\label{Prop1}
For every  torse-forming vector field $V$ that satisfies (\ref{TFVF}),  we have
\begin{equation}\label{Theta}
		\theta ={\rm d}\rho - f {\rm e}^{-2 \rho} \omega.
	\end{equation}

It is well known that generalizing mathematical concepts is a prerequisite for most researchers. It is also often an important means of developing mathematics and answering many questions that have occupied the minds of researchers in many fields. For example, the notion of a Ricci soliton is a natural generalization of an Einstein metric; at the same time, it is a stationary solution of a famous PDE for Riemannian metrics, known under the name of the Ricci flow equation. Without forgetting that Ricci soliton itself has had many generalizations made recently in many works.

Generally, a mathematical concept is initially defined by a simple expression that satisfies the need to include it. Then questions begin: What if we replace the constant with a function? What if we add a differential one-form? What happens if we add a certain term? ... And so, little by little, the simple expression begins to take on longer and more complex forms.

In principle, if these generalizations fail to answer any outstanding mathematical problems or fail to impact other concepts, then they can be considered an intellectual luxury.

In this paper we present a step in the reverse direction of generalization; as a subject of study, we will discuss the concept of the torse-forming vector field as a generalization of both the  torqued vector field, the concircular vector field, the concurrent vector field, the recurrent vector field, and the parallel vector field.

 Torse-forming vector fields  demand special attention due to their applications not only in relativity and cosmology but in theory of submanifolds also. On a Riemannian manifold $(M^n, g)$, a the torse-forming vector field is a vector field $V$ that satisfies
\begin{equation}\label{TFVF}
		\nabla_X V = fX  + \theta(X)V,
	\end{equation}
for any vector field $X$ on $M$ where $\nabla$ is the Levi-Civita connection of $g$, $\theta$ is a one-form and $f$ is a smooth function on $(M^n, g)$. The 1-form $\theta$ is called the generating form and the function $f$ is called the conformal scalar.  If $\omega$ is the corresponding  $1$-form of $V$ , i.e., $\omega(X) = g(X,V)$ then for all $X, Y$ vector fields on $M$
\begin{equation}\label{TFVF2}
 (\nabla_X \omega)Y = g(\nabla_X V , Y)= fg(X,Y)+ \theta(X)\omega(Y).
	\end{equation}
 Some special types of torse-forming vector fields have been considered in various studies. A torse-forming vector field $V$ is called:
\begin{itemize}
\item[1)] proper torse-forming if $f \neq 0$ and the 1-form $\theta$ is nowhere zero on a dense open subset of $M$.
\item[2)] torqued vector field if $V$ satisfying (\ref{TFVF}) with $\theta(V)=0$ (see \cite{CBY2, CBY3}).
\item[3)] concircular vector field  if $\theta$ is identically zero \cite{YK2} .
\item[4)] concurrent vector field if $\theta = 0$ and $f=1$.
\item[5)] reccurent vector field if $\theta \neq 0$ and $f=0$.
\item[6)] geodesic vector field if $\nabla_V V=0$.
\item[7)] parallel vector field if $\theta = 0$ and $f=0$.
\end{itemize}

\textbf{Agreement:} Through the rest of this paper, $(M,g)$ always denotes a Riemannian manifold.  $V$ denotes a proper torse-forming vector field on $M$ that satisfies (\ref{TFVF}) with $\Vert V \Vert = {\rm e}^{\rho}$, and  $\omega $  the  $1$-form corresponding to $V$, i.e.,  for all $X$ vector field on $M$,
$$ \omega(X)=g(V , X)\qquad and \qquad  \omega(V)={\rm e}^{2\rho}.$$

It is well known that generalizing mathematical concepts is a prerequisite for most researchers. It is also often an important means of developing mathematics and answering many questions that have occupied the minds of researchers in many fields. For example, the notion of a Ricci soliton is a natural generalization of an Einstein metric; at the same time, it is a stationary solution of a famous PDE for Riemannian metrics, known under the name of the Ricci flow equation. Without forgetting that Ricci soliton itself has had many generalizations made recently in many works.

Generally, a mathematical concept is initially defined by a simple expression that satisfies the need to include it. Then questions begin: What if we replace the constant with a function? What if we add a differential one-form? What happens if we add a certain term? ... And so, little by little, the simple expression begins to take on longer and more complex forms.

In this paper we present a step in the reverse direction of generalization; as a subject of study, we will discuss the concept of the torse-forming vector field as a generalization of both the torqued vector field, the concircular vector field, the concurrent vector field, the recurrent vector field, and the parallel vector field.

Torse-forming vector fields demand special attention due to their applications not only in relativity and cosmology but in theory of submanifolds also. On a Riemannian manifold $(M^n, g)$, a the torse-forming vector field is a vector field $V$ that satisfies
\begin{equation}\label{TFVF}
 \nabla_X V = fX + \theta(X)V,
 \end{equation}
for any vector field $X$ on $M$ where $\nabla$ is the Levi-Civita connection of $g$, $\theta$ is a one-form and $f$ is a smooth function on $(M^n, g)$. The 1-form $\theta$ is called the generating form and the function $f$ is called the conformal scalar. If $\omega$ is the corresponding $1$-form of $V$ , i.e., $\omega(X) = g(X,V)$ then for all $X, Y$ vector fields on $M$
\begin{equation}\label{TFVF2}
 (\nabla_X \omega)Y = g(\nabla_X V , Y)= fg(X,Y)+ \theta(X)\omega(Y).
 \end{equation}
 Some special types of torse-forming vector fields have been considered in various studies. A torse-forming vector field $V$ is called:
\begin{itemize}
\item[1)] proper torse-forming if $f \neq 0$ and the 1-form $\theta$ is nowhere zero on a dense open subset of $M$.
\item[2)] torqued vector field if $V$ satisfying (\ref{TFVF}) with $\theta(V)=0$ (see \cite{CBY2, CBY3}).
\item[3)] concircular vector field if $\theta$ is identically zero \cite{YK2} .
\item[4)] concurrent vector field if $\theta = 0$ and $f=1$.
\item[5)] reccurent vector field if $\theta \neq 0$ and $f=0$.
\item[6)] geodesic vector field if $\nabla_V V=0$.
\item[7)] parallel vector field if $\theta = 0$ and $f=0$.
\end{itemize}

\textbf{Agreement:} Through the rest of this paper, $(M,g)$ always denotes a Riemannian manifold. $V$ denotes a proper torse-forming vector field on $M$ that satisfies (\ref{TFVF}) with $\Vert V \Vert = {\rm e}^{\rho}$, and $\omega $ the $1$-form corresponding to $V$, i.e., for all $X$ vector field on $M$,
$$ \omega(X)=g(V , X)\qquad and \qquad \omega(V)={\rm e}^{2\rho}.$$

\begin{equation}\label{TFVF}
		\nabla_X V = fX  + \theta(X)V,
	\end{equation}

For a conformal transformation, it is well known that the Levi-Civita connection $\nabla$ and $\tilde{\nabla}$ associated with the metrics $g$ and $\tilde{g}$ respectively are connected by
\begin{eqnarray}\label{TransfConf}
\tilde{\nabla}_{X}Y= \nabla_{X}Y + X(\sigma)Y + Y(\sigma)X - g(X,Y){\rm grad}\sigma.
 \end{eqnarray}
By using (\ref{TFVF}) and Proposition \ref{Prop1}, we have
\begin{eqnarray}\label{TildNabV}
\tilde{\nabla}_{X}V &=& \nabla_{X}V + X(\sigma)V + V(\sigma)X - g(X,V){\rm grad}\sigma \notag\\
&=& f X + \big( X(\rho) - f {\rm e}^{-2\rho} \omega(X) \big)V + X(\sigma)V + V(\sigma)X - \omega(X){\rm grad}\sigma \notag\\
&=& \big(f + V(\sigma) \big) X + \big( X(\rho) + X(\sigma) \big)V - \omega(X) \big( f {\rm e}^{-2\rho} V +{\rm grad}\sigma \big).
\end{eqnarray}
It is clear that $V$ is a torse-formming vector field on $(M,\tilde{g})$ if and only if
$${\rm grad}\sigma = -f {\rm e}^{-2\rho} V.$$
Also, under this condition, we get $ f + V(\sigma)=0$ and (\ref{TildNabV}) gives
$$\tilde{\nabla}_{X}V = \theta(X)V.$$
Consequently, $V$ is recurrent with respect to $\tilde{g}$. Hence, we conclude the following:
\begin{theorem}\label{Th1}
Let $V$ be a proper torse-formming vector field on $(M,g)$ which satisfying (\ref{TFVF}). $V$ is recurrent with respect to $\tilde{g}={\rm e}^{2\sigma}g$ where $\sigma \in \mathcal{C}^{\infty}(M)$ if and only if ${\rm grad}\sigma = -f {\rm e}^{-2\rho} V$.
\end{theorem}
\begin{example}
It is convenient to have some sort of hyperspherical coordinates on the unit sphere $\mathbb{S}^3$ in analogy to the usual spherical coordinates on $\mathbb{S}^2$. One such choice is to use $(x_1, x_2,x_3)$ where
\begin{equation}\label{parametrisationS3}
 \left\{
 \begin{array}{lll}
x_1 = \cos x_1,\\
x_2 = \sin x_1 \cos x_2,\\
x_3 = \sin x_1 \sin x_2 \cos x_3,\\
x_4 = \sin x_1 \sin x_2 \sin x_3,
 \end{array}
 \right.
\end{equation}
where $x_1$ and $x_2$ run over the range $0$ to $\pi$, and $x_3$ runs over $0$ to $2 \pi$.

