\label{main_thm} If $0<p<3/4$ then there exists positive constants $C_1$, $C_2$, $C_3$ and $C_4$ such that, for all $N \ge 2$ and $t \geq 1$, the following bound holds:

    \begin{equation*}
        C_1 \exp \left( -C_2 \frac{t}{N^2} \right) \leq \mathbb{P} \left( \tau_N > t \right) \le C_3 \exp \left(-C_4\frac{t}{N^2}\right).
    \end{equation*}

\begin{theorem}\label{main_thm} If $0<p<3/4$ then there exists positive constants $C_1$, $C_2$, $C_3$ and $C_4$ such that, for all $N \ge 2$ and $t \geq 1$, the following bound holds:

    \begin{equation*}
        C_1 \exp \left( -C_2 \frac{t}{N^2} \right) \leq \P \left( \tau_N > t \right) \le C_3 \exp \left(-C_4\frac{t}{N^2}\right).
    \end{equation*}
\end{theorem}

\begin{proposition}\label{prop_1}
    If $0<p<3/4$ then there exists a constant $\theta$ such that: $$\E[\tau_N] \underset{N \rightarrow\infty}{\sim} \theta N^2. $$
\end{proposition}

The gambler’s ruin may be the oldest problem in probability theory, dating back to a letter from Blaise Pascal to Pierre de Fermat in 1656. In the classical gambler's ruin, a gambler is betting on the outcome of a possibly biased coin, winning or loosing one dollar at each step, with probability $p$ and $1-p$ respectively, independently of what happened up to the current step. Assume that the gambler starts with fortune $N$ and stop either when the totality of its fortune has been lost, or when it has doubled. For simplicity we adopt the representation in which $S_n$ represents the gambler's gains (when $S_n \geq 0$) or losses (when $S_n \leq 0$) with $S_0=0$. Thus $(S_n)_{n \ge 0}$ is a Simple Random Walk (SRW) on the one-dimensional integer lattice, starting at the origin. Moreover we denote the game's duration by $\sigma_N = \inf\left\{n\ge 1: \left|S_n\right| = N \right\}$. One of the main object of interest is the expected duration of the game, and, in this standard (Markovian and time-homogeneous) setting the problem has long been fully and analytically solved. Classical results\footnote{See for example \cite{feller}, pages 348 and 349.} tell us that $\mathbb{E}[\sigma_N]$ is of quadratic order in the symmetric case, while it is of linear order in the asymmetric case. More precisely, if $p=1/2$, then $\mathbb{E} \left[\sigma_N \right] = N^2$, otherwise: $$\mathbb{E} \left[\sigma_N \right] = \frac{N}{2p - 1} \left( \frac{1 - \left( \frac{1-p}{p}\right)^N}{1 + \left( \frac{1-p}{p}\right)^N}\right).$$
The proofs of these closed-form formulas are highly dependent on the Markovian and time-homogeneous nature of the dynamic. In the present work, we develop an approach allowing the analysis of a case in which the random walk has a past-dependent dynamic. Namely, we consider the problem above when the classical simple random walk is replaced by the elephant's random walk.

\vspace{0.2 cm}

The Elephant's Random Walk (ERW) is a discrete time stochastic process introduced by Schütz and Trimper \cite{schutz2004elephants} as a paradigmatic example of past-dependent random walks. It is both quantitatively tractable and qualitatively rich---exhibiting in particular a phase transition from diffusive to super-diffusive behavior---and has therefore recently become the standard choice for testing classical results outside the traditional Markovian setting, when memory is allowed to play a role. It can be defined as follows. Let $(Z_n)_{n \geq 0}$ be a stochastic chain with value in $\mathbb{Z}$, making jumps of unit size---either upward or downward---at each time step. It can be defined, like classical random walks, as the partial sum of a sequence $(X_n)_{n \geq 0}$ of random variables: $$Z_n = \sum_{k=0}^n X_k,$$ with the crucial difference that the $(X_n)_{n \geq 0}$ are not independent. Conventionally $X_0 = 0$, and the first jump leads to either $X_1 = 1$ or $X_1 = -1$, with probabilities that are unimportant for our purposes, since in either cases $|Z_1|=1$. Then, for any $n \geq 1$, the jump at time $n+1$ is given by:
$$X_{n+1} = \begin{cases} X_{K(n)} &\text{ with probability } p,\\
-X_{K(n)} &\text{ with probability } 1-p,
\end{cases}$$
where $(K(n))_{n \geq 1}$ is a sequence of independent random variables, $K(n)$ having uniform distribution on $\{1, \ldots, n\}$. In words, at each time step, the ERW selects a step uniformly at random from its entire past and repeats it with probability $p$, or takes the opposite step with probability $1 - p$. The parameter $p$ is often referred to as the \textit{memory} of the elephant: when $p$ is close to $1$, the dependence on the past tends to persist over a long period, whereas it fades rapidly when $p$ is close to $0$.

\vspace{0.2 cm}

Much effort has been done during the last two decades to understand the effect of the memory of the elephant on its asymptotic behavior, as compared to classical Markovian random walks, with a particular focus on laws of large numbers, central limit theorems and functional limit theorems---see for example \cite{baur2016elephant, bercu, coletticlt, coletticlt2}. Results on hitting times have been obtained, but mostly from the view point of \textit{return times} and recurrence/transience---see \cite{bertoinZeros, colettiRec}. Surprisingly, such natural questions as the gambler's ruin problem---in other words, the question of \textit{escape times}---has, to the best of our knowledge, never been addressed for the ERW, even though it was briefly mentioned in \cite{stadtmuller} and \cite{pozdnyakov}.

\vspace{0.2 cm}

The question is whether or not the classical behavior mentioned at the beginning of this section remains when $(S_n)_{n \ge 0}$ is replaced by $(Z_n)_{n \ge 0}$. For concreteness, one can imagine a gambling game in which the coin is replaced by Pólya's urn\footnote{See \cite{baur2016elephant} for details on the connection between ERW and Pólya's urn.}. More precisely, suppose an urn contains initially a unique ball, which is either green or red. Then at any time $n \geq 1$ a ball is drawn at random (uniformly) from the urn, and then put back in with another ball, which has either the same color or the other color, with probability $p$ and $1-p$ respectively. Then replacing $(S_n)_{n \geq 0}$ by $(Z_n)_{n \geq 0}$ is the same as saying that the gambler is betting on the outcome of the urn sampling, and win or loose one dollar at time $n$ depending on whether the ball drawn at time $n$ is green or not. Define:

\begin{equation*}
    \tau_N = \inf\left\{k\ge 1: |Z_k| = N\right\}.
\end{equation*}

\vspace{0.2 cm}

It is well-known (see \cite{schutz2004elephants}) that the behavior of the ERW critically depends on the value of $p$, with a phase transition at $3/4$. $(Z_n)_{n \geq 0}$ is said to be in the \textit{diffusive regime} when $p < 3/4$, and in the \textit{super-diffusive} regime when $p > 3/4$. We prove that, in the diffusive regime, the expectation of the game duration exhibits similar quadratic behavior as in the classical symmetric gambler's ruin for big $N$. More precisely, we prove the following.

\begin{proposition}\label{prop_1}
    If $0<p<3/4$ then there exists a constant $\theta$ such that: $$\mathbb{E}[\tau_N] \underset{N \rightarrow\infty}{\sim} \theta N^2. $$
\end{proposition}

\vspace{0.2 cm}

The above result is obtained as a corollary of a much stronger one, which might actually be considered as the main contribution of this work. Namely we obtain tight bounds on the tail of the game's duration of the elephant's ruin.

The gambler’s ruin may be the oldest problem in probability theory, dating back to a letter from Blaise Pascal to Pierre de Fermat in 1656. In the classical gambler's ruin, a gambler is betting on the outcome of a possibly biased coin, winning or loosing one dollar at each step, with probability $p$ and $1-p$ respectively, independently of what happened up to the current step. Assume that the gambler starts with fortune $N$ and stop either when the totality of its fortune has been lost, or when it has doubled. For simplicity we adopt the representation in which $S_n$ represents the gambler's gains (when $S_n \geq 0$) or losses (when $S_n \leq 0$) with $S_0=0$. Thus $(S_n)_{n \ge 0}$ is a Simple Random Walk (SRW) on the one-dimensional integer lattice, starting at the origin. Moreover we denote the game's duration by $\sigma_N = \inf\left\{n\ge 1: \left|S_n\right| = N \right\}$. One of the main object of interest is the expected duration of the game, and, in this standard (Markovian and time-homogeneous) setting the problem has long been fully and analytically solved. Classical results\footnote{See for example \cite{feller}, pages 348 and 349.} tell us that $\mathbb{E}[\sigma_N]$ is of quadratic order in the symmetric case, while it is of linear order in the asymmetric case. More precisely, if $p=1/2$, then $\mathbb{E} \left[\sigma_N \right] = N^2$, otherwise: $$\mathbb{E} \left[\sigma_N \right] = \frac{N}{2p - 1} \left( \frac{1 - \left( \frac{1-p}{p}\right)^N}{1 + \left( \frac{1-p}{p}\right)^N}\right).$$
The proofs of these closed-form formulas are highly dependent on the Markovian and time-homogeneous nature of the dynamic. In the present work, we develop an approach allowing the analysis of a case in which the random walk has a past-dependent dynamic. Namely, we consider the problem above when the classical simple random walk is replaced by the elephant's random walk.

The Elephant's Random Walk (ERW) is a discrete time stochastic process introduced by Schütz and Trimper \cite{schutz2004elephants} as a paradigmatic example of past-dependent random walks. It is both quantitatively tractable and qualitatively rich---exhibiting in particular a phase transition from diffusive to super-diffusive behavior---and has therefore recently become the standard choice for testing classical results outside the traditional Markovian setting, when memory is allowed to play a role. It can be defined as follows. Let $(Z_n)_{n \geq 0}$ be a stochastic chain with value in $\mathbb{Z}$, making jumps of unit size---either upward or downward---at each time step. It can be defined, like classical random walks, as the partial sum of a sequence $(X_n)_{n \geq 0}$ of random variables: $$Z_n = \sum_{k=0}^n X_k,$$ with the crucial difference that the $(X_n)_{n \geq 0}$ are not independent. Conventionally $X_0 = 0$, and the first jump leads to either $X_1 = 1$ or $X_1 = -1$, with probabilities that are unimportant for our purposes, since in either cases $|Z_1|=1$. Then, for any $n \geq 1$, the jump at time $n+1$ is given by:
$$X_{n+1} = \begin{cases} X_{K(n)} &\text{ with probability } p,\\
-X_{K(n)} &\text{ with probability } 1-p,
\end{cases}$$
where $(K(n))_{n \geq 1}$ is a sequence of independent random variables, $K(n)$ having uniform distribution on $\{1, \ldots, n\}$. In words, at each time step, the ERW selects a step uniformly at random from its entire past and repeats it with probability $p$, or takes the opposite step with probability $1 - p$. The parameter $p$ is often referred to as the \textit{memory} of the elephant: when $p$ is close to $1$, the dependence on the past tends to persist over a long period, whereas it fades rapidly when $p$ is close to $0$.

\begin{equation*}
 \tau_N = \inf\left\{k\ge 1: |Z_k| = N\right\}.
\end{equation*}

\begin{proposition}\label{prop_1}
 If $0<p<3/4$ then there exists a constant $\theta$ such that: $$\mathbb{E}[\tau_N] \underset{N \rightarrow\infty}{\sim} \theta N^2. $$
\end{proposition}

\vspace{0.2 cm}

The above result is obtained as a corollary of a much stronger one, which might actually be considered as the main contribution of this work. Namely we obtain tight bounds on the tail of the game's duration of the elephant's ruin.

The above result is obtained as a corollary of a much stronger one, which might actually be considered as the main contribution of this work. Namely we obtain tight bounds on the tail of the game's duration of the elephant's ruin.

\begin{remark}
 Even though formally the result above is stated only for $0 < p < 3/4$, the upper bound actually holds for all $0 < p < 1$. Moreover, some of the constants above may depend on $p$.
\end{remark}

\begin{lemma}\label{lemma:sym} There exists positive absolute constants $C_1$, $C_2$, $C_3$ and $C_4$ such that
\begin{equation*}
 C_1 \exp \left(-C_2\frac{t}{N^2}\right) \le \mathbb{P}(\sigma_N > t) \le C_3 \exp \left(-C_4\frac{t}{N^2}\right),
\end{equation*}
for all $N\ge 2$ and $t \ge 1$.
\end{lemma}

\begin{enumerate}[(i)]
 \item \textbf{Case $\bm{p \ge 1/2}$.}
 Let $\tilde{S}_0 = 0$ and, for $k\ge 1$, define recursively $\xi_k$ and $\tilde{S}_k$ by:
 \begin{equation*}
 \xi_{k} = \mathbbm{1}_{\left\{\tilde{S}_{k-1} = 0 \right\}} + \mathbbm{1}_{\left\{\tilde{S}_{k-1} \neq 0 \right\}} \left( \mathbbm{1}_{\left\{U_{k} < \frac{1}{2} \right\}} - \mathbbm{1}_{\left\{U_{k} \ge \frac{1}{2}\right\}}\right),
 \end{equation*}
 and
 \begin{equation}\label{srw-coupling}
 \tilde{S}_{k} = \sum_{j=1}^{k}\xi_j.
 \end{equation}
 Then $\tilde{S}_k$ has the same distribution as $|S_k|$, where $(S_k)_{k \ge 0}$ is still the symmetric simple random walk defined in the previous section. Moreover, $(\tilde{S}_k)_{k\ge 0}$ and $(\tilde{Z}_{k} )_{k\ge 0}$ are constructed in such a way that the following holds:
 \begin{equation}\label{enq_1}
 \tilde{Z}_k \ge \tilde{S}_k, \, \text{ for all } k\ge 0.
 \end{equation}
 Notice that the only way inequality \eqref{enq_1} could be broken would be if at some point $\tilde{Z}_k = 1$ and $\tilde{S}_k = 0$ (and $\tilde{Z}_{k}$ jumps to the left between $k$ and $k+1$). Fortunately this never happens since $\tilde{Z}_k$ and $\tilde{S}_k$ have the same parity. Below, we deliberately abuse notation, redefining $\tau_N$ and $\sigma_N$ in terms of $(\tilde{S}_k)_{k\ge 0}$ and $(\tilde{Z}_{k} )_{k\ge 0}$. By inequality \eqref{enq_1} and Lemma \ref{lemma:sym}, for all $N\ge 1$ and $t\ge 1$, one indeed has:

