text:
\newpage
\appendix
\section{Proofs for \S\ref{sec:proof}-\ref{sec:extension}}
For notational simplicity, we denote by $\EE_{\cD}$ the expectation with respect to $x \sim \cD$ and $\EE_{\tilde \cD}$ the expectation with respect to $(x, r, x') \sim \tilde \cD$. Also, with a slight abuse of notations, we write $\theta^{(m)} = \{\theta_i\}_{i=1}^m$.
\subsection{Proof of Lemma \ref{lem:rho-star}} \label{sec:pf-rho-star}
\begin{proof}

	\noindent{\bf Existence and uniqueness of $Q^*$.}
	To establish the existence of the fixed point solution $Q^*$ to the projected Bellman equation $ Q = \Pi_\cF\cT^\pi Q $, it suffices to show that $\Pi_\cF\cT^\pi: \cF\rightarrow \cF$ is a contraction mapping. It holds for any $Q^1, Q^2 \in \cF$ that
	\begin{align*}

	\norm{\Pi_\cF\cT^\pi Q^1 - \Pi_\cF\cT^\pi Q^2}_{\cL_2(\cD)}^2 &\le  \gamma^2\cdot \EE_{\tilde \cD}\Bigl[ \bigl(Q^1(x') - Q^2(x')  \bigr)^2 \Bigr] \nonumber \\
	&= \gamma^2\cdot \norm[\big]{Q^1 - Q^2}_{\cL_2(\cD)}^2,
	\end{align*}
	where the last equality follows from the fact that $\cD$ is the stationary distribution. Thus, $\Pi_\cF\cT^\pi: \cF\rightarrow \cF$ is a contraction mapping. Note that $\cF$ is complete.
	Following from the Banach fixed point theorem \citep{conway2019course}, there exists a unique $Q^* \in \cF$ that solves the projected Bellman equation $Q = \Pi_\cF\cT^\pi Q$. Moreover, by the definition of $\cF$ in \eqref{eq:func-class}, there exists $\bar\rho \in \sP_2(\RR^D)$ such that
	\begin{align*}

	Q^*(x) = \int \sigma(x; \theta) \, \rd\bar\rho( \theta).
	\end{align*}

	\vskip4pt

	\noindent{\bf Proof of (i) in Lemma \ref{lem:rho-star}.}
	We define
	\begin{align}
	\label{eq:rho-bar}
	\rho^* = \rho_0 + \alpha^{-1} \cdot (\bar \rho - \rho_0).
	\end{align}
	By the definition of $Q(\cdot ; \rho)$ in \eqref{eq:nn-infty} and the fact that $Q(x; \rho_0) = 0$, we have that $Q(x; \rho^*) = Q^*(x)$, which completes the proof of (i) in Lemma \ref{lem:rho-star}.

	\vskip4pt

	\noindent{\bf Proof of (ii) in Lemma \ref{lem:rho-star}.}
	For (ii) of Lemma \ref{lem:rho-star}, note that $Q(\cdot; \rho^*) = \Pi_\cF\cT^\pi Q(\cdot; \rho^*)$. Thus, we have that
	\begin{align*}

	\inp[\big]{ Q(\cdot; \rho^*) - \cT^\pi Q(\cdot; \rho^*) }{f(\cdot ) - Q(\cdot; \rho^*)}_{\cD} \ge 0, \quad \forall f\in \cF,
	\end{align*}
	which further implies that
	\begin{align}
	\label{eq:pf-rh2}
	\EE_{\tilde \cD}\Bigl[ \bigl( Q(x; \rho^*) - r - \gamma\cdot  Q(x'; \rho^*) \bigr) \cdot \int \sigma(x; \theta) \,\rd ( \rho - \bar \rho)(\theta) \Bigr] \ge 0, \quad \forall \rho\in \sP_2(\RR^D).
	\end{align}
	Let $\rho = (\textrm{id} + h \cdot v)_\sharp \bar \rho$ for a sufficiently small scaling parameter $h\in \RR_+$ and any Lipschitz-continuous mapping $v : \RR^D \rightarrow \RR^D$. Then, following from \eqref{eq:pf-rh2}, we have that
	\begin{align}
	\label{eq:pf-rh3}
	\int \EE_{\tilde \cD}\biggl[ \bigl( Q(x; \rho^*) - r - \gamma \cdot Q(x'; \rho^*) \bigr) \cdot \Bigl(\sigma\bigl(x; \theta+ h\cdot v(\theta)\bigr) - \sigma(x;\theta) \Bigr) \biggr]  \,\rd \bar \rho(\theta) \ge 0
	\end{align}
	for any $ v : \RR^D \rightarrow \RR^D$.
	Dividing the both sides of \eqref{eq:pf-rh3} by $h$ and letting $h\rightarrow 0^+$, we have for any $ v: \RR^D \rightarrow \RR^D$ that
	\begin{align*}

	0 &\le \int \EE_{\tilde \cD}\Bigl[ \bigl( Q(x; \rho^*) - r - \gamma \cdot Q(x'; \rho^*) \bigr) \cdot \inp[\big]{\nabla_\theta \sigma(x; \theta)}{v(\theta)} \Bigr]  \,\rd \bar \rho(\theta) \nonumber\\
	& = -\alpha^{-1} \cdot \int \inp[\big]{g(\theta; \rho^*)}{v(\theta)}\, \rd \bar \rho(\theta),
	\end{align*}
	where the equality follows from the definition of $g$ in \eqref{eq:g-rho}.
	Thus, we have that $g(\theta; \rho^*) = 0$ for $\bar \rho$-a.e., which completes the proof of (ii) in Lemma \ref{lem:rho-star}.

	\vskip4pt

	\noindent{\bf Proof of (iii) in Lemma \ref{lem:rho-star}.}
	Following from the definition of $\rho^*$ in \eqref{eq:rho-bar}, we have that
	\begin{align*}

	&D_{\chi^2}(\rho^* \,\|\, \rho_0) \nonumber \\
	&\quad = \int  \biggl( \frac{\rho^*(\theta)}{\rho_0(\theta)} - 1 \biggr)^2 \,\rd \rho_0(\theta) = \int \biggl( \frac{(1-\alpha^{-1}) \cdot \rho_0(\theta) + \alpha^{-1}\cdot \bar \rho(\theta)}{\rho_0(\theta)} - 1\biggr) \,\rd\rho_0(\theta)   = \alpha^{-2} \cdot \bar D^2,
	\end{align*}
	where $\bar{D} = D_{\chi^2}(\bar{\rho} \,\|\, \rho_0)^{1/2}$. By Lemma \ref{lem:talagrand}, we have that
	\begin{align*}

	\cW_2(\rho^*, \rho_0) \le D_{\rm KL}(\rho^*\,\|\,\rho_0)^{1/2} \le  D_{\chi^2}(\rho^* \,\|\, \rho_0)^{1/2} \le \alpha^{-1} \cdot \bar D,
	\end{align*}
	which completes the proof of (iii) in Lemma \ref{lem:rho-star}.
\end{proof}

\subsection{Proof of Lemma \ref{lem:descent}}\label{sec:pf-descent}
We first introduce the following lemmas. The first lemma establishes the one-point monotonicity of $g(\cdot; \beta_t)$ along a curve $\beta: [0,1 ] \rightarrow \sP_2(\RR^D)$ on the Wasserstein space.

\begin{lemma}
	\label{lem:td-q}
	Let $\beta: [0, 1] \rightarrow \sP_2(\RR^D)$ be a curve such that $\partial_t \beta_t = -\Div(\beta_t \cdot v_t)$ for a vector field $v$. We have that
	\begin{align*}

	\inp[\big]{\partial_t g(\cdot; \beta_t)}{v_t}_{\beta_t} \le -(1-\gamma) \cdot \EE_\cD\Bigl[ \bigl( \partial_t Q(x; \beta_t) \bigr)^2 \Bigr].
	\end{align*}
	Furthermore, we have that
	\begin{align}
	\label{eq:td-q1}
	\int_{0}^{1}\inp[\big]{\partial_s g(\cdot; \beta_s)}{v_s}_{\beta_s} \, \rd s \le - (1-\gamma) \cdot \EE_\cD\Bigl[ \bigl( Q(x; \beta_0) - Q(x; \beta_1)\bigr)^2 \Bigr].
	\end{align}
\end{lemma}

\begin{proof}
	Following from the definition of $g$ in \eqref{eq:g-rho}, we have that
	\begin{align*}
	\partial_t g(\theta; \beta_t) = -\alpha \cdot \EE_{\tilde \cD} \Bigl[ \partial_t \bigl(Q(x; \beta_t) - \gamma \cdot Q(x'; \beta_t)\bigr) \cdot \nabla_\theta \sigma(x; \theta) \Bigr].
	\end{align*}
	Thus, following from integration by parts and the continuity equation $\partial_t \beta_t =- \Div(\beta_t \cdot v_t)$, we have that
	\begin{align}
	\label{eq:pf-td-q1}
	\inp[\big]{\partial_t g(\cdot; \beta_t)}{ v_t}_{\beta_t}
	&= -\int \inp[\bigg]{\alpha \cdot \EE_{\tilde \cD} \Bigl[ \partial_t \bigl(Q(x; \beta_t) - \gamma \cdot Q(x'; \beta_t)\bigr) \cdot \nabla_\theta \sigma(x; \theta) \Bigr]}{v_t(\theta) \cdot \beta_t(\theta)}\,\rd \theta \nonumber \\
	&= - \int \alpha \cdot \EE_{\tilde \cD} \Bigl[ \partial_t \bigl(Q(x; \beta_t) - \gamma \cdot Q(x'; \beta_t)\bigr) \cdot \sigma(x; \theta)\Bigr] \cdot \partial_t \beta_t(\theta) \, \rd \theta \nonumber \\
	&= - \EE_{\tilde \cD} \Bigl[ \partial_t \bigl(Q(x; \beta_t) - \gamma \cdot Q(x'; \beta_t)\bigr) \cdot \partial_t Q(x; \beta_t)\Bigr],
	\end{align}
	where the last equality follows from the definition of $Q$ in \eqref{eq:nn-infty}. Applying the Cauchy-Schwartz inequality to \eqref{eq:pf-td-q1}, we have that
	\begin{align}
	\label{eq:pf-td-q2}
	\inp[\big]{\partial_t g(\cdot; \beta_t)}{ v_t}_{\beta_t} &= - \EE_{\tilde \cD} \Bigl[ \bigl(\partial_t Q(x; \beta_t)\bigr)^2\Bigr] + \gamma \cdot \EE_{\tilde \cD}\bigl[ \partial_t Q(x'; \beta_t) \cdot \partial_t Q(x; \beta_t)\bigr] \nonumber \\
	&\le - \EE_{\tilde \cD} \Bigl[ \bigl(\partial_t Q(x; \beta_t)\bigr)^2\Bigr] + \gamma \cdot \EE_{\tilde \cD} \Bigl[ \bigl(\partial_t Q(x; \beta_t)\bigr)^2\Bigr]^{1/2} \cdot \EE_{\tilde \cD} \Bigl[ \bigl(\partial_t Q(x'; \beta_t)\bigr)^2\Bigr]^{1/2} \nonumber \\
	& = -(1-\gamma) \cdot \EE_{\cD} \Bigl[ \bigl(\partial_t Q(x; \beta_t)\bigr)^2\Bigr],
	\end{align}
	where the last equality follows from the fact that the marginal distributions of $\tilde \cD$ with respect to $x$ and $x'$ are $\cD$, since $\cD$ is the stationary distribution.
	Furthermore, we have that
	\begin{align*}
	\int_0^1 \inp[\big]{\partial_s g(\cdot; \beta_s)}{ v_s}_{\beta_s} \, \rd s &\le - (1-\gamma) \cdot \int_0^1 \EE_\cD\Bigl[ \bigl( \partial_s Q(x; \beta_s) \bigr)^2 \Bigr] \, \rd s \\
	& \le - (1-\gamma) \cdot \EE_\cD\biggl[ \Bigl( \int_0^1 \partial_s Q(x; \beta_s) \, \rd s\Bigr)^2 \biggr] \\
	&= - (1-\gamma) \cdot \EE_\cD\Bigl[ \bigl( Q(x; \beta_1) - Q(x; \beta_0)\bigr)^2 \Bigr],
	\end{align*}
	which completes the proof of Lemma \ref{lem:td-q}.
\end{proof}

The following lemma upper bounds the norms of $Q$ and $\nabla_\theta g$.
\begin{lemma}
	\label{lem:boundness}
	Under Assumptions \ref{asp:data} and \ref{asp:activation}, it holds for any $\rho\in \sP_2(\RR^D)$ that
	\begin{align}
	\label{eq:lem-boundness1}
	\sup_{x\in \cX} \bigl| Q(x; \rho)\bigr| &\le \alpha\cdot \min \bigl\{B_1 \cdot \cW_2(\rho, \rho_0),\, B_0\bigr\}, \\
	\label{eq:lem-boundness2}
	\sup_{\theta\in \RR^D}\norm[\big]{\nabla_\theta g(\theta; \rho)}_{\rm F} &\le \alpha \cdot B_2 \cdot \min\bigl\{ 2\alpha \cdot B_1 \cdot \cW_2(\rho, \rho_0) + B_r,\, 2\alpha\cdot B_0 + B_r \bigr\}.
	\end{align}
\end{lemma}
\begin{proof}
	We introduce the Wasserstein-1 distance, which is defined as
	\begin{align*}