One can define the Riemannian metric $g$ in these coordinates as follows:
$$ g = d x_1^2 + \sin^2 x_1 ( dx_2^2 + \sin^2 x_2 \;d x_3^2).$$
where
$$\left\lbrace e_1 = \frac{\partial}{\partial x_1},\quad e_2 = \frac{1}{\sin x_1}\frac{\partial}{\partial x_2},\quad
e_3=\frac{1}{\sin x_1 \sin x_2}\frac{\partial}{\partial x_3}\right\rbrace ,$$
is an orthonormal basis for $g$. By Kozsul's formula, the covariant derivatives of the basis elements are as follows:
$$\begin{tabular}{lll}
$ \nabla_{e_1} e_1 = 0,$ & $\nabla_{e_1} e_2 =0,$ & $\nabla_{e_1} e_3 = 0$, \\
$\nabla_{e_2} e_1= \cot x_1 e_2,$ & $\nabla_{e_2} e_2= -\cot x_1 e_1,$ & $\nabla_{e_2} e_3 =0,$ \\
$\nabla_{e_3} e_1= \cot x_1 e_3,$& $\nabla_{e_3} e_2 = \frac{\cot x_2 }{\sin x_1} e_3,$ & $\nabla_{e_3} e_3 = -\cot x_1 e_1 - \frac{\cot x_2 }{\sin x_1} e_2.$ \\
\end{tabular}
$$
Easily, we can see that for $ V = h(x_3) \sin x_1 e_1$ we have
\begin{equation}\label{NabV1}
 \left\{
 \begin{array}{lll}
\nabla_{e_1} V =h(x_3) \cos x_1 e_1 ,\\
\nabla_{e_2} V = h(x_3) \cos x_1 e_2,\\
\nabla_{e_3} V =\frac{h'}{\sin x_2 } e_1 + h(x_3) \cos x_1 e_3,
 \end{array}
 \right.
\end{equation}
where $h' = \frac{\partial h}{\partial x_3}$. Then, for $\theta = \frac{h'}{h} dx_3$ we get
$$\nabla_{e_i} V = h(x_3) \cos x_1 e_i + \theta(e_i) V,$$
hence, according to (\ref{TFVF}), $V$ is a proper torse-forming vector field on $(\mathbb{S}^3, g)$ with $f =h(x_3) \cos x_1$ and $\theta = \frac{h'}{h} dx_3$.

\begin{proposition}
 Let $\nabla$ and $\hat{\nabla} $ denote the Riemannian connections of $ g$ and $\hat{g}$ respectively. Then, for all $X,Y $ vector fields on $M$, we have the relation:
\begin{eqnarray}\label{NablaHat}
\hat{\nabla}_{X}Y &=& \nabla_X Y + X(\sigma)Y + Y(\sigma)X \notag\\
&-& g(X,Y){\rm grad} \sigma - \omega(X)\omega(Y){\rm grad}(\sigma +\rho)\notag\\
&+& \frac{1}{1+{\rm e}^{2\rho}} \Big( \big(f + V(\sigma)\big) g(X,Y) +X(\rho) \omega(Y) + \omega(X)Y(\rho)\notag\\
&&\qquad\qquad +\big(V(\rho)+V(\sigma)-f {\rm e}^{-2\rho}\big)\omega(X)\omega(Y)\Big)V.
 \end{eqnarray}
\end{proposition}
\begin{proof}
Using (\ref{TransfConf}), we get
\begin{eqnarray}\label{TransfDConf}
\hat{\nabla}_{X}Y= \overline{\nabla}_{X}Y + X(\sigma)Y + Y(\sigma)X - \overline{g}(X,Y)\overline{{\rm grad}}\sigma.
 \end{eqnarray}
For all $X$ vector field on $M$, we compute
\begin{eqnarray}\label{Eq300}
\overline{g}(\overline{{\rm grad}}\sigma, X) = X(\sigma)
 &=& g({\rm grad} \sigma, X) \notag\\
&=& \overline{g}( {\rm grad} \sigma, X) - V(\sigma)\omega(X) \notag\\
&=& \overline{g}\Big( {\rm grad} \sigma -\frac{ V(\sigma)}{1+ {\rm e}^{2\rho}} V,X\Big),
\end{eqnarray}
then,
\begin{equation}\label{Eq310}
\overline{{\rm grad}}\sigma = {\rm grad} \sigma -\frac{ V(\sigma)}{1+ {\rm e}^{2\rho}} V.
\end{equation}
Now, by substituting (\ref{NablaOverline}) and (\ref{Eq310}) in (\ref{TransfDConf}) taking into account $\overline{g} = g + \omega \otimes \omega$ we get our formula.
\end{proof}
Let's replace $Y$ by $V$ in (\ref{NablaHat}) and using (\ref{TFVF}), we get
\begin{eqnarray}\label{NablaHat2}
\hat{\nabla}_{X}V &=& \big(f + V(\sigma)\big) X + \theta(X)V + X(\sigma)V \notag\\
&-& (1+{\rm e}^{2 \rho})\omega(X){\rm grad} \sigma - {\rm e}^{2 \rho}\omega(X){\rm grad}(\rho)\notag\\
&+& \frac{1}{1+{\rm e}^{2\rho}} \Big( {\rm e}^{2 \rho}X(\rho) +(1+{\rm e}^{2 \rho})\big(V(\rho)+V(\sigma)\big)\omega(X)\Big)V.
 \end{eqnarray}
 Reorganize this equation using (\ref{Theta}), we find

\begin{eqnarray}\label{Eq400}
\hat{\nabla}_{X}V &=& \big(f + V(\sigma)\big) X \notag\\
&&+ \Big( X(\sigma) + \big(V(\rho)+V(\sigma)- f {\rm e}^{-2 \rho}\big)\omega(X) + \frac{1+2{\rm e}^{2 \rho}}{1+{\rm e}^{2\rho}} X(\rho) \Big)V \notag\\
&&- (1+{\rm e}^{2 \rho})\omega(X){\rm grad} \sigma - {\rm e}^{2 \rho}\omega(X){\rm grad}(\rho).
 \end{eqnarray}
For $V$ to be a torse forming vector field with respect to $\hat{g}$, it is sufficient for it to be
\begin{equation}\label{Eq410}
 {\rm grad} \sigma= - \frac{{\rm e}^{2 \rho}}{1+{\rm e}^{2\rho}}{\rm grad}\rho.
\end{equation}
From this equation, we get
\begin{equation}\label{Eq420}
X(\sigma) = - \frac{{\rm e}^{2 \rho}}{1+{\rm e}^{2\rho}}X(\rho)\qquad and \qquad V(\sigma) = - \frac{{\rm e}^{2 \rho}}{1+{\rm e}^{2\rho}}V(\rho).
\end{equation}
By (\ref{Eq410}) and (\ref{Eq420}), (\ref{Eq400}) becomes
\begin{equation}\label{430}
\hat{\nabla}_{X}V = \Big(f - \frac{{\rm e}^{2 \rho}}{1+{\rm e}^{2\rho}} \Big) X + \Big( X(\rho) + \Big( \frac{V(\rho)}{1+{\rm e}^{2\rho}} - f {\rm e}^{-2 \rho}\Big)\omega(X) \Big)V.
 \end{equation}
Hence, $V$ is a torse-forming with respect to $\hat{g}$. Therfore,
we can extract two important cases:\\
(1):\quad For $f = \frac{{\rm e}^{2 \rho}}{1+{\rm e}^{2\rho}}$ we get
$ \hat{\nabla}_{X}V = X(\rho)V$ this means that $V$ is reccurent on $(M, \hat{g})$.\\
(2):\quad For $ X(\rho) = \Big( f {\rm e}^{-2 \rho} -\frac{V(\rho)}{1+{\rm e}^{2\rho}}\Big)\omega(X)$, by taking $X=V$ we get $ f =\frac{1+2{\rm e}^{2\rho}}{1+{\rm e}^{2\rho}}V(\rho)$ then we obtain $ \hat{\nabla}_{X}V = V(\rho)X$.
By combining the arguments of the above discussion, we obtain the following theorem:
\begin{theorem}
Let $V$ be a proper torse-forming vector field on $(M,g)$ with $V^{\flat}=\omega$ and $\omega(V)={\rm e}^{2\rho}$. If
\begin{equation}\label{Eq450}
 {\rm grad} \sigma= - \frac{{\rm e}^{2 \rho}}{1+{\rm e}^{2\rho}}{\rm grad}\rho.
\end{equation}
then $V$ is a torse-forming with respect to $\hat{g}$. Moreover,\\
\textbf{(i)}\quad if $f = \frac{{\rm e}^{2 \rho}}{1+{\rm e}^{2\rho}}$ then
 $V$ is reccurent vector field on $(M, \hat{g})$.\\
\textbf{(ii)}\quad if $ {\rm d}\rho = \Big( f {\rm e}^{-2 \rho} -\frac{V(\rho)}{1+{\rm e}^{2\rho}}\Big)\omega$ then $V$ is a concircular vector field on $(M, \hat{g})$.

\begin{proof}
Using (\ref{TFVF}), for all $X$ vector field on $M$, we have
\begin{eqnarray}\label{Eq100}
g(\nabla_X V, V) &=& f g(X,V)  + \theta(X)g(V,V) \notag\\
&=& f \omega(X) + {\rm e}^{2\rho} \theta(X).
\end{eqnarray}
On the other hand, we have
\begin{eqnarray}\label{Eq110}
g(\nabla_X V, V)= X g(V,V)-g(V , \nabla_X V)  &\Leftrightarrow &   g(\nabla_X V, V) = \frac{1}{2} X({\rm e}^{2\rho}) \notag\\
&\Leftrightarrow & g(\nabla_X V, V) = {\rm e}^{2\rho} X(\rho).
\end{eqnarray}
Then, from (\ref{Eq100}) and (\ref{Eq110}) we get
$$ \theta(X) =  X(\rho) - f {\rm e}^{-2\rho} \omega(X).$$
This is what is required to be proven,

\end{proof}

From (\ref{TFVF}), compute for any vector field $X$:
\begin{eqnarray}
 g(\nabla_X V, V) &=& f\, g(X,V)+\theta(X)g(V,V)= f\,\omega(X)+\mathrm{e}^{2\rho}\theta(X). \tag{\ref{Eq100}}
\end{eqnarray}
On the other hand, using $g(\nabla_X V,V)=\tfrac12 X(g(V,V))$ and $g(V,V)=\mathrm{e}^{2\rho}$ gives
\begin{eqnarray}
 g(\nabla_X V, V)=\frac12 X(\mathrm{e}^{2\rho})=\mathrm{e}^{2\rho}X(\rho). \tag{\ref{Eq110}}
\end{eqnarray}
Equating (\ref{Eq100}) and (\ref{Eq110}) yields
$$\theta(X)=X(\rho)-f\,\mathrm{e}^{-2\rho}\omega(X),$$
i.e. $\theta=\mathrm{d}\rho-f\,\mathrm{e}^{-2\rho}\omega$, which is (\ref{Theta}).

