has strong solution and strong uniqueness. Equation (1.1) is weakly well-posed if it has weak solution and weak uniqueness. Equation (1.1) is well-posed if it is both strongly and weakly well-posed.

In Section 2, we state our main results about existence, Hölder regularity, and uniqueness of a solution of (1.1). In Section 3, we remind some facts about optimal transport. Also, we recall and establish estimates of marginal density/distribution of classical SDEs. We explicitly mention the dependence of an estimate on parameters of a given assumption. This is crucial in ensuring that those estimates are stable in our mollifying argument. We prove our theorems in Section 4 and Section 5 respectively. To facilitate the navigation of the readers, we summarize the results in our papers as follows:

<table><thead><tr><th>Result</th><th>Content</th><th>Assumption</th></tr></thead><tbody><tr><td>Theorem 2.2</td><td>Main results</td><td>Assumption 2.1</td></tr><tr><td>Theorem 2.3</td><td></td><td></td></tr><tr><td>Lemma 3.1</td><td>Properties of W<sub>p</sub></td><td></td></tr><tr><td>Lemma 3.2</td><td></td><td></td></tr><tr><td>Theorem 3.4</td><td>Operator norms of semigroups</td><td></td></tr><tr><td>Lemma 3.5</td><td>Heat kernel estimates</td><td>Assumption 3.3</td></tr><tr><td>Proposition 3.8</td><td>Krylov's and Khasminskii's estimates</td><td></td></tr><tr><td>Theorem 3.9</td><td>Moment estimates</td><td>Assumption 3.6</td></tr><tr><td>Theorem 3.10</td><td>L<sup>P</sup> estimates</td><td></td></tr><tr><td>Theorem 3.12</td><td>Hölder continuity in space</td><td></td></tr><tr><td>Corollary 3.13</td><td>Duhamel representation</td><td>Assumption 3.11</td></tr><tr><td>Theorem 3.14</td><td>Hölder continuity in time</td><td></td></tr></tbody></table>

Above, Assumption 3.11 implies both Assumption 3.6 and Assumption 3.3. On the other hand, Assumption 3.6 does not imply Assumption 3.3, and vice versa. Throughout this paper, we use the following conventions:

(a) The set $\mathbb{R}^m \otimes \mathbb{R}^n$ is the space of matrices of size $m \times n$ with real entries. For $x \in \mathbb{R}^m \otimes \mathbb{R}^n$ and $y \in \mathbb{R}^n \otimes \mathbb{R}^k$, let $xy$ be their matrix product. For $x, y \in \mathbb{R}^m \otimes \mathbb{R}^n$, let $\langle x, y \rangle$ be their Frobenius inner product and $|x|$ the induced Frobenius norm of $x$.

(b) Let $\mathbb{R}_+ := \{x \in \mathbb{R} : x \ge 0\}$. We denote by $\mathcal{B}(\mathbb{R}^d)$ the Borel $\sigma$-algebra on $\mathbb{R}^d$. For brevity, we write $\infty$ for $+\infty$. We denote $x \vee y := \max\{x, y\}$ and $x \wedge y := \min\{x, y\}$ for $x, y \in \mathbb{R}$.

(c) We denote by $\nabla, \nabla^2$ the gradient and the Hessian with respect to (w.r.t.) the spatial variable. We denote by $\partial_t$ the derivative w.r.t. time. For $\alpha \in (0, 1]$ and a function $f : \mathbb{R}^d \to \mathbb{R}$, we denote by $[f]_\alpha$ the best $\alpha$-Hölder constant of $f$, i.e.,

$$[f]_\alpha := \sup_{\substack{x,y \in \mathbb{R}^d \\ x \neq y}} \frac{|f(x) - f(y)|}{|x - y|^\alpha}.$$
1. Equation (1.1) has strong uniqueness if any two strong solutions on a given probability space $(\Omega, \mathcal{A}, \mathbb{P})$ for a given Brownian motion and a given admissible filtration coincide $\mathbb{P}$-a.s. on the path space $C(\mathbb{T}; \mathbb{R}^d)$. Equation (1.1) has weak uniqueness if any weak solution induces the same distribution on $C(\mathbb{T}; \mathbb{R}^d)$.

2. Equation (1.1) is strongly well-posed if it has strong solution and strong uniqueness. Equation (1.1) is weakly well-posed if it has weak solution and weak uniqueness. Equation (1.1) is well-posed if it is both strongly and weakly well-posed.

In Section 2, we state our main results about existence, Hölder regularity, and uniqueness of a solution of (1.1). In Section 3, we remind some facts about optimal transport. Also, we recall and establish estimates of marginal density/distribution of classical SDEs. We explicitly mention the dependence of an estimate on parameters of a given assumption. This is crucial in ensuring that those estimates are stable in our mollifying argument. We prove our theorems in Section 4 and Section 5 respectively. To facilitate the navigation of the readers, we summarize the results in our papers as follows:

<table><thead><tr><th>Result</th><th>Content</th><th>Assumption</th></tr></thead><tbody><tr><td>Theorem 2.2</td><td>Main results</td><td>Assumption 2.1</td></tr><tr><td>Theorem 2.3</td><td></td><td></td></tr><tr><td>Lemma 3.1</td><td>Properties of W<sub>p</sub></td><td></td></tr><tr><td>Lemma 3.2</td><td></td><td></td></tr><tr><td>Theorem 3.4</td><td>Operator norms of semigroups</td><td>Assumption 3.3</td></tr><tr><td>Lemma 3.5</td><td>Heat kernel estimates</td><td></td></tr><tr><td>Proposition 3.8</td><td>Krylov's and Khasminskii's estimates</td><td></td></tr><tr><td>Theorem 3.9</td><td>Moment estimates</td><td>Assumption 3.6</td></tr><tr><td>Theorem 3.10</td><td>L<sup>P</sup> estimates</td><td></td></tr><tr><td>Theorem 3.12</td><td>Hölder continuity in space</td><td></td></tr><tr><td>Corollary 3.13</td><td>Duhamel representation</td><td>Assumption 3.11</td></tr><tr><td>Theorem 3.14</td><td>Hölder continuity in time</td><td></td></tr></tbody></table>

Above, Assumption 3.11 implies both Assumption 3.6 and Assumption 3.3. On the other hand, Assumption 3.6 does not imply Assumption 3.3, and vice versa. Throughout this paper, we use the following conventions:

(a) The set $\mathbb{R}^m \otimes \mathbb{R}^n$ is the space of matrices of size $m \times n$ with real entries. For $x \in \mathbb{R}^m \otimes \mathbb{R}^n$ and $y \in \mathbb{R}^n \otimes \mathbb{R}^k$, let $xy$ be their matrix product. For $x, y \in \mathbb{R}^m \otimes \mathbb{R}^n$, let $\langle x, y \rangle$ be their Frobenius inner product and $|x|$ the induced Frobenius norm of $x$.

(b) Let $\mathbb{R}_+ := \{x \in \mathbb{R} : x \ge 0\}$. We denote by $\mathcal{B}(\mathbb{R}^d)$ the Borel $\sigma$-algebra on $\mathbb{R}^d$. For brevity, we write $\infty$ for $+\infty$. We denote $x \vee y := \max\{x, y\}$ and $x \wedge y := \min\{x, y\}$ for $x, y \in \mathbb{R}$.

(c) We denote by $\nabla, \nabla^2$ the gradient and the Hessian with respect to (w.r.t.) the spatial variable. We denote by $\partial_t$ the derivative w.r.t. time. For $\alpha \in (0, 1]$ and a function $f : \mathbb{R}^d \to \mathbb{R}$, we denote by $[f]_\alpha$ the best $\alpha$-Hölder constant of $f$, i.e.,

$$[f]_\alpha := \sup_{\substack{x,y \in \mathbb{R}^d \\ x \neq y}} \frac{|f(x) - f(y)|}{|x - y|^\alpha}.$$
(d) Let $L^0(\mathbb{R}^d)$ be the space of real-valued measurable functions on $\mathbb{R}^d$. Let $L_+^0(\mathbb{R}^d)$ be the subset of $L^0(\mathbb{R}^d)$ that consists of non-negative functions. Let $L_b^0(\mathbb{R}^d)$ be the subset of $L^0(\mathbb{R}^d)$ that consists of bounded functions.

(e) Let $\mathcal{P}(\mathbb{R}^d)$ be the space of Borel probability measures on $\mathbb{R}^d$. The weak topology (and thus weak convergence $\rightarrow^*$) of $\mathcal{P}(\mathbb{R}^d)$ is the topology induced by $C_b(\mathbb{R}^d)$. The weak* topology (and thus weak convergence $\rightarrow_{*}$) of $\mathcal{P}(\mathbb{R}^d)$ is the topology induced by $C_c(\mathbb{R}^d)$.

## 2 Main results

Let $p, q \in [1, \infty]$. Let $L^p(\mathbb{R}^d) \subset L^0(\mathbb{R}^d)$ be the usual Lebesgue space. The localized version $\tilde{L}^p(\mathbb{R}^d)$ of $L^p(\mathbb{R}^d)$ is defined by the norm

$$ \|f\|_{\tilde{L}^p} := \sup_{x \in \mathbb{R}^d} \|1_{B(x,1)}f\|_{L^p}, $$

where $B(x, r)$ is the open ball centered at $x$ with radius $r$. We define the Bochner space

$$ L_q^p(t_0, t_1) := L^q([t_0, t_1]; L^p(\mathbb{R}^d)), \quad 0 \le t_0 < t_1 \le T. $$

The localized version $\tilde{L}_q^p(t_0, t_1)$ of $L_q^p(t_0, t_1)$ is defined by the norm

$$ \|g\|_{\tilde{L}_q^p(t_0, t_1)} := \sup_{x \in \mathbb{R}^d} \|1_{B(x,1)}g\|_{L_q^p(t_0, t_1)}. $$

In particular, $\|g\|_{\tilde{L}_\infty^p(t_0, t_1)} = \|g\|_{L_\infty^p(t_0, t_1)}$ and we have for $p, q \in [1, \infty)$:

$$ \|g\|_{L_q^p(t_0, t_1)} = \left( \int_{t_0}^{t_1} \left( \int_{\mathbb{R}^d} |g(s, y)|^p dy \right)^{\frac{q}{p}} ds \right)^{\frac{1}{q}}, $$

$$ \|g\|_{\tilde{L}_q^p(t_0, t_1)} = \sup_{x \in \mathbb{R}^d} \left( \int_{t_0}^{t_1} \left( \int_{B(x,1)} |g(s, y)|^p dy \right)^{\frac{q}{p}} ds \right)^{\frac{1}{q}}. $$

For brevity, we denote

$$ L_q^p(t) := L_q^p(0, t), \quad \tilde{L}_q^p(t) := \tilde{L}_q^p(0, t), \quad L_q^p := L_q^p(0, T), \quad \tilde{L}_q^p := \tilde{L}_q^p(0, T). $$

The class $\mathcal{K}$ of exponent parameter is defined by

$$ \mathcal{K} := \left\{ (p, q) \in (2, \infty]^2 : \frac{d}{p} + \frac{2}{q} < 1 \right\}. $$

For $\alpha \in (0, 1)$, let $C_b^\alpha(\mathbb{R}^d)$ be the Hölder space of functions $f: \mathbb{R}^d \to \mathbb{R}$ with the norm

$$ \|f\|_{C_b^\alpha} := \|f\|_{\infty} + [f]_\alpha. $$

We denote by $M_p(\rho)$ the $p$-th moment of $\rho \in \mathcal{P}(\mathbb{R}^d)$, i.e.,

$$ M_p(\rho) := \int_{\mathbb{R}^d} |x|^p d\rho(x). $$

Below, we introduce the main assumption about the initial distribution and the coefficients of (1.1). Let $a := \sigma\sigma^\top$. We denote $b_t(x, r, \rho) := b(t, x, r, \rho)$, $\sigma_t(x, \rho) := \sigma(t, x, \rho)$ and $a_t(x, \rho) := a(t, x, \rho)$. We consider the set of assumptions:
(d) Let $L^0(\mathbb{R}^d)$ be the space of real-valued measurable functions on $\mathbb{R}^d$. Let $L_+^0(\mathbb{R}^d)$ be the subset of $L^0(\mathbb{R}^d)$ that consists of non-negative functions. Let $L_b^0(\mathbb{R}^d)$ be the subset of $L^0(\mathbb{R}^d)$ that consists of bounded functions.

(e) Let $\mathcal{P}(\mathbb{R}^d)$ be the space of Borel probability measures on $\mathbb{R}^d$. The weak topology (and thus weak convergence $\rightarrow^*$) of $\mathcal{P}(\mathbb{R}^d)$ is the topology induced by $C_b(\mathbb{R}^d)$. The weak* topology (and thus weak convergence $\rightarrow_{*}$) of $\mathcal{P}(\mathbb{R}^d)$ is the topology induced by $C_c(\mathbb{R}^d)$.

## 2 Main results

Let $p, q \in [1, \infty]$. Let $L^p(\mathbb{R}^d) \subset L^0(\mathbb{R}^d)$ be the usual Lebesgue space. The localized version $\tilde{L}^p(\mathbb{R}^d)$ of $L^p(\mathbb{R}^d)$ is defined by the norm

$$ \|f\|_{\tilde{L}^p} := \sup_{x \in \mathbb{R}^d} \|1_{B(x,1)}f\|_{L^p}, $$

where $B(x, r)$ is the open ball centered at $x$ with radius $r$. We define the Bochner space

$$ L_q^p(t_0, t_1) := L^q([t_0, t_1]; L^p(\mathbb{R}^d)), \quad 0 \le t_0 < t_1 \le T. $$

The localized version $\tilde{L}_q^p(t_0, t_1)$ of $L_q^p(t_0, t_1)$ is defined by the norm

$$ \|g\|_{\tilde{L}_q^p(t_0, t_1)} := \sup_{x \in \mathbb{R}^d} \|1_{B(x,1)}g\|_{L_q^p(t_0, t_1)}. $$

In particular, $\|g\|_{\tilde{L}_\infty^p(t_0, t_1)} = \|g\|_{L_\infty^p(t_0, t_1)}$ and we have for $p, q \in [1, \infty)$:

$$ \|g\|_{L_q^p(t_0, t_1)} = \left( \int_{t_0}^{t_1} \left( \int_{\mathbb{R}^d} |g(s, y)|^p dy \right)^{\frac{q}{p}} ds \right)^{\frac{1}{q}}, $$

$$ \|g\|_{\tilde{L}_q^p(t_0, t_1)} = \sup_{x \in \mathbb{R}^d} \left( \int_{t_0}^{t_1} \left( \int_{B(x,1)} |g(s, y)|^p dy \right)^{\frac{q}{p}} ds \right)^{\frac{1}{q}}. $$

For brevity, we denote

$$ L_q^p(t) := L_q^p(0, t), \quad \tilde{L}_q^p(t) := \tilde{L}_q^p(0, t), \quad L_q^p := L_q^p(0, T), \quad \tilde{L}_q^p := \tilde{L}_q^p(0, T). $$

The class $\mathcal{K}$ of exponent parameter is defined by

$$ \mathcal{K} := \left\{ (p, q) \in (2, \infty]^2 : \frac{d}{p} + \frac{2}{q} < 1 \right\}. $$

For $\alpha \in (0, 1)$, let $C_b^\alpha(\mathbb{R}^d)$ be the Hölder space of functions $f: \mathbb{R}^d \rightarrow \mathbb{R}$ with the norm

$$ \|f\|_{C_b^\alpha} := \|f\|_{\infty} + [f]_\alpha. $$

We denote by $M_p(\rho)$ the $p$-th moment of $\rho \in \mathcal{P}(\mathbb{R}^d)$, i.e.,

$$ M_p(\rho) := \int_{\mathbb{R}^d} |x|^p d\rho(x). $$

Below, we introduce the main assumption about the initial distribution and the coefficients of (1.1). Let $a := \sigma\sigma^\top$. We denote $b_t(x, r, \rho) := b(t, x, r, \rho)$, $\sigma_t(x, \rho) := \sigma(t, x, \rho)$ and $a_t(x, \rho) := a(t, x, \rho)$. We consider the set of assumptions:
**Assumption 2.1.** There exist constants $\alpha \in (0, 1)$, $\beta \in (0, 1)$, $p \in [1, \infty)$, $C > 0$ such that for $t \in \mathbb{T}$; $x, y \in \mathbb{R}^d$; $r, \tilde{r} \in \mathbb{R}_+$ and $\rho, \tilde{\rho} \in \mathcal{P}_p(\mathbb{R}^d)$:

(A1) $a_t$ is invertible and $\sigma_t$ is differentiable w.r.t. the spatial variable.

(A2) There exists $1 \le f_0 \in \tilde{L}_{q_0}^{p_0}$ with $(p_0, q_0) \in \mathcal{K}$ such that $|b_t(x, r, \rho)| \le f_0(t, x)$.

(A3) $\alpha \in (0, 1 - \frac{d}{p_0} - \frac{2}{q_0})$ and $\nu \in \mathcal{P}_{p+\alpha}(\mathbb{R}^d)$ has a density $\ell_\nu \in C_b^\alpha(\mathbb{R}^d)$.

(A4) The following conditions hold:

$$ \begin{aligned} & |b_t(x, r, \rho) - b_t(x, \tilde{r}, \tilde{\rho})| \le C\{|r - \tilde{r}| + W_p(\rho, \tilde{\rho})\}, \\ & \|\nabla\sigma_t\|_{\infty} + \|\sigma_t\|_{\infty} + \|a_t^{-1}\|_{\infty} \le C, \\ & |\sigma_t(x, \rho) - \sigma_t(x, \tilde{\rho})| \le CW_p(\rho, \tilde{\rho}), \\ & |\nabla\sigma_t(\cdot, \rho)(x) - \nabla\sigma_t(\cdot, \rho)(y)| \le C(|x - y| + |x - y|^{\beta}). \end{aligned} $$

We gather parameters in Assumption 2.1:

$$ \Theta_1 := (d, T, \alpha, \beta, C, p_0, q_0, f_0, p). $$

There is no continuity condition on the spatial variable of the drift. If $b$ is bounded, then it satisfies Assumption 2.1[A2]. Assumption 2.1[A4] implies for $t \in \mathbb{T}$, $\gamma \in (0, 1)$; $x, y \in \mathbb{R}^d$ and $\rho \in \mathcal{P}_p(\mathbb{R}^d)$ that $|\sigma_t(x, \rho) - \sigma_t(y, \rho)| \le 2C|x - y|^{\gamma}$. Assumption 2.1[A2] means that the marginal density and marginal distribution do not affect the local integrability of the drift. In Assumption 2.1[A3], we requires $\nu \in \mathcal{P}_{p+\alpha}(\mathbb{R}^d)$ instead of just $\nu \in \mathcal{P}_p(\mathbb{R}^d)$. This gives us a uniform control on the tail behavior of the marginal distribution in (3.17) of Theorem 3.9.

Our main results are the following:

**Theorem 2.2 (Existence and Regularity).** Let Assumption 2.1 hold.

1. Equation (1.1) has a strong solution whose marginal distribution is denoted by $(\mu_t, t \in \mathbb{T})$ and marginal density is denoted by $(\ell_t, t \in \mathbb{T})$.

2. There exist constants $c > 0$ (depending only on $\Theta_1$), $c_1, c_2 > 0$ (depending only on $\Theta_1, \nu$), $\delta_1 \in (0, \frac{1}{2})$ (depending only on $d, p_0, q_0, \alpha$), and $\delta_2 \in (0, \frac{1}{2})$ (depending only on $q_0$) such that

$$ \begin{aligned} & \sup_{t \in \mathbb{T}} \|\ell_t\|_{C_b^\alpha} \le c\|\ell_\nu\|_{C_b^\alpha}, \\ & \|\ell_t - \ell_s\|_{\infty} \le c_1|t - s|^{\delta_1}, \\ & W_p(\mu_s, \mu_t) \le c_2|t - s|^{\delta_2}, \quad s, t \in \mathbb{T}. \end{aligned} $$

**Theorem 2.3 (Uniqueness).** Let Assumption 2.1 hold. Assume in addition that $p = 1$, $\|b\|_{\infty} \le C$ and that $\sigma$ does not depend on marginal distribution. Let $X^1, X^2$ be two weak solutions of (1.1). Let $\nu_1, \nu_2$ be their initial distributions and $\ell^1, \ell^2$ their marginal densities. We assume that $\nu_1, \nu_2$ satisfy Assumption 2.1[A3] for the same parameters.
**Assumption 2.1.** There exist constants $\alpha \in (0, 1)$, $\beta \in (0, 1)$, $p \in [1, \infty)$, $C > 0$ such that for $t \in \mathbb{T}$; $x, y \in \mathbb{R}^d$; $r, \tilde{r} \in \mathbb{R}_+$ and $\rho, \tilde{\rho} \in \mathcal{P}_p(\mathbb{R}^d)$:

(A1) $a_t$ is invertible and $\sigma_t$ is differentiable w.r.t. the spatial variable.

(A2) There exists $1 \le f_0 \in \tilde{L}_{q_0}^{p_0}$ with $(p_0, q_0) \in \mathcal{K}$ such that $|b_t(x, r, \rho)| \le f_0(t, x)$.

(A3) $\alpha \in (0, 1 - \frac{d}{p_0} - \frac{2}{q_0})$ and $\nu \in \mathcal{P}_{p+\alpha}(\mathbb{R}^d)$ has a density $\ell_\nu \in C_b^\alpha(\mathbb{R}^d)$.

(A4) The following conditions hold:

$$ \begin{aligned} & |b_t(x, r, \rho) - b_t(x, \tilde{r}, \tilde{\rho})| \le C\{|r - \tilde{r}| + W_p(\rho, \tilde{\rho})\}, \\ & \|\nabla \sigma_t\|_{\infty} + \|\sigma_t\|_{\infty} + \|a_t^{-1}\|_{\infty} \le C, \\ & |\sigma_t(x, \rho) - \sigma_t(x, \tilde{\rho})| \le CW_p(\rho, \tilde{\rho}), \\ & |\nabla \sigma_t(\cdot, \rho)(x) - \nabla \sigma_t(\cdot, \rho)(y)| \le C(|x - y| + |x - y|^{\beta}). \end{aligned} $$

We gather parameters in Assumption 2.1:

$$ \Theta_1 := (d, T, \alpha, \beta, C, p_0, q_0, f_0, p). $$

There is no continuity condition on the spatial variable of the drift. If $b$ is bounded, then it satisfies Assumption 2.1[A2]. Assumption 2.1[A4] implies for $t \in \mathbb{T}$, $\gamma \in (0, 1)$; $x, y \in \mathbb{R}^d$ and $\rho \in \mathcal{P}_p(\mathbb{R}^d)$ that $|\sigma_t(x, \rho) - \sigma_t(y, \rho)| \le 2C|x - y|^{\gamma}$. Assumption 2.1[A2] means that the marginal density and marginal distribution do not affect the local integrability of the drift. In Assumption 2.1[A3], we requires $\nu \in \mathcal{P}_{p+\alpha}(\mathbb{R}^d)$ instead of just $\nu \in \mathcal{P}_p(\mathbb{R}^d)$. This gives us a uniform control on the tail behavior of the marginal distribution in (3.17) of Theorem 3.9.

Our main results are the following:

**Theorem 2.2 (Existence and Regularity).** Let Assumption 2.1 hold.

1. Equation (1.1) has a strong solution whose marginal distribution is denoted by $(\mu_t, t \in \mathbb{T})$ and marginal density is denoted by $(\ell_t, t \in \mathbb{T})$.

2. There exist constants $c > 0$ (depending only on $\Theta_1$), $c_1, c_2 > 0$ (depending only on $\Theta_1, \nu$), $\delta_1 \in (0, \frac{1}{2})$ (depending only on $d, p_0, q_0, \alpha$), and $\delta_2 \in (0, \frac{1}{2})$ (depending only on $q_0$) such that

$$ \begin{aligned} & \sup_{t \in \mathbb{T}} \|\ell_t\|_{C_b^\alpha} \le c \|\ell_\nu\|_{C_b^\alpha}, \\ & \|\ell_t - \ell_s\|_{\infty} \le c_1 |t - s|^{\delta_1}, \\ & W_p(\mu_s, \mu_t) \le c_2 |t - s|^{\delta_2}, \quad s, t \in \mathbb{T}. \end{aligned} $$

**Theorem 2.3 (Uniqueness).** Let Assumption 2.1 hold. Assume in addition that $p = 1$, $\|b\|_{\infty} \le C$ and that $\sigma$ does not depend on marginal distribution. Let $X^1, X^2$ be two weak solutions of (1.1). Let $\nu_1, \nu_2$ be their initial distributions and $\ell^1, \ell^2$ their marginal densities. We assume that $\nu_1, \nu_2$ satisfy Assumption 2.1[A3] for the same parameters.
1. There exists an increasing function $\Lambda : \mathbb{R}_+ \to \mathbb{R}_+$ (depending only on $\Theta_1$) such that

$$ \sup_{t \in \mathbb{T}} \int_{\mathbb{R}^d} (|x| + 1) |\ell_t^1(x) - \ell_t^2(x)| dx \\ \le \Lambda(\|\ell_{\nu_1}\|_{\infty} + M_1(\nu_1)) \int_{\mathbb{R}^d} (|x| + 1) |\ell_{\nu_1}(x) - \ell_{\nu_2}(x)| dx. $$

2. Equation (1.1) has both weak and strong uniqueness.

# 3 Preliminaries

## 3.1 Some facts from optimal transport

Let $\mu, \nu \in \mathcal{P}(\mathbb{R}^d)$. We denote by $|\mu - \nu|$ the variation of the signed measure $\mu - \nu$ as in [37, Section 6.1]. By [37, Theorems 6.2 and 6.4], $|\mu - \nu|$ is a non-negative finite measure. The set of transport plans (or couplings) between $\mu$ and $\nu$ is defined by

$$ \Gamma(\mu, \nu) := \{ \xi \in \mathcal{P}(\mathbb{R}^d \times \mathbb{R}^d) : \pi_{\#}^1 \xi = \mu \text{ and } \pi_{\#}^2 \xi = \nu \}, $$

where $\pi^i$ is the projection of $\mathbb{R}^d \times \mathbb{R}^d$ onto its $i$-th coordinate and $\pi_{\#}^i \xi$ is the push-forward of $\xi$ by $\pi^i$. Their extended Wasserstein metric is defined by

$$ W_p(\mu, \nu) := \inf_{\xi \in \Gamma(\mu, \nu)} \left( \int_{\mathbb{R}^d} |x - y|^p d\xi(x, y) \right)^{1/p}. $$

By [38, Theorem 6.18], $(\mathcal{P}_p(\mathbb{R}^d), W_p)$ is a Polish space. By [38, Theorem 6.9], we have for $\mu_n, \mu \in \mathcal{P}_p(\mathbb{R}^d)$ that $W_p(\mu_n, \mu) \to 0$ if and only if $\mu_n \rightharpoonup \mu$ and $M_p(\mu_n) \to M_p(\mu)$. For more information about optimal transport, we refer to [39, 38, 40, 41, 42, 43, 44].

Let $\Phi_p$ be the collection of all $(\varphi, \psi) \in C_b(\mathbb{R}^d) \times C_b(\mathbb{R}^d)$ such that $\varphi(x) + \psi(y) \le |x - y|^p$ for $x, y \in \mathbb{R}^d$. We recall results needed for the proofs of our theorems:

### Lemma 3.1.

1. [39, Theorem 1.3] We have for $\mu, \nu \in \mathcal{P}(\mathbb{R}^d)$ and $p \in [1, \infty)$ that

$$ (W_p(\mu, \nu))^p = \sup \left\{ \int_{\mathbb{R}^d} \varphi \, d\mu + \int_{\mathbb{R}^d} \psi \, d\nu : (\varphi, \psi) \in \Phi_p \right\}. $$

2. [39, Theorem 1.14] We have for $\mu, \nu \in \mathcal{P}_1(\mathbb{R}^d)$ that

$$ W_1(\mu, \nu) = \sup \left\{ \int_{\mathbb{R}^d} f \, d(\mu - \nu) : f \in L^1(|\mu - \nu|) \text{ with } [f]_1 \le 1 \right\}. $$

3. [39, Remark 7.1.2] We have for $\mu, \nu \in \mathcal{P}(\mathbb{R}^d)$ and $1 \le p \le q < \infty$ that $W_p(\mu, \nu) \le W_q(\mu, \nu)$.

Above, the first claim is called Kantorovich duality while the second one is called Kantorovich-Rubinstein theorem. The next result states that the extended Wasserstein metric is controlled by the weighted $L^1$-metric between the corresponding p.d.f.'s.
1. There exists an increasing function $\Lambda : \mathbb{R}_+ \to \mathbb{R}_+$ (depending only on $\Theta_1$) such that

$$ \sup_{t \in \mathbb{T}} \int_{\mathbb{R}^d} (|x| + 1) |\ell_t^1(x) - \ell_t^2(x)| dx \\ \le \Lambda (\|\ell_{\nu_1}\|_{\infty} + M_1(\nu_1)) \int_{\mathbb{R}^d} (|x| + 1) |\ell_{\nu_1}(x) - \ell_{\nu_2}(x)| dx. $$

2. Equation (1.1) has both weak and strong uniqueness.

# 3 Preliminaries

## 3.1 Some facts from optimal transport

Let $\mu, \nu \in \mathcal{P}(\mathbb{R}^d)$. We denote by $|\mu - \nu|$ the variation of the signed measure $\mu - \nu$ as in [37, Section 6.1]. By [37, Theorems 6.2 and 6.4], $|\mu - \nu|$ is a non-negative finite measure. The set of transport plans (or couplings) between $\mu$ and $\nu$ is defined by

$$ \Gamma(\mu, \nu) := \{ \xi \in \mathcal{P}(\mathbb{R}^d \times \mathbb{R}^d) : \pi_{\#}^1 \xi = \mu \text{ and } \pi_{\#}^2 \xi = \nu \}, $$

where $\pi^i$ is the projection of $\mathbb{R}^d \times \mathbb{R}^d$ onto its $i$-th coordinate and $\pi_{\#}^i \xi$ is the push-forward of $\xi$ by $\pi^i$. Their extended Wasserstein metric is defined by

$$ W_p(\mu, \nu) := \inf_{\xi \in \Gamma(\mu, \nu)} \left( \int_{\mathbb{R}^d} |x - y|^p d\xi(x, y) \right)^{1/p}. $$

By [38, Theorem 6.18], $(\mathcal{P}_p(\mathbb{R}^d), W_p)$ is a Polish space. By [38, Theorem 6.9], we have for $\mu_n, \mu \in \mathcal{P}_p(\mathbb{R}^d)$ that $W_p(\mu_n, \mu) \to 0$ if and only if $\mu_n \rightharpoonup \mu$ and $M_p(\mu_n) \to M_p(\mu)$. For more information about optimal transport, we refer to [39, 38, 40, 41, 42, 43, 44].