	\cW_1(\mu^1, \mu^2) = \inf\Bigl\{ \EE\bigl[\norm{X-Y}\bigr] \Biggiven {\rm law}(X) = \mu^1, {\rm law}(Y) = \mu^2 \Bigr\}
	\end{align*}
	for any $\mu^1, \mu^2 \in\sP(\RR^D)$ with finite first moments. Thus, we have that $\cW_1(\mu^1, \mu^2) \le \cW_2(\mu^1, \mu^2)$. The Wasserstein-1 distance has the following dual representation \citep{ambrosio2008gradient},
	\begin{align}
	\label{eq:w-1-dual}
	\cW_1(\mu^1, \mu^2) = \sup\biggl\{ \int f(x) \,\rd (\mu^1-\mu^2)(x) \bigggiven {\rm continuous}~f: \RR^D \rightarrow \RR, \lip(f) \le 1 \biggr\}.
	\end{align}
	Following from Assumptions \ref{asp:data} and \ref{asp:activation}, we have that $\norm{\nabla_\theta\sigma(x; \theta)} \le B_1$ for any $x\in \cX$ and $\theta\in \RR^D$, which implies that $\lip(\sigma(x; \cdot)/B_1) \le 1$ for any $x \in \cX$. Note that $Q(x; \rho_0) = 0$ for any $x \in \cX$. Thus, by \eqref{eq:w-1-dual} we have for any  $ \rho\in \sP_2(\RR^D) $ and $x\in \cX$ that
	\begin{align}
	\label{eq:pf-bo1}
	\bigl| Q(x; \rho) \bigr| &= \alpha\cdot \biggl| \int \sigma(x; \theta) \cdot  \,\rd (\rho - \rho_0)(\theta) \biggr|  \le \alpha\cdot B_1 \cdot \cW_1(\rho, \rho_0) \le \alpha \cdot B_1 \cdot \cW_2(\rho, \rho_0).
	\end{align}
	Meanwhile, following from Assumptions \ref{asp:data} and \ref{asp:activation}, we have for any $x\in \cX$ and $\rho\in \sP_2(\RR^D)$ that
	\begin{align}
	\label{eq:pf-bo3}
	\bigl| Q(x; \rho) \bigr| = \alpha\cdot \biggl| \int \sigma(x; \theta)   \,\rd \rho(\theta )\biggr| \le \alpha\cdot B_0.
	\end{align}
	Combining \eqref{eq:pf-bo1} and \eqref{eq:pf-bo3}, we have for any $\rho \in \sP_2(\RR^D)$ that
	\begin{align}\label{eq:pf-bo4}
	\sup_{x\in \cX} \bigl| Q(x; \rho)\bigr| &\le \alpha\cdot \min \bigl\{B_1 \cdot \cW_2(\rho, \rho_0),\, B_0\bigr\},
	\end{align}
	which completes the proof of \eqref{eq:lem-boundness1} in Lemma \ref{lem:boundness}.
	Following from the definition of $g$ in \eqref{eq:g-rho}, we have for any $x\in \cX$ and $\rho\in \sP_2(\RR^D)$ that
	\begin{align*}

	\bigl\|\nabla_\theta g(\theta; \rho) \bigr\|_{\rm F} &\le  \alpha \cdot \EE_{\tilde \cD}\Bigl[ \bigl| Q(x; \rho) - r - \gamma\cdot Q(x'; \rho)\bigr|\cdot \norm[\big]{\nabla_{\theta\theta}^2 \sigma(x; \theta)}_{\rm F} \Bigr] \nonumber\\
	&\le \alpha  \cdot \min\bigl\{ 2\alpha \cdot B_1 \cdot \cW_2(\rho, \rho_0) + B_r,\, 2\alpha\cdot B_0 + B_r \bigr\} \cdot B_2.
	\end{align*}
	Here the last inequality follows from \eqref{eq:pf-bo4} and the fact that $ \norm{\nabla_{\theta\theta}^2 \sigma(x; \theta)}_{\rm F} \le B_2 $ for any $x\in \cX$ and $\rho\in \sP_2(\RR^D)$, which follows from Assumptions \ref{asp:data} and \ref{asp:activation}. Thus, we complete the proof of Lemma \ref{lem:boundness}.
\end{proof}

We are now ready to present the proof of Lemma \ref{lem:descent}.
\begin{proof}
	Recall that $\rho_t$ is the PDE solution in \eqref{eq:pde-fixed}, that is,
	\begin{align*}
	\partial_t \rho_t = - \eta \cdot \Div \bigl(\rho_t \cdot g(\cdot; \rho_t) \bigr),
	\end{align*}
	where
	\begin{align*}
	g(\theta; \rho) = -\alpha \cdot \EE_{\tilde \cD} \Bigl[ \bigl(Q(x; \rho) - r - \gamma\cdot Q(x'; \rho)\bigr) \cdot \nabla_\theta \sigma(x; \theta) \Bigr].
	\end{align*}
	We fix a $t \in [0, T]$. We denote by $\beta:[0, 1]\rightarrow \sP_2(\RR^D)$ the geodesic connecting $\rho_t$ and $\rho^*$. Specifically, $\beta$ satisfies that $\beta'_s = - \Div(\beta_s \cdot  v_s)$ for a vector field $v$. Following from Lemma \ref{lem:diff}, we have that
	\begin{align}
	\label{eq:pf-ds1}
	\frac{\rd }{\rd t}\frac{\cW_2(\rho_t, \rho^*)^2}{2}  &= -\eta\cdot \inp[\big]{ g(\cdot ;\rho_t) }{ v_0 }_{\rho_t} \nonumber \\
	& = \eta\cdot \int_{0}^{1} \partial_s \inp[\big]{ g(\cdot ;\beta_s) }{ v_s }_{\beta_s} \,\rd s - \eta\cdot \inp[\big]{ g(\cdot ;\rho^*) }{ v_1 }_{\rho^*} \nonumber \\
	& = \eta \cdot \underbrace{\int_0^1 \inp[\big]{ \partial_s g(\cdot; \beta_s) }{ v_s}_{\beta_s} \,\rd s}_{\displaystyle {\rm (i)}} + \eta \cdot \underbrace{ \int_0^1 \int \inp[\big]{ g(\theta; \beta_s) }{\partial_s(v_s \cdot \beta_s)(\theta) } \,\rd \theta\,\rd s}_{\displaystyle {\rm (ii)}},
	\end{align}
	where the last equality follows from (ii) of Lemma \ref{lem:rho-star}.

	For term (i) of \eqref{eq:pf-ds1}, following from \eqref{eq:td-q1} of Lemma \ref{lem:td-q}, we have that
	\begin{align}
	\label{eq:pf-ds2}
	\int_{0}^{1}\inp[\big]{\partial_s g(\cdot; \beta_s)}{v_s}_{\beta_s} \, \rd s &\le - (1-\gamma) \cdot \EE_\cD\Bigl[ \bigl( Q(x; \beta_0) - Q(x; \beta_1)\bigr)^2 \Bigr] \nonumber\\
	&= - (1-\gamma) \cdot \EE_\cD\Bigl[ \bigl( Q(x; \rho_t) - Q^*(x)\bigr)^2 \Bigr].
	\end{align}
	For term (ii) of \eqref{eq:pf-ds2}, we have that
	\begin{align*}
	\int \Bigl| \inp[\big]{ g(\theta; \beta_s) }{\partial_s(v_s \cdot  \beta_s)(\theta) } \Bigr| \,\rd \theta &= \int \Bigl| \inp[\big]{\nabla_\theta g(\theta; \beta_s)}{ \beta_s(\theta)\cdot v_s(\theta) \otimes v_s(\theta) }\Bigr| \,\rd \theta \\
	&\le \sup_{\theta\in \RR^D} \norm[\big]{\nabla_\theta g(\theta; \beta_s)}_{\rm F} \cdot \norm{v_s}_{\beta_s}^2,
	\end{align*}
	where the equality follows from integration by parts and Lemma \ref{lem:euler}. Since $\beta$ is the geodesic connecting $\rho_t$ and $\rho^*$, \eqref{eq:constant-speed} implies that $\norm{v_s}_{\beta_s}^2 = \cW_2(\beta_0, \beta_1)^2 = \cW_2(\rho_t, \rho^*)^2$ for any $s\in [0,1]$.
	Applying \eqref{eq:lem-boundness2} of Lemma \ref{lem:boundness}, we have that
	\begin{align}\label{eq:pf-ds3}
	\int \Bigl| \inp[\big]{ g(\theta; \beta_s) }{\partial_s(v_s \cdot \beta_s)(\theta) } \Bigr| \,\rd \theta
	&\le \alpha \cdot B_2 \cdot \bigl( 2\alpha \cdot B_1 \cdot \cW_2(\rho_t, \rho_0) + B_r \bigr) \cdot \cW_2(\rho_t, \rho^*)^2 \nonumber\\
	&\le 4\alpha \cdot B_2 \cdot \bigl( 6\alpha \cdot B_1 \cdot \cW_2(\rho_0, \rho^*) + B_r \bigr) \cdot \cW_2(\rho_0, \rho^*)^2 ,
	\end{align}
	where the last inequality follows from the condition of Lemma \ref{lem:descent} that $\cW_2(\rho_t, \rho^*) \le 2\cW_2(\rho_0, \rho^*)$ and the fact that $\cW_2(\rho_t, \rho_0) \le \cW_2(\rho_t, \rho^*) + \cW_2(\rho_0, \rho^*)$. Then, applying (iii) of Lemma \ref{lem:rho-star} to \eqref{eq:pf-ds3}, we have that
	\begin{align}
	\label{eq:pf-ds4}
	\int_0^1 \int \Bigl| \inp[\big]{ g(\theta; \beta_s) }{\partial_s(v_s \cdot \beta_s)(\theta) } \Bigr| \,\rd \theta \,\rd s &\le 4\alpha^{-1} \cdot B_2 \cdot \bar D^2 \cdot ( 6  B_1 \cdot \bar D+ B_r ) \nonumber \\
	&=C_* \cdot \alpha^{-1},
	\end{align}
	where $C_*>0$ is a constant depending on $\bar D$, $B_1$, $B_2$, and $B_r$.

	Finally, plugging \eqref{eq:pf-ds2} and \eqref{eq:pf-ds4} into \eqref{eq:pf-ds1}, we have that
	\begin{align*}

	\frac{\rd }{\rd t}\frac{\cW_2(\rho_t, \rho^*)^2}{2} \le - (1-\gamma) \cdot \eta\cdot \EE_\cD\Bigl[ \bigl( Q(x; \rho_t) - Q^*(x)\bigr)^2 \Bigr] + C_* \cdot \alpha^{-1} \cdot \eta,
	\end{align*}
	which completes the proof of Lemma \ref{lem:descent}.
\end{proof}

\subsection{Proof of Theorem \ref{th:q-convergence}} \label{sec:pf-q-convergenve}
\begin{proof}
	In parallel to the proof of Lemma \ref{lem:rho-star} in \S\ref{sec:pf-rho-star}, to establish the existence and uniqueness of the fixed point solution to the projected Bellman optimality equation $Q = \Pi_\cF \cT^* Q$, it suffices to show that $\Pi_\cF\cT^*:\cF\rightarrow \cF$ is a contraction mapping. In particular, it holds for any $Q^1, Q^2 \in \cF$ that
	\begin{align*}
	\norm{\Pi_\cF \cT^*Q^1 - \Pi_\cF \cT^* Q^2}^2_{\cL_2(\cD_{\rE})} &\le \gamma^2 \cdot \EE_{\tilde \cD_\rE}\Bigl[\bigl( \max_{a\in \cA} Q^{1}(s', \underline a) - \max_{a\in \cA} Q^{2}(s', \underline a) \bigr)^2 \Bigr] \\
	& = \gamma^2 \cdot \EE_{\cD_\rE}\Bigl[\bigl( \max_{a\in \cA} Q^{1}(s, \underline a) - \max_{a\in \cA} Q^{2}(s, \underline a) \bigr)^2 \Bigr] \\
	&\le \frac{\gamma^2}{(\gamma+\kappa)^2} \cdot  \EE_{\cD_\rE}\Bigl[\bigl( Q^{1}(s, a) -  Q^{2}(s, a) \bigr)^2 \Bigr],
	\end{align*}
	where the equality follows from the fact that $\cD_{\rE}$ is the stationary distribution and the last inequality follows from Assumption \ref{asp:exp}. Thus, $\Pi_\cF\cT^*:\cF \rightarrow \cF$ is a contraction mapping. Following from the Banach fixed point theorem \citep{conway2019course}, there exists a unique fixed point solution $Q^\dagger\in \cF$ to the projected Bellman optimality equation $Q = \Pi_\cF \cT^* Q$. Moreover, in parallel to the proof of Lemma \ref{lem:rho-star} in \S\ref{sec:pf-rho-star}, there exists $\nu^\dagger \in \sP_2(\RR^D)$ such that $Q(x; \nu^\dagger) = Q^\dagger(x)$, $h(x; \nu^\dagger) = 0$, and $\cW_2(\nu^\dagger, \nu_0) \le \alpha^{-1} \cdot \bar D$, where $\bar D = D_{\chi^2}(\bar \nu \,\|\, \nu_0)^{1/2}$.

