text:
\section{Introduction }
Probability theory  has played a very important role in the study of partial differential equations (PDEs). Starting with the work of A. N. Kolmogorov \cite{KL}, the probabilistic approach  has been   successfully applied  in deducing scaling laws of turbulence in fluid (see \cite{FF, DS} and references within). As a funder of stochastic calculus,   K. It\^o \cite{Ito} developed stochastic differential equations (SDEs) to construct Markov processes characterizing linear Kolmogorov-Fokker-Planck equations. Since then several probability models have been introduced to solve nonlinear PDEs, which include the backward SDE initiated by Bismut \cite{Bismut}  for the linear case and    Pardoux-Peng \cite{PP} for the nonlinear case  (see \cite{CSTV} and references within), the SDE driven by $G$-Brownian motion introduced by Peng \cite{Peng0}
(see \cite{Peng} and references within), and the distribution dependent (McKean-Vlasov) SDE introduced by McKean \cite{MC} (see \cite{BR} and references within). The first two models describe   the viscosity solution, while the third one corresponds to the weak solution (in the sense of integration by parts) of nonlinear PDEs. See also  \cite{CI, Qian}  and references within
for characterizations on the incompressible Navier-Stokes equation    using SDEs.

 In this paper, we investigate a class of new type distribution dependent SDEs and establish correspondence to nonlinear PDEs,  such that the classical existence and uniqueness  of singular nonlinear PDEs are derived.

Let $m,d \in \mathbb N$,    and let $E= \R^d$ or $\T^d:= \R^d/\mathbb Z^d.$  For fixed $T\in (0,\infty)$, we consider   the following equation  for
$u: [0,T]\times E\rightarrow} \def\gg{\gamma \R^m$:
\beq\label{E3} \pp_t u_t= (L_t +V_t)u_t+ F_t(\cdot, u_t,\nabla} \def\pp{\partial} \def\E{\mathbb E u_t)\cdot \nabla} \def\pp{\partial} \def\E{\mathbb E  u_t + g_t(\cdot, u_t,\nabla} \def\pp{\partial} \def\E{\mathbb E u_t,\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u_t),\ \ t\in [0,T], \end{equation}
where
$$ L_t:= {\rm tr}\{a_t\nabla} \def\pp{\partial} \def\E{\mathbb E^2\}+b_t\cdot\nabla} \def\pp{\partial} \def\E{\mathbb E=\sum_{i,j=1}^d a_t^{ij}\pp_i\pp_j+\sum_{i=1}^d b_t^i\pp_i$$
is a  time dependent second order differentiable operator on $E$ with measurable coefficients
$$a: [0,T]\times E\rightarrow} \def\gg{\gamma \R^{d\otimes d},\ \ b: [0,T]\times E\rightarrow} \def\gg{\gamma\R^d,$$
$\nabla} \def\pp{\partial} \def\E{\mathbb E u_t=(\pp_i u_t^j)_{1\le i\le d, 1\le j\le m}$ takes values in $\R^{d\otimes m},$   $\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u_t=(\pp_i\pp_j u_t^k)_{1\le i,j\le d, 1\le k\le m}$ takes values in $\R^{d\otimes d\otimes m},$ and
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*}& g: [0,T]\times E\times \R^m\times \R^{d\otimes m}\times\R^{d\otimes d\otimes m} \rightarrow} \def\gg{\gamma \R^m,\\
&F: [0,T]\times E\times \R^m\times \R^{d\otimes m}  \rightarrow} \def\gg{\gamma \R^d,\ \ \ \   \ V: [0,T]\times E\rightarrow} \def\gg{\gamma \R\end{align*}
are measurable.
Not that formally one may put  $V_tu_t$ and $F_t \cdot \nabla} \def\pp{\partial} \def\E{\mathbb E  u_t$ into $g_t$, but  conditions we will use for these two terms  are not covered by that for $g_t$,    see Remark 2.1 below for details.

Besides the Schr\"odinger equation for $F=0$ and $g_t(x,r_1,r_2,r_3)=g_t(x)$,   \eqref{E3} includes the following KPZ and Navier-Stokes type equations as typical examples. We will use $|\cdot|$ to denote the Euclidean norm for vectors or matrices.

 \ \newline
{\bf KPZ equation.}
 In \cite{KPZ},  Kardar, Parisi and Zhang (KPZ)  introduced a nonlinear PDE
  to describe the fluctuation   of interfaces for fluids on $E$. After a Taylor expansion for small size of the
gradient, the    KPZ equation  is formulated as
\beq\label{E1} \pp_t u_t = \DD u_t + \bb |\nabla} \def\pp{\partial} \def\E{\mathbb E u_t|^2+ g_t,\ \ t\ge 0  \end{equation} for
$u: [0,\infty)\times E\rightarrow} \def\gg{\gamma \R$,
 where $\bb $ is a constant,  and $g: [0,\infty)\times E\rightarrow} \def\gg{\gamma \R$ is measurable.
 By   the Cole-Hopf transformation
 $$v_t=\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-\bb u_t},\ \ t\ge 0,$$
 \eqref{E1} is equivalent to the linear (Schr\"odinger) equation
 $$\pp_t v_t = \DD v_t -\bb   g_t v_t,\ \ t\ge 0.$$

Some extensions of \eqref{E1} have been studied in the literature,  for instances,  see \cite{ADP} for
 $\bb$ being replaced by a function $\bb(u_t)$,   see \cite{KS,Gi} for constant $\bb$ but with
   with $|\nabla} \def\pp{\partial} \def\E{\mathbb E u_t|^2$ being generalized as   $|\nabla} \def\pp{\partial} \def\E{\mathbb E u_t|^p$  for some $p>0$, and see \cite{APP} for $\DD$ being replaced by a fractional Laplacian $-(-\DD)^s$ for $s\in (0,1)$. In the SPDE setup, the function $g$ is replaced by a time-space noise (see \cite{H}),   which will not be  considered in the present paper. Obviously, when $V=0$,  $L_t=\DD, m=1,$
   $g_t(x,r_2,r_2,r_3)=g_t(x)$ and $F_t(x,u,\nabla} \def\pp{\partial} \def\E{\mathbb E u,\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u)=\bb (u) |\nabla} \def\pp{\partial} \def\E{\mathbb E u|^{p-2} \nabla} \def\pp{\partial} \def\E{\mathbb E u$ for some function $\bb: \R\rightarrow} \def\gg{\gamma\R,$   \eqref{E3} reduces to the KPZ type equation
$$\pp_t u_t= \DD u_t + \bb(u_t)|\nabla} \def\pp{\partial} \def\E{\mathbb E u_t|^p+g_t.$$
\newline
{\bf     Navier-Stokes type equation.} Recently, the author \cite{22} introduced  distribution dependent SDEs to solve the equation
 \beq\label{E1'} \pp_t u_t = L_t u_t -(u_t\cdot\nabla} \def\pp{\partial} \def\E{\mathbb E) u_t-\nabla} \def\pp{\partial} \def\E{\mathbb E p_t\end{equation}
 for $u: [0,\infty)\times E\rightarrow} \def\gg{\gamma \R^d.$
When $L_t=\kk \DD$ for some constant $\kk>0$, \eqref{E1'} together with the divergence free condition
$\nabla} \def\pp{\partial} \def\E{\mathbb E\cdot u_t=0$ is known as the incompressible Navier-Stokes equation describing the  turbulence of fluids, for which   a long standing open problem is to prove (or disprove) the existence and smoothness of global solutions  for $d=3$.  Obviously, when $V=0$,   $L_t=\kk\DD$ for some constant $\kk>0$, $m=d$, $F_t(x,u,\nabla} \def\pp{\partial} \def\E{\mathbb E u)=- u$ and $g_t(x,r_1,r_2,r_3) =-\nabla} \def\pp{\partial} \def\E{\mathbb E p_t(x)$, \eqref{E3} becomes \eqref{E1'}.

\

We will state the main results of the paper in Section 2,   explain   the   idea of solving the new type SDE and present some lemmas in Section 3,
and finally  prove the main results   in Sections 4 and 5.

\section{Main results}

We first state the main result on   \eqref{E3}, then introduce a new type distribution dependent SDEs and make correspondence  to nonlinear PDEs.

\subsection{Well-posedness   of \eqref{E3}}

As we mentioned above that the coefficients of  \eqref{E3} may be singular in $(t,x)$. To measure the singularity,   we recall some functional spaces introduced  in \cite{XXZZ}.

 For any $p,q>1$ and $0\le t<s$, we write $f\in \tilde}\def\[{\lfloor} \def\]{\rfloor L_q^p(t,s)$ if   $f$ is a (real or vector valued) measurable function on $[t,s]\times E$ such that
 $$\|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_q^p(t,s)}:=\sup_{z\in E} \bigg(\int_t^s\|f_r1_{B(z,1)}\|_{L^p}^q\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r\bigg)^{\ff 1 q}<\infty,$$
 where $B(z,1)$ is the unit ball at $z$, and   $\|\cdot\|_{L^p} $ is the $L^p$-norm for the Lebesgue measure.
When $f_t(x)=f(x)$ does not depend on $t$, let
  $$\|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L^p}:=\sup_{z\in E}\|f1_{B(z,1)}\|_{L^p}.$$ When $E=\T^d$ which is compact, we may drop $1_{B(z,1)}$ such that $\tilde}\def\[{\lfloor} \def\]{\rfloor L^p$ and $\tilde}\def\[{\lfloor} \def\]{\rfloor L_q^p$ reduce to the usual
  $L^p$ and $L_q^p$ spaces.

 We will take $(p,q)$ from the following class:
 $$\scr K:=\Big\{(p,q): \ p,q\in (2,\infty), \      \ff {d}p +\ff 2 q<1\Big\}.$$
Let $\|\cdot\|_\infty$ denote the uniform norm, and let $\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1$ be the class of (real or vector valued) weakly differentiable functions $f$ on $E$ such that
$$\|f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}:=\|f\|_\infty+\|\nabla} \def\pp{\partial} \def\E{\mathbb E f\|_\infty<\infty,$$
where $\|\nabla} \def\pp{\partial} \def\E{\mathbb E f\|_\infty$ coincides with the Lipschitz constant of $f$.
Let
$$k(u_0,g,V):= \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_0^T \|V_t\|_\infty\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t} \bigg(\|u_0\|_\infty+ \int_0^T \|g_t\|_\infty\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t\bigg),$$ which might be infinite, and define
$$\bar F_t(x):= \sup_{|r_1|\le k(u_0,g,V)}\|F_t(x,r_1,\cdot)\|_\infty,\ \ \bar g_t(x)=|g_t(x,0,0,0)|,\ \ \ (t,x)\in [0,T]\times E.$$
When $k(u_0,g,V)<\infty$, $\bar F$ is finite provided $F_t(x,r_1,r_2)$ is bounded in $r_2$ but locally bounded in $r_1$.
We assume that $V, u_0, a,b, F$ and $g$ satisfy the following  conditions.