\begin{equation}\label{enq_2}
 \tilde{S}_k - 2\sum_{j=1}^k\mathbbm{1}_{\left\{\frac{1}{2} + \frac{(2p-1)\tilde{Z}_{j +dN^{2}}}{j + dN^2}\le U_{j + dN^{2}} < \frac{1}{2}\right\}} \le \tilde{Z}_{k + dN^{2}} , \, \text{ for all } k\ge 0.
 \end{equation}
 Let $c > 12 + d$. For any $x < N$ we have
 \begin{align*}\label{eq:boundonethird+}
 \P_x \left(\tau_N > cN^2\right) &= \P_x\left(\tau_N > cN^{2}, \sigma_{2N} > 3(2N)^2\right) + \P_x \left(\tau_N > cN^{2}, \sigma_{2N} \le 3(2N)^2)\right)\\
 &\le \P_x \left(\sigma_{2N} > 3(2N)^2\right) + \P_x \left(\tau_N > cN^{2}, \sigma_{2N} \le 3(2N)^2\right)
 \end{align*}
 But $\P_x \left(\sigma_{2N} > 3(2N)^2 \right) \leq 1/3$ by Markov inequality (see inequality $(\ref{eq:markov_sym})$ in the proof of Lemma \ref{lemma:sym}), hence:
 \begin{equation}\label{eq:boundonethird+}
 \P_x \left(\tau_N > cN^2 \right) \le \frac{1}{3} + \P_x \left(\tau_N > cN^{2}, \, \sigma_{2N} \le 3(2N)^2 \right).
 \end{equation}
 Now observe that $\tau_N > cN^{2}$ means that the elephant has not reached $N$ yet at time $c N^2$ while $\sigma_{2N} \le 3(2N)^2$ means that the symmetric random walk reached $2N$ in the meantime, even though it was exactly at the same point $12N^2$ time-steps earlier (or more). Now assume that both hold. Then $\tilde{Z}_k < N$ for any $k \le dN^2 + \sigma_{2N}$, where $dN^2 + \sigma_{2N}$ represents the time at which $(\tilde{S}_k)_{k \geq 0}$ reaches $2N$, but on the $(\tilde{Z}_k)_{k \geq 0}$ time-scale. Then the inequality $\tilde{Z}_{dN^2 + \sigma_{2N}}< N$ together with \eqref{enq_2} implies that
 $$2N - 2 \sum_{j=1}^{\sigma_{2N}}\mathbbm{1}_{\big\{\frac{1}{2} + \frac{(2p-1)\tilde{Z}_{j + dN^{2}}}{j + dN^{2}}\le U_{j + dN^{2}} < \frac{1}{2}\big\}} < N.$$
 Moreover, since $\sigma_{2N} \le 3(2N)^2$ and $\tilde{Z}_{j + dN^2} < N$ for any of the indices $j$ in the sum above, the above inequality implies that
 $$ 2N - 2\sum_{j=1}^{3 (2 N)^2}\mathbbm{1}_{\{\frac{1}{2} + \frac{(2p - 1)}{dN}\le U_{j + dN^{2}} < \frac{1}{2} \} } < N.$$

\begin{equation*}
 \big\{ \tau_N > cN^{2}, \sigma_{2N} \le 3(2N)^2 \big\} \subset \Big\{B_N > \frac{N}{2} \Big\},
 \end{equation*}
 where: $$B_N = \sum_{j=1}^{12 N^2}\mathbbm{1}_{\left\{\frac{1}{2} + \frac{(2p - 1)}{dN}\le U_{j + dN^{2}} < \frac{1}{2} \right\}} \sim \text{Binomial}\left(12 N^2, \, \frac{1-2p}{dN}\right).$$
 Now, by letting $d > \lceil 24(1-2p)\rceil$, we have:
 \begin{equation}\label{eq_2}
 \mathbb{E}[B_N] = \frac{12(1 - 2p)N}{d} < \frac{N}{2},
 \end{equation}
 and thus
 \begin{equation*}
 \P\Big(B_N > \frac{N}{2}\Big) \le \frac{1}{2}.
 \end{equation*}
 Hence, for our choice of $d$, it holds that:
 \begin{equation} \label{eq:onehalfpart}
 \P_x \left(\tau_N > cN^{2}, \sigma_{2N} \le 3(2N)^2 \right) \le \frac{1}{2}.
 \end{equation}
 Finally, inequality \eqref{eq:onehalfpart} together with inequality \eqref{eq:boundonethird+} leads to:
 \begin{equation*}
 \P_x \left(\tau_N > cN^2 \right) \le \frac{5}{6}.
 \end{equation*}
 Then the same line of reasoning as in the end of the proof of Lemma \ref{lemma:sym} allows us to conclude:

Notice that, since ${\left\{U_j < p \right\} = \left\{U_j < \frac{1}{2} + \frac{2p-1}{2} \right\}}$, one has $\tilde{Z}_k \le \hat{S}_k$, for all $k\ge 0$. Therefore, if
\begin{equation*}
 \hat{\sigma}_N = \inf\{k\ge 1: \hat{S}_k = N\},
\end{equation*}
then, for all $N \ge 2$ and $t\ge 1$, one has:
\begin{equation*}
 \mathbb{P}(\tau_N > t) \ge \mathbb{P}(\hat{\sigma}_N > t).
\end{equation*}
Observing that $\{U_1 > p , \, \ldots , \, U_t > p\}\subset \{\hat{\sigma}_N > t\}$, we obtain the following---very crude, but good enough---bound:
\begin{equation*}
 \mathbb{P}(\hat{\sigma}_N > t) \ge (1-p)^t = e^{\log(1 - p) t} \ge e^{-\log \left(\frac{1}{1-p}\right)C_A^2 \frac{t}{N^2}},
\end{equation*}
for all $t\ge 1$ and $2\le N\le C_A$. Hence taking $C_2 = \max \left\{\log \left(\frac{1}{1-p}\right)C_A^2, \tilde{C}_2 \right \}$, we can indeed conclude:
\begin{equation*}
 \mathbbm{P}(\tau_N \ge t) \ge C_1e^{-C_2\frac{t}{N^2}},
\end{equation*}
for all $N\ge 2$ and $t\ge 1$.
\end{enumerate}

We start with the upper bound and then proceed to the lower bound. Fix some $c > 1$. For any $x \in \llbracket -N, N \rrbracket$, a classical martingale argument (see for example Theorem 4.8.7 in \cite{durrett2019probability}) gives us:
 \begin{equation*}
 \E_x \left[\sigma_N \right] = (N-x)(x+N) \le N^2.
 \end{equation*}
 Hence, by Markov inequality:
 \begin{equation}\label{eq:markov_sym}
 \P_x \left(\sigma_N > cN^2 \right) \le \frac{\E_x \left[ \sigma_N \right]}{cN^2}\le \frac{1}{c}.
 \end{equation}
 For $t \geq 0$, it follows that:
 \begin{equation*}
 \P(\sigma_N > t) \leq \P \left(\sigma_N \ge \left\lfloor \frac{t}{cN^2}\right\rfloor cN^2 \right) \le \left(\frac{1}{c}\right)^{\left\lfloor \frac{t}{cN^2}\right\rfloor} \le \left(\frac{1}{c}\right)^{\frac{t}{cN^2} - 1} = C_3e^{-C_4\frac{t}{N^2}},
 \end{equation*}
 with $C_3 = c$ and $C_4 = \log(c)/c$. Notice that, for $t \geq cN^2$, the second inequality above is a consequence of \eqref{eq:markov_sym} together with the strong Markov property, whereas it is trivial for $t < c N^2$, since in that case $\left(\frac{1}{c}\right)^{\left\lfloor \frac{t}{cN^2} \right\rfloor} = 1$.

\begin{remark}
    Even though formally the result above is stated only for $0 < p < 3/4$, the upper bound actually holds for all $0 < p < 1$. Moreover, some of the constants above may depend on $p$.
\end{remark}

The main idea behind the proof of Theorem \ref{main_thm} is to first establish analogous bounds for the SRW, and then demonstrate via couplings that, on a quadratic time scale, the ERW does not deviate excessively from the SRW. These couplings rely on a well-known time-inhomogeneous Markovian description of the ERW. Given a sequence of i.i.d. random variables with uniform distribution on $(0,1)$, we can construct both the ERW and the SRW on the same probability space. Each walk determines its next jump based on the value of the current uniform random variable relative to a specified threshold. For the SRW this threshold is always $1/2$, while for the ERW it depends on both time and the walk’s current state—though it is always either greater or less than $1/2$, depending on whether $p>1/2$ or not. This construction yields joint realizations in which the ERW either dominates or is dominated by the SRW, according to the value of $p$. By leveraging the SRW's bounds, this approach already resolves half of the cases. To address the remaining cases and thus complete the proof of Theorem \ref{main_thm}, one then needs to control the distance between the SRW and the ERW on the appropriate time scale. Specifically, for $t = cN^2$, we identify some scaling constant $A$ such that, whenever the ERW or the SRW reaches the boundaries of $[-AN, AN]$, the other one has vanishingly small probability of not having reached $\{-N,N\}$ yet. This allows us prove Theorem \ref{main_thm} on the quadratic time scale $t = cN^2$. Then, by partitioning the interval $[0, t]$ into sub-intervals of length $t/cN^2$, we extend the result to any $t \geq 1$. Finally, Proposition \ref{prop_1} follows directly from the functional limit theorem in \cite{baur2016elephant} and uniform integrability.

\vspace{0.2 cm}

The paper is organized as follows. In Section 2 we derive a well-known (time inhomogeneous) Markovian description of the ERW allowing a coupling with the simple random walks. In Section 3 we prove the exponential bounds on the tail of the simple random walk. In Section 4 we define a coupling between ERW and the classical random walks and prove our main result, Theorem \ref{main_thm}. Finally, in Section 5, we prove Proposition \ref{prop_1}.

\vspace{0.2 cm}

The proof of Theorem \ref{main_thm} proceeds as follows:

- First, “establish analogous bounds for the SRW, and then demonstrate via couplings that, on a quadratic time scale, the ERW does not deviate excessively from the SRW.”

- The couplings use “a well-known time-inhomogeneous Markovian description of the ERW.” Using i.i.d. uniforms on $(0,1)$, “we can construct both the ERW and the SRW on the same probability space,” where each walk’s next jump is determined by comparing the current uniform to a threshold: for SRW the threshold is always $1/2$, while for ERW it “depends on both time and the walk’s current state.” This produces joint realizations where “the ERW either dominates or is dominated by the SRW, according to the value of $p$,” and “by leveraging the SRW's bounds, this approach already resolves half of the cases.”

- For the remaining cases, “one then needs to control the distance between the SRW and the ERW on the appropriate time scale.” Concretely, for $t=cN^2$, choose a scaling constant $A$ such that whenever one walk hits the boundary of $[-AN,AN]$, “the other one has vanishingly small probability of not having reached $\{-N,N\}$ yet.” This yields Theorem \ref{main_thm} “on the quadratic time scale $t=cN^2$.”

- To pass from $t=cN^2$ to general $t\ge 1$, “partitioning the interval $[0,t]$ into sub-intervals of length $t/cN^2$,” thereby extending the bound to “any $t\ge 1$.”

The proof of the main theorem proceeds as follows:

- First, “establish analogous bounds for the SRW, and then demonstrate via couplings that, on a quadratic time scale, the ERW does not deviate excessively from the SRW.”

- The couplings use “a well-known time-inhomogeneous Markovian description of the ERW.” Using i.i.d. uniforms on $(0,1)$, “we can construct both the ERW and the SRW on the same probability space,” where each walk’s next jump is determined by comparing the current uniform to a threshold: for SRW the threshold is always $1/2$, while for ERW it “depends on both time and the walk’s current state.” This produces joint realizations where “the ERW either dominates or is dominated by the SRW, according to the value of $p$,” and “by leveraging the SRW's bounds, this approach already resolves half of the cases.”

- For the remaining cases, “one then needs to control the distance between the SRW and the ERW on the appropriate time scale.” Concretely, for $t=cN^2$, choose a scaling constant $A$ such that whenever one walk hits the boundary of $[-AN,AN]$, “the other one has vanishingly small probability of not having reached $\{-N,N\}$ yet.” This yields the main theorem “on the quadratic time scale $t=cN^2$.”

- To pass from $t=cN^2$ to general $t\ge 1$, “partitioning the interval $[0,t]$ into sub-intervals of length $t/cN^2$,” thereby extending the bound to “any $t\ge 1$.”

Existential–Universal
Inequality or Bound

Question:
Consider the elephant random walk $(Z_n)_{n\ge 0}$ on $\mathbb Z$, defined by $Z_n=\sum_{k=0}^n X_k$ with $X_0=0$, the first step $X_1\in\{-1,1\}$, and for each $n\ge 1$, after choosing $K(n)$ uniformly from $\{1,\dots,n\}$ independently of the past,
\[
X_{n+1}=\begin{cases}
X_{K(n)} & \text{with probability } p,\\
- X_{K(n)} & \text{with probability } 1-p.
\end{cases}
\]
For $N\ge 2$, define the hitting time of the two-sided boundary $\{\pm N\}$ by
\[
\tau_N=\inf\{k\ge 1: |Z_k|=N\}.
\]
Assume $0<p<3/4$. Which quantitative estimate holds for the tail of $\tau_N$ for all $N\ge 2$ and $t\ge 1$?

There exist positive constants $C_1,C_2,C_3,C_4$ such that, for every $N\ge 2$ and $t\ge 1$,
\[
C_1\exp\!\left(-C_2\frac{t}{N^2}\right)\le \mathbb P(\tau_N>t)\le C_3\exp\!\left(-C_4\frac{t}{N^2}\right).
\]

Choices:
B.
There exist positive constants $C_1,C_2,C_3,C_4$ such that, for every $N\ge 2$ and $t\ge 1$,
\[
C_1\exp\!\left(-C_2\frac{t}{N}\right)\le \mathbb P(\tau_N>t)\le C_3\exp\!\left(-C_4\frac{t}{N}\right).
\]
C.
There exist positive constants $C_3,C_4$ such that, for every $N\ge 2$ and $t\ge 1$,
\[
\mathbb P(\tau_N>t)\le C_3\exp\!\left(-C_4\frac{t}{N^2}\right).
\]
D.
For every $N\ge 2$ there exist positive constants $C_1(N),C_2(N),C_3(N),C_4(N)$ such that, for every $t\ge 1$,
\[
C_1(N)\exp\!\left(-C_2(N)\frac{t}{N^2}\right)\le \mathbb P(\tau_N>t)\le C_3(N)\exp\!\left(-C_4(N)\frac{t}{N^2}\right).
\]
E.
There exist positive constants $C_1,C_2,C_3,C_4$ such that, for every $N\ge 2$ and $t\ge 1$,
\[
C_1\exp\!\left(-C_2\frac{t}{N^2}\right)\le \mathbb P(\tau_N=t)\le C_3\exp\!\left(-C_4\frac{t}{N^2}\right).
\]


Question:
Consider the elephant's random walk \((Z_n)_{n\ge 0}\) on \(\mathbb Z\), defined by \(Z_n=\sum_{k=0}^n X_k\) with \(X_0=0\), \(X_1\in\{\pm1\}\), and for each \(n\ge 1\), after choosing \(K(n)\) uniformly from \(\{1,\dots,n\}\), setting
\[
X_{n+1}=\begin{cases}
X_{K(n)} & \text{with probability } p,\\
- X_{K(n)} & \text{with probability } 1-p.
\end{cases}
\]
For \(N\ge 2\), define the hitting time
\[
\tau_N=\inf\{k\ge 1: |Z_k|=N\}.
\]
When the memory parameter satisfies \(0<p<3/4\), which statement holds?