Let $\Phi_p$ be the collection of all $(\varphi, \psi) \in C_b(\mathbb{R}^d) \times C_b(\mathbb{R}^d)$ such that $\varphi(x) + \psi(y) \le |x - y|^p$ for $x, y \in \mathbb{R}^d$. We recall results needed for the proofs of our theorems:

**Lemma 3.1.**

1. [39, Theorem 1.3] We have for $\mu, \nu \in \mathcal{P}(\mathbb{R}^d)$ and $p \in [1, \infty)$ that

$$ (W_p(\mu, \nu))^p = \sup \left\{ \int_{\mathbb{R}^d} \varphi \, d\mu + \int_{\mathbb{R}^d} \psi \, d\nu : (\varphi, \psi) \in \Phi_p \right\}. $$

2. [39, Theorem 1.14] We have for $\mu, \nu \in \mathcal{P}_1(\mathbb{R}^d)$ that

$$ W_1(\mu, \nu) = \sup \left\{ \int_{\mathbb{R}^d} f \, d(\mu - \nu) : f \in L^1(|\mu - \nu|) \text{ with } [f]_1 \le 1 \right\}. $$

3. [39, Remark 7.1.2] We have for $\mu, \nu \in \mathcal{P}(\mathbb{R}^d)$ and $1 \le p \le q < \infty$ that $W_p(\mu, \nu) \le W_q(\mu, \nu)$.

Above, the first claim is called Kantorovich duality while the second one is called Kantorovich-Rubinstein theorem. The next result states that the extended Wasserstein metric is controlled by the weighted $L^1$-metric between the corresponding p.d.f.'s.
**Lemma 3.2.** Let $p \in [1, \infty)$ and $\mu, \nu \in \mathcal{P}(\mathbb{R}^d)$ be absolutely continuous with $\ell_\mu, \ell_\nu$ being their p.d.f.'s respectively. Then

$$ (W_p(\mu, \nu))^p \le (1 \vee 2^{p-1}) \int_{\mathbb{R}^d} |x|^p |\ell_\mu - \ell_\nu|(x) \, dx. $$

*Proof.* For $B \in \mathcal{B}(\mathbb{R}^d)$, we denote by $\Pi(B)$ the collection of all finite measurable partitions of $B$. This means $(B_1, \dots, B_n) \in \Pi(B)$ if and only if $\{B_1, \dots, B_n\} \subset \mathcal{B}(\mathbb{R}^d)$ are pairwise disjoint and $B = \cup_{k=1}^n B_k$. We have

$$ \begin{aligned} |\mu - \nu|(B) &= \sup \left\{ \sum_{k=1}^n |(\mu - \nu)(B_k)| : (B_1, \dots, B_n) \in \Pi(B) \right\} \\ &= \sup \left\{ \sum_{k=1}^n \left| \int_{B_k} (\ell_\mu - \ell_\nu)(x) \, dx \right| : (B_1, \dots, B_n) \in \Pi(B) \right\} \\ &\le \sup \left\{ \sum_{k=1}^n \int_{B_k} |\ell_\mu - \ell_\nu|(x) \, dx : (B_1, \dots, B_n) \in \Pi(B) \right\} \\ &= \int_B |\ell_\mu - \ell_\nu|(x) \, dx. \end{aligned} $$

On the other hand, we have from [39, Proposition 7.10] that

$$ (W_p(\mu, \nu))^p \le (1 \vee 2^{p-1}) \int_{\mathbb{R}^d} |x|^p d|\mu - \nu|(x). $$

The claim then follows.

<figure><img src="image_2.png" /></figure>

## 3.2 Heat kernel estimates of transition density

In the rest of Section 3, we consider measurable functions

$$ \begin{aligned} b: \mathbb{T} \times \mathbb{R}^d &\to \mathbb{R}^d, \\ \sigma: \mathbb{T} \times \mathbb{R}^d &\to \mathbb{R}^d \otimes \mathbb{R}^m. \end{aligned} $$

Let $a := \sigma\sigma^\top$. We denote $b_t := b(t, \cdot)$, $\sigma_t := \sigma(t, \cdot)$ and $a_t := a(t, \cdot)$. We consider the following set of assumptions:

**Assumption 3.3.** There exist constants $\beta \in (0, 1)$ and $C > 0$ such that for $t \in \mathbb{T}$ and $x, y \in \mathbb{R}^d$:

(A1) $a_t$ is invertible.

(A2) The following conditions hold:

$$ \begin{aligned} \|b_t\|_{\infty} + \|\sigma_t\|_{\infty} + \|a_t^{-1}\|_{\infty} &\le C, \\ |b_t(x) - b_t(y)| &\le C(|x - y| + |x - y|^{\beta}), \\ |\sigma_t(x) - \sigma_t(y)| &\le C|x - y|^{\beta}. \end{aligned} $$
**Lemma 3.2.** Let $p \in [1, \infty)$ and $\mu, \nu \in \mathcal{P}(\mathbb{R}^d)$ be absolutely continuous with $\ell_\mu, \ell_\nu$ being their p.d.f.'s respectively. Then

$$ (W_p(\mu, \nu))^p \le (1 \vee 2^{p-1}) \int_{\mathbb{R}^d} |x|^p |\ell_\mu - \ell_\nu|(x) \, dx. $$

*Proof.* For $B \in \mathcal{B}(\mathbb{R}^d)$, we denote by $\Pi(B)$ the collection of all finite measurable partitions of $B$. This means $(B_1, \dots, B_n) \in \Pi(B)$ if and only if $\{B_1, \dots, B_n\} \subset \mathcal{B}(\mathbb{R}^d)$ are pairwise disjoint and $B = \cup_{k=1}^n B_k$. We have

$$ \begin{aligned} |\mu - \nu|(B) &= \sup \left\{ \sum_{k=1}^{n} |(\mu - \nu)(B_k)| : (B_1, \dots, B_n) \in \Pi(B) \right\} \\ &= \sup \left\{ \sum_{k=1}^{n} \left| \int_{B_k} (\ell_\mu - \ell_\nu)(x) \, dx \right| : (B_1, \dots, B_n) \in \Pi(B) \right\} \\ &\le \sup \left\{ \sum_{k=1}^{n} \int_{B_k} |\ell_\mu - \ell_\nu|(x) \, dx : (B_1, \dots, B_n) \in \Pi(B) \right\} \\ &= \int_B |\ell_\mu - \ell_\nu|(x) \, dx. \end{aligned} $$

On the other hand, we have from [39, Proposition 7.10] that

$$ (W_p(\mu, \nu))^p \le (1 \vee 2^{p-1}) \int_{\mathbb{R}^d} |x|^p d|\mu - \nu|(x). $$

The claim then follows.

<figure><img src="image_2.png" /></figure>

## 3.2 Heat kernel estimates of transition density

In the rest of Section 3, we consider measurable functions

$$ \begin{aligned} b: \mathbb{T} \times \mathbb{R}^d &\to \mathbb{R}^d, \\ \sigma: \mathbb{T} \times \mathbb{R}^d &\to \mathbb{R}^d \otimes \mathbb{R}^m. \end{aligned} $$

Let $a := \sigma\sigma^\top$. We denote $b_t := b(t, \cdot)$, $\sigma_t := \sigma(t, \cdot)$ and $a_t := a(t, \cdot)$. We consider the following set of assumptions:

**Assumption 3.3.** There exist constants $\beta \in (0, 1)$ and $C > 0$ such that for $t \in \mathbb{T}$ and $x, y \in \mathbb{R}^d$:

(A1) $a_t$ is invertible.

(A2) The following conditions hold:

$$ \begin{aligned} \|b_t\|_{\infty} + \|\sigma_t\|_{\infty} + \|a_t^{-1}\|_{\infty} &\le C, \\ |b_t(x) - b_t(y)| &\le C(|x - y| + |x - y|^{\beta}), \\ |\sigma_t(x) - \sigma_t(y)| &\le C|x - y|^{\beta}. \end{aligned} $$
We gather parameters in Assumption 3.3:

$$ \Theta_2 := (d, T, \beta, C). $$

Throughout the remaining of Section 3.2, we assume that $(b, \sigma)$ satisfies Assumption 3.3.

For any $(s, x) \in [0, T) \times \mathbb{R}^d$, the SDE

$$ dX_{s,t}^x = b(t, X_{s,t}^x) dt + \sigma(t, X_{s,t}^x) dB_t, \quad t \in [s, T], X_{s,s}^x = x, \quad (3.1) $$

is weakly well-posed by [45, Theorem 1.2] and has transition density denoted by $(p_{s,t}^{b,\sigma})_{0\le s<t\le T}$, i.e., $p_{s,t}^{b,\sigma}(x, \cdot)$ is the density of $X_{s,t}^x$. The associated semigroup $(P_{s,t}^{b,\sigma})_{0\le s<t\le T}$ is defined for $x \in \mathbb{R}^d$ and $f \in L_+^0(\mathbb{R}^d) \cup L_b^0(\mathbb{R}^d)$ by

$$ P_{s,t}^{b,\sigma} f(x) := \mathbb{E}[f(X_{s,t}^x)] = \int_{\mathbb{R}^d} p_{s,t}^{b,\sigma}(x, y)f(y) dy. \quad (3.2) $$

As in [45, Section 1.2], we construct a family $(\psi_{s,t})_{s,t \in \mathbb{T}}$ of $C^\infty$-diffeomorphisms on $\mathbb{R}^d$. Let $\rho: \mathbb{R}^d \to \mathbb{R}$ be a smooth symmetric p.d.f. whose support is contained in the unit ball of $\mathbb{R}^d$. We define $\underline{b}: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ by $\underline{b}(t, \cdot) := b(t, \cdot) * \rho$ where * is the convolution operator. By [45, Inequalities 1.9 and 1.10],

$$ \sup_{t \in \mathbb{T}} \{ \| \nabla^n \underline{b}(t, \cdot) \|_{\infty} + \| b(t, \cdot) - \underline{b}(t, \cdot) \|_{\infty} \} < \infty, \quad n \in \mathbb{N}. $$

For a fixed $(s, x) \in \mathbb{T} \times \mathbb{R}^d$, we consider the ODE

$$ \begin{cases} \frac{d}{dt}\psi_{s,t}(x) = \underline{b}(t, \psi_{s,t}(x)), & t \in \mathbb{T}, \\ \psi_{s,s}(x) = x. & \end{cases} $$

For $\kappa > 0$, we consider the Gaussian heat kernel

$$ p_t^\kappa(x) := \frac{1}{(\kappa \pi t)^{\frac{d}{2}}} \exp \left( -\frac{|x|^2}{\kappa t} \right), \quad t > 0, x \in \mathbb{R}^d. $$

The following results give density and gradient estimates for (3.1):

**Theorem 3.4.** Let Assumption 3.3 hold.

1. [46, Inequalities 2.7 and 2.8] There exist constants $c, \kappa > 0$ (depending only on $\Theta_2$) such that for $i \in \{0, 1\}, 0 \le s < t \le T$ and $x, y \in \mathbb{R}^d$:

$$ |\nabla_x^i p_{s,t}^{b,\sigma}(x, y)| \le c(t-s)^{-\frac{i}{2}} p_{t-s}^{\kappa}(x-y). $$

2. [45, Lemma A.1] For $\alpha \in (0, \beta)$, there exist constants $c, \kappa > 0$ (depending only on $\Theta_2, \alpha$) such that for $0 \le s < t \le T$ and $x, y, y' \in \mathbb{R}^d$:

$$ \begin{aligned} & |\nabla_x p_{s,t}^{b,\sigma}(x, y) - \nabla_x p_{s,t}^{b,\sigma}(x, y')| \\ & \le c|y - y'|^{\alpha}(t - s)^{-\frac{1+\alpha}{2}} \{p_{t-s}^{\kappa}(\psi_{s,t}(x) - y) + p_{t-s}^{\kappa}(\psi_{s,t}(x) - y')\}. \end{aligned} $$
We gather parameters in Assumption 3.3:

$$ \Theta_2 := (d, T, \beta, C). $$

Throughout the remaining of Section 3.2, we assume that $(b, \sigma)$ satisfies Assumption 3.3.

For any $(s, x) \in [0, T) \times \mathbb{R}^d$, the SDE

$$ dX_{s,t}^x = b(t, X_{s,t}^x) dt + \sigma(t, X_{s,t}^x) dB_t, \quad t \in [s, T], X_{s,s}^x = x, \quad (3.1) $$

is weakly well-posed by [45, Theorem 1.2] and has transition density denoted by $(p_{s,t}^{b,\sigma})_{0\le s<t\le T}$, i.e., $p_{s,t}^{b,\sigma}(x, \cdot)$ is the density of $X_{s,t}^x$. The associated semigroup $(P_{s,t}^{b,\sigma})_{0\le s<t\le T}$ is defined for $x \in \mathbb{R}^d$ and $f \in L_+^0(\mathbb{R}^d) \cup L_b^0(\mathbb{R}^d)$ by

$$ P_{s,t}^{b,\sigma} f(x) := \mathbb{E}[f(X_{s,t}^x)] = \int_{\mathbb{R}^d} p_{s,t}^{b,\sigma}(x, y)f(y) dy. \quad (3.2) $$

As in [45, Section 1.2], we construct a family $(\psi_{s,t})_{s,t \in \mathbb{T}}$ of $C^\infty$-diffeomorphisms on $\mathbb{R}^d$. Let $\rho: \mathbb{R}^d \to \mathbb{R}$ be a smooth symmetric p.d.f. whose support is contained in the unit ball of $\mathbb{R}^d$. We define $\underline{b}: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ by $\underline{b}(t, \cdot) := b(t, \cdot) * \rho$ where * is the convolution operator. By [45, Inequalities 1.9 and 1.10],

$$ \sup_{t \in \mathbb{T}} \{ \| \nabla^n \underline{b}(t, \cdot) \|_{\infty} + \| b(t, \cdot) - \underline{b}(t, \cdot) \|_{\infty} \} < \infty, \quad n \in \mathbb{N}. $$

For a fixed $(s, x) \in \mathbb{T} \times \mathbb{R}^d$, we consider the ODE

$$ \begin{cases} \frac{d}{dt}\psi_{s,t}(x) = \underline{b}(t, \psi_{s,t}(x)), & t \in \mathbb{T}, \\ \psi_{s,s}(x) = x. & \end{cases} $$

For $\kappa > 0$, we consider the Gaussian heat kernel

$$ p_t^\kappa(x) := \frac{1}{(\kappa \pi t)^{\frac{d}{2}}} \exp \left( -\frac{|x|^2}{\kappa t} \right), \quad t > 0, x \in \mathbb{R}^d. $$

The following results give density and gradient estimates for (3.1):

**Theorem 3.4.** Let Assumption 3.3 hold.

1. [46, Inequalities 2.7 and 2.8] There exist constants $c, \kappa > 0$ (depending only on $\Theta_2$) such that for $i \in \{0, 1\}$, $0 \le s < t \le T$ and $x, y \in \mathbb{R}^d$:

$$ |\nabla_x^i p_{s,t}^{b,\sigma}(x, y)| \le c(t-s)^{-\frac{i}{2}} p_{t-s}^{\kappa}(x-y). $$

2. [45, Lemma A.1] For $\alpha \in (0, \beta)$, there exist constants $c, \kappa > 0$ (depending only on $\Theta_2, \alpha$) such that for $0 \le s < t \le T$ and $x, y, y' \in \mathbb{R}^d$:

$$ \begin{aligned} & |\nabla_x p_{s,t}^{b,\sigma}(x, y) - \nabla_x p_{s,t}^{b,\sigma}(x, y')| \\ & \le c|y - y'|^{\alpha}(t-s)^{-\frac{1+\alpha}{2}} \{p_{t-s}^{\kappa}(\psi_{s,t}(x) - y) + p_{t-s}^{\kappa}(\psi_{s,t}(x) - y')\}. \end{aligned} $$
We define for $\kappa > 0, f \in L_+^0(\mathbb{R}^d) \cup L_b^0(\mathbb{R}^d), x \in \mathbb{R}^d$ and $0 \le s < t \le T$:

$$ P_t^\kappa f(x) := \int_{\mathbb{R}^d} p_t^\kappa(x-y)f(y) \, dy, \qquad (3.3) $$

$$ \hat{P}_{s,t}^{\kappa} f(x) := \int_{\mathbb{R}^d} p_{t-s}^{\kappa}(\psi_{s,t}(x) - y)f(y) \, dy, $$

$$ \tilde{P}_{s,t}^{\kappa} f(x) := \int_{\mathbb{R}^d} p_{t-s}^{\kappa}(\psi_{s,t}(y) - x)f(y) \, dy. \qquad (3.4) $$

For brevity, we denote $\frac{q-p}{pq} := \frac{1}{p} - \frac{1}{q}$ for $p, q \in [1, \infty]$. By Young's inequality for convolution, there exists a constant $c > 0$ (depending only on $d, \kappa$) such that for $t > 0$ and $1 \le p \le \bar{p} \le \infty$:

$$ \|P_t^\kappa\|_{L^p \to L^{\bar{p}}} := \sup_{\|f\|_{L^p} \le 1} \|P_t^\kappa f\|_{L^{\bar{p}}} \le c t^{-\frac{d(\bar{p}-p)}{2p\bar{p}}}. $$

We recall an essential generalization for dealing with unbounded drift:

**Lemma 3.5.** [35] Let Assumption 3.3 hold. There exists a constant $c > 0$ (depending only on $\Theta_2$) such that for $0 \le s < t \le T$ and $1 \le p \le \bar{p} \le \infty$:

$$ \|P_{t-s}^{\kappa}\|_{\tilde{L}^p \to \tilde{L}^{\bar{p}}} + \|\hat{P}_{s,t}^{\kappa}\|_{\tilde{L}^p \to \tilde{L}^{\bar{p}}} + \|\tilde{P}_{s,t}^{\kappa}\|_{\tilde{L}^p \to \tilde{L}^{\bar{p}}} \le c(t-s)^{-\frac{d(\bar{p}-p)}{2p\bar{p}}}. $$

For the sake of being self-contained, we also include the proof.

*Proof.* Let $z \in \mathbb{R}^d, f \in L_+^0(\mathbb{R}^d)$ and $\mathbf{B}_n := \{v = (v_1, \dots, v_d) \in \mathbb{Z}^d : \sum_{i=1}^d |v_i| = n\}$ for $n \in \mathbb{N}$. We write $M_1 \lesssim M_2$ if there exists a constant $c > 0$ (depending only on $\Theta_2$) such that $M_1 \le cM_2$. By [35, Inequality 3.11], there exists a constant $c_1 > 0$ (depending only on $\Theta_2$) such that

$$ \|1_{B(z,1)} \hat{P}_{s,t}^{\kappa} f\|_{L^{\bar{p}}} \lesssim (t-s)^{-\frac{d(\bar{p}-p)}{2p\bar{p}}} \sum_{n=0}^{\infty} \sum_{v \in \mathbf{B}_n} e^{-\frac{n^2}{c_1(t-s)}} \|1_{B(z+v,d)} f\|_{L^p}. $$

We have $\text{card}(\mathbf{B}_n) \le (2n+1)^d$, so

$$ \sum_{n=0}^{\infty} \sum_{v \in \mathbf{B}_n} e^{-\frac{n^2}{c_1(t-s)}} \le \sum_{n=0}^{\infty} (2n+1)^d e^{-\frac{n^2}{c_1 T}} \lesssim 1. $$

On the other hand, $\|1_{B(z+v,d)} f\|_{L^p} \lesssim \|f\|_{\tilde{L}^p}$. Then

$$ \|\hat{P}_{s,t}^{\kappa} f\|_{\tilde{L}^{\bar{p}}} \lesssim (t-s)^{-\frac{d(\bar{p}-p)}{2p\bar{p}}} \|f\|_{\tilde{L}^{\bar{p}}}. $$

Clearly, $P_{t-s}^\kappa$ is a special case of $\hat{P}_{s,t}^\kappa$ in which $\psi_{s,t}$ is the identity map on $\mathbb{R}^d$. Then

$$ \|P_{t-s}^{\kappa} f\|_{\tilde{L}^{\bar{p}}} \lesssim (t-s)^{-\frac{d(\bar{p}-p)}{2p\bar{p}}} \|f\|_{\tilde{L}^{\bar{p}}}. $$

It remains to prove for $\tilde{P}_{s,t}^\kappa$. By [35, Inequality 3.2], there exists a constant $c_2 \ge 1$ (depending only on $\Theta_2$) such that

$$ \sup_{0 \le s \le t \le T} \{\|\nabla \psi_{s,t}\|_{\infty} + \|\nabla \psi_{s,t}^{-1}\|_{\infty}\} \le c_2. \qquad (3.5) $$
We define for $\kappa > 0, f \in L_+^0(\mathbb{R}^d) \cup L_b^0(\mathbb{R}^d), x \in \mathbb{R}^d$ and $0 \le s < t \le T$:

$$P_t^\kappa f(x) := \int_{\mathbb{R}^d} p_t^\kappa(x-y)f(y) \, dy, \qquad (3.3)$$

$$\hat{P}_{s,t}^{\kappa} f(x) := \int_{\mathbb{R}^d} p_{t-s}^{\kappa}(\psi_{s,t}(x) - y)f(y) \, dy,$$

$$\tilde{P}_{s,t}^{\kappa} f(x) := \int_{\mathbb{R}^d} p_{t-s}^{\kappa}(\psi_{s,t}(y) - x)f(y) \, dy. \qquad (3.4)$$

For brevity, we denote $\frac{q-p}{pq} := \frac{1}{p} - \frac{1}{q}$ for $p, q \in [1, \infty]$. By Young's inequality for convolution, there exists a constant $c > 0$ (depending only on $d, \kappa$) such that for $t > 0$ and $1 \le p \le \bar{p} \le \infty$:

$$\|P_t^\kappa\|_{L^p \to L^{\bar{p}}} := \sup_{\|f\|_{L^p} \le 1} \|P_t^\kappa f\|_{L^{\bar{p}}} \le c t^{-\frac{d(\bar{p}-p)}{2p\bar{p}}}.$$

We recall an essential generalization for dealing with unbounded drift:

**Lemma 3.5.** [35] Let Assumption 3.3 hold. There exists a constant $c > 0$ (depending only on $\Theta_2$) such that for $0 \le s < t \le T$ and $1 \le p \le \bar{p} \le \infty$:

$$\|P_{t-s}^{\kappa}\|_{\tilde{L}^p \to \tilde{L}^{\bar{p}}} + \|\hat{P}_{s,t}^{\kappa}\|_{\tilde{L}^p \to \tilde{L}^{\bar{p}}} + \|\tilde{P}_{s,t}^{\kappa}\|_{\tilde{L}^p \to \tilde{L}^{\bar{p}}} \le c(t-s)^{-\frac{d(\bar{p}-p)}{2p\bar{p}}}.$$

For the sake of being self-contained, we also include the proof.

*Proof.* Let $z \in \mathbb{R}^d, f \in L_+^0(\mathbb{R}^d)$ and $\mathbf{B}_n := \{v = (v_1, \dots, v_d) \in \mathbb{Z}^d : \sum_{i=1}^d |v_i| = n\}$ for $n \in \mathbb{N}$. We write $M_1 \lesssim M_2$ if there exists a constant $c > 0$ (depending only on $\Theta_2$) such that $M_1 \le cM_2$. By [35, Inequality 3.11], there exists a constant $c_1 > 0$ (depending only on $\Theta_2$) such that

$$\|1_{B(z,1)} \hat{P}_{s,t}^{\kappa} f\|_{L^{\bar{p}}} \lesssim (t-s)^{-\frac{d(\bar{p}-p)}{2p\bar{p}}} \sum_{n=0}^{\infty} \sum_{v \in \mathbf{B}_n} e^{-\frac{n^2}{c_1(t-s)}} \|1_{B(z+v,d)} f\|_{L^p}.$$

We have $\text{card}(\mathbf{B}_n) \le (2n+1)^d$, so

$$\sum_{n=0}^{\infty} \sum_{v \in \mathbf{B}_n} e^{-\frac{n^2}{c_1(t-s)}} \le \sum_{n=0}^{\infty} (2n+1)^d e^{-\frac{n^2}{c_1 T}} \lesssim 1.$$

On the other hand, $\|1_{B(z+v,d)} f\|_{L^p} \lesssim \|f\|_{\tilde{L}^p}$. Then

$$\|\hat{P}_{s,t}^{\kappa} f\|_{\tilde{L}^{\bar{p}}} \lesssim (t-s)^{-\frac{d(\bar{p}-p)}{2p\bar{p}}} \|f\|_{\tilde{L}^{\bar{p}}}.$$

Clearly, $P_{t-s}^{\kappa}$ is a special case of $\hat{P}_{s,t}^{\kappa}$ in which $\psi_{s,t}$ is the identity map on $\mathbb{R}^d$. Then

$$\|P_{t-s}^{\kappa} f\|_{\tilde{L}^{\bar{p}}} \lesssim (t-s)^{-\frac{d(\bar{p}-p)}{2p\bar{p}}} \|f\|_{\tilde{L}^{\bar{p}}}.$$

It remains to prove for $\tilde{P}_{s,t}^{\kappa}$. By [35, Inequality 3.2], there exists a constant $c_2 \ge 1$ (depending only on $\Theta_2$) such that

$$\sup_{0 \le s \le t \le T} \{\|\nabla \psi_{s,t}\|_{\infty} + \|\nabla \psi_{s,t}^{-1}\|_{\infty}\} \le c_2. \qquad (3.5)$$
We have

$$
\begin{align}
\tilde{P}_{s,t}^{\kappa} f(x) &= \int_{\mathbb{R}^d} p_{t-s}^{\kappa} (\psi_{s,t}(y) - x) f(y) \, dy \quad \text{by (3.4)} \\
&\le \int_{\mathbb{R}^d} p_{t-s}^{\kappa} \left( \frac{\psi_{s,t}^{-1}(x) - y}{c_2} \right) f(y) \, dy \quad \text{by (3.5)} \\
&\lesssim \int_{\mathbb{R}^d} p_{t-s}^{\bar{\kappa}} (\psi_{s,t}^{-1}(x) - y) f(y) \, dy \quad \text{where } \bar{\kappa} := \kappa |c_2|^2 \\
&= (P_{t-s}^{\bar{\kappa}} f) \circ \psi_{s,t}^{-1}(x). \tag{3.6}
\end{align}
$$

It suffices to consider $\bar{p} < \infty$. We have

$$
\begin{align}
\|1_{B(z,1)} \tilde{P}_{s,t}^{\kappa} f\|_{L^{\bar{p}}}^{\bar{p}} &\lesssim \int_{B(z,1)} |(P_{t-s}^{\bar{\kappa}} f) \circ \psi_{s,t}^{-1}(x)|^{\bar{p}} dx \quad \text{by (3.6)} \\
&= \int_{\psi_{s,t}^{-1}(B(z,1))} |(P_{t-s}^{\bar{\kappa}} f)(x)|^{\bar{p}} |\det \nabla \psi_{s,t}(x)| dx \tag{3.7}
\end{align}
$$

$$
\lesssim \int_{\psi_{s,t}^{-1}(B(z,1))} |(P_{t-s}^{\bar{\kappa}} f)(x)|^{\bar{p}} dx \quad (3.8)
$$

$$
= \|1_{\psi_{s,t}^{-1}(B(z,1))} P_{t-s}^{\bar{\kappa}} f\|_{L^{\bar{p}}}^{\bar{p}} \\
\lesssim \|P_{t-s}^{\bar{\kappa}} f\|_{\bar{L}^{\bar{p}}}^{\bar{p}} \quad \text{by (3.5).}
$$

Above, (3.7) is due to change of variables formula and (3.8) due to Hadamard's inequality for
determinants. It follows that

$$
\|1_{B(z,1)} \tilde{P}_{s,t}^{\kappa} f\|_{L^{\bar{p}}} \lesssim \|P_{t-s}^{\bar{\kappa}} f\|_{\bar{L}^{\bar{p}}}.
$$

This completes the proof.

<figure><img src="image_3.png" /></figure>

3.3 Moment and L$^p$ estimates of marginal density

We consider another set of assumption:

**Assumption 3.6.** There exist constants $\beta \in (0, 1)$, $C > 0$, $l \in \mathbb{N}^*$ such that for $t \in \mathbb{T}$ and $x, y \in \mathbb{R}^d$:

(A1) $\nu \in \mathcal{P}_1(\mathbb{R}^d)$ has a density $\ell_\nu$.

(A2) $a_t$ is invertible and $\sigma_t$ is weakly differentiable.