From
\begin{equation}\label{TFVF}
		\nabla_X V = fX  + \theta(X)V,
	\end{equation}
compute for any vector field $X$:
\begin{eqnarray}\label{Eq100}
g(\nabla_X V, V) &=& f g(X,V)  + \theta(X)g(V,V) \\notag\\
&=& f \omega(X) + {\rm e}^{2\rho} \theta(X).
\end{eqnarray}
On the other hand, using $g(\nabla_X V,V)=\tfrac12 X(g(V,V))$ and $g(V,V)=\mathrm{e}^{2\rho}$ gives
\begin{eqnarray}\label{Eq110}
g(\nabla_X V, V)= X g(V,V)-g(V , \nabla_X V)  &\Leftrightarrow &   g(\nabla_X V, V) = \\frac{1}{2} X({\rm e}^{2\rho}) \\notag\\
&\Leftrightarrow & g(\nabla_X V, V) = {\rm e}^{2\rho} X(\rho).
\end{eqnarray}
Equating the two displayed identities yields
$$\theta(X)=X(\rho)-f\,\mathrm{e}^{-2\rho}\omega(X),$$
i.e. $\theta=\mathrm{d}\rho-f\,\mathrm{e}^{-2\rho}\omega$, which is (\ref{Theta}).

\label{Prop1}
For every  torse-forming vector field $V$ that satisfies
\begin{equation}\label{TFVF}
		\nabla_X V = fX  + \theta(X)V,
	\end{equation}
we have
\begin{equation}\label{Theta}
		\theta ={\rm d}\rho - f {\rm e}^{-2 \rho} \omega.
	\end{equation}

Question: Let \((M,g)\) be a Riemannian manifold, and let \(V\) be a torse-forming vector field, meaning that there exist a smooth function \(f\) and a 1-form \(\theta\) such that for every vector field \(X\), \(\nabla_X V = fX + \theta(X)V\). Let \(\omega = V^{\flat}\) be the metric dual 1-form of \(V\), so \(\omega(X)=g(V,X)\), and write \(\|V\|=e^{\rho}\) (equivalently, \(\omega(V)=e^{2\rho}\)). Which statement holds for every such vector field \(V\)?

The generating 1-form satisfies \(\theta = d\rho - f e^{-2\rho}\,\omega\); equivalently, for every vector field \(X\), \(\theta(X)=X(\rho)-f e^{-2\rho}\omega(X)\).

Choices:
B. The generating 1-form satisfies \(\theta = d\rho + f e^{-2\rho}\,\omega\); equivalently, for every vector field \(X\), \(\theta(X)=X(\rho)+f e^{-2\rho}\omega(X)\).
C. For every vector field \(X\) orthogonal to \(V\) (that is, \(\omega(X)=0\)), one has \(\theta(X)=X(\rho)\).
D. The generating 1-form satisfies \(\theta = d\rho - f e^{-\rho}\,\omega\); equivalently, for every vector field \(X\), \(\theta(X)=X(\rho)-f e^{-\rho}\omega(X)\).
E. There exists a smooth function \(\lambda\) on \(M\), depending only on \(\rho\), such that \(\theta = d\rho - \lambda\,\omega\); equivalently, for every vector field \(X\), \(\theta(X)=X(\rho)-\lambda\omega(X)\).

sign from equating the two expressions for g(\nabla_XV,V)

dropped the full formula by restricting to the case \omega(X)=0

power of e^\rho arising from dividing by g(V,V)=e^{2\rho}

specific dependence of the coefficient on f and e^{-2\rho}

The stem gives the definition of a torse-forming vector field and relevant notation, but it does not state or strongly telegraph the target identity. The correct formula for \(\theta\) must still be derived.

This is not a mere restatement of the definition in the stem. The correct choice is a derived consequence relating \(\theta\), \(\rho\), \(f\), and \(\omega\), and the options present genuinely competing conclusions.

The item requires a short but real derivation: compare \(X(g(V,V))\) with \(2g(\nabla_XV,V)\) and solve for \(\theta(X)\). This is more than recall, but the reasoning is fairly direct rather than deeply generative.

The distractors are plausible and mathematically targeted: sign error (B), wrong exponent (D), a weaker-but-true statement (C), and an overgeneralized existential form (E). These reflect realistic failure modes.

A strong MCQ: it avoids answer leakage, is not tautological, and uses well-designed distractors. The reasoning required is meaningful but relatively short, so the generative demand is moderate rather than high.


\begin{cases}\label{pde}
\partial_t u = \Delta(e^{-v}u+ u), \qquad &x\in\R^2,~ t>0,\\
-\Delta v + v =\chi u, \qquad &x\in\R^2,~ t>0,\\
u(x,0) = u_0(x), \qquad &x\in\R^2,
\end{cases}

\begin{theorem}\label{propagation of chaos}
The assumptions of Lemma \ref{weak solution} still hold. Let $\{X_{N,i}^\varepsilon\}_{1\le i\le N}$ and $\{\overline{X}_i^\varepsilon\}_{1\le i\le N}$ be the solutions to systems \eqref{sde} and \eqref{mfs}, respectively. Then
for any $m,N\in \mathbb{N}^+$ and parameters $0<\theta<\frac{1}{2},\quad 0<\alpha<\frac{\theta}{2},\quad m>\frac{\theta+1}{1-2\theta}$, there exist a constant $C(m,T)>0$ and a parameter $\gamma,\eta>0$ satisfying the bounds
\begin{align}
&0<\gamma<\min\big\{\frac{\alpha}{3},\frac{-2\alpha +m(1-2\theta)-1}{4m+4}\big\},\label{gamma} \\
&0<\eta\le\min\{\theta-2\alpha, -(4m+4)\gamma-2\alpha +m(1-2\theta)-1\}\label{eta}
\end{align}
such that for all $0\le t\le T$,
\begin{align}\label{Pe}
\mathbb{P}\big(\max_{i=1,\cdots,N}\big|(X_{N,i}^\varepsilon - \overline{X}_i^\varepsilon)(t)\big|> N^{-\alpha}\big)\le C(m,T)N^{-\eta},
\end{align}
where the cut-off parameter satisfies
$\varepsilon\sim N^{-\gamma} $.
\end{theorem}

\begin{align}\label{Ps111}
\|u_{N,r}^{\varepsilon}(t) - u^{\varepsilon\otimes r}(t)\|_{L^\infty(0,T^*;L^1(\R^{2r}))} \le C(r,m,T)\varepsilon^{\beta},
  \end{align}

\begin{cases}\label{rpde}
\partial_t u^\varepsilon = \Delta(e^{-v^\varepsilon}u^\varepsilon+ u^\varepsilon),\qquad &x\in\R^2,~ t>0,\\
-\Delta v^\varepsilon + v^\varepsilon =\chi u^\varepsilon * j^\varepsilon,\qquad &x\in\R^2,~ t>0,\\
u^\varepsilon(x,0) = u_0*j^\varepsilon(x), \qquad &x\in\R^2.
\end{cases}

In this work, we provide a rigorous derivation of the two-dimensional Keller–Segel-type system with signal-dependent sensitivity expressed as
\begin{align}
\begin{cases}\label{pde}
\partial_t u = \Delta(e^{-v}u+ u), \qquad &x\in\mathbb{R}^2,~ t>0,\\
-\Delta v + v =\chi u, \qquad &x\in\mathbb{R}^2,~ t>0,\\
u(x,0) = u_0(x), \qquad &x\in\mathbb{R}^2,
\end{cases}
\end{align}
where $u(x,t)$ denotes cell density and $v(x,t)$ represents the concentration of the chemical signal. The coefficient $\chi>0$ quantifies the signal-dependent strength of chemotactic sensitivity.

The formulation of the model is grounded in a robust biological rationale. This model describes the formation of stripe patterns through a self-trapping mechanism. Referring
to \cite{fu2012stripe}, we know that this process, which has been extensively studied using synthetic biological experimental methods, involves Escherichia coli secreting the signaling molecule acyl-homoserine lactone (AHL). At low concentrations of AHL, the bacteria exhibit high motility, characterized by random motion driven by typical swimming and tumbling behavior with little external interference. However, as the AHL concentration increases, the behavior of the bacterial population changes significantly, eventually leading to a macroscopically static state.
The partial differential equation (PDE) \eqref{pde} is derived from the general signal-dependent Keller–Segel model proposed
\begin{align}
\begin{cases}\label{pde1}
\partial_t u = \Delta(\gamma(v)u),\\
\tau\partial_t v-\Delta v + v = u,
\end{cases}
\end{align}
where $\tau$ takes the value of $0$ or $1$. Here, $\gamma(v)$ represents the signal-dependent motility, which satisfies $\gamma'(v)\le 0$.
A substantial body of research has been devoted to the existence of solutions for the model \eqref{pde1}, particularly in cases where the signal-dependent function $\gamma(v)$ is subject to different conditions.
In 2017, under the assumptions of upper and lower bounds for both $\gamma(v)$ and its derivative $\gamma'(v)$, Tao and Winkler \cite{tao2017effects} proved the existence of global classical solutions for the two-dimensional case, while demonstrating that the system admits only weak solutions in higher dimensions. Fujie and Jiang \cite{fujie2020global} established that when $\gamma(v)$ satisfies the following conditions:
\begin{equation*}
0<\gamma(v)\in C^3[0,\infty),\quad \gamma'(v)\leq 0 \text{ on } [0,\infty),\quad \lim_{v\to\infty}\gamma(v)=0,
\end{equation*}
global classical solutions exist for arbitrary initial data. Their subsequent work \cite{fujie2021boundedness} showed that if the function $\gamma(v)$ satisfies the asymptotic condition $\lim_{v\to\infty} e^{\alpha v}\gamma(v) = +\infty$ for any $\alpha > 0$, then there exists a globally bounded classical solution.
For the case $\gamma(v) = e^{-\alpha v}$ ($\alpha > 0$), the authors in \cite{fujie2020global,fujie2021comparison,jin2020critical} derived a critical mass threshold: solutions remain uniformly bounded when the initial cell mass is below this critical value. In contrast, studies in \cite{burger2021delayed, fujie2020global,fujie2021comparison} indicate that solutions  blow up as time approaches infinity when the initial cell mass exceeds the critical mass. Nevertheless, it is impossible to blow-up in finite time, which is a difference from the Keller-Segel model.