	For notational simplicity, we define $Q^\cA(x) = \max_{\underline a\in \cA} Q(s, \underline a)$.
	In parallel to \eqref{eq:pf-ds1} in the proof of Lemma \ref{lem:descent} in \S\ref{sec:pf-descent}, we have that
	\begin{align}
	\label{eq:pf-qd1}
	\frac{\rd }{\rd t}\frac{\cW_2(\nu_t, \nu^\dagger)^2}{2} = \eta \cdot \underbrace{\int_0^1 \inp[\big]{ \partial_s h(\cdot; \beta_s) }{ v_s}_{\beta_s} \,\rd s}_{\displaystyle {\rm (i)}} + \eta \cdot \underbrace{ \int_0^1 \int \inp[\big]{ h(\theta; \beta_s) }{\partial_s(v_s \cdot \beta_s)(\theta) } \,\rd \theta \,\rd s}_{\displaystyle {\rm (ii)}},
	\end{align}
	where $\beta: [0, 1] \rightarrow \sP_2(\RR^D)$ is the geodesic connecting $\nu_t$ and $\nu^\dagger$ with $\partial_s \beta_s = -\Div(\beta_s\cdot  v_s)$.

	\vskip4pt

	\noindent{\bf Upper bounding term (i) of \eqref{eq:pf-qd1}.} In parallel to \eqref{eq:pf-td-q1} and \eqref{eq:pf-td-q2} in the proof of Lemma \ref{lem:td-q}, we have that
	\begin{align}
	\label{eq:pf-qd2}
	\inp[\big]{\partial_s h(\cdot; \beta_s)}{ v_s}_{\beta_s} &= - \EE_{\tilde \cD_\rE} \Bigl[ \partial_s \bigl(Q(x; \beta_s) - \gamma \cdot Q^\cA(x'; \beta_s)\bigr) \cdot \partial_s Q(x; \beta_s)\Bigr] \\
	&\le -\EE_{\cD_\rE} \Bigl[ \bigl(\partial_s Q(x; \beta_s)\bigr)^2 \Bigr] + \gamma \cdot \EE_{\cD_\rE} \Bigl[ \bigl(\partial_s Q(x; \beta_s)\bigr)^2 \Bigr]^{1/2} \cdot \EE_{\cD_\rE} \Bigl[ \bigl(\partial_s Q^\cA(x; \beta_s)\bigr)^2 \Bigr]^{1/2}. \nonumber
	\end{align}
	For the second term on the right-hand side of \eqref{eq:pf-qd2}, we have that
	\begin{align}
	\label{eq:pf-qd3}
	\EE_{\cD_\rE} \Bigl[ \bigl(\partial_s Q^\cA(x; \beta_s)\bigr)^2 \Bigr]
	& = \lim_{u\rightarrow 0}  \EE_{\cD_{\rE}}\biggl[\Bigl( u^{-1} \cdot \bigl( Q^\cA(x; \beta_{s+u}) - Q^\cA(x; \beta_{s}) \bigr)\Bigr)^2 \biggr] \nonumber \\
	& \le (\gamma + \kappa)^{-2} \cdot  \lim_{u\rightarrow 0} u^{-2} \cdot \EE_{\cD_\rE}\Bigl[ \bigl( Q(x; \beta_{s+u}) - Q(x; \beta_{s}) \bigr)^2 \Bigr] \nonumber \\
	& = (\gamma + \kappa)^{-2} \cdot \EE_{\cD_\rE}\Bigl[ \bigl( \partial_s Q(x; \beta_{s}) \bigr)^2 \Bigr],
	\end{align}
	where the inequality follows from Assumption \ref{asp:exp} and the fact that $Q(\cdot; \nu) \in \alpha \cdot \cF$. Plugging \eqref{eq:pf-qd3} into \eqref{eq:pf-qd2}, we have that
	\begin{align*}
	\inp[\big]{\partial_s h(\cdot; \beta_s)}{ v_s}_{\beta_s} \le -\frac{\kappa}{\gamma + \kappa} \cdot \EE_{\cD_\rE}\Bigl[ \bigl( \partial_s Q(x; \beta_{s}) \bigr)^2 \Bigr],
	\end{align*}
	which further implies that
	\begin{align}
	\label{eq:pf-qd4}
	\int_0^1 \inp[\big]{\partial_s h(\cdot; \beta_s)}{ v_s}_{\beta_s} \, \rd s &\le -\frac{\kappa}{\gamma + \kappa} \cdot \int_0^1 \EE_{\cD_\rE}\Bigl[ \bigl( \partial_s Q(x; \beta_{s}) \bigr)^2 \Bigr] \,\rd s \nonumber \\
	& \le -\frac{\kappa}{\gamma + \kappa} \cdot \EE_{\cD_\rE}\biggl[ \Bigl( \int_0^1 \partial_s Q(x; \beta_{s}) \,\rd s \Bigr)^2 \biggr] \nonumber\\
	&= -\frac{\kappa}{\gamma + \kappa} \cdot \EE_{\cD_\rE}\Bigl[ \bigl( Q(x; \nu_t) - Q(x; \nu^\dagger) \bigr)^2 \Bigr].
	\end{align}
	\noindent{\bf Upper bounding term (ii) of \eqref{eq:pf-qd1}.} In parallel to the proof of Lemma \ref{lem:boundness} in \S\ref{sec:pf-descent}, noting that $|Q^\cA(x; \nu)| \le \sup_{x\in \cX} |Q(x; \nu)|$ for any $\nu \in \sP_2(\RR^D)$,  we have that
	\begin{align*}
	\norm[\big]{\nabla_\theta h(\theta;\nu_t)}_{\rm F} \le \alpha \cdot B_2 \cdot \bigl( 2\alpha \cdot B_1 \cdot \cW_2(\nu_t, \nu_0) + B_r \bigr).
	\end{align*}
	In parallel to \eqref{eq:pf-ds3} and \eqref{eq:pf-ds4}, we have that
	\begin{align}\label{eq:pf-qd5}
	\int_0^1 \int \Bigl| \inp[\big]{ h(\theta; \beta_s) }{\partial_s(v_s \cdot  \beta_s)(\theta) }\Bigr| \,\rd \theta\,\rd s \le  C_* \cdot \alpha^{-1},
	\end{align}
	where $C_*>0$ is a constant that depends on $\bar D$, $B_1$, $B_2$, and $B_r$.

	Plugging \eqref{eq:pf-qd4} and \eqref{eq:pf-qd5} into \eqref{eq:pf-qd1}, we have that
	\begin{align*}

	\frac{\rd }{\rd t} \frac{\cW_2(\nu_t, \nu^\dagger)^2}{2} \le -\frac{\eta \cdot \kappa}{\gamma + \kappa} \cdot \EE_{\cD_\rE}\Bigl[ \bigl( Q(x; \nu_t) - Q(x; \nu^\dagger) \bigr)^2 \Bigr] + C_* \cdot \eta \cdot \alpha^{-1}.
	\end{align*}
	Thus, in parallel to the proof of Theorem \ref{th:convergence-fix} in \S\ref{sec:proof}, we have that
	\begin{align*}

	\inf_{t\in[0, T]}\EE_\cD\Bigl[ \bigl( Q(x; \nu_t) - Q^\dagger(x)\bigr)^2 \Bigr] \le \frac{ (\kappa+\gamma) \cdot  D_{\chi^2}(\bar \nu \,\|\, \nu_0)}{2\kappa \cdot T} + C_*\cdot \alpha^{-1} \cdot \frac{\kappa + \gamma}{\kappa},
	\end{align*}
	which completes the proof of \eqref{eq:q-convergence} in Theorem \ref{th:q-convergence}. Meanwhile, in parallel to the proof of Lemma \ref{lem:opt-dis} in \S\ref{sec:pf-opt-dis}, we  upper bound the error of approximating $\hat \nu_k$ by $\nu_{k\epsilon}$, which further implies \eqref{eq:conv-q} of Theorem \ref{th:q-convergence}.
\end{proof}

\section{Mean-Field Limit of Neural Networks}
\label{sec:discretization}
In this section, we  prove Proposition \ref{prop:discretization0}, whose formal version is presented as follows. Recall that $\rho_t$ is the PDE solution in \eqref{eq:pde-fixed} and  $\hat \rho_k = m^{-1}\cdot \sum_{i=1}^m \theta_i(k)$ is the empirical distribution of $\theta^{(m)}(k) = \{\theta_i(k)\}_{i=1}^m$. Note that we omit the dependence of $\hat \rho_k$ on $m$ and $\epsilon$ for notational simplicity.
\begin{proposition}[Formal Version of Proposition \ref{prop:discretization0}]
	\label{prop:discretization}
	Let $f: \RR^D\rightarrow \RR$ be any continuous function such that $\norm{f}_\infty \le 1$ and $\lip(f) \le 1$. Under Assumptions \ref{asp:data} and \ref{asp:activation}, it holds that
	\begin{align*}
	&\sup_{\substack{ k\le T/\epsilon \\ (k\in\NN)}}\biggl| \int f(\theta) \,\rd \rho_{k\epsilon}(\theta) - \int f(\theta) \,\rd \hat \rho_{k}(\theta) \biggr| \nonumber\\
	&\quad \le B\cdot e^{BT} \cdot \Bigl( \sqrt{\log(m/\delta)/m}  + \sqrt{\epsilon\cdot \bigl(D + \log(m/\delta)\bigr)}\Bigr)
	\end{align*}
	with probability at least $1-\delta$. Here $B$ is a constant that depends on $\alpha$, $\eta$, $\gamma$, $B_r$, and $B_j \ (j\in \{0, 1, 2\})$.
\end{proposition}
The proof of Proposition \ref{prop:discretization} is based on \cite{mei2018mean, mei2019mean,araujo2019mean}, which  utilizes the propagation of chaos \citep{sznitman1991topics}. Recall that $g(\cdot; \rho)$ is a vector field defined as follows,
\begin{align*}
g(\theta; \rho) = -\alpha \cdot \EE_{\tilde \cD} \Bigl[ \bigl(Q(x; \rho) - r - \gamma \cdot Q(x'; \rho) \bigr) \cdot \nabla_\theta \sigma(x; \theta)\Bigr].
\end{align*}
Correspondingly, we define the finite-width and stochastic counterparts of $g(\theta; \rho)$ as follows,
\begin{align}
\label{eq:g-hat}
\hat g(\theta; \theta^{(m)}) &= -\alpha \cdot \EE_{\tilde \cD} \Bigl[ \bigl(\hat Q(x; \theta^{(m)}) - r -\gamma \cdot \hat Q(x'; \theta^{(m)}) \bigr) \cdot \nabla_\theta \sigma(x; \theta)\Bigr], \\
\label{eq:G}
\hat G_k(\theta; \theta^{(m)}) &= -\alpha\cdot \bigl( \hat Q(x_k; \theta^{(m)}) - r_k -  \gamma\cdot \hat Q(x'_k; \theta^{(m)})\bigr) \cdot\nabla_\theta \sigma(x_k; \theta),
\end{align}
where $(x_k, r_k, x_k') \sim \tilde \cD$.
Following from \cite{mei2019mean,araujo2019mean}, we consider the following four dynamics.
\begin{itemize}[align=left, leftmargin=*]
	\item {\bf Temporal-difference (TD).} We consider the following TD dynamics $\theta^{(m)}(k)$, where $k\in \NN$, with $\theta_i(0) \overset{\iid}{\sim} \rho_0 \ (i\in [m])$  as its initialization,
	\begin{align}
	\label{eq:td}
	\theta_i(k+1) &= \theta_i(k) - \eta\epsilon \cdot \alpha\cdot \Bigl( \hat Q\bigl(x_k; \theta^{(m)}(k)\bigr) - r_k -  \gamma\cdot \hat Q\bigl(x_k'; \theta^{(m)}(k)\bigr)\Bigr) \cdot\nabla_\theta \sigma\bigl(x_k; \theta_i(k) \bigr) \nonumber \\
	&= \theta_i(k) + \eta\epsilon\cdot \hat G_k\bigl(\theta_i(k); \theta^{(m)}(k)\bigr),
	\end{align}
	where $(x_k, r_k, x'_k) \sim \tilde \cD$. Note that this definition is equivalent to \eqref{eq:sgd}.

	\item {\bf Expected temporal-difference (ETD).}  We consider the following expected TD dynamics $\breve \theta^{(m)}(k)$, where $k\in \NN$, with $\breve \theta_i(0) = \theta_i(0) \ (i\in [m])$ as its initialization,
	\begin{align}\label{eq:ptd}
	\breve\theta_i(k+1) &= \breve\theta_i(k) - \eta\epsilon \cdot \alpha\cdot \EE_{\tilde \cD}\biggl[\Bigl( \hat Q\bigl(x; \breve\theta^{(m)}(k)\bigr) - r -  \gamma\cdot \hat Q\bigl(x'; \breve\theta^{(m)}(k)\bigr)\Bigr) \cdot\nabla_\theta \sigma\bigl(x; \breve\theta_i(k) \bigr)\biggr] \nonumber \\
	&= \breve \theta_i(k) + \eta\epsilon\cdot \hat g\bigl(\breve \theta_i(k); \breve\theta^{(m)}(k)\bigr).
	\end{align}

	\item {\bf Continuous-time temporal-difference (CTTD).} We consider the following continuous-time TD dynamics $\tilde \theta^{(m)}(t)$, where $t\in \RR_+$, with $\tilde \theta_i(0) =\theta_i(0) \ (i\in [m])$ as its initialization,
	\begin{align}
	\label{eq:cttd}
	\frac{\rd }{\rd t} \tilde \theta_i(t) &= -\eta \cdot \alpha \cdot \EE_{\tilde \cD}\biggl[\Bigl( \hat Q\bigl(x; \tilde \theta^{(m)}(t)\bigr) - r -  \gamma\cdot \hat Q\bigl(x'; \tilde \theta^{(m)}(t)\bigr)\Bigr) \cdot\nabla_\theta \sigma\bigl(x; \tilde \theta_i(t) \bigr)\biggr] \nonumber \\
	& = \eta \cdot \hat g\bigl(\tilde\theta_i(t) ; \tilde \theta^{(m)}(t) \bigr ).
	\end{align}

	\item {\bf Ideal particle (IP).} We consider the following ideal particle dynamics $\bar \theta^{(m)}(t)$, where $t\in \RR_+$, with $\bar \theta_i(0) =\theta_i(0) \ (i\in [m])$ as its initialization,
	\begin{align}
	\label{eq:idp}
	\frac{\rd }{\rd t} \bar \theta_i(t) &= -\eta \cdot \alpha \cdot \EE_{\tilde \cD}\Bigl[ \bigl( Q(x; \rho_t ) - r -  \gamma\cdot  Q(x'; \rho_t )\bigr) \cdot\nabla_\theta \sigma\bigl(x; \bar \theta_i(t) \bigr) \Bigr] \nonumber \\
	& = \eta \cdot g\bigl(\bar \theta_i(t) ; \rho_t\bigr),
	\end{align}
	where $\rho_t$ is the PDE solution in \eqref{eq:pde-fixed}.