(A3) There exist measurable maps $b^{(0)} : \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ and $b^{(1)} : \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ such that $b_t(x) = b_t^{(0)}(x) + b_t^{(1)}(x)$.

(A4) There exists $1 \le f_0 \in \tilde{L}_{q_0}^{p_0}$ with $(p_0, q_0) \in \mathcal{K}$ such that $|b_t^{(0)}(x)| \le f_0(t, x)$.

(A5) For $i \in \{1, 2, \dots, l\}$, there exists $1 \le f_i \in \tilde{L}_{q_i}^{p_i}$ with $(p_i, q_i) \in \mathcal{K}$ such that $|\nabla \sigma_t(x)| \le \sum_{i=1}^l f_i(t, x)$.
We have

$$
\begin{align}
\tilde{P}_{s,t}^{\kappa} f(x) &= \int_{\mathbb{R}^d} p_{t-s}^{\kappa} (\psi_{s,t}(y) - x) f(y) \, dy \quad \text{by (3.4)} \\
&\le \int_{\mathbb{R}^d} p_{t-s}^{\kappa} \left( \frac{\psi_{s,t}^{-1}(x) - y}{c_2} \right) f(y) \, dy \quad \text{by (3.5)} \\
&\lesssim \int_{\mathbb{R}^d} p_{t-s}^{\bar{\kappa}} (\psi_{s,t}^{-1}(x) - y) f(y) \, dy \quad \text{where } \bar{\kappa} := \kappa |c_2|^2 \\
&= (P_{t-s}^{\bar{\kappa}} f) \circ \psi_{s,t}^{-1}(x). \tag{3.6}
\end{align}
$$

It suffices to consider $\bar{p} < \infty$. We have

$$
\begin{align}
\|1_{B(z,1)} \tilde{P}_{s,t}^{\kappa} f\|_{L^{\bar{p}}}^{\bar{p}} &\lesssim \int_{B(z,1)} |(P_{t-s}^{\bar{\kappa}} f) \circ \psi_{s,t}^{-1}(x)|^{\bar{p}} dx \quad \text{by (3.6)} \\
&= \int_{\psi_{s,t}^{-1}(B(z,1))} |(P_{t-s}^{\bar{\kappa}} f)(x)|^{\bar{p}} |\det \nabla \psi_{s,t}(x)| dx \tag{3.7}
\end{align}
$$

$$
\lesssim \int_{\psi_{s,t}^{-1}(B(z,1))} |(P_{t-s}^{\bar{\kappa}} f)(x)|^{\bar{p}} dx \quad (3.8)
$$

$$
= \|1_{\psi_{s,t}^{-1}(B(z,1))} P_{t-s}^{\bar{\kappa}} f\|_{L^{\bar{p}}}^{\bar{p}} \\
\lesssim \|P_{t-s}^{\bar{\kappa}} f\|_{\bar{L}^{\bar{p}}}^{\bar{p}} \quad \text{by (3.5).}
$$

Above, (3.7) is due to change of variables formula and (3.8) due to Hadamard's inequality for
determinants. It follows that

$$
\|1_{B(z,1)} \tilde{P}_{s,t}^{\kappa} f\|_{L^{\bar{p}}} \lesssim \|P_{t-s}^{\bar{\kappa}} f\|_{\bar{L}^{\bar{p}}}.
$$

This completes the proof.

<figure><img src="image_3.png" /></figure>

3.3 Moment and L^p estimates of marginal density

We consider another set of assumption:

**Assumption 3.6.** There exist constants $\beta \in (0, 1)$, $C > 0$, $l \in \mathbb{N}^*$ such that for $t \in \mathbb{T}$ and $x, y \in \mathbb{R}^d$:

(A1) $\nu \in \mathcal{P}_1(\mathbb{R}^d)$ has a density $\ell_\nu$.

(A2) $a_t$ is invertible and $\sigma_t$ is weakly differentiable.

(A3) There exist measurable maps $b^{(0)} : \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ and $b^{(1)} : \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ such that $b_t(x) = b_t^{(0)}(x) + b_t^{(1)}(x)$.

(A4) There exists $1 \le f_0 \in \tilde{L}_{q_0}^{p_0}$ with $(p_0, q_0) \in \mathcal{K}$ such that $|b_t^{(0)}(x)| \le f_0(t, x)$.

(A5) For $i \in \{1, 2, \dots, l\}$, there exists $1 \le f_i \in \tilde{L}_{q_i}^{p_i}$ with $(p_i, q_i) \in \mathcal{K}$ such that $|\nabla \sigma_t(x)| \le \sum_{i=1}^l f_i(t, x)$.
(A6) The following conditions hold:

$$
\begin{align*}
& |b_t^{(1)}(x) - b_t^{(1)}(y)| \le C|x - y|, \\
& |b_t^{(1)}(0)| + \|\sigma_t\|_{\infty} + \|a_t^{-1}\|_{\infty} \le C, \\
& |\sigma_t(x) - \sigma_t(y)| \le C|x - y|^{\beta}.
\end{align*}
$$

Assumption 3.6 is appealing because it is a general but sufficient condition to obtain Krylov’s
and Khasminskii’s estimates, which are a key ingredient for establishing the other estimates in the
remaining of Section 3. We gather parameters in Assumption 3.6:

$$
\Theta_3 := (d, T, \beta, C, l, (p_i, q_i, f_i)_{i=0}^l).
$$

The class $\bar{\mathcal{K}}$ of exponent parameter is defined by

$$
\bar{\mathcal{K}} := \left\{ (p, q) \in (1, \infty)^2 : \frac{d}{p} + \frac{2}{q} < 2 \right\}.
$$

**Remark 3.7.** If $f \in \tilde{L}_q^p$ for some $(p, q) \in \mathcal{K}$ then there exists $(\bar{p}, \bar{q}) \in \bar{\mathcal{K}}$ such that $|f|^2 \in \tilde{L}_{\bar{q}}^{\bar{p}}$.

Let $\nu \in \mathcal{P}(\mathbb{R}^d)$ and $\ell_\nu$ be its density. We consider the SDE

$$
dX_t = b(t, X_t) dt + \sigma(t, X_t) dB_t, \quad t \in \mathbb{T}, \tag{3.9}
$$

where the distribution of $X_0$ is $\nu$. By [10, Theorem 1.1(1)], (3.9) is well-posed under Assumption 3.6.
First, we establish an essential result for proving Theorem 3.9:

**Proposition 3.8.** Let $(b, \sigma)$ satisfy Assumption 3.6 and $(X_t, t \in \mathbb{T})$ be the solution of (3.9).

1. (Khasminskii's estimate) For each $(p, q) \in \bar{\mathcal{K}}$, there exist constants $c > 0, k > 1$ (depending only on $\Theta_3, p, q$) such that for $0 \le t_0 < t_1 \le T$ and $g \in \tilde{L}_q^p(t_0, t_1)$:

$$
\mathbb{E} \left[ \exp \left( \int_{t_0}^{t_1} |g(s, X_s)| \, ds \right) \middle| \mathcal{F}_{t_0} \right] \le \exp \left( c \left( 1 + \|g\|_{\tilde{L}_q^p(t_0, t_1)}^k \right) \right). \quad (3.10)
$$

2. (Krylov's estimate) For each $(p, q) \in \bar{\mathcal{K}}$ and $j \ge 1$, there exists a constant $c > 0$ (depending only on $\Theta_3, p, q, j$) such that for $0 \le t_0 < t_1 \le T$ and $g \in \tilde{L}_q^p(t_0, t_1)$:

$$
\mathbb{E} \left[ \left( \int_{t_0}^{t_1} |g(s, X_s)| \, ds \right)^j \middle| \mathcal{F}_{t_0} \right] \le c \|g\|_{\tilde{L}_q^p(t_0, t_1)}^j. \quad (3.11)
$$

Proof. We fix $(p, q) \in \bar{K}$.

1. There exists $\bar{q} \in (1, q)$ such that $(p, \bar{q}) \in \bar{\mathcal{K}}$. By [47, Theorem 3.1], there exists a constant $c_1 > 0$ (depending only on $\Theta_3, p, \bar{q}$) such that for $0 \le t_0 < t_1 \le T$, a stopping time $\tau$ and $g \in \tilde{L}_{\bar{q}}^p(t_0, t_1)$:

$$
\mathbb{E} \left[ \left[ \int_{t_0}^{t_1 \wedge \tau} |g(s, X_s)| \, ds \right] \middle| \mathcal{F}_{t_0} \right] \le c_1 \|g\|_{\tilde{L}_{\bar{q}}^p(t_0, t_1)}
$$
(A6) The following conditions hold:

$$
\begin{align*}
& |b_t^{(1)}(x) - b_t^{(1)}(y)| \le C|x - y|, \\
& |b_t^{(1)}(0)| + \|\sigma_t\|_{\infty} + \|a_t^{-1}\|_{\infty} \le C, \\
& |\sigma_t(x) - \sigma_t(y)| \le C|x - y|^{\beta}.
\end{align*}
$$

Assumption 3.6 is appealing because it is a general but sufficient condition to obtain Krylov’s
and Khasminskii’s estimates, which are a key ingredient for establishing the other estimates in the
remaining of Section 3. We gather parameters in Assumption 3.6:

$$
\Theta_3 := (d, T, \beta, C, l, (p_i, q_i, f_i)_{i=0}^l).
$$

The class $\bar{\mathcal{K}}$ of exponent parameter is defined by

$$
\bar{\mathcal{K}} := \left\{ (p, q) \in (1, \infty)^2 : \frac{d}{p} + \frac{2}{q} < 2 \right\}.
$$

**Remark 3.7.** If $f \in \tilde{L}_q^p$ for some $(p, q) \in \mathcal{K}$ then there exists $(\bar{p}, \bar{q}) \in \bar{\mathcal{K}}$ such that $|f|^2 \in \tilde{L}_{\bar{q}}^{\bar{p}}$.

Let $\nu \in \mathcal{P}(\mathbb{R}^d)$ and $\ell_\nu$ be its density. We consider the SDE

$$
dX_t = b(t, X_t) dt + \sigma(t, X_t) dB_t, \quad t \in \mathbb{T}, \tag{3.9}
$$

where the distribution of $X_0$ is $\nu$. By [10, Theorem 1.1(1)], (3.9) is well-posed under Assumption 3.6.
First, we establish an essential result for proving Theorem 3.9:

**Proposition 3.8.** Let $(b, \sigma)$ satisfy Assumption 3.6 and $(X_t, t \in \mathbb{T})$ be the solution of (3.9).

1. (Khasminskii's estimate) For each $(p, q) \in \bar{\mathcal{K}}$, there exist constants $c > 0, k > 1$ (depending only on $\Theta_3, p, q$) such that for $0 \le t_0 < t_1 \le T$ and $g \in \tilde{L}_q^p(t_0, t_1)$:

$$
\mathbb{E} \left[ \exp \left( \int_{t_0}^{t_1} |g(s, X_s)| \, ds \right) \middle| \mathcal{F}_{t_0} \right] \le \exp \left( c \left( 1 + \|g\|_{\tilde{L}_q^p(t_0, t_1)}^k \right) \right). \quad (3.10)
$$

2. (Krylov's estimate) For each $(p, q) \in \bar{\mathcal{K}}$ and $j \ge 1$, there exists a constant $c > 0$ (depending only on $\Theta_3, p, q, j$) such that for $0 \le t_0 < t_1 \le T$ and $g \in \tilde{L}_q^p(t_0, t_1)$:

$$
\mathbb{E} \left[ \left( \int_{t_0}^{t_1} |g(s, X_s)| \, ds \right)^j \middle| \mathcal{F}_{t_0} \right] \le c \|g\|_{\tilde{L}_q^p(t_0, t_1)}^j. \quad (3.11)
$$

Proof. We fix $(p, q) \in \bar{K}$.

1. There exists $\bar{q} \in (1, q)$ such that $(p, \bar{q}) \in \bar{\mathcal{K}}$. By [47, Theorem 3.1], there exists a constant $c_1 > 0$ (depending only on $\Theta_3, p, \bar{q}$) such that for $0 \le t_0 < t_1 \le T$, a stopping time $\tau$ and $g \in \tilde{L}_{\bar{q}}^p(t_0, t_1)$:

$$
\mathbb{E} \left[ \left[ \int_{t_0}^{t_1 \wedge \tau} |g(s, X_s)| \, ds \right] \middle| \mathcal{F}_{t_0} \right] \le c_1 \|g\|_{\tilde{L}_{\bar{q}}^p(t_0, t_1)}
$$
Let $\delta := \frac{1}{\bar{q}} - \frac{1}{q} \in (0, 1)$. By Hölder's inequality, we have for $0 \le t_0 < t_1 \le T$ and $g \in \tilde{L}_q^p(t_0, t_1)$:

$$ \|g\|_{\tilde{L}_{\bar{q}}^p(t_0, t_1)} \le (t_1 - t_0)^{\delta} \|g\|_{\tilde{L}_{\bar{q}}^p(t_0, t_1)}. \quad (3.12) $$

We denote by $I_j^n$ the open interval $(\frac{(j-1)(t_1-t_0)}{n}, \frac{j(t_1-t_0)}{n})$ for $j = 1, \dots, n$. We fix $g \in \tilde{L}_q^p(t_0, t_1) \subset \tilde{L}_{\bar{q}}^p(t_0, t_1)$. Let $n \ge 2$ be the smallest integer such that

$$ \|g\|_{\tilde{L}_{\bar{q}}^p(I_j^n)} \le \frac{1}{2c_1}, \quad j = 1, \dots, n. \quad (3.13) $$

By [48, Lemma 3.5],

$$ \mathbb{E} \left[ \exp \left( \int_{t_0}^{t_1} |g(s, X_s)| \, ds \right) \middle| \mathcal{F}_{t_0} \right] \le 2^n. $$

By (3.13), there exists $\bar{j} \in \{1, \dots, n-1\}$ such that

$$ \|g\|_{\tilde{L}_{\bar{q}}^p(I_j^{n-1})} > \frac{1}{2c_1}. \quad (3.14) $$

It follows from (3.12) and (3.14) that

$$ \left( \frac{t_1 - t_0}{n - 1} \right)^{\delta} \|g\|_{\tilde{L}_{\bar{q}}^p(I_j^{n-1})} > \frac{1}{2c_1}, $$

which implies

$$ n < 1 + T(2c_1)^{-\frac{1}{\delta}} \|g\|_{\tilde{L}_{\bar{q}}^p(t_0, t_1)}^{1/\delta}. $$

The estimate (3.10) then follows with $k := \frac{1}{\delta}$.

2. We fix $j \ge 1$. We adapt an elegant idea from [10, Lemma 2.3] into our simpler setting. Let $C_j := e^{j-1}$. Then $h(r) := |\log(C_j + r)|^j$ is concave for $r \in \mathbb{R}_+$. We have

$$ \begin{aligned} & \mathbb{E} \left[ \left( \int_{t_0}^{t_1} |g(s, X_s)| \, ds \right)^j \middle| \mathcal{F}_{t_0} \right] \\ & \le \mathbb{E} \left[ \left\{ \log \left( C_j + \exp \left( \int_{t_0}^{t_1} |g(s, X_s)| \, ds \right) \right) \right\}^j \middle| \mathcal{F}_{t_0} \right] \\ & \le \left\{ \log \left( C_j + \mathbb{E} \left[ \exp \left( \int_{t_0}^{t_1} |g(s, X_s)| \, ds \right) \middle| \mathcal{F}_{t_0} \right] \right) \right\}^j \quad \text{by Jensen's inequality} \\ & \le \left\{ \log[C_j + \exp(c(1 + \|g\|_{\tilde{L}_{\bar{q}}^p(t_0, t_1)}^k))] \right\}^j, \end{aligned} $$

where the constants $c, k > 0$ are given by (3.10). As a result, there exists a constant $\bar{C}_j > 0$ (depending only on $c, j$) such that

$$ \mathbb{E} \left[ \left( \int_{t_0}^{t_1} |g(s, X_s)| \, ds \right)^j \middle| \mathcal{F}_{t_0} \right] \le \bar{C}_j (1 + \|g\|_{\tilde{L}_{\bar{q}}^p(t_0, t_1)}^k)^j. $$
Let $\delta := \frac{1}{\bar{q}} - \frac{1}{q} \in (0, 1)$. By Hölder's inequality, we have for $0 \le t_0 < t_1 \le T$ and $g \in \tilde{L}_q^p(t_0, t_1)$:

$$ \|g\|_{\tilde{L}_{\bar{q}}^p(t_0, t_1)} \le (t_1 - t_0)^{\delta} \|g\|_{\tilde{L}_{\bar{q}}^p(t_0, t_1)}. \quad (3.12) $$

We denote by $I_j^n$ the open interval $(\frac{(j-1)(t_1-t_0)}{n}, \frac{j(t_1-t_0)}{n})$ for $j = 1, \dots, n$. We fix $g \in \tilde{L}_q^p(t_0, t_1) \subset \tilde{L}_{\bar{q}}^p(t_0, t_1)$. Let $n \ge 2$ be the smallest integer such that

$$ \|g\|_{\tilde{L}_{\bar{q}}^p(I_j^n)} \le \frac{1}{2c_1}, \quad j = 1, \dots, n. \quad (3.13) $$

By [48, Lemma 3.5],

$$ \mathbb{E} \left[ \exp \left( \int_{t_0}^{t_1} |g(s, X_s)| \mathrm{d}s \right) \middle| \mathcal{F}_{t_0} \right] \le 2^n. $$

By (3.13), there exists $\bar{j} \in \{1, \dots, n-1\}$ such that

$$ \|g\|_{\tilde{L}_{\bar{q}}^p(I_j^{n-1})} > \frac{1}{2c_1}. \quad (3.14) $$

It follows from (3.12) and (3.14) that

$$ \left( \frac{t_1 - t_0}{n - 1} \right)^{\delta} \|g\|_{\tilde{L}_{\bar{q}}^p(I_j^{n-1})} > \frac{1}{2c_1}, $$

which implies

$$ n < 1 + T(2c_1)^{-\frac{1}{\delta}} \|g\|_{\tilde{L}_{\bar{q}}^p(t_0, t_1)}^{1/\delta}. $$

The estimate (3.10) then follows with $k := \frac{1}{\delta}$.

2. We fix $j \ge 1$. We adapt an elegant idea from [10, Lemma 2.3] into our simpler setting. Let $C_j := e^{j-1}$. Then $h(r) := |\log(C_j + r)|^j$ is concave for $r \in \mathbb{R}_+$. We have

$$ \begin{aligned} & \mathbb{E} \left[ \left( \int_{t_0}^{t_1} |g(s, X_s)| \mathrm{d}s \right)^j \middle| \mathcal{F}_{t_0} \right] \\ & \le \mathbb{E} \left[ \left\{ \log \left( C_j + \exp \left( \int_{t_0}^{t_1} |g(s, X_s)| \mathrm{d}s \right) \right) \right\}^j \middle| \mathcal{F}_{t_0} \right] \\ & \le \left\{ \log \left( C_j + \mathbb{E} \left[ \exp \left( \int_{t_0}^{t_1} |g(s, X_s)| \mathrm{d}s \right) \middle| \mathcal{F}_{t_0} \right] \right) \right\}^j \quad \text{by Jensen's inequality} \\ & \le \{\log[C_j + \exp(c(1 + \|g\|_{\tilde{L}_{\bar{q}}^p(t_0, t_1)}^k))]\}^j, \end{aligned} $$

where the constants $c, k > 0$ are given by (3.10). As a result, there exists a constant $\bar{C}_j > 0$ (depending only on $c, j$) such that

$$ \mathbb{E} \left[ \left( \int_{t_0}^{t_1} |g(s, X_s)| \mathrm{d}s \right)^j \middle| \mathcal{F}_{t_0} \right] \le \bar{C}_j (1 + \|g\|_{\tilde{L}_{\bar{q}}^p(t_0, t_1)}^k)^j. $$
Replacing $g$ with $\frac{g}{\|g\|_{\tilde{L}_q^p(t_0, t_1)}}$ in the above inequality, we obtain

$$ \mathbb{E} \left[ \left( \int_{t_0}^{t_1} |g(s, X_s)| ds \right)^j \middle| \mathcal{F}_{t_0} \right] \le \bar{C}_j 2^j \|g\|_{\tilde{L}_q^p(t_0, t_1)}^j. $$

The estimate (3.11) then follows. This completes the proof.

□

For another proof of Proposition 3.8, see [49, Lemma 4.1]. Second, we establish moment estimates:

**Theorem 3.9.** Let $p \in [1, \infty)$ and $(b, \sigma)$ satisfy Assumption 3.6. Let $\mu_t$ be the distribution of $X_t$ in (3.9).

1. $\mu_t$ is absolutely continuous w.r.t. Lebesgue measure on $\mathbb{R}^d$.

2. There exist constants $c > 0$ (depending only on $\Theta_3, p$) and $\delta \in (0, \frac{1}{2})$ (depending only on $q_0$) such that for $\alpha > 0$ and $0 \le u \le t \le T$:

$$ \mathbb{E} \left[ \sup_{s \in [u, t]} |X_s|^p \right] \le c(1 + \mathbb{E}(|X_u|^p)), \quad (3.15) $$

$$ \mathbb{E} \left[ \sup_{s \in [u, t]} |X_s - X_u|^p \right] \le c|t - u|^{\delta p} (1 + \mathbb{E}(|X_u|^p)), \quad (3.16) $$

$$ \sup_{s \in [u, t]} \int_{B_R^c} |x|^p d\mu_s(x) \le \frac{c(1 + \mathbb{E}(|X_u|^{p+\alpha}))}{R}, \quad R > 0, \quad (3.17) $$

where $B_R^c := \mathbb{R}^d \setminus B(0, R)$. Estimate (3.17) also holds for $p = 0$.

In particular, (3.17) is essential in proving Lemma 4.3 in Section 4.2.

*Proof.* By Assumption 3.6[A3], there exist measurable maps $b^{(0)}: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ and $b^{(1)}: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ such that $b_t(x) = b_t^{(0)}(x) + b_t^{(1)}(x)$. By Assumption 3.6[A4], there exists $f_0 \in \tilde{L}_{q_0}^{p_0}$ with $(p_0, q_0) \in \mathcal{K}$ such that $|b_t^{(0)}(x)| \le f_0(t, x)$. We consider the SDE

$$ d\bar{X}_t = b^{(1)}(t, \bar{X}_t) dt + \sigma(t, \bar{X}_t) dB_t, \quad t \in \mathbb{T}, \quad (3.18) $$

where the distribution of $\bar{X}_0$ is $\nu$. Clearly, $(b^{(1)}, \sigma)$ satisfies Assumption 3.6, so (3.18) is well-posed. We define

$$ \xi_t := \{\sigma_t^\top a_t^{-1} b_t^{(0)}\}(\bar{X}_t), $$

$$ \bar{B}_t := B_t - \int_0^t \xi_s \, ds, $$

$$ R_t := \exp \left( \int_0^t \xi_t^\top dB_t - \frac{1}{2} \int_0^t |\xi_t|^2 dt \right), $$

$$ I_t := \mathbb{E} \left[ \exp \left( \frac{1}{2} \int_0^t |\xi_t|^2 dt \right) \right], \quad t \in \mathbb{T}. $$
Replacing $g$ with $\frac{g}{\|g\|_{\tilde{L}_q^p(t_0, t_1)}}$ in the above inequality, we obtain

$$ \mathbb{E} \left[ \left( \int_{t_0}^{t_1} |g(s, X_s)| ds \right)^j \left| \mathcal{F}_{t_0} \right| \right] \leq \bar{C}_j 2^j \|g\|_{\tilde{L}_q^p(t_0, t_1)}^j. $$

The estimate (3.11) then follows. This completes the proof.

□

For another proof of Proposition 3.8, see [49, Lemma 4.1]. Second, we establish moment estimates:

**Theorem 3.9.** Let $p \in [1, \infty)$ and $(b, \sigma)$ satisfy Assumption 3.6. Let $\mu_t$ be the distribution of $X_t$ in (3.9).

1. $\mu_t$ is absolutely continuous w.r.t. Lebesgue measure on $\mathbb{R}^d$.

2. There exist constants $c > 0$ (depending only on $\Theta_3, p$) and $\delta \in (0, \frac{1}{2})$ (depending only on $q_0$) such that for $\alpha > 0$ and $0 \le u \le t \le T$:

$$ \mathbb{E} \left[ \sup_{s \in [u, t]} |X_s|^p \right] \le c(1 + \mathbb{E}(|X_u|^p)), \quad (3.15) $$

$$ \mathbb{E} \left[ \sup_{s \in [u, t]} |X_s - X_u|^p \right] \le c|t - u|^{\delta p} (1 + \mathbb{E}(|X_u|^p)), \quad (3.16) $$

$$ \sup_{s \in [u, t]} \int_{B_R^c} |x|^p d\mu_s(x) \le \frac{c(1 + \mathbb{E}(|X_u|^{p+\alpha}))}{R}, \quad R > 0, \quad (3.17) $$

where $B_R^c := \mathbb{R}^d \setminus B(0, R)$. Estimate (3.17) also holds for $p = 0$.

In particular, (3.17) is essential in proving Lemma 4.3 in Section 4.2.

*Proof.* By Assumption 3.6[A3], there exist measurable maps $b^{(0)}: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ and $b^{(1)}: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ such that $b_t(x) = b_t^{(0)}(x) + b_t^{(1)}(x)$. By Assumption 3.6[A4], there exists $f_0 \in \tilde{L}_{q_0}^{p_0}$ with $(p_0, q_0) \in \mathcal{K}$ such that $|b_t^{(0)}(x)| \le f_0(t, x)$. We consider the SDE

$$ d\bar{X}_t = b^{(1)}(t, \bar{X}_t) dt + \sigma(t, \bar{X}_t) dB_t, \quad t \in \mathbb{T}, \quad (3.18) $$

where the distribution of $\bar{X}_0$ is $\nu$. Clearly, $(b^{(1)}, \sigma)$ satisfies Assumption 3.6, so (3.18) is well-posed. We define

$$ \xi_t := \{\sigma_t^\top a_t^{-1} b_t^{(0)}\}(\bar{X}_t), $$

$$ \bar{B}_t := B_t - \int_0^t \xi_s \, ds, $$

$$ R_t := \exp \left( \int_0^t \xi_t^\top dB_t - \frac{1}{2} \int_0^t |\xi_t|^2 dt \right), $$

$$ I_t := \mathbb{E} \left[ \exp \left( \frac{1}{2} \int_0^t |\xi_t|^2 dt \right) \right], \quad t \in \mathbb{T}. $$
By the uniform boundedness of $\sigma_t^\top a_t^{-1}$, Remark 3.7 and Proposition 3.8[1], we get $I_T < \infty$. So $R_T$ is an exponential martingale with $\mathbb{E}[R_T] = 1$. By Girsanov's theorem, the process $(\bar{B}_t, t \in \mathbb{T})$ is an $m$-dimensional Brownian motion under the probability measure $\bar{\mathbb{P}} := R_T \bar{\mathbb{P}}$. We denote by $\bar{\mathbb{E}}$ the expectation w.r.t. $\bar{\mathbb{P}}$. Clearly, (3.18) can be written under $\bar{\mathbb{P}}$ as

$$d\bar{X}_t = b(t, \bar{X}_t) dt + \sigma(t, \bar{X}_t) d\bar{B}_t, \quad t \in \mathbb{T}. \qquad (3.19)$$

1. By Assumption 3.6[A6] and [45, Theorem 1.2], the distribution of $\bar{X}_t$ under $\bar{\mathbb{P}}$ admits a density. Notice that $\bar{\mathbb{P}}$ and $\bar{\mathbb{P}}$ are equivalent, so the distribution of $\bar{X}_t$ under $\bar{\mathbb{P}}$ also admits a density. Because $\bar{X}_0$ is $\mathcal{F}_0$-measurable, we have for each $\varphi \in C_c^\infty(\mathbb{R}^d)$ that

$$\bar{\mathbb{E}}[\varphi(\bar{X}_0)] = \mathbb{E}[\varphi(\bar{X}_0)R_0] = \mathbb{E}[\varphi(\bar{X}_0)].$$

It follows that the distribution of $\bar{X}_0$ under $\bar{\mathbb{P}}$ is also $\nu$. By weak uniqueness of (3.9) and (3.19), the distribution of $X_t$ under $\bar{\mathbb{P}}$ is the same as that of $\bar{X}_t$ under $\bar{\mathbb{P}}$. As a result, the distribution of $X_t$ under $\bar{\mathbb{P}}$ admits a density.