In their seminal 2025 work on the model \eqref{pde} (with $\gamma(v)=e^{-v}+1$), the reference \cite{BOL2026113712}
rigorously established the well-posedness of solutions to the PDE \eqref{pde} and gave error estimates between \eqref{pde} and its regularized version. The discussion is already so comprehensive that we can cite it directly. They also conducted an in-depth study in which the convergences for the stochastic particle systems of \eqref{pde} inspired us to refine their methodologies.
Our focus lies particularly on the stochastic particle equations and the mean field limit corresponding to  \eqref{pde}. By introducing the concept of stopping time (achieving convergence in probability in Section 2), we obtained results with faster convergence rates than those in \cite{BOL2026113712}.
In Section 3, we establish the propagation of chaos in strong senses for stochastic differential equations (SDEs), (where we apply the relative entropy method and derive a higher convergence rate), which build a foundational bridge between the macroscopic and microscopic systems.
In the following, we propose that the SDE corresponding to \eqref{pde} for $N\in\mathbb{N}$ interacting particles $\{X_{N,i}^\varepsilon(\cdot)\}_{1\le i\le N}$ in $\mathbb{R}^2$ reads as
\begin{align}\label{sde}
\begin{cases}
dX_{N,i}^\varepsilon(t) = \Big(2\exp\Big(-\frac{1}{N}\sum_{j=1}^N\Phi^\varepsilon(X_{N,i}^\varepsilon(t) - X_{N,j}^\varepsilon(t))\Big)+ 2\Big)^{1/2}dB_i(t),\\
X_{N,i}^\varepsilon(0) =\zeta_i,\qquad 1\le i\le N,
\end{cases}
\end{align}
where $0<\varepsilon<1$. Here, we consider a filtered probability space defined by $(\Omega,\mathcal{F}, (\mathcal{F})_{t\ge0}, \mathbb{P})$ and introduce $\{B_i\}_{1\le i\le N}$, a collection of independent $\mathcal{F}_t$-Brownian motions. The initial data $\zeta_1,\zeta_2,\cdots,\zeta_N$ are assumed to be random variables independent and identically distributed (i.i.d.) with the common probability density function $u_0$. The potential $\Phi^\varepsilon$ is given by
$$\Phi^\varepsilon:=\Phi*j^\varepsilon,\quad \Phi:=\chi \tilde{\Phi},\quad j^\varepsilon(x):=\frac{1}{\varepsilon^2}j(\frac{x}{\varepsilon}),$$
where $\chi>0$ describes the strength rate of the signals in the model \eqref{pde} and $\tilde{\Phi}$ is the Yukawa potential \cite{lieb2001analysis}, which is defined for any $\mu > 0$, by
\begin{align*}
&\tilde{\Phi}(x) = \int_0^\infty(4\pi t)^{-1}\exp\Big\{-\frac{|x|^2}{4t}-\mu^2t\Big\}dt.
\end{align*}
Based on \cite[Appendix B]{li2023optimal}, the following properties are observed
\begin{align}\label{Phi}
&\|\tilde{\Phi}\|_{L^p(\mathbb{R}^2)}<\infty,~\|\nabla\tilde{\Phi}\|_{L^q(\mathbb{R}^2)}<\infty,\qquad p\in[1,\infty),~q\in[1,2).
\end{align}
Through calculation, the following results are obtained that for any constant $C$ independent of $\varepsilon$,
\begin{align}
&\|\Phi^\varepsilon\|_{W^{1,1}(\mathbb{R}^2)}\le\|\Phi\|_{W^{1,1}(\mathbb{R}^2)}\|j^\varepsilon\|_{L^1(\mathbb{R}^2)}\le C,\label{D1phi}\\
&\|\Phi^\varepsilon\|_{W^{1,\infty}(\mathbb{R}^2)}\le\|\Phi\|_{W^{1,1}(\mathbb{R}^2)}\|j^\varepsilon\|_{L^\infty(\mathbb{R}^2)}\le \frac{C}{\varepsilon^2},\label{phi}\\
&\|D^2\Phi^\varepsilon\|_{L^\infty(\mathbb{R}^2)}\le \|\nabla\Phi\|_{L^1(\mathbb{R}^2)}\|\nabla j^\varepsilon\|_{L^\infty(\mathbb{R}^2)}\le \frac{C}{\varepsilon^3}\label{D2phi}.
\end{align}
The rigorous derivation of PDEs from stochastic particle systems represents a fundamental challenge in mathematical physics and applied analysis.
A particularly influential framework for such derivations is the theory of moderately interacting particles, introduced in the pioneering works of Oelschl\"ager  \cite{oelschlager1985law, oelschlager1987fluctuation, oelschlager1989derivation},
where the Law of Large Numbers for interacting diffusions was established, together with central-limit-type fluctuation results.
This approach, characterized by a specific scaling of the interaction potential that preserves nonlocality while weakening pointwise strength, has since been extended to various biological and physical models. For example, Stevens \cite{stevens2000derivation} derived the Keller–Segel chemotaxis model from a moderately interacting system, overcoming the lack of ellipticity, a common assumption in earlier mean field limits, by introducing novel analytical techniques.

In recent years, significant progress has been made in handling singular interactions, such as Coulomb and Riesz potentials, which arise naturally in models of collective behavior. Lazarovici and Pickl \cite{lazarovici2017mean} analyzed a particle system incorporating a regularized potential and random initial conditions. Serfaty \cite{serfaty2020mean} developed the modulated energy method to treat Coulomb-type flows, providing a powerful tool for the analysis of the mean field without confinement.
Subsequent studies on related models employing diverse regularization strategies have further expanded this line of inquiry, as documented in \cite{boers2016mean,bolley2011stochastic,chen2017mean,chen2020combined}.
Currently, the relative entropy method, advanced by Jabin and Wang \cite{jabin2017mean} and further applied by Bresch et al.\cite{bresch2023mean} and Chen et al.\cite{chen2025quantitative,chen2025mean}, has enabled quantitative estimates of convergence for singular attractive kernels. These techniques have been instrumental in the establishment of various modes of convergence, including convergence in the Wasserstein distance (Carrillo et al.\cite{carrillo2019propagation}) and convergence in probability Pickl et al. \cite{lazarovici2017mean,huang2020mean,huang2017error}, strong $L^1$ convergence (Chen et al. \cite{chen2025quantitative} and \cite{olivera2020quantitative,olivera2023quantitative}). More recently, the moderate interaction framework has been instrumental in the derivation of cross-diffusion models; see, for example, \cite{carrillo2024interacting,chen2021rigorous, chen2019rigorous,li2024convergence}.

To study the relationship between the macroscopic system \eqref{pde} and its microscopic system \eqref{sde},
we need to introduce an intermediate particle system (Mean-Field equation). A key object in the mean-field limit is the empirical measure, a random probability measure defined as
$$\mu^\varepsilon_N(t)=\frac{1}{N}\sum_{i=1}^N\delta_{X_{N,i}^\varepsilon(t)},\quad t>0,$$
where $\delta$ is the Dirac delta distribution. The work \cite{olivera2021quantitative} shows that $\mu^\varepsilon_N(t)$ converges to the following PDE solution $u^\varepsilon$. For fixed $\varepsilon>0$, the particle system \eqref{sde} propagates chaos in the many-particle limit $N\to\infty$ towards the non-linear SDE system:
\begin{align}\label{mfs}
\begin{cases}
d\overline{X}_i^\varepsilon(t) = \big(2\exp\big(-\Phi^\varepsilon * u^\varepsilon( \overline{X}_i^\varepsilon,t)\big)+2\big)^{1/2} dB_i(t),\\
\overline{X}_i^\varepsilon(0) = \zeta_i,\qquad 1\le i\le N,
\end{cases}
\end{align}
where
$u^\varepsilon(\cdot,t)$ is the density function of i.i.d. random processes
$\overline{X}_1^\varepsilon(t),\cdots, \overline{X}_N^\varepsilon(t)$.
And the initial data $\{\zeta_i\}_{i=1}^N$
are subject to the same conditions as in \eqref{sde}. Using It\^{o}'s formula, the density function $u^\varepsilon$ satisfies the so-called intermediate nonlocal problem, namely,
\begin{align}
\begin{cases}\label{rpde}
\partial_t u^\varepsilon = \Delta(e^{-v^\varepsilon}u^\varepsilon+ u^\varepsilon),\qquad &x\in\mathbb{R}^2,~ t>0,\\
-\Delta v^\varepsilon + v^\varepsilon =\chi u^\varepsilon * j^\varepsilon,\qquad &x\in\mathbb{R}^2,~ t>0,\\
u^\varepsilon(x,0) = u_0*j^\varepsilon(x), \qquad &x\in\mathbb{R}^2.
\end{cases}
\end{align}

In this paper, our argument relies on the Lemma \ref{weak solution} from \cite{BOL2026113712}, which establishes the existence and uniqueness of the weak solution $u^\varepsilon$ to the above system \eqref{rpde} and provides the $L^\infty$ estimate for $\nabla\log u^\varepsilon$.
\begin{lemma}\label{weak solution}
\textup{(Existence of weak solution, \cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.
Let $u_0\ge0$ be a initial probability density that satisfies
\begin{eqnarray*}
&u_0\log u_0\in L^1(\mathbb{R}^2), \quad \nabla\log u_0\in W^{1,q}(\mathbb{R}^2)~(q>2),\\
&u_0\in L^1(\mathbb{R}^2,|x|^2dx)\cap L^p(\mathbb{R}^2)~(1\le p\le \infty).
\end{eqnarray*}
Furthermore, assume $\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\|\omega\|_{L^4(\mathbb{R}^2)}^4\le c_*\|\omega\|_{L^2(\mathbb{R}^2)}^2\|\nabla\omega\|_{L^2(\mathbb{R}^2)}^2$. Then for any $T>0$ and $t\in[0,T]$, the problem \eqref{rpde} possesses weak solutions $(u^\varepsilon, v^\varepsilon)$ in $\mathbb{R}^2\times(0,T)$, and there exists a time $T^*\in(0,T)$ such that
\begin{eqnarray}
&\|u^\varepsilon\|_{{L^2(0,T;H^1(\mathbb{R}^2))}\cap L^\infty(0,T;L^p(\mathbb{R}^2))}\le C,\label{uniformweaksolution}\\
&\||x|^2u^\varepsilon\|_{L^\infty(0, T; L^1(\mathbb{R}^2))}\le C,\quad\|\nabla\log u^\varepsilon\|_{L^\infty(0, T^*; W^{1,q}(\mathbb{R}^2))}\le C.\label{uniformweaksolution1}
\end{eqnarray}
where $C$ is a constant independent of $\varepsilon$.
\end{lemma}

Using the above lemma, we obtain our first main result: convergence in probability of the mean-field limit under algebraic scaling.

In this work, we provide a rigorous derivation of the two-dimensional Keller–Segel-type system with signal-dependent sensitivity expressed as
\begin{align}
\begin{cases}\label{pde}
\partial_t u = \Delta(e^{-v}u+ u), \qquad &x\in\mathbb{R}^2,~ t>0,\\
-\Delta v + v =\chi u, \qquad &x\in\mathbb{R}^2,~ t>0,\\
u(x,0) = u_0(x), \qquad &x\in\mathbb{R}^2,
\end{cases}
\end{align}
where $u(x,t)$ denotes cell density and $v(x,t)$ represents the concentration of the chemical signal. The coefficient $\chi>0$ quantifies the signal-dependent strength of chemotactic sensitivity.