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{enumerate} \item[$(H_{V,u_0})$]
 $u_0\in \scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1,$ and there exists $(p_0,q_0)\in \scr K$ such that $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty.$
  \item[$(H_{a,b})$]   $a$ is invertible and positive definite,  $b$ is locally bounded,
  $$ \sup_{t\in [0,T]}\bigg\{\sup_{x\ne y} \ff{|b_t(x)-b_t(y)|}{|x-y|}+  \|a_t\|_\infty + \|a_t^{-1}\|_\infty\bigg\} <\infty,$$
$$ \lim_{\vv\rightarrow} \def\gg{\gamma 0} \sup_{|x-y|\le \vv, t\in [0,T]} |a_t(x)-a_t(y)|=0,$$
 and there exist $l\in \mathbb N$, $ (p_i,q_i) \in\scr K$ and $1\le f_i\in \tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_i}^{p_i}(0,T), 0\le i\le l$, such that
$$ |\nabla} \def\pp{\partial} \def\E{\mathbb E a_t(x)|\le \sum_{i=1}^l f_i (t,x),\ \ (t,x)\in [0,T]\times E. $$
 \item[$(H_{F,g})$]
 $\|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+ \|\bar g\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty$  for $(p_0,q_0)$ in $(H_{V,u_0})$, and there exist a constant $\aa>0$ and a map $K: \mathbb N\rightarrow} \def\gg{\gamma (0,\infty)$ such that
 \begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &  |F_t(x,r_1,r_2)- F_t(x,\tilde}\def\[{\lfloor} \def\]{\rfloor r_1,\tilde}\def\[{\lfloor} \def\]{\rfloor r_2)| +   |g_t(x,r_1,r_2, r_3)- g_t(x,\tilde}\def\[{\lfloor} \def\]{\rfloor r_1,\tilde}\def\[{\lfloor} \def\]{\rfloor r_2,\tilde}\def\[{\lfloor} \def\]{\rfloor r_3)|  \\
&  \le K_n(|r_1-\tilde}\def\[{\lfloor} \def\]{\rfloor r_1|+|r_2-\tilde}\def\[{\lfloor} \def\]{\rfloor r_2|) +\aa |r_3-\tilde}\def\[{\lfloor} \def\]{\rfloor r_3|,\\
   &\qquad (t,x)\in [0,T]\times E,\ |r_1|\lor |r_2|\lor |\tilde}\def\[{\lfloor} \def\]{\rfloor r_1|\lor |\tilde}\def\[{\lfloor} \def\]{\rfloor r_2|\le n,\ r_3,\tilde}\def\[{\lfloor} \def\]{\rfloor r_3\in \R^{d\otimes d\otimes m}.\end{align*}
  \end{enumerate}

 Under   these assumptions, we will   prove the existence and uniqueness of \eqref{E3}   in the following sense.

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{defn} We call $(u_t)_{t\in [0,T^*)}$ a solution of \eqref{E3} in the class $\scr U (p_0,q_0)$, if
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{enumerate}
\item[$(1)$] $T^*\in (0, T]$,
\beq\label{*0} \sup_{t\in [0,s]}  \|u_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1} +\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,s)}<\infty,\ \ s\in (0,T^*),\end{equation}
and for any $t \in [0,T^*),$
$$u_t =u_0  +\int_0^t \Big\{(L_s+V_s) u_s  +  F_s(\cdot,u_s,\nabla} \def\pp{\partial} \def\E{\mathbb E u_s)\cdot \nabla} \def\pp{\partial} \def\E{\mathbb E u_s + g_s(\cdot, u_s,\nabla} \def\pp{\partial} \def\E{\mathbb E u_s,\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u_s)\Big\}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s.$$
\item[$(2)$] When $T^*<T$,
$\limsup_{t\uparrow T^*} \|u_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}=\infty.$
\end{enumerate} \end{defn}

     To ensure the non-explosion (i.e.   $T^*=T$), we need the following assumption.

 \begin} \def\beq{\begin{equation}}  \def\F{\scr F{enumerate} \item[$(H_{F,g}')$] There exist   constants $C,\aa >0$ and $ \theta\in (0,\ff 1 2)$ such that
  \begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*}& \sup_{(t,x)\in [0,T]\times  E} |g_{t}(x, r_1,r_2, r_3)-g_t(x,0,0,0)|\\
 & \qquad\le C (\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+|r_1|+|r_2|) \{\log (\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega +|r_1|+|r_2|)\}^\theta+\aa|r_3|,\\
 &\qquad  \qquad\qquad (r_1, r_2,r_3)\in\R^m\times \R^{d\otimes m}\times\R^{d\otimes d\otimes m}.
\end{align*}
 \end{enumerate}

We have the following result.

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{thm}\label{T1} Assume $(H_{V,u_0})$, $(H_{a,b})$ and $(H_{F,g})$ with small enough $\aa>0$ determined by $(H_{a,b})$ and $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+
\|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$.
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{enumerate} \item[$(1)$]    $\eqref{E3}$ has a unique solution
in $\scr U(p_0,q_0)$.
\item[$(2)$]
 If   $(H_{F,g}')$   holds with small enough $\aa>0$ determined by $(H_{a,b})$ and $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+
\|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$, then $T^*=T$ and
\beq\label{NN} \sup_{t\in [0,T]} \|u_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty.\end{equation} \end{enumerate}
\end{thm}

\paragraph{Remark 2.1.} We do not combine $Vu$ and $F\cdot\nabla} \def\pp{\partial} \def\E{\mathbb E u$ with $g$, since   in Theorem \ref{T1} conditions on $V$ and $F$ are not covered by that on $g$.
 To see this, we assume   $(H_{a,b})$, let $F_t(x,r_1,r_2)$ and $g_t(x,r_1,r_2,r_3)=  g_t(x,r_1,r_2)$ be locally Lipschitz continuous in $(r_1,r_2)$, and  let $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty$. We consider the following two situations.
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{enumerate}
\item[(a)] Let for instance  $F$ and $g$ be  bounded.  Then  Theorem \ref{T1} implies   the well-posedness and non-explosion of \eqref{E3}.
However, since $|V_t|$ may be unbounded,   $(H_{F,g}')$ does not hold if we regard $V_t r_1$ as a part of $g_t(\cdot, r_1,r_2).$
\item[(b)]  Let for instance $ \|g\|_\infty+  \int_0^T \|V_t\|_\infty<\infty$ and $F_t(\cdot,r_1,r_2)=\bb(r_1)h_t$ for some locally bounded function $\bb$ on $\R^m$ and some
$\R^d$-valued $h\in \tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)$.  Then $k(u_0,V,g)<\infty$, so that
$$\|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}\le   \|h\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)} \sup_{|r_1|\le k(u_0,V,g)}|\bb(r_1) |  <\infty.$$   Hence,  Theorem \ref{T1} implies the the well-posedness and non-explosion of \eqref{E3}. However, since $|\bb(r_1)|$ may have arbitrary growth in $|r_1|$, $(H_{F,g}')$ does not hold if we regard $\bb_t(r_1)h_t\cdot     r_2$
as a part of $g_t(\cdot, r_1,r_2)$.
 \end{enumerate}

 \

To illustrate Theorem \ref{T1}, we consider  the following  equation \eqref{RE}. When $F=0$ it reduces to the KPZ type equation
 $$\pp_t u_t= L_t u_t +\bb \<a_t\nabla} \def\pp{\partial} \def\E{\mathbb E u_t,\nabla} \def\pp{\partial} \def\E{\mathbb E u_t\>+\bar V_t,$$  and
 for $\bb=0$ it becomes the Navier-Stokes type equation
 $$\pp_t u_t= L_t u_t + F_t(\cdot, u_t)\cdot \nabla} \def\pp{\partial} \def\E{\mathbb E u_t+\bar V_t.$$

 \paragraph{Example 2.1.}  Assume $(H_{a,b})$,   let $\bb\in \R$, and  let
 $$F: [0,\infty)\times E\times \R^m \rightarrow} \def\gg{\gamma \R^d,\ \ V: [0,\infty)\times E\rightarrow} \def\gg{\gamma\R $$ be measurable such that
 $$\tilde}\def\[{\lfloor} \def\]{\rfloor F_t(x,r):=\sup_{x\in E}|F_t(x,r)-F_t(x,0)|$$ is locally bounded in $(t,r)\in [0,\infty)\times \R^m$,
   and  for some $(p_0,q_0)\in \scr K$
 $$\|F(\cdot, 0)\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty,\ \ \ T\in (0,T).$$  Consider the PDE
 \beq\label{RE} \pp_t u_t= L_t u_t + \bb \<a_t\nabla} \def\pp{\partial} \def\E{\mathbb E u_t, \nabla} \def\pp{\partial} \def\E{\mathbb E u_t\> +  F_t\big(\cdot, \varphi_\bb(u_t)\big) \cdot\nabla} \def\pp{\partial} \def\E{\mathbb E u_t + \bar V_t,\ \ t\ge 0,\end{equation}
 where $\<a_t\nabla} \def\pp{\partial} \def\E{\mathbb E u_t, \nabla} \def\pp{\partial} \def\E{\mathbb E u_t\>:=(\<a_t\nabla} \def\pp{\partial} \def\E{\mathbb E u_t^i,\nabla} \def\pp{\partial} \def\E{\mathbb E u_t^i\>)_{1\le i\le m}$, $\bar V_t(x):= (V_t(x),\cdots, V_t(x))\in \R^m$, and
 $$\varphi_\bb(u_t):=  (\bb^{-1} (1- \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-\bb u_t^i}))_{1\le i\le m}$$ which reduces to $u_t$ when $\bb=0.$
If either $\|V_t\|_{\infty}$ or $\|\tilde}\def\[{\lfloor} \def\]{\rfloor F_t\|_\infty$ is locally integrable in $t\ge 0$, then for any $u_0\in \scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1$, \eqref{RE}  has a unique solution $u: [0,\infty)\times E\rightarrow} \def\gg{\gamma \R^m$ satisfying
 $$\sup_{t\in [0,T]} \|\varphi_\bb(u_t)\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 \varphi_\bb(u)\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty,\ \ T\in (0,\infty).$$

 \begin} \def\beq{\begin{equation}}  \def\F{\scr F{proof}
 It suffices to prove the assertion up to  an arbitrarily fixed time $T\in [0,\infty)$.
 We only consider $\bb\ne 0$, as when $\bb=0$ the desired assertion follows from Theorem \ref{T1} for $V=0$ and $g=\bar V$.  Let $v_t=\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-\bb u_t}:=(\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-\bb u_t^i})_{1\le i\le m}.$  Then \eqref{RE} up totime $T$ s equivalent to
\beq\label{WN}  \pp_t v_t= (L_t -\bb V_t) v_t +F_t(\cdot, \bb^{-1}(1-v_t))\cdot\nabla} \def\pp{\partial} \def\E{\mathbb E v_t,\ \ t\in [0,T],\ v_0=\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-\bb u_0}.\end{equation}
If  either  $\|V_t\|_\infty$ or $\|\tilde}\def\[{\lfloor} \def\]{\rfloor F_t\|_\infty$ is locally integrable in $t\ge 0$,
 we have $\|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty$ for
 $$\bar F_t(x):= \sup_{|r|\le k_0} |F_t(x,\bb^{-1}(1-r))|,\ \ k_0=k(u_0,-\bb V,0):=\|u_0\|_\infty\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_0^T \|\bb V_t\|_\infty\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t}.$$
 By Theorem \ref{T1} with $g=0$ and $-\bb V_t$ replacing $V_t$,  \eqref{WN}  has a unique solution satisfying
 $$\sup_{t\in [0,T]} \|v_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+ \|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 v_t\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty.$$
 Then the proof is finished.
 \end{proof}

\subsection{Distribution dependent SDE for nonlinear PDE}

In this part, we let $E=\R^d,$ and assume that   $g_t(x,r_1,r_2,r_3)=g_t(x)$ depends only on $(t,x)$.
Then the PDE \eqref{E3} reduces to
\beq\label{E2} \pp_t u_t= (L_t +V_t)u_t+ F_t(x, u_t,\nabla} \def\pp{\partial} \def\E{\mathbb E u_t)\cdot \nabla} \def\pp{\partial} \def\E{\mathbb E  u_t+ g_t,\ \ \ t\in [0,T].  \end{equation}
To solve this PDE using SDE, we  simply denote
$$\sigma} \def\ess{\text{\rm{ess}}}\def\s{{\bf s}_t(x)=\ss{2a_{T-t}(x)},\ \  t\in [0,T], x\in \R^d,$$ let  $W_t$ be a $d$-dimensional Brownian motion with respect to a complete filtration probability space $(\OO, \{\F_t\}_{t\ge 0}, \P)$, and  consider the following   SDE  on $\R^d$ where differentials are in $s\in [t,T]$:
 \beq\label{S1}\begin} \def\beq{\begin{equation}}  \def\F{\scr F{split}&\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D X_{t,s}^x=\sigma} \def\ess{\text{\rm{ess}}}\def\s{{\bf s}_s(X_{t,s}^x)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D W_s
   +\big\{b_{T-s}+ F_{T-s}(\cdot,\psi_{s}, \nabla} \def\pp{\partial} \def\E{\mathbb E\psi_{s})\big\}(X_{t,s}^x) \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\\
    &\psi_{s}(y):= \E\bigg[u_0(X_{s,T}^y)\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_s^T V_{T-r}(X_{t,r}^y)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}+ \int_s^T g_{T-r}(X_{s,r}^y)\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_s^r V_{T-r'}(X_{t,r'}^y)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r'}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r\bigg],\ \ y\in\R^d,\\
 &\qquad  \   s\in [t,T], \  X_{t,t}^x=x\in\R^d, \ t\in [0,T].\end{split}\end{equation}
 Since $\psi_{s}$ depends on the distribution of $\{X_{s,r}^x\}_{(r,x)\in [s,T]\times E}$,
 the SDE \eqref{S1} is distribution dependent for the future and for all initial values. This is essentially different from the usual McKean-Vlasov SDEs.