2. We combine the localization argument (see, for instance, [50, Theorem 9.1]) with Krylov's estimate.

(a) We fix $p \in [1, \infty)$ and $u \in [0, T)$ such that $\mathbb{E}[|X_u|^p] < \infty$. For $R > 0$, let $\tau_R := \inf\{t \in [u, T] : |X_t| \ge R\}$ be the exit time of $X_t$ from the open ball $B(0, R)$. We adopt the convention that $\tau_R = T$ if $|X_t| < R$ for all $t \in [u, T]$. We denote $X_R(t) := X_{t \wedge \tau_R}$. We have for $t \in [u, T]$:

$$\begin{aligned}
X_R(t) &= X_u + \int_u^{t \wedge \tau_R} b(r, X_r) dr + \int_u^{t \wedge \tau_R} \sigma(r, X_r) dB_r \\
&= X_u + \int_u^t b(r, X_r) 1_{\{r < \tau_R\}} dr + \int_u^t \sigma(r, X_r) 1_{\{r < \tau_R\}} dB_r \\
&= X_u + \int_u^t b^{(1)}(r, X_R(r)) 1_{\{r < \tau_R\}} dr + \int_u^t b^{(0)}(r, X_r) 1_{\{r < \tau_R\}} dr \\
&\qquad + \int_u^t \sigma(r, X_R(r)) 1_{\{r < \tau_R\}} dB_r.
\end{aligned}$$

By Hardy's inequality, we have for $n \in \mathbb{N}^*$, $p \ge 1$ and $x_1, \dots, x_n \in \mathbb{R}^d$:

$$|x_1 + \dots + x_n|^p \le n^p (|x_1|^p + \dots + |x_n|^p).$$

We write $M_1 \lesssim M_2$ if there exists a constant $c > 0$ (depending only on $\Theta_3, p$) such that $M_1 \le cM_2$. Then

$$\begin{aligned}
\mathbb{E}[\sup_{s \in [u,t]} |X_R(s)|^p] &\lesssim \mathbb{E}[|X_u|^p] + \mathbb{E}\left[\left(\int_u^t f_0(r, X_r) dr\right)^p\right] \\
&\quad + \mathbb{E}\left[\left(\int_u^t |b^{(1)}(r, X_R(r))| dr\right)^p\right] \\
&\quad + \mathbb{E}\left[\sup_{s \in [u,t]} \left|\int_u^s \sigma(r, X_R(r)) 1_{\{r < \tau_R\}} dB_r\right|^p\right] \\
&=: \mathbb{E}[|X_u|^p] + I_1 + I_2 + I_3.
\end{aligned}$$
By the uniform boundedness of $\sigma_t^\top a_t^{-1}$, Remark 3.7 and Proposition 3.8[1], we get $I_T < \infty$. So $R_T$ is an exponential martingale with $\mathbb{E}[R_T] = 1$. By Girsanov's theorem, the process $(\bar{B}_t, t \in \mathbb{T})$ is an $m$-dimensional Brownian motion under the probability measure $\bar{\mathbb{P}} := R_T \bar{\mathbb{P}}$. We denote by $\bar{\mathbb{E}}$ the expectation w.r.t. $\bar{\mathbb{P}}$. Clearly, (3.18) can be written under $\bar{\mathbb{P}}$ as

$$d\bar{X}_t = b(t, \bar{X}_t) dt + \sigma(t, \bar{X}_t) d\bar{B}_t, \quad t \in \mathbb{T}. \qquad (3.19)$$

1. By Assumption 3.6[A6] and [45, Theorem 1.2], the distribution of $\bar{X}_t$ under $\bar{\mathbb{P}}$ admits a density. Notice that $\bar{\mathbb{P}}$ and $\bar{\mathbb{P}}$ are equivalent, so the distribution of $\bar{X}_t$ under $\bar{\mathbb{P}}$ also admits a density. Because $\bar{X}_0$ is $\mathcal{F}_0$-measurable, we have for each $\varphi \in C_c^\infty(\mathbb{R}^d)$ that

$$\bar{\mathbb{E}}[\varphi(\bar{X}_0)] = \mathbb{E}[\varphi(\bar{X}_0)R_0] = \mathbb{E}[\varphi(\bar{X}_0)].$$

It follows that the distribution of $\bar{X}_0$ under $\bar{\mathbb{P}}$ is also $\nu$. By weak uniqueness of (3.9) and (3.19), the distribution of $X_t$ under $\bar{\mathbb{P}}$ is the same as that of $\bar{X}_t$ under $\bar{\mathbb{P}}$. As a result, the distribution of $X_t$ under $\bar{\mathbb{P}}$ admits a density.

2. We combine the localization argument (see, for instance, [50, Theorem 9.1]) with Krylov's estimate.

(a) We fix $p \in [1, \infty)$ and $u \in [0, T)$ such that $\mathbb{E}[|X_u|^p] < \infty$. For $R > 0$, let $\tau_R := \inf\{t \in [u, T] : |X_t| \ge R\}$ be the exit time of $X_t$ from the open ball $B(0, R)$. We adopt the convention that $\tau_R = T$ if $|X_t| < R$ for all $t \in [u, T]$. We denote $X_R(t) := X_{t \wedge \tau_R}$. We have for $t \in [u, T]$:

$$\begin{aligned}
X_R(t) &= X_u + \int_u^{t \wedge \tau_R} b(r, X_r) dr + \int_u^{t \wedge \tau_R} \sigma(r, X_r) dB_r \\
&= X_u + \int_u^t b(r, X_r) 1_{\{r < \tau_R\}} dr + \int_u^t \sigma(r, X_r) 1_{\{r < \tau_R\}} dB_r \\
&= X_u + \int_u^t b^{(1)}(r, X_R(r)) 1_{\{r < \tau_R\}} dr + \int_u^t b^{(0)}(r, X_r) 1_{\{r < \tau_R\}} dr \\
&\qquad + \int_u^t \sigma(r, X_R(r)) 1_{\{r < \tau_R\}} dB_r.
\end{aligned}$$

By Hardy's inequality, we have for $n \in \mathbb{N}^*$, $p \ge 1$ and $x_1, \dots, x_n \in \mathbb{R}^d$:

$$|x_1 + \dots + x_n|^p \le n^p (|x_1|^p + \dots + |x_n|^p).$$

We write $M_1 \lesssim M_2$ if there exists a constant $c > 0$ (depending only on $\Theta_3, p$) such that $M_1 \le cM_2$. Then

$$\begin{aligned}
\mathbb{E}[\sup_{s \in [u,t]} |X_R(s)|^p] &\lesssim \mathbb{E}[|X_u|^p] + \mathbb{E}\left[\left(\int_u^t f_0(r, X_r) dr\right)^p\right] \\
&\quad + \mathbb{E}\left[\left(\int_u^t |b^{(1)}(r, X_R(r))| dr\right)^p\right] \\
&\quad + \mathbb{E}\left[\sup_{s \in [u,t]} \left|\int_u^s \sigma(r, X_R(r)) 1_{\{r < \tau_R\}} dB_r\right|^p\right] \\
&=: \mathbb{E}[|X_u|^p] + I_1 + I_2 + I_3.
\end{aligned}$$
By Proposition 3.8[2], $I_1 \lesssim 1$. We have $|b^{(1)}(r, x)| \lesssim 1 + |x|$, so

$$
\begin{align*}
I_2 &\le |t-u|^{p-1} \mathbb{E} \left[ \int_u^t |b^{(1)}(r, X_R(r))|^p dr \right] && \text{by Hölder's inequality} \\
&\lesssim |t-u|^{p-1} \mathbb{E} \left[ \int_u^t (1+|X_R(r)|^p) dr \right] \\
&\lesssim 1 + |t-u|^{p-1} \mathbb{E} \left[ \int_u^t |X_R(r)|^p dr \right].
\end{align*}
$$

By Burkholder-Davis-Gundy inequality (see, for instance, [51, Theorem 19.20]) and the boundedness of $\sigma$ (from Assumption 3.6[A6]),

$$
I_3 \lesssim \mathbb{E} \left[ \left( \int_u^t |\sigma(r, X_R(r))|^2 dr \right)^{p/2} \right] \lesssim |t-u|^{p/2}.
$$

As a result,

$$
\begin{align*}
\eta_R(t) &:= \mathbb{E} \left[ \sup_{s \in [u,t]} |X_R(s)|^p \right] \\
&\lesssim 1 + \mathbb{E} [ |X_u|^p ] + \int_u^t \mathbb{E} [ |X_R(r)|^p ] dr, \quad t \in [u,T].
\end{align*}
$$

By construction, $|X_R(s)| \le |X_u| \vee R$ for $s \in [u, t]$. This implies $\eta_R(t) \le \mathbb{E}[|X_u|^p \vee R^p] < \infty$ for $t \in [u, T]$. We have

$$
\eta_R(t) \lesssim 1 + \mathbb{E}[|X_u|^p] + \int_u^t \eta_R(r) dr, \quad t \in [u, T].
$$

By Grönwall's lemma,

$$
\eta_R(t) \lesssim 1 + \mathbb{E}[|X_u|^p], \quad t \in [u, T], R > 0.
$$

Because $X$ has continuous sample paths, $\tau_R \uparrow T$ a.s. as $R \to \infty$. Hence

$$
\sup_{s \in [u,t]} |X_R(s)|^p \uparrow \sup_{s \in [u,t]} |X_s|^p \quad \text{a.s. as } R \to \infty.
$$

The estimate (3.15) then follows from an application of monotone convergence theorem.

(b) We have

$$
\begin{align*}
\mathbb{E} \left[ \sup_{s \in [u,t]} |X_s - X_u|^p \right] &\lesssim \mathbb{E} \left[ \left( \int_u^t f_0(r, X_r) dr \right)^p \right] \\
&\quad + \mathbb{E} \left[ \left( \int_u^t |b^{(1)}(r, X_r)| dr \right)^p \right] \\
&\quad + \mathbb{E} \left[ \sup_{s \in [u,t]} \left| \int_u^s \sigma(r, X_r) dB_r \right|^p \right] \\
&=: J_1 + J_2 + J_3.
\end{align*}
$$
By Proposition 3.8[2], $I_1 \lesssim 1$. We have $|b^{(1)}(r, x)| \lesssim 1 + |x|$, so

$$ 
\begin{aligned}
I_2 &\le |t-u|^{p-1} \mathbb{E} \left[ \int_u^t |b^{(1)}(r, X_R(r))|^p dr \right] && \text{by Hölder's inequality} \\
&\lesssim |t-u|^{p-1} \mathbb{E} \left[ \int_u^t (1+|X_R(r)|^p) dr \right] \\
&\lesssim 1 + |t-u|^{p-1} \mathbb{E} \left[ \int_u^t |X_R(r)|^p dr \right].
\end{aligned}
$$

By Burkholder-Davis-Gundy inequality (see, for instance, [51, Theorem 19.20]) and the boundedness of $\sigma$ (from Assumption 3.6[A6]),

$$ I_3 \lesssim \mathbb{E} \left[ \left( \int_u^t |\sigma(r, X_R(r))|^2 dr \right)^{p/2} \right] \lesssim |t-u|^{p/2}. $$

As a result,

$$ 
\begin{aligned}
\eta_R(t) &:= \mathbb{E} \left[ \sup_{s \in [u,t]} |X_R(s)|^p \right] \\
&\lesssim 1 + \mathbb{E} [ |X_u|^p ] + \int_u^t \mathbb{E} [ |X_R(r)|^p ] dr, \quad t \in [u,T].
\end{aligned}
$$

By construction, $|X_R(s)| \le |X_u| \vee R$ for $s \in [u, t]$. This implies $\eta_R(t) \le \mathbb{E}[|X_u|^p \vee R^p] < \infty$ for $t \in [u, T]$. We have

$$ \eta_R(t) \lesssim 1 + \mathbb{E}[|X_u|^p] + \int_u^t \eta_R(r) dr, \quad t \in [u, T]. $$

By Grönwall's lemma,

$$ \eta_R(t) \lesssim 1 + \mathbb{E}[|X_u|^p], \quad t \in [u, T], R > 0. $$

Because $X$ has continuous sample paths, $\tau_R \uparrow T$ a.s. as $R \to \infty$. Hence

$$ \sup_{s \in [u,t]} |X_R(s)|^p \uparrow \sup_{s \in [u,t]} |X_s|^p \quad \text{a.s. as } R \to \infty. $$

The estimate (3.15) then follows from an application of monotone convergence theorem.

(b) We have

$$ 
\begin{aligned}
\mathbb{E} \left[ \sup_{s \in [u,t]} |X_s - X_u|^p \right] &\lesssim \mathbb{E} \left[ \left( \int_u^t f_0(r, X_r) dr \right)^p \right] \\
&\quad + \mathbb{E} \left[ \left( \int_u^t |b^{(1)}(r, X_r)| dr \right)^p \right] \\
&\quad + \mathbb{E} \left[ \sup_{s \in [u,t]} \left| \int_u^s \sigma(r, X_r) dB_r \right|^p \right] \\
&=: J_1 + J_2 + J_3.
\end{aligned}
$$
There exists $\bar{q}_0 \in (2, q_0)$ such that $(p_0, \bar{q}_0) \in \bar{K}$. Let $\delta := \frac{1}{\bar{q}_0} - \frac{1}{q_0} \in (0, \frac{1}{2})$. By Hölder's inequality,

$$ \|f_0\|_{\tilde{L}_{\bar{q}_0}^{p_0}(u,t)} \le (t-u)^\delta \|f_0\|_{\tilde{L}_{q_0}^{p_0}(u,t)}. $$

By Proposition 3.8[2],

$$ J_1 \lesssim \|f_0\|_{\tilde{L}_{\bar{q}_0}^{p_0}(u,t)}^p \le (t-u)^{\delta p} \|f_0\|_{\tilde{L}_{q_0}^{p_0}(u,t)}^p \lesssim (t-u)^{\delta p}. $$

As for $I_2$ and $I_3$, we get

$$ \begin{aligned} J_2 & \lesssim |t-u|^{p-1} \mathbb{E} \left[ \int_u^t (1+|X_r|^p) dr \right] \\ & \lesssim |t-u|^p (1+\mathbb{E}[|X_u|^p]) \quad \text{by (3.15)}, \\ J_3 & \lesssim |t-u|^{\frac{p}{2}}. \end{aligned} $$

It follows that

$$ \mathbb{E} \left[ \sup_{s \in [u,t]} |X_s - X_u|^p \right] \lesssim |t-u|^{\delta p} (1 + \mathbb{E}[|X_u|^p]). $$

The estimate (3.16) then follows.

(c) We have

$$ \begin{aligned} \sup_{s \in [u,t]} \int_{B_R^c} |x|^p d\mu_s(x) & \le \frac{1}{R} \sup_{s \in [u,t]} \int_{\mathbb{R}^d} |x|^{p+\alpha} d\mu_s(x) \quad \text{by Markov's inequality} \\ & \lesssim \frac{1+\mathbb{E}[|X_u|^{p+\alpha}]}{R} \quad \text{by (3.15)}. \end{aligned} $$

The estimate (3.17) then follows. This completes the proof.

□

Third, we recall a Duhamel presentation and an $L^p$ estimate of marginal density:

**Theorem 3.10.** [35, Lemma 4.1] Let $(b, \sigma)$ satisfy Assumption 3.6. Let $\ell_t$ be the p.d.f. of $X_t$ in (3.9). Then

1. Let $v: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ be another drift such that $(v, \sigma)$ satisfies Assumption 3.3. We assume there exists $1 \le g \in \tilde{L}_{\bar{q}}^{\bar{p}}$ with $(\bar{p}, \bar{q}) \in \bar{K}$ such that $|b_t(x) - v_t(x)| \le g(t, x)$ for $t \in \mathbb{T}$ and $x \in \mathbb{R}^d$. Then we have for $t \in \mathbb{T}$ and $x \in \mathbb{R}^d$:

$$ \ell_t(x) = \int_{\mathbb{R}^d} p_{0,t}^{v,\sigma}(y,x) \ell_\nu(y) dy \\ + \int_0^t \int_{\mathbb{R}^d} \ell_s(y) \langle b_s(y) - v_s(y), \nabla_y p_{s,t}^{v,\sigma}(y,x) \rangle dy ds. $$

2. There exists a constant $c > 0$ (depending only on $\Theta_3$) such that

$$ \sup_{t \in \mathbb{T}} \|\ell_t\|_{\tilde{L}^k} \le c \|\ell_\nu\|_{\tilde{L}^k}, \quad k \in [p_0^*, \infty]. $$

We remind that $p_0$ is a parameter in $\Theta_3$. Also, $p_0^*$ is the Hölder conjugates of $p_0$, i.e., $1 = \frac{1}{p_0} + \frac{1}{p_0^*}$. Although the statement of Theorem 3.10[1] does not appear in [35], its proof is contained in part (c) of the proof of [35, Lemma 4.1].
There exists $\bar{q}_0 \in (2, q_0)$ such that $(p_0, \bar{q}_0) \in \bar{K}$. Let $\delta := \frac{1}{\bar{q}_0} - \frac{1}{q_0} \in (0, \frac{1}{2})$. By Hölder's inequality,

$$ \|f_0\|_{\tilde{L}_{\bar{q}_0}^{p_0}(u,t)} \le (t-u)^{\delta} \|f_0\|_{\tilde{L}_{q_0}^{p_0}(u,t)}. $$

By Proposition 3.8[2],

$$ J_1 \lesssim \|f_0\|_{\tilde{L}_{\bar{q}_0}^{p_0}(u,t)}^p \le (t-u)^{\delta p} \|f_0\|_{\tilde{L}_{q_0}^{p_0}(u,t)}^p \lesssim (t-u)^{\delta p}. $$

As for $I_2$ and $I_3$, we get

$$ \begin{aligned} J_2 &\lesssim |t-u|^{p-1} \mathbb{E} \left[ \int_u^t (1+|X_r|^p) dr \right] \\ &\lesssim |t-u|^p (1+\mathbb{E}[|X_u|^p]) \quad \text{by (3.15)}, \\ J_3 &\lesssim |t-u|^{\frac{p}{2}}. \end{aligned} $$

It follows that

$$ \mathbb{E} \left[ \sup_{s \in [u,t]} |X_s - X_u|^p \right] \lesssim |t-u|^{\delta p} (1 + \mathbb{E}[|X_u|^p]). $$

The estimate (3.16) then follows.

(c) We have

$$ \begin{aligned} \sup_{s \in [u,t]} \int_{B_R^c} |x|^p d\mu_s(x) &\le \frac{1}{R} \sup_{s \in [u,t]} \int_{\mathbb{R}^d} |x|^{p+\alpha} d\mu_s(x) \quad \text{by Markov's inequality} \\ &\lesssim \frac{1+\mathbb{E}[|X_u|^{p+\alpha}]}{R} \quad \text{by (3.15)}. \end{aligned} $$

The estimate (3.17) then follows. This completes the proof.

□

Third, we recall a Duhamel presentation and an $L^p$ estimate of marginal density:

**Theorem 3.10.** [35, Lemma 4.1] Let $(b, \sigma)$ satisfy Assumption 3.6. Let $\ell_t$ be the p.d.f. of $X_t$ in (3.9). Then

1. Let $v: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ be another drift such that $(v, \sigma)$ satisfies Assumption 3.3. We assume there exists $1 \le g \in \tilde{L}_{\bar{q}}^{\bar{p}}$ with $(\bar{p}, \bar{q}) \in \bar{K}$ such that $|b_t(x) - v_t(x)| \le g(t, x)$ for $t \in \mathbb{T}$ and $x \in \mathbb{R}^d$. Then we have for $t \in \mathbb{T}$ and $x \in \mathbb{R}^d$:

$$ \ell_t(x) = \int_{\mathbb{R}^d} p_{0,t}^{v,\sigma}(y,x) \ell_{\nu}(y) dy \\ + \int_0^t \int_{\mathbb{R}^d} \ell_s(y) \langle b_s(y) - v_s(y), \nabla_y p_{s,t}^{v,\sigma}(y,x) \rangle dy ds. $$

2. There exists a constant $c > 0$ (depending only on $\Theta_3$) such that

$$ \sup_{t \in \mathbb{T}} \|\ell_t\|_{\tilde{L}^k} \le c \|\ell_\nu\|_{\tilde{L}^k}, \quad k \in [p_0^*, \infty]. $$

We remind that $p_0$ is a parameter in $\Theta_3$. Also, $p_0^*$ is the Hölder conjugates of $p_0$, i.e., $1 = \frac{1}{p_0} + \frac{1}{p_0^*}$. Although the statement of Theorem 3.10[1] does not appear in [35], its proof is contained in part (c) of the proof of [35, Lemma 4.1].
## 3.4 Hölder estimates of marginal density

For Hölder continuity of $\ell_t$, we need another set of stronger assumptions:

**Assumption 3.11.** There exist constants $\beta \in (0, 1)$ and $C > 0$ such that for $t \in \mathbb{T}$ and $x \in \mathbb{R}^d$:

(A1) $\nu \in \mathcal{P}_1(\mathbb{R}^d)$ has a density $\ell_\nu$.

(A2) $a_t$ is invertible and $\sigma_t$ is weakly differentiable.

(A3) There exists $1 \le f_0 \in \tilde{L}_{q_0}^{p_0}$ with $(p_0, q_0) \in \mathcal{K}$ such that $|b_t(x)| \le f_0(t, x)$.

(A4) The following conditions hold:

$$ \begin{aligned} & \|\nabla \sigma_t\|_{\infty} + \|\sigma_t\|_{\infty} + \|a_t^{-1}\|_{\infty} \le C, \\ & |\nabla \sigma_t(x) - \nabla \sigma_t(y)| \le C(|x - y| + |x - y|^{\beta}). \end{aligned} $$

We gather parameters in **Assumption 3.11**:

$$ \Theta_4 := (d, T, \beta, C, p_0, q_0, f_0). $$

**Assumption 3.11** is stronger than **Assumption 3.6** in two aspects: $b$ is not allowed to have linear growth and $\nabla \sigma_t$ is uniformly bounded and uniformly continuous (in space). **Assumption 3.11**[A4] implies for $t \in \mathbb{T}$, $\gamma \in (0, 1)$ and $x, y \in \mathbb{R}^d$ that $|\sigma_t(x) - \sigma_t(y)| \le 2C|x - y|^{\gamma}$.

First, we recall an estimate about Hölder continuity in space:

**Theorem 3.12.** [35, Lemma 5.1] Let $(b, \sigma)$ satisfy **Assumption 3.11** and $\alpha \in (0, 1 - \frac{d}{p_0} - \frac{2}{q_0})$. Let $\ell_t$ be the p.d.f. of $X_t$ in (3.9). There exists a constant $c > 0$ (depending only on $\Theta_4$, $\alpha$) such that

$$ \sup_{t \in \mathbb{T}} \|\ell_t\|_{C_b^\alpha} \le c \|\ell_\nu\|_{C_b^\alpha}. $$

Second, we define $\bar{b} : \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ by

$$ (\bar{b}_t)^i := \frac{1}{2} \sum_{j=1}^{d} \frac{\partial(a_t)^{i,j}}{\partial x_j}, \quad i \in \{1, 2, \dots, d\}, \qquad (3.20) $$

where

1. $(\bar{b}_t)^i$ is the entry in the $i$-th row of $\bar{b}_t$.

2. $(a_t)^{i,j}$ is the entry in the $i$-th row and $j$-th column of $a_t$.

By construction, $\bar{b}$ depends only on $\sigma$. By [52, Equation 1.6], $p_{s,t}^{\bar{b},\sigma}$ is symmetric, i.e., $p_{s,t}^{\bar{b},\sigma}(x, y) = p_{s,t}^{\bar{b},\sigma}(y, x)$. Clearly, if $(b, \sigma)$ satisfies **Assumption 3.11**, then $(\bar{b}, \sigma)$ satisfies both **Assumption 3.3** and **Assumption 3.11**. We apply **Theorem 3.10**[1] to the pair $(b, \sigma)$ and $v = \bar{b}$. As a consequence, we get

**Corollary 3.13.** Let $(b, \sigma)$ satisfy **Assumption 3.11**. Let $\ell_t$ be the p.d.f. of $X_t$ in (3.9). Let $\bar{b}$ be defined by (3.20). We have for $t \in \mathbb{T}$ and $x \in \mathbb{R}^d$:

$$ \ell_t(x) = P_{0,t}^{\bar{b},\sigma} \ell_\nu(x) + \int_0^t \int_{\mathbb{R}^d} \ell_s(y) \langle b_s(y) - \bar{b}_s(y), \nabla_y p_{s,t}^{\bar{b},\sigma}(y, x) \rangle dy ds, $$

where $P_{0,t}^{\bar{b},\sigma} \ell_\nu$ is defined by (3.2).
## 3.4 Hölder estimates of marginal density

For Hölder continuity of $\ell_t$, we need another set of stronger assumptions:

**Assumption 3.11.** There exist constants $\beta \in (0, 1)$ and $C > 0$ such that for $t \in \mathbb{T}$ and $x \in \mathbb{R}^d$:

(A1) $\nu \in \mathcal{P}_1(\mathbb{R}^d)$ has a density $\ell_\nu$.

(A2) $a_t$ is invertible and $\sigma_t$ is weakly differentiable.

(A3) There exists $1 \le f_0 \in \tilde{L}_{q_0}^{p_0}$ with $(p_0, q_0) \in \mathcal{K}$ such that $|b_t(x)| \le f_0(t, x)$.

(A4) The following conditions hold:

$$ \begin{aligned} & \|\nabla \sigma_t\|_{\infty} + \|\sigma_t\|_{\infty} + \|a_t^{-1}\|_{\infty} \le C, \\ & |\nabla \sigma_t(x) - \nabla \sigma_t(y)| \le C(|x - y| + |x - y|^{\beta}). \end{aligned} $$

We gather parameters in **Assumption 3.11**:

$$ \Theta_4 := (d, T, \beta, C, p_0, q_0, f_0). $$

**Assumption 3.11** is stronger than **Assumption 3.6** in two aspects: $b$ is not allowed to have linear growth and $\nabla \sigma_t$ is uniformly bounded and uniformly continuous (in space). **Assumption 3.11**[A4] implies for $t \in \mathbb{T}$, $\gamma \in (0, 1)$ and $x, y \in \mathbb{R}^d$ that $|\sigma_t(x) - \sigma_t(y)| \le 2C|x - y|^\gamma$.

First, we recall an estimate about Hölder continuity in space:

**Theorem 3.12.** [35, Lemma 5.1] Let $(b, \sigma)$ satisfy **Assumption 3.11** and $\alpha \in (0, 1 - \frac{d}{p_0} - \frac{2}{q_0})$. Let $\ell_t$ be the p.d.f. of $X_t$ in (3.9). There exists a constant $c > 0$ (depending only on $\Theta_4$, $\alpha$) such that

$$ \sup_{t \in \mathbb{T}} \|\ell_t\|_{C_b^\alpha} \le c \|\ell_\nu\|_{C_b^\alpha}. $$

Second, we define $\bar{b} : \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ by

$$ (\bar{b}_t)^i := \frac{1}{2} \sum_{j=1}^{d} \frac{\partial(a_t)^{i,j}}{\partial x_j}, \quad i \in \{1, 2, \dots, d\}, \qquad (3.20) $$

where

1. $(\bar{b}_t)^i$ is the entry in the $i$-th row of $\bar{b}_t$.

2. $(a_t)^{i,j}$ is the entry in the $i$-th row and $j$-th column of $a_t$.

By construction, $\bar{b}$ depends only on $\sigma$. By [52, Equation 1.6], $p_{s,t}^{\bar{b},\sigma}$ is symmetric, i.e., $p_{s,t}^{\bar{b},\sigma}(x, y) = p_{s,t}^{\bar{b},\sigma}(y, x)$. Clearly, if $(b, \sigma)$ satisfies **Assumption 3.11**, then $(\bar{b}, \sigma)$ satisfies both **Assumption 3.3** and **Assumption 3.11**. We apply **Theorem 3.10**[1] to the pair $(b, \sigma)$ and $v = \bar{b}$. As a consequence, we get

**Corollary 3.13.** Let $(b, \sigma)$ satisfy **Assumption 3.11**. Let $\ell_t$ be the p.d.f. of $X_t$ in (3.9). Let $\bar{b}$ be defined by (3.20). We have for $t \in \mathbb{T}$ and $x \in \mathbb{R}^d$:

$$ \ell_t(x) = P_{0,t}^{\bar{b},\sigma} \ell_\nu(x) + \int_0^t \int_{\mathbb{R}^d} \ell_s(y) \langle b_s(y) - \bar{b}_s(y), \nabla_y p_{s,t}^{\bar{b},\sigma}(y, x) \rangle dy ds, $$

where $P_{0,t}^{\bar{b},\sigma} \ell_\nu$ is defined by (3.2).
Third, we establish an essential estimate about Hölder continuity in time:

**Theorem 3.14.** Let $(b, \sigma)$ satisfy Assumption 3.11 and $\ell_\nu \in C_b^\alpha(\mathbb{R}^d)$ for some $\alpha \in (0, 1 - \frac{d}{p_0} - \frac{2}{q_0})$. Let $\ell_t$ be the p.d.f. of $X_t$ in (3.9). Then there exist a constant $\delta \in (0, \frac{1}{2})$ (depending only on $d, p_0, q_0, \alpha$) and a constant $c > 0$ (depending only on $\Theta_4, \alpha, \nu$) such that

$$ \|\ell_t - \ell_s\|_{\infty} \le c|t - s|^{\delta}, \quad s, t \in \mathbb{T}. $$

We remind that the constants $C, p_0, q_0$ are parameters in $\Theta_4$. Theorem 3.12 and Theorem 3.14 are prerequisite for applying Arzelà-Ascoli theorem in Section 4.2. The proof of Theorem 3.14 is by extending an idea from [53, Corollary 2.5(iii)].