There exist positive constants \(C_1,C_2,C_3,C_4\) such that for all \(N\ge 2\) and all \(t\ge 1\),
\[
C_1\exp\!\left(-C_2\frac{t}{N^2}\right)\le \mathbb P(\tau_N>t)\le C_3\exp\!\left(-C_4\frac{t}{N^2}\right).
\]

Choices:
B.
There exist positive constants \(C_1,C_2,C_3,C_4\) such that for all \(N\ge 2\) and all \(t\ge 1\),
\[
C_1\exp\!\left(-C_2\frac{t}{N}\right)\le \mathbb P(\tau_N>t)\le C_3\exp\!\left(-C_4\frac{t}{N}\right).
\]
C.
There exist positive constants \(C_3,C_4\) such that for all \(N\ge 2\) and all \(t\ge 1\),
\[
\mathbb P(\tau_N>t)\le C_3\exp\!\left(-C_4\frac{t}{N^2}\right).
\]
D.
For each \(N\ge 2\) there exist positive constants \(C_1(N),C_2(N),C_3(N),C_4(N)\) such that for all \(t\ge 1\),
\[
C_1(N)\exp\!\left(-C_2(N)\frac{t}{N^2}\right)\le \mathbb P(\tau_N>t)\le C_3(N)\exp\!\left(-C_4(N)\frac{t}{N^2}\right).
\]
E.
There exist positive constants \(C_1,C_2,C_3,C_4\) such that for all \(N\ge 2\) and all \(t\ge 1\),
\[
C_1\exp\!\left(-C_2\frac{t}{N^2}\right)\le \mathbb P(\tau_N=t)\le C_3\exp\!\left(-C_4\frac{t}{N^2}\right).
\]


The Prym maps
$$
\mathcal{P}^{iso}_3 :  \cRH_3^{iso} \rightarrow  \cA_6^{(1,1,1,2,2,4)} \qquad \mathcal{P}^{ni}_3 :  \cRH_3^{ni} \rightarrow  \cA_6^{(1,1,1,1,4,4)}
$$
are injective on each irreducible component of the source moduli space.

Let $H$ be a hyperelliptic smooth projective curve over $\mathbb C$ of genus 3. We consider Klein coverings $f:\widetilde{C} \rightarrow H$, that is,
$\widetilde{C}$ admits an action by a group of fixed point free automorphisms $V_4$ isomorphic to $\ZZ_2 \times \ZZ_2$ such that $H = \widetilde{C} / V_4$.
One can associate to such a covering a polarised abelian sixfold in the following way. Let $\Nm_f : J\widetilde{C} \rightarrow JH$ be the norm map, sending the divisor class $[\sum_i n_ip_i]$ to $[\sum_i n_if(p_i)]$,
with $p_i \in \widetilde{C}$ and $n_i\in \mathbb Z$. The Prym variety $P(f)$ of the covering $f$ is defined as the connected component of 0 of the kernel of $\Nm_f$. It is an abelian subvariety of $J\widetilde{C}$ of dimension the difference of the genera of the curves, which in this case is 6 and the restriction of the principal polarisation on $J\widetilde{C}$ induces a non-principal one on $P(f)$.

Prym varieties of étale double covers have been extensively studied in the past and more recently it has been proven that
the Prym map is injective for double covers ramified in at least six points (\cite{NO22}). For étale cyclic coverings over a hyperelliptic curve, the (generic) injectivity
of the Prym has been shown for infinitely many degrees (\cite{NOS24},
\cite{NOPS25}). There are also new results on the fiber of the Prym map for non-cyclic coverings in low genera (\cite{S25}).

Klein coverings can be defined from the curve $H$ by choosing a subgroup
$\langle \eta, \xi\rangle $ of order four of the group $JH[2]$ of 2-torsion points in $JH$.
We distinguish two types of Klein coverings depending whether the subgroup is isotropic or not with respect to the Weil pairing on $JH[2]$.
Let $\cRH_3^{iso}$, respectively $\cRH_3^{ni}$, denote the moduli space parametrising isotropic, respectively non-isotropic, Klein
coverings on hyperelliptic curves of genus 3.
Each of these moduli spaces consists of two irreducible components (see Lemma \ref{4components}).

Consider now the Prym map that sends $[f:\widetilde{C} \rightarrow H] \in \cRH_3^{V_4}=\cRH_3^{iso} \sqcup \cRH_3^{ni} $ to the
polarised Prym variety $(P,\Xi)$. In this article we continue the study of the injectivity of the Prym map for Klein
coverings of genus $3$ curves, which was started in \cite{BO24}, where we prove the injectivity on the locus of hyperelliptic coverings $f:\widetilde{C} \rightarrow H $ (i.e. $\widetilde{C}$ is also hyperelliptic)
corresponding to the case I.1 below. Let us denote by
$\cA_g^{\delta}$ the moduli space of polarised abelian varieties of dimension $g$ with polarisation type $\delta$.
The main theorem of the paper is the following.

\begin{lem}\label{4components}
The moduli space $\cRH_3^{V_4}$ consists of four irreducible components, corresponding to the configuration of the Weierstrass points in the generators $ \eta, \ \xi$ as shown
in I.1, I.2, II.1 and II.2. Moreover, the degree of the forgetful
map  $\cRH_3^{V_4} \ra \cH_3$ on each component is 56, 280, 105, 210, respectively.
\end{lem}

Klein coverings can be defined from the curve $H$ by choosing a subgroup
$\langle \eta, \xi\rangle $ of order four of the group $JH[2]$ of 2-torsion points in $JH$.
We distinguish two types of Klein coverings depending whether the subgroup is isotropic or not with respect to the Weil pairing on $JH[2]$.
Let $\cRH_3^{iso}$, respectively $\cRH_3^{ni}$, denote the moduli space parametrising isotropic, respectively non-isotropic, Klein
coverings on hyperelliptic curves of genus 3.
Each of these moduli spaces consists of two irreducible components (see Lemma \ref{4components}).

Consider now the Prym map that sends $[f:\tC \ra H] \in \cRH_3^{V_4}=\cRH_3^{iso} \sqcup \cRH_3^{ni} $ to the
polarised Prym variety $(P,\Xi)$. In this article we continue the study of the injectivity of the Prym map for Klein
coverings of genus $3$ curves, which was started in \cite{BO24}, where we prove the injectivity on the locus of hyperelliptic coverings $f:\tC \ra H $ (i.e. $\tC$ is also hyperelliptic)
corresponding to the case I.1 below. Let us denote by
$\cA_g^{\delta}$ the moduli space of polarised abelian varieties of dimension $g$ with polarisation type $\delta$.
The main theorem of the paper is the following.

Actually, the injectivity also holds on the entire moduli space $\cRH_3^{V_4}$ since the dimensions of the factors appearing in the isotypical decomposition of the associated Prym variety, depends on the type of the Klein covering and it is different for each irreducible component.

\begin{thm}\label{thmI.2}
The Prym map
$\cP^{ni}_3 : \cRH_3^{ni} \ra \cA_6^{(1,1,1,1,4,4)}$ is injective on the irreducible component consisting of Klein coverings of type I.2.

\begin{thm}
The Prym map
$\cP^{iso}_3 : \cRH_3^{iso} \ra \cA_6^{(1,1,1,2,2,4)}$ is injective on the irreducible component consisting of Klein coverings of type II.1.
\end{thm}
\begin{proof}
Let $(P, \Xi)$ be the image of a Klein covering of type $II.1$ under $\cP^{iso}_3$. First note that, by Proposition \ref{autoofP}, $(P, \Xi)$ determines the group of involutions $B\simeq \ZZ^4_2$ on $P$.
One then look at the fixed loci of the involutions together with the restricted polarisation types to distinguish $j,j\s,j\tau,j\s\tau$ and $JC_j,JC_{j\s},JC_{j\tau},JC_{j\s\tau}$ respectively. Now, one chooses any two of these, say $j$ and $j\s$ and by Torelli Theorem one recovers $C_j$ and $C_{j\s}$.
By Lemma \ref{intersection_Jacobians_II.1} and Lemma \ref{key_lemma} we obtain the maps $C_j\to E$ and $C_{j\s}\to E$, so we have the following fibered product.

\begin{thm}
The Prym map
$\cP^{iso}_3 : \cRH_3^{iso} \ra \cA_6^{(1,1,1,2,2,4)}$ is injective on the irreducible component consisting of Klein coverings of type II.2.
\end{thm}
\begin{proof}
Let $(P, \Xi)$ be an element on the image of $\cP^{ni}_3$. Then the group $\langle \sigma, \tau,j, -1 \rangle \simeq \ZZ_2^4$ acts on $P$ and according to Proposition \ref{isotyp_II.2}, this action induces the isotypical decomposition
$$
P = P^*(C_{\s}/H) \boxplus P^*(C_{\tau}/H) \boxplus E_j \boxplus F
$$
with a distinguished elliptic curve $F$ of exponent $(4)$.
Moreover, by construction
$$
P^*(C_{\s}/H)+F \sim (JC_{j\s}, 2\Theta_{C_{j\s}}), \quad P^*(C_{\tau}/H)+F \sim (JC_{j\tau}, 2\Theta_{C_{j\tau}}).
$$
So the isotypical components determine uniquely the
polarised Jacobians called $JC_{j\s}$ and $JD_{j\tau}$ and
by Torelli Theorem the genus 3 curves $C_{j\s}$ and $ C_{j\tau}$ are also determined. Since $F$ is embedded in the Jacobians by Lemma \ref{key_lemma}, we consider the fibered product
\begin{equation}\label{fibred_product_II.2}
\xymatrix@R=.9cm@C=1cm{
 \widetilde{C_{j\s} \times_{F} C_{j\tau}}\ar[d]_{2:1} \ar[r]^{ \ 2:1} & C_{j\s} \ar[d]^{h_\s} \\
C_{j\tau} \ar[r]_{h_\tau} & F
}
\end{equation}

\begin{thm}\label{FullKlein}
 The Klein Prym maps $$
\cP^{iso}_3 : \cR_3^{iso} \ra \cA_6^{(1,1,1,2,2,4)} \qquad \cP^{ni}_3 : \cR_3^{ni} \ra \cA_6^{(1,1,1,1,4,4)}
$$ are generically finite. Moreover they are of degree $1$ on images of coverings of hyperelliptic curves of types $I.2, II.1, II.2$.
\end{thm}
\begin{proof}
 Let us consider the non-isotropic case. By Theorems \ref{thmI.1} and \ref{thmI.2} the Prym map ${\cP^{ni}_3}_{|\cRH_3^{ni}}$ is injective on each component. Moreover, images of respective components are disjoint as the isotypical decomposition yields components of different dimensions. Hence, the image of the Prym map, being an irreducible variety and containing $2$ disjoint subvarieties of dimension $5$ has to be of dimension at least $6=\dim(\cR_3^{ni})$. This shows that the map is generically finite.

The above theorem is an evidence to a positive answer of the following question.
\begin{question}
Are the following Prym maps
$$
\cP^{iso}_3 : \cR_3^{iso} \ra \cA_6^{(1,1,1,2,2,4)} \qquad \cP^{ni}_3 : \cR_3^{ni} \ra \cA_6^{(1,1,1,1,4,4)}
$$
 injective?
\end{question}

\begin{lem}\label{4components}
The moduli space $\cRH_3^{V_4}$ consists of four irreducible components, corresponding to the configuration of the Weierstrass points in the generators $ \eta, \ \xi$ as shown
in I.1, I.2, II.1 and II.2. Moreover, the degree of the forgetful
map  $\cRH_3^{V_4} \ra \cH_3$ on each component is 56, 280, 105, 210, respectively.
\end{lem}

\begin{prop}\label{autoofP}
Let $(P, \Xi)$ be an element in the image of the Prym map
$\cP^{iso}_3$, respectively in $\cP^{ni}_3$ Case I.2.  Then it holds
$$
B:=\{\psi \in \Aut (P,\Xi) \ : \ \psi^2=id, \
\psi_{|K(\Xi)}= \pm id \} = \langle \sigma, \tau, j, -1 \rangle \simeq \ZZ_2^4
$$
where $j$ is any lift of the hyperelliptic involution from $H$.
\end{prop}

Actually, the injectivity also holds on the entire moduli space $\cRH_3^{V_4}$ since the dimensions of the factors appearing in the isotypical decomposition of the associated Prym variety,  depends on the type of the Klein covering and it is different for each irreducible component.

The general idea of the proof, with some variations in each case, is the following. A Prym variety $(P,\Xi)$ in the image of the Prym map determines a group of automorphisms of $P$ isomorphic to
$\ZZ_2^4$ (see Proposition \ref{autoofP}). This gives us the isotypical decomposition of $P$ and one can identify the Jacobians by means of the type of the restricted polarisation and using the action of the involutions on the kernel of the polarisation.  With this information we can reconstruct the quotients curves and finally recover $\widetilde{C}$ as a fibered product.

As an application, we can show the following theorem for Klein coverings over {\it any} curve of genus 3, see Theorem \ref{FullKlein}.
\begin{thm}
    The Klein Prym maps $$
\mathcal{P}^{iso}_3 :  \cR_3^{iso} \rightarrow  \cA_6^{(1,1,1,2,2,4)} \qquad \mathcal{P}^{ni}_3 :  \cR_3^{ni} \rightarrow  \cA_6^{(1,1,1,1,4,4)}
$$ are generically finite. Moreover they are of degree $1$ on images of coverings of hyperelliptic curves of types $I.2, II.1, II.2$.
\end{thm}

The structure of the paper is as follows. In Section 1, we state basic definitions and key lemmas and show that the moduli space have 4 components, called I.1, I.2, II.1, II.2. Each one of the Sections 2,\ 3,\ 4 and 5 is devoted to one of these cases and to the proof of the injectivity of the corresponding  Prym map. Finally, Section 6 is devoted to Klein coverings of any genus 3 curve.