 Let   $D_T:= \{(t,s): 0\le t\le s\le T\}$. We define the solution   of \eqref{S1} as follows.

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{defn}  A family $X:=(X_{t,s}^x)_{(t,s,x)\in D_T\times \R^d}$  of random variables on $\R^d$ is called a solution of \eqref{S1}, if   $X_{t,s}^x$ is $\F_s$-measurable,
$(\psi_{s},\nabla} \def\pp{\partial} \def\E{\mathbb E \psi_{s})$ exists such that
 $$\E\int_t^T  \Big\{\big|\sigma} \def\ess{\text{\rm{ess}}}\def\s{{\bf s}_s|^2+ \big|b_{T-s}+ F_{T-s}(\cdot, \psi_{s}, \nabla} \def\pp{\partial} \def\E{\mathbb E \psi_{s})\big|\Big\}(X_{t,s}^x) \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s  <\infty$$  holds  for $ (t,x)\in [0,T]\times \R^d,$  and  $\P$-a.s.
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*}& X_{t,s}^x= x+   \int_t^s\sigma} \def\ess{\text{\rm{ess}}}\def\s{{\bf s}_r(X_{t,r}^x)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D W_r
 + \int_t^s \Big\{b_{T-r} + F_{T-s}\big(\cdot,\psi_{r},\nabla} \def\pp{\partial} \def\E{\mathbb E\psi_{r}\big)\Big\}(X_{t,r}^x)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r,\\
 &\ \ (t,s,x)\in D_T\times\R^d.\end{align*}  \end{defn}

To prove the well-posedness of \eqref{S1}, we strengthen $(H_{F,g})$ as follows.

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{enumerate}\item[$(H_{F,g}'')$]    $\| F(\cdot,0,0) \|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+\|g\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty$ for   $(p_0,q_0)\in \scr K$ in $(H_{V,u_0})$,
and there exists a constant $K>0$ such that
 \begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*}   &|F_t(x,r_1,r_2)- F_t(x,\tilde}\def\[{\lfloor} \def\]{\rfloor r_1,\tilde}\def\[{\lfloor} \def\]{\rfloor r_2)|
  \le K\big\{1\land  (|r_1-\tilde}\def\[{\lfloor} \def\]{\rfloor r_1|+|r_2-\tilde}\def\[{\lfloor} \def\]{\rfloor r_2|)\big\},\\
     &\qquad (t,x)\in [0,T]\times \R^d,\ |r_1|\lor |\tilde}\def\[{\lfloor} \def\]{\rfloor r_1| \le k(u_0,g,V),\ r_2,\tilde}\def\[{\lfloor} \def\]{\rfloor r_2\in \R^{d\otimes m}.\end{align*}
 \end{enumerate}

Under  $(H_{V,u_0})$, $(H_{a,b})$ and  $(H_{F,g}''),$   we will prove the well-posedness of \eqref{S1}.
 In this case,    let
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &P_{t,s}f(x):= \E[f(X_{t,s}^x)],\ \ f\in \B(\R^d),\\
&P_{t,s}^V f(x):= \E\bigg[f(X_{t,s}^x)\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^s V_{T-r}(X_{t,r}^x)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\bigg],  \ (t,s)\in D_T,\ x\in\R^d\ f\in \B(\R^d),\\
& u_t^V(f):= \int_t^T P_{t,s}^V f_s\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\ \ \ f\in \B([0,T]\times \R^d) \end{align*} provided the expectations and integrals exist, where $\B(\cdot)$ is the class of measurable functions
for a measurable space.

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{thm}\label{T1'} Assume $(H_{V,u_0})$, $(H_{a,b})$  and   $(H_{F,g}'')$. Then the following assertions hold.
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{enumerate} \item[$(1)$] The SDE \eqref{S1} has a unique solution $X:=(X_{t,s}^x)_{(t,s,x)\in D_T\times \R^d}.$
\item[$(2)$] For any $p\in [1,\infty)$ and $(t,x)\in [0,T]\times \R^d$, the derivative
$$\nabla} \def\pp{\partial} \def\E{\mathbb E_v X_{t,s}^x:= \lim_{\vv\downarrow 0} \ff{X_{t,s}^{x+\vv v}-X_{t,s}^x}\vv,\ \  v\in\R^d, s\in [t,T]$$
exists in $L^p(\OO\rightarrow} \def\gg{\gamma C([t,T]; \R^d), \P),$ and there exists a constant $c(p)>0$  such that
\beq\label{A*1} \sup_{(t,x)\in [0,T]\times \R^d} \E\bigg[\sup_{s\in [t,T]} |\nabla} \def\pp{\partial} \def\E{\mathbb E_v X_{t,s}^{x}|^p\bigg]\le c(p)|v|^p,\ \ v\in \R^d.\end{equation}
\item[$(3)$] For any $0\le t<s\le T,$   $v\in\R^d$ and $f\in \B_b(\R^d)$,
\beq\label{BS} \begin} \def\beq{\begin{equation}}  \def\F{\scr F{split} &\nabla} \def\pp{\partial} \def\E{\mathbb E_v P_{t,s} f(x):=\lim_{\vv\downarrow 0} \ff{P_{t,s} f(x+\vv v)-P_{t,s} f(x)}\vv \\
&= \ff 1 {s-t}\E\bigg[f(X_{t,s}^x)\int_t^s \Big\<\sigma} \def\ess{\text{\rm{ess}}}\def\s{{\bf s}_r^{-1}(X_{t,r}^x) \nabla} \def\pp{\partial} \def\E{\mathbb E_v X_{t,r}^x,\ \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D W_r\Big\>\bigg].\end{split}\end{equation}
\item[$(4)$] For any $p\in (1,\infty]$, there exists a constant $k_p>0$ such that for any $0\le t<s\le T$ and $f\in C_b^1(\R^d)$,
\beq\label{GRD}  |\nabla} \def\pp{\partial} \def\E{\mathbb E P_{t,s}f| \le k_p \min\Big\{(t-s)^{-\ff 1 2}\big(P_{t,s}|f|^p\big)^{\ff 1 p},\
(P_{t,s}|\nabla} \def\pp{\partial} \def\E{\mathbb E f|^p)^{\ff 1 p}\Big\}.\end{equation}
\item[$(5)$] For any $p\in (1,\infty]$ and $V\in \tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)$,  there exists a constant $c>0$ determined by $ (H_{a,b})$ and $\|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+
\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$, such that
 \beq\label{GRDD}  \begin} \def\beq{\begin{equation}}  \def\F{\scr F{split} &\| P_{t,s}^V f\|_{ \scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1} \le c \|f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1},\ \ f\in \scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1,\ (t,s)\in D_T,\\
&\|\nabla} \def\pp{\partial} \def\E{\mathbb E P_{t,s}^V f\|_\infty\le c (s-t)^{-\ff 1 2}  \|f\|_\infty,\ \ 0\le t<s\le T, \ f\in \B_b(\R^d),\\
& \|u_t^V(f)\|_\infty+\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u^V(f)\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)}\le c \|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)},\ \ t\in [0,T], \ f\in \tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T).\end{split} \end{equation}
\end{enumerate} \end{thm}

The next result provides a correspondence between solutions  of \eqref{E2} and \eqref{S1}.

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{thm}\label{CC}  Assume $(H_{V,u_0})$, $(H_{a,b})$  and   $(H_{F,g}'')$.
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{enumerate}
\item[$(1)$]  If $(u_t)_{t\in [0,T]}$ solves \eqref{E2} in the class  $\scr U(p_0,q_0)$,   then
 \beq\label{SL}\begin} \def\beq{\begin{equation}}  \def\F{\scr F{split}& u_t(x)=  \E\Big[u_0(X_{T-t,T}^x)\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_{T-t}^TV_{T-r}(X_{T-t,r}^x)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\Big]\\
 &\qquad +\E \int_{T-t}^T g_{T-s}(X_{T-t,s}^x)\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_{T-t}^sV_{T-r}(X_{T-t,r}^x)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\ \
    (t,x)\in [0,T]\times \R^d.\end{split}\end{equation}
 \item[$(2)$] On the other hand,     $u_t$ given by $\eqref{SL}$ solves $\eqref{E2}$ such that
$$  \|u_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+\|u\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)}\le c\big(\|u_0\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+\|g\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)}\big),\ \ t\in [0,T]$$
holds for some constant $c>0$ determined by $(H_{a,b})$ and $\|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$.

\end{enumerate}
\end{thm}

 \paragraph{Remark 2.2.} Comparing with the backward SDE  and the SDE  driven by $G$-Brownian motion  characterizing the viscosity solution, as well as the McKean-Vlasov SDE
 describing the weak solution of nonlinear PDEs, the   present SDE \eqref{S1} corresponds to  the classical solution, and allows the coefficients $F,G$ and $V$ to be singular in the space-time variable.

\section{Some lemmas}

We first explain the idea of solving \eqref{S1} by fixed point theorem.

Let  $\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_T:=C(D_T; \R^d)$ be the space of all  maps
 $$\xi=(\xi_{t,s})_{(t,s)\in D_T}: D_T\rightarrow} \def\gg{\gamma \R^d;\ \ \xi_{t,s}\ \text{is \ continuous\ in \ }(t,s)\in D_T.$$ It is a Banach space under the uniform norm $\|\cdot\|_\infty$.