*Proof.* We write $M_1 \lesssim M_2$ if there exists a constant $c > 0$ (depending only on $\Theta_4, \alpha, \nu$) such that $M_1 \le cM_2$. Let $\bar{b}$ be defined by (3.20). Recall that $(\bar{b}, \sigma)$ satisfies Assumption 3.3. By Theorem 3.4,

(a) There exists a constant $\bar{\kappa} > 0$ (depending only on $\Theta_4$) such that for $i \in \{0, 1\}, 0 \le s < t \le T$ and $x, y \in \mathbb{R}^d$:

$$ |\nabla_y^i p_{s,t}^{\bar{b},\sigma}(y,x)| \lesssim (t-s)^{-\frac{i}{2}} p_{t-s}^{\bar{\kappa}}(y-x). \quad (3.21) $$

(b) There exist a constant $\kappa > 0$ (depending only on $\Theta_4, \alpha$) and a family $(\psi_{s,t})_{0 \le s < t \le T}$ of $C^\infty$-diffeomorphisms on $\mathbb{R}^d$ such that for $0 \le s < t \le T$ and $x, y, z \in \mathbb{R}^d$:

$$ |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,z) - \nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| \lesssim |z-x|^\alpha (t-s)^{-\frac{1+\alpha}{2}} \{ p_{t-s}^\kappa(\psi_{s,t}(y)-z) + p_{t-s}^\kappa(\psi_{s,t}(y)-x) \}. \quad (3.22) $$

Recall that $p_{r,t}^{\bar{b},\sigma}$ is symmetric, so

$$ \int_{\mathbb{R}^d} p_{s,t}^{\bar{b},\sigma}(z,x) \, dz = \int_{\mathbb{R}^d} p_{s,t}^{\bar{b},\sigma}(x,z) \, dz = 1. \quad (3.23) $$

WLOG, we assume $s < t$. We have for $i \in \{0, 1\}$ that

$$ \nabla_y^i p_{r,t}^{\bar{b},\sigma}(y,x) - \nabla_y^i p_{r,s}^{\bar{b},\sigma}(y,x) = \nabla_y^i \int_{\mathbb{R}^d} p_{r,s}^{\bar{b},\sigma}(y,z) p_{s,t}^{\bar{b},\sigma}(z,x) \, dz - \nabla_y^i p_{r,s}^{\bar{b},\sigma}(y,x) \quad (3.24) $$

$$ = \nabla_y^i \int_{\mathbb{R}^d} p_{r,s}^{\bar{b},\sigma}(y,z) p_{s,t}^{\bar{b},\sigma}(z,x) \, dz - \nabla_y^i \int_{\mathbb{R}^d} p_{r,s}^{\bar{b},\sigma}(y,x) p_{s,t}^{\bar{b},\sigma}(z,x) \, dz \quad (3.25) $$

$$ = \int_{\mathbb{R}^d} \{ \nabla_y^i p_{r,s}^{\bar{b},\sigma}(y,z) - \nabla_y^i p_{r,s}^{\bar{b},\sigma}(y,x) \} p_{s,t}^{\bar{b},\sigma}(z,x) \, dz. \quad (3.26) $$

Above, (3.24) is due to Chapman-Kolmogorov equation, (3.25) due to (3.23), and (3.26) due to Leibniz integral rule. Recall that $|\bar{b}| \lesssim 1 \le f_0$ and $|b| \le f_0$, so

$$ |b - \bar{b}| \lesssim f_0. \quad (3.27) $$

By Corollary 3.13,

$$ \ell_t(x) = P_{0,t}^{\bar{b},\sigma} \ell_\nu(x) + \int_0^t \int_{\mathbb{R}^d} \ell_r(y) \langle b_r(y) - \bar{b}_r(y), \nabla_y p_{r,t}^{\bar{b},\sigma}(y,x) \rangle \, dy \, dr, \quad (3.28) $$

$$ \ell_s(x) = P_{0,s}^{\bar{b},\sigma} \ell_\nu(x) + \int_0^s \int_{\mathbb{R}^d} \ell_r(y) \langle b_r(y) - \bar{b}_r(y), \nabla_y p_{r,s}^{\bar{b},\sigma}(y,x) \rangle \, dy \, dr. \quad (3.29) $$
Third, we establish an essential estimate about Hölder continuity in time:

**Theorem 3.14.** Let $(b, \sigma)$ satisfy Assumption 3.11 and $\ell_\nu \in C_b^\alpha(\mathbb{R}^d)$ for some $\alpha \in (0, 1 - \frac{d}{p_0} - \frac{2}{q_0})$. Let $\ell_t$ be the p.d.f. of $X_t$ in (3.9). Then there exist a constant $\delta \in (0, \frac{1}{2})$ (depending only on $d, p_0, q_0, \alpha$) and a constant $c > 0$ (depending only on $\Theta_4, \alpha, \nu$) such that

$$ \|\ell_t - \ell_s\|_{\infty} \le c|t-s|^\delta, \quad s, t \in \mathbb{T}. $$

We remind that the constants $C, p_0, q_0$ are parameters in $\Theta_4$. Theorem 3.12 and Theorem 3.14 are prerequisite for applying Arzelà–Ascoli theorem in Section 4.2. The proof of Theorem 3.14 is by extending an idea from [53, Corollary 2.5(iii)].

*Proof.* We write $M_1 \lesssim M_2$ if there exists a constant $c > 0$ (depending only on $\Theta_4, \alpha, \nu$) such that $M_1 \le cM_2$. Let $\bar{b}$ be defined by (3.20). Recall that $(\bar{b}, \sigma)$ satisfies Assumption 3.3. By Theorem 3.4,

(a) There exists a constant $\bar{\kappa} > 0$ (depending only on $\Theta_4$) such that for $i \in \{0, 1\}, 0 \le s < t \le T$ and $x, y \in \mathbb{R}^d$:

$$ |\nabla_y^i p_{s,t}^{\bar{b},\sigma}(y,x)| \lesssim (t-s)^{-\frac{i}{2}} p_{t-s}^{\bar{\kappa}}(y-x). \quad (3.21) $$

(b) There exist a constant $\kappa > 0$ (depending only on $\Theta_4, \alpha$) and a family $(\psi_{s,t})_{0\le s<t\le T}$ of $C^\infty$-diffeomorphisms on $\mathbb{R}^d$ such that for $0 \le s < t \le T$ and $x, y, z \in \mathbb{R}^d$:

$$ |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,z) - \nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| \lesssim |z-x|^\alpha (t-s)^{-\frac{1+\alpha}{2}} \{ p_{t-s}^\kappa(\psi_{s,t}(y)-z) + p_{t-s}^\kappa(\psi_{s,t}(y)-x) \}. \quad (3.22) $$

Recall that $p_{r,t}^{\bar{b},\sigma}$ is symmetric, so

$$ \int_{\mathbb{R}^d} p_{s,t}^{\bar{b},\sigma}(z,x) \, dz = \int_{\mathbb{R}^d} p_{s,t}^{\bar{b},\sigma}(x,z) \, dz = 1. \quad (3.23) $$

WLOG, we assume $s < t$. We have for $i \in \{0, 1\}$ that

$$ \nabla_y^i p_{r,t}^{\bar{b},\sigma}(y,x) - \nabla_y^i p_{r,s}^{\bar{b},\sigma}(y,x) = \nabla_y^i \int_{\mathbb{R}^d} p_{r,s}^{\bar{b},\sigma}(y,z) p_{s,t}^{\bar{b},\sigma}(z,x) \, dz - \nabla_y^i p_{r,s}^{\bar{b},\sigma}(y,x) \quad (3.24) $$

$$ = \nabla_y^i \int_{\mathbb{R}^d} p_{r,s}^{\bar{b},\sigma}(y,z) p_{s,t}^{\bar{b},\sigma}(z,x) \, dz - \nabla_y^i \int_{\mathbb{R}^d} p_{r,s}^{\bar{b},\sigma}(y,x) p_{s,t}^{\bar{b},\sigma}(z,x) \, dz \quad (3.25) $$

$$ = \int_{\mathbb{R}^d} \{ \nabla_y^i p_{r,s}^{\bar{b},\sigma}(y,z) - \nabla_y^i p_{r,s}^{\bar{b},\sigma}(y,x) \} p_{s,t}^{\bar{b},\sigma}(z,x) \, dz. \quad (3.26) $$

Above, (3.24) is due to Chapman–Kolmogorov equation, (3.25) due to (3.23), and (3.26) due to Leibniz integral rule. Recall that $|\bar{b}| \lesssim 1 \le f_0$ and $|b| \le f_0$, so

$$ |b - \bar{b}| \lesssim f_0. \quad (3.27) $$

By Corollary 3.13,

$$ \ell_t(x) = P_{0,t}^{\bar{b},\sigma} \ell_\nu(x) + \int_0^t \int_{\mathbb{R}^d} \ell_r(y) \langle b_r(y) - \bar{b}_r(y), \nabla_y p_{r,t}^{\bar{b},\sigma}(y,x) \rangle \, dy \, dr, \quad (3.28) $$

$$ \ell_s(x) = P_{0,s}^{\bar{b},\sigma} \ell_\nu(x) + \int_0^s \int_{\mathbb{R}^d} \ell_r(y) \langle b_r(y) - \bar{b}_r(y), \nabla_y p_{r,s}^{\bar{b},\sigma}(y,x) \rangle \, dy \, dr. \quad (3.29) $$
By (3.27) and (3.28),

$$
\begin{align*}
|\ell_t(x) - \ell_s(x)| &\lesssim \left| \int_{\mathbb{R}^d} \ell_\nu(y) \{p_{0,t}^{\bar{b},\sigma}(y,x) - p_{0,s}^{\bar{b},\sigma}(y,x)\} \mathrm{d}y \right| \\
&\quad + \int_0^s \int_{\mathbb{R}^d} \ell_r(y) f_0(r,y) |\nabla_y p_{r,t}^{\bar{b},\sigma}(y,x) - \nabla_y p_{r,s}^{\bar{b},\sigma}(y,x)| \mathrm{d}y \mathrm{d}r \\
&\quad + \int_s^t \int_{\mathbb{R}^d} \ell_r(y) f_0(r,y) |\nabla_y p_{r,t}^{\bar{b},\sigma}(y,x)| \mathrm{d}y \mathrm{d}r \\
&=: I_1(x) + I_2(x) + I_3(x).
\end{align*}
$$

We are going to upper bound $I_1(x)$, $I_2(x)$ and $I_3(x)$ separately.

1. We consider the SDE

$$
dY_t = \bar{b}(t, Y_t) dt + \sigma(t, Y_t) dB_t, \quad t \in \mathbb{T},
$$

where the distribution of $Y_0$ is $\nu$. Recall that $(\bar{b}, \sigma)$ satisfies Assumption 3.11. Let $\bar{\ell}_t$ be the p.d.f. of $Y_t$. Then

$$
\left| \int_{\mathbb{R}^d} \ell_\nu(y) \{p_{0,s}^{\bar{b},\sigma}(y,z) - p_{0,s}^{\bar{b},\sigma}(y,x)\} \mathrm{d}y \right| = |\bar{\ell}_s(z) - \bar{\ell}_s(x)| \\
\lesssim |z-x|^\alpha \quad \text{by Theorem 3.12.} \tag{3.29}
$$

By (3.26) with $i=r=0$,

$$
\begin{align*}
I_1(x) &\le \int_{\mathbb{R}^d} \left| \int_{\mathbb{R}^d} \ell_\nu(y) \{p_{0,s}^{\bar{b},\sigma}(y,z) - p_{0,s}^{\bar{b},\sigma}(y,x)\} \mathrm{d}y \right| p_{s,t}^{\bar{b},\sigma}(z,x) \mathrm{d}z \\
&\lesssim \int_{\mathbb{R}^d} |z-x|^\alpha p_{s,t}^{\bar{b},\sigma}(z,x) \mathrm{d}z \quad \text{by (3.29)} \\
&=: J(x) \\
&\lesssim \int_{\mathbb{R}^d} |z-x|^\alpha p_{t-s}^{\bar{\kappa}}(z-x) \mathrm{d}z \quad \text{by (3.21)} \\
&= \int_{\mathbb{R}^d} |z|^\alpha p_{t-s}^{\bar{\kappa}}(z) \mathrm{d}z \\
&\lesssim |t-s|^{\frac{\alpha}{2}}.
\end{align*}
$$

2. By (3.26) with $i=1$ and (3.22),

$$
I_2(x) \lesssim \int_0^s (s-r)^{-\frac{1+\alpha}{2}} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \ell_r(y) f_0(r, y) |z-x|^\alpha \{p_{s-r}^\kappa(\psi_{r,s}(y) - z) \\
+ p_{s-r}^\kappa(\psi_{r,s}(y) - x)\} p_{s,t}^{\bar{b},\sigma}(z, x) \mathrm{d}z \mathrm{d}y \mathrm{d}r. \quad (3.30)
$$

By Theorem 3.10[2],

$$
\sup_{r \in \mathbb{T}} \|\ell_r\|_{\infty} \lesssim \|\ell_{\nu}\|_{\infty}. \qquad (3.31)
$$
By (3.27) and (3.28),

$$
\begin{aligned}
|\ell_t(x) - \ell_s(x)| &\lesssim \left|\int_{\mathbb{R}^d} \ell_\nu(y) \{p_{0,t}^{\bar{b},\sigma}(y,x) - p_{0,s}^{\bar{b},\sigma}(y,x)\} \mathrm{d}y\right| \\
&\quad + \int_0^s \int_{\mathbb{R}^d} \ell_r(y) f_0(r,y) |\nabla_y p_{r,t}^{\bar{b},\sigma}(y,x) - \nabla_y p_{r,s}^{\bar{b},\sigma}(y,x)| \mathrm{d}y \mathrm{d}r \\
&\quad + \int_s^t \int_{\mathbb{R}^d} \ell_r(y) f_0(r,y) |\nabla_y p_{r,t}^{\bar{b},\sigma}(y,x)| \mathrm{d}y \mathrm{d}r \\
&=: I_1(x) + I_2(x) + I_3(x).
\end{aligned}
$$

We are going to upper bound $I_1(x)$, $I_2(x)$ and $I_3(x)$ separately.

1. We consider the SDE

$$dY_t = \bar{b}(t, Y_t) dt + \sigma(t, Y_t) dB_t, \quad t \in \mathbb{T},$$

where the distribution of $Y_0$ is $\nu$. Recall that $(\bar{b}, \sigma)$ satisfies Assumption 3.11. Let $\bar{\ell}_t$ be the p.d.f. of $Y_t$. Then

$$ \left| \int_{\mathbb{R}^d} \ell_\nu(y) \{ p_{0,s}^{\bar{b},\sigma}(y,z) - p_{0,s}^{\bar{b},\sigma}(y,x) \} \mathrm{d}y \right| = |\bar{\ell}_s(z) - \bar{\ell}_s(x)| \lesssim |z-x|^\alpha \quad \text{by Theorem 3.12.} \quad (3.29) $$

By (3.26) with $i=r=0$,

$$
\begin{aligned}
I_1(x) &\le \int_{\mathbb{R}^d} \left| \int_{\mathbb{R}^d} \ell_\nu(y) \{ p_{0,s}^{\bar{b},\sigma}(y,z) - p_{0,s}^{\bar{b},\sigma}(y,x) \} \mathrm{d}y \right| p_{s,t}^{\bar{b},\sigma}(z,x) \mathrm{d}z \\
&\lesssim \int_{\mathbb{R}^d} |z-x|^\alpha p_{s,t}^{\bar{b},\sigma}(z,x) \mathrm{d}z \quad \text{by (3.29)} \\
&=: J(x) \\
&\lesssim \int_{\mathbb{R}^d} |z-x|^\alpha p_{t-s}^{\bar{\kappa}}(z-x) \mathrm{d}z \quad \text{by (3.21)} \\
&= \int_{\mathbb{R}^d} |z|^\alpha p_{t-s}^{\bar{\kappa}}(z) \mathrm{d}z \\
&\lesssim |t-s|^{\frac{\alpha}{2}}.
\end{aligned}
$$

2. By (3.26) with $i=1$ and (3.22),

$$ I_2(x) \lesssim \int_0^s (s-r)^{-\frac{1+\alpha}{2}} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \ell_r(y) f_0(r,y) |z-x|^\alpha \{ p_{s-r}^\kappa(\psi_{r,s}(y)-z) \\ + p_{s-r}^\kappa(\psi_{r,s}(y)-x) \} p_{s,t}^{\bar{b},\sigma}(z,x) \mathrm{d}z \mathrm{d}y \mathrm{d}r. \quad (3.30) $$

By Theorem 3.10[2],

$$ \sup_{r \in \mathbb{T}} \|\ell_r\|_{\infty} \lesssim \|\ell_\nu\|_{\infty}. \quad (3.31) $$
By Lemma 3.5 with $(p, \bar{p}) = (p_0, \infty)$,

$$ \| \tilde{P}_{r,s}^{\kappa} \{ (s-r)^{\frac{d}{2p_0}} f_0(r, \cdot) \} \|_{\infty} \lesssim \| f_0(r, \cdot) \|_{\bar{L}^{p_0}}. \quad (3.32) $$

Then

$$
\begin{align*}
& \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \ell_r(y) f_0(r, y) |z - x|^\alpha p_{s-r}^\kappa (\psi_{r,s}(y) - z) p_{s,t}^{\bar{b},\sigma}(z, x) \, dz \, dy \\
& \lesssim \int_{\mathbb{R}^d} \left( \int_{\mathbb{R}^d} f_0(r, y) p_{s-r}^\kappa (\psi_{r,s}(y) - z) \, dy \right) |z - x|^\alpha p_{s,t}^{\bar{b},\sigma}(z, x) \, dz \quad \text{by (3.31)} \\
& = (s-r)^{-\frac{d}{2p_0}} \int_{\mathbb{R}^d} |z - x|^\alpha p_{s,t}^{\bar{b},\sigma}(z, x) \tilde{P}_{r,s}^\kappa \{ (s-r)^{\frac{d}{2p_0}} f_0(r, \cdot) \} (z) \, dz \quad \text{by (3.4)} \\
& \lesssim (s-r)^{-\frac{d}{2p_0}} \| f_0(r, \cdot) \|_{\bar{L}^{p_0}} \int_{\mathbb{R}^d} |z - x|^\alpha p_{s,t}^{\bar{b},\sigma}(z, x) \, dz \quad \text{by (3.32)} \\
& = (s-r)^{-\frac{d}{2p_0}} \| f_0(r, \cdot) \|_{\bar{L}^{p_0}} J(x). \tag{3.33}
\end{align*}
$$

Similarly,

$$ \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \ell_r(y) f_0(r, y) |z - x|^\alpha p_{s-r}^\kappa (\psi_{r,s}(y) - x) p_{s,t}^{\bar{b},\sigma}(z, x) \, dz \, dy \quad (3.34) \\ \lesssim (s-r)^{-\frac{d}{2p_0}} \| f_0(r, \cdot) \|_{\bar{L}^{p_0}} J(x). $$

It follows from (3.30), (3.33) and (3.34) that

$$ I_2(x) \lesssim J(x) \int_0^s (s-r)^{-(\frac{1+\alpha}{2} + \frac{d}{2p_0})} \|f_0(r, \cdot)\|_{\bar{L}^{p_0}} dr. $$

By Hölder's inequality,

$$
\begin{align*}
& \int_0^s (s-r)^{-(\frac{1+\alpha}{2} + \frac{d}{2p_0})} \|f_0(r, \cdot)\|_{\bar{L}^{p_0}} dr \\
& \le \left( \int_0^s (s-r)^{-q_0^* (\frac{1+\alpha}{2} + \frac{d}{2p_0})} dr \right)^{\frac{1}{q_0^*}} \left( \int_0^s \|f_0(r, \cdot)\|_{\bar{L}^{p_0}}^{q_0} dr \right)^{\frac{1}{q_0}} \\
& = \left( \int_0^s (s-r)^{-q_0^* (\frac{1+\alpha}{2} + \frac{d}{2p_0})} dr \right)^{\frac{1}{q_0^*}} \|f_0\|_{\bar{L}^{p_0}(s)}.
\end{align*}
$$

Above, $q_0^* \in [1, 2)$ is the Hölder conjugate of $q_0$, i.e., $\frac{1}{q_0} + \frac{1}{q_0^*} = 1$. Notice that $\alpha \in (0, 1 - \frac{d}{p_0} - \frac{2}{q_0})$ implies $q_0^* (\frac{1+\alpha}{2} + \frac{d}{2p_0}) < 1$ and thus

$$ \sup_{s \in \mathbb{T}} \int_0^s (s-r)^{-q_0^* (\frac{1+\alpha}{2} + \frac{d}{2p_0})} dr < \infty. $$

Recall from (1) that $J(x) \lesssim |t-s|^{\frac{\alpha}{2}}$, so $I_2(x) \lesssim |t-s|^{\frac{\alpha}{2}}$.
By Lemma 3.5 with $(p, \bar{p}) = (p_0, \infty)$,

$$ \| \tilde{P}_{r,s}^{\kappa} \{ (s-r)^{\frac{d}{2p_0}} f_0(r, \cdot) \} \|_{\infty} \lesssim \| f_0(r, \cdot) \|_{\bar{L}^{p_0}}. \quad (3.32) $$

Then

$$
\begin{align*}
& \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \ell_r(y) f_0(r, y) |z - x|^\alpha p_{s-r}^\kappa (\psi_{r,s}(y) - z) p_{s,t}^{\bar{b},\sigma}(z, x) \, dz \, dy \\
& \lesssim \int_{\mathbb{R}^d} \left( \int_{\mathbb{R}^d} f_0(r, y) p_{s-r}^\kappa (\psi_{r,s}(y) - z) \, dy \right) |z - x|^\alpha p_{s,t}^{\bar{b},\sigma}(z, x) \, dz \quad \text{by (3.31)} \\
& = (s-r)^{-\frac{d}{2p_0}} \int_{\mathbb{R}^d} |z - x|^\alpha p_{s,t}^{\bar{b},\sigma}(z, x) \tilde{P}_{r,s}^\kappa \{ (s-r)^{\frac{d}{2p_0}} f_0(r, \cdot) \} (z) \, dz \quad \text{by (3.4)} \\
& \lesssim (s-r)^{-\frac{d}{2p_0}} \| f_0(r, \cdot) \|_{\bar{L}^{p_0}} \int_{\mathbb{R}^d} |z - x|^\alpha p_{s,t}^{\bar{b},\sigma}(z, x) \, dz \quad \text{by (3.32)} \\
& = (s-r)^{-\frac{d}{2p_0}} \| f_0(r, \cdot) \|_{\bar{L}^{p_0}} J(x). \tag{3.33}
\end{align*}
$$

Similarly,

$$ \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \ell_r(y) f_0(r, y) |z - x|^\alpha p_{s-r}^\kappa (\psi_{r,s}(y) - x) p_{s,t}^{\bar{b},\sigma}(z, x) \, dz \, dy \quad (3.34) \\ \lesssim (s-r)^{-\frac{d}{2p_0}} \|f_0(r, \cdot)\|_{\bar{L}^{p_0}} J(x). $$

It follows from (3.30), (3.33) and (3.34) that

$$ I_2(x) \lesssim J(x) \int_0^s (s-r)^{-(\frac{1+\alpha}{2} + \frac{d}{2p_0})} \|f_0(r, \cdot)\|_{\bar{L}^{p_0}} dr. $$

By Hölder's inequality,

$$
\begin{align*}
& \int_0^s (s-r)^{-(\frac{1+\alpha}{2} + \frac{d}{2p_0})} \|f_0(r, \cdot)\|_{\bar{L}^{p_0}} dr \\
\le & \left( \int_0^s (s-r)^{-q_0^* (\frac{1+\alpha}{2} + \frac{d}{2p_0})} dr \right)^{\frac{1}{q_0^*}} \left( \int_0^s \|f_0(r, \cdot)\|_{\bar{L}^{p_0}}^{q_0} dr \right)^{\frac{1}{q_0}} \\
= & \left( \int_0^s (s-r)^{-q_0^* (\frac{1+\alpha}{2} + \frac{d}{2p_0})} dr \right)^{\frac{1}{q_0^*}} \|f_0\|_{\bar{L}^{p_0}(s)}.
\end{align*}
$$

Above, $q_0^* \in [1, 2)$ is the Hölder conjugate of $q_0$, i.e., $\frac{1}{q_0} + \frac{1}{q_0^*} = 1$. Notice that $\alpha \in (0, 1 - \frac{d}{p_0} - \frac{2}{q_0})$ implies $q_0^* (\frac{1+\alpha}{2} + \frac{d}{2p_0}) < 1$ and thus

$$ \sup_{s \in \mathbb{T}} \int_0^s (s-r)^{-q_0^* (\frac{1+\alpha}{2} + \frac{d}{2p_0})} dr < \infty. $$

Recall from (1) that $J(x) \lesssim |t-s|^{\frac{\alpha}{2}}$, so $I_2(x) \lesssim |t-s|^{\frac{\alpha}{2}}$.
3. By Lemma 3.5 with $(p, \bar{p}) = (p_0, \infty)$,

$$ \|P_{t-r}^{\bar{\kappa}}\{(t-r)^{\frac{d}{2p_0}}f_0(r, \cdot)\}\|_{\infty} \lesssim \|f_0(r, \cdot)\|_{\bar{L}^{p_0}}. \quad (3.35) $$

We have

$$
\begin{align*}
I_3(x) &\lesssim \int_s^t (t-r)^{-\frac{1}{2}} \left( \int_{\mathbb{R}^d} \ell_r(y) f_0(r, y) p_{t-r}^{\bar{\kappa}}(y-x) \, dy \right) \, dr \quad \text{by (3.21)} \\
&\lesssim \int_s^t (t-r)^{-\frac{1}{2}} \left( \int_{\mathbb{R}^d} f_0(r, y) p_{t-r}^{\bar{\kappa}}(y-x) \, dy \right) \, dr \quad \text{by (3.31)} \\
&= \int_s^t (t-r)^{-\left(\frac{1}{2} + \frac{d}{2p_0}\right)} P_{t-r}^{\bar{\kappa}}\{(t-r)^{\frac{d}{2p_0}} f_0(r, \cdot)\}(x) \, dr \quad \text{by (3.3)} \\
&\lesssim \int_s^t (t-r)^{-\left(\frac{1}{2} + \frac{d}{2p_0}\right)} \|f_0(r, \cdot)\|_{\bar{L}^{p_0}} \, dr \quad \text{by (3.35)} \\
&\le \left( \int_s^t (t-r)^{-q_0^* \left(\frac{1}{2} + \frac{d}{2p_0}\right)} \, dr \right)^{\frac{1}{q_0^*}} \|f_0\|_{\bar{L}^{p_0}(s,t)} \quad \text{by Hölder's inequality.}
\end{align*}
$$

Notice that $(p_0, q_0) \in \mathcal{K}$ implies $\varepsilon := q_0^*(\frac{1}{2} + \frac{d}{2p_0}) \in (0, 1)$. Then

$$ \int_s^t (t-r)^{-\varepsilon} dr = \frac{(t-s)^{1-\varepsilon}}{1-\varepsilon}. $$

It follows that

$$ I_3(x) \lesssim |t-s|^{\frac{1-\varepsilon}{q_0^*}}. $$

The claim then follows by picking

$$ \delta := \frac{\alpha}{2} \wedge \frac{1-\varepsilon}{q_0^*}. $$

□

# 4 Proof of Theorem 2.2

We recall that $\Theta_1 = (d, T, \alpha, \beta, C, p_0, q_0, f_0, p)$ contains the parameters in Assumption 2.1. We write $M_1 \lesssim M_2$ if there exists a constant $c > 0$ (depending only on $\Theta_1, \nu$) such that $M_1 \le cM_2$. We construct a sequence $(\rho^n)$ of mollifiers as follows. We fix a smooth p.d.f. $\rho: \mathbb{R}^d \to \mathbb{R}$ whose support is contained in the unit ball of $\mathbb{R}^d$. For each $n \in \mathbb{N}^*$, we define $\rho^n: \mathbb{R}^d \to \mathbb{R}$ by $\rho^n(x) := n^d\rho(nx)$ and consider the SDE

$$ dX_t^n = b(t, X_t^n, (\rho^n * \mu_t^n)(X_t^n), \mu_t^n) dt + \sigma(t, X_t^n, \mu_t^n) dB_t, \quad t \in \mathbb{T}, \quad (4.1) $$

where the distribution of $X_0^n$ is $\nu$ and that of $X_t^n$ is $\mu_t^n$. Above, $*$ is the convolution operator, i.e.,

$$ (\rho^n * \mu_t^n)(x) := \int_{\mathbb{R}^d} \rho^n(x-y) d\mu_t^n(y). $$

Clearly, (4.1) is the mollified version of (1.1).
3. By Lemma 3.5 with $(p, \bar{p}) = (p_0, \infty)$,

$$ \|P_{t-r}^{\bar{\kappa}}\{(t-r)^{\frac{d}{2p_0}}f_0(r, \cdot)\}\|_{\infty} \lesssim \|f_0(r, \cdot)\|_{\bar{L}^{p_0}}. \quad (3.35) $$

We have

$$
\begin{align*}
I_3(x) &\lesssim \int_s^t (t-r)^{-\frac{1}{2}} \left( \int_{\mathbb{R}^d} \ell_r(y) f_0(r, y) p_{t-r}^{\bar{\kappa}}(y-x) \, dy \right) \, dr \quad \text{by (3.21)} \\
&\lesssim \int_s^t (t-r)^{-\frac{1}{2}} \left( \int_{\mathbb{R}^d} f_0(r, y) p_{t-r}^{\bar{\kappa}}(y-x) \, dy \right) \, dr \quad \text{by (3.31)} \\
&= \int_s^t (t-r)^{-\left(\frac{1}{2} + \frac{d}{2p_0}\right)} P_{t-r}^{\bar{\kappa}}\{(t-r)^{\frac{d}{2p_0}} f_0(r, \cdot)\}(x) \, dr \quad \text{by (3.3)} \\
&\lesssim \int_s^t (t-r)^{-\left(\frac{1}{2} + \frac{d}{2p_0}\right)} \|f_0(r, \cdot)\|_{\bar{L}^{p_0}} \, dr \quad \text{by (3.35)} \\
&\le \left( \int_s^t (t-r)^{-q_0^* \left(\frac{1}{2} + \frac{d}{2p_0}\right)} \, dr \right)^{\frac{1}{q_0^*}} \|f_0\|_{\bar{L}^{p_0}(s,t)} \quad \text{by Hölder's inequality.}
\end{align*}
$$

Notice that $(p_0, q_0) \in \mathcal{K}$ implies $\varepsilon := q_0^*(\frac{1}{2} + \frac{d}{2p_0}) \in (0, 1)$. Then

$$ \int_s^t (t-r)^{-\varepsilon} \, dr = \frac{(t-s)^{1-\varepsilon}}{1-\varepsilon}. $$

It follows that

$$ I_3(x) \lesssim |t-s|^{\frac{1-\varepsilon}{q_0^*}}. $$

The claim then follows by picking

$$ \delta := \frac{\alpha}{2} \wedge \frac{1-\varepsilon}{q_0^*}. $$

□

# 4 Proof of Theorem 2.2

We recall that $\Theta_1 = (d, T, \alpha, \beta, C, p_0, q_0, f_0, p)$ contains the parameters in Assumption 2.1. We write $M_1 \lesssim M_2$ if there exists a constant $c > 0$ (depending only on $\Theta_1, \nu$) such that $M_1 \le cM_2$. We construct a sequence $(\rho^n)$ of mollifiers as follows. We fix a smooth p.d.f. $\rho: \mathbb{R}^d \to \mathbb{R}$ whose support is contained in the unit ball of $\mathbb{R}^d$. For each $n \in \mathbb{N}^*$, we define $\rho^n: \mathbb{R}^d \to \mathbb{R}$ by $\rho^n(x) := n^d\rho(nx)$ and consider the SDE

$$ dX_t^n = b(t, X_t^n, (\rho^n * \mu_t^n)(X_t^n), \mu_t^n) \, dt + \sigma(t, X_t^n, \mu_t^n) \, dB_t, \quad t \in \mathbb{T}, \quad (4.1) $$

where the distribution of $X_0^n$ is $\nu$ and that of $X_t^n$ is $\mu_t^n$. Above, $*$ is the convolution operator, i.e.,

$$ (\rho^n * \mu_t^n)(x) := \int_{\mathbb{R}^d} \rho^n(x-y) \, d\mu_t^n(y). $$

Clearly, (4.1) is the mollified version of (1.1).
## 4.1 Stability estimates for mollified SDEs

We define the map $b^n: \mathbb{T} \times \mathbb{R}^d \times \mathcal{P}_p(\mathbb{R}^d) \to \mathbb{R}^d$ by $b^n(t, x, \mu) := b(t, x, (\rho^n * \mu)(x), \mu)$. Then

$$|b^n(t, x, \mu) - b^n(t, x, \tilde{\mu})| \lesssim \left| \int_{\mathbb{R}^d} \rho^n(x-y) \, d(\mu - \tilde{\mu})(y) \right| + W_p(\mu, \tilde{\mu}) \quad (4.2)$$

$$\le \|\nabla \rho^n\|_{\infty} W_1(\mu, \tilde{\mu}) + W_p(\mu, \tilde{\mu}) \quad (4.3)$$

$$\le (1 + \|\nabla \rho^n\|_{\infty}) W_p(\mu, \tilde{\mu}). \quad (4.4)$$

Above, (4.2) is due to Assumption 2.1[A4], (4.3) due to Lemma 3.1[2], and (4.4) due to Lemma 3.1[3]. It follows that $b^n$ is Lipschitz in distribution. We consider the McKean-Vlasov SDE

$$dY_t = b^n(t, Y_t, \xi_t) dt + \sigma(t, Y_t, \xi_t) dB_t, \quad t \in \mathbb{T}, \quad (4.5)$$

where the distribution of $Y_0$ is $\nu$ and that of $Y_t$ is $\xi_t$. It follows from [10, Theorem 1.1(1)] that (4.5) is well-posed. Clearly, $(X_t^n, t \in \mathbb{T})$ satisfies (4.5). As a consequence, (4.1) is well-posed. We define the maps $\bar{b}^n: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ and $\bar{\sigma}^n: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d \times \mathbb{R}^m$ by

$$\bar{b}^n(t, x) := b^n(t, x, \mu_t^n),$$

$$\bar{\sigma}^n(t, x) := \sigma(t, x, \mu_t^n).$$

Clearly,

$$dX_t^n = \bar{b}^n(t, X_t^n) dt + \bar{\sigma}^n(t, X_t^n) dB_t. \quad (4.6)$$

Let $\bar{a}^n := \bar{\sigma}^n(\bar{\sigma}^n)^\top$. We denote $\bar{b}_t^n := \bar{b}^n(t, \cdot)$, $\bar{\sigma}_t^n := \bar{\sigma}^n(t, \cdot)$ and $\bar{a}_t^n := \bar{a}^n(t, \cdot)$.