\subsection*{Acknowledgments}
The authors would like to thank Anatoli Shatsila for finding a reference that shortened considerably one of the proofs. The first author has been supported by the Polish National Science Centre project number 2024/54/E/ST1/00330. Some results of the paper were obtained during his visit to Humboldt University in Berlin. He would like to thank the university for hospitality.

Actually, the injectivity also holds on the entire moduli space $\cRH_3^{V_4}$ since the dimensions of the factors appearing in the isotypical decomposition of the associated Prym variety depends on the type of the Klein covering and it is different for each irreducible component.

The general idea of the proof (``with some variations in each case'') is:
\begin{itemize}
\item A Prym variety $(P,\Xi)$ in the image of the Prym map determines a group of automorphisms of $P$ isomorphic to $\ZZ_2^4$ (Proposition \ref{autoofP}).
\item This yields the isotypical decomposition of $P$.
\item ``One can identify the Jacobians" using the type of the restricted polarisation and the action of the involutions on the kernel of the polarisation.
\item From this data, ``we can reconstruct the quotients curves" and ``finally recover $\widetilde{C}$ as a fibered product."
\end{itemize}

Actually, the injectivity also holds on the entire moduli space $\cRH_3^{V_4}$ since the dimensions of the factors appearing in the isotypical decomposition of the associated Prym variety depends on the type of the Klein covering and it is different for each irreducible component.

The general idea of the proof (``with some variations in each case'') is:
\begin{itemize}
\item A Prym variety $(P,\Xi)$ in the image of the Prym map determines a group of automorphisms of $P$ isomorphic to $\ZZ_2^4$. More precisely:
\begin{prop}\label{autoofP}
Let $(P, \Xi)$ be an element in the image of the Prym map
$\cP^{iso}_3$, respectively in $\cP^{ni}_3$ Case I.2.  Then it holds
$$
B:=\{\psi \in \Aut (P,\Xi) \ : \ \psi^2=id, \
\psi_{|K(\Xi)}= \pm id \} = \langle \sigma, \tau, j, -1 \rangle \simeq \ZZ_2^4
$$
where $j$ is any lift of the hyperelliptic involution from $H$.
\end{prop}
\item This yields the isotypical decomposition of $P$.
\item ``One can identify the Jacobians" using the type of the restricted polarisation and the action of the involutions on the kernel of the polarisation.
\item From this data, ``we can reconstruct the quotients curves" and ``finally recover $\widetilde{C}$ as a fibered product."
\end{itemize}

Question:
Let \(\cRH_3^{iso}\) (respectively \(\cRH_3^{ni}\)) be the moduli space of Klein coverings \(f:\widetilde C\to H\) of hyperelliptic smooth projective curves \(H\) of genus \(3\) over \(\mathbb C\), where \(\widetilde C\) admits a fixed-point-free action of \(V_4\simeq \mathbb Z_2\times \mathbb Z_2\) with quotient \(H=\widetilde C/V_4\), and where the associated order-four subgroup \(\langle \eta,\xi\rangle\subset JH[2]\) is isotropic (respectively non-isotropic) with respect to the Weil pairing. For such a covering, let \(P(f)\) be the Prym variety, namely the connected component of \(0\) in \(\ker(\operatorname{Nm}_f:J\widetilde C\to JH)\), endowed with the induced polarization. Consider the Prym maps
\[
\mathcal P^{iso}_3: \cRH_3^{iso}\to \cA_6^{(1,1,1,2,2,4)},\qquad
\mathcal P^{ni}_3: \cRH_3^{ni}\to \cA_6^{(1,1,1,1,4,4)},
\]
where \(\cA_6^{\delta}\) denotes the moduli space of polarized abelian sixfolds of polarization type \(\delta\). Which statement holds for every irreducible component of the source moduli spaces?

For both maps, the restriction to any irreducible component of the source is injective; that is, on each irreducible component of \(\cRH_3^{iso}\) and on each irreducible component of \(\cRH_3^{ni}\), two Klein coverings with the same polarized Prym variety represent the same point of that component.

Choices:
B. For both maps, the Prym variety determines the Klein covering globally on the whole source moduli space; that is, \(\mathcal P^{iso}_3\) is injective on all of \(\cRH_3^{iso}\) and \(\mathcal P^{ni}_3\) is injective on all of \(\cRH_3^{ni}\), so two Klein coverings with the same polarized Prym variety necessarily coincide even if they lie in different irreducible components.
C. For both maps, the restriction to each irreducible component has finite fibers; equivalently, on every irreducible component of \(\cRH_3^{iso}\) and of \(\cRH_3^{ni}\), only finitely many Klein coverings can have the same polarized Prym variety.
D. For both maps, there is a single reconstruction procedure from the polarized Prym variety that is uniform across all irreducible components, in the sense that the same isotypical decomposition and the same pair of quotient Jacobians recover the Klein covering in every isotropic and non-isotropic case.
E. For both maps, the polarized Prym variety determines the underlying hyperelliptic base curve \(H\) on each irreducible component, but not necessarily the full Klein covering; in other words, equality of polarized Prym varieties forces the same quotient curve \(H\), although the corresponding order-four subgroup \(\langle\eta,\xi\rangle\subset JH[2]\) may still vary.

componentwise injectivity versus global injectivity across distinct irreducible components

dropped injective and kept finite fibers on each irreducible component

case-dependent isotypical decomposition and reconstruction data

reconstruction of the full covering from quotient curves and fibered product, not merely the base curve


\begin{appendices}
	\section{Poincar\'e inequality: local to global}\label{A}
		\begin{lemma}\label{p3.10}
	Let $x_0\in S$, $0<r_0\leq1$ and  $B=B(x_0,r_0)\subseteq S$,  $\rho_B=\frac{\rho\restr{B}}{\rho(B)}$.  Then the following strong local $(1,1)$-Poincar\'e inequality holds for $f\in \Lip(B,  \d)$:
			\begin{equation}
			\int_{B}|f(x)-\E_{\rho_B}(f)|\,\d\rho_B(x)\leq C_4 r_0\int_B |\nabla_x f|\,\d\rho_B,
		\end{equation}
		where $C_4$ depends on $K,N,a_1,a_2$.
	\end{lemma}

	\begin{proof}
		By \cite[Chapter 9]{zbMATH01474795} and \cite[Remark 3.3]{zbMATH06043352},  $\mm$ satisfies the  $(1,1)$-Poincar\'e inequality:
		\begin{equation}
			\int_{B} |f-\E_{\mm_B}(f)|\,\d\mm\leq c_1 r_0 \int_B |\nabla_x f|\,\d\mm,
		\end{equation}
		where $c_1$ depends on $K,N$, $\mm_B=\frac{\mm\restr{B}}{\mm(B)}$. Since $a_1\mm \leq \rho\leq a_2 \mm$, we have
		\begin{equation}
			\begin{aligned}
				\int_{B} |f-\E_{\rho_B}(f)|\,\d\rho\leq &\int_{B} |f-\E_{\mm_B}(f)|\,\d\rho+\int_{B} |\E_{\mm_B}(f)-\E_{\rho_B}(f)|\,\d\rho\\
				\leq & 2a_2\int_{B} |f-\E_{\mm_B}(f)|\,\d\mm\\
				\leq &2a_2 c_1 r_0 \int_B |\nabla_x f|\,\d\mm
				\leq  2\frac{a_2}{a_1}c_1 r_0 \int_B |\nabla_x f|\,\d\rho,\\
			\end{aligned}
		\end{equation}
	which is the thesis.
	\end{proof}

	Next we prove an $L^1$-variant of the gluing Lemma in \cite[Lemma 3.3]{arXiv:2411.04908},   following the same Boman-chain decomposition,  utilizing the doubling property directly instead of the maximal function estimates.

		\begin{definition}[Boman chain condition]\label{Boman chain condition}
		We say that a probability measure $\rho$ on an open set $S$ of a metric space satisfies  \emph{Boman chain condition} with parameters \(E, F,G>1\) if there is a covering \(\mathcal{F}\) of $S$ by open balls  $B \in \mathcal{F}$ such that:

		\begin{enumerate}
				\item For any \(x \in S\),
				\begin{equation*}
						\sum_{B \in \mathcal{F}} \chi_{2B}(x) \leq E \chi_{S}(x).
					\end{equation*}
				\item For some fixed ball \(B_0\) in \(\mathcal{F}\), called the \emph{central ball}, and for every \(B \in \mathcal{F}\), there exists a chain \(B_0, B_1, \ldots, B_N = B\) of distinct balls from \(\mathcal{F}\) such that
				\begin{equation*}
					B \subset F B_j,~~~\forall j \in \{0, \ldots, N-1\}.
					\end{equation*}
			\item Consecutive balls of the above chain overlap quantitatively:
				\begin{equation*}
						\rho(B_j \cap B_{j+1}) \geq G^{-1} \max(\rho(B_j), \rho(B_{j+1})),~~~\forall j \in \{0, \ldots, N-1\}.
					\end{equation*}
			\end{enumerate}
	\end{definition}

	\begin{lemma}\label{3.1}
	For any $f\in L^1(\rho)$,   it holds that
	\begin{equation}\label{A.5}
		\int_S |f(x)-\E_{\rho}(f)|\,\d\rho(x)\leq C_5 \sum_{B\in\mathcal{F}}\rho(B)\int_B |f(x)-\E_{\rho_B}(f)|\,\d\rho_B(x),
	\end{equation}
	where $C_5$ depends on $K,N,\diam(S),S$.
	\end{lemma}

    \begin{proof}
    	Since $S$ is a John domain, and $\rho$ is a doubling measure on $S$, by  \cite[Proposition 3.7]{arXiv:2504.05412}, $\rho$ satisfies the Boman chain condition. Hence there exists a covering $\mathcal{F}$ of $S$ satisfying Definition \ref{Boman chain condition}.

    	For the central ball $B_0$, note that
    	\begin{equation}\label{A.2}
    		\begin{aligned}
    			\int_S |f(x)-\E_{\rho}(f)|\,\d\rho(x)
    			\leq & \int_S |f(x)-\E_{\rho_{B_0}}(f)|\,\d\rho(x)+\int_S |\E_{\rho_{B_0}}(f)-\E_{\rho}(f)|\,\d\rho(x)\\
    			\leq & 2\int_S |f(x)-\E_{\rho_{B_0}}(f)|\,\d\rho(x).
    		\end{aligned}
    	\end{equation}

   For  $B\in\mathcal{F}$,  	denote $a_B=\int_B |f-\E_{\rho_{B}}(f)|\,\d\rho=\rho(B)\int_B |f-\E_{\rho_{B}}(f)|\,\d\rho_B$. Since $\mathcal{F}$ is a covering of $S$, we have
    	\begin{equation}\label{A.3}
    		\begin{aligned}
    			&\int_S |f(x)-\E_{\rho_{B_0}}(f)|\,\d\rho(x)
    			\leq  \sum_{B \in \mathcal{F}} \int_B |f(x)-\E_{\rho_{B_0}}(f)|\,\d\rho(x)\\
    			\leq & \sum_{B \in \mathcal{F}} \left(\int_B |f(x)-\E_{\rho_{B}}(f)|\,\d\rho(x)+\int_B |\E_{\rho_{B}}(f)-\E_{\rho_{B_0}}(f)|\,\d\rho(x)\right)\\
    			\leq &  \sum_{B \in \mathcal{F}} \left(a_B+\rho(B) |\E_{\rho_{B}}(f)-\E_{\rho_{B_0}}(f)|\right).
    		\end{aligned}
    	\end{equation}

    	For any $B\in\mathcal{F}$, by Boman chain condition, there exists a chain $B_0, B_1,\dots, B_N=B$ of distinct balls from \(\mathcal{F}\), such that for any \(j \in \{0, \ldots, N-1\}\),
    	\begin{equation}
    		\begin{aligned}
    		|\E_{\rho_{B_j}}(f)-\E_{\rho_{B_{j+1}}}(f)|= &\left|\frac{1}{\rho(B_j\cap B_{j+1})}\int_{B_j\cap B_{j+1}}\left(\E_{\rho_{B_j}}(f)-\E_{\rho_{B_{j+1}}}(f)\right)\,\d\rho\right|\\
    		\leq &\frac{1}{\rho(B_j\cap B_{j+1})}\int_{B_j\cap B_{j+1}}\left|\E_{\rho_{B_j}}(f)-\E_{\rho_{B_{j+1}}}(f)\right|\,\d\rho\\
    		\leq &\frac{1}{\rho(B_j\cap B_{j+1})}\left(\int_{B_j\cap B_{j+1}}\left|f-\E_{\rho_{B_j}}(f)\right|+\left|f-\E_{\rho_{B_{j+1}}}(f)\right|\,\d\rho\right)\\
    		\leq & \frac{a_{B_j}+a_{B_{j+1}}}{\rho(B_j\cap B_{j+1})}
    		\overset{*}{\leq}  G \left(\frac{a_{B_j}}{\rho(B_j)}+\frac{a_{B_{j+1}}}{\rho(B_{j+1})}\right),
    		\end{aligned}
    	\end{equation}
    	where $(*)$ follows from the quantitative chain overlap of Boman chain condition $3$.

    	 Thus, we have
    	 \begin{equation}
    	 |\E_{\rho_{B}}(f)-\E_{\rho_{B_0}}(f)|\leq \sum_{j=0}^{N-1}|\E_{\rho_{B_j}}(f)-\E_{\rho_{B_{j+1}}}(f)|\leq 2G \sum_{j=0}^{N}\frac{a_{B_j}}{\rho(B_j)}\overset{*}{\leq} 2G \sum_{B\subset F \bar{B}}\frac{a_{\bar{B}}}{\rho(\bar{B})},
    	 \end{equation}
    	 where $ \sum_{B\subset F \bar{B}}$ means that the sum runs over all  $\bar{B}\in\mathcal{F}$ satisfying $B\subset F \bar{B}$, and $(*)$ follows from the Boman chain condition $2$.
    	 Then by Fubini--Tonelli theorem,
    	 \begin{equation}\label{A.6}
    	 	\begin{aligned}
    	 	 	  \sum_{B \in \mathcal{F}} \rho(B) |\E_{\rho_{B}}(f)-\E_{\rho_{B_0}}(f)|
    	 	 	  \leq & 2G\sum_{B \in \mathcal{F}} \rho(B) \sum_{B\subset F \overline{B}}\frac{a_{\overline{B}}}{\rho(\overline{B})}
    	 	 	  \leq 2G\sum_{\bar{B} \in \mathcal{F}} \frac{a_{\bar{B}}}{\rho(\bar{B})}\sum_{B\subset F \bar{B}}\rho(B).
    	 	\end{aligned}
    	 \end{equation}

By Boman chain condition $1$ and  the doubling property of $\rho$, we have
    	 \begin{equation}\label{aa}
       \sum_{B\subset F \bar{B}}\rho(B)\leq E\rho(F\bar{B})\leq E\beta^2F^\frac{\log \beta}{\log 2}\rho(\bar{B}),
    	 \end{equation}
where  $\beta=\beta(K,N,\diam(S))$ is the doubling constant.