 Let $\scr P_{T}$ be the space of all  probability measures   on $\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_T$ equipped with the weak topology.
 We   define
  $\GG$ as the set of all measurable maps
$$\gg=(\gg^x)_{x\in\R^d}:   \R^d\rightarrow} \def\gg{\gamma \scr P_T$$
such that
\beq\label{PGG} \begin} \def\beq{\begin{equation}}  \def\F{\scr F{split} &\psi_{s}^\gg(x):= \int_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_T} u_0(\xi_{s,T}) \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_s^T V_{T-s}(\xi_{s,r})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}
\gg^x(\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \xi)\\
&\qquad + \int_s^T \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r \int_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_T} g_{T-r}(\xi_{s,r}) \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_s^r V_{T-r'}(\xi_{s,r'})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r'} \gg^x(\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \xi)
\end{split}\end{equation} is well-defined  and the weak gradient $\nabla} \def\pp{\partial} \def\E{\mathbb E \psi_{s}^\gg$ exists for any  $s \in [0,T].$

We will see  that for any $\gg\in \GG,$ the classical  SDE
\beq\label{GG} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D X_{t,s}^{\gg,x}= \big\{b_{T-s}+F_{T-s}(\cdot,\psi_{s}^\gg, \nabla} \def\pp{\partial} \def\E{\mathbb E \psi_{s}^\gg)\big\}(X_{t,s}^{\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s + \sigma} \def\ess{\text{\rm{ess}}}\def\s{{\bf s}_s(X_{t,s}^{\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D W_s,\ \ s\in [t,T], X_{t,t}^{\gg,x}=x \end{equation}
is well-posed. In this case,  we define  a map
\beq\label{PHI} \gg\mapsto \Phi(\gg);\ x\mapsto \Phi^x(\gg):=\scr L}\def\Tt{\tt} \def\TT{\tt}\def\II{\mathbb I_{X^{\gg,x}},  \end{equation}
where $\scr L}\def\Tt{\tt} \def\TT{\tt}\def\II{\mathbb I_{X^{\gg,x}}$ is the distribution of the $\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_T$-valued  random variable $X^{\gg,x}:=(X_{t,s}^{\gg,x})_{(t,s)\in D_T}.$
It is clear that   $X_{t,s}^{\gg,x}$ solves \eqref{S1} if and only if $\gg$  is a fixed point of $\Phi$, i.e. $\Phi(\gg)=\gg$.
So, for the well-posedness of \eqref{S1}, it suffices to show that $\Phi$ has a unique fixed point.

To this end, we   construct a non-empty subspace $\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$ of $\GG$, such  that $\Phi$  is a contractive map on $\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$ under a complete metric.
According to   the definition of $\psi_t^\gg$,  the   space $\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$ should be determined by the  following functionals:
for any $(t,s,x)\in D_T\times\R^d$ and $\gg\in \GG$,   let
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*}
 &\gg_{t,s}^V(f)(x):= \int_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_T} f(\xi_{t,s})\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^sV_{T-r}(\xi_{t,r})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}  \gg^x(\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \xi),\ \ f\in\B(\R^d),\\
 &u_t^{\gg,V}(f)(x):= \int_t^T \gg_{t,s}^V(f_s)(x)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s \\
 &= \int_t^T\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s \int_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_T} f(\xi_{t,s})\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^sV_{T-r}(\xi_{t,r})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}  \gg^x(\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \xi),\ \ f\in \B([t,T]\times\R^d),\end{align*} provided the integrals exist.
  The class $\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$ is defined as follows.

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{defn}\label{D*}
Let $\tilde}\def\[{\lfloor} \def\]{\rfloor \GG$ be the set of  $\gg\in\GG$ satisfying
  $\gg_{t,t}^x=\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_x$ (the Dirac measure at $x$) for $(t,x) \in [0,T]\times  \R^d$,  and
  $$\sup_{t\in [0,T]} \|\gg_t\|_V<\infty,\ \ \|\gg_t\|_V:= \sup_{x\in \R^d} \int_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_T} \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^T |V_{T-r}(\xi_{t,r})|\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r} \gg^x(\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\xi) <\infty,$$
  $$   \sup_{t\in [0,T]} \bigg\{\sup_{\|f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\le 1} \|\nabla} \def\pp{\partial} \def\E{\mathbb E\gg_{t,T}^V(f)\|_{\infty}+
   \sup_{\|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}\le 1}  \|u_t^{\gg,V}(f)\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\bigg\}  <\infty.$$

\end{defn}

\

To apply the fixed point theorem to $\Phi$, we introduce a complete metric on $\tilde}\def\[{\lfloor} \def\]{\rfloor \GG$, which consists of three parts: one is the weighted   variation distance
induced by
$$\eta_t^V(\xi):= \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^T |V_{T-r}(\xi_{t,r})|\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r},\ \ \xi\in C([t,T];\R^d),$$
and the other two are induced by the above functionals  $\gg_{t,T}^V$ and $u_t^{\gg,V}$. More precisely,
for any $\gg\in\GG$ and $t\in [0,T]$, let $\gg_t=(\gg_t^x)_{x\in \R^d} $  be the marginal distribution of $\gg=(\gg^x)_{x\in\R^d}$ for
$$\xi\mapsto\xi_t:=(\xi_{t,s})_{s\in [t,T]}\in C([t,T];\R^d),$$ i.e. $\gg_t^x:= \gg^x\circ \xi_t^{-1}, x\in\R^d.$
Then $$\gg_t(f)(x):=\int_{C([t,T];\R^d)} f(\xi)\gg_t^x(\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \xi)=\int_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_T} f(\xi_{t}) \gg^x(\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\xi),\ \ x\in\R^d,\ f\in \B_b(C([t,T];\R^d)).$$
For any $\gg,\tilde}\def\[{\lfloor} \def\]{\rfloor\gg\in \tilde}\def\[{\lfloor} \def\]{\rfloor \GG$ and $t\in [0,T]$,  let
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &\|\gg_{t}-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_{t}\|_{V}:=\sup_{f\in \B_b(C([t,T];\R^d),\|f\|_\infty\le \eta_t^V } \|\gg_{t}   (f)- \tilde}\def\[{\lfloor} \def\]{\rfloor\gg_{t}(f)\|_\infty,\\
&\|\gg_{t}-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_{t}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}:=\sup_{\|f|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\le 1 } \|\nabla} \def\pp{\partial} \def\E{\mathbb E\{\gg_{t,T}^V(f)-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_{t,T}^V(f)\}\|_{\infty},\\
&\|\gg_{t}-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_{t}\|_{p_0,q_0}:= \sup_{\|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}\le 1}  \|u_t^{\gg,V}(f)- u_t^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg,V}(f)\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}.\end{align*}

We will prove that for any $\ll\ge 0$, $\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$ is complete under the metric
$$\rr_\ll(\gg,\tilde}\def\[{\lfloor} \def\]{\rfloor\gg):= \sup_{(t,s)\in D_T  } \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-\ll(T-t)} \big\{\|\gg_{t}-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_{t}\|_{V}+\|\gg_{t}-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_{t}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+
\|\gg_{t}-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_{t}\|_{p_0,q_0} \big\},$$
and that $\Phi$ is contractive in $\rr_\ll$ for large $\ll>0$, so that it has a unique fixed point in $\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$. To this end, we present below some lemmas.

We first extend \cite[Theorem 2.1]{YZ} with $V=0$ to the the present setting.
For any $p,q\ge 1$, let $\tilde}\def\[{\lfloor} \def\]{\rfloor H_{q}^{2,p}(0,T)$ be the space of
$f: [0,T]\times \R^d\rightarrow} \def\gg{\gamma \R^d$ such that
$$\|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q}^p(0,T)}+\|\nabla} \def\pp{\partial} \def\E{\mathbb E f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q}^p(0,T)}+\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q}^p(0,T)}<\infty.$$

 \begin} \def\beq{\begin{equation}}  \def\F{\scr F{lem}\label{LN3} Assume $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty$ and $(H_{a,b})$ but without the condition on $|\nabla} \def\pp{\partial} \def\E{\mathbb E a|$.
 Let  $b^{(0)}: [0,T]\times\R^d\rightarrow} \def\gg{\gamma\R^d$  with $\|b^{(0)}\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q}^{p}(0,T)}<\infty$ for some  $(p,q)\in \scr K$, and let
 $$ L_t^V := L_{T-t}+b_t^{(0)}\cdot\nabla} \def\pp{\partial} \def\E{\mathbb E+V_{T-t},\ \ t\in [0,T].$$
   Then for any $\ll\ge 0$ and $f\in \tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T),$   the PDE
 \beq\label{POE} (\pp_t +   L_t^V-\ll)u_t^\ll= f_t,\ \ t\in [0,T], u_T^\ll=0\end{equation}
 has a  unique  solution in $\tilde}\def\[{\lfloor} \def\]{\rfloor H_{q_0}^{2,p_0}(0,T)$.
 Moreover,
 for any  $0<\vv< \ff 1 2 (1-\ff d {p_0}-\ff 2 {q_0})$, there exists a constant
 $c>0$ increasing in $\|b^{(0)}\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q}^{p}(0,T)}$ such that
 \beq\label{UES}\begin} \def\beq{\begin{equation}}  \def\F{\scr F{split}& (1+\ll)^{\vv}(\|u_t^\ll\|_\infty+\|\nabla} \def\pp{\partial} \def\E{\mathbb E u_t^\ll\|_\infty)+\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u^\ll\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)} \le c \|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)},\\
 &\ \ \ll\ge 0, \ f\in \tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T),\ t\in [0,T].\end{split}\end{equation}
 \end{lem}

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{proof} (a) Let $L^0_t= L_{T-t}+b_t^{(0)}\cdot\nabla} \def\pp{\partial} \def\E{\mathbb E.$ By \cite[Theorem 2.1]{YZ}, for any $\ll\ge 0$ and $f\in \tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)$, the PDE
\beq\label{PO0} (\pp_t + L_t^0-\ll)\hat u_t^\ll = f_t,\ \ \hat u_T^\ll=0\end{equation}
has a unique solution in the $\tilde}\def\[{\lfloor} \def\]{\rfloor H_{q_0}^{2,p_0}(0,T)$,
and   for any $0<\vv<\ff 1 2 (1-\ff d {p_0}-\ff 2 {q_0})$, there exists a constant
 $c_1>0$ increasing in $\|b^{(0)}\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q}^{p}(0,T)}$ such that
 \beq\label{UES'}\begin} \def\beq{\begin{equation}}  \def\F{\scr F{split} &(1+\ll)^{\vv} \big\{\|\hat u_t^\ll\|_\infty+\|\nabla} \def\pp{\partial} \def\E{\mathbb E \hat u_t^\ll\|_\infty\big\}+\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 \hat u^\ll\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)} \\
 &\le c_1 \|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)},\ \ \ll\ge 0,\ f\in \tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T),\ t\in [0,T].\end{split}\end{equation}
 Note that in  \cite[Theorem 2.1]{YZ}  the class $\tilde}\def\[{\lfloor} \def\]{\rfloor H_{q_0}^{2,p_0}(0,T)$ is replaced by
 $\tilde}\def\[{\lfloor} \def\]{\rfloor W_{1,q_0}^{2,p_0,w}(0,T)$, which consists of $f\in\tilde}\def\[{\lfloor} \def\]{\rfloor H_{q_0}^{2,p_0}(0,T)$ such that for $w(x):=(1+|x|)^{-1}$ (note that $w(x)$ therein should be $(1+|x|^2)^{-\ff p 2}$ according to
\cite[Lemma 2.3]{YZ}),
\beq\label{W} |w f|+|w\nabla} \def\pp{\partial} \def\E{\mathbb E f|+|w\nabla} \def\pp{\partial} \def\E{\mathbb E^2 f|+ |w \pp_t f|\in \tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T).\end{equation}
 Since $w\le 1$ and $(H_{a,b})$ implies that $a$ is bounded and $|b_t(x)|\le k(1+|x|) $ for some constant $k>0$,
   a solution of \eqref{PO0} in $\tilde}\def\[{\lfloor} \def\]{\rfloor H_{q_0}^{p_0}(0,T)$  must satisfy \eqref{W}, so we may replace  $\tilde}\def\[{\lfloor} \def\]{\rfloor W_{1,q_0}^{2,p_0,w}(0,T)$ by $ \tilde}\def\[{\lfloor} \def\]{\rfloor H_{q_0}^{2,p_0}(t,T)$.