**Remark 4.1.** We emphasize that every pair $(\bar{b}^n, \bar{\sigma}^n)$ satisfies Assumption 3.11 for the same set of parameters. More precisely, we have for $n \in \mathbb{N}^*, t \in \mathbb{T}$ and $x, y \in \mathbb{R}^d$:

$$|\bar{b}_t^n(x)| \le f_0(t, x), \\ \|\nabla \bar{\sigma}_t^n\|_{\infty} + \|\bar{\sigma}_t^n\|_{\infty} + \|(\bar{a}_t^n)^{-1}\|_{\infty} \le C, \\ |\nabla \bar{\sigma}_t^n(x) - \nabla \bar{\sigma}_t^n(y)| \le C(|x - y| + |x - y|^{\beta}).$$

By Theorem 3.9[1], each $X_t^n$ admits a p.d.f. denoted by $\ell_t^n$. By Theorem 3.12,

$$\sup_{n \in \mathbb{N}^*} \sup_{t \in \mathbb{T}} \|\ell_t^n\|_{C_b^\alpha} \lesssim 1. \quad (4.7)$$

By Theorem 3.14, there exists a constant $\delta_1 \in (0, \frac{1}{2})$ (depending only on $d, p_0, q_0, \alpha$) such that

$$\sup_{n \in \mathbb{N}^*} \|\ell_t^n - \ell_s^n\|_{\infty} \lesssim |t - s|^{\delta_1}, \quad s, t \in \mathbb{T}. \quad (4.8)$$

By Theorem 3.9[2], there exists a constant $\delta_2 \in (0, \frac{1}{2})$ (depending only on $q_0$) such that

$$\sup_{n \in \mathbb{N}^*} \sup_{t \in \mathbb{T}} M_p(\mu_t^n) \lesssim 1, \quad (4.9)$$

$$\sup_{n \in \mathbb{N}^*} W_p(\mu_s^n, \mu_t^n) \lesssim |t - s|^{\delta_2}, \quad s, t \in \mathbb{T}, \quad (4.10)$$

$$\sup_{n \in \mathbb{N}^*} \sup_{t \in \mathbb{T}} \int_{B_R^c} |x|^p d\mu_t^n(x) \lesssim \frac{1}{R}, \quad (4.11)$$

$$\sup_{n \in \mathbb{N}^*} \sup_{t \in \mathbb{T}} \mu_t^n(B_R^c) \lesssim \frac{1}{R}, \quad R > 0. \quad (4.12)$$

By (4.10), the map $\mathbb{T} \to \mathcal{P}_p(\mathbb{R}^d)$, $t \mapsto \mu_t^n$ is continuous.
## 4.1 Stability estimates for mollified SDEs

We define the map $b^n: \mathbb{T} \times \mathbb{R}^d \times \mathcal{P}_p(\mathbb{R}^d) \to \mathbb{R}^d$ by $b^n(t, x, \mu) := b(t, x, (\rho^n * \mu)(x), \mu)$. Then

$$|b^n(t, x, \mu) - b^n(t, x, \tilde{\mu})| \lesssim \left| \int_{\mathbb{R}^d} \rho^n(x-y) d(\mu - \tilde{\mu})(y) \right| + W_p(\mu, \tilde{\mu}) \quad (4.2)$$

$$\leq \|\nabla \rho^n\|_{\infty} W_1(\mu, \tilde{\mu}) + W_p(\mu, \tilde{\mu}) \quad (4.3)$$

$$\leq (1 + \|\nabla \rho^n\|_{\infty}) W_p(\mu, \tilde{\mu}). \quad (4.4)$$

Above, (4.2) is due to Assumption 2.1[A4], (4.3) due to Lemma 3.1[2], and (4.4) due to Lemma 3.1[3]. It follows that $b^n$ is Lipschitz in distribution. We consider the McKean-Vlasov SDE

$$dY_t = b^n(t, Y_t, \xi_t) dt + \sigma(t, Y_t, \xi_t) dB_t, \quad t \in \mathbb{T}, \quad (4.5)$$

where the distribution of $Y_0$ is $\nu$ and that of $Y_t$ is $\xi_t$. It follows from [10, Theorem 1.1(1)] that (4.5) is well-posed. Clearly, $(X_t^n, t \in \mathbb{T})$ satisfies (4.5). As a consequence, (4.1) is well-posed. We define the maps $\bar{b}^n: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ and $\bar{\sigma}^n: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d \otimes \mathbb{R}^m$ by

$$\bar{b}^n(t, x) := b^n(t, x, \mu_t^n),$$

$$\bar{\sigma}^n(t, x) := \sigma(t, x, \mu_t^n).$$

Clearly,

$$dX_t^n = \bar{b}^n(t, X_t^n) dt + \bar{\sigma}^n(t, X_t^n) dB_t. \quad (4.6)$$

$$\text{Let } \bar{a}^n := \bar{\sigma}^n(\bar{\sigma}^n)^\top. \text{ We denote } \bar{b}_t^n := \bar{b}^n(t, \cdot), \bar{\sigma}_t^n := \bar{\sigma}^n(t, \cdot) \text{ and } \bar{a}_t^n := \bar{a}^n(t, \cdot).$$

**Remark 4.1.** We emphasize that every pair $(\bar{b}^n, \bar{\sigma}^n)$ satisfies Assumption 3.11 for the same set of parameters. More precisely, we have for $n \in \mathbb{N}^*, t \in \mathbb{T}$ and $x, y \in \mathbb{R}^d$:

$$|\bar{b}_t^n(x)| \leq f_0(t, x), \\ \|\nabla \bar{\sigma}_t^n\|_{\infty} + \|\bar{\sigma}_t^n\|_{\infty} + \|(\bar{a}_t^n)^{-1}\|_{\infty} \leq C, \\ |\nabla \bar{\sigma}_t^n(x) - \nabla \bar{\sigma}_t^n(y)| \leq C(|x - y| + |x - y|^{\beta}).$$

By Theorem 3.9[1], each $X_t^n$ admits a p.d.f. denoted by $\ell_t^n$. By Theorem 3.12,

$$\sup_{n \in \mathbb{N}^*} \sup_{t \in \mathbb{T}} \|\ell_t^n\|_{C_b^\alpha} \lesssim 1. \quad (4.7)$$

By Theorem 3.14, there exists a constant $\delta_1 \in (0, \frac{1}{2})$ (depending only on $d, p_0, q_0, \alpha$) such that

$$\sup_{n \in \mathbb{N}^*} \|\ell_t^n - \ell_s^n\|_{\infty} \lesssim |t - s|^{\delta_1}, \quad s, t \in \mathbb{T}. \quad (4.8)$$

By Theorem 3.9[2], there exists a constant $\delta_2 \in (0, \frac{1}{2})$ (depending only on $q_0$) such that

$$\sup_{n \in \mathbb{N}^*} \sup_{t \in \mathbb{T}} M_p(\mu_t^n) \lesssim 1, \quad (4.9)$$

$$\sup_{n \in \mathbb{N}^*} W_p(\mu_s^n, \mu_t^n) \lesssim |t - s|^{\delta_2}, \quad s, t \in \mathbb{T}, \quad (4.10)$$

$$\sup_{n \in \mathbb{N}^*} \sup_{t \in \mathbb{T}} \int_{B_R^c} |x|^p d\mu_t^n(x) \lesssim \frac{1}{R}, \quad (4.11)$$

$$\sup_{n \in \mathbb{N}^*} \sup_{t \in \mathbb{T}} \mu_t^n(B_R^c) \lesssim \frac{1}{R}, \quad R > 0. \quad (4.12)$$

By (4.10), the map $\mathbb{T} \to \mathcal{P}_p(\mathbb{R}^d)$, $t \mapsto \mu_t^n$ is continuous.
## 4.2 Convergence of marginal densities of mollified SDEs

By (4.7), (4.8) and Arzelà–Ascoli theorem, there exist a sub-sequence (also denoted by $(\ell^n)$ for simplicity) and a continuous function $\ell: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}_+$ such that

$$ \limsup_n \sup_{x \in B(0,R)} |\ell_t^n(x) - \ell_t(x)| = 0, \quad R > 0. \quad (4.13) $$

where $\ell_t := \ell(t, \cdot)$. Clearly, $\ell_0 = \ell_\nu$. Taking the limit $n \to \infty$ in (4.7) and (4.8), we get

$$ \sup_{t \in \mathbb{T}} \|\ell_t\|_{C_b^\alpha} \lesssim \|\ell_\nu\|_{C_b^\alpha}, \quad (4.14) $$

$$ \|\ell_t - \ell_s\|_{\infty} \lesssim |t - s|^{\delta_1}, \quad s, t \in \mathbb{T}. $$

We remark that the constant in (4.14) depends only on $\Theta_1$. Next we verify that $\ell_t$ is indeed a p.d.f. for $t \in \mathbb{T}$. We have

$$ \int_{B(0,R)} \ell_t^n(x) \, dx = 1 - \int_{B_R^c} \ell_t^n(x) \, dx \\ \gtrsim 1 - \frac{1}{R} \quad \text{by (4.12).} $$

By (4.13), (4.14) and dominated convergence theorem (DCT),

$$ \int_{B(0,R)} \ell_t(x) \, dx = \lim_{n \to \infty} \int_{B(0,R)} \ell_t^n(x) \, dx. $$

It follows that

$$ 1 - \frac{1}{R} \lesssim \int_{B(0,R)} \ell_t(x) \, dx \le 1, $$

which implies

$$ \int_{\mathbb{R}^d} \ell_t(x) \, dx = \lim_{R \to \infty} \int_{B(0,R)} \ell_t(x) \, dx = 1. $$

Let $\mu_t \in \mathcal{P}(\mathbb{R}^d)$ be the probability measure induced by $\ell_t$, i.e.,

$$ \mu_t(B) := \int_B \ell_t(x) \, dx, \quad B \in \mathcal{B}(\mathbb{R}^d). $$

**Lemma 4.2.** We have for each $t \in \mathbb{T}$ that $\mu_t^n \to \mu_t$ as $n \to \infty$.

*Proof.* By (4.13),

$$ \mu_t^n \stackrel{*}{\to} \mu_t \quad \text{as} \quad n \to \infty. \quad (4.15) $$

Let $f \in C_b(\mathbb{R}^d)$ and $g \in C_c(\mathbb{R}^d)$ such that $0 \le g \le 1$. Then $gf \in C_c(\mathbb{R}^d)$ and $f = (1-g)f + gf$. We have

$$ \left| \int_{\mathbb{R}^d} f \, d(\mu_t^n - \mu_t) \right| \le \|f\|_{\infty} \int_{\mathbb{R}^d} (1-g) \, d(\mu_t^n + \mu_t) + \left| \int_{\mathbb{R}^d} gf \, d(\mu_t^n - \mu_t) \right|. $$

By (4.15),

$$ \limsup_n \int_{\mathbb{R}^d} gf \, d(\mu_t^n - \mu_t) = 0, $$
## 4.2 Convergence of marginal densities of mollified SDEs

By (4.7), (4.8) and Arzelà–Ascoli theorem, there exist a sub-sequence (also denoted by $(\ell^n)$ for simplicity) and a continuous function $\ell: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}_+$ such that

$$ \limsup_n \sup_{x \in B(0,R)} |\ell_t^n(x) - \ell_t(x)| = 0, \quad R > 0. \quad (4.13) $$

where $\ell_t := \ell(t, \cdot)$. Clearly, $\ell_0 = \ell_\nu$. Taking the limit $n \to \infty$ in (4.7) and (4.8), we get

$$ \sup_{t \in \mathbb{T}} \|\ell_t\|_{C_b^\alpha} \lesssim \|\ell_\nu\|_{C_b^\alpha}, \quad (4.14) $$

$$ \|\ell_t - \ell_s\|_{\infty} \lesssim |t - s|^{\delta_1}, \quad s, t \in \mathbb{T}. $$

We remark that the constant in (4.14) depends only on $\Theta_1$. Next we verify that $\ell_t$ is indeed a p.d.f. for $t \in \mathbb{T}$. We have

$$ \int_{B(0,R)} \ell_t^n(x) \, dx = 1 - \int_{B_R^c} \ell_t^n(x) \, dx \\ \gtrsim 1 - \frac{1}{R} \quad \text{by (4.12).} $$

By (4.13), (4.14) and dominated convergence theorem (DCT),

$$ \int_{B(0,R)} \ell_t(x) \, dx = \lim_{n \to \infty} \int_{B(0,R)} \ell_t^n(x) \, dx. $$

It follows that

$$ 1 - \frac{1}{R} \lesssim \int_{B(0,R)} \ell_t(x) \, dx \le 1, $$

which implies

$$ \int_{\mathbb{R}^d} \ell_t(x) \, dx = \lim_{R \to \infty} \int_{B(0,R)} \ell_t(x) \, dx = 1. $$

Let $\mu_t \in \mathcal{P}(\mathbb{R}^d)$ be the probability measure induced by $\ell_t$, i.e.,

$$ \mu_t(B) := \int_B \ell_t(x) \, dx, \quad B \in \mathcal{B}(\mathbb{R}^d). $$

**Lemma 4.2.** We have for each $t \in \mathbb{T}$ that $\mu_t^n \to \mu_t$ as $n \to \infty$.

*Proof.* By (4.13),

$$ \mu_t^n \stackrel{*}{\to} \mu_t \quad \text{as} \quad n \to \infty. \quad (4.15) $$

Let $f \in C_b(\mathbb{R}^d)$ and $g \in C_c(\mathbb{R}^d)$ such that $0 \le g \le 1$. Then $gf \in C_c(\mathbb{R}^d)$ and $f = (1-g)f + gf$. We have

$$ \left| \int_{\mathbb{R}^d} f \, d(\mu_t^n - \mu_t) \right| \le \|f\|_{\infty} \int_{\mathbb{R}^d} (1-g) \, d(\mu_t^n + \mu_t) + \left| \int_{\mathbb{R}^d} gf \, d(\mu_t^n - \mu_t) \right|. $$

By (4.15),

$$ \limsup_n \int_{\mathbb{R}^d} gf \, d(\mu_t^n - \mu_t) = 0, $$
which implies

$$ \limsup_n \left| \int_{\mathbb{R}^d} f \, d(\mu_t^n - \mu_t) \right| \le \|f\|_{\infty} \limsup_n \int_{\mathbb{R}^d} (1-g) \, d(\mu_t^n + \mu_t). $$

Notice that

$$ \begin{align*} \limsup_n \int_{\mathbb{R}^d} (1-g) \, d\mu_t^n &= 1 - \liminf_n \int_{\mathbb{R}^d} g \, d\mu_t^n \\ &= 1 - \int_{\mathbb{R}^d} g \, d\mu_t \quad \text{by (4.15)} \\ &= \int_{\mathbb{R}^d} (1-g) \, d\mu_t, \end{align*} $$

which implies

$$ \limsup_n \left| \int_{\mathbb{R}^d} f \, d(\mu_t^n - \mu_t) \right| \le 2 \|f\|_{\infty} \int_{\mathbb{R}^d} (1-g) \, d\mu_t. \quad (4.16) $$

Because $\mu_t$ is a probability measure,

$$ \sup \left\{ \int_{\mathbb{R}^d} g \, d\mu_t : g \in C_c(\mathbb{R}^d) \text{ and } 0 \le g \le 1 \right\} = 1. \quad (4.17) $$

The claim then follows from (4.16) and (4.17). $\square$

<figure><img src="image_4.png" /></figure>

By monotone convergence theorem (MCT),

$$ \begin{align*} \int_{B_R^c} |x|^p \, d\mu_t(x) &= \lim_{K \to \infty} \int_{B_R^c \cap B(0,K)} |x|^p \, d\mu_t(x) \\ &= \lim_{K \to \infty} \lim_{n \to \infty} \int_{B_R^c \cap B(0,K)} |x|^p \, d\mu_t^n \quad \text{by (4.13)} \\ &\lesssim \frac{1}{R} \quad \text{by (4.11)}, \end{align*} $$

which implies

$$ \sup_{t \in \mathbb{T}} \int_{B_R^c} |x|^p \, d\mu_t(x) \lesssim \frac{1}{R}. \quad (4.18) $$

Clearly, $|\cdot|^p$ is continuous and bounded from below. By Lemma 4.2 and Portmanteau's theorem,

$$ \begin{align*} \sup_{t \in \mathbb{T}} M_p(\mu_t) &\le \sup_{t \in \mathbb{T}} \liminf_n M_p(\mu_t^n) \\ &\lesssim 1 \quad \text{by (4.9)}, \end{align*} $$
which implies

$$ \limsup_n \left| \int_{\mathbb{R}^d} f \, d(\mu_t^n - \mu_t) \right| \le \|f\|_{\infty} \limsup_n \int_{\mathbb{R}^d} (1-g) \, d(\mu_t^n + \mu_t). $$

Notice that

$$ \begin{align*} \limsup_n \int_{\mathbb{R}^d} (1-g) \, d\mu_t^n &= 1 - \liminf_n \int_{\mathbb{R}^d} g \, d\mu_t^n \\ &= 1 - \int_{\mathbb{R}^d} g \, d\mu_t \quad \text{by (4.15)} \\ &= \int_{\mathbb{R}^d} (1-g) \, d\mu_t, \end{align*} $$

which implies

$$ \limsup_n \left| \int_{\mathbb{R}^d} f \, d(\mu_t^n - \mu_t) \right| \le 2 \|f\|_{\infty} \int_{\mathbb{R}^d} (1-g) \, d\mu_t. \quad (4.16) $$

Because $\mu_t$ is a probability measure,

$$ \sup \left\{ \int_{\mathbb{R}^d} g \, d\mu_t : g \in C_c(\mathbb{R}^d) \text{ and } 0 \le g \le 1 \right\} = 1. \quad (4.17) $$

The claim then follows from (4.16) and (4.17). $\square$

<figure><img src="image_4.png" /></figure>

By monotone convergence theorem (MCT),

$$ \begin{align*} \int_{B_R^c} |x|^p \, d\mu_t(x) &= \lim_{K \to \infty} \int_{B_R^c \cap B(0,K)} |x|^p \, d\mu_t(x) \\ &= \lim_{K \to \infty} \lim_{n \to \infty} \int_{B_R^c \cap B(0,K)} |x|^p \, d\mu_t^n \quad \text{by (4.13)} \\ &\lesssim \frac{1}{R} \quad \text{by (4.11)}, \end{align*} $$

which implies

$$ \sup_{t \in \mathbb{T}} \int_{B_R^c} |x|^p \, d\mu_t(x) \lesssim \frac{1}{R}. \quad (4.18) $$

Clearly, $|\cdot|^p$ is continuous and bounded from below. By Lemma 4.2 and Portmanteau's theorem,

$$ \begin{align*} \sup_{t \in \mathbb{T}} M_p(\mu_t) &\le \sup_{t \in \mathbb{T}} \liminf_n M_p(\mu_t^n) \\ &\lesssim 1 \quad \text{by (4.9)}, \end{align*} $$
which implies $\mu_t \in \mathcal{P}_p(\mathbb{R}^d)$ for $t \in \mathbb{T}$. We have

$$ (W_p(\mu_s, \mu_t))^p = \sup \left\{ \int_{\mathbb{R}^d} \varphi \, d\mu_s + \int_{\mathbb{R}^d} \psi \, d\mu_t : (\varphi, \psi) \in \Phi_p \right\} \quad (4.19) $$

$$ = \sup \left\{ \lim_n \left( \int_{\mathbb{R}^d} \varphi \, d\mu_s^n + \int_{\mathbb{R}^d} \psi \, d\mu_t^n \right) : (\varphi, \psi) \in \Phi_p \right\} \quad (4.20) $$

$$ \leq \limsup_n \sup \left\{ \int_{\mathbb{R}^d} \varphi \, d\mu_s^n + \int_{\mathbb{R}^d} \psi \, d\mu_t^n : (\varphi, \psi) \in \Phi_p \right\} \quad (4.21) $$

$$ = \limsup_n (W_p(\mu_s^n, \mu_t^n))^p \quad (4.21) $$

$$ \lesssim |t-s|^{\delta_2 p}. \quad (4.22) $$

Above, (4.19) and (4.21) are due to Lemma 3.1[1], (4.20) due to Lemma 4.2, and (4.22) due to (4.10). It follows that

$$ W_p(\mu_s, \mu_t) \lesssim |t-s|^{\delta_2}, \quad s, t \in \mathbb{T}. $$

Next we establish an essential result about convergence:

**Lemma 4.3.** We have

$$ \limsup_n \sup_{t \in \mathbb{T}} W_p(\mu_t^n, \mu_t) = 0. $$

*Proof.* Let $t \in \mathbb{T}$, $n \in \mathbb{N}^*$ and $R > 0$. We have

$$ \begin{aligned} (W_p(\mu_t^n, \mu_t))^p &\lesssim \int_{\mathbb{R}^d} |x|^p |\ell_t^n(x) - \ell_t(x)| \, dx && \text{by Lemma 3.2} \\ &\leq \int_{B(0,R)} |x|^p |\ell_t^n(x) - \ell_t(x)| \, dx + \int_{B_R^c} |x|^p (\ell_t^n(x) + \ell_t(x)) \, dx \\ &=: J(t, n, R) + K(t, n, R). \end{aligned} $$

By (4.13),

$$ \limsup_n J(t, n, R) = 0. $$

By (4.11) and (4.18),

$$ \sup_{n \in \mathbb{N}^*} \sup_{t \in \mathbb{T}} K(t, n, R) \lesssim \frac{1}{R}. $$

As such,

$$ \begin{aligned} & \limsup_n \sup_{t \in \mathbb{T}} (W_p(\mu_t^n, \mu_t))^p \\ & \lesssim \limsup_n \sup_{t \in \mathbb{T}} J(t, n, R) + \limsup_n \sup_{t \in \mathbb{T}} K(t, n, R) \\ & \lesssim \frac{1}{R}. \end{aligned} $$

The claim then follows by taking the limit $R \to \infty$. $\square$
which implies $\mu_t \in \mathcal{P}_p(\mathbb{R}^d)$ for $t \in \mathbb{T}$. We have

$$ (W_p(\mu_s, \mu_t))^p = \sup \left\{ \int_{\mathbb{R}^d} \varphi \, d\mu_s + \int_{\mathbb{R}^d} \psi \, d\mu_t : (\varphi, \psi) \in \Phi_p \right\} \quad (4.19) $$

$$ = \sup \left\{ \lim_n \left( \int_{\mathbb{R}^d} \varphi \, d\mu_s^n + \int_{\mathbb{R}^d} \psi \, d\mu_t^n \right) : (\varphi, \psi) \in \Phi_p \right\} \quad (4.20) $$

$$ \leq \limsup_n \sup \left\{ \int_{\mathbb{R}^d} \varphi \, d\mu_s^n + \int_{\mathbb{R}^d} \psi \, d\mu_t^n : (\varphi, \psi) \in \Phi_p \right\} \quad (4.21) $$

$$ = \limsup_n (W_p(\mu_s^n, \mu_t^n))^p \quad (4.21) $$

$$ \lesssim |t-s|^{\delta_2 p}. \quad (4.22) $$

Above, (4.19) and (4.21) are due to Lemma 3.1[1], (4.20) due to Lemma 4.2, and (4.22) due to (4.10). It follows that

$$ W_p(\mu_s, \mu_t) \lesssim |t-s|^{\delta_2}, \quad s, t \in \mathbb{T}. $$

Next we establish an essential result about convergence:

**Lemma 4.3.** We have

$$ \limsup_n \sup_{t \in \mathbb{T}} W_p(\mu_t^n, \mu_t) = 0. $$

*Proof.* Let $t \in \mathbb{T}$, $n \in \mathbb{N}^*$ and $R > 0$. We have

$$ \begin{aligned} (W_p(\mu_t^n, \mu_t))^p &\lesssim \int_{\mathbb{R}^d} |x|^p |\ell_t^n(x) - \ell_t(x)| \, dx && \text{by Lemma 3.2} \\ &\leq \int_{B(0,R)} |x|^p |\ell_t^n(x) - \ell_t(x)| \, dx + \int_{B_R^c} |x|^p (\ell_t^n(x) + \ell_t(x)) \, dx \\ &=: J(t, n, R) + K(t, n, R). \end{aligned} $$

By (4.13),

$$ \limsup_n J(t, n, R) = 0. $$

By (4.11) and (4.18),

$$ \sup_{n \in \mathbb{N}^*} \sup_{t \in \mathbb{T}} K(t, n, R) \lesssim \frac{1}{R}. $$

As such,

$$ \begin{aligned} & \limsup_n \sup_{t \in \mathbb{T}} (W_p(\mu_t^n, \mu_t))^p \\ & \lesssim \limsup_n \sup_{t \in \mathbb{T}} J(t, n, R) + \limsup_n \sup_{t \in \mathbb{T}} K(t, n, R) \\ & \lesssim \frac{1}{R}. \end{aligned} $$

The claim then follows by taking the limit $R \to \infty$. $\square$
## 4.3 Existence of a weak solution