\end{itemize}

We aim to prove that $\hat \rho_{k} = m^{-1} \cdot \sum_{i=1}^{m} \delta_{\theta_i(k)}$ weakly converges to $\rho_{k\epsilon}$. For any continuous function $f: \RR^D \rightarrow \RR$ such that $\norm{f}_{\infty} \le 1$ and $\lip(f) \le 1$,
we use the IP, CTTD, and ETD dynamics as the interpolating dynamics,
\begin{align}
\label{eq:dis}
&\overbrace{\biggl| \int f(\theta) \,\rd \rho_{k\epsilon}(\theta) - \int f(\theta) \,\rd \hat \rho_{k}(\theta)\biggr|}^{\displaystyle{\rm PDE-TD}} \nonumber \\
&\quad \le \biggl|\int f(\theta) \,\rd \rho_{k\epsilon}(\theta) - m^{-1} \cdot \sum_{i=1}^{m} f\bigl(\bar \theta_i(k\epsilon) \bigr)\biggr| + \biggl| m^{-1} \cdot \sum_{i=1}^{m} f\bigl(\bar \theta_i(k\epsilon) \bigr) - m^{-1} \cdot \sum_{i=1}^{m} f\bigl(\tilde \theta_i(k\epsilon) \bigr) \biggr| \nonumber \\
&\qquad + \biggl| m^{-1} \cdot  \sum_{i=1}^{m} f\bigl(\tilde \theta_i(k\epsilon) \bigr) - m^{-1} \cdot  \sum_{i=1}^{m} f\bigl(\breve \theta_i(k) \bigr) \biggr| + \biggl| m^{-1} \cdot  \sum_{i=1}^{m} f\bigl(\breve \theta_i(k) \bigr) - m^{-1} \cdot \sum_{i=1}^{m} f\bigl(\theta_i(k) \bigr) \biggr| \nonumber \\
&\quad \le \underbrace{\biggl|\int f(\theta) \,\rd \rho_{k\epsilon}(\theta) - m^{-1} \cdot \sum_{i=1}^{m} f\bigl(\bar \theta_i(k\epsilon) \bigr)\biggr|}_{\displaystyle{\rm PDE-IP}} + \underbrace{\norm[\big]{\bar \theta^{(m)}(k\epsilon) - \tilde\theta^{(m)}(k\epsilon)}_{(m)}}_{\displaystyle{\rm IP-CTTD}} \nonumber \\
&\qquad + \underbrace{\norm[\big]{\tilde \theta^{(m)}(k\epsilon) - \breve\theta^{(m)}(k)}_{(m)}}_{\displaystyle{\rm CTTD-ETD}} + \underbrace{\norm[\big]{\breve \theta^{(m)}(k) - \theta^{(m)}(k)}_{(m)}}_{\displaystyle{\rm ETD-TD}},
\end{align}
where the last inequality follows from the the fact that $\lip(f) \le 1$. Here the norm $\norm{\cdot}_{(m)}$ of $ \theta^{(m)} = \{\theta_i\}_{i=1}^m $ is defined as follows,
\begin{align}
\label{eq:norm-m}
\norm{\theta^{(m)}}_{(m)} = \sup_{i\in[m]} \norm{\theta_i}.
\end{align}
In what follows, we define $B > 0$ as a constant that depends on $\alpha$, $\eta$, $\gamma$, $B_r$, and $B_j \ (j\in \{0,1,2\})$, whose value varies from line to line. We establish the following lemmas to upper bound the terms on the right-hand side of \eqref{eq:norm-m}.

\begin{lemma}[Upper Bound of PDE -- IP]
	\label{lem:pde-ipd}
	Let $f$ be any continuous function such that $\norm{f}_\infty \le 1$ and $\lip(f) \le 1$.
	Under Assumptions \ref{asp:data} and \ref{asp:activation}, it holds for any $f$ that
	\begin{align*}
	\sup_{t\in [0, T]}\Bigl|\int f(\theta)  \,\rd \rho_{t}(\theta) - m^{-1} \cdot \sum_{i=1}^{m} f\bigl(\bar \theta_i(t) \bigr) \Bigr| &\le B\cdot \sqrt{\log(mT/\delta) / m}
	\end{align*}
	with probability at least $1-\delta$.
\end{lemma}
\begin{proof}
	See \S\ref{sec:pf-pde-ipd} for a detailed proof.
\end{proof}

\begin{lemma}[Upper Bound of IP -- CTTD]
	\label{lem:ipd-cttd}
	Under Assumptions \ref{asp:data} and \ref{asp:activation}, it holds that
	\begin{align*}
	\sup_{t\in [0, T]}\norm[\big]{\bar \theta^{(m)}(t) - \tilde \theta^{(m)}(t)}_{(m)} \le B\cdot e^{BT} \cdot \sqrt{\log(m/\delta) / m}
	\end{align*}
	with probability at least $1-\delta$.
\end{lemma}
\begin{proof}
	See \S\ref{sec:pf-ipd-cttd} for a detailed proof.
\end{proof}

\begin{lemma}[Upper Bound of CTTD -- ETD]
	\label{lem:cttd-ptd}
	Under Assumptions \ref{asp:data} and \ref{asp:activation}, it holds that
	\begin{align*}
	\sup_{\substack{k \le T/\epsilon \\ (k \in \NN)}} \norm[\big]{\tilde \theta^{(m)}(k\epsilon) - \breve\theta^{(m)}(k)}_{(m)} \le B \cdot e^{BT} \cdot \epsilon .
	\end{align*}
\end{lemma}
\begin{proof}
	See \S\ref{sec:pf-cttd-ptd} for a detailed proof.
\end{proof}

\begin{lemma}[Upper Bound of ETD -- TD]
	\label{lem:ptd-td}
	Under Assumptions \ref{asp:data} and \ref{asp:activation}, it holds that
	\begin{align*}
	\sup_{\substack{k \le T/\epsilon \\ (k \in \NN)}} \norm[\big]{\breve\theta^{(m)}(k) - \theta^{(m)}(k)}_{(m)} \le B\cdot e^{BT}\cdot \sqrt{\epsilon\cdot \bigl(D + \log(m/\delta) \bigr)}
	\end{align*}
	with probability at least $1-\delta$
\end{lemma}
\begin{proof}
	See \S\ref{sec:pf-ptd-td} for a detailed proof.
\end{proof}
We are now ready to present the proof of Proposition \ref{prop:discretization}.
\begin{proof}
	Plugging Lemmas \ref{lem:pde-ipd}-\ref{lem:ptd-td} into \eqref{eq:dis}, we have that
	\begin{align*}
    &\sup_{\substack{k \le T/\epsilon \\ (k \in \NN)}}\biggl| \int f(\theta) \,\rd \rho_{k\epsilon}(\theta) - \int f(\theta) \,\rd\hat \rho_{k}(\theta) \biggr| \nonumber\\
    &\quad \le B\cdot e^{BT} \cdot \Bigl( \sqrt{\log(m/\delta)/m}  + \sqrt{\epsilon\cdot \bigl(D + \log(m/\delta)\bigr)}\Bigr)
	\end{align*}
	with probability at least $1-\delta$. Thus, we complete the proof of Proposition \ref{prop:discretization}.
\end{proof}

\subsection{Proofs of Lemmas \ref{lem:pde-ipd}-\ref{lem:ptd-td}}
In this section, we present the proofs of Lemmas \ref{lem:pde-ipd}-\ref{lem:ptd-td}, which are based on \cite{mei2018mean,mei2019mean,araujo2019mean}. We include the required technical lemmas in \S\ref{sec:tech}. Recall that $B > 0$ is a constant that depends on $\alpha$, $\eta$, $\gamma$, $B_r$, and $B_j \ (j\in \{0,1,2\})$, whose value varies from line to line.
\subsubsection{Proof of Lemma \ref{lem:pde-ipd}} \label{sec:pf-pde-ipd}
\begin{proof}
For the IP dynamics in \eqref{eq:idp}, it holds that $\bar\theta_i(t) \sim \rho_t \ (i\in [m])$ (Proposition 8.1.8 in \cite{ambrosio2008gradient}). Furthermore, since the randomness of $\bar\theta_i(t)$ comes from $\theta_i(0)$ while $\theta_i(0)\ (i\in [m])$ are independent, we have that $\bar\theta_i(t) \overset\iid\sim \rho_t \ (i\in [m])$.
Thus, we have that
\begin{align*}
\EE_{\rho_{t}} \Bigl[ m^{-1} \cdot \sum_{i=1}^{m} f\bigl(\bar \theta_i(t) \bigr) \Bigr] = \int f(\theta)  \,\rd \rho_{t}(\theta).
\end{align*}
Let $\theta^{1, (m)} = \{\theta_1, \ldots, \theta_i^1, \ldots, \theta_m\}$ and $\theta^{2, (m)} = \{\theta_1, \ldots, \theta_i^2, \ldots, \theta_m\}$ be two sets that only differ in the $i$-th element. Then, by the condition of Lemma \ref{lem:pde-ipd} that $\norm{f}_\infty \le 1$, we have that
\begin{align*}

\Bigl| m^{-1}\cdot \sum_{j=1}^{m} f(\theta^1_j ) - m^{-1} \cdot \sum_{j=1}^{m} f(\theta^2_j )\Bigr| = m^{-1} \cdot \bigl| f(\theta^1_i ) - f(\theta^2_i) \bigr| \le 2/m.
\end{align*}
Applying McDiarmid's inequality \citep{wainwright2019high}, we have for a fixed $t\in [0, T]$ that
\begin{align}
\label{eq:pf-pi3}
\PP\biggl(\Bigl| m^{-1} \cdot \sum_{i=1}^{m} f\bigl(\bar \theta_i(t) \bigr) - \int f(\theta)  \,\rd \rho_{t}(\theta) \Bigr| \ge p \biggr) \le \exp( -mp^2 / 4 ).
\end{align}
Moreover, we have for any $s, t \in [0, T]$ that
\begin{align*}
&\biggl| \Bigl| m^{-1} \cdot  \sum_{i=1}^{m} f\bigl(\bar \theta_i(t) \bigr) - \int f(\theta)  \,\rd \rho_{t}(\theta) \Bigr| - \Bigl| m^{-1} \cdot \sum_{i=1}^{m} f\bigl(\bar \theta_i(s) \bigr) - \int f(\theta)  \,\rd \rho_{s}(\theta) \Bigr|   \biggr|\nonumber\\
&\quad \le \Bigl| m^{-1} \cdot  \sum_{i=1}^{m} f\bigl(\bar \theta_i(t) \bigr) - m^{-1} \cdot \sum_{i=1}^{m} f\bigl(\bar \theta_i(s) \bigr)\Bigr| + \Bigl| \int f(\theta)  \,\rd \rho_{t}(\theta) - \int f(\theta)  \,\rd \rho_{s}(\theta) \Bigr| \nonumber \\
&\quad \le \norm[\big]{\bar\theta^{(m)}(t) - \bar\theta^{(m)}(s)}_{(m)} + \cW_1(\rho_t, \rho_s)  \nonumber\\
&\quad \le \norm[\big]{\bar\theta^{(m)}(t) - \bar\theta^{(m)}(s)}_{(m)} + \cW_2(\rho_t, \rho_s),
\end{align*}
where the second inequality follows from the fact that $\lip(f) \le 1$ and \eqref{eq:w-1-dual}.
Applying \eqref{eq:lip-ip} and \eqref{eq:lip-pde} of Lemma \ref{lem:lip-theta}, we have for any $s, t\in [0, T]$ that
\begin{align*}
\biggl| \Bigl| m^{-1} \cdot  \sum_{i=1}^{m} f\bigl(\bar \theta_i(t) \bigr) - \int f(\theta)  \,\rd \rho_{t}(\theta) \Bigr| - \Bigl| m^{-1} \cdot \sum_{i=1}^{m} f\bigl(\bar \theta_i(s) \bigr) - \int f(\theta)  \,\rd \rho_{s}(\theta) \Bigr|   \biggr| \le B\cdot |t - s|.
\end{align*}
Applying the union bound to \eqref{eq:pf-pi3} for $t \in \iota \cdot \{ 0, 1, \ldots, \lfloor T/\iota \rfloor \} $, we have that
\begin{align*}
\PP\biggl(\sup_{t\in [0,T]}\Bigl| m^{-1} \cdot \sum_{i=1}^{m} f\bigl(\bar \theta_i(t) \bigr) - \int f(\theta)  \,\rd \rho_{t}(\theta) \Bigr| \ge p + B \cdot \iota\biggr) \le (T/\iota + 1)\cdot \exp( -mp^2 / 4 ).
\end{align*}
Setting $\iota = m^{-1/2} $ and $p = B\cdot \sqrt{\log(mT/\delta) / m}$, we have that
\begin{align*}