To study the relationship between the macroscopic system \eqref{pde} and its microscopic system \eqref{sde},
we need to introduce an intermediate particle system (Mean-Field equation). A key object in the mean-field limit is the empirical measure, a random probability measure defined as
$$\mu^\varepsilon_N(t)=\frac{1}{N}\sum_{i=1}^N\delta_{X_{N,i}^\varepsilon(t)},\quad t>0,$$
where $\delta$ is the Dirac delta distribution. The work \cite{olivera2021quantitative} shows that $\mu^\varepsilon_N(t)$ converges to the following PDE solution $u^\varepsilon$. For fixed $\varepsilon>0$, the particle system \eqref{sde} propagates chaos in the many-particle limit $N\to\infty$ towards the non-linear SDE system:
\begin{align}\label{mfs}
\begin{cases}
d\overline{X}_i^\varepsilon(t) = \big(2\exp\big(-\Phi^\varepsilon * u^\varepsilon( \overline{X}_i^\varepsilon,t)\big)+2\big)^{1/2} dB_i(t),\\
\overline{X}_i^\varepsilon(0) = \zeta_i,\qquad 1\le i\le N,
\end{cases}
\end{align}
where
$u^\varepsilon(\cdot,t)$ is the density function of i.i.d. random processes
$\overline{X}_1^\varepsilon(t),\cdots, \overline{X}_N^\varepsilon(t)$.
And the initial data $\{\zeta_i\}_{i=1}^N$
are subject to the same conditions as in \eqref{sde}. Using It\^{o}'s formula, the density function $u^\varepsilon$ satisfies the so-called intermediate nonlocal problem, namely,
\begin{align}
\begin{cases}\label{rpde}
\partial_t u^\varepsilon = \Delta(e^{-v^\varepsilon}u^\varepsilon+ u^\varepsilon),\qquad &x\in\mathbb{R}^2,~ t>0,\\
-\Delta v^\varepsilon + v^\varepsilon =\chi u^\varepsilon * j^\varepsilon,\qquad &x\in\mathbb{R}^2,~ t>0,\\
u^\varepsilon(x,0) = u_0*j^\varepsilon(x), \qquad &x\in\mathbb{R}^2.
\end{cases}
\end{align}

In this paper, our argument relies on the Lemma \ref{weak solution} from \cite{BOL2026113712}, which establishes the existence and uniqueness of the weak solution $u^\varepsilon$ to the above system \eqref{rpde} and provides the $L^\infty$ estimate for $\nabla\log u^\varepsilon$.
\begin{lemma}\label{weak solution}
\textup{(Existence of weak solution, \cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.
Let $u_0\ge0$ be a initial probability density that satisfies
\begin{eqnarray*}
&u_0\log u_0\in L^1(\mathbb{R}^2), \quad \nabla\log u_0\in W^{1,q}(\mathbb{R}^2)~(q>2),\\
&u_0\in L^1(\mathbb{R}^2,|x|^2dx)\cap L^p(\mathbb{R}^2)~(1\le p\le \infty).
\end{eqnarray*}
Furthermore, assume $\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\|\omega\|_{L^4(\mathbb{R}^2)}^4\le c_*\|\omega\|_{L^2(\mathbb{R}^2)}^2\|\nabla\omega\|_{L^2(\mathbb{R}^2)}^2$. Then for any $T>0$ and $t\in[0,T]$, the problem \eqref{rpde} possesses weak solutions $(u^\varepsilon, v^\varepsilon)$ in $\mathbb{R}^2\times(0,T)$, and there exists a time $T^*\in(0,T)$ such that
\begin{eqnarray}
&\|u^\varepsilon\|_{{L^2(0,T;H^1(\mathbb{R}^2))}\cap L^\infty(0,T;L^p(\mathbb{R}^2))}\le C,\label{uniformweaksolution}\\
&\||x|^2u^\varepsilon\|_{L^\infty(0, T; L^1(\mathbb{R}^2))}\le C,\quad\|\nabla\log u^\varepsilon\|_{L^\infty(0, T^*; W^{1,q}(\mathbb{R}^2))}\le C.\label{uniformweaksolution1}
\end{eqnarray}
where $C$ is a constant independent of $\varepsilon$.
\end{lemma}

Using the above lemma, we obtain our first main result: convergence in probability of the mean-field limit under algebraic scaling.

Using the above lemma, we obtain our first main result: convergence in probability of the mean-field limit under algebraic scaling.

The proof of Theorem \ref{propagation of chaos}, given in Section 2, follows the approach of \cite{lazarovici2017mean}, adopting a superior algebraic scaling with respect to the maximum norm of the trajectory, which \cite{BOL2026113712} only achieved results under logaritheoremic scaling.
Since the diffusion coefficients in the stochastic models \eqref{sde} and \eqref{mfs} given the nonlinear dependence of the diffusion coefficients on inter-particle interactions, the use of the Burkholder-Davis-Gundy inequality becomes indispensable. The key ingredient of our proof is the introduction of stopping times, a technique that facilitates the application of the Law of Large Numbers.

Therefore, plugging \eqref{I12} and \eqref{I11} into \eqref{EI1i} gives
\begin{align}\label{I1i}
&\E\Big(N^{2\alpha }\int_0^{t\land\tau_\alpha}\max_{i=1,\cdots,N}|I_{1,i}(s)|^2 ds\Big)\nonumber\\
\le\ & C(T) (1+\varepsilon^{-6}N^{-2\alpha })\int_0^{t\land\tau_\alpha}\E(S_\alpha^2(s))ds + C(m, T)\varepsilon^{-4m-4}N^{-m+1}.
\end{align}
Substituting \eqref{I2i} and \eqref{I1i} into \eqref{ES} yields the inequality
\begin{align}\label{initial ineq}
\E(S_\alpha^2(t))
\le\ & C(T) (1+\varepsilon^{-6}N^{-2\alpha })\int_0^{t\land\tau_\alpha}\E(S_\alpha^2(s))ds \nonumber\\
&+ C(m, T)(\varepsilon^{-4m-4}N^{-m+1} +N^{(2\alpha-\theta)} + \varepsilon^{-4m-4}N^{2\alpha +m(2\theta -1)+1}).
\end{align}
We now show that the terms multiplying the constants decay as a negative power of $N$. Recall the scaling $\varepsilon\sim N^{-\gamma},\gamma>0$. To bound the error terms, we require a positive exponent
$\eta>0$ such that the following bounds hold
\begin{align*}
\varepsilon^{-6}N^{-2\alpha } = N^{6\gamma -2\alpha}\le N^{0}&\Longleftrightarrow\gamma\le\frac{\alpha}{3},\\
N^{2\alpha-\theta}\le N^{-\eta}&\Longleftrightarrow\eta\le\theta-2\alpha, \\
\varepsilon^{-4m-4}N^{-m+1} = N^{(4m+4)\gamma -m+1}\le N^{-\eta}
&\Longleftrightarrow\eta\le-(4m+4)\gamma+m-1,\\
\varepsilon^{-4m-4}N^{2\alpha +m(2\theta -1)+1}=N^{(4m+4)\gamma+2\alpha +m(2\theta -1)+1}\le N^{-\eta}&\Longleftrightarrow\eta\le-(4m+4)\gamma-2\alpha +m(1-2\theta)-1.
\end{align*}
The condition $\eta>0$ implies that $\theta$, $\alpha$, $m$, and $\gamma$ need to be selected to meet the following relationships
\begin{align*}
0<\theta<\frac{1}{2},\quad 0<\alpha<\frac{\theta}{2},\quad m>\frac{\theta+1}{1-2\theta},\quad 0<\gamma<\frac{-2\alpha+m(1-2\theta)-1}{4m+4}.
\end{align*}
Thus choose $\gamma$ satisfying
$$
0<\gamma<\min\big\{\frac{\alpha}{3},\frac{-2\alpha +m(1-2\theta)-1}{4m+4}\big\},
$$
and $\eta$ satisfying
$$
0<\eta\le\min\{\theta-2\alpha, -(4m+4)\gamma-2\alpha +m(1-2\theta)-1\}.
$$
Then the inequality \eqref{initial ineq} simplifies the estimate
\begin{align}\label{Sak}
\E(S_\alpha^2(t))
\le\ & C\int_0^{t\land\tau_\alpha}\E(S_\alpha^2(s))ds + C(m, T)N^{-\eta}.
\end{align}
Using Gr\"{o}nwall's inequality for \eqref{Sak}, we obtain for any $t\in[0,T]$
\begin{align} \label{Egronwall}
\E(S_\alpha^2(t))\le C(m,T)N^{-\eta} (1+CTe^{CT}).
\end{align}
By inserting \eqref{MK} into \eqref{Egronwall}, we derive
\begin{align}
\mathbb{P}\big(\max_{i=1,\cdots,N}\big|(X_{N,i}^\varepsilon - \overline{X}_i^\varepsilon)(t)\big|> N^{-\alpha}\big)\le C(m,T)N^{-\eta} (1+CTe^{CT}).
\end{align}
The proof of this theorem is completed.