For convenience, we write $\ell^n(t, x) := \ell_t^n(x)$. Notice that $\rho^n * \mu_t^n = \rho^n * \ell_t^n$. The Fokker-Planck equation (in distributional sense) associated with (4.1) is

$$ \partial_t \ell^n(t, x) = - \sum_{i=1}^{d} \frac{\partial}{\partial x_i} \{ b(t, x, (\rho^n * \ell_t^n)(x), \mu_t^n) \ell^n(t, x) \} \\ + \frac{1}{2} \sum_{i,j=1}^{d} \frac{\partial^2}{\partial x_i \partial x_j} \{ a^{i,j}(t, x, \mu_t^n) \ell^n(t, x) \}, $$

which means for each $(\varphi, \psi) \in C_c^\infty(\mathbb{T}) \times C_c^\infty(\mathbb{R}^d)$ that

$$ \begin{aligned} & - \int_{\mathbb{T}} \int_{\mathbb{R}^d} \varphi'(t) \psi(x) \, d\mu_t^n(x) \, dt \\ &= \sum_{i=1}^{d} \int_{\mathbb{T}} \int_{\mathbb{R}^d} b(t, x, (\rho^n * \ell_t^n)(x), \mu_t^n) \varphi(t) \frac{\partial \psi}{\partial x_i}(x) \, d\mu_t^n(x) \, dt \quad (4.23) \\ &\quad + \frac{1}{2} \sum_{i,j=1}^{d} \int_{\mathbb{T}} \int_{\mathbb{R}^d} a^{i,j}(t, x, \mu_t^n) \varphi(t) \frac{\partial^2 \psi}{\partial x_i \partial x_j}(x) \, d\mu_t^n(x) \, dt. \end{aligned} $$

Above, $a^{i,j}$ is the entry in the $i$-th row and $j$-th column of $a$. We recall from Lemma 4.3 and (4.13) that

$$ \sup_{t \in \mathbb{T}} W_p(\mu_t^n, \mu_t) \xrightarrow{n \to \infty} 0, \quad (4.24) $$

$$ \sup_{t \in \mathbb{T}} \sup_{x \in B(0, R)} |\ell_t^n(x) - \ell_t(x)| \xrightarrow{n \to \infty} 0, \quad R > 0. \quad (4.25) $$

We fix $(\varphi, \psi) \in C_c^\infty(\mathbb{T}) \times C_c^\infty(\mathbb{R}^d)$. By (4.24), the boundedness of $a$, and the continuity of $a$ w.r.t. the distribution variable, we get for $t \in \mathbb{T}$:

$$ \int_{\mathbb{R}^d} \psi(x) \, d\mu_t^n(x) \xrightarrow{n \to \infty} \int_{\mathbb{R}^d} \psi(x) \, d\mu_t(x), \\ \int_{\mathbb{R}^d} a^{i,j}(t, x, \mu_t^n) \frac{\partial^2 \psi}{\partial x_i \partial x_j}(x) \, d\mu_t^n(x) \xrightarrow{n \to \infty} \int_{\mathbb{R}^d} a^{i,j}(t, x, \mu_t) \frac{\partial^2 \psi}{\partial x_i \partial x_j}(x) \, d\mu_t(x). $$

Let $S := B(0, 1) + \text{supp } \psi$. By triangle inequality,

$$ \begin{aligned} \|1_S\{(\rho^n * \ell_t^n) - \ell_t\}\|_{\infty} &\le \|1_S\{\rho^n * (\ell_t^n - \ell_t)\}\|_{\infty} + \|1_S(\rho^n * \ell_t - \ell_t)\|_{\infty} \\ &\le \|\rho^n * \{1_S(\ell_t^n - \ell_t)\}\|_{\infty} + \|1_S(\rho^n * \ell_t - \ell_t)\|_{\infty} \\ &\le \|1_S(\ell_t^n - \ell_t)\|_{\infty} + \|1_S(\rho^n * \ell_t - \ell_t)\|_{\infty}. \end{aligned} $$

By (4.25), $\|1_S(\ell_t^n - \ell_t)\|_{\infty} \to 0$ as $n \to \infty$. By [54, Proposition 4.21], $\|1_S(\rho^n * \ell_t - \ell_t)\|_{\infty} \to 0$ as $n \to \infty$. It follows that $\|1_S\{(\rho^n * \ell_t^n) - \ell_t\}\|_{\infty} \to 0$ as $n \to \infty$. Hence we have for $t \in \mathbb{T}$:

$$ \|1_S\{b(t, \cdot, (\rho^n * \ell_t^n)(\cdot), \mu_t^n) - b(t, \cdot, \ell_t(\cdot), \mu_t)\}\|_{\infty} \xrightarrow{n \to \infty} 0. \quad (4.26) $$
## 4.3 Existence of a weak solution

For convenience, we write $\ell^n(t, x) := \ell_t^n(x)$. Notice that $\rho^n * \mu_t^n = \rho^n * \ell_t^n$. The Fokker-Planck equation (in distributional sense) associated with (4.1) is

$$ \partial_t \ell^n(t, x) = - \sum_{i=1}^{d} \frac{\partial}{\partial x_i} \{ b(t, x, (\rho^n * \ell_t^n)(x), \mu_t^n) \ell^n(t, x) \} \\ + \frac{1}{2} \sum_{i,j=1}^{d} \frac{\partial^2}{\partial x_i \partial x_j} \{ a^{i,j}(t, x, \mu_t^n) \ell^n(t, x) \}, $$

which means for each $(\varphi, \psi) \in C_c^\infty(\mathbb{T}) \times C_c^\infty(\mathbb{R}^d)$ that

$$ \begin{aligned} & - \int_{\mathbb{T}} \int_{\mathbb{R}^d} \varphi'(t) \psi(x) \, d\mu_t^n(x) \, dt \\ &= \sum_{i=1}^{d} \int_{\mathbb{T}} \int_{\mathbb{R}^d} b(t, x, (\rho^n * \ell_t^n)(x), \mu_t^n) \varphi(t) \frac{\partial \psi}{\partial x_i}(x) \, d\mu_t^n(x) \, dt \quad (4.23) \\ &\quad + \frac{1}{2} \sum_{i,j=1}^{d} \int_{\mathbb{T}} \int_{\mathbb{R}^d} a^{i,j}(t, x, \mu_t^n) \varphi(t) \frac{\partial^2 \psi}{\partial x_i \partial x_j}(x) \, d\mu_t^n(x) \, dt. \end{aligned} $$

Above, $a^{i,j}$ is the entry in the $i$-th row and $j$-th column of $a$. We recall from Lemma 4.3 and (4.13) that

$$ \sup_{t \in \mathbb{T}} W_p(\mu_t^n, \mu_t) \xrightarrow{n \to \infty} 0, \quad (4.24) $$

$$ \sup_{t \in \mathbb{T}} \sup_{x \in B(0, R)} |\ell_t^n(x) - \ell_t(x)| \xrightarrow{n \to \infty} 0, \quad R > 0. \quad (4.25) $$

We fix $(\varphi, \psi) \in C_c^\infty(\mathbb{T}) \times C_c^\infty(\mathbb{R}^d)$. By (4.24), the boundedness of $a$, and the continuity of $a$ w.r.t. the distribution variable, we get for $t \in \mathbb{T}$:

$$ \begin{aligned} & \int_{\mathbb{R}^d} \psi(x) \, d\mu_t^n(x) \xrightarrow{n \to \infty} \int_{\mathbb{R}^d} \psi(x) \, d\mu_t(x), \\ & \int_{\mathbb{R}^d} a^{i,j}(t, x, \mu_t^n) \frac{\partial^2 \psi}{\partial x_i \partial x_j}(x) \, d\mu_t^n(x) \xrightarrow{n \to \infty} \int_{\mathbb{R}^d} a^{i,j}(t, x, \mu_t) \frac{\partial^2 \psi}{\partial x_i \partial x_j}(x) \, d\mu_t(x). \end{aligned} $$

Let $S := B(0, 1) + \text{supp } \psi$. By triangle inequality,

$$ \begin{aligned} \|1_S\{(\rho^n * \ell_t^n) - \ell_t\}\|_{\infty} &\le \|1_S\{\rho^n * (\ell_t^n - \ell_t)\}\|_{\infty} + \|1_S(\rho^n * \ell_t - \ell_t)\|_{\infty} \\ &\le \|\rho^n * \{1_S(\ell_t^n - \ell_t)\}\|_{\infty} + \|1_S(\rho^n * \ell_t - \ell_t)\|_{\infty} \\ &\le \|1_S(\ell_t^n - \ell_t)\|_{\infty} + \|1_S(\rho^n * \ell_t - \ell_t)\|_{\infty}. \end{aligned} $$

By (4.25), $\|1_S(\ell_t^n - \ell_t)\|_{\infty} \to 0$ as $n \to \infty$. By [54, Proposition 4.21], $\|1_S(\rho^n * \ell_t - \ell_t)\|_{\infty} \to 0$ as $n \to \infty$. It follows that $\|1_S\{(\rho^n * \ell_t^n) - \ell_t\}\|_{\infty} \to 0$ as $n \to \infty$. Hence we have for $t \in \mathbb{T}$:

$$ \|1_S\{b(t, \cdot, (\rho^n * \ell_t^n)(\cdot), \mu_t^n) - b(t, \cdot, \ell_t(\cdot), \mu_t)\}\|_{\infty} \xrightarrow{n \to \infty} 0. \quad (4.26) $$
Recall that $|b| \le f_0$ and $f_0 \in \tilde{L}_{q_0}^{p_0}$. It follows from (4.24), (4.26) and DCT that

$$ \int_{\mathbb{R}^d} b(t, x, (\rho^n * \ell_t^n)(x), \mu_t^n) \frac{\partial \psi}{\partial x_i}(x) d\mu_t^n(x) \xrightarrow{n \to \infty} \int_{\mathbb{R}^d} b(t, x, \ell_t(x), \mu_t) \frac{\partial \psi}{\partial x_i}(x) d\mu_t(x). $$

Taking the limit $n \to \infty$ in (4.23), we get

$$ \begin{aligned} & - \int_{\mathbb{T}} \int_{\mathbb{R}^d} \varphi'(t) \psi(x) \, d\mu_t(x) \, dt \\ &= \sum_{i=1}^{d} \int_{\mathbb{T}} \int_{\mathbb{R}^d} b(t, x, \ell_t(x), \mu_t) \varphi(t) \frac{\partial \psi}{\partial x_i}(x) \, d\mu_t(x) \, dt \\ &\quad + \frac{1}{2} \sum_{i,j=1}^{d} \int_{\mathbb{T}} \int_{\mathbb{R}^d} a^{i,j}(t, x, \mu_t) \varphi(t) \frac{\partial^2 \psi}{\partial x_i \partial x_j}(x) \, d\mu_t(x) \, dt, \quad (\varphi, \psi) \in C_c^\infty(\mathbb{T}) \times C_c^\infty(\mathbb{R}^d). \end{aligned} $$

So $\ell$ satisfies the Fokker-Planck equation

$$ \begin{aligned} \partial_t \ell(t, x) = & - \sum_{i=1}^{d} \frac{\partial}{\partial x_i} \{b(t, x, \ell_t(x), \mu_t) \ell(t, x)\} \\ & + \frac{1}{2} \sum_{i,j=1}^{d} \frac{\partial^2}{\partial x_i \partial x_j} \{a^{i,j}(t, x, \mu_t) \ell(t, x)\}. \end{aligned} $$

Moreover, $\ell$ satisfies the following integrability estimate:

**Lemma 4.4.** There exists a constant $c > 0$ (depending only on $\Theta_1$) such that

$$ \int_{\mathbb{T}} \int_{\mathbb{R}^d} \{|b(t, x, \ell_t(x), \mu_t)| + |a(t, x, \mu_t)|\} d\mu_t(x) \, dt \le c(1 + \|f_0\|_{\tilde{L}_{q_0}^{p_0}}). $$

*Proof.* By (4.6), $dX_t^n = \bar{b}^n(t, X_t^n) dt + \bar{\sigma}^n(t, X_t^n) dB_t$. By Remark 4.1, every pair $(\bar{b}^n, \bar{\sigma}^n)$ satisfies Assumption 3.6 for the **same** set of parameters. Then

$$ \begin{aligned} \int_{\mathbb{T}} \int_{\mathbb{R}^d} f_0(t, x) \, d\mu_t^n(x) \, dt &= \mathbb{E} \left[ \int_0^T f_0(t, X_t^n) \, dt \right] \quad \text{by Tonelli's theorem} \\ &\lesssim 1 + \|f_0\|_{\tilde{L}_{q_0}^{p_0}} \quad \text{by Proposition 3.8[2]}. \end{aligned} \quad (4.27) $$

We have

$$ \int_{\mathbb{T}} \int_{\mathbb{R}^d} f_0(t, x) \, d\mu_t(x) \, dt = \int_{\mathbb{T}} \lim_{k} \int_{\mathbb{R}^d} 1_{B(0,k)}(x) f_0(t, x) \ell_t(x) \, dx \, dt \quad (4.28) $$

$$ \le \liminf_k \int_{\mathbb{T}} \int_{\mathbb{R}^d} 1_{B(0,k)}(x) f_0(t, x) \ell_t(x) \, dx \, dt \quad (4.29) $$

$$ = \liminf_k \int_{\mathbb{T}} \lim_n \int_{\mathbb{R}^d} 1_{B(0,k)}(x) f_0(t, x) \ell_t^n(x) \, dx \, dt \quad (4.30) $$

$$ \le \liminf_k \liminf_n \int_{\mathbb{T}} \int_{\mathbb{R}^d} 1_{B(0,k)}(x) f_0(t, x) \ell_t^n(x) \, dx \, dt \quad (4.31) $$

$$ \le \liminf_n \int_{\mathbb{T}} \int_{\mathbb{R}^d} f_0(t, x) \ell_t^n(x) \, dx \, dt \\ \lesssim 1 + \|f_0\|_{\tilde{L}_{q_0}^{p_0}} \quad \text{by (4.27).} \quad (4.32) $$
Recall that $|b| \le f_0$ and $f_0 \in \tilde{L}_{q_0}^{p_0}$. It follows from (4.24), (4.26) and DCT that

$$ \int_{\mathbb{R}^d} b(t, x, (\rho^n * \ell_t^n)(x), \mu_t^n) \frac{\partial \psi}{\partial x_i}(x) d\mu_t^n(x) \xrightarrow{n \to \infty} \int_{\mathbb{R}^d} b(t, x, \ell_t(x), \mu_t) \frac{\partial \psi}{\partial x_i}(x) d\mu_t(x). $$

Taking the limit $n \to \infty$ in (4.23), we get

$$ \begin{aligned} & - \int_{\mathbb{T}} \int_{\mathbb{R}^d} \varphi'(t) \psi(x) \, d\mu_t(x) \, dt \\ &= \sum_{i=1}^{d} \int_{\mathbb{T}} \int_{\mathbb{R}^d} b(t, x, \ell_t(x), \mu_t) \varphi(t) \frac{\partial \psi}{\partial x_i}(x) \, d\mu_t(x) \, dt \\ &\quad + \frac{1}{2} \sum_{i,j=1}^{d} \int_{\mathbb{T}} \int_{\mathbb{R}^d} a^{i,j}(t, x, \mu_t) \varphi(t) \frac{\partial^2 \psi}{\partial x_i \partial x_j}(x) \, d\mu_t(x) \, dt, \quad (\varphi, \psi) \in C_c^\infty(\mathbb{T}) \times C_c^\infty(\mathbb{R}^d). \end{aligned} $$

So $\ell$ satisfies the Fokker-Planck equation

$$ \begin{aligned} \partial_t \ell(t, x) = & - \sum_{i=1}^{d} \frac{\partial}{\partial x_i} \{b(t, x, \ell_t(x), \mu_t) \ell(t, x)\} \\ & + \frac{1}{2} \sum_{i,j=1}^{d} \frac{\partial^2}{\partial x_i \partial x_j} \{a^{i,j}(t, x, \mu_t) \ell(t, x)\}. \end{aligned} $$

Moreover, $\ell$ satisfies the following integrability estimate:

**Lemma 4.4.** There exists a constant $c > 0$ (depending only on $\Theta_1$) such that

$$ \int_{\mathbb{T}} \int_{\mathbb{R}^d} \{|b(t, x, \ell_t(x), \mu_t)| + |a(t, x, \mu_t)|\} d\mu_t(x) \, dt \le c(1 + \|f_0\|_{\tilde{L}_{q_0}^{p_0}}). $$

*Proof.* By (4.6), $dX_t^n = \bar{b}^n(t, X_t^n) dt + \bar{\sigma}^n(t, X_t^n) dB_t$. By Remark 4.1, every pair $(\bar{b}^n, \bar{\sigma}^n)$ satisfies Assumption 3.6 for the **same** set of parameters. Then

$$ \begin{aligned} \int_{\mathbb{T}} \int_{\mathbb{R}^d} f_0(t, x) \, d\mu_t^n(x) \, dt &= \mathbb{E} \left[ \int_0^T f_0(t, X_t^n) \, dt \right] \quad \text{by Tonelli's theorem} \\ &\lesssim 1 + \|f_0\|_{\tilde{L}_{q_0}^{p_0}} \quad \text{by Proposition 3.8[2].} \end{aligned} \quad (4.27) $$

We have

$$ \int_{\mathbb{T}} \int_{\mathbb{R}^d} f_0(t, x) \, d\mu_t(x) \, dt = \int_{\mathbb{T}} \lim_{k} \int_{\mathbb{R}^d} 1_{B(0,k)}(x) f_0(t, x) \ell_t(x) \, dx \, dt \quad (4.28) $$

$$ \le \liminf_k \int_{\mathbb{T}} \int_{\mathbb{R}^d} 1_{B(0,k)}(x) f_0(t, x) \ell_t(x) \, dx \, dt \quad (4.29) $$

$$ = \liminf_k \int_{\mathbb{T}} \lim_n \int_{\mathbb{R}^d} 1_{B(0,k)}(x) f_0(t, x) \ell_t^n(x) \, dx \, dt \quad (4.30) $$

$$ \le \liminf_k \liminf_n \int_{\mathbb{T}} \int_{\mathbb{R}^d} 1_{B(0,k)}(x) f_0(t, x) \ell_t^n(x) \, dx \, dt \quad (4.31) $$

$$ \le \liminf_n \int_{\mathbb{T}} \int_{\mathbb{R}^d} f_0(t, x) \ell_t^n(x) \, dx \, dt \\ \lesssim 1 + \|f_0\|_{\tilde{L}_{q_0}^{p_0}} \quad \text{by (4.27).} \quad (4.32) $$
Above, (4.28) is due to MCT; (4.29) and (4.31) are due to Fatou's lemma. We will justify how (4.30) follows from $f_0 \in \tilde{L}_{q_0}^{p_0}$ and DCT:

1. From (4.7), we get $1_{B(0,k)}(x)f_0(t,x)\ell_t^n(x) \lesssim 1_{B(0,k)}(x)f_0(t,x)$.

2. From (4.25), we get $1_{B(0,k)}(x)f_0(t,x)\ell_t^n(x) \to 1_{B(0,k)}(x)f_0(t,x)\ell_t(x)$ (as $n \to \infty$) for all $x \in \mathbb{R}^d$.

We denote by $I$ the LHS of the inequality in the statement of Lemma 4.4. Then

$$ 
\begin{aligned}
I & \lesssim 1 + \int_{\mathbb{T}} \int_{\mathbb{R}^d} f_0(t,x) \, d\mu_t(x) \, dt \\
& \lesssim 1 + \|f_0\|_{\tilde{L}_{q_0}^{p_0}} \quad \text{by (4.32).}
\end{aligned}
$$

This completes the proof.

□

Clearly,

1. The maps $(t, x) \mapsto b(t, x, \ell_t(x), \mu_t)$ and $(t, x) \mapsto a(t, x, \mu_t)$ are measurable.

2. By Lemma 4.4,

$$ \int_{\mathbb{T}} \int_{\mathbb{R}^d} \{|b(t, x, \ell_t(x), \mu_t)| + |a(t, x, \mu_t)|\} d\mu_t(x) \, dt < \infty. $$

3. The map $\mathbb{T} \to \mathcal{P}_p(\mathbb{R}^d)$, $t \mapsto \mu_t$ is continuous by (4.22).

By the same application of the superposition principle [55, 56, 57] as in [14, Section 2], (1.1) has a weak solution whose marginal distribution is exactly $(\mu_t, t \in \mathbb{T})$.

## 4.4 Existence of a strong solution

By the previous sub-section, there exists a probability space $(\bar{\Omega}, \bar{\mathcal{A}}, \bar{\mathbb{P}})$ on which there exist an $m$-dimensional Brownian motion $(\bar{B}_t, t \ge 0)$, an admissible filtration $\mathbb{F} := (\bar{\mathcal{F}}_t)_{t \ge 0}$, and a continuous $\mathbb{F}$-adapted process $(\bar{X}_t, t \in \mathbb{T})$ such that

$$ d\bar{X}_t = b(t, \bar{X}_t, \ell_t(\bar{X}_t), \bar{\mu}_t) \, dt + \sigma(t, \bar{X}_t, \bar{\mu}_t) \, dB_t, \quad t \in \mathbb{T}, $$

where the distribution of $\bar{X}_0$ is $\nu$, that of $\bar{X}_t$ is $\bar{\mu}_t$, and the p.d.f. of $\bar{X}_t$ is $\ell_t$. We define the map $\bar{b}: \mathbb{T} \times \mathbb{R}^d \times \mathcal{P}_p(\mathbb{R}^d) \to \mathbb{R}^d$ by $\bar{b}(t, x, \tilde{\rho}) := b(t, x, \ell_t(x), \tilde{\rho})$. We consider the SDE

$$ dY_t = \bar{b}(t, Y_t, \mu_t) \, dt + \sigma(t, Y_t, \mu_t) \, dB_t, \quad t \in \mathbb{T}, \quad (4.33) $$

where the distribution of $Y_0$ is $\nu$ and that of $Y_t$ is $\mu_t$. We recall that $(B_t, t \ge 0)$ is the fixed $m$-dimensional Brownian motion on the fixed probability space $(\Omega, \mathcal{A}, \mathbb{P})$ introduced in Section 1. By [10, Theorem 1.1(1)], (4.33) is well-posed. On the other hand,

$$ d\bar{X}_t = \bar{b}(t, \bar{X}_t, \bar{\mu}_t) \, dt + \sigma(t, \bar{X}_t, \bar{\mu}_t) \, dB_t. $$

It follows that $\mu_t = \bar{\mu}_t$ and thus the p.d.f. of $Y_t$ is $\ell_t$. In particular,

$$ dY_t = b(t, Y_t, \ell_t(Y_t), \mu_t) \, dt + \sigma(t, Y_t, \mu_t) \, dB_t. $$

This completes the proof.
Above, (4.28) is due to MCT; (4.29) and (4.31) are due to Fatou's lemma. We will justify how (4.30) follows from $f_0 \in \tilde{L}_{q_0}^{p_0}$ and DCT:

1. From (4.7), we get $1_{B(0,k)}(x)f_0(t,x)\ell_t^n(x) \lesssim 1_{B(0,k)}(x)f_0(t,x)$.

2. From (4.25), we get $1_{B(0,k)}(x)f_0(t,x)\ell_t^n(x) \to 1_{B(0,k)}(x)f_0(t,x)\ell_t(x)$ (as $n \to \infty$) for all $x \in \mathbb{R}^d$.

We denote by $I$ the LHS of the inequality in the statement of Lemma 4.4. Then

$$ 
\begin{aligned}
I & \lesssim 1 + \int_{\mathbb{T}} \int_{\mathbb{R}^d} f_0(t,x) \, d\mu_t(x) \, dt \\
& \lesssim 1 + \|f_0\|_{\tilde{L}_{q_0}^{p_0}} \quad \text{by (4.32).}
\end{aligned}
$$

This completes the proof.

□

Clearly,

1. The maps $(t, x) \mapsto b(t, x, \ell_t(x), \mu_t)$ and $(t, x) \mapsto a(t, x, \mu_t)$ are measurable.

2. By Lemma 4.4,

$$ \int_{\mathbb{T}} \int_{\mathbb{R}^d} \{|b(t, x, \ell_t(x), \mu_t)| + |a(t, x, \mu_t)|\} d\mu_t(x) \, dt < \infty. $$

3. The map $\mathbb{T} \to \mathcal{P}_p(\mathbb{R}^d)$, $t \mapsto \mu_t$ is continuous by (4.22).

By the same application of the superposition principle [55, 56, 57] as in [14, Section 2], (1.1) has a weak solution whose marginal distribution is exactly $(\mu_t, t \in \mathbb{T})$.

## 4.4 Existence of a strong solution

By the previous sub-section, there exists a probability space $(\bar{\Omega}, \bar{\mathcal{A}}, \bar{\mathbb{P}})$ on which there exist an $m$-dimensional Brownian motion $(\bar{B}_t, t \ge 0)$, an admissible filtration $\mathbb{F} := (\bar{\mathcal{F}}_t)_{t \ge 0}$, and a continuous $\mathbb{F}$-adapted process $(\bar{X}_t, t \in \mathbb{T})$ such that

$$ d\bar{X}_t = b(t, \bar{X}_t, \ell_t(\bar{X}_t), \bar{\mu}_t) \, dt + \sigma(t, \bar{X}_t, \bar{\mu}_t) \, dB_t, \quad t \in \mathbb{T}, $$

where the distribution of $\bar{X}_0$ is $\nu$, that of $\bar{X}_t$ is $\bar{\mu}_t$, and the p.d.f. of $\bar{X}_t$ is $\ell_t$. We define the map $\bar{b}: \mathbb{T} \times \mathbb{R}^d \times \mathcal{P}_p(\mathbb{R}^d) \to \mathbb{R}^d$ by $\bar{b}(t, x, \tilde{\rho}) := b(t, x, \ell_t(x), \tilde{\rho})$. We consider the SDE

$$ dY_t = \bar{b}(t, Y_t, \mu_t) \, dt + \sigma(t, Y_t, \mu_t) \, dB_t, \quad t \in \mathbb{T}, \quad (4.33) $$

where the distribution of $Y_0$ is $\nu$ and that of $Y_t$ is $\mu_t$. We recall that $(B_t, t \ge 0)$ is the fixed $m$-dimensional Brownian motion on the fixed probability space $(\Omega, \mathcal{A}, \mathbb{P})$ introduced in Section 1. By [10, Theorem 1.1(1)], (4.33) is well-posed. On the other hand,

$$ d\bar{X}_t = \bar{b}(t, \bar{X}_t, \bar{\mu}_t) \, dt + \sigma(t, \bar{X}_t, \bar{\mu}_t) \, dB_t. $$

It follows that $\mu_t = \bar{\mu}_t$ and thus the p.d.f. of $Y_t$ is $\ell_t$. In particular,

$$ dY_t = b(t, Y_t, \ell_t(Y_t), \mu_t) \, dt + \sigma(t, Y_t, \mu_t) \, dB_t. $$

This completes the proof.
# 5 Proof of Theorem 2.3

For $k \in \{1, 2\}$, we consider

$$dX_t^k = b(t, X_t^k, \ell_t^k(X_t^k), \mu_t^k) dt + \sigma(t, X_t^k) dB_t^k, \quad t \in \mathbb{T}, \qquad (5.1)$$

where the distribution of $X_0^k$ is $\nu_k$, that of $X_t^k$ is $\mu_t^k$, the p.d.f. of $X_t^k$ is $\ell_t^k$, and $(B_t^k, t \ge 0)$ is a $d$-dimensional Brownian motion on a probability space $(\Omega^k, \mathcal{A}^k, \mathbb{P}^k)$ with an admissible filtration $\mathbb{F}^k := (\mathcal{F}_t^k)_{t \ge 0}$. We define measurable maps $b^k: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ by $b^k(t, x) := b(t, x, \ell_t^k(x), \mu_t^k)$.

## 5.1 Uniqueness of marginal density

Clearly, $(b^k, \sigma)$ satisfies Assumption 3.11. Let $\bar{b}$ be defined by (3.20). We denote $b_t^k(x) := b^k(t, x)$. By Corollary 3.13,

$$\ell_t^k(x) = P_{0,t}^{\bar{b},\sigma} \ell_{\nu_k}(x) + \int_0^t \int_{\mathbb{R}^d} \ell_s^k(y) \langle b_s^k(y) - \bar{b}_s(y), \nabla_y p_{s,t}^{\bar{b},\sigma}(y,x) \rangle dy ds,$$

which implies

$$\begin{aligned} |\ell_t^2(x) - \ell_t^1(x)| &\le P_{0,t}^{\bar{b},\sigma} \{|\ell_{\nu_1}(x) - \ell_{\nu_2}|\}(x) \tag{5.2} \\ &\quad + \int_0^t \int_{\mathbb{R}^d} |b_s^2(y) - \bar{b}_s(y)| |\ell_s^2(y) - \ell_s^1(y)| |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| dy ds \\ &\quad + \int_0^t \int_{\mathbb{R}^d} \ell_s^1(y) |b_s^2(y) - b_s^1(y)| |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| dy ds. \end{aligned}$$

We write $M_1 \preceq M_2$ if there exists a constant $c > 0$ (depending only on $\Theta_1$) such that $M_1 \preceq cM_2$. We have $\|b^k\|_{\infty} + \|\bar{b}\|_{\infty} < \infty$, so

$$|b^k - \bar{b}| \preceq 1. \qquad (5.3)$$

By (5.2) and (5.3),

$$\begin{aligned} |\ell_t^2(x) - \ell_t^1(x)| &\preceq P_{0,t}^{\bar{b},\sigma} \{|\ell_{\nu_1}(x) - \ell_{\nu_2}|\}(x) \tag{5.4} \\ &\quad + \int_0^t \int_{\mathbb{R}^d} |\ell_s^2(y) - \ell_s^1(y)| |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| dy ds \\ &\quad + \int_0^t \int_{\mathbb{R}^d} \ell_s^1(y) |b_s^2(y) - b_s^1(y)| |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| dy ds. \end{aligned}$$

By Theorem 3.10[2],

$$\sup_{t \in \mathbb{T}} \|\ell_t^1\|_{\infty} \preceq \|\ell_{\nu_1}\|_{\infty}. \qquad (5.5)$$

By Assumption 2.1[A4],

$$|b_s^2(y) - b_s^1(y)| \preceq |\ell_s^2(y) - \ell_s^1(y)| + W_p(\mu_s^2, \mu_s^1). \qquad (5.6)$$
# 5 Proof of Theorem 2.3

For $k \in \{1, 2\}$, we consider

$$dX_t^k = b(t, X_t^k, \ell_t^k(X_t^k), \mu_t^k) dt + \sigma(t, X_t^k) dB_t^k, \quad t \in \mathbb{T}, \qquad (5.1)$$

where the distribution of $X_0^k$ is $\nu_k$, that of $X_t^k$ is $\mu_t^k$, the p.d.f. of $X_t^k$ is $\ell_t^k$, and $(B_t^k, t \ge 0)$ is a $d$-dimensional Brownian motion on a probability space $(\Omega^k, \mathcal{A}^k, \mathbb{P}^k)$ with an admissible filtration $\mathbb{F}^k := (\mathcal{F}_t^k)_{t \ge 0}$. We define measurable maps $b^k: \mathbb{T} \times \mathbb{R}^d \to \mathbb{R}^d$ by $b^k(t, x) := b(t, x, \ell_t^k(x), \mu_t^k)$.