\sup_{t\in [0, T]}\Bigl| m^{-1} \cdot \sum_{i=1}^{m} f\bigl(\bar \theta_i(t) \bigr) - \int f(\theta)  \,\rd \rho_{t}(\theta) \Bigr| \le B\cdot \sqrt{\log(mT/\delta) / m}
\end{align*}
with probability at least $1-\delta$. Thus, we complete the proof of Lemma \ref{lem:pde-ipd}.
\end{proof}

\subsubsection{Proof of Lemma \ref{lem:ipd-cttd}} \label{sec:pf-ipd-cttd}
\begin{proof}
Recall that $g$ and $\hat g$ are defined in \eqref{eq:g-rho} and \eqref{eq:g-hat}, respectively, that is,
\begin{align*}
g(\theta; \rho) &= -\alpha \cdot \EE_{\tilde \cD} \Bigl[ \bigl(Q(x; \rho) - r - \gamma \cdot Q(x'; \rho) \bigr) \cdot \nabla_\theta \sigma(x; \theta)\Bigr], \\
\hat g(\theta; \theta^{(m)}) &= -\alpha \cdot \EE_{\tilde \cD} \Bigl[ \bigl(\hat Q(x; \theta^{(m)}) - r -\gamma \cdot \hat Q(x'; \theta^{(m)}) \bigr) \cdot \nabla_\theta \sigma(x; \theta)\Bigr].
\end{align*}
Following from the definition of $\tilde \theta_i(t)$ and $\bar\theta_i(t)$ in \eqref{eq:cttd} and \eqref{eq:idp}, respectively, we have for any $i \in [m]$ and $t\in[0,T]$ that
\begin{align}
\label{eq:pf-ic1}
&\norm[\big]{\bar \theta_i(t) - \tilde \theta_i(t)} \nonumber\\
&\quad \le \int_0^t \norm[\bigg]{ \frac{\rd \tilde\theta_i(s)}{\rd s} - \frac{\rd \bar \theta_i(s)}{\rd s} }\,\rd s \nonumber \\
& \quad = \eta \cdot \int_0^t \norm[\Big]{ \hat g\bigl(\tilde \theta_i(s); \tilde\theta^{(m)}(s)\bigr) - g\bigl(\bar \theta_i(s); \rho_s\bigr)}\,\rd s \nonumber\\
&\quad \le \eta \cdot \int_0^t \norm[\Big]{ \hat g\bigl(\tilde \theta_i(s); \tilde\theta^{(m)}(s)\bigr) - \hat g\bigl(\bar \theta_i(s); \bar \theta^{(m)}(s)\bigr)}\,\rd s + \eta  \cdot \int_0^t \norm[\Big]{ \hat g\bigl(\bar \theta_i(s); \bar \theta^{(m)}(s)\bigr) - g\bigl(\bar \theta_i(s); \rho_s\bigr)}\,\rd s \nonumber\\
&\quad \le B\cdot \int_0^t \norm[\big]{\tilde \theta^{(m)}(s) - \bar \theta^{(m)}(s)}_{(m)}\,\rd s + \eta \cdot \int_0^t \norm[\Big]{ \hat g\bigl(\bar \theta_i(s); \bar \theta^{(m)}(s)\bigr) - g\bigl(\bar \theta_i(s); \rho_s\bigr)}\,\rd s,
\end{align}
where the last inequality follows from \eqref{eq:lip-gh} of Lemma \ref{lem:g-hat}.
We now upper bound the second term on the right-hand side of \eqref{eq:pf-ic1}.
Following from the definition of $\hat Q$, $Q$, and $\hat g$ in \eqref{eq:nn-fin}, \eqref{eq:nn-infty}, and \eqref{eq:g-hat}, respectively, we have for any $s \in [0, T]$ and $i \in [m]$ that
\begin{align}
\label{eq:pf-ic2}
&\norm[\Big]{ \hat g\bigl(\bar \theta_i(s); \bar \theta^{(m)}(s)\bigr) - g\bigl(\bar \theta_i(s); \rho_s\bigr)} = \alpha^2 \cdot \norm[\Big]{m^{-1} \cdot \sum_{j=1}^m Z_i^j(s)},
\end{align}
where
\begin{align*}
Z_i^j(s) = \EE_{\tilde \cD}\biggl[\Bigl(\sigma\bigl(x; \bar\theta_j(s)\bigr) -\int \sigma(x; \theta) \,\rd \rho_s(\theta) - \gamma \cdot  \sigma\bigl(x'; \bar\theta_j(s)\bigr) + \gamma \cdot \int \sigma(x'; \theta) \,\rd \rho_s(\theta) \Bigr) \cdot \nabla_\theta \sigma\bigl(x; \bar \theta_i(s) \bigr)\biggr].
\end{align*}
Following from Assumptions \ref{asp:data} and \ref{asp:activation}, we have that $\norm{Z_i^j(s)} \le B$. When $i \neq j$, following from the fact that $\bar \theta_i(s) \overset{\iid}\sim \rho_s \ (i\in [m]) $, it holds that $\EE[Z_i^j(s) \given \bar \theta_i(s)] = 0$. Following from Lemma \ref{lem:mc}, we have for fixed $s \in [0,T]$ and $i \in [m]$ that
\begin{align}
\label{eq:pf-ic3}
\PP\biggl( \norm[\Big]{m^{-1} \cdot \sum_{j \neq i} Z_i^j(s)} \ge B\cdot (m^{-1/2} + p) \biggr) &= \EE\Biggl[ \PP\biggl( \norm[\Big]{m^{-1} \cdot \sum_{j \neq i} Z_i^j(s)} \ge B\cdot (m^{-1/2} + p)\bigggiven \bar\theta_i(s) \biggr) \Biggr] \nonumber\\&\le \exp(-mp^2).
\end{align}
By \eqref{eq:w-1-dual}, we have that
\begin{align*}
\sup_{x\in \cX} \Bigl|\int\sigma(x;\theta) \,\rd \rho_s(\theta) - \int\sigma(x;\theta) \,\rd \rho_t(\theta) \Bigr| \le B \cdot \cW_1(\rho_s, \rho_t) \le B \cdot \cW_2(\rho_s, \rho_t) \le B\cdot |s - t|,
\end{align*}
where the last inequality follows from \eqref{eq:lip-pde} of Lemma \ref{lem:lip-theta}.
Thus, following from Assumptions \ref{asp:data} and \ref{asp:activation}, Lemma \ref{lem:lip-theta}, and the fact that $\lip(f g) \le \norm{f}_\infty \cdot \lip(g) + \norm{g}_\infty\cdot \lip(f)$ for any functions $f$ and $g$, we have for any $s, t\in [0, T]$ that
\begin{align*}
\biggl| \norm[\Big]{m^{-1} \cdot \sum_{j \neq i} Z_i^j(s)}  - \norm[\Big]{m^{-1} \cdot \sum_{j \neq i} Z_i^j(t)}\biggr| \le B\cdot |t-s|.
\end{align*}
Applying the union bound to \eqref{eq:pf-ic3} for $ i \in [m]$ and $t\in \iota \cdot \{ 0, 1, \ldots, \lfloor T/\iota \rfloor \}$, we have that
\begin{align*}
\PP\biggl( \sup_{\substack{i\in[m],\\ s\in [0, T]}} \norm[\Big]{m^{-1} \cdot \sum_{j \neq i} Z_i^j(s)} \ge B\cdot (m^{-1/2} + p) + B\iota \biggr) \le m\cdot (T/\iota + 1)\cdot \exp(-mp^2).
\end{align*}
Setting $\iota = m^{-1/2}$ and $p = B\cdot \sqrt{\log(mT/\delta) / m}$, we have that
\begin{align}
\label{eq:pf-ic4}
\sup_{\substack{i\in[m],\\ s\in [0, T]}} \norm[\Big]{m^{-1} \cdot  \sum_{j \neq i} Z_i^j(s)} \le B\cdot \sqrt{\log(mT/\delta) / m}
\end{align}
with probability at least $1-\delta$. When $i=j$, it holds that $\norm{m^{-1}\cdot Z_i^i(s)} \le B / m$ in \eqref{eq:pf-ic2}, which follows from Assumptions \ref{asp:data} and \ref{asp:activation}. Thus, plugging \eqref{eq:pf-ic4} into \eqref{eq:pf-ic2}, we have that
\begin{align}
\label{eq:pf-ic5}
\sup_{\substack{i\in[m],\\ s\in [0, T]}} \norm[\Big]{ \hat g\bigl(\bar \theta_i(s); \bar \theta^{(m)}(s)\bigr) - g\bigl(\bar \theta_i(s); \rho_s\bigr)} &\le \sup_{\substack{i\in[m],\\ s\in [0, T]}} \alpha^2 \cdot \biggl( \norm[\big]{m^{-1} \cdot Z_i^i(s)} + \norm[\Big]{m^{-1} \cdot \sum_{j \neq i} Z_i^j(s)} \biggr) \nonumber \\
&\le B\cdot \sqrt{\log(mT/\delta) / m}
\end{align}
with probability at least $1-\delta$.

Conditioning on the event in \eqref{eq:pf-ic5}, we obtain from \eqref{eq:pf-ic1} that
\begin{align*}

\norm[\big]{\tilde \theta^{(m)}(t) - \bar \theta^{(m)}(t)}_{(m)} \le B\cdot \int_0^t \norm[\big]{\tilde \theta^{(m)}(s) - \bar \theta^{(m)}(s)}_{(m)}\,\rd s + BT \cdot \sqrt{\log(mT/\delta) / m}
\end{align*}
for any $t\in [0, T]$.
Following from Gronwall's Lemma \citep{holte2009discrete}, we have that
\begin{align*}

\norm[\big]{\tilde \theta^{(m)}(t) - \bar \theta^{(m)}(t)}_{(m)} &\le B\cdot e^{Bt} \cdot BT \cdot \sqrt{\log(mT/\delta) / m} \nonumber \\
&\le B\cdot e^{BT} \cdot \sqrt{\log(m/\delta) / m},  \qquad \forall t \in [0,T]
\end{align*}
with probability at least $1-\delta$.
Here the last inequality holds since we allow the value of $B$ to vary from line to line.
Thus, we complete the proof of Lemma \ref{lem:ipd-cttd}
\end{proof}

\subsubsection{Proof of Lemma \ref{lem:cttd-ptd}} \label{sec:pf-cttd-ptd}
\begin{proof}
By the definition of $ \hat g $, $\breve\theta_i(t)$, and $\tilde \theta_i(t)$ in \eqref{eq:g-hat}, \eqref{eq:ptd}, and \eqref{eq:cttd}, respectively,
it holds that
\begin{align*}

\norm[\big]{\tilde \theta_i(k\epsilon) - \breve\theta_i(k)} &\le \eta \cdot \int_0^{k\epsilon} \norm[\Big]{ \hat g\bigl(\tilde \theta_i(s) ; \tilde \theta^{(m)}(s) \bigr) - \hat g\bigl(\breve \theta_i(\lfloor s/\epsilon\rfloor) ; \breve \theta^{(m)}(\lfloor s/\epsilon \rfloor) \bigr) }\,\rd s \nonumber\\
&\le \eta \cdot \int_0^{k\epsilon} \norm[\Big]{ \hat g\bigl(\tilde \theta_i(s) ; \tilde \theta^{(m)}(s) \bigr) - \hat g\bigl(\tilde \theta_i(\lfloor s/\epsilon\rfloor \cdot \epsilon) ; \tilde \theta^{(m)}(\lfloor s/\epsilon \rfloor \cdot \epsilon )  \bigr) }\,\rd s  \nonumber\\
&\quad + \eta \cdot \sum_{\ell = 0}^{k-1} \norm[\Big]{ \hat g\bigl(\tilde \theta_i(\ell\epsilon) ; \tilde \theta^{(m)}(\ell\epsilon) \bigr) - \hat g\bigl(\breve \theta_i(\ell) ; \breve \theta^{(m)}(\ell) \bigr) } \nonumber\\
&\le B\cdot k\cdot \epsilon^2 + B \cdot \sum_{\ell = 0}^{k-1} \norm[\big]{\tilde \theta^{(m)}(\ell\epsilon) - \breve\theta^{(m)}(\ell)}_{(m)},
\end{align*}
where the last inequality follows from \eqref{eq:lip-gh} of Lemma \ref{lem:g-hat} and \eqref{eq:lip-cttd} of Lemma \ref{lem:lip-theta}. Following from the definition of $\norm{\cdot}_{(m)}$ in \eqref{eq:norm-m}, it holds for any $k \le T/\epsilon \ (k\in \NN)$ that
\begin{align*}

\norm[\big]{\tilde \theta^{(m)}(k\epsilon) - \breve\theta^{(m)}(k)}_{(m)} \le
B\cdot T\cdot \epsilon + B \cdot \sum_{\ell = 0}^{k-1} \norm[\big]{\tilde \theta^{(m)}(\ell\epsilon) - \breve\theta^{(m)}(\ell)}_{(m)}.
\end{align*}
Following from the discrete Gronwall's lemma \citep{holte2009discrete}, we have that
\begin{align*}