    	 Combining \eqref{A.2}, \eqref{A.3},  \eqref{A.6} and \eqref{aa}, we obtain
    	 \begin{equation}\label{A_A}
    	 	\begin{aligned}
    	 		 \int_S |f(x)-\E_{\rho}(f)|\,\d\rho(x)\leq & 2 \sum_{B \in \mathcal{F}} \left(a_B+\rho(B) |\E_{\rho_{B}}(f)-\E_{\rho_{B_0}}(f)|\right)\\
    	 		 \leq &2(1+2\beta^2 EF^\frac{\log \beta}{\log 2}G) \sum_{B \in \mathcal{F}} \rho(B)\int_B |f-\E_{\rho_{B}}(f)|\,\d\rho_B,
    	 	\end{aligned}
    	 \end{equation}
  which is the thesis.
    \end{proof}

	\end{appendices}

\begin{theorem}\label{thm0.1}
		Let $(X,\d,\mm)$ be an ${\rm RCD}(K,N)$ metric measure space. Let $S\subseteq X$ be a John domain and $Y\subseteq X$ be compact with $\mm(Y)>0$. Let $\rho\in \mathcal{P}(S)$  be with $a_1\mm\restr{S} \leq \rho\leq a_2\mm\restr{S}$ for some positive constants $a_1,a_2$. Then there exists a constant $C>0$, depending on $K,N,a_1,a_2, S, \diam(S\cup Y)$, such that for any $\mu,\nu\in\mathcal{P}(Y)$,
		\begin{equation}
			\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}\leq C W_1^\frac{1}{2}(\mu,\nu),
		\end{equation}
		where $\phi_\mu$ and $\phi_\nu$ are the Kantorovich potentials from $\rho$ to $\mu$ and $\rho$ to $\nu$ respectively.
	\end{theorem}

\subsection{Motivation and setting}
	\paragraph{Optimal transport.}Optimal transport, initiated by Monge  \cite{monge1781} and reframed by  Kantorovich \cite{zbMATH03099866},  seeks the most efficient way to redistribute mass between two probability distributions.   Precisely, given two probability measures $\rho$ and $\mu$ defined on Polish spaces $X$ and $Y$ respectively, and a  cost function $c:X\times Y\rightarrow \mathbb{R}$, the Monge optimal transport problem aims to find a minimizer  $T$,   called an optimal transport map,   of the following optimization problem among all measurable maps $H: X \to Y$ pushing $\rho$ forward to $\mu$:
	\begin{equation}\label{2.1}
		\int_X c(x, T(x))  \,{\mathrm d}\rho(x)=\inf_{H_\sharp \rho = \mu}\int_X c(x, H(x))  \,{\mathrm d}\rho(x), \tag{MP}
	\end{equation}
 The   Monge problem  prevents mass splitting, sometimes making the problem ill-posed.  Its relaxation,  called Kantorovich problem, optimizes over joint couplings instead of deterministic maps. A fundamental result known as Kantorovich duality, asserts that the Kantorovich problem is equivalent to the  following dual problem:
	\begin{equation}
		\sup_{ \phi(x) + \psi(y) \leq c(x, y)} \left\{ \int_X \phi(x) \, {\mathrm d}\rho(x) + \int_Y \psi(y) \, {\mathrm d}\mu(y)  \right\}.\tag{KD}
	\end{equation}
	The optimal functions $\phi,\psi$,  which always exist under standard assumptions,   are called  Kantorovich potentials \cite{AG-U, V-O}.

	In the Euclidean space,   for $c(x,y)=\frac{1}{2}|x-y|^2$, Brenier's landmark result \cite{brenier1991polar} established that, for absolutely continuous source measures, the optimal transport map is unique and takes the form $T(x)=\nabla u(x)$ for a convex function $u$. This was later extended to Riemannian manifolds by McCann \cite{mccann2001polar}, who showed that for $c(x,y)=\frac{1}{2}{\mathrm d}^2(x,y)$, the optimal transport map is given almost everywhere by $T(x)=\exp_x(-\nabla\varphi(x))$ for a  Kantorovich potential $\varphi$.

	 A fundamental question in both theoretical and applied contexts of optimal transport is the quantitative stability of optimal transport maps and Kantorovich potentials under perturbations of the target measure (see Letrouit's lecture note \cite{LetrouitLecture} and the references therein).   Based on recent breakthroughs in the quantitative stability of optimal transport  on  Euclidean spaces \cite{zbMATH07794624, arXiv:2411.04908},    on boundaries of convex bodies \cite{Kitagawa12,  Kitagawa25}  and  on Riemannian manifolds  \cite{arXiv:2504.05412},
Kitagawa--Letrouit--M\'erigot   \cite[\S 1.2]{arXiv:2504.05412} conjecture that:
	\begin{center}
		\emph{the quantitative stability results are also true in more general metric measure spaces with synthetic curvature bounds.}
	\end{center}

	\paragraph{General setting.}
	In this paper, we confirm the conjecture of Kitagawa-- Letrouit--M\'erigot for the following spaces:
	\begin{itemize}
		\item  RCD$(K,N)$  spaces (metric measure spaces with synthetic Ricci curvature lower bound):  we prove quantitative $L^1$-stability of Kantorovich potentials with respect to Wasserstein perturbations of the target measure;
		\item Alexandrov spaces with (sectional) curvature lower bound:   we establish stability estimates for optimal transport maps in terms of the $L^2$-norm of the potential difference.
	\end{itemize}

	An Alexandrov space is a  geodesic  space of finite Hausdorff dimension and of curvature bounded from below (cf.  \cite{zbMATH01626771}).   An ${\rm RCD}(K,N)$ space  is  a metric measure spaces verifying the synthetic Riemannian curvature-dimension condition  \cite{zbMATH05049052,zbMATH05578758, zbMATH06303881, G-O}.   An $n$-dimensional Alexandrov space with curvature bounded from below by $k$,  equipped with its $n$-Hausdorff measure,   is an ${\rm RCD}(k(n-1), n)$ space  \cite{zbMATH06032507,zbMATH05962558}.
		 After \cite{brenier1991polar, mccann2001polar},   Gigli--Rajala--Sturm \cite[Theorem 1.1]{ gigli2016optimal}  proved the existence and uniqueness of the optimal transport map on an $\rm{RCD}(K,N)$ space $(X,{\mathrm d},\mathfrak m)$,  for the quadratic cost $c(x,y)=\frac{1}{2}{\mathrm d}^2(x,y)$ and the source measure $\rho\ll \mathfrak m$.

	  John domains,    named after  F. John \cite{zbMATH03166766},   encompass many cases of interest, such as  bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \cite{zbMATH00858510} for more discussions).
	\begin{definition}[John domain]
		A bounded open subset $S$ of a metric space is called a \emph{John domain} if there is a distinguished point $x_0 \in S$ and a constant $\eta > 0$ such that, for every $x \in S$, there is a rectifiable curve $\gamma \colon [0, \ell(\gamma)] \to S$ parametrized by arc length, such that $\gamma(0) = x$, $\gamma(\ell(\gamma)) = x_0$, and
		\begin{equation*}
			{\mathrm d}(\gamma(t), S^c) \geq \eta t,~~~\forall t \in [0, \ell(\gamma)],
		\end{equation*}
		where $S^c$ denotes the complement of $S$.
	\end{definition}
	In  \cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\rho$ is the uniform density on some non-John domain  $S\subset \mathbb{R}^n$, then no quantitative stability estimates of the form $\|\phi_\mu-\phi_\nu\|_{L^2(\rho)}\leq C W_p^\alpha(\mu,\nu)$
	can hold for any $C, \alpha, p>0$.
	Moreover, examples in \cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of    $S$ may cause  instability  of optimal transport maps.

	 In this paper,  we always assume that the Kantorovich potential  $\phi_\mu$ from $\rho$ to $\mu$ satisfies $\E_\rho(\phi_\mu)=\int_S\phi_\mu\,{\mathrm d}\rho=0$.  Together with the uniqueness of the optimal transport map, such  $\phi_\mu$ is  unique and thus makes sense to study its stability.

	\subsection{Main results}
	Our main theorem concerns the quantitative  $L^1$ stability of Kantorovich potentials on ${\rm RCD}(K,N)$  spaces.

\subsection{Motivation and setting}
 \paragraph{Optimal transport.}Optimal transport, initiated by Monge \cite{monge1781} and reframed by Kantorovich \cite{zbMATH03099866}, seeks the most efficient way to redistribute mass between two probability distributions. Precisely, given two probability measures $\rho$ and $\mu$ defined on Polish spaces $X$ and $Y$ respectively, and a cost function $c:X\times Y\rightarrow \mathbb{R}$, the Monge optimal transport problem aims to find a minimizer $T$, called an optimal transport map, of the following optimization problem among all measurable maps $H: X \to Y$ pushing $\rho$ forward to $\mu$:
 \begin{equation}\label{2.1}
 \int_X c(x, T(x)) \,{\mathrm d}\rho(x)=\inf_{H_\sharp \rho = \mu}\int_X c(x, H(x)) \,{\mathrm d}\rho(x), \tag{MP}
 \end{equation}
 The Monge problem prevents mass splitting, sometimes making the problem ill-posed. Its relaxation, called Kantorovich problem, optimizes over joint couplings instead of deterministic maps. A fundamental result known as Kantorovich duality, asserts that the Kantorovich problem is equivalent to the following dual problem:
 \begin{equation}
 \sup_{ \phi(x) + \psi(y) \leq c(x, y)} \left\{ \int_X \phi(x) \, {\mathrm d}\rho(x) + \int_Y \psi(y) \, {\mathrm d}\mu(y) \right\}.\tag{KD}
 \end{equation}
 The optimal functions $\phi,\psi$, which always exist under standard assumptions, are called Kantorovich potentials \cite{AG-U, V-O}.

A fundamental question in both theoretical and applied contexts of optimal transport is the quantitative stability of optimal transport maps and Kantorovich potentials under perturbations of the target measure (see Letrouit's lecture note \cite{LetrouitLecture} and the references therein). Based on recent breakthroughs in the quantitative stability of optimal transport on Euclidean spaces \cite{zbMATH07794624, arXiv:2411.04908}, on boundaries of convex bodies \cite{Kitagawa12, Kitagawa25} and on Riemannian manifolds \cite{arXiv:2504.05412},
Kitagawa--Letrouit--M\'erigot \cite[\S 1.2]{arXiv:2504.05412} conjecture that:
 \begin{center}
 \emph{the quantitative stability results are also true in more general metric measure spaces with synthetic curvature bounds.}
 \end{center}

An Alexandrov space is a geodesic space of finite Hausdorff dimension and of curvature bounded from below (cf. \cite{zbMATH01626771}). An ${\rm RCD}(K,N)$ space is a metric measure spaces verifying the synthetic Riemannian curvature-dimension condition \cite{zbMATH05049052,zbMATH05578758, zbMATH06303881, G-O}. An $n$-dimensional Alexandrov space with curvature bounded from below by $k$, equipped with its $n$-Hausdorff measure, is an ${\rm RCD}(k(n-1), n)$ space \cite{zbMATH06032507,zbMATH05962558}.
 After \cite{brenier1991polar, mccann2001polar}, Gigli--Rajala--Sturm \cite[Theorem 1.1]{ gigli2016optimal} proved the existence and uniqueness of the optimal transport map on an $\rm{RCD}(K,N)$ space $(X,{\mathrm d},\mathfrak m)$, for the quadratic cost $c(x,y)=\frac{1}{2}{\mathrm d}^2(x,y)$ and the source measure $\rho\ll \mathfrak m$.

John domains, named after F. John \cite{zbMATH03166766}, encompass many cases of interest, such as bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \cite{zbMATH00858510} for more discussions).
 \begin{definition}[John domain]
 A bounded open subset $S$ of a metric space is called a \emph{John domain} if there is a distinguished point $x_0 \in S$ and a constant $\eta > 0$ such that, for every $x \in S$, there is a rectifiable curve $\gamma \colon [0, \ell(\gamma)] \to S$ parametrized by arc length, such that $\gamma(0) = x$, $\gamma(\ell(\gamma)) = x_0$, and
 \begin{equation*}
 {\mathrm d}(\gamma(t), S^c) \geq \eta t,~~~\forall t \in [0, \ell(\gamma)],
 \end{equation*}
 where $S^c$ denotes the complement of $S$.
 \end{definition}
 In \cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\rho$ is the uniform density on some non-John domain $S\subset \mathbb{R}^n$, then no quantitative stability estimates of the form $\|\phi_\mu-\phi_\nu\|_{L^2(\rho)}\leq C W_p^\alpha(\mu,\nu)$
 can hold for any $C, \alpha, p>0$.
 Moreover, examples in \cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of $S$ may cause instability of optimal transport maps.

In this paper, we always assume that the Kantorovich potential $\phi_\mu$ from $\rho$ to $\mu$ satisfies $\E_\rho(\phi_\mu)=\int_S\phi_\mu\,{\mathrm d}\rho=0$. Together with the uniqueness of the optimal transport map, such $\phi_\mu$ is unique and thus makes sense to study its stability.

\subsection{Main results}
 Our main theorem concerns the quantitative $L^1$ stability of Kantorovich potentials on ${\rm RCD}(K,N)$ spaces.

John domains, named after F. John \cite{zbMATH03166766}, encompass many cases of interest, such as bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \cite{zbMATH00858510} for more discussions).
 \begin{definition}[John domain]
 A bounded open subset $S$ of a metric space is called a \emph{John domain} if there is a distinguished point $x_0 \in S$ and a constant $\eta > 0$ such that, for every $x \in S$, there is a rectifiable curve $\gamma \colon [0, \ell(\gamma)] \to S$ parametrized by arc length, such that $\gamma(0) = x$, $\gamma(\ell(\gamma)) = x_0$, and
 \begin{equation*}
 \d(\gamma(t), S^c) \geq \eta t,~~~\forall t \in [0, \ell(\gamma)],
 \end{equation*}
 where $S^c$ denotes the complement of $S$.
 \end{definition}
 In \cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\rho$ is the uniform density on some non-John domain $S\subset \R^n$, then no quantitative stability estimates of the form $\|\phi_\mu-\phi_\nu\|_{L^2(\rho)}\leq C W_p^\alpha(\mu,\nu)$
 can hold for any $C, \alpha, p>0$.
 Moreover, examples in \cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of $S$ may cause instability of optimal transport maps.