\setcounter{theorem}{1}
\begin{theorem}\label{Propagation of chaos in the strong sense}\textup{(Propagation of chaos in the strong sense).}
Under the assumptions of Lemma \ref{weak solution}, let
$u_{N}^\varepsilon(t,x_1,\cdots,x_N)$ and $u^{\varepsilon\otimes N}(t,x_1,\cdots,x_N)$ be the solutions of systems \eqref{lpde1} and \eqref{lpde2}, respectively. For any
$r\in\mathbb{N^+}$, denote by $u_{N,r}^\varepsilon(t,x_1,\cdots,x_r)$ and $u^{\varepsilon\otimes r}(t,x_1,\cdots,x_r)$ their corresponding
$r$-th marginal distributions.
Then for parameters $0<\theta<1/2$, $0 <\alpha<\frac{\theta}{2}$, and $m\in \mathbb{N}^+$ satisfying $m \ge \frac{1+\theta}{1-2\theta}$, there exist a time $T^*\in(0,T)$, a constant $C(m,r,T)>0$, and a parameter $\beta>0$ satisfying the bound $1<\beta\le\min\{\frac{2\alpha}{\gamma}-6,\frac{\eta}{\gamma}-4\}$ such that
\begin{align}\label{Ps}
\|u_{N,r}^{\varepsilon}(t) - u^{\varepsilon\otimes r}(t)\|_{L^\infty(0,T^*;L^1(\R^{2r}))} \le C(r,m,T)\varepsilon^{\beta},
 \end{align}
where the cut-off parameter satisfies
$\varepsilon\sim N^{-\gamma} $,
and the parameter $\gamma$ defined by
$$0<\gamma<\min\{\frac{2\alpha}{7},\frac{-2\alpha + m(1-2\theta)-1}{4m+4}\},$$
and the parameter $\eta$ satisfies
$$5\gamma<\eta<\min\{\theta-2\alpha,-(4m+4)\gamma-2\alpha +m(1-2\theta)-1\}.$$
\end{theorem}

Plugging \eqref{J1} and \eqref{J2} into \eqref{K21} produces the following
\begin{align}
K_2\leq &-\int_{\R^{2N}}\frac{1}{N}\sum_{i=1}^Nu_N^\varepsilon\Big[\exp\big(-\frac{1}{N}\sum_{j=1}^N\Phi^\varepsilon(x_i - x_j)\big) \Big|\nabla_{x_i}\log\frac{u_N^\varepsilon}{u^{\varepsilon\otimes N}}\Big|^2\Big]d_{x_1}
\cdots d_{x_N}\nonumber\\
&+ C(m, T)(N^{-2\alpha}\varepsilon^{-6} + (N^{-1} + N^{-\eta})\varepsilon^{-4}).\label{K_2}
\end{align}
Combining \eqref{K1} and \eqref{H1}, we derive
\begin{align}
\frac{d}{dt}\mathcal{H}(u_N^\varepsilon|u^{\varepsilon\otimes N})
\leq& -\frac{1}{2}\int_{\R^{2N}}\frac{1}{N}\sum_{i=1}^Nu_N^\varepsilon\Big|\nabla_{x_i}\log\frac{u_N^\varepsilon}{u^{\varepsilon\otimes N}}\Big|^2d_{x_1}\cdots d_{x_N}\nonumber\\
&-\frac{1}{2}\int_{\R^{2N}}\frac{1}{N}\sum_{i=1}^Nu_N^\varepsilon\Big[\exp\big(-\frac{1}{N}\sum_{j=1}^N\Phi^\varepsilon(x_i - x_j)\big) \Big|\nabla_{x_i}\log\frac{u_N^\varepsilon}{u^{\varepsilon\otimes N}}\Big|^2\Big]d_{x_1}
\cdots d_{x_N}\nonumber\\
& + C(m, T)(N^{-2\alpha}\varepsilon^{-6} + (N^{-1} + N^{-\eta})\varepsilon^{-4}).\label{entropy}
\end{align}
Noticing that the initial relative entropy is zero, it can be obtained from (\ref{entropy}) that for any $0<t<T^*<T$,
$$
\mathcal{H}(u_{N}^{\varepsilon}|u^{\varepsilon\otimes N})(t)\le C(m, T)(N^{-2\alpha}\varepsilon^{-6} + (N^{-1} + N^{-\eta})\varepsilon^{-4}).
$$
Combing \eqref{relative entropy} and the C-K-P inequality \eqref{CKP}, we have for $r\in\mathbb{N}$,
\begin{align}
\sup_{t\in(0,T^*)}\|u_{N,r}^{\varepsilon}(t) - u^{\varepsilon\otimes r}(t)\|_{L^1(\R^{2r})}
&\le \sup_{t\in(0,T^*)}2\mathcal{H}_r(u_{N,r}^{\varepsilon}|u^{\varepsilon\otimes r})(t)\nonumber\\
&\le \sup_{t\in(0,T^*)}4r\mathcal{H}(u_{N}^{\varepsilon}|u^{\varepsilon\otimes N})(t)\nonumber\\
&\le C(m, T)(N^{-2\alpha}\varepsilon^{-6} + (N^{-1} + N^{-\eta})\varepsilon^{-4}).
\end{align}
According to Theorem \ref{propagation of chaos}, we know that
\begin{align}
&0<\eta\le\min\{\theta-2\alpha,-(4m+4)\gamma-2\alpha +m(1-2\theta)-1\},\label{eta1}\\
&0<\gamma\le\min\{\frac{\alpha}{3},\frac{-2\alpha + m(1-2\theta)-1}{4m+4}\}. \label{gamma1}
\end{align}
Based on the previous assumption $\varepsilon\sim N^{-\gamma}$, it holds that there exists $\beta>0$ such that
\begin{align}
N^{-2\alpha}\varepsilon^{-6} + (N^{-1} + N^{- \eta})\varepsilon^{-4}=\varepsilon^\frac{2\alpha}{\gamma}\varepsilon^{-6}+(\varepsilon^\frac{1}{\gamma}+\varepsilon^\frac{\eta}{\gamma})\varepsilon^{-4}\le \varepsilon^\beta,
\end{align}
where $\beta$ satisfies
$$1<\beta\le\min\{\frac{2\alpha}{\gamma}-6,\frac{\eta}{\gamma}-4\}.$$
Here we have used $\eta<1$.
This further requires that $0<\gamma<\frac{2\alpha}{7}$, $\eta>5\gamma$. Building on the assumptions in \eqref{eta1} and \eqref{gamma1}, we assume
\begin{align*}
&0<\gamma<\min\{\frac{2\alpha}{7},\frac{-2\alpha + m(1-2\theta)-1}{4m+4}\},\\
&5\gamma<\eta<\min\{\theta-2\alpha,-(4m+4)\gamma-2\alpha +m(1-2\theta)-1\}.
\end{align*}
Then we obtain that for $0<t<T^*<T$,
\begin{align}
\sup_{t\in(0,T^*)}\|u_{N,r}^{\varepsilon}(t) - u^{\varepsilon\otimes r}(t)\|_{L^1(\R^{2r})} \le C(m,T)\varepsilon^\beta,\quad \beta>1.
\end{align}
Thus, we complete the proof of propagation of chaos.
\end{proof}

\begin{remark}
To see the value of $\gamma$ and $\eta$ more clearly, we provide a special case. When $m\in\mathbb{N}^+$, $ m\ge\frac{1+\theta}{1-2\theta}$ and $0<\alpha<\frac{\theta}{2}<\frac{1}{2}$, we take $\alpha=0.1$, $\theta=0.3$ and $m=4$ such that
$0<\gamma<\min\{\frac{\alpha}{3},\frac{-2\alpha +m(1-2\theta)-1}{2(2+2m)}\}=\min\{0.033,0.02\}=0.02$. we may assume that $\gamma=0.019$, then $0<\eta\le\min\{\theta-2\alpha,-2\gamma(2+2m)-2\alpha -m(2\theta -1)-1\}=\min\{0.1, 0.02\}=0.02$.
\end{remark}

The  proof of Theorem \ref{propagation of chaos}, given in Section 2, follows the approach of \cite{lazarovici2017mean}, adopting a superior algebraic scaling with respect to the maximum norm of the trajectory, which \cite{BOL2026113712} only achieved results under logaritheoremic scaling.
Since the diffusion coefficients in the stochastic models \eqref{sde} and \eqref{mfs} given the nonlinear dependence of the diffusion coefficients on inter-particle interactions, the use of the Burkholder-Davis-Gundy inequality becomes indispensable. The key ingredient of our proof is the introduction of stopping times, a technique that facilitates the application of the Law of Large Numbers.

Based on the uniform $L^\infty(0,T^*;H^2(\mathbb{R}^2))$ bound for $\nabla\log u_\varepsilon$ in Lemma \ref{weak solution} and the propagation of chaos established in Theorem \ref{propagation of chaos}, our next main result addresses the strong $L^1$ convergence for the propagation of chaos. The proof is mainly presented in Section 3, employing the relative entropy method \cite{jabin2018quantitative} as recently refined in \cite{chen2025quantitative}.

\begin{theorem}\label{Propagation of chaos in the strong sense}\textup{(Propagation of chaos in the strong sense).}
Under the assumptions of Theorem \ref{propagation of chaos}, let $r\in \mathbb{N^+}$, $u_{N,r}^\varepsilon(t, x_1,\cdots, x_r)$ be the $r$-th marginal density of the joint density $u_N^\varepsilon(t, x_1,\cdots, x_N)$ of $\{X_{N,i}^\varepsilon\}_{1\le i\le N}$, and $u^{\varepsilon \otimes r}(t, x_1,\cdots, x_r)$ be the tensor product of the solutions $u^{\varepsilon}$ to the model \eqref{rpde}. Then for parameters $0<\theta<1/2$, $0 <\alpha<\frac{\theta}{2}$, and $m\in \mathbb{N}^+$ satisfying $m \ge \frac{1+\theta}{1-2\theta}$, there exist a time $T^*\in(0,T)$, a constant $C(m,r,T)>0$, and a parameter $\beta>0$ satisfying the bound $1<\beta\le\min\{\frac{2\alpha}{\gamma}-6,\frac{\eta}{\gamma}-4\}$ such that
\begin{align}\label{Ps111}
\|u_{N,r}^{\varepsilon}(t) - u^{\varepsilon\otimes r}(t)\|_{L^\infty(0,T^*;L^1(\mathbb{R}^{2r}))} \le C(r,m,T)\varepsilon^{\beta},
  \end{align}
where the cut-off parameter satisfies
$\varepsilon\sim N^{-\gamma} $,
and the parameter $\gamma$ defined by
$$0<\gamma<\min\{\frac{2\alpha}{7},\frac{-2\alpha + m(1-2\theta)-1}{4m+4}\},$$
and the parameter $\eta$ satisfies
$$5\gamma<\eta<\min\{\theta-2\alpha,-(4m+4)\gamma-2\alpha +m(1-2\theta)-1\}.$$
\end{theorem}
\begin{remark}
To illustrate the feasible range of parameters, we give an explicit example. Choose $\theta = 0.4,\ \alpha = 0.1$ and $m\ge \frac{1+\theta}{1-2\theta}=7$ such that $m=7$.
Then we obtain
$$0<\gamma<\min\{\frac{2\alpha}{7},\frac{-2\alpha + m(1-2\theta)-1}{4m+4}\}=\min\{0.029,0.006\}=0.006.$$
Taking $\gamma=0.0025$ gives
$$0.0125=5\gamma<\eta<\min\{\theta-2\alpha,-(4m+4)\gamma-2\alpha +m(1-2\theta)-1\}=\min\{0.2,0.12\}=0.12.$$
so we pick $\eta=0.03$. Finally, $$1<\beta\le\min\{\frac{2\alpha}{\gamma}-6,\frac{\eta}{\gamma}-4\}=\min\{74,8\}=8$$
we take $\beta = 8$.
With these choices, the propagation of chaos estimate becomes
\begin{align*}
\max_{i=1,\cdots,N}\mathbb{E}\big(\sup_{t\in[0,T]}\big|(X_{N,i}^\varepsilon - \overline{X}_i^\varepsilon)(t)\big|^2\big)\le C\varepsilon^8,
\end{align*}
\end{remark}
We comment here that the convergence of the mean-field limit have already been discussed in \cite{BOL2026113712}. It is therefore especially gratifying that our results accelerate the convergence rates  for the solutions to the SDE \eqref{sde} and \eqref{mfs} across different norms. Specifically, Chen et al. in \cite[Theorem 3]{BOL2026113712} presented as
\begin{align*}
\max_{i=1,\cdots,N}\mathbb{E}\big(\sup_{t\in[0,T]}\big|(X_{N,i}^\varepsilon - \overline{X}_i^\varepsilon)(t)\big|^2\big)\le C\varepsilon^2,
\end{align*}
where the cut-off parameter $\varepsilon=(\lambda\log N)^{-\frac{1}{4}}$. However, in \eqref{Pe}, we prove algebraic convergence in contrast to logarithmic convergence established in \cite{BOL2026113712}. This represents an  improvement in the convergence rate. Furthermore, in \cite[Theorem 5]{BOL2026113712}, they obtained
\begin{align*}
\|u_{N,r}^{\varepsilon}(t) - u^{\varepsilon\otimes r}(t)\|_{L^\infty(0,T^*;L^1(\mathbb{R}^{2r}))}^2 \le C(r,m,T)\varepsilon.
\end{align*}
By contrast, our estimate in \eqref{Ps111} yields a  faster convergence rate than that bound.