(b) To solve \eqref{POE},    for any $\eta\in \B_b([0,T]\times\R^d)$, consider the PDE
\beq\label{PLT} (\pp_t + L_t^0-\ll)  u_t^{\ll,\eta} = f_t- V_{T-t} \eta_t,\ \   u_T^{\ll,\eta}=0.\end{equation}
Let $\aa(V)=\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}.$ We have
 $$\|f- V_{T-\cdot} \eta\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}\le \|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)} +  \aa(V) \|\eta\|_\infty.$$
 Then  (a) implies that \eqref{PLT} has a unique solution in $\tilde}\def\[{\lfloor} \def\]{\rfloor H_{q_0}^{2, p_0}(0,T)\cap \B_b([0,T]\times \R^d)$. Thus, we obtain a map
 $$\B_b([0,T]\times \R^d)\ni \eta \mapsto u^{\ll,\eta}\in \B_b([0,T]\times \R^d).$$
 So, to see that \eqref{POE} has a unique solution in $\tilde}\def\[{\lfloor} \def\]{\rfloor H_{q_0}^{p_0}(0,T)\cap \B_b([0,T]\times \R^d)$,
 it suffices to prove that this map has a unique fixed point.

 To this end,  for any $\eta,\tilde}\def\[{\lfloor} \def\]{\rfloor\eta\in \B_b([0,T]\times\R^d)$, let $u^{\ll,\eta,\tilde}\def\[{\lfloor} \def\]{\rfloor\eta}:= u^{\ll,\eta}- u^{\ll, \tilde}\def\[{\lfloor} \def\]{\rfloor\eta}.$ Then
 $$(\pp_t + L_t^0-\ll) u_t^{\ll,\eta,\tilde}\def\[{\lfloor} \def\]{\rfloor\eta} = V_{T-t} (\tilde}\def\[{\lfloor} \def\]{\rfloor\eta_t-\eta_t),\ \ u_T^{\ll, \eta,\tilde}\def\[{\lfloor} \def\]{\rfloor\eta}=0.$$
 Applying \eqref{UES'} to this equation, by (a) we find a constant $c_1>0$ such that
$$  \|u_t^{\ll,\eta}- u_t^{\ll,\eta}\|_\infty= \| u_t^{\ll,\eta,\tilde}\def\[{\lfloor} \def\]{\rfloor\eta}\|_\infty\\
 \le c_1 \|V_{T-t} (\tilde}\def\[{\lfloor} \def\]{\rfloor\eta-\eta)\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)}.$$
 Letting
 $$\|\eta-\tilde}\def\[{\lfloor} \def\]{\rfloor\eta\|_{\infty,n}:= \sup_{t\in [0,T]}\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-n(T-t)} \|\eta_t-\tilde}\def\[{\lfloor} \def\]{\rfloor\eta_t\|_\infty,\ \ n\ge 1,$$
 this implies
 $$ \|u^{\ll,\eta}- u^{\ll,\eta}\|_{\infty,n}\le c_1 \|\eta-\tilde}\def\[{\lfloor} \def\]{\rfloor\eta\|_{\infty,n}
 \sup_{t\in [0,T]}   |V_{T-\cdot} \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-n(\cdot-t)}\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)}.$$
 Hence, when $n$ is large enough, the map $u^{\ll, \eta}$ is contractive in the complete metric $\|\eta-\tilde}\def\[{\lfloor} \def\]{\rfloor\eta\|_{\infty,n}$. Therefore, it has a unique fixed point as desired.

(c) It remains to prove the estimate \eqref{UES}.  Let $u^\ll$ be the unique solution of \eqref{POE} in $\tilde}\def\[{\lfloor} \def\]{\rfloor H_{q_0}^{p_0}(0,T)$. By \eqref{UES'} we obtain
 \beq\label{AI} \begin} \def\beq{\begin{equation}}  \def\F{\scr F{split}& (1+\ll)^{\vv}(\|u_t^\ll\|_\infty+\|\nabla} \def\pp{\partial} \def\E{\mathbb E  u_t^\ll\|_\infty)+\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2  u^\ll\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)} \\
 &\le c_1 \|f-V_{T-\cdot} u^\ll\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_q^p(t,T)}\le c_1 \|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)} + c_1 \|V_{T-\cdot}u^\ll\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)},\ \ t\in [0,T].\end{split}\end{equation}
 Letting
 $$\|u^\ll\|_{\infty,n}:= \sup_{s\in [0,T]} \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-n(T-s)} \|u_s^\ll\|_\infty,\ \ n\ge 1,$$
 we derive
 $$    \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-n(T-t)}\|u_t^\ll\|_\infty\le c_1  \|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)}+ c_1
\|u^\ll\|_{\infty,n} \|V_{T-\cdot}\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-n(\cdot-t)}\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)}.$$
 When $n\ge 1$ is large enough, this implies
 $$ \|u^\ll\|_{\infty,n}\le c_1   \|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)}+\ff 1 2 \|u^\ll\|_{\infty,n},\ \ t\in [0,T].$$
 Therefore, there exists a constant $c_2>0$ such that
 $\|u^\ll_t\|_\infty\le c_2  \|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)}, t\in [0,T].$
 Combining this with \eqref{AI} and $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty$, we finish the proof.
\end{proof}

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{lem} \label{N1} Assume $(H_{V,u_0})$, $(H_{a,b})$  and    $(H_{F,g}'')$. Then for any $\gg\in \GG$, the SDE  $\eqref{GG}$ is well-posed,
  and $\Phi(\gg)$ defined in $\eqref{PHI}$ is a measurable map from $\R^d$ to $\scr P_T$.  Moreover: \begin} \def\beq{\begin{equation}}  \def\F{\scr F{enumerate}
\item[$(1)$]   For any $v\in\R^d$,
$$\nabla} \def\pp{\partial} \def\E{\mathbb E_v X_{t,s}^{\gg,x}:= \lim_{\vv\downarrow 0} \ff{X_{t,s}^{\gg, x+\vv v}-X_{t,s}^x}{\vv},\ \ s\in [t,T]$$
exists in $L^p(\OO\rightarrow} \def\gg{\gamma C([t,T];\R^d),\P)$ for any $p\in (1,\infty)$, there exists a constant $k_p>0$  determined by $p$, $(H_{a,b})$ and $ \|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$, such that
\beq\label{DX}\sup_{\gg\in\GG} \sup_{(t,x)\in [0,T]\times\R^d}   \E \Big[\sup_{s\in [t,T]}|\nabla} \def\pp{\partial} \def\E{\mathbb E_v X_{t,s}^{\gg,x}|^p\Big]\le k_p |v|^p,\ \ v\in\R^d,\end{equation}
\item[$(2)$] For any $p\in (1,\infty]$ there exits a constant $k_p>0$ $k_p>0$  determined by $p$, $(H_{a,b})$ and $  \|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$,  such that
 $$P_{t,s}^\gg f(x):= \E[f(X_{t,s}^{\gg,x})],\ \ (t,s)\in D_T, x\in \R^d, f\in \B_b(\R^d)$$ satisfies
  \beq\label{GRG}  \begin} \def\beq{\begin{equation}}  \def\F{\scr F{split}&|\nabla} \def\pp{\partial} \def\E{\mathbb E P_{t,s}^\gg f|\le k_p \min\big\{(s-t)^{-\ff 1 2}(P_{t,s}^\gg |f|^p)^{\ff 1 p},\  (P_{t,s}^\gg |\nabla} \def\pp{\partial} \def\E{\mathbb E f|^p)^{\ff 1 p}\big\},\\
  & \ \gg\in \GG, 0\le t<s\le T, f\in C_b^1(\R^d),\end{split}\end{equation}
and  for any $v,x\in\R^d$, any $0\le t<s\le T$,
\beq\label{BSM} \nabla} \def\pp{\partial} \def\E{\mathbb E_v P_{t,s}^{\gg}f(x)= \ff 1 {s-t} \E\bigg[f(X_{t,s}^{\gg,x}) \int_t^s \Big\<\sigma} \def\ess{\text{\rm{ess}}}\def\s{{\bf s}_r^{-1}   (X_{t,r}^{\gg,x}) \nabla} \def\pp{\partial} \def\E{\mathbb E_vX_{t,r}^{\gg,x},  \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D W_r\Big\> \bigg],\ \ f\in \B_b(\R^d).\end{equation}
 \end{enumerate} \end{lem}

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{proof}  By   $(H_{F,g}'')$,  $b_t^\gg:= F_{T-t}(\cdot, \psi_t^\gg,\nabla} \def\pp{\partial} \def\E{\mathbb E\psi_t^\gg)$ satisfies
\beq\label{B0} \sup_{\gg\in\GG} \|b^\gg\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}\le \|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty.\end{equation}
According to \cite[Theorem 2.1]{W21e}, $(H_{a,b})$ and \eqref{B0} imply that the SDE \eqref{GG}  is well-posed, and  \eqref{DX}, \eqref{GRG} and \eqref{BSM} hold.
Noting that the Borel $\sigma} \def\ess{\text{\rm{ess}}}\def\s{{\bf s}$-field on $\scr P_T$ is induced by the maps
$$\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_T\ni \mu\mapsto \mu_{t,s}(f):= \int_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_{T}} f(\xi_{t,s})\mu(\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\xi),\ \ (t,s)\in D_T, f\in C_b(\R^d),$$
by combining \eqref{DX} with the continuity of $X_{t,s}^{\gg,x}$ in $(t,s)\in D_T$,
we see that $\Phi(\gg)$ is a measurable map from $ \R^d$ to $\scr P_T$.
\end{proof}

In the following, we intend to prove $\Phi\GG\subset \tilde}\def\[{\lfloor} \def\]{\rfloor\GG$, so that  we may apply the fixed point theorem to
the map $\Phi:\tilde}\def\[{\lfloor} \def\]{\rfloor\GG\rightarrow} \def\gg{\gamma\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$. To this end, we introduce the Feynman-Kac semigroup
$$P_{t,s}^{\gg,V}f(x):=\E\Big[f(X_{t,s}^{\gg,x})\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^s V_{T-r}(X_{t,r}^{\gg, x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\Big],\ \ (t,s,x)\in D_T\times \R^d, f\in\B_b(\R^d),$$ where $X_{t,s}^{\gg,x}$ solves \eqref{GG} with generator
\beq\label{LG} \tilde}\def\[{\lfloor} \def\]{\rfloor L_t^\gg:= L_{T-t}+F_{T-t}(\cdot,\psi_t^\gg,\nabla} \def\pp{\partial} \def\E{\mathbb E\psi_t^\gg)\cdot\nabla} \def\pp{\partial} \def\E{\mathbb E,\ \ t\in [0,T].\end{equation}

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{lem} \label{N1'} Assume $(H_{V,u_0})$, $(H_{a,b})$ and     $(H_{F,g}'')$. Then the following assertions hold.
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{enumerate}
\item[$(1)$] There exists a constant $c>0$ determined by $(H_{a,b})$ and $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+ \|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$, such that for any $\gg\in \GG,$ any $f: [0,T]\times\R^d \rightarrow} \def\gg{\gamma \R$ with $\|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty$,
$$u_t^{\Phi(\gg),V}(f) :=\E \int_t^T f_s(X_{t,s}^x) \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^s V_{T-r}(X_{t,r}^x)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s=\int_t^T P_{t,s}^{\gg,V} f_{s}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\ \ t\in [0,T]$$
satisfies
\beq\label{BE} (\pp_t +\tilde}\def\[{\lfloor} \def\]{\rfloor L_t^\gg+V_{T-t})u_t^{\Phi(\gg),V}(f)= - f_{t},\ \ t\in [0,T],\end{equation}
\beq\label{B2} \|u_t^{\Phi(\gg),f}\|_\infty+\|\nabla} \def\pp{\partial} \def\E{\mathbb E u_t^{\Phi(\gg),f}\|_\infty+ \|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u^{\Phi(\gg),f}\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)}\le c \|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)},\ \ t\in [0,T].\end{equation}
\item[$(2)$] For any  $f\in \B_b(\R^d)$ and $\gg\in\GG$,
\beq\label{KK} \pp_t P_{t,s}^{\gg,V}f = - (\tilde}\def\[{\lfloor} \def\]{\rfloor L_{t}^\gg +V_{T-t})P_{t,s}^{\gg,V}f,\ \ t\in [0,s], s\in (0,T],\end{equation}
and there exists a constant $c>0$ determined by $(H_{a,b})$ and $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+ \|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$, such that
\beq\label{KK2}
\sup_{\gg\in\GG}\sup_{(t,s)\in D_T} \|P_{t,s}^{\gg,V}f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1} \le c  \|f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}, \ \ f\in \scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1,\end{equation}
\beq\label{KK3} \|\nabla} \def\pp{\partial} \def\E{\mathbb E P_{t,s}^{\gg,V}f\|_\infty\le c (s-t)^{-\ff 1 2} \|f\|_\infty,\ \ 0\le t<s\le T, f\in \B_b(\R^d).\end{equation}
\end{enumerate}\end{lem}