\sup_{\substack{k\le T/\epsilon \\ (k\in \NN)}} \norm[\big]{\tilde \theta^{(m)}(k\epsilon) - \breve\theta^{(m)}(k)}_{(m)} \le B^2 \cdot T \cdot \epsilon \cdot e^{BT} \le B \cdot e^{BT} \cdot \epsilon,
\end{align*}
where the last inequality holds since we allow the value of $B$ to vary from line to line.
Thus, we complete the proof of Lemma \ref{lem:cttd-ptd}.
\end{proof}

\subsubsection{Proof of Lemma \ref{lem:ptd-td}} \label{sec:pf-ptd-td}
\begin{proof}
Let $\cG_k = \sigma(\theta^{(m)}(0), z_0, \ldots, z_{k})$ be the $\sigma$-algebra generated by $\theta^{(m)}(0)$ and $z_\ell = (x_\ell, r_\ell, x'_\ell)\ (\ell\le k)$. Recall that $\hat g$ and $\hat G_k$ are defined in \eqref{eq:g-hat} and \eqref{eq:G}, respectively. We have for any $i \in [m]$ and $k\in \NN_+$ that
\begin{align*}

\EE\Bigl[ \hat G_k\bigl(\theta_i(k); \theta^{(m)}(k)\bigr) \Biggiven \cG_{k-1} \Bigr] = \hat g\bigl(\theta_i(k); \theta^{(m)}(k)\bigr).
\end{align*}
Recall that $\theta^{(m)}(k)$ and $\breve\theta^{(m)}(k)$ are the TD and ETD dynamics defined in \eqref{eq:td} and \eqref{eq:ptd}, respectively.
Thus, we have for any $i \in [m]$ and $k \in \NN_+$ that
\begin{align}
\label{eq:pf-pt2}
\norm[\big]{\breve\theta_i(k) - \theta_i(k)} &= \eta\epsilon\cdot \norm[\Big]{ \sum_{\ell=0}^{k-1} \hat G_\ell\bigl(\theta_i(\ell); \theta^{(m)}(\ell)\bigr) - \sum_{\ell=0}^{k-1} \hat g\bigl(\breve \theta_i(\ell); \breve \theta^{(m)}(\ell)\bigr) } \nonumber\\
&\le \eta\epsilon \cdot \norm[\Big]{\sum_{\ell=0}^{k-1} X_i(\ell)} + \eta\epsilon\cdot \sum_{\ell=0}^{k-1}\norm[\Big]{\hat g\bigl(\breve \theta_i(\ell); \breve \theta^{(m)}(\ell)\bigr) - \hat g\bigl(\theta_i(\ell); \theta^{(m)}(\ell)\bigr)} \nonumber\\
&\le \eta\epsilon \cdot \norm[\big]{A_i(k)} + B \epsilon \cdot \sum_{\ell = 0}^{k-1}\norm[\big]{\breve \theta^{(m)}(\ell) - \theta^{(m)}(\ell)}_{(m)},
\end{align}
where the last inequality follows from \eqref{eq:lip-gh} of Lemma \ref{lem:g-hat}, and $X_i(\ell)$ and $A_i(k)$ are defined as
\begin{align*}
X_i(0) &= 0, \\ X_i(\ell) &= \hat G_\ell\bigl(\theta_i(\ell) ;  \theta^{(m)}(\ell)\bigr) - \EE\Bigl[ \hat G_\ell\bigl(\theta_i(\ell); \theta^{(m)}(\ell)\bigr) \Biggiven \cG_{\ell-1} \Bigr] \quad \forall \ell\ge 1, \\
A_i(k) &= \sum_{\ell=0}^{k-1} X_i(\ell).
\end{align*}
Following from \eqref{eq:bound-G} of Lemma \ref{lem:g-hat}, we have that $\norm{X_i(\ell)}\le B$. Thus, the stochastic process $\{A_i(k)\}_{k\in \NN_+}$ is a martingale with $\|A_i(k) - A_i(k-1) \|\le B$.
Applying Lemma \ref{lem:azuma}, we have that
\begin{align}
\label{eq:pf-pt3}
\PP\Bigl( \max_{\substack{k\le T/\epsilon \\ (k\in \NN_+)}} \norm[\big]{A_i(k)} \ge B \cdot \sqrt{T/\epsilon} \cdot (\sqrt{D} + p)\Bigr) \le \exp(-p^2) .
\end{align}
Applying the union bound to \eqref{eq:pf-pt3} for $i \in [m]$, we have that
\begin{align*}

\PP\Bigl( \max_{\substack{i\in [m], \\k\le T/\epsilon \ (k\in \NN_+)}} \norm[\big]{A_i(k)} \ge B \cdot \sqrt{T/\epsilon} \cdot (\sqrt{D} + p)\Bigr) \le m \cdot \exp(-p^2).
\end{align*}
By setting $p = \sqrt{\log(m/\delta)}$, we have that
\begin{align}
\label{eq:pf-pt5}
\norm[\big]{A_i(k)} \le B \cdot \sqrt{T/\epsilon} \cdot \bigl(\sqrt{D} + \sqrt{\log(m/\delta)} \bigr), \quad \forall i\in [m], k \le T/\epsilon \ (k\in \NN_+)
\end{align}
with probability at least $1-\delta$. By \eqref{eq:pf-pt2} and \eqref{eq:pf-pt5}, we have that
\begin{align*}

&\norm[\big]{\breve\theta^{(m)}(k) - \theta^{(m)}(k)}_{(m)} \nonumber\\ &\quad \le B\cdot \sqrt{T\epsilon}\cdot (\sqrt{D} + \sqrt{\log(m/\delta)}) + B\epsilon \cdot \sum_{\ell = 0}^{k-1}\norm[\big]{\breve \theta^{(m)}(\ell) - \theta^{(m)}(\ell)}_{(m)}, \quad
\forall k\le T/\epsilon \ (k\in \NN)
\end{align*}
with probability at least $1-\delta$.
Applying the discrete Gronwall's Lemma \citep{holte2009discrete}, we have that
\begin{align*}

\norm[\big]{\breve\theta^{(m)}(k) - \theta^{(m)}(k)}_{(m)} &\le B\cdot e^{BT}\cdot B\cdot \sqrt{T\epsilon}\cdot \bigl(\sqrt{D} + \sqrt{\log(m/\delta)} \bigr) \nonumber\\
&\le B\cdot e^{BT}\cdot \sqrt{\epsilon\cdot \bigl(D + \log(m/\delta) \bigr)}, \quad \forall k\le T/\epsilon \ (k\in \NN)
\end{align*}
with probability at least $1-\delta$. Here the last inequality holds since we allow the value of $B$ to vary from line to line. Thus, we complete the proof of Lemma \ref{lem:ptd-td}.
\end{proof}

\subsection{Proof of Lemma \ref{lem:opt-dis}} \label{sec:pf-opt-dis}

\begin{proof}
	Recall that $\hat Q$ and $Q(\cdot; \rho)$ are defined in \eqref{eq:nn-fin} and \eqref{eq:nn-infty}, respectively.
	For notational simplicity, we denote the optimality gaps for $\theta^{(m)} = \{\theta_i\}_{i=1}^m$ and $\rho \in \sP_2(\RR^D)$ by
	\begin{align}
	L( \theta^{(m)}) &= \EE_\cD \Bigl[ \bigl( \hat Q(x; \theta^{(m)}) - Q^*(x) \bigr)^2 \Bigr], \label{eq:l} \\
	\bar L(\rho) &=\EE_\cD\Bigl[ \bigl( Q(x; \rho) - Q^*(x)\bigr)^2 \Bigr]. \label{eq:l-bar}
	\end{align}
	Recall that $\theta^{(m)}(k)$, $\bar\theta^{(m)}(k\epsilon)$, and $\rho_t$ are the TD dynamics, the IP dynamics, and the PDE solution defined in \eqref{eq:td}, \eqref{eq:idp}, and \eqref{eq:pde-fixed}, respectively.
	It holds for any $k \in \NN$ that
	\begin{align}
	\label{eq:pf-od1}
	\Bigl| L\bigl(\theta^{(m)}(k)\bigr) - \bar L(\rho_{k\epsilon}) \Bigr| \le \underbrace{ \Bigl| L\bigl(\theta^{(m)}(k)\bigr) - L\bigl(\bar \theta^{(m)}(k\epsilon)\bigr) \Bigr|}_{\displaystyle{\rm (i)}} + \underbrace{ \Bigl| L\bigl(\bar \theta^{(m)}(k\epsilon)\bigr) - \bar L(\rho_{k\epsilon}) \Bigr|}_{\displaystyle{\rm (ii)}}.
	\end{align}
	In what follows, we upper bound the two terms on the right-hand side of \eqref{eq:pf-od1}.

	\vskip4pt

	\noindent{\bf Upper bounding term (i) of \eqref{eq:pf-od1}.} Following from the definition of $L$ in \eqref{eq:l}, it holds for any $k \in \NN$ that
	\begin{align}
	&\Bigl| L\bigl(\theta^{(m)}(k)\bigr) - L\bigl(\bar \theta^{(m)}(k\epsilon)\bigr) \Bigr|  \nonumber\\
	&\quad = \Biggl| \EE_\cD\biggl[ \Bigl(\hat Q\bigl(x; \theta^{(m)}(k)  \bigr) + \hat Q\bigl(x;\bar \theta_i(k\epsilon) \bigr) - 2Q^*(x) \Bigr) \cdot \Bigl(\hat Q\bigl(x; \theta^{(m)}(k)  \bigr) - \hat Q\bigl(x;\bar \theta_i(k\epsilon) \bigr) \Bigr) \biggr] \Biggr| . \label{eq:ubi1}
	\end{align}
	Following from \eqref{eq:bound-qh}, \eqref{eq:lip-qh}, and \eqref{eq:bound-q} of Lemma \ref{lem:g-hat}, we have for any $k \in \NN$ that
	\begin{align}
	\label{eq:ubi2}
	&\sup_{x\in \cX}\Bigl|\hat Q\bigl(x; \theta^{(m)}(k)  \bigr) + \hat Q\bigl(x;\bar \theta_i(k\epsilon) \bigr) - 2Q^*(x) \Bigr| \le B, \\
	&\sup_{x\in \cX}\Bigl|\hat Q\bigl(x; \theta^{(m)}(k)  \bigr) - \hat Q\bigl(x;\bar \theta_i(k\epsilon) \bigr) \Bigr| \le B\cdot \norm[\big]{\theta^{(m)}(k) - \bar \theta^{(m)}(k\epsilon)}_{(m)}. \label{eq:ubi3}
	\end{align}
	Thus, we have that
	\begin{align}\label{eq:pf-od2}
	&\Bigl| L\bigl(\theta^{(m)}(k)\bigr) - L\bigl(\bar \theta^{(m)}(k\epsilon)\bigr) \Bigr| \nonumber \\
	&\quad \le B \cdot \norm[\big]{\theta^{(m)}(k) - \bar \theta^{(m)}(k\epsilon)}_{(m)} \nonumber\\
	&\quad \le B\cdot e^{BT} \cdot \Bigl( \sqrt{\log(m/\delta)/m}  + \sqrt{\epsilon\cdot \bigl(D + \log(m/\delta)\bigr)}\Bigr), \quad \forall k \le T/\epsilon \ (k\in \NN)
	\end{align}
	with probability at least $1-\delta$. Here the last inequality follows from Lemmas \ref{lem:ipd-cttd}-\ref{lem:ptd-td}.