\subsection{Main results}
 Our main theorem concerns the quantitative $L^1$ stability of Kantorovich potentials on ${\rm RCD}(K,N)$ spaces.

If $(S, \d, \mm)$ is a compact $\rcdkn$ space, it supports a global $(1,1)$-Poincar\'e inequality (cf.\cite{zbMATH01474795, zbMATH06043352}) and we can remove the John domain assumption:
 \begin{corollary}\label{cor}
 Let $(X,\d,\mm)$ be a compact ${\rm RCD}(K,N)$ metric measure space. Let $\rho\in \mathcal{P}(X)$ be with $a_1\mm \leq \rho\leq a_2\mm$ for some positive constants $a_1,a_2$. Then the conclusion of Theorem \ref{thm0.1} holds.
 \end{corollary}

Adapting the strategy of \cite{arXiv:2504.05412}, we can also prove the stability of optimal transport maps on Alexandrov spaces.
 \begin{theorem}\label{thm0.2}
 Let $(X, \d)$ be an $n$-dimensional Alexandrov space, $\mm$ be the $n$-Hausdorff measure. Under the same assumption for $S, Y$ and $\rho$ as in Theorem \ref{thm0.1}, and if $S$ additionally has finite perimeter, then there exists a constant $C>0$, depending on $k,n,a_1, a_2, \diam(S\cup Y), {\rm{Per}}(S), S$, such that for any $\mu,\nu\in\mathcal{P}_2(Y)$, we have
 \begin{equation}
 \int_S \d^2(T_\mu(x), T_\nu(x))\,\d \rho(x) \leq CW_1^{1/6}(\mu, \nu),
 \end{equation}
 where $T_\mu$ and $T_\nu$ are the optimal transport maps from $\rho$ to $\mu$ and $\rho$ to $\nu$ respectively.
 \end{theorem}

\begin{proposition}\label{2.11}
 It holds that
 \begin{equation}
 \int_{S}\big| \E_{\mu_x^t[\psi_*]}(v)-\E_{\mu^t[\psi_*]}(v))\big|\,\d\rho(x)\leq \frac{\kappa}{\sqrt{t}}\left(\int_S {\rm{Var}}_{ \mu_x^t[\psi_*]}(v)\,\d\rho(x)\right)^\frac{1}{2},
 \end{equation}
 where $\mu^t[\psi_*]$ is defined as in \eqref{mu} and $\kappa$ depends on $K,N,\Lambda_\psi,a_1,a_2, S$.
 \end{proposition}
 \begin{proof}
 Since $a_1\mm\leq \rho\leq a_2\mm$, by \cite[Corollary 2.4]{zbMATH05049052}, $\rho$ is a doubling measure. Moreover,
 since $S$ is a John domain, by \cite[Proposition 3.7]{arXiv:2504.05412}, $\rho$ satisfies the Boman chain condition (see Definition \ref{Boman chain condition}) and we can choose a covering $\mathcal{F}$, such that for any $B\in \mathcal{F}$, $r_B\leq 1$. Then
 \begin{equation}
 \begin{aligned}
 &\int_{S}\big| \E_{\mu_x^t[\psi_*]}(v)-\E_{\mu^t[\psi_*]}(v))\big|\,\d\rho(x)\\
 \mathop{\leq}^{\text {Lemma \ref{3.1}}} &C_5 \sum_{B\in\mathcal{F}}\rho(B)\int_B \big| \E_{\mu_x^t[\psi_*]}(v)-{\rho(B)}^{-1}\E_{\mu^t[\psi_*]}(v))\big|\,\d\rho_B(x)\\
 \mathop{\leq }^{\text{Lemma \ref{p3.10}}}&C_5 \sum_{B\in\mathcal{F}}\rho(B)C_4 r_B\int_B \big|\nabla_x \E_{\mu_x^t[\psi_*]}(v)\big|\,\d\rho_B(x)\\
 \mathop{\leq}^{\text{Proposition \ref{3.9}} }& \frac{C_3C_4C_5}{\sqrt{t}}\sum_{B\in\mathcal{F}}\sqrt{\mm(B_{2r_B})}\sqrt{r_B}\left(\int_B {\rm{Var}}_{ \mu_x^t[\psi_*]}(v)\,\d\rho(x)\right)^\frac{1}{2}\\
 \mathop{\leq}^{\text{H\"{o}lder}~~}&\frac{C_3C_4C_5}{\sqrt{t}}\left(\sum_{B\in\mathcal{F}}\mm(B_{2r_B})\right)^\frac{1}{2}\left(\sum_{B\in\mathcal{F}}\int_B {\rm{Var}}_{ \mu_x^t[\psi_*]}(v)\,\d\rho(x)\right)^\frac{1}{2}\\
 \overset{\text{Boman chain~}}{\leq}&\frac{\kappa}{\sqrt{t}}\left(\int_S {\rm{Var}}_{ \mu_x^t[\psi_*]}(v)\,\d\rho(x)\right)^\frac{1}{2}
 \end{aligned}
 \end{equation}
 which is the thesis.
 \end{proof}

\begin{theorem}\label{thm2}
 Let $(X, \d)$ be an Alexandrov space with no boundary. Then under the same assumptions of Theorem \ref{thm0.2}, there exists a constant $\bar{C}>0$, depending on $k,n,a_1, a_2, \diam(S\cup Y), {\rm{Per}}(S), S$, such that for any $\mu,\nu\in\mathcal{P}_2(Y)$, we have
 \begin{equation}
 \int_S |\nabla\phi_\mu(x)-\nabla\phi_\nu(x)|^2\,\d\rho(x)\leq \bar{C} \left(\int_S|\phi_\mu(x)-\phi_\nu(x)|^2\,\d\rho(x)\right)^\frac{1}{3},
 \end{equation}
 where $\phi_\mu$ and $\phi_\nu$ are the Kantorovich potentials from $\rho$ to $\mu$ and $\rho$ to $\nu$ respectively.
 \end{theorem}

\begin{lemma}[\cite{zbMATH07794624}, Lemma 5.1]\label{l0.18}
 Let $I\subseteq \R$ be a compact segment and let $u,v: I\rightarrow\R$ be two convex functions such that $|u'|$ and $|v'|$ (defined a.e. on $I$) are uniformly bounded on $I$. Then
 \begin{equation*}
 \|u'-v'\|_{L^2(I)}^2\leq 8 (\|u'\|_{L^\infty(I)}+\|v'\|_{L^\infty(I)})^\frac{4}{3}\|u-v\|_{L^2(I)}^\frac{2}{3}.
 \end{equation*}
 \end{lemma}
 Therefore, applying Lemma \ref{l0.18} to the functions $\zeta|s|^2-u_\mu^{(x,v)}$ and $\zeta|s|^2-u_\nu^{(x,v)}$ on each compact segment $[\alpha_i(x,v),\beta_i(x,v)]$ and using H\"{o}lder inequality, we obtain
 \begin{equation}\label{0.55}
 \begin{aligned}
 &\int_S |\nabla\phi_\mu(x)-\nabla\phi_\nu(x)|^2\,\d\rho(x)\\
 \mathop{\leq}^{\eqref{0.54}}& \frac{a_2}{c_nd_n}\int_{{\rm S}X}\sum_{i\in I_S(x,v)}\int_{\alpha_i(x,v)}^{\beta_i(x,v)}\langle\nabla\phi_\mu(b_s(x,v))-\nabla\phi_\nu(b_s(x,v)),t_s(x,v)\rangle^2\,\d s \,\d\mm_S(x,v)\\
 \leq&C_1 \int_{{\rm S}X}\sum_{i\in I_S(x,v)}\left(\int_{\alpha_i(x,v)}^{\beta_i(x,v)}|\phi_\mu(b_s(x,v))-\phi_\nu(b_s(x,v))|^2\,\d s\right)^\frac{1}{3}\,\d\mm_S(x,v)\\
 \leq&C_1 \int_{{\rm S}X}\left(\# I_S\right)^\frac{2}{3}\left(\int_0^1|\phi_\mu(b_s)-\phi_\nu(b_s)|^2\chi_S(b_s)\,\d s\right)^\frac{1}{3}\d\mm_S\\
 \leq&C_1 \left(\int_{{\rm S}X}\# I_S\,\d\mm_S\right)^\frac{2}{3}\left(\int_{{\rm S}X}\int_0^1|\phi_\mu(b_s)-\phi_\nu(b_s)|^2\chi_S(b_s)\,\d s \,\d\mm_S\right)^\frac{1}{3}\\
 \leq&C_1 \left(\frac{c_n}{a_1}\right)^\frac{1}{3}\left(\int_{{\rm S}X}\# I_S\,\d\mm_S\right)^\frac{2}{3}\left(\int_S|\phi_\mu-\phi_\nu|^2\,\d\rho\right)^\frac{1}{3},
 \end{aligned}
 \end{equation}
 where $C_1$ depends on $k,n,a_2, \diam(S\cup Y)$, and the last inequality follows from the invariance of $\mm_S$ under the geodesic flow.

Let $\phi_\mu$ and $\phi_\nu$ be the Kantorovich potentials from $\rho$ to $\mu$ and $\rho$ to $\nu$ respectively. From \cite{zbMATH05349267,zbMATH06032507} we know that $\nabla \phi_\mu(x), \nabla \phi_\nu(x) \in {\rm{T}}_xX$ and
 $T_\mu(x)= \exp_x(-\nabla \phi_\mu(x))$, $T_\nu(x)= \exp_x(-\nabla \phi_\nu(x))$ are well-defined for almost every $x\in S$. By triangle comparison condition (cf. \cite{zbMATH05342782}), it holds
 \begin{equation}\label{exp}
 \d \big(T_\mu(x), T_\nu(x) \big) \leq C_2|\nabla \phi_\mu(x)-\nabla \phi_\nu(x)|,
 \end{equation}
 for some constant $C_2>0$ which depends on $k, \diam(S\cup Y)$.
 \medskip

If $(S,  {\mathrm d},  \mathfrak m)$  is a compact ${\rm RCD}(K, N)$ space,   it  supports a global $(1,1)$-Poincar\'e inequality (cf.\cite{zbMATH01474795, zbMATH06043352}) and we can remove the John domain assumption:
	\begin{corollary}\label{cor}
		Let $(X,{\mathrm d},\mathfrak m)$ be a compact ${\rm RCD}(K,N)$ metric measure space.  Let $\rho\in \mathcal{P}(X)$ be with $a_1\mathfrak m \leq \rho\leq a_2\mathfrak m$ for some positive constants $a_1,a_2$. Then the conclusion of Theorem \ref{thm0.1} holds.
	\end{corollary}

	Adapting the  strategy of \cite{arXiv:2504.05412}, we can also prove the stability of optimal transport maps on Alexandrov spaces.
		\begin{theorem}\label{thm0.2}
		Let $(X,  {\mathrm d})$ be an $n$-dimensional Alexandrov space, $\mathfrak m$ be the $n$-Hausdorff measure. Under the same assumption for $S, Y$ and $\rho$ as in Theorem \ref{thm0.1}, and if $S$ additionally has finite perimeter, then there exists a constant $C>0$, depending on $k,n,a_1, a_2, \mathop{\rm diam}\nolimits(S\cup Y), {\rm{Per}}(S), S$, such that for any $\mu,\nu\in\mathcal{P}_2(Y)$, we have
		\begin{equation}
			\int_S {\mathrm d}^2(T_\mu(x), T_\nu(x))\,{\mathrm d} \rho(x) \leq CW_1^{1/6}(\mu, \nu),
		\end{equation}
		where $T_\mu$ and $T_\nu$ are the optimal transport maps from $\rho$ to $\mu$ and $\rho$ to $\nu$ respectively.
	\end{theorem}

	\subsection{Strategy: heat kernel-regularized \texorpdfstring{$c$}{c}-transform}

Motivated by   regularized $c$-transforms  using  Gibbs kernels $e^{-c(x, y)/\varepsilon}$ \cite{arXiv:2504.05412},  entropic optimal transport \cite{GTJEMS, GTT25}  and Varadhan's formula
\[
\lim_{t \to 0} -t \log p_{t/2}(x, y) = \frac{1}{2} {\mathrm d}^2(x, y)=c(x,y),
\]
  we make use of {heat kernel-regularized $c$-transform} (Hopf--Cole transform):
\[
\mathop{\rm Lip}\nolimits(X, {\mathrm d}) \ni \psi  \mapsto \Phi_t[\psi](x) = -t \log \int_X e^{\frac{\psi(y)}{t}} p_{t/2}(x, y) \, {\mathrm d}\mathfrak m(y).
\]
This approach  allows us to bypass the low regularity of Kantorovich potentials in the non-smooth setting.

We define the heat kernel regularized Kantorovich functional as
	\begin{equation*}
		\K_t[\varphi]:=\int_S \Phi_t[\varphi]\,{\mathrm d}\rho.
	\end{equation*}
Similar to \cite{arXiv:2504.05412},  the key in our proof is the strong concavity of the functional $\K_t$.
To achieve this,  we  first derive a local strong concavity estimate using heat kernel estimate,    then globalize the estimate on the support of the source measure using a Boman chain argument for John domains.

Unlike the regularized $c$-transform used in \cite{arXiv:2504.05412},  the existence of the boundary of $Y$ may lead to the failure in our heat kernel regularization argument.  This possibility is ruled out by using the measure concentration property of the heat kernel and by making a careful choice of Lipschitz extension.

\medskip

	\noindent {\bf Organization.}  This paper is structured as follows. In Section~\ref{sec3}, we prove quantitative stability of Kantorovich potentials on $\mathrm{RCD}(K,N)$ spaces. In Section \ref{sec4}, we establish stability of optimal transport maps on Alexandrov spaces. The Appendix \ref{A} contains technical lemmas about Poincar\'e inequalities.

	\bigskip

\noindent {\bf Declaration.} The authors declare no conflict of interest and that the manuscript has no associated data.

\medskip

\noindent {\bf Acknowledgement.} This work is supported by the Ministry of Science \& Technology of China (2021YFA1000900, 2021YFA1002200),  National Natural Science Foundation of China  (12201596)   and Shandong Provincial Natural Science Foundation (ZR2025QB05).  The authors  thank Nicola Gigli, Jun Kitagawa, Nan Li,  Cyril Letrouit, Quentin M\'erigot, Luca Tamanini for  helpful discussions and  suggestions on the bibliography.