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{proof} (1) By Lemma \ref{LN3}, $(H_{a,b})$, \eqref{B0} and $\|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty$ imply that the PDE
\beq\label{BE*} (\pp_t +\tilde}\def\[{\lfloor} \def\]{\rfloor L_t^\gg+V_{T-t})\hat u_t^\gg= - f_t,\ \ t\in [0,T],\ \hat u_T^\gg=0\end{equation} has a unique solution
satisfying
\beq\label{B2*} \|\hat u_t^\gg\|_\infty+\|\nabla} \def\pp{\partial} \def\E{\mathbb E \hat u_t^\gg\|_\infty+ \|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 \hat u^\gg\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)}\le c_0\|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_q^p(t,T)}\end{equation}
for some constant $c_0>0$ determined by $(H_{a,b})$ and $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+ \|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$.

Next, by $(H_{a,b})$ and \eqref{B0}, the Krylov estimate in \cite[Lemma 3.2(2)]{YZ} holds,
so that as shown in the proof of \cite[Lemma 4.1(2)]{XXZZ}, we obtain Khasminskii's estimate: for some function $c: (0,\infty)\rightarrow} \def\gg{\gamma (0,\infty)$ determined by $(H_{a,b})$ and $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+ \|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$,
\beq\label{*DD} \sup_{\gg\in \GG}\sup_{(t,x)\in [0,T]\times \R^d}  \E \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\ll \int_0^t|V_{T-s}(X_{t,s}^{\gg,x})|\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s}\le c(\ll),\ \ \ll\in (0,\infty).\end{equation}

Moreover, by \eqref{BE*} and   It\^o's  formula in \cite[Lemma 3.3]{YZ},
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \Big[\hat u_{s}^\gg(X_{t,s}^{\gg,x})\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^s V_{T-r}(X_{t,r}^{\gg, x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\Big]
= \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^s V_{T-r}(X_{t,r}^{\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r} (\pp_s + \tilde}\def\[{\lfloor} \def\]{\rfloor L_{s}^\gg +V_{T-s}) \tilde}\def\[{\lfloor} \def\]{\rfloor u_s^\gg (X_{t,s}^{\gg,x}) \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s +\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D M_s\\
&= -f_s(X_{t,s}^{\gg,x})\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^s V_{T-r}(X_{t,r}^{\gg, x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s + \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D M_s,\ \ s\in [t,T]\end{align*}
holds for some martingale $M_s$. This together with $\hat u_T^\gg=0$, \eqref{B2*} and \eqref{*DD} yields
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} & 0=\E \Big[\hat u_T^\gg(X_{t,T}^{\gg,x})\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^T V_{T-r}(X_{t,r}^{\gg, x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\Big]\\
&= \hat u_t^\gg(x) - \E \int_t^T f_s(X_{t,s}^{\gg,x})\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^s V_{T-r}(X_{t,r}^{\gg, x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s=\hat u_t^\gg(x)-u_t^{\Phi(\gg),V}(f).\end{align*}
Therefore,  \eqref{BE*} and \eqref{B2*} imply  \eqref{BE} and \eqref{B2}.

(2)   By \eqref{*DD}, there exists a constant $c_0>0$ determined by $(H_{a,b})$ and $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+ \|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$, such that
\beq\label{F0} \sup_{(t,s)\in D_T, \gg\in \GG} \|P_{t,s}^{\gg, V}f\|_\infty\le c_0\|f\|_\infty,\ \ f\in\B_b(\R^d).\end{equation}
By \cite[Theorem 2.1]{YZ}, or Lemma \ref{LN3} for $V=0$, $(H_{a,b})$ and \eqref{B0} imply that for any $s\in (0,T]$, the PDE
\beq\label{F1} (\pp_t +\tilde}\def\[{\lfloor} \def\]{\rfloor L_t^\gg)\tilde}\def\[{\lfloor} \def\]{\rfloor u_t= - P_{t,s}^\gg f,\ \ \tilde}\def\[{\lfloor} \def\]{\rfloor u_s=0, t\in [0,s]\end{equation}
has a unique solution satisfying
\beq\label{F2}\sup_{t\in [0,s]}\big\{ \|\tilde}\def\[{\lfloor} \def\]{\rfloor u_t\|_\infty+\|\nabla} \def\pp{\partial} \def\E{\mathbb E \tilde}\def\[{\lfloor} \def\]{\rfloor u_t\|_\infty\big\}+\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 \tilde}\def\[{\lfloor} \def\]{\rfloor u\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,s)}\le c_1
\|f\|_\infty\end{equation}
for some constant $c_1>0$ determined by $(H_{a,b})$ and $ \|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$. By It\^o's formula in \cite[Lemma 3.3]{YZ},
we find a martingale $(M_r)_{r\in [t,s]}$ such that \eqref{F1} implies
$$\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \tilde}\def\[{\lfloor} \def\]{\rfloor u_r(X_{t,r}^{\gg,x})= \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D M_r- P_{r,s}^\gg f(X_{t,r}^{\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r,\ \ r\in [t,s],$$
so that
$$0=\E[\tilde}\def\[{\lfloor} \def\]{\rfloor u_T(X_{t,s}^{\gg,x})]= \tilde}\def\[{\lfloor} \def\]{\rfloor u_t(x) -\int_t^s \E[P_{r,s}^\gg f(X_{t,r}^{\gg,x})]\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r=\tilde}\def\[{\lfloor} \def\]{\rfloor u_t(x)- (s-t)P_{t,s}^\gg f.$$
Then $P_{t,s}^\gg f= \ff{\tilde}\def\[{\lfloor} \def\]{\rfloor u_t}{s-t}$ for $t\in [0,s)$, which together with \eqref{F1} and \eqref{F2} implies
  \beq\label{F4}(\pp_t+\tilde}\def\[{\lfloor} \def\]{\rfloor L_t^\gg)P_{t,s}^\gg f=0,\ \ t\in [0,s],\end{equation}
and
$$\sup_{t\in [0,r)} \|P_{t,s}^\gg f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1} +  \|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 P_{\cdot,s}^\gg f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,r)}<\infty,\ \ r\in [0,s).$$
By It\^o's formula in \cite[Lemma 3.3]{YZ}, we find a martingale $\tilde}\def\[{\lfloor} \def\]{\rfloor M_r$ such that \eqref{F4} implies
$$\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \big\{\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^rV_{T-r'}(X_{t,r'}^{\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r'} P_{r,s}^\gg f(X_{t,r}^{\gg,x})\big\}=\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \tilde}\def\[{\lfloor} \def\]{\rfloor M_r+
\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^rV_{T-r'}(X_{t,r'}^{\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r'}\big\{V_{T-r}P_{r,s}^\gg f\big\}(X_{t,r}^{\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r,\ \ r\in [t,s).$$
Thus,
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &P_{t,s}^{\gg, V}f(x)= \lim_{r\uparrow s} \E\big[\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^rV_{T-r'}(X_{t,r}^{\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r'} P_{r,s}^\gg f(X_{t,r}^{\gg,x} ) \big]\\
&=\lim_{r\uparrow s}  \E\bigg[ P_{t,s}^\gg f(x)   +\int_t^r \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^{s'} V_{T-r'}(X_{t,r'}^{\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r'}\big\{ V_{T-s'}P_{s',s}^\gg f\big\}(X_{t,s'}^{\gg,x})  \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s'  \bigg]  \\
&= P_{t,s}^\gg f(x)+ \int_t^s P_{t,r}^{\gg,V} \big\{V_{T-r} P_{r,s}^\gg f\big\}(x)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r,\ \ f\in C_b(\R^d).\end{align*}
By a standard approximation argument, this implies
\beq\label{F5} P_{t,s}^{\gg,V}f= P_{t,s}^\gg f+ u_t^{\Phi(\gg),V}(\tilde}\def\[{\lfloor} \def\]{\rfloor f),\  \ t\in [0,s],\ f\in \B_b(\R^d),\end{equation}
where $\tilde}\def\[{\lfloor} \def\]{\rfloor f_r:= 1_{[0,s]}(r)V_{T-r} P_{r,s}^\gg f $ satisfies
$$\|\tilde}\def\[{\lfloor} \def\]{\rfloor f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,s)}\le \|f\|_\infty\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,s)}.$$
Combining this with  \eqref{GRG} for  $p=\infty$ and  \eqref{B2}, we find a constant $c_2>0$ determined by $(H_{a,b})$ and $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+ \|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$,  such that
$$\sup_{(t,s)\in D_T,\gg\in\GG} \|\nabla} \def\pp{\partial} \def\E{\mathbb E P_{t,s}^{\gg,V}f\|_\infty\le c_2 \|f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1},\ \ f\in \scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1, $$
$$\sup_{\gg\in\GG} \|\nabla} \def\pp{\partial} \def\E{\mathbb E P_{t,s}^{\gg,V}f\|_\infty\le c_2 (s-t)^{-\ff 1 2},\ \ f\in \B_b(\R^d), \ 0\le t<s\le T.$$
These together with \eqref{F0} implies  \eqref{KK2} and \eqref{KK3}. Finally,  by \eqref{BE}, \eqref{F4} and \eqref{F5},
we prove \eqref{KK}.
 \end{proof}

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{lem}\label{N2} Assume $(H_{V,u_0})$, $(H_{a,b})$  and    $(H_{F,g}'')$.
Then $\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$ is non-empty, $\rr_\ll$-complete and satisfies  $\Phi\tilde}\def\[{\lfloor} \def\]{\rfloor\GG \subset \tilde}\def\[{\lfloor} \def\]{\rfloor\GG$.  \end{lem}

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{proof}  (a) We first prove $\Phi\GG\subset\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$.   Let $\gg\in \GG$. By   the definition  of $\Phi$ in \eqref{PHI},   we have $\Phi_{t,t}^x(\gg)=\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_x$.
Next,
\eqref{KK2} implies that $\{\Phi(\gg)\}_{t,s}^V(f)=P_{t,s}^{\gg,V} f$  satisfies
$$ \sup_{\gg\in \GG } \| \{\Phi(\gg)\}_{t,s}^V (f) \|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\le c   \| f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}.$$
Moreover,   by   \eqref{B2} we obtain
$$\sup_{\gg\in \GG  }\sup_{t\in [0,T]} \|u_t^{\Phi(\gg),V}(f)\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}
\le c\|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}.$$
Thus,    $(\Phi\tilde}\def\[{\lfloor} \def\]{\rfloor\GG \subset ) \Phi\GG  \subset \tilde}\def\[{\lfloor} \def\]{\rfloor\GG.$

On the other hand, let
$\gg_X$ be the distribution of the solution $X=(X_{t,s}^x)_{(t,s)\in D_T, x\in\R^d} $ to \eqref{GG} with $F=0$.   Then the above argument implies $\gg\in\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$. So, $\tilde}\def\[{\lfloor} \def\]{\rfloor\GG\ne \emptyset$.