## 5.1 Uniqueness of marginal density

Clearly, $(b^k, \sigma)$ satisfies Assumption 3.11. Let $\bar{b}$ be defined by (3.20). We denote $b_t^k(x) := b^k(t, x)$. By Corollary 3.13,

$$\ell_t^k(x) = P_{0,t}^{\bar{b},\sigma} \ell_{\nu_k}(x) + \int_0^t \int_{\mathbb{R}^d} \ell_s^k(y) \langle b_s^k(y) - \bar{b}_s(y), \nabla_y p_{s,t}^{\bar{b},\sigma}(y,x) \rangle dy ds,$$

which implies

$$\begin{aligned} |\ell_t^2(x) - \ell_t^1(x)| &\le P_{0,t}^{\bar{b},\sigma} \{|\ell_{\nu_1}(x) - \ell_{\nu_2}|\}(x) \tag{5.2} \\ &\quad + \int_0^t \int_{\mathbb{R}^d} |b_s^2(y) - \bar{b}_s(y)| |\ell_s^2(y) - \ell_s^1(y)| |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| dy ds \\ &\quad + \int_0^t \int_{\mathbb{R}^d} \ell_s^1(y) |b_s^2(y) - b_s^1(y)| |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| dy ds. \end{aligned}$$

We write $M_1 \preceq M_2$ if there exists a constant $c > 0$ (depending only on $\Theta_1$) such that $M_1 \preceq cM_2$. We have $\|b^k\|_{\infty} + \|\bar{b}\|_{\infty} < \infty$, so

$$|b^k - \bar{b}| \preceq 1. \qquad (5.3)$$

By (5.2) and (5.3),

$$\begin{aligned} |\ell_t^2(x) - \ell_t^1(x)| &\preceq P_{0,t}^{\bar{b},\sigma} \{|\ell_{\nu_1}(x) - \ell_{\nu_2}|\}(x) \tag{5.4} \\ &\quad + \int_0^t \int_{\mathbb{R}^d} |\ell_s^2(y) - \ell_s^1(y)| |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| dy ds \\ &\quad + \int_0^t \int_{\mathbb{R}^d} \ell_s^1(y) |b_s^2(y) - b_s^1(y)| |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| dy ds. \end{aligned}$$

By Theorem 3.10[2],

$$\sup_{t \in \mathbb{T}} \|\ell_t^1\|_{\infty} \preceq \|\ell_{\nu_1}\|_{\infty}. \qquad (5.5)$$

By Assumption 2.1[A4],

$$|b_s^2(y) - b_s^1(y)| \preceq |\ell_s^2(y) - \ell_s^1(y)| + W_p(\mu_s^2, \mu_s^1). \qquad (5.6)$$
By (5.4), (5.5) and (5.6),

$$
\begin{aligned}
& |\ell_t^2(x) - \ell_t^1(x)| \le P_{0,t}^{\bar{b},\sigma}\{|\ell_{\nu_1}(x) - \ell_{\nu_2}|\}(x) \\
& \qquad + (1 + \|\ell_{\nu_1}\|_{\infty}) \int_0^t \int_{\mathbb{R}^d} |\ell_s^2(y) - \ell_s^1(y)| |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| \, dy \, ds \\
& \qquad + \int_0^t W_p(\mu_s^2, \mu_s^1) \int_{\mathbb{R}^d} \ell_s^1(y) |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| \, dy \, ds \\
& =: I_1(t,x) + (1 + \|\ell_{\nu_1}\|_{\infty}) I_2(t,x) + I_3(t,x).
\end{aligned}
$$

The pair $(\bar{b}, \sigma)$ satisfies Assumption 3.3. By Theorem 3.4[1], there exists a constant $\kappa > 0$ (depending only on $\Theta_1$) such that for $i \in \{0,1\}, 0 \le s < t \le T$ and $x, y \in \mathbb{R}^d$:

$$ |\nabla_y^i p_{s,t}^{\bar{b},\sigma}(y,x)| \le (t-s)^{-\frac{i}{2}} p_{t-s}^{\kappa}(y-x). \quad (5.7) $$

Then

$$
\begin{aligned}
& \int_{\mathbb{R}^d} (|x|^p + 1) |\nabla_y^i p_{s,t}^{\bar{b},\sigma}(y,x)| \, dx \\
& \le (t-s)^{-\frac{i}{2}} \int_{\mathbb{R}^d} (|x|^p + 1) p_{t-s}^{\kappa}(y-x) \, dx \quad \text{by (5.7)} \\
& \le (t-s)^{-\frac{i}{2}} (|y|^p + 1).
\end{aligned}
\qquad (5.8)
$$

We define a measurable map $f: \mathbb{T} \to \mathbb{R}_+$ by

$$ f(s) := \int_{\mathbb{R}^d} (|x|^p + 1) |\ell_s^2(x) - \ell_s^1(x)| \, dx, \quad s \in \mathbb{T}. $$

By (3.15), $f$ is bounded. First,

$$
\begin{aligned}
& \int_{\mathbb{R}^d} (|x|^p + 1) I_1(t, x) \, dx \\
& = \int_0^t \int_{\mathbb{R}^d} |\ell_{\nu_1}(y) - \ell_{\nu_2}(y)| \int_{\mathbb{R}^d} (|x|^p + 1) |p_{0,t}^{\bar{b},\sigma}(y,x)| \, dx \, dy \, ds \\
& \le \int_0^t \int_{\mathbb{R}^d} |\ell_{\nu_1}(y) - \ell_{\nu_2}(y)| (|y|^p + 1) \, dy \, ds \quad \text{by (5.8)} \\
& \le \int_{\mathbb{R}^d} (|y|^p + 1) |\ell_{\nu_1}(y) - \ell_{\nu_2}(y)| \, dy = f(0).
\end{aligned}
$$

Second,

$$
\begin{aligned}
& \int_{\mathbb{R}^d} (|x|^p + 1) I_2(t, x) \, dx \\
& = \int_0^t \int_{\mathbb{R}^d} |\ell_s^2(y) - \ell_s^1(y)| \int_{\mathbb{R}^d} (|x|^p + 1) |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| \, dx \, dy \, ds \\
& \le \int_0^t (t-s)^{-\frac{1}{2}} \int_{\mathbb{R}^d} |\ell_s^2(y) - \ell_s^1(y)| (|y|^p + 1) \, dy \, ds \quad \text{by (5.8)} \\
& = \int_0^t (t-s)^{-\frac{1}{2}} f(s) \, ds.
\end{aligned}
$$
By (5.4), (5.5) and (5.6),

$$
\begin{aligned}
& |\ell_t^2(x) - \ell_t^1(x)| \le P_{0,t}^{\bar{b},\sigma}\{|\ell_{\nu_1}(x) - \ell_{\nu_2}|\}(x) \\
& \qquad + (1 + \|\ell_{\nu_1}\|_{\infty}) \int_0^t \int_{\mathbb{R}^d} |\ell_s^2(y) - \ell_s^1(y)| |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| \, dy \, ds \\
& \qquad + \int_0^t W_p(\mu_s^2, \mu_s^1) \int_{\mathbb{R}^d} \ell_s^1(y) |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| \, dy \, ds \\
& =: I_1(t,x) + (1 + \|\ell_{\nu_1}\|_{\infty}) I_2(t,x) + I_3(t,x).
\end{aligned}
$$

The pair $(\bar{b}, \sigma)$ satisfies Assumption 3.3. By Theorem 3.4[1], there exists a constant $\kappa > 0$ (depending only on $\Theta_1$) such that for $i \in \{0,1\}, 0 \le s < t \le T$ and $x, y \in \mathbb{R}^d$:

$$ |\nabla_y^i p_{s,t}^{\bar{b},\sigma}(y,x)| \le (t-s)^{-\frac{i}{2}} p_{t-s}^{\kappa}(y-x). \quad (5.7) $$

Then

$$
\begin{aligned}
& \int_{\mathbb{R}^d} (|x|^p + 1) |\nabla_y^i p_{s,t}^{\bar{b},\sigma}(y,x)| \, dx \\
& \le (t-s)^{-\frac{i}{2}} \int_{\mathbb{R}^d} (|x|^p + 1) p_{t-s}^{\kappa}(y-x) \, dx \quad \text{by (5.7)} \\
& \le (t-s)^{-\frac{i}{2}} (|y|^p + 1).
\end{aligned}
\qquad (5.8)
$$

We define a measurable map $f: \mathbb{T} \to \mathbb{R}_+$ by

$$ f(s) := \int_{\mathbb{R}^d} (|x|^p + 1) |\ell_s^2(x) - \ell_s^1(x)| \, dx, \quad s \in \mathbb{T}. $$

By (3.15), $f$ is bounded. First,

$$
\begin{aligned}
& \int_{\mathbb{R}^d} (|x|^p + 1) I_1(t, x) \, dx \\
& = \int_0^t \int_{\mathbb{R}^d} |\ell_{\nu_1}(y) - \ell_{\nu_2}(y)| \int_{\mathbb{R}^d} (|x|^p + 1) |p_{0,t}^{\bar{b},\sigma}(y,x)| \, dx \, dy \, ds \\
& \le \int_0^t \int_{\mathbb{R}^d} |\ell_{\nu_1}(y) - \ell_{\nu_2}(y)| (|y|^p + 1) \, dy \, ds \quad \text{by (5.8)} \\
& \le \int_{\mathbb{R}^d} (|y|^p + 1) |\ell_{\nu_1}(y) - \ell_{\nu_2}(y)| \, dy = f(0).
\end{aligned}
$$

Second,

$$
\begin{aligned}
& \int_{\mathbb{R}^d} (|x|^p + 1) I_2(t, x) \, dx \\
& = \int_0^t \int_{\mathbb{R}^d} |\ell_s^2(y) - \ell_s^1(y)| \int_{\mathbb{R}^d} (|x|^p + 1) |\nabla_y p_{s,t}^{\bar{b},\sigma}(y,x)| \, dx \, dy \, ds \\
& \le \int_0^t (t-s)^{-\frac{1}{2}} \int_{\mathbb{R}^d} |\ell_s^2(y) - \ell_s^1(y)| (|y|^p + 1) \, dy \, ds \quad \text{by (5.8)} \\
& = \int_0^t (t-s)^{-\frac{1}{2}} f(s) \, ds.
\end{aligned}
$$
Third,

$$
\begin{align*}
& \int_{\mathbb{R}^d} (|x|^p + 1) I_3(t, x) \, dx \\
&= \int_0^t W_p(\mu_s^2, \mu_s^1) \int_{\mathbb{R}^d} \ell_s^1(y) \int_{\mathbb{R}^d} (|x|^p + 1) |\nabla_y p_{s,t}^{\bar{b},\sigma}(y, x)| \, dx \, dy \, ds \\
&\lesssim \int_0^t (t-s)^{-\frac{1}{2}} W_p(\mu_s^2, \mu_s^1) \int_{\mathbb{R}^d} \ell_s^1(y) (|y|^p + 1) \, dy \, ds \quad \text{by (5.8)} \\
&\lesssim (1 + M_p(\nu_1)) \int_0^t (t-s)^{-\frac{1}{2}} W_p(\mu_s^2, \mu_s^1) \, ds \quad \text{by (3.15)} \\
&\lesssim (1 + M_p(\nu_1)) \int_0^t (t-s)^{-\frac{1}{2}} |f(s)|^{\frac{1}{p}} \, ds \quad \text{by Lemma 3.2.}
\end{align*}
$$

To sum up,

$$
f(t) \asymp f(0) + (1 + \|\ell_{\nu_1}\|_{\infty} + M_1(\nu_1)) \int_0^t (T-s)^{-\frac{1}{2}} (f(s) + |f(s)|^{\frac{1}{p}}) ds.
$$

Assuming $p=1$, we get

$$
f(t) \asymp f(0) + (1 + \|\ell_{\nu_1}\|_{\infty} + M_1(\nu_1)) \int_0^t (T-s)^{-\frac{1}{2}} f(s) \, ds.
$$

By Grönwall’s lemma,

$$
\sup_{t \in \mathbb{T}} f(t) \asymp f(0) \exp \{2\sqrt{T}(1 + \|\ell_{\nu_1}\|_{\infty} + M_1(\nu_1))\}, \quad (5.9)
$$

which implies the existence of the function Λ as required in Theorem 2.3[1].

5.2 Weak and strong uniqueness of a solution

By (5.1),

$$
dX_t^k = b^k(t, X_t^k) dt + \sigma(t, X_t^k) dB_t^k.
$$

Now we let $\nu := \nu_1 = \nu_2$. By (5.9), $\ell_t^1 = \ell_t^2$ and $\mu_t^1 = \mu_t^2$ for $t \in \mathbb{T}$. Then $\mathbf{b} := b^1 = b^2$. We consider the SDE

$$
dY_t = \mathbf{b}(t, Y_t) dt + \sigma(t, Y_t) dB_t, \quad t \in \mathbb{T}, \qquad (5.10)
$$

where the distribution of $Y_0$ is $\nu$. By [10, Theorem 1.1(1)], (5.10) is well-posed. On the other hand, ($X_t^1, t \in \mathbb{T}$) and ($X_t^2, t \in \mathbb{T}$) satisfy (5.10). It follows that (1.1) has both weak and strong uniqueness.

6 Acknowledgment

The author is grateful to Professor Stéphane Villeneuve for his guidance and encouragement, and to Professor Sébastien Gadat for his generous funding during the author’s PhD. The author is grateful to Professor Feng-Yu Wang for kindly addressing all questions about his work [35]. The author would like to thank Professor Michael Röckner and Dr. Zimo Hao for their useful comments and suggestions.
Third,

$$
\begin{align*}
& \int_{\mathbb{R}^d} (|x|^p + 1) I_3(t, x) \, dx \\
&= \int_0^t W_p(\mu_s^2, \mu_s^1) \int_{\mathbb{R}^d} \ell_s^1(y) \int_{\mathbb{R}^d} (|x|^p + 1) |\nabla_y p_{s,t}^{\bar{b},\sigma}(y, x)| \, dx \, dy \, ds \\
&\lesssim \int_0^t (t-s)^{-\frac{1}{2}} W_p(\mu_s^2, \mu_s^1) \int_{\mathbb{R}^d} \ell_s^1(y) (|y|^p + 1) \, dy \, ds \quad \text{by (5.8)} \\
&\lesssim (1 + M_p(\nu_1)) \int_0^t (t-s)^{-\frac{1}{2}} W_p(\mu_s^2, \mu_s^1) \, ds \quad \text{by (3.15)} \\
&\lesssim (1 + M_p(\nu_1)) \int_0^t (t-s)^{-\frac{1}{2}} |f(s)|^{\frac{1}{p}} \, ds \quad \text{by Lemma 3.2.}
\end{align*}
$$

To sum up,

$$
f(t) \asymp f(0) + (1 + \|\ell_{\nu_1}\|_{\infty} + M_1(\nu_1)) \int_0^t (T-s)^{-\frac{1}{2}} (f(s) + |f(s)|^{\frac{1}{p}}) \, ds.
$$

Assuming $p=1$, we get

$$
f(t) \asymp f(0) + (1 + \|\ell_{\nu_1}\|_{\infty} + M_1(\nu_1)) \int_0^t (T-s)^{-\frac{1}{2}} f(s) \, ds.
$$

By Grönwall’s lemma,

$$
\sup_{t \in \mathbb{T}} f(t) \asymp f(0) \exp \{2\sqrt{T}(1 + \|\ell_{\nu_1}\|_{\infty} + M_1(\nu_1))\}, \quad (5.9)
$$

which implies the existence of the function Λ as required in Theorem 2.3[1].

5.2 Weak and strong uniqueness of a solution

By (5.1),

$$
dX_t^k = b^k(t, X_t^k) dt + \sigma(t, X_t^k) dB_t^k.
$$

Now we let $\nu := \nu_1 = \nu_2$. By (5.9), $\ell_t^1 = \ell_t^2$ and $\mu_t^1 = \mu_t^2$ for $t \in \mathbb{T}$. Then $\mathbf{b} := b^1 = b^2$. We consider the SDE

$$
dY_t = \mathbf{b}(t, Y_t) dt + \sigma(t, Y_t) dB_t, \quad t \in \mathbb{T}, \qquad (5.10)
$$

where the distribution of $Y_0$ is $\nu$. By [10, Theorem 1.1(1)], (5.10) is well-posed. On the other hand, ($X_t^1, t \in \mathbb{T}$) and ($X_t^2, t \in \mathbb{T}$) satisfy (5.10). It follows that (1.1) has both weak and strong uniqueness.

6 Acknowledgment

The author is grateful to Professor Stéphane Villeneuve for his guidance and encouragement, and to Professor Sébastien Gadat for his generous funding during the author’s PhD. The author is grateful to Professor Feng-Yu Wang for kindly addressing all questions about his work [35]. The author would like to thank Professor Michael Röckner and Dr. Zimo Hao for their useful comments and suggestions.
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[25] Viorel Barbu and Michael Röckner. Probabilistic representation for solutions to nonlinear fokker-planck equations. *SIAM Journal on Mathematical Analysis*, 50(4):4246-4260, 2018.

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[27] Viorel Barbu and Michael Röckner. Uniqueness for nonlinear fokker-planck equations and weak uniqueness for McKean-Vlasov sdes. *Stochastics and Partial Differential Equations: Analysis and Computations*, 9:702-713, 2021.

[28] Viorel Barbu and Michael Röckner. The evolution to equilibrium of solutions to nonlinear Fokker-Planck equation. *Indiana University Mathematics Journal*, 72(1):89-131, 2023.

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[18] Philippe Blanchard, Michael Röckner, and Francesco Russo. Probabilistic representation for solutions of an irregular porous media type equation. *The Annals of Probability*, 38(5), September 2010.

[19] Viorel Barbu, Michael Röckner, and Francesco Russo. Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case. *Probability Theory and Related Fields*, 151(1-2):1-43, October 2011.

[20] Nadia Belaribi and Francesco Russo. Uniqueness for Fokker-Planck equations with measurable coefficients and applications to the fast diffusion equation. *Electronic Journal of Probability*, 17(none), January 2012.

[21] Nadia Belaribi, François Cuvelier, and Francesco Russo. Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation. *Stochastic Partial Differential Equations: Analysis and Computations*, 1(1):3-62, March 2013.

[22] B Jourdain and S Meleard. Propagation of chaos and fluctuations for a moderate model with smooth initial data. *Annales de l'Institut Henri Poincare (B) Probability and Statistics*, 34(6):727-766, 1998.

[23] Benjamin Jourdain and Julien Reygner. Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation. *Stochastic Partial Differential Equations: Analysis and Computations*, 1(3):455-506, September 2013.

[24] Mireille Bossy and Jean Francois Jabir. On the wellposedness of some McKean models with moderated or singular diffusion coefficient, September 2018. arXiv:1809.01742 [math].

[25] Viorel Barbu and Michael Röckner. Probabilistic representation for solutions to nonlinear fokker-planck equations. *SIAM Journal on Mathematical Analysis*, 50(4):4246-4260, 2018.

[26] Viorel Barbu and Michael Röckner. Solutions for nonlinear fokker-planck equations with measures as initial data and McKean-Vlasov equations. *Journal of Functional Analysis*, 280(7):108926, 2021.

[27] Viorel Barbu and Michael Röckner. Uniqueness for nonlinear fokker-planck equations and weak uniqueness for McKean-Vlasov sdes. *Stochastics and Partial Differential Equations: Analysis and Computations*, 9:702-713, 2021.

[28] Viorel Barbu and Michael Röckner. The evolution to equilibrium of solutions to nonlinear Fokker-Planck equation. *Indiana University Mathematics Journal*, 72(1):89-131, 2023.

[29] Viorel Barbu and Michael Röckner. Uniqueness for nonlinear Fokker-Planck equations and for McKean-Vlasov SDEs: The degenerate case. *Journal of Functional Analysis*, 285(4):109980, August 2023.
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[33] Viorel Barbu and Michael Röckner. Nonlinear fokker-planck equations with time-dependent coefficients. *SIAM Journal on Mathematical Analysis*, 55(1):1-18, 2023.

[34] Zimo Hao, Michael Röckner, and Xicheng Zhang. Strong Convergence of Propagation of Chaos for McKean-Vlasov SDEs with Singular Interactions. *SIAM Journal on Mathematical Analysis*, 56(2):2661-2713, April 2024.

[35] Feng-Yu Wang. Singular density dependent stochastic differential equations. *Journal of Differential Equations*, 361:562-589, 2023.

[36] Lucas Izydorczyk, Nadia Oudjane, and Francesco Russo. McKean feynman-kac probabilistic representations of non-linear partial differential equations. In *International Conference on Random Transformations and Invariance in Stochastic Dynamics*, pages 187-212. Springer, 2019.

[37] Walter Rudin. *Real and complex analysis*. McGraw-Hill, New York, 3rd edition, 1987.

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[41] Filippo Santambrogio. Optimal transport for applied mathematicians. *Birkhäuser, NY*, 55(58-63):94, 2015.

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[43] Luigi Ambrosio, Elia Brué, and Daniele Semola. *Lectures on optimal transport*. Springer, 2021.

[44] Francesco Maggi. *Optimal Mass Transport on Euclidean Spaces*. Cambridge University Press, 2023.

[45] Stéphane Menozzi, Antonello Pesce, and Xicheng Zhang. Density and gradient estimates for non-degenerate brownian sdes with unbounded measurable drift. *Journal of Differential Equations*, 272:330-369, 2021.

[46] Seiichiro Kusuoka. Continuity and gaussian two-sided bounds of the density functions of the solutions to path-dependent stochastic differential equations via perturbation. *Stochastic Processes and their Applications*, 127(2):359-384, 2017.
[30] Marco Rehmeier. Weighted $L^1$-semigroup approach for nonlinear Fokker-Planck equations and generalized Ornstein-Uhlenbeck processes, August 2023. arXiv:2308.09420 [math].

[31] Sebastian Grube. Strong solutions to McKean-Vlasov SDEs with coefficients of Nemytskii-type. *Electronic Communications in Probability*, 28(none), January 2023.

[32] Sebastian Grube. Strong solutions to McKean-Vlasov SDEs with coefficients of Nemytskii-type: the time-dependent case, August 2022. arXiv:2203.09576 [math].

[33] Viorel Barbu and Michael Röckner. Nonlinear fokker-planck equations with time-dependent coefficients. *SIAM Journal on Mathematical Analysis*, 55(1):1-18, 2023.

[34] Zimo Hao, Michael Röckner, and Xicheng Zhang. Strong Convergence of Propagation of Chaos for McKean-Vlasov SDEs with Singular Interactions. *SIAM Journal on Mathematical Analysis*, 56(2):2661-2713, April 2024.

[35] Feng-Yu Wang. Singular density dependent stochastic differential equations. *Journal of Differential Equations*, 361:562-589, 2023.

[36] Lucas Izydorczyk, Nadia Oudjane, and Francesco Russo. McKean feynman-kac probabilistic representations of non-linear partial differential equations. In *International Conference on Random Transformations and Invariance in Stochastic Dynamics*, pages 187-212. Springer, 2019.

[37] Walter Rudin. *Real and complex analysis*. McGraw-Hill, New York, 3rd edition, 1987.

[38] Cédric Villani. *Optimal transport: old and new*, volume 338. Springer, 2009.

[39] Cédric Villani. *Topics in Optimal Transportation*. Graduate studies in mathematics. American Mathematical Society, 2003.

[40] Luigi Ambrosio and Nicola Gigli. *A User's Guide to Optimal Transport*, pages 1-155. Springer Berlin Heidelberg, Berlin, Heidelberg, 2013.

[41] Filippo Santambrogio. Optimal transport for applied mathematicians. *Birkhäuser, NY*, 55(58-63):94, 2015.

[42] Alessio Figalli and Federico Glaudo. *An invitation to optimal transport, Wasserstein distances, and gradient flows*. EMS Textbooks in Mathematics. EMS Press, Berlin, 2021.

[43] Luigi Ambrosio, Elia Brué, and Daniele Semola. *Lectures on optimal transport*. Springer, 2021.

[44] Francesco Maggi. *Optimal Mass Transport on Euclidean Spaces*. Cambridge University Press, 2023.

[45] Stéphane Menozzi, Antonello Pesce, and Xicheng Zhang. Density and gradient estimates for non-degenerate brownian sdes with unbounded measurable drift. *Journal of Differential Equations*, 272:330-369, 2021.

[46] Seiichiro Kusuoka. Continuity and gaussian two-sided bounds of the density functions of the solutions to path-dependent stochastic differential equations via perturbation. *Stochastic Processes and their Applications*, 127(2):359-384, 2017.
[47] Shao-Qin Zhang and Chenggui Yuan. A zvonkin's transformation for stochastic differential equations with singular drift and applications. *Journal of Differential Equations*, 297:277-319, 2021.

[48] Longjie Xie and Xicheng Zhang. Ergodicity of stochastic differential equations with jumps and singular coefficients. *Annales de l'Institut Henri Poincaré, Probabilités et Statistiques*, 56(1):175-229, 2020.

[49] Pengcheng Xia, Longjie Xie, Xicheng Zhang, and Guohuan Zhao. Lq (lp)-theory of stochastic differential equations. *Stochastic Processes and their Applications*, 130(8):5188-5211, 2020.

[50] Paolo Baldi. *Stochastic Calculus: An Introduction Through Theory and Exercises*. Universitext. Springer International Publishing, 1st edition, 2017.

[51] René L. Schilling. *Brownian Motion: A Guide to Random Processes and Stochastic Calculus*. De Gruyter, September 2021.

[52] Nawaf Bou-Rabee, Aleksandar Donev, and Eric Vanden-Eijnden. Metropolis integration schemes for self-adjoint diffusions. *Multiscale modeling & simulation*, 12(2):781-831, 2014.

[53] Zimo Hao, Michael Röckner, and Xicheng Zhang. Euler scheme for density dependent stochastic differential equations. *Journal of Differential Equations*, 274:996-1014, 2021.

[54] Haim Brezis. *Functional Analysis, Sobolev Spaces and Partial Differential Equations*. Springer New York, New York, NY, 2011.

[55] Alessio Figalli. Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. *Journal of Functional Analysis*, 254(1):109-153, January 2008.

[56] Dario Trevisan. Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients. *Electronic Journal of Probability*, 21(none), January 2016.

[57] Vladimir I. Bogachev, Michael Röckner, and Stanislav V. Shaposhnikov. On the Ambrosio-Figalli-Trevisan Superposition Principle for Probability Solutions to Fokker-Planck-Kolmogorov Equations. *Journal of Dynamics and Differential Equations*, 33(2):715-739, June 2021.
[47] Shao-Qin Zhang and Chenggui Yuan. A zvonkin's transformation for stochastic differential equations with singular drift and applications. *Journal of Differential Equations*, 297:277-319, 2021.

[48] Longjie Xie and Xicheng Zhang. Ergodicity of stochastic differential equations with jumps and singular coefficients. *Annales de l'Institut Henri Poincaré, Probabilités et Statistiques*, 56(1):175-229, 2020.

[49] Pengcheng Xia, Longjie Xie, Xicheng Zhang, and Guohuan Zhao. Lq (lp)-theory of stochastic differential equations. *Stochastic Processes and their Applications*, 130(8):5188-5211, 2020.

[50] Paolo Baldi. *Stochastic Calculus: An Introduction Through Theory and Exercises*. Universitext. Springer International Publishing, 1st edition, 2017.

[51] René L. Schilling. *Brownian Motion: A Guide to Random Processes and Stochastic Calculus*. De Gruyter, September 2021.

[52] Nawaf Bou-Rabee, Aleksandar Donev, and Eric Vanden-Eijnden. Metropolis integration schemes for self-adjoint diffusions. *Multiscale modeling & simulation*, 12(2):781-831, 2014.

[53] Zimo Hao, Michael Röckner, and Xicheng Zhang. Euler scheme for density dependent stochastic differential equations. *Journal of Differential Equations*, 274:996-1014, 2021.

[54] Haim Brezis. *Functional Analysis, Sobolev Spaces and Partial Differential Equations*. Springer New York, New York, NY, 2011.

[55] Alessio Figalli. Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. *Journal of Functional Analysis*, 254(1):109-153, January 2008.

[56] Dario Trevisan. Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients. *Electronic Journal of Probability*, 21(none), January 2016.

[57] Vladimir I. Bogachev, Michael Röckner, and Stanislav V. Shaposhnikov. On the Ambrosio-Figalli-Trevisan Superposition Principle for Probability Solutions to Fokker-Planck-Kolmogorov Equations. *Journal of Dynamics and Differential Equations*, 33(2):715-739, June 2021.