The article is organized as follows. In section 2, we establish the propagation of chaos, which corresponds to the convergence in probability of solutions to the stochastic differential equations \eqref{sde} and \eqref{mfs}. Section 3 further derive quantitative propagation of chaos result in the strong sense by applying the relative entropy method.

The proof of Theorem~\ref{propagation of chaos} (given in Section 2) “follows the approach of \cite{lazarovici2017mean},” and “adopt[s] a superior algebraic scaling with respect to the maximum norm of the trajectory” (contrasting with \cite{BOL2026113712}, which “only achieved results under logarithmic scaling”). Because “the diffusion coefficients … [have] the nonlinear dependence … on inter-particle interactions,” the argument says “the use of the Burkholder-Davis-Gundy inequality becomes indispensable.” The “key ingredient” is “the introduction of stopping times,” which “facilitates the application of the Law of Large Numbers.”

The proof (given later) “follows the approach of Lazarovici and Pickl, \emph{A mean field limit for the Vlasov--Poisson system} (2017),” and “adopt[s] a superior algebraic scaling with respect to the maximum norm of the trajectory” (contrasting with \cite{BOL2026113712}, which “only achieved results under logarithmic scaling”). Because “the diffusion coefficients … [have] the nonlinear dependence … on inter-particle interactions,” the argument says “the use of the Burkholder-Davis-Gundy inequality becomes indispensable.” The “key ingredient” is “the introduction of stopping times,” which “facilitates the application of the Law of Large Numbers.”

\label{propagation of chaos}
The assumptions of the following lemma still hold.
\begin{lemma}\label{weak solution}
\textup{(Existence of weak solution, \cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.
Let $u_0\ge0$ be a initial probability density that satisfies
\begin{eqnarray*}
&u_0\log u_0\in L^1(\R^2), \quad \nabla\log u_0\in W^{1,q}(\R^2)~(q>2),\\
&u_0\in L^1(\R^2,|x|^2dx)\cap L^p(\R^2)~(1\le p\le \infty).
\end{eqnarray*}
Furthermore, assume $\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\|\omega\|_{L^4(\R^2)}^4\le c_*\|\omega\|_{L^2(\R^2)}^2\|\nabla\omega\|_{L^2(\R^2)}^2$. Then for any $T>0$ and $t\in[0,T]$, the problem \eqref{rpde} possesses weak solutions $(u^\varepsilon, v^\varepsilon)$ in $\R^2\times(0,T)$, and there exists a time $T^*\in(0,T)$ such that
\begin{eqnarray}
&\|u^\varepsilon\|_{{L^2(0,T;H^1(\R^2))}\cap L^\infty(0,T;L^p(\R^2))}\le C,\label{uniformweaksolution}\\
&\||x|^2u^\varepsilon\|_{L^\infty(0, T; L^1(\R^2))}\le C,\quad\|\nabla\log u^\varepsilon\|_{L^\infty(0, T^*; W^{1,q}(\R^2))}\le C.\label{uniformweaksolution1}
\end{eqnarray}
where $C$ is a constant independent of $\varepsilon$.
\end{lemma}
Let $\{X_{N,i}^\varepsilon\}_{1\le i\le N}$ and $\{\overline{X}_i^\varepsilon\}_{1\le i\le N}$ be the solutions to systems
\begin{align}\label{sde}
\begin{cases}
dX_{N,i}^\varepsilon(t) = \Big(2\exp\Big(-\frac{1}{N}\sum_{j=1}^N\Phi^\varepsilon(X_{N,i}^\varepsilon(t) - X_{N,j}^\varepsilon(t))\Big)+ 2\Big)^{1/2}dB_i(t),\\
X_{N,i}^\varepsilon(0) =\zeta_i,\qquad 1\le i\le N,
\end{cases}
\end{align}
and
\begin{align}\label{mfs}
\begin{cases}
d\overline{X}_i^\varepsilon(t) = \big(2\exp\big(-\Phi^\varepsilon * u^\varepsilon( \overline{X}_i^\varepsilon,t)\big)+2\big)^{1/2} dB_i(t),\\
\overline{X}_i^\varepsilon(0) = \zeta_i,\qquad 1\le i\le N,
\end{cases}
\end{align}
respectively. Then
for any $m,N\in \mathbb{N}^+$ and parameters $0<\theta<\frac{1}{2},\quad 0<\alpha<\frac{\theta}{2},\quad m>\frac{\theta+1}{1-2\theta}$, there exist a constant $C(m,T)>0$ and a parameter $\gamma,\eta>0$ satisfying the bounds
\begin{align}
&0<\gamma<\min\big\{\frac{\alpha}{3},\frac{-2\alpha +m(1-2\theta)-1}{4m+4}\big\},\label{gamma} \\
&0<\eta\le\min\{\theta-2\alpha, -(4m+4)\gamma-2\alpha +m(1-2\theta)-1\}\label{eta}
\end{align}
such that for all $0\le t\le T$,
\begin{align}\label{Pe}
\mathbb{P}\big(\max_{i=1,\cdots,N}\big|(X_{N,i}^\varepsilon - \overline{X}_i^\varepsilon)(t)\big|> N^{-\alpha}\big)\le C(m,T)N^{-\eta},
\end{align}
where the cut-off parameter satisfies
$\varepsilon\sim N^{-\gamma} $.

Existential–Universal
Inequality or Bound

Question:
Let $u_0\ge 0$ be an initial probability density on $\mathbb{R}^2$ such that
$$u_0\log u_0\in L^1(\mathbb{R}^2),\qquad \nabla\log u_0\in W^{1,q}(\mathbb{R}^2)\ \text{for some }q>2,$$
and
$$u_0\in L^1(\mathbb{R}^2,|x|^2dx)\cap L^p(\mathbb{R}^2)\qquad (1\le p\le \infty).$$
Assume also that $\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo--Nirenberg inequality
$$\|\omega\|_{L^4(\mathbb{R}^2)}^4\le c_*\|\omega\|_{L^2(\mathbb{R}^2)}^2\|\nabla\omega\|_{L^2(\mathbb{R}^2)}^2.$$
Let $(u^\varepsilon,v^\varepsilon)$ be the weak solution of the intermediate nonlocal problem
$$\partial_t u^\varepsilon=\Delta(e^{-v^\varepsilon}u^\varepsilon+u^\varepsilon),\qquad -\Delta v^\varepsilon+v^\varepsilon=\chi \,u^\varepsilon * j^\varepsilon,\qquad u^\varepsilon(\cdot,0)=u_0*j^\varepsilon,$$
on $\mathbb{R}^2\times(0,T)$, where $*$ denotes convolution.
For $1\le i\le N$, let $X_{N,i}^\varepsilon$ and $\overline X_i^\varepsilon$ solve, with the same Brownian motions $B_i$ and the same initial data $\zeta_i$,
$$dX_{N,i}^\varepsilon(t)=\Big(2\exp\Big(-\frac1N\sum_{j=1}^N\Phi^\varepsilon(X_{N,i}^\varepsilon(t)-X_{N,j}^\varepsilon(t))\Big)+2\Big)^{1/2}dB_i(t),$$
$$d\overline X_i^\varepsilon(t)=\Big(2\exp\big(-\Phi^\varepsilon*u^\varepsilon(\overline X_i^\varepsilon,t)\big)+2\Big)^{1/2}dB_i(t).$$
Fix $m,N\in\mathbb N^+$ and parameters $0<\theta<\tfrac12$, $0<\alpha<\tfrac\theta2$, and $m>\frac{\theta+1}{1-2\theta}$. Under these assumptions, which quantitative estimate holds for the probability that the interacting particle system and the mean-field system differ by more than $N^{-\alpha}$ in maximum norm at time $t$?