(b) The proof of $\rr_\ll$-completeness is more or less standard, which is  addressed below for readers' convenience.

Let $\{\gg^{(n)}\}_{n\ge 1}\subset \tilde}\def\[{\lfloor} \def\]{\rfloor\GG$ such that
\beq\label{MN} \lim_{n,m\rightarrow} \def\gg{\gamma\infty} \rr_\ll(\gg^{(n)},\gg^{(m)})=0.\end{equation}
Then there exists a constant $c>0$ such that
\beq\label{MN0} \sup_{n\ge 1,t\in [0,T]}\bigg\{\|\gg_t^{(n)}\|_V+ \sup_{\|f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\le 1}   \|\nabla} \def\pp{\partial} \def\E{\mathbb E(\gg^{(n)})_{t,T}^V(f)\|_{\infty}+\sup_{\|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}\le 1}  \|u_t^{\gg^{(n)},V}(f)\|_{p_0,q_0}\bigg\}\le c.\end{equation}
Moreover,
$\{\gg^{(n)}\}_{n\ge 1}$ is a Cauchy sequence under the weighted variation distance
$$\|\gg -\tilde}\def\[{\lfloor} \def\]{\rfloor\gg\|_{V}:=\sup_{(t,x)\in [0,T]\times \R^d} \|\gg_t-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_t\|_{V},$$
which is complete.
So,   there exists a unique measurable map $\gg: \R^d\rightarrow} \def\gg{\gamma\scr P_T$ such that
\beq\label{MN2}  \lim_{n\rightarrow} \def\gg{\gamma\infty}\sup_{t\in [0,T]} \|\gg^{(n)}_t- \gg_t\|_{V}=0.\end{equation}In particular,
\beq\label{MN4} \gg^V_{t,T} (f)(x)= \int_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_T} f(\xi_{t,T})\eta^V_t(\xi)  \gg_t^x(\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \xi)= \lim_{n\rightarrow} \def\gg{\gamma\infty} (\gg^{(n)})_{t,T}^V(f)(x),\ \ f\in \B_b(\R^d).\end{equation}
We aim to prove that $\gg\in\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$ and $\rr_\ll(\gg^{(n)},\gg)\rightarrow} \def\gg{\gamma 0$ as $n\rightarrow} \def\gg{\gamma\infty$.

By \eqref{MN0} and \eqref{MN2}, we obtain
\beq\label{MN3}\sup_{t\in [0,T]}   \|\gg_t\|_V = \sup_{t\in [0,T]}   \lim_{n\rightarrow} \def\gg{\gamma\infty} \|(\gg^{(n)})_{t}\|_V <\infty.\end{equation}
By
\eqref{MN0}, we find a   constant $c_1>0$ such that
$$ \sup_{t\in [0,T]}\sup_{\|f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\le 1} | (\gg^{(n)})_{t,T}^V(f)(x)- (\gg^{(n)})_{t,T}^V(f)(y)|\le c_1|x-y|,\ \ x,y\in\R^d, n\ge 1.$$
Combining this with \eqref{MN4}, we obtain
$$ \sup_{t\in [0,T]}\sup_{\|f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\le 1}| \gg_{t,T}^V(f)(x)- \gg_{t,T}^V(f)(y)|\le c_1|x-y|,\ \ x,y\in\R^d.$$
Hence,
$$ \sup_{t\in [0,T]}\sup_{\|f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\le 1} \|\nabla} \def\pp{\partial} \def\E{\mathbb E\gg_{t,T}^V(f)\|_{\infty}\le c_1<\infty.$$
Similarly, we can  prove
 $$\sup_{t\in [0,T]}\sup_{\|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}\le 1} \|u_t^{\gg,V}(f) \|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}<\infty.$$  Therefore, by Definition \ref{D*}, we have $\gg\in\tilde}\def\[{\lfloor} \def\]{\rfloor \GG$.

Next,  let $\|f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\le 1$. By \eqref{MN} and \eqref{MN4},
we find  positive constants $\{\vv_{m,n}\}_{m,n\ge 1}$   uniformly in $f$ with $\vv_{m,n}\rightarrow} \def\gg{\gamma 0$ as $m,n\rightarrow} \def\gg{\gamma\infty$, such that
  for any $(t,x)\in\R^d$ and $v\in\R^d$ with $|v|=1$,
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*}&\ff 1\vv \big|\gg_{t,T}^V (f) (x+\vv v)- \gg_{t,T}^V(f)(x)-(\gg^{(n)})_{t,T}^V(f)(x+\vv v)+(\gg^{(n)})_{t,T}^V(f)(x)\big|\\
&=\lim_{m\rightarrow} \def\gg{\gamma\infty} \ff 1\vv \big|(\gg^{(m)})_{t,T}^V (f) (x+\vv v)- (\gg^{(m)})_{t,T}^V(f)(x)-(\gg^{(n)})_{t,T}^V(f)(x+\vv v)+ (\gg^{(n)})_{t,T}^V(f)(x)\big|\\
&\le \limsup_{m\rightarrow} \def\gg{\gamma\infty}\ff 1 \vv \int_0^\vv \big|\nabla} \def\pp{\partial} \def\E{\mathbb E_v \big[(\gg_{t,T}^{(m)})_{t,T}^V(f)-(\gg^{(n)})_{t,T}^V(f)\big](x+ r v)\big|\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r\\
&\le \limsup_{m\rightarrow} \def\gg{\gamma\infty}\vv_{m,n},\ \ n\ge 1,\ \vv>0, \ t\in [0,T].\end{align*}
Letting $\vv\downarrow 0$ and taking sup in $x,v\in\R^d$ with $|v|=1$,  we derive
$$\sup_{\|f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\le 1}\|\nabla} \def\pp{\partial} \def\E{\mathbb E \{\gg_{t,T}^V (f)-   (\gg^{(n)})_{t,T}^V (f)\}\|_{\infty} \le \limsup_{m\rightarrow} \def\gg{\gamma\infty}\vv_{m,n},\ \ n\ge 1,\ t\in [0,T],$$
so that  $\vv_{m,n}\rightarrow} \def\gg{\gamma 0$ as $m,n\rightarrow} \def\gg{\gamma\infty$ yields
$$\lim_{n\rightarrow} \def\gg{\gamma\infty} \sup_{t\in [0,T]}\|\gg_{t}-\gg_{t}^{(n)}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}=0.$$
Similarly, the same holds for $\|\gg_t-\gg_t^{(n)}\|_{p_0,p_0}$. Combining these with \eqref{MN2}, we prove $\rr_\ll(\gg^{(n)},\gg)\rightarrow} \def\gg{\gamma 0$ as $n\rightarrow} \def\gg{\gamma\infty.$
\end{proof}

\section{Proofs of Theorem  \ref{T1'} and Theorem \ref{CC}}

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{proof}[Proof of Theorem \ref{T1'}]    We intend to prove that $\Phi$ has a unique fixed point $ \gg$ in $\tilde}\def\[{\lfloor} \def\]{\rfloor\GG,$    so that \eqref{E2} has a  unique solution  given by $X_{t,s}^x=X_{t,s}^{\gg},$ and \eqref{A*1}, \eqref{BS},  \eqref{GRD} and \eqref{GRDD}   follow  from \eqref{DX}, \eqref{BSM},  \eqref{GRG}, \eqref{B2}, \eqref{KK2} and \eqref{KK3}.

Let $\gg,\tilde}\def\[{\lfloor} \def\]{\rfloor\gg\in \tilde}\def\[{\lfloor} \def\]{\rfloor\GG$.
By    $(H_{F,g}'')$, we find a constant $K>0$ such that
\beq\label{PO} \big\|F_{T-r}(\cdot,\psi_{r}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg},\nabla} \def\pp{\partial} \def\E{\mathbb E\psi_{r}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg}) -F_{T-r}(\cdot,\psi_{r}^\gg,\nabla} \def\pp{\partial} \def\E{\mathbb E\psi_{r}^\gg)\big\|_\infty\le K\big\{1\land \|\psi_{r}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg}-\psi_{r}^{\gg}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\big\},
 \ \ \gg,\tilde}\def\[{\lfloor} \def\]{\rfloor\gg\in\tilde}\def\[{\lfloor} \def\]{\rfloor\GG. \end{equation}
By   $\|u_0\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+\|g\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty$,                                         there exists a constant $c_2>0$ such that
\beq\label{P*} \begin} \def\beq{\begin{equation}}  \def\F{\scr F{split}&\|\psi_{r,T}^\gg-\psi_{r,T}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}
 \le c_2\sup_{s\in [r,T]}  \big\{  \|\gg_{s}-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_{s}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1} +\|\gg_{s}-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_{s}\|_{p_0,q_0}\big\}.\end{split}\end{equation}
 Let
 $$h_r^x:= \sigma} \def\ess{\text{\rm{ess}}}\def\s{{\bf s}_r^{-1}(X_{t,r}^{\gg,x})\big\{F_{T-r}(\cdot,\psi_{r}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg},\nabla} \def\pp{\partial} \def\E{\mathbb E\psi_{r}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg}) -F_{T-r}(\cdot,\psi_{r}^\gg,\nabla} \def\pp{\partial} \def\E{\mathbb E\psi_{r}^\gg)\big\}(X_{t,r}^{\gg,x}),\  \ r\in [t,T].$$
 By $(H_{a,b})$, \eqref{PO} and \eqref{P*}, we find a constant $c_3>0$ such that
 \beq\label{CB} |h_r^x|\le c_2 \sup_{s\in [r,T]} 1\land \big\{\|\gg_s-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+\|\gg_s-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_s\|_{p_0,q_0}\big\}=:\zeta_r.\end{equation}
 Let
 $$R^x_t:= \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^T \< h_r^x,\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D W_r\>-\ff 1 2\int_t^T | h_r^x|^2\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}.$$
 By Girsanov's theorem,
 $$\tilde}\def\[{\lfloor} \def\]{\rfloor W_r:= W_r- \int_t^r h_s^x\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\ \ r\in [t,T]$$
 is a Brownian motion under probability $\Q_t^x:=R_t^x\P$. Reformulating \eqref{GG} as
 $$ \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D X_{t,s}^{\gg,x}= \big\{b_{T-s}+F_{T-s}(\cdot,\psi_{s}^{\tilde}\def\[{\lfloor} \def\]{\rfloor \gg}, \nabla} \def\pp{\partial} \def\E{\mathbb E \psi_{s}^{\tilde}\def\[{\lfloor} \def\]{\rfloor \gg})\big\}(X_{t,s}^{\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s + \sigma} \def\ess{\text{\rm{ess}}}\def\s{{\bf s}_s(X_{t,s}^{\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \tilde}\def\[{\lfloor} \def\]{\rfloor W_s,\ \ s\in [t,T], X_{t,t}^{\gg,x}=x,$$
 by the weak uniqueness we see that the law of $X_{t,\cdot}^x:=(X_{t,s}^{\gg,x})_{s\in [t,T]}$ under $\Q_t^x$ coincides with that of $(X_{t,s}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg,x})_{s\in [t,T]}$ under $\P$. So, by \eqref{*DD}, we find a constant $c_3>0$ such that
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &\|\Phi_t(\gg)-\Phi_t(\tilde}\def\[{\lfloor} \def\]{\rfloor\gg)\|_{V}= \sup_{\|f\|_\infty\le \eta_t^V}\sup_{x\in\R^d}  \big|\E[(R_t^x-1)f(X_{t,\cdot}^{\gg,x})]\big|\\
&\le \sup_{x\in\R^d} \E[\eta_t^V|R_t^x-1|]\le \big(\E |\eta_t^V|^2\big)^{\ff 1 2} \big(\E |R_t^x|^2-1\big)^{\ff 1 2}\le c_3 \big(\E |R_t^x|^2-1\big)^{\ff 1 2}.\end{align*}
Noting that \eqref{CB} implies
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*}&\E |R_t^x|^2 \le \E\big[\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{  2\int_t^T \< h_r^x,\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D W_r\>-  2\int_t^T |h_r^x|^2\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r+ \int_t^T |h_r^x|^2\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\big]
\le  \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^T |\zeta_r|^2\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\\
&\le 1+ \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^T |\zeta_r|^2\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\int_t^T |\zeta_r|^2\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r\le 1  +c_2^2  \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{Tc_2^2} \int_t^T \big\{\|\gg_s-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+\|\gg_s-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_s\|_{p_0,q_0}\big\}^2\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\end{align*} we find a constant $c_4>0$ such that
$$\|\Phi_t(\gg)-\Phi_t(\tilde}\def\[{\lfloor} \def\]{\rfloor\gg)\|_{V}\le c_4 \bigg(\int_t^T \big\{\|\gg_s-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+\|\gg_s-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_s\|_{p_0,q_0}\big\}^2\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\bigg)^{\ff 1 2}.$$
  Consequently,
\beq\label{P1} \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-\ll (T-t)} \|\Phi_{t}(\gg)- \Phi_{t}(\tilde}\def\[{\lfloor} \def\]{\rfloor\gg)\|_{V} \le
  c_4 \rr_{\ll}(\gg,\tilde}\def\[{\lfloor} \def\]{\rfloor\gg)\bigg(\int_t^T \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-2\ll (r-t)}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r\bigg)^{\ff 1 2},\ \ t\in [0,T]. \end{equation}