Motivated by “regularized $c$-transforms … entropic optimal transport … and Varadhan's formula”, the proof of Theorem~\ref{thm0.1} uses a **heat kernel-regularized $c$-transform** (Hopf--Cole transform)
\[
\psi\mapsto \Phi_t[\psi](x)=-t\log\int_X e^{\frac{\psi(y)}{t}}\,p_{t/2}(x,y)\,\mathrm d\mathfrak m(y),
\]
which “allows us to bypass the low regularity of Kantorovich potentials in the non-smooth setting.” One then defines the “heat kernel regularized Kantorovich functional”
\[
\mathcal K_t[\varphi]:=\int_S \Phi_t[\varphi]\,\mathrm d\rho.
\]
“Similar to \cite{arXiv:2504.05412}, the key in our proof is the strong concavity of the functional $\mathcal K_t$.” To obtain this strong concavity, they “first derive a local strong concavity estimate using heat kernel estimate, then globalize the estimate on the support of the source measure using a Boman chain argument for John domains.”

They also note a technical issue specific to this regularization: “the existence of the boundary of $Y$ may lead to the failure in our heat kernel regularization argument,” and this is “ruled out by using the measure concentration property of the heat kernel and by making a careful choice of Lipschitz extension.”

Motivated by “regularized $c$-transforms … entropic optimal transport … and Varadhan's formula”, in establishing the main theorem, the proof uses a **heat kernel-regularized $c$-transform** (Hopf--Cole transform)
\[
\psi\mapsto \Phi_t[\psi](x)=-t\log\int_X e^{\frac{\psi(y)}{t}}\,p_{t/2}(x,y)\,\mathrm d\mathfrak m(y),
\]
which “allows us to bypass the low regularity of Kantorovich potentials in the non-smooth setting.” One then defines the “heat kernel regularized Kantorovich functional”
\[
\mathcal K_t[\varphi]:=\int_S \Phi_t[\varphi]\,\mathrm d\rho.
\]
“Similar to arXiv:2504.05412, the key in our proof is the strong concavity of the functional $\mathcal K_t$.” To obtain this strong concavity, they “first derive a local strong concavity estimate using heat kernel estimate, then globalize the estimate on the support of the source measure using a Boman chain argument for John domains.”

They also note a technical issue specific to this regularization: “the existence of the boundary of $Y$ may lead to the failure in our heat kernel regularization argument,” and this is “ruled out by using the measure concentration property of the heat kernel and by making a careful choice of Lipschitz extension.”,

\label{thm0.1}
		Let $(X,{\mathrm d},\mathfrak m)$ be an ${\rm RCD}(K,N)$ metric measure space. Let $S\subseteq X$ be a John domain and $Y\subseteq X$ be compact with $\mathfrak m(Y)>0$. Let $\rho\in \mathcal{P}(S)$  be with $a_1\mathfrak m\restr{S} \leq \rho\leq a_2\mathfrak m\restr{S}$ for some positive constants $a_1,a_2$. Then there exists a constant $C>0$, depending on $K,N,a_1,a_2, S, \mathop{\rm diam}\nolimits(S\cup Y)$, such that for any $\mu,\nu\in\mathcal{P}(Y)$,
		\begin{equation}
			\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}\leq C W_1^\frac{1}{2}(\mu,\nu),
		\end{equation}
		where $\phi_\mu$ and $\phi_\nu$ are the Kantorovich potentials from $\rho$ to $\mu$ and $\rho$ to $\nu$ respectively.,

Question:
Let \((X,{\mathrm d},\mathfrak m)\) be an \({\rm RCD}(K,N)\) metric measure space. Let \(S\subseteq X\) be a John domain, meaning that \(S\) is a bounded open set for which there exist \(x_0\in S\) and \(\eta>0\) such that every \(x\in S\) can be joined to \(x_0\) by a rectifiable curve \(\gamma:[0,\ell(\gamma)]\to S\), parametrized by arc length, with \(\gamma(0)=x\), \(\gamma(\ell(\gamma))=x_0\), and \({\mathrm d}(\gamma(t),X\setminus S)\ge \eta t\) for all \(t\in[0,\ell(\gamma)]\). Let \(Y\subseteq X\) be compact with \(\mathfrak m(Y)>0\). Assume \(\rho\in \mathcal P(S)\) satisfies \(a_1\,\mathfrak m\!\restriction_S \le \rho \le a_2\,\mathfrak m\!\restriction_S\) for some constants \(a_1,a_2>0\). For each \(\mu\in\mathcal P(Y)\), let \(\phi_\mu\) denote a Kantorovich potential for the quadratic cost \(c(x,y)=\tfrac12{\mathrm d}(x,y)^2\) from \(\rho\) to \(\mu\), normalized by \(\int_S \phi_\mu\,{\mathrm d}\rho=0\). Also let
\[
W_1(\mu,\nu):=\inf_{\pi\in\Pi(\mu,\nu)}\int_{Y\times Y}{\mathrm d}(y,z)\,{\mathrm d}\pi(y,z)
\]
be the 1-Wasserstein distance. Which quantitative estimate holds for all \(\mu,\nu\in\mathcal P(Y)\)?

There exists a constant \(C>0\), depending only on \(K,N,a_1,a_2,S\), and \(\operatorname{diam}(S\cup Y)\), such that for every \(\mu,\nu\in\mathcal P(Y)\),
\[
\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}=\int_S |\phi_\mu-\phi_\nu|\,{\mathrm d}\rho \le C\,W_1(\mu,\nu)^{1/2}.
\]

Choices:
B.
There exists a constant \(C>0\), depending only on \(K,N,a_1,a_2,S\), and \(\operatorname{diam}(S\cup Y)\), such that for every \(\mu,\nu\in\mathcal P(Y)\),
\[
\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}=\int_S |\phi_\mu-\phi_\nu|\,{\mathrm d}\rho \le C\,W_1(\mu,\nu).
\]
C.
There exists a constant \(C>0\), depending only on \(K,N,a_1,a_2,S\), and \(\operatorname{diam}(S\cup Y)\), such that for every \(\mu,\nu\in\mathcal P(Y)\),
\[
\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}=\int_S |\phi_\mu-\phi_\nu|\,{\mathrm d}\rho \le C.
\]
D.
There exists a constant \(C>0\), depending only on \(K,N,a_1,a_2\), such that for every John domain \(S\subseteq X\), every compact \(Y\subseteq X\) with \(\mathfrak m(Y)>0\), and every \(\mu,\nu\in\mathcal P(Y)\),
\[
\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}=\int_S |\phi_\mu-\phi_\nu|\,{\mathrm d}\rho \le C\,W_1(\mu,\nu)^{1/2}.
\]
E.
There exists a constant \(C>0\), depending only on \(K,N,a_1,a_2,S\), and \(\operatorname{diam}(S\cup Y)\), such that for every \(\mu,\nu\in\mathcal P(Y)\),
\[
\|\phi_\mu-\phi_\nu\|_{L^\infty(S)}\le C\,W_1(\mu,\nu)^{1/2}.
\]


Question:
Let $(X,\mathrm d,\mathfrak m)$ be an ${\rm RCD}(K,N)$ metric measure space. Let $S\subseteq X$ be a John domain, meaning a bounded open set for which there exist $x_0\in S$ and $\eta>0$ such that every $x\in S$ can be joined to $x_0$ by a rectifiable curve $\gamma:[0,\ell(\gamma)]\to S$, parametrized by arc length, with $\gamma(0)=x$, $\gamma(\ell(\gamma))=x_0$, and $\mathrm d(\gamma(t),S^c)\ge \eta t$ for all $t\in[0,\ell(\gamma)]$. Let $Y\subseteq X$ be compact with $\mathfrak m(Y)>0$, and let $\rho\in\mathcal P(S)$ satisfy
\[
a_1\,\mathfrak m\!\restriction_S\le \rho\le a_2\,\mathfrak m\!\restriction_S
\]
for some constants $a_1,a_2>0$. For each $\mu\in\mathcal P(Y)$, let $\phi_\mu$ denote the Kantorovich potential from $\rho$ to $\mu$ for the quadratic transport cost $c(x,y)=\tfrac12\mathrm d(x,y)^2$, normalized by $\int_S \phi_\mu\,d\rho=0$. Which explicit uniform stability statement holds for all target measures $\mu,\nu\in\mathcal P(Y)$?

There exists a constant $C>0$, depending only on $K,N,a_1,a_2,S$, and $\operatorname{diam}(S\cup Y)$, such that for every $\mu,\nu\in\mathcal P(Y)$,
\[
\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}\le C\,W_1(\mu,\nu)^{1/2},
\]
where $W_1$ is the $1$-Wasserstein distance between $\mu$ and $\nu$.

Choices:
B.
There exists a constant $C>0$, depending only on $K,N,a_1,a_2,S$, and $\operatorname{diam}(S\cup Y)$, such that for every $\mu,\nu\in\mathcal P(Y)$,
\[
\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}\le C\,W_1(\mu,\nu),
\]
where $W_1$ is the $1$-Wasserstein distance between $\mu$ and $\nu$.
C.
There exists a constant $C>0$, depending only on $K,N,a_1,a_2,S$, and $\operatorname{diam}(S\cup Y)$, such that for every $\mu,\nu\in\mathcal P(Y)$,
\[
\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}\le C,
\]
where $\phi_\mu$ and $\phi_\nu$ are the normalized Kantorovich potentials from $\rho$ to $\mu$ and $\rho$ to $\nu$, respectively.
D.
For every $\mu,\nu\in\mathcal P(Y)$ there exists a constant $C_{\mu,\nu}>0$, depending on $K,N,a_1,a_2,S$, $\operatorname{diam}(S\cup Y)$, and on the pair $(\mu,\nu)$, such that
\[
\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}\le C_{\mu,\nu}\,W_1(\mu,\nu)^{1/2}.
\]
E.
There exists a constant $C>0$, depending only on $K,N,a_1,a_2,S$, and $\operatorname{diam}(S\cup Y)$, such that for every $\mu,\nu\in\mathcal P(Y)$,
\[
\|\phi_\mu-\phi_\nu\|_{L^\infty(\rho)}\le C\,W_1(\mu,
\nu)^{1/2},
\]
where $W_1$ is the $1$-Wasserstein distance between $\mu$ and $\nu$.


The systematic study of multiple zeta values (MZVs) started
in the early 1990s with the work of Hoffman \cite{[H-PJM]}
and Zagier \cite{[Z-FECM]}. Let $r\ge 1$ be an integer. For any multiple index
$\mathbf{s}=(s_1,s_2,\cdots,s_r)\in \mathbb{Z}_{\geq 1}^r$ with
$s_r\textgreater 1$, the {\it multiple zeta value (MZV for short)}
is defined by
\begin{align*}
\zeta(s_1,s_2,\cdots,s_r):=\sum_{1\leq k_1<k_2<\cdots<k_r}
\frac{1}{k_1^{s_1}k_2^{s_2}\cdots k_r^{s_r}}.
\end{align*}
The {\it weight} of MZV is defined to be the number
$|\mathbf{s}|:=s_1+\cdots+s_r$, and its {\it depth} is defined
by $l(\mathbf{s}):=r$. For brevity, we let $\{a_1,\cdots,a_k\}^d$
denote the sequence formed by repeating the sequence
$\{a_1,\cdots,a_k\}$ exactly $d$ times. The simplest precise
evaluations of MZVs and MZSVs are given by
\begin{eqnarray*}
\zeta(\{2\}^d)=\frac{\pi^{2d}}{(2d+1)!}.
\end{eqnarray*}

A central problem in MZV is to investigate the algebraic structure
of the $\mathbb{Q}$-vector space $\mathfrak{Z}_k$ spanned by MZVs
of weight $k$. In this direction, Hoffman \cite{[H-JA]} proposed a
conjecture that stats that every MZV of weight $k$ can be expressed as a
$\mathbb{Q}$-linear combination of those MZVs whose entries are only
$2$'s and $3$'s. This conjecture was proved by Brown \cite{[B-ANN]}
using motivic method. In particular, Brown showed that
$H(a,b):=\zeta(\{2\}^a,3,\{2\}^b)$ can be expressed as a $\mathbb{Q}$-linear
combination of the products $\pi^{2m}\zeta(2n+1)$ with $m+n=a+b+1$.
In 2012, Zagier \cite{[Z-AM]} found surprisingly an elegant and explicit
formula as follows.

\section{Introduction}
The systematic study of multiple zeta values (MZVs) started
in the early 1990s with the work of Hoffman \cite{[H-PJM]}
and Zagier \cite{[Z-FECM]}. Let $r\ge 1$ be an integer. For any multiple index
$\mathbf{s}=(s_1,s_2,\cdots,s_r)\in \mathbb{Z}_{\geq 1}^r$ with
$s_r\textgreater 1$, the {\it multiple zeta value (MZV for short)}
is defined by
\begin{align*}
\zeta(s_1,s_2,\cdots,s_r):=\sum_{1\leq k_1<k_2<\cdots<k_r}
\frac{1}{k_1^{s_1}k_2^{s_2}\cdots k_r^{s_r}}.
\end{align*}
The {\it weight} of MZV is defined to be the number
$|\mathbf{s}|:=s_1+\cdots+s_r$, and its {\it depth} is defined
by $l(\mathbf{s}):=r$. For brevity, we let $\{a_1,\cdots,a_k\}^d$
denote the sequence formed by repeating the sequence
$\{a_1,\cdots,a_k\}$ exactly $d$ times. The simplest precise
evaluations of MZVs and MZSVs are given by
\begin{eqnarray*}
\zeta(\{2\}^d)=\frac{\pi^{2d}}{(2d+1)!}.
\end{eqnarray*}

A central problem in MZV is to investigate the algebraic structure
of the $\mathbb{Q}$-vector space $\mathfrak{Z}_k$ spanned by MZVs
of weight $k$. In this direction, Hoffman \cite{[H-JA]} proposed a
conjecture that stats that every MZV of weight $k$ can be expressed as a
$\mathbb{Q}$-linear combination of those MZVs whose entries are only
$2$'s and $3$'s. This conjecture was proved by Brown \cite{[B-ANN]}
using motivic method. In particular, Brown showed that
$H(a,b):=\zeta(\{2\}^a,3,\{2\}^b)$ can be expressed as a $\mathbb{Q}$-linear
combination of the products $\pi^{2m}\zeta(2n+1)$ with $m+n=a+b+1$.
In 2012, Zagier \cite{[Z-AM]} found surprisingly an elegant and explicit
formula as follows.