	\vskip4pt
	\noindent{\bf Upper bounding term (ii) of \eqref{eq:pf-od1}.} Let $t =k\epsilon$. It holds for any $t \in [0,T]$ that
	\begin{align}
	\label{eq:pf-od3}
		\Bigl| L\bigl(\bar \theta^{(m)}(t)\bigr) - \bar L(\rho_{t}) \Bigr| \le \biggl| L\bigl(\bar \theta^{(m)}(t)\bigr) - \EE_{\rho_t}\Bigl[L\bigl(\bar \theta^{(m)}(t)\bigr)\Bigr] \biggr| +  \biggl| \EE_{\rho_t}\Bigl[L\bigl(\bar \theta^{(m)}(t)\bigr)\Bigr] - \bar L(\rho_{t}) \biggr|,
	\end{align}
	where the expectation is with respect to $\bar\theta_i(t) \overset\iid\sim \rho_t \ (i\in [m])$.
	For the second term on the right-hand side of \eqref{eq:pf-od3}, following from the fact that $\EE_{\rho_t}[\hat Q(x;\bar\theta^{(m)}(t))] = Q(x; \rho_t)$ for any $x\in \cX$, we have that
	\begin{align}
	\label{eq:pf-od31}
	\biggl| \EE_{\rho_t}\Bigl[L\bigl(\bar \theta^{(m)}(t)\bigr)\Bigr] - \bar L(\rho_{t}) \biggr| &=\biggl| \int \EE_{\rho_t}\Bigl[ \hat Q\bigl(x;\bar \theta^{(m)}(t) \bigr)^2 - Q(x; \rho_t)^2 \Bigr] \,\rd \cD(x) \biggr| \nonumber \\
	&= \biggl| \int {\rm Var}_{\rho_t}\Bigl[  \hat Q\bigl(x;\bar\theta^{(m)}(t) \bigr) \Bigr] \,\rd \cD(x) \biggr| \nonumber\\
	&\le B / m,
	\end{align}
	where the inequality follows from the fact that $\norm{\sigma} \le B$ in Assumption \ref{asp:activation} and the independence of $\bar\theta_i(t) \ (i\in [m])$. Let $\theta^{1, (m)} = \{\theta_1, \ldots, \theta_i^1,\ldots, \theta_m\}$ and $\theta^{2, (m)} =\{ \theta_1, \ldots, \theta_i^2,\ldots, \theta_m\}$ be two sets that only differ in the $i$-th element. It holds that
	\begin{align*}
	\bigl| L(\theta^{1, (m)}) - L(\theta^{2, (m)}) \bigr| \le B \cdot m^{-1} \cdot \EE_\cD\Bigl[\bigl| \sigma(x; \theta_i^1) - \sigma(x; \theta_i^2) \bigr| \Bigr] \le B/ m,
	\end{align*}
	where the first inequality follows from \eqref{eq:ubi1} and \eqref{eq:ubi2} and the second inequality follows from Assumption \ref{asp:activation}. Applying McDiarmid's inequality \citep{wainwright2019high}, we have for a fixed $t\in [0, T]$ that
	\begin{align}
	\label{eq:pf-od32}
	\PP\Biggl( \biggl| L\bigl(\bar \theta^{(m)}(t)\bigr) - \EE_{\rho_t}\Bigl[L\bigl(\bar \theta^{(m)}(t)\bigr)\Bigr] \biggr| \ge p \Biggr) \le \exp(-mp^2 / B).
	\end{align}
	It holds for any $s, t\in [0, T]$ that
	\begin{align*}
	&\Biggl|\biggl| L\bigl(\bar \theta^{(m)}(t)\bigr) - \EE_{\rho_t}\Bigl[L\bigl(\bar \theta^{(m)}(t)\bigr)\Bigr] \biggr| - \biggl| L\bigl(\bar \theta^{(m)}(s)\bigr) - \EE_{\rho_t}\Bigl[L\bigl(\bar \theta^{(m)}(s)\bigr)\Bigr] \biggr| \Biggr| \\&\quad \le B \cdot \norm[\big]{\bar \theta^{(m)}(t) - \bar \theta^{(m)}(s)}_{(m)} \le B\cdot |t-s|,
	\end{align*}
	where the first inequality follows from \eqref{eq:ubi1}, \eqref{eq:ubi2}, and \eqref{eq:ubi3} and the second inequality follows from \eqref{eq:lip-ip} of Lemma \ref{lem:lip-theta}.
	Applying the union bound to \eqref{eq:pf-od32} for $t \in \iota \cdot \{0, 1, \ldots, \lfloor T/\iota \rfloor \}$, we have that
	\begin{align*}
	\PP\Biggl( \sup_{t\in [0, T]}\biggl| L\bigl(\bar \theta^{(m)}(t)\bigr) - \EE_{\rho_t}\Bigl[L\bigl(\bar \theta^{(m)}(t)\bigr)\Bigr] \biggr| \ge p + B\iota \Biggr) \le (T/\iota + 1)\cdot \exp(-mp^2 / B),
	\end{align*}
	Setting $\iota = m^{-1/2}$ and $p = B\cdot \sqrt{\log(mT\delta)/m}$, we have that
	\begin{align}
	\label{eq:pf-od33}
	\sup_{t\in [0, T]}\biggl| L\bigl(\bar \theta^{(m)}(t)\bigr) - \EE_{\rho_t}\Bigl[L\bigl(\bar \theta^{(m)}(t)\bigr)\Bigr] \biggr| \le  B\cdot \sqrt{\log(mT\delta)/m}
	\end{align}
	with probability at least $1-\delta$. Plugging \eqref{eq:pf-od31} and \eqref{eq:pf-od33} into  \eqref{eq:pf-od3}, noting that $t = k\epsilon$, we have that
	\begin{align} \label{eq:pf-od30}
	\Bigl| L\bigl(\bar \theta^{(m)}(k\epsilon)\bigr) - \bar L(\rho_{k\epsilon}) \Bigr| \le B\cdot \sqrt{\log(mT\delta)/m}, \quad \forall k \le T/\epsilon \ (k\in \NN)
	\end{align}
	with probability at least $1-\delta$.

	Plugging \eqref{eq:pf-od2} and \eqref{eq:pf-od30} into \eqref{eq:pf-od1}, we have that
	\begin{align*}
	\Bigl| L\bigl(\theta^{(m)}(k)\bigr) - \bar L(\rho_{k\epsilon}) \Bigr| \le B\cdot e^{BT} \cdot \Bigl( \sqrt{\log(m/\delta)/m}  + \sqrt{\epsilon\cdot \bigl(D + \log(m/\delta)\bigr)}\Bigr), \quad \forall k\le T/\epsilon \ (k\in \NN)
	\end{align*}
	with probability at least $1-\delta$. Thus, we complete the proof of Lemma \ref{lem:opt-dis}.
\end{proof}

\subsection{Technical Lemmas for \S\ref{sec:discretization}} \label{sec:tech}
In what follows, we present the technical lemmas used in \S\ref{sec:discretization}. Recall that $\hat Q$, $\hat g$, and $\hat G_k$ are defined in \eqref{eq:nn-fin}, \eqref{eq:g-hat}, and \eqref{eq:G}, respectively. Let $B > 0$ be a constant depending on $\alpha$, $ \eta $, $ \gamma $, $ B_r $, and $ B_j \ (j\in \{0,1,2\}) $, whose value varies from line to line.
\begin{lemma}
	\label{lem:g-hat}
	Under Assumptions \ref{asp:data} and \ref{asp:activation}, it holds for any $\theta^{(m)} = \{\theta_i\}_{i=1}^m$ and $\underline \theta^{(m)} = \{\underline \theta_i\}_{i=1}^m$ that
	\begin{align}
	\sup_{x\in \cX} \bigl| \hat Q(x; \theta^{(m)}) \bigr| &\le B, \label{eq:bound-qh}\\
	\sup_{x\in \cX}\bigl| \hat Q(x; \theta^{(m)}) - \hat Q(x; \underline\theta^{(m)}) \bigr| &\le B\cdot \norm{\theta^{(m)} - \underline\theta^{(m)} }_{(m)}, \label{eq:lip-qh}\\
	\norm[\big]{\hat G_k(\theta_i; \theta^{(m)})} &\le B, \label{eq:bound-G} \\
	\norm[\big]{ \hat G_k(\theta_i; \theta^{(m)}) - \hat G_k(\underline\theta_i; \underline\theta^{(m)}) } &\le B\cdot \norm{\theta^{(m)} - \underline\theta^{(m)} }_{(m)}, \quad   \forall k\in \NN,\label{eq:lip-G}\\
	\norm[\big]{\hat g(\theta_i; \theta^{(m)})} &\le B, \label{eq:bound-gh} \\
	\norm[\big]{ \hat g(\theta_i; \theta^{(m)}) - \hat g (\underline\theta_i; \underline\theta^{(m)}) } &\le B\cdot \norm{\theta^{(m)} - \underline\theta^{(m)} }_{(m)}. \label{eq:lip-gh}
	\end{align}
	Meanwhile, for any $Q \in \cF$, it holds that
	\begin{align}
	\sup_{x\in \cX} \norm[\big]{Q(x)} \le B. \label{eq:bound-q}
	\end{align}
	For any $\rho \in \sP_2(\RR^D)$, it holds that
	\begin{align}
	\norm[\big]{g(\theta; \rho)} &\le B. \label{eq:bound-g}
	\end{align}
\end{lemma}
\begin{proof}
	For \eqref{eq:bound-qh} and \eqref{eq:lip-qh} of Lemma \ref{lem:g-hat},
	following from Assumptions \ref{asp:data} and \ref{asp:activation} and the definition of $\hat Q$ in \eqref{eq:nn-fin}, we have for any $x\in \cX$, $\theta^{(m)}$, and $\underline \theta^{(m)}$ that
	\begin{align*}
	&\bigl| \hat Q(x; \theta^{(m)}) \bigr| \le \alpha \cdot m^{-1} \sum_{i=1}^{m} \bigl| \sigma(x; \theta_i) \bigr| \le B, \\
	&\bigl| \hat Q(x; \theta^{(m)}) - \hat Q(x; \underline\theta^{(m)}) \bigr| \le \alpha \cdot m^{-1} \sum_{i=1}^{m} \bigl| \sigma(x; \theta_i) - \sigma(x; \underline \theta_i) \bigr| \le B\cdot \norm{\theta^{(m)} - \underline \theta^{(m)}}_{(m)}.
	\end{align*}
	For \eqref{eq:bound-G} and \eqref{eq:lip-G} of Lemma \ref{lem:g-hat},
	following from the definition of $\hat G_k$ in \eqref{eq:G}, we have for any $\theta^{(m)}$ and $\underline \theta^{(m)}$  that
	\begin{align*}
	&\bigl\| \hat G_k(\theta_i; \theta^{(m)})\bigr\| = \alpha \cdot \bigl|\hat Q(x_k; \theta^{(m)}) - r_k- \gamma \cdot \hat Q(x_k'; \theta^{(m)})\bigr| \cdot \norm[\big]{ \nabla_\theta \sigma(x_k; \theta_i)} \le B, \\
	&\norm[\big]{ \hat G_k(\theta_i; \theta^{(m)}) - \hat G_k(\underline\theta_i; \underline\theta^{(m)}) } \\&\quad \le \alpha \cdot \sup_{\theta^{(m)}} \bigl|\hat Q(x_k; \theta^{(m)}) - r_k- \gamma \cdot \hat Q(x_k'; \theta^{(m)})\bigr| \cdot \norm[\big]{ \nabla_\theta \sigma(x_k; \theta_i) - \nabla_\theta \sigma(x_k; \underline \theta_i)} \nonumber\\
	&\qquad + \alpha \cdot \bigl|\hat Q(x_k; \theta^{(m)}) - \gamma \cdot \hat Q(x_k'; \theta^{(m)}) - \hat Q(x_k; \underline \theta^{(m)}) + \gamma \cdot\hat Q(x_k'; \theta^{(m)})  \bigr| \cdot \sup_{\theta_i \in \RR^D}\norm[\big]{ \nabla_\theta \sigma(x_k; \theta_i)} \\
	&\quad \le B \cdot \norm{\theta^{(m)} - \underline \theta^{(m)}}_{(m)}.
	\end{align*}
	The inequalities in \eqref{eq:bound-gh} and \eqref{eq:lip-gh} of Lemma \ref{lem:g-hat} for $\hat g$ follow from the fact that
	\begin{align*}
	\hat g(\theta_i; \theta^{(m)}) = \EE_{(x_k, r_k, x_k') \sim \tilde \cD}\bigl[G_k(\theta_i; \theta^{(m)})\bigr].
	\end{align*}
	The inequalities in \eqref{eq:bound-q} and \eqref{eq:bound-g} follow from the definition of $\cF$ and $g$ in \eqref{eq:func-class} and \eqref{eq:g-rho}, respectively. Thus, we complete the proof of Lemma \ref{lem:g-hat}.
\end{proof}

Recall that $\rho_t$ is the PDE solution in \eqref{eq:pde-fixed} and $\tilde\theta^{(m)}(t)$ and $\bar \theta^{(m)}(t)$ are the CTTD and IP dynamics defined in \eqref{eq:cttd} and \eqref{eq:idp}, respectively.
\begin{lemma}
	\label{lem:lip-theta}
	Under Assumptions \ref{asp:data} and \ref{asp:activation},
	it holds for any $s, t \in [0, T]$ that
	\begin{align}
	\norm[\big]{\bar \theta^{(m)}(t) - \bar \theta^{(m)}(s)}_{(m)} &\le B\cdot |t-s|, \label{eq:lip-ip} \\
	\norm[\big]{\tilde \theta^{(m)}(t) - \tilde \theta^{(m)}(s)}_{(m)} &\le B\cdot |t-s|, \label{eq:lip-cttd}\\
	\cW_2(\rho_t, \rho_s) &\le B \cdot |t-s|. \label{eq:lip-pde}
	\end{align}
\end{lemma}
\begin{proof}
	For \eqref{eq:lip-ip} of Lemma \ref{lem:lip-theta},
	by the definition of $\bar \theta_i(t)$ in \eqref{eq:idp} and \eqref{eq:bound-g} of Lemma \ref{lem:g-hat}, we have for any $s, t \in [0, T]$ and $i \in [m]$ that
	\begin{align*}
	\norm[\big]{\bar \theta_i(t) - \bar \theta_i(s)} = \eta\cdot \int_s^t \norm[\Big]{g\bigl(\bar \theta_i(\tau); \rho_\tau \bigr)} \, \rd \tau \le B\cdot |t-s|.
	\end{align*}
	Similarly, for \eqref{eq:lip-cttd} of Lemma \ref{lem:lip-theta}, by the definition of $\tilde \theta_i(t)$ in \eqref{eq:cttd} and \eqref{eq:bound-gh} of Lemma \ref{lem:g-hat}, we have for any $i \in [m]$ and $s, t\in [0, T]$ that $ \norm{\tilde \theta_i(t) - \tilde \theta_i(s)}\le B\cdot |t-s| $.