There exist a constant $C(m,T)>0$ and parameters $\gamma,\eta>0$ such that
$$0<\gamma<\min\Big\{\frac\alpha3,\frac{-2\alpha+m(1-2\theta)-1}{4m+4}\Big\},$$
$$0<\eta\le \min\Big\{\theta-2\alpha,
-(4m+4)\gamma-2\alpha+m(1-2\theta)-1\Big\},$$
and, if the cutoff parameter satisfies $\varepsilon\sim N^{-\gamma}$, then for every $t\in[0,T]$,
$$\mathbb P\Big(\max_{i=1,\dots,N}|X_{N,i}^\varepsilon(t)-\overline X_i^\varepsilon(t)|>N^{-\alpha}\Big)\le C(m,T)N^{-\eta}.$$

Choices:
B.
There exist a constant $C(m,T)>0$ and parameters $\gamma,\eta>0$ such that
$$0<\gamma<\min\Big\{\frac\alpha3,\frac{-2\alpha+m(1-2\theta)-1}{4m+4}\Big\},$$
$$0<\eta\le \min\Big\{\theta-2\alpha,
-(4m+4)\gamma-2\alpha+m(1-2\theta)-1\Big\},$$
and, if the cutoff parameter satisfies $\varepsilon\sim N^{-\gamma}$, then for every $t\in[0,T]$,
$$\mathbb P\Big(\max_{i=1,\dots,N}|X_{N,i}^\varepsilon(t)-\overline X_i^\varepsilon(t)|>N^{-\alpha}\Big)\le C(m,T)N^{-2\eta}.$$
C.
There exist a constant $C(m,T)>0$ and parameters $\gamma,\eta>0$ such that
$$0<\gamma<\min\Big\{\frac\alpha3,\frac{-2\alpha+m(1-2\theta)-1}{4m+4}\Big\},$$
$$0<\eta\le \min\Big\{\theta-2\alpha,
-(4m+4)\gamma-2\alpha+m(1-2\theta)-1\Big\},$$
and, if the cutoff parameter satisfies $\varepsilon\sim N^{-\gamma}$, then for every $t\in[0,T]$,
$$\mathbb P\Big(\max_{i=1,\dots,N}|X_{N,i}^\varepsilon(t)-\overline X_i^\varepsilon(t)|>N^{-\alpha}\Big)\le C(m,T).$$
D.
For every choice of parameters $\gamma,\eta>0$ satisfying
$$0<\gamma<\min\Big\{\frac\alpha3,\frac{-2\alpha+m(1-2\theta)-1}{4m+4}\Big\},$$
$$0<\eta\le \min\Big\{\theta-2\alpha,
-(4m+4)\gamma-2\alpha+m(1-2\theta)-1\Big\},$$
there exists a constant $C(m,T)>0$ such that, if the cutoff parameter satisfies $\varepsilon\sim N^{-\gamma}$, then for every $t\in[0,T]$,
$$\mathbb P\Big(\max_{i=1,\dots,N}|X_{N,i}^\varepsilon(t)-\overline X_i^\varepsilon(t)|>N^{-\alpha}\Big)\le C(m,T)N^{-\eta},$$
with the same constant $C(m,T)$ working uniformly for all such admissible pairs $(\gamma,\eta)$.
E.
There exist a constant $C(m,T)>0$ and parameters $\gamma,\eta>0$ such that
$$0<\gamma\le\min\Big\{\frac\alpha3,\frac{-2\alpha+m(1-2\theta)-1}{4m+4}\Big\},$$
$$0<\eta\le \min\Big\{\theta-2\alpha,
-(4m+4)\gamma-2\alpha+m(1-2\theta)-1\Big\},$$
and, if the cutoff parameter satisfies $\varepsilon\sim N^{-\gamma}$, then for every $t\in[0,T]$,
$$\mathbb P\Big(\max_{i=1,\dots,N}|X_{N,i}^\varepsilon(t)-\overline X_i^\varepsilon(t)|>N^{-\alpha}\Big)\le C(m,T)N^{-\eta}.$$

algebraic decay rate from stopping-time/BDG argument

dropped the quantitative decay factor $N^{-\eta}$

dependence of the constant on the admissible pair $(\gamma,\eta)$

strict upper bound on $\gamma$ replaced by an attained endpoint

The stem does not reveal the correct option directly. It gives hypotheses and asks which conclusion holds, without explicitly signaling the exact probability estimate or the precise quantifier structure.

The item is close to a theorem-recall question: the correct choice is essentially the exact theorem conclusion under the listed assumptions. It is not a pure restatement in the stem, but it mainly tests recognition of the formal statement rather than deriving a new consequence.

Some reasoning is needed to distinguish subtle changes in rates, strict vs. non-strict bounds, pointwise-in-time vs. supremum-in-time estimates, and explicit rate vs. mere convergence. However, this is more precision/recall of a known result than substantial generative mathematical reasoning.

The distractors are strong: they are mathematically close to the target statement and differ in realistic ways that reflect common failure modes, such as weakening the conclusion, changing the rate variable, altering endpoint admissibility, or strengthening time uniformity improperly.

A technically strong MCQ with little answer leakage and high-quality distractors, but it leans more toward theorem-statement recognition than genuine generative reasoning.


Question:
Let $u_0\ge 0$ be an initial probability density on $\mathbb{R}^2$ such that
$$u_0\log u_0\in L^1(\mathbb{R}^2),\qquad \nabla\log u_0\in W^{1,q}(\mathbb{R}^2)\ \text{for some }q>2,$$
and
$$u_0\in L^1(\mathbb{R}^2,|x|^2dx)\cap L^p(\mathbb{R}^2)\qquad (1\le p\le \infty).$$
Assume also that $\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo--Nirenberg inequality
$$\|\omega\|_{L^4(\mathbb{R}^2)}^4\le c_*\|\omega\|_{L^2(\mathbb{R}^2)}^2\|\nabla\omega\|_{L^2(\mathbb{R}^2)}^2.$$
Let $(u^\varepsilon,v^\varepsilon)$ be the weak solution of the intermediate nonlocal problem
$$\partial_t u^\varepsilon=\Delta(e^{-v^\varepsilon}u^\varepsilon+u^\varepsilon),\qquad -\Delta v^\varepsilon+v^\varepsilon=\chi \,u^\varepsilon * j^\varepsilon,\qquad u^\varepsilon(\cdot,0)=u_0*j^\varepsilon,$$
on $\mathbb{R}^2\times(0,T)$, where $*$ denotes convolution.
For $1\le i\le N$, let $X_{N,i}^\varepsilon$ and $\overline X_i^\varepsilon$ solve, with the same Brownian motions $B_i$ and the same initial data $\zeta_i$,
$$dX_{N,i}^\varepsilon(t)=\Big(2\exp\Big(-\frac1N\sum_{j=1}^N\Phi^\varepsilon(X_{N,i}^\varepsilon(t)-X_{N,j}^\varepsilon(t))\Big)+2\Big)^{1/2}dB_i(t),$$
$$d\overline X_i^\varepsilon(t)=\Big(2\exp\big(-\Phi^\varepsilon*u^\varepsilon(\overline X_i^\varepsilon,t)\big)+2\Big)^{1/2}dB_i(t).$$
Fix $m,N\in\mathbb N^+$ and parameters $0<\theta<\tfrac12$, $0<\alpha<\tfrac\theta2$, and $m>\frac{\theta+1}{1-2\theta}$. Under these assumptions, which quantitative estimate holds for the probability that the interacting particle system and the mean-field system differ by more than $N^{-\alpha}$ in maximum norm at time $t$?

Choices:
B.
There exist a constant $C(m,T)>0$ and parameters $\gamma,\eta>0$ such that
$$0<\gamma<\min\Big\{\frac\alpha3,\frac{-2\alpha+m(1-2\theta)-1}{4m+4}\Big\},$$
$$0<\eta\le \min\Big\{\theta-2\alpha,
-(4m+4)\gamma-2\alpha+m(1-2\theta)-1\Big\},$$
and, if the cutoff parameter satisfies $\varepsilon\sim N^{-\gamma}$, then for every $t\in[0,T]$,
$$\mathbb P\Big(\max_{i=1,\dots,N}|X_{N,i}^\varepsilon(t)-\overline X_i^\varepsilon(t)|>N^{-\alpha}\Big)\le C(m,T)N^{-2\eta}.$$
C.
There exist a constant $C(m,T)>0$ and parameters $\gamma,\eta>0$ such that
$$0<\gamma<\min\Big\{\frac\alpha3,\frac{-2\alpha+m(1-2\theta)-1}{4m+4}\Big\},$$
$$0<\eta\le \min\Big\{\theta-2\alpha,
-(4m+4)\gamma-2\alpha+m(1-2\theta)-1\Big\},$$
and, if the cutoff parameter satisfies $\varepsilon\sim N^{-\gamma}$, then for every $t\in[0,T]$,
$$\mathbb P\Big(\max_{i=1,\dots,N}|X_{N,i}^\varepsilon(t)-\overline X_i^\varepsilon(t)|>N^{-\alpha}\Big)\le C(m,T).$$
D.
For every choice of parameters $\gamma,\eta>0$ satisfying
$$0<\gamma<\min\Big\{\frac\alpha3,\frac{-2\alpha+m(1-2\theta)-1}{4m+4}\Big\},$$
$$0<\eta\le \min\Big\{\theta-2\alpha,
-(4m+4)\gamma-2\alpha+m(1-2\theta)-1\Big\},$$
there exists a constant $C(m,T)>0$ such that, if the cutoff parameter satisfies $\varepsilon\sim N^{-\gamma}$, then for every $t\in[0,T]$,
$$\mathbb P\Big(\max_{i=1,\dots,N}|X_{N,i}^\varepsilon(t)-\overline X_i^\varepsilon(t)|>N^{-\alpha}\Big)\le C(m,T)N^{-\eta},$$
with the same constant $C(m,T)$ working uniformly for all such admissible pairs $(\gamma,\eta)$.
E.
There exist a constant $C(m,T)>0$ and parameters $\gamma,\eta>0$ such that
$$0<\gamma\le\min\Big\{\frac\alpha3,\frac{-2\alpha+m(1-2\theta)-1}{4m+4}\Big\},$$
$$0<\eta\le \min\Big\{\theta-2\alpha,
-(4m+4)\gamma-2\alpha+m(1-2\theta)-1\Big\},$$
and, if the cutoff parameter satisfies $\varepsilon\sim N^{-\gamma}$, then for every $t\in[0,T]$,
$$\mathbb P\Big(\max_{i=1,\dots,N}|X_{N,i}^\varepsilon(t)-\overline X_i^\varepsilon(t)|>N^{-\alpha}\Big)\le C(m,T)N^{-\eta}.$$

The stem does not reveal the conclusion outright; it states hypotheses and asks for the resulting quantitative estimate. The correct answer is not explicitly leaked.

This is essentially a direct theorem-recall item: the hypotheses are given in full and the task is to pick the exact stated conclusion. It functions as a restatement of a proposition rather than a conceptually restructured problem.

There is some reasoning pressure in distinguishing subtle variants involving decay rate, endpoint admissibility, and quantifier dependence, but the item mainly tests precise recall/recognition of the theorem statement rather than genuine derivation.

The distractors are mathematically close and plausible: one weakens the bound, one changes the decay exponent, one alters endpoint strictness, and one changes quantifier/constant dependence. These reflect realistic theorem-statement failure modes.

A technically well-constructed theorem-recognition MCQ with strong distractors, but it is largely tautological and only moderately tests generative reasoning.