In a similar way, Hoffman \cite{[H-CNTP]} defined the multiple
$t$-values (``odd variant'' of MZVs). For any multiple index
$\mathbf{s}=(s_1,s_2,...,s_r)\in \mathbb{Z}_{\geq 1}^r$, the
{\it multiple $t$-value } is defined by
\begin{align*}
t(s_1,s_2,...,s_r):=\sum_{1\leq k_1<k_2<...<k_r}
\frac{1}{(2k_1-1)^{s_1}(2k_2-1)^{s_2}...(2k_r-1)^{s_r}},
\end{align*}
where if the index is empty, then we define the value $t(\emptyset):=1$.
As we stated above, first let us recall the the analogue formula
for the simplest multiple zeta value (\cite{[MK-PSPM]},\cite{[Z-FM]})
\begin{align*}
t(\mathop{\underbrace{2,2,...,2}}_{n})=\frac{\pi^{2n}}{2^{2n}(2n)!}.
\end{align*}

By the method of establishing the generating functions, Murakami \cite{[M-MA]}
obtained an equivalent version of Theorem \ref{theorem1.01}. Define
$$\widetilde{t}(k_1,k_2,...,k_r):=2^{|k|}t(k_1,k_2,...,k_r)$$
and set
$$K(a,b):=\widetilde{t}(\{2\}^a,3,\{2\}^b),K(n)=\widetilde{t}(\{2\}^n)$$
for any $a,b,n\in \mathbb{N}_{\geq 0}$. Then for all integers $a,b\geq 0$,
Murakami \cite{[M-MA]} provided the following formula
\begin{align}\label{eq1.1}
K(a,b)=\sum_{k=1}^{a+b+1}(-1)^{k-1}d_{a,b}K(a+b-k+1)\zeta(2k+1),
\end{align}
where
$$d_{a,b}:=\binom{2k}{2a+1}+\binom{2k}{2b+1}\Big(1-\frac{1}{2^{2k}}\Big).$$
Moreover, let $T(a,b):=t(\{2\}^a,3,\{2\}^b)$. Then \eqref{eq1.1} is equivalent to
\begin{align}
T(a,b)=2\sum_{k=1}^{r+s+1}(-1)^{k-1}d_{a,b}\frac{1}{2^{2k}}\zeta(2k+1)
t(\{2\}^{a+b-k+1}),
\end{align}

\begin{conjecture}\label{con2}
For nonnegative integers $a$ and $b$, we have
\begin{align}
T(a,b)=\frac{-2}{(2a+1)!}\left(\frac{\pi}{2}\right)^{2a+2b+2}
\sum_{n=0}^{\infty}\frac{\zeta(2n)}{(2n+2a+1)(2n+2a+2)\cdots(2n+2a+2b+2)2^{2n}}.
\end{align}
\end{conjecture}
\noindent Notice that Conjectures 1.2 and 1.3 are shown in \cite{[L-MZ]}
to be true when $b=0$. But for the general case $b>0$, these two conjectures
are still kept open so far.

\begin{theorem}\label{theorem1.1}
For nonnegative integers $a$ and $b$, we have $H(a,b)=L(a,b)$, where
\begin{align*}
L(a,b):=\frac{-4\pi^{2a+2b+2}}{(2a+2)!}\sum_{n=0}^{\infty}
\frac{\zeta(2n)}{(2n+2a+2)(2n+2a+3)\cdots(2n+2a+2b+3)2^{2n}}.
\end{align*}
\end{theorem}

\begin{theorem}\label{theorem1.2}
For nonnegative integers $a$ and $b$, we have $T(a,b)=\hat L(a,b)$, where
\begin{align*}
\hat L(a,b):=\frac{-2}{(2a+1)!}\left(\frac{\pi}{2}\right)^{2a+2b+2}
\sum_{n=0}^{\infty}\frac{\zeta(2n)}{(2n+2a+1)(2n+2a+2)\cdots(2n+2a+2b+2)2^{2n}}.
\end{align*}
\end{theorem}

First, we consider the integral
$$\int_{0}^{\frac{\pi}{2}} x^{2a + 2}\left(1 - \frac{2}{\pi} x\right)^{2b + 1} \cot(x)\mathrm{d}x.$$
By using the well-known identity (see, for example, Theorem 1.2.4 of \cite{[AAR]})
\begin{align}\label{eq3.1'}
x\cot(x)=-2\sum_{n=0}^{\infty}\frac{\zeta(2n)}{\pi^{2n}}x^{2n},
\end{align}
we derive that
\begin{align}\label{eq3.2}
&\int_{0}^{\frac{\pi}{2}}x^{2a+2}\Big(1-\frac{2}{\pi} x\Big)^{2b+1}
\cot(x)\mathrm{d}x\nonumber\\
=&-2\int_{0}^{\frac{\pi}{2}}x^{2a+1}\Big(1-\frac{2}{\pi} x\Big)^{2b+1}
\sum_{n=0}^{\infty}\frac{\zeta(2n)}{\pi^{2n}}x^{2n} \mathrm{d}x \nonumber\\
=&-2\sum_{n=0}^{\infty}\frac{\zeta(2n)}{\pi^{2n}}
\int_{0}^{\frac{\pi}{2}}x^{2n+2a+1}\Big(1-\frac{2}{\pi} x\Big)^{2b+1}\mathrm{d}x.
\end{align}

\begin{theorem}\label{theorem1.01}
(Zagier \cite{[Z-AM]}) For non-negative integers $a$ and $b$,
\begin{align}
H(a,b)=2\sum_{k=1}^{a+b+1}(-1)^kc_{a,b}\zeta(2k+1)
\zeta(\{2\}^{a+b+1-k}),
\end{align}
where
$$c_{a,b}:=\binom{2k}{2a+2}-\Big(1-\frac{1}{2^{2k}}\Big)\binom{2k}{2b+1}.$$
\end{theorem}

\noindent The proof of Theorem \ref{theorem1.01} is given by computing the associated
generating functions of both sides in a closed form, and then showing
they are entire functions of exponential growth that agree at
sufficiently many points to force their equality. Later on, his proof
was simplified by Li \cite{[L-MRL]}. In 2017, Hessami Pilehrood
and Hessami Pilehrood \cite{[HHT-JMAA]} gave an alternative proof of Theorem
\ref{theorem1.01}.

In a similar way, Hoffman \cite{[H-CNTP]} defined the multiple
$t$-values  (``odd variant'' of MZVs). For any multiple index
$\mathbf{s}=(s_1,s_2,...,s_r)\in \mathbb{Z}_{\geq 1}^r$, the
{\it multiple $t$-value } is defined by
\begin{align*}
t(s_1,s_2,...,s_r):=\sum_{1\leq k_1<k_2<...<k_r}
\frac{1}{(2k_1-1)^{s_1}(2k_2-1)^{s_2}...(2k_r-1)^{s_r}},
\end{align*}
where if the index is empty, then we define the value $t(\emptyset):=1$.
As we stated above, first let us recall the the analogue formula
for the simplest multiple zeta value (\cite{[MK-PSPM]},\cite{[Z-FM]})
\begin{align*}
t(\mathop{\underbrace{2,2,...,2}}_{n})=\frac{\pi^{2n}}{2^{2n}(2n)!}.
\end{align*}

By the method of establishing the generating functions, Murakami \cite{[M-MA]}
obtained an equivalent version of Theorem \ref{theorem1.01}. Define
$$\widetilde{t}(k_1,k_2,...,k_r):=2^{|k|}t(k_1,k_2,...,k_r)$$
and set
$$K(a,b):=\widetilde{t}(\{2\}^a,3,\{2\}^b),K(n)=\widetilde{t}(\{2\}^n)$$
for any $a,b,n\in \mathbb{N}_{\geq 0}$. Then for all integers $a,b\geq 0$,
Murakami \cite{[M-MA]} provided the following formula
\begin{align}\label{eq1.1}
K(a,b)=\sum_{k=1}^{a+b+1}(-1)^{k-1}d_{a,b}K(a+b-k+1)\zeta(2k+1),
\end{align}
where
$$d_{a,b}:=\binom{2k}{2a+1}+\binom{2k}{2b+1}\Big(1-\frac{1}{2^{2k}}\Big).$$
Moreover, let $T(a,b):=t(\{2\}^a,3,\{2\}^b)$. Then \eqref{eq1.1} is equivalent to
\begin{align}
T(a,b)=2\sum_{k=1}^{r+s+1}(-1)^{k-1}d_{a,b}\frac{1}{2^{2k}}\zeta(2k+1)
t(\{2\}^{a+b-k+1}),
\end{align}

In a different approach, Lupu \cite{[L-MZ]} provided elementary proofs of
Zagier's formula for the special case $b=0$, and also established a Zagier-type
formula for multiple $t$-values when $b=0$. For the general case $b>0$, Lupu
\cite{[L-MZ]} proposed the following two conjectures.

\begin{conjecture}\label{con1}
For nonnegative integers $a$ and $b$, we have
\begin{align}
H(a,b)=\frac{-4\pi^{2a+2b+2}}{(2a+2)!}\sum_{n=0}^{\infty}
\frac{\zeta(2n)}{(2n+2a+2)(2n+2a+3)\cdots(2n+2a+2b+3)2^{2n}}.
\end{align}
\end{conjecture}

\begin{conjecture}\label{con2}
For nonnegative integers $a$ and $b$, we have
\begin{align}
T(a,b)=\frac{-2}{(2a+1)!}\left(\frac{\pi}{2}\right)^{2a+2b+2}
\sum_{n=0}^{\infty}\frac{\zeta(2n)}{(2n+2a+1)(2n+2a+2)\cdots(2n+2a+2b+2)2^{2n}}.
\end{align}
\end{conjecture}
\noindent Notice that Conjectures 1.2 and 1.3 are shown in \cite{[L-MZ]}
to be true when $b=0$. But for the general case $b>0$, these two conjectures
are still kept open so far.

In 2026, Lai, Lupu, and Orr \cite{[LLO-PAMS]} gave elementary and direct proofs
for both $H(a, b)$ and $T(a, b)$. In fact, they \cite{[LLO-PAMS]} first provided
an integral expression for $H(a, b)$, and then showed that this integral expression
is equal to Zagier's result. The same proof method was applied to $T(a, b)$.

In this paper, our main goal is to investigate Conjectures \ref{con1} and \ref{con2}.
We will prove that both conjectures of Lupu are true. Actually, by employing the integral
expressions presented in \cite{[LLO-PAMS]} for $H(a,b)$ and $T(a,b)$ and using
the properties of Beta and Gamma functions, we show the following main results
of this paper.

\begin{theorem}\label{theorem1.1}
For nonnegative integers $a$ and $b$, we have $H(a,b)=L(a,b)$, where
\begin{align*}
L(a,b):=\frac{-4\pi^{2a+2b+2}}{(2a+2)!}\sum_{n=0}^{\infty}
\frac{\zeta(2n)}{(2n+2a+2)(2n+2a+3)\cdots(2n+2a+2b+3)2^{2n}}.
\end{align*}
\end{theorem}

\begin{theorem}\label{theorem1.2}
For nonnegative integers $a$ and $b$, we have $T(a,b)=\hat L(a,b)$, where
\begin{align*}
\hat L(a,b):=\frac{-2}{(2a+1)!}\left(\frac{\pi}{2}\right)^{2a+2b+2}
\sum_{n=0}^{\infty}\frac{\zeta(2n)}{(2n+2a+1)(2n+2a+2)\cdots(2n+2a+2b+2)2^{2n}}.
\end{align*}
\end{theorem}

This paper is organized as follows. In Section 2, we supply several preliminary lemmas
that are needed in the proofs of our main results. Sections 3 and 4 are devoted to
the proofs of Theorems \ref{theorem1.1} and \ref{theorem1.2}, respectively.

The post-theorem introduction sketches Zagier's strategy for proving Theorem~\ref{theorem1.01}: compute the associated generating functions of both sides "in a closed form," then show these generating functions are "entire functions of exponential growth" and that they "agree at sufficiently many points to force their equality." It is also noted that "his proof was simplified by Li" and that Hessami Pilehrood and Hessami Pilehrood gave "an alternative proof" of Theorem~\ref{theorem1.01}.

The post-theorem introduction sketches Zagier's strategy for proving the main theorem: compute the associated generating functions of both sides "in a closed form," then show these generating functions are "entire functions of exponential growth" and that they "agree at sufficiently many points to force their equality." It is also noted that "his proof was simplified by Li" and that Hessami Pilehrood and Hessami Pilehrood gave "an alternative proof" of the main theorem.

Question: Let \(H(a,b):=\zeta(\{2\}^a,3,\{2\}^b)\), where \(a,b\in \mathbb Z_{\ge 0}\), \(\{2\}^m\) denotes the string consisting of \(m\) copies of \(2\), and \(\zeta(s_1,\dots,s_r)\) is the multiple zeta value. Which explicit identity holds for \(H(a,b)\)?

For all nonnegative integers \(a\) and \(b\),
\[
H(a,b)=2\sum_{k=1}^{a+b+1}(-1)^k c_{a,b}\,\zeta(2k+1)\,\zeta(\{2\}^{a+b+1-k}),
\]
where
\[
c_{a,b}=\binom{2k}{2a+2}-\left(1-\frac{1}{2^{2k}}\right)\binom{2k}{2b+1}.
\]

Choices:
B.
For all nonnegative integers \(a\) and \(b\),
\[
H(a,b)=2\sum_{k=0}^{a+b+1}(-1)^k c_{a,b}\,\zeta(2k+1)\,\zeta(\{2\}^{a+b+1-k}),
\]
where
\[
c_{a,b}=\binom{2k}{2a+2}-\left(1-\frac{1}{2^{2k}}\right)\binom{2k}{2b+1}.
\]
C.
For all nonnegative integers \(a\) and \(b\), \(H(a,b)\) is a \(\mathbb{Q}\)-linear combination of products
\[
\zeta(2k+1)\,\zeta(\{2\}^{a+b+1-k}) \qquad (1\le k\le a+b+1).
\]
D.
For all nonnegative integers \(a\) and \(b\),
\[
H(a,b)=2\sum_{k=1}^{a+b+1}(-1)^k c_{a,b}\,\zeta(2k+1)\,\zeta(\{2\}^{a+b+1-k}),
\]
where
\[
c_{a,b}=\binom{2k}{2a+2}-\binom{2k}{2b+1}.
\]
E.
For all nonnegative integers \(a\) and \(b\),
\[
H(a,b)=2\sum_{k=1}^{a+b+1}(-1)^k c_{a,b}\,\zeta(2k+1)\,\zeta(\{2\}^{a+b+1-k}),
\]
where
\[
c_{a,b}=\binom{2k}{2a+1}-\left(1-\frac{1}{2^{2k}}\right)\binom{2k}{2b+1}.
\]

dropped the explicit coefficient formula and exact equality

specific coefficient factor \((1-2^{-2k})\) in the closed-form generating-function identity

dependence of the first binomial entry on \(2a+2\) rather than \(2a+1\)