	For \eqref{eq:lip-pde} of Lemma \ref{lem:lip-theta}, following from the fact that $\bar\theta_i(t) \overset\iid\sim \rho_t\ (i\in [m])$ and the definition of $\cW_2$ in \eqref{eq:w2-1}, it holds for any $s, t\in [0, T]$ that
	\begin{align*}
	\cW_2(\rho_t, \rho_s) \le \EE\Bigl[ \norm[\big]{\bar \theta_i(t) - \bar \theta_i(s)}^2 \Bigr]^{1/2} \le B\cdot |t-s|.
	\end{align*}
	Thus, we complete the proof of Lemma \ref{lem:lip-theta}.
\end{proof}

\begin{lemma}[Lemma 30 in \cite{mei2019mean}]
	\label{lem:mc}
	Let $\{X_i\}_{i=1}^m$ be i.i.d.\ random variables with $\norm{X_i} \le \xi$ and $\EE[X_i] = 0$. Then, it  holds for any $p>0$ that
	\begin{align*}
	\PP\bigg( \norm[\Big]{m^{-1} \cdot \sum_{i=1}^{m} X_i} \ge C\xi\cdot (m^{-1/2} + p)\biggr) \le \exp(-mp^2),
	\end{align*}
	where $C > 0$ is an absolute constant.
\end{lemma}

\begin{lemma}[Lemma 31 in \cite{mei2019mean} and Lemma A.3 in \cite{araujo2019mean}]
	\label{lem:azuma}
	Let $X_k\in \RR^D \ (k\in \NN)$ be a martingale with respect to the filtration $\cG_k\ (k\ge 0)$ with $X_0 = 0$. We assume for $\xi > 0$ and any $\lambda \in \RR^D$ that
	\begin{align*}
	\EE\Bigl[ \exp\bigl( \inp{\lambda}{X_k-X_{k-1}} \bigr) \Biggiven \cG_{k-1} \Bigr] \le \exp\bigl(\xi^2\cdot \norm{\lambda}^2 / 2 \bigr).
	\end{align*}
	Then, it holds that
	\begin{align*}
	\PP\Bigl(\max_{\substack{k\le n\\(k\in \NN)}} \norm{X_k} \ge C \xi \cdot\sqrt{n}\cdot (\sqrt{D} + p)\Bigr) \le \exp(-p^2),
	\end{align*}
	where $C>0$ is an absolute constant.
\end{lemma}

\section{Auxiliary Lemmas}
We use the definition of absolutely continuous curves in $\sP_2(\RR^D)$ in \cite{ambrosio2008gradient}.
\begin{definition}[Absolutely Continuous Curve] \label{def:ac-curve}
	Let $\beta: [a, b] \rightarrow \sP_2(\RR^D)$ be a curve. Then, we say $\beta$ is an absolutely continuous curve if there exists a square-integrable function $f:[a, b] \rightarrow \RR$ such that
	\begin{align*}
	\cW_2( \beta_s, \beta_t) \le \int_{s}^{t} f(\tau) \,\rd \tau
	\end{align*}
	for any $a\le s< t\le b$.
\end{definition}

Then, we have the following first variation formula.
\begin{lemma}[First Variation Formula, Theorem 8.4.7 in \cite{ambrosio2008gradient}]
	\label{lem:diff}
	Given $\nu \in \sP_2(\RR^D)$ and an absolutely continuous curve $\mu: [0, T] \rightarrow \sP_2(\RR^D)$, let $\beta: [0,1] \rightarrow \sP_2(\RR^D)$ be the geodesic connecting $\mu_t$ and $\nu$. It holds that
	\begin{align*}

	\frac{\rd }{\rd t}\frac{ \cW_2(\mu_t, \nu)^2}{2} = -\inp{\mu'_t}{\beta'_0}_{\mu_t},
	\end{align*}
	where $\mu'_t = \partial_t \mu_t$, $\beta'_0 = \partial_t \beta_t\given_{t=0}$, and the inner product is defined in \eqref{eq:w-inner}.
\end{lemma}

\begin{lemma}[Talagrand's Inequality, Corollary 2.1 in \cite{otto2000generalization}]
	\label{lem:talagrand}
	Let $\nu$ be $N(0, \kappa \cdot I_D)$. It holds for any $\mu \in \sP_2(\RR^D)$ that
	\begin{align*}
	\cW_2(\mu, \nu)^2 \le 2 D_{\rm KL}(\mu \,\|\, \nu) / \kappa.
	\end{align*}
\end{lemma}

\begin{lemma}[Eulerian Representation of Geodesics, Proposition 5.38 in \cite{villani2003topics}]
	\label{lem:euler}
	Let $\beta: [0, 1] \rightarrow \sP_2(\RR^D)$ be a geodesic and $v$ be the corresponding vector field such that $\partial_t \beta_t = - \Div(\beta_t \cdot v_t)$. It holds that
	\begin{align*}

	\partial_t(\beta_t \cdot v_t) = - \Div(\beta_t \cdot v_t\otimes v_t).
	\end{align*}
\end{lemma}


TITLE: Well-posedness for equations of the form $u_t = grad[V(u)]$ and $u_{tt}=grad[V(u)]$?
QUESTION [0 upvotes]: Let $V \in C^{1}(\mathbb{R}^n, \mathbb{R})$ consider the following PDE:
$$u_t = grad[V(u)]$$
For $u \in C^{1}([0,1]^n\times [0,T),\mathbb{R}^n)$, with boundary conditions specified on the $n$-dimensional faces of the $n+1$-th cube $[0,1]^{n} \times [0,T)$. Where by "$grad$" I mean the gradient w.r.t. to the spatial coordinates (excluding the last coordinate).
Are there necessary and sufficient conditions one can put on $V$ for which the above problem will always be well-posed for some choice of $T$? (for initial conditions in reasonable function spaces).
Are there similar conditions (perhaps more complicated) for the second order equation:
$$u_{tt} = grad[V(u)]$$
With the same type of boundary conditions?

REPLY [2 votes]: The first equation (first order) is a quasilinear transport equation. Developping, it writes $u_t+(Z\cdot\nabla_x)u=0$ where $Z:=-(\nabla_uV)\circ u$. The method of characteristics tells you that $u$ is constant along the integral curves of $Z$. Since $Z$ depends only on $u$, these curves are straight lines.
The boundary conditions that you mention are not appropriate her. Your equation is hyperbolic. A Cauchy problem (initial data prescribed at $t=0$) is well-suited. If your spatial domain is bounded, you must impose a Dirichlet condition only along the parts of the boundary where the characteristics are incoming. Mind however that because of nonlinearity, shock waves will usually develop in finite time (think that the characteristic lines are not parallel, and must meet  after some time interval). Therefore you have to consider entropy solution. The Cauchy problem is uniquely solvable (Kruzhkov, 1970). An appropriate initial-boundary value problem is, too (F. Otto ?).
As for the second problem, the Cauchy problem is usually ill-posed, because the order of space derivatives is less than that of the time derivative. This is already true in the linear case if $n=1$, where your equation is a "rotated" heat equation $u_{tt}=u_x$. The pure Cauchy problem (domain ${\mathbb R}^n$) is ill-posed in every reasonable space ($C^\infty$, Sobolev spaces) because it displays growing modes $e^{\alpha t|\xi|}$ ($|\xi|$ the space frequency variable), where $\alpha=e^{i\pi/4}$ ; their modulus $\exp(t\frac{|\xi|\sqrt2}2)$ grows arbitrarily fast. This, with the Uniform Boundedness Principle, proves ill-posedness.


TITLE: Doubt about find probability
QUESTION [0 upvotes]: I need help with this exercise:

The river pollution in the United States is a problem of many years. Consider the events below:
$A$ = {The river is polluted}
$B$ = {A tested sample water detects pollution}
$C$ = {Fishing is allowed}

Assume:

$P(A) = 0.3$
$P(B|A) = 0.75$
$P(B|A^c) = 0.20$
$P(C|A\cap B) = 0.20$
$P(C|A^c \cap B) = 0.15$
$P(C|A \cap B^c) = 0.80$
$P(C|A^c \cap B^c) = 0.90$

(a) Determine $P(A \cap B \cap C)$.
(b) Determine $P(B^c \cap C)$.
(c) Determine $P(C)$.
(d) Determine the probability of the river be polluted given the fishing is allowed and the tested sample didn't detect pollution.
My Try:
(a) $P(A \cap B \cap C) = P(C|A \cap B) \cdot P(A \cap B) \\ P(A \cap B \cap C) = 0.2 * P(B|A) * P(A) \\ P(A \cap B \cap C) = 0.2 * 0.75 * 0.3 = 0.045 = 4.5\%$
(b)I'm stuck at this point because I can't find any algebric manipulation in the given formulas and I don't understand the meaning if I draw the Venn Diagram.
Please, help.

REPLY [0 votes]: (a) $$P(A\cap B\cap C)=(0.3)(0.75)(0.2)$$
(b) $$P(B^c\cap C)=(0.3)(1-0.75)(0.8)+(1-0.3)(1-0.2)(0.9)$$
(c) $$P(C)=(0.3)(0.75)(0.2)+(0.3)(1-0.75)(0.8)+(1-0.3)(0.2)(0.15)+(1-0.3)(1-0.2)(0.9)$$
(d) $$P(A|B^c\cap C)=(0.3)(1-0.75)(0.8)\div\text{answer to part (b)}$$


TITLE: How can Particle Accelerators even crash particles?
QUESTION [1 upvotes]: In quantum mechanics there is a law that states we cannot know a position and the momentum of a particle and so in a particle accelerator, we need to know the position of the particles we are going to crash into each other because or else would they not basically  miss each other because we do not know the location of these. Next, if we do not know the particle's position how can we operate the magnetic fields without crashing these particles into the walls of the particle accelerator. As even a millimeter of inaccuracy might extrapolate and lead to a massive diversion from it desired path?
Furthermore, if we do not know the velocity of these particles assuming we know a very high accuracy of the particles position how can we even crash them into each other if they technically could be left behind or already past a location?
Or is it just done on basis of statistics and likeliness? If so is it impossible to collide ONLY 2 particles together at high energy levels without have certain amounts of pass-bys rather than a crash?

REPLY [1 votes]: With the uncertainty priciple, you cannot know the position and momentum of a particle (or anything for that matter) beyond a maximum precision.  You can know where and how fast a particle is moving, just not with infinite precision in both variables.  It's not just a measurement issue, but a fundamental fact.  So a particle accelerator can still focus a beam to a fairly fine point with a fairly well know velocity. It is just impossible to know both with absolute precision. In practice, measurement is the real issue as the uncertainty principle limit is very small.  They do require a bit of luck to get collisions.


TITLE: Prove that $f$ is onto if and only if $g$ is onto
QUESTION [0 upvotes]: For any function $f:A\rightarrow B$, define a new function $g:P(A)\rightarrow P(B)$ as follows: for every $S \subseteq A$, define $g(S)=\{f(x) \mid x\in S\}$. Prove that $f$ is onto if and only if $g$ is onto.
I'm not sure how to begin, and I'm particularly confused about what exactly $g(S)$ means. Is there any insight that may help me on my way to solve this?  How can I show $f$ is onto $ \leftrightarrow g$ is onto?
Thank you!

REPLY [1 votes]: A picture like this one may help a bit:

The black arrows show the function $f:A\to B$; the colored blobs show elements of $\wp(A)$ on the left and their images under $g$ in $\wp(B)$ on the right. The colored arrows show the action of $g$ on these sets.
Added: I’ll do one direction and leave the other to you. Suppose that $f$ maps $A$ onto $B$. To show that $g$ is onto, I must show that for each $Y\subseteq B$ there is some $X\subseteq A$ such that $g(X)=Y$, so let $Y$ be an arbitrary subset of $B$. Let $X=\{x\in A:f(x)\in Y\}$. (In other words, $X=f^{-1}[Y]$.) I claim that $g(X)=Y$.
To see this, recall that $g(X)=\{f(x):x\in X\}$; clearly $f(x)\in Y$ for each $x\in X$, since we chose $X$ to make that true, so $g(X)\subseteq Y$. On the other hand, the map $f$ is onto, so for each $y\in Y$ there is an $x\in A$ such that $f(x)=y$. Clearly this $x_y\in X$, so for each $y\in Y$ there is an $x\in X$ such that $f(x)=y$. Thus, $g(X)\supseteq Y$. Combining the inclusions, we see that $g(X)=Y$, so $Y$ is in the range of $g$. $Y$ was an arbitrary subset of $B$, so every subset of $B$ is in the range of $g$, and therefore $g$ must map $\wp(A)$ onto $\wp(B)$.
Now see if you can show that if $g$ is onto, so is $f$.


TITLE: $\lim_{x \to 0} \frac{e^{-x}\sin(x^2)}{x^4}$
QUESTION [0 upvotes]: Is there an easy way to solve this or would I have to painstakingly use l'hopitals rule many times over? $$\lim_{x \to 0} \frac{e^{-x}\sin(x^2)}{x^4}$$

REPLY [0 votes]: Just break the limit down.
$$\lim_{x \to 0} \frac{e^{-x}\sin (x^2)}{x^4} = \lim_{x \to 0}\frac{\sin (x^2)}{x^2} \cdot \frac{1}{e^x} \cdot \frac{1}{x^2}$$
Now, evaluate the limits as $x \to 0$.


TITLE: Where can I find the following S. Shelah's paper?
QUESTION [7 upvotes]: I've been trying to find the following article: "S. Shelah, Remarks on cardinal invariants in topology, General topology Appl. 7(3) (1977), 251-259". I tried to go directly to the journal page, but it turns out that Issue 3 isn't registered there (see: General Topology and its Applications).
Is there any other place where I can find this paper?

REPLY [9 votes]: please get the pdf here.
note that the text itself starts on page 5.