Next,
 by  \eqref{KK} and \eqref{KK2}, for any $f\in \scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1(\R^d)$,  we may apply It\^o's formula in \cite[Lemma 3.3]{YZ} to deduce
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \Big\{\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^r V_{T-r'}(X_{t, r'}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r'} P_{r,s}^{\gg,V} f(X_{t,r}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg,x}) \Big\}= \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D M_r\\
&+\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^r V_{T-r'}(X_{t, r'}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r'}\big\{\big[F_{T-r}(\cdot,\psi_{r,T}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg},\nabla} \def\pp{\partial} \def\E{\mathbb E\psi_{r,T}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg}) -F_{T-r}(\cdot,\psi_{r,T}^{\gg},\nabla} \def\pp{\partial} \def\E{\mathbb E\psi_{r,T}^{\gg})\big]\cdot \nabla} \def\pp{\partial} \def\E{\mathbb E P_{r,s}^{\gg,V} f\big\}(X_{t,r}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r
\end{align*} for some martingale $M_r, r\in [0,s]$.
Then
\beq\label{RP0} \begin} \def\beq{\begin{equation}}  \def\F{\scr F{split} &P_{t,s}^{\tilde}\def\[{\lfloor} \def\]{\rfloor \gg,V } f(x)- P_{t,s}^{\gg,V} f(x)=\E\big[\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^s V_{T-r}(X_{t, r}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}P_{s,s}^{\gg,V} f(X_{t,s}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg,x})\big] - \E\big[\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^t V_{T-r}(X_{t, r}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg,x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}P_{t,s}^{\gg,V} f(X_{t,t}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg,x})\big]\\
&=\int_t^s P_{t,r}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg,V} \big\{\big[F_{T-r}(\cdot,\psi_{r}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg},\nabla} \def\pp{\partial} \def\E{\mathbb E\psi_{r}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg}) -F_{T-r}(\cdot,\psi_{r}^\gg,\nabla} \def\pp{\partial} \def\E{\mathbb E\psi_{r}^\gg)\big]\cdot \nabla} \def\pp{\partial} \def\E{\mathbb E P_{r,s}^{\gg,V} f\big\}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r.\end{split} \end{equation}
Combining this with \eqref{KK2}, \eqref{KK3}, \eqref{PO}   and \eqref{P*},  we find constants
    $c_5,c_6>0$ such that
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &\sup_{\|f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\le 1}\|\nabla} \def\pp{\partial} \def\E{\mathbb E\{\Phi(\gg)\}_{t,T}^V(f)- \nabla} \def\pp{\partial} \def\E{\mathbb E\{\Phi (\tilde}\def\[{\lfloor} \def\]{\rfloor\gg)\}_{t,T}^V(f)\|_{\infty}  \\
 &=\sup_{\|f\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\le 1}\|\nabla} \def\pp{\partial} \def\E{\mathbb E P_{t,T}^{\gg,V} f-\nabla} \def\pp{\partial} \def\E{\mathbb E P_{t,T}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg,V} f\|_{\infty}
 \le c_5\int_t^T(r-t)^{-\ff 1 2}  \|\psi_{r}^\gg-\psi_{r}^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}  \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r \\
 & \le c_6 \int_t^T(r-t)^{-\ff 1 2}\sup_{\theta\in [r,T]} \big\{\|\gg_{r}-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_{r}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1} +\|\gg_{r}-\tilde}\def\[{\lfloor} \def\]{\rfloor\gg_{r}\|_{p_0,q_0}\big\}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r.\end{align*}
 Then
 \beq\label{P2} \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-\ll (T-t)} \| \Phi_{t}(\gg) -  \Phi_{t}(\tilde}\def\[{\lfloor} \def\]{\rfloor\gg) \|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1} \le
 c_6 \rr_\ll(\gg,\tilde}\def\[{\lfloor} \def\]{\rfloor\gg) \int_t^s(r-t)^{-\ff 1 2}\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-\ll(r-t)}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r,\ \ t\in [0,T].\end{equation}

Finally, let $\|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}\le 1$. By \eqref{BE}, for any $r\in [0,T)$, we have
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &(\pp_t + \tilde}\def\[{\lfloor} \def\]{\rfloor L_t^\gg+V_{T-t}) \big\{u_t^{\Phi(\gg),V}(f)-u_t^{\Phi(\tilde}\def\[{\lfloor} \def\]{\rfloor\gg),V}(f)\big\}= \tilde}\def\[{\lfloor} \def\]{\rfloor f_t,\ \ t\in [0,T], u_T^{\Phi(\gg), f}- u_T^{\Phi(\tilde}\def\[{\lfloor} \def\]{\rfloor\gg),f}=0,\\
&\tilde}\def\[{\lfloor} \def\]{\rfloor f_t:= \big\{F_{T-t}(\cdot,\psi_t^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg}, \nabla} \def\pp{\partial} \def\E{\mathbb E \psi_t^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg})- F_{T-t}(\cdot,\psi_t^\gg, \nabla} \def\pp{\partial} \def\E{\mathbb E \psi_t^\gg)\big\}\cdot\nabla} \def\pp{\partial} \def\E{\mathbb E u_t^{\Phi(\tilde}\def\[{\lfloor} \def\]{\rfloor\gg),V}(f).\end{align*}
By $\|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}\le 1$, \eqref{B2} and \eqref{PO}, there exists a constant $c>0$ such that
$$\|\tilde}\def\[{\lfloor} \def\]{\rfloor f_t\|_\infty\le c  \|\psi_t^\gg-\psi_t^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1},\ \ t\in [0,T].$$
So, by Lemma \ref{LN3}, there exist  constants $c_7, c_8>0$ such that
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*}  \sup_{t\in [r,T]}\ \|u_t^{\Phi(\gg),V}(f)-u_t^{\Phi(\tilde}\def\[{\lfloor} \def\]{\rfloor\gg),V}(f)\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}
 \le c_7 \|\tilde}\def\[{\lfloor} \def\]{\rfloor f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(r,T)}
 \le c_8 \bigg(\int_r^T \|\psi_t^\gg-\psi_t^{\tilde}\def\[{\lfloor} \def\]{\rfloor\gg}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}^{q_0}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t\bigg)^{\ff 1 {q_0}}.\end{align*}
Combining this with \eqref{P*}, we find a constant $c_9>0$ such that
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-\ll (T-t)}  \|\{\Phi(\gg)\}_t -\{\Phi(\tilde}\def\[{\lfloor} \def\]{\rfloor\gg)\}_t \|_{p_0,q_0}
 \le c_9 \rr_{\ll}(\gg,\tilde}\def\[{\lfloor} \def\]{\rfloor\gg) \bigg(\int_t^T \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-q_0\ll(s-t)}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\bigg)^{\ff 1 {q_0}},\ \ t\in [0,T].\end{align*}
By taking large enough $\ll>0$, this and \eqref{P1}-\eqref{P2} yield that
 $$\rr_\ll(\Phi(\gg),\Phi(\tilde}\def\[{\lfloor} \def\]{\rfloor\gg))\le \ff 1 2 \rr_\ll(\gg,\tilde}\def\[{\lfloor} \def\]{\rfloor\gg),\ \ \gg,\tilde}\def\[{\lfloor} \def\]{\rfloor\gg\in \tilde}\def\[{\lfloor} \def\]{\rfloor\GG.$$
 Therefore, $\Phi$ has a unique fixed point in $\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$ as desired.

  Since for any solution  $X=(X_{t,s}^x)_{(t,s,x)\in D_T\times \R^d}$  of \eqref{E2},   Lemmas \ref{N1} and  \ref{N1'} imply   that the law $\gg_X$    of $X$ is in $\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$, so that $\gg_X$ is a fixed point of $\Phi$ in $\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$. By the uniqueness of the fixed point, as well as the well-posedness of \eqref{GG}, we prove the uniqueness of \eqref{S1}.
   \end{proof}

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{proof}[Theorem \ref{CC}]  (1) Let $u$ solve \eqref{E2} for $t\in [0,T]$  and satisfy \eqref{*0}, we intend to prove that $u$ satisfies \eqref{SL}.

Note that $\psi_t=\psi_t^{\gg}$ for the unique fixed point $\gg\in\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$ of $\Phi$, \eqref{KK2} for $P_{T-t,T}^{\gg,V} u_0$ and
\eqref{B2} for $f=g_{T-\cdot}$ imply that
\beq\label{VV} \sup_{t\in [0,T]} (\| \psi_t\|_\infty+\|\nabla} \def\pp{\partial} \def\E{\mathbb E \psi_t\|_\infty)<\infty.\end{equation}
Next, let
$$\tilde}\def\[{\lfloor} \def\]{\rfloor f_s:= F_{T-s} (\cdot,\psi_s,\nabla} \def\pp{\partial} \def\E{\mathbb E \psi_s)-F_{T-s}(\cdot,u_{T-s},\nabla} \def\pp{\partial} \def\E{\mathbb E u_{T-s}).$$
By $(H_{F,g}'')$, we find a constant $c_1>0$ such that
\beq\label{*88} |\tilde}\def\[{\lfloor} \def\]{\rfloor f_s|\le c_1 \|\psi_s- u_{T-s}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1},\ \ s\in [0,T]. \end{equation}
By \eqref{S1} and It\^o's formula in   \cite[Lemma 3.3]{YZ},  we obtain
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \big\{u_{T-s}(X_{t,s}^x)\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^sV_{T-r}(X_{t,r}^x)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\big\}= \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D M_s
 +\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^sV_{T-r}(X_{t,r}^x)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\big[ \tilde}\def\[{\lfloor} \def\]{\rfloor f_s\cdot \nabla} \def\pp{\partial} \def\E{\mathbb E u_{T-s} -g_{T-s}\big]\big(X_{t,s}^x\big)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\ \ s\in [t,T]\end{align*}
for  some martingale $M_s$.
Combining this with the definition of $\psi_t$ in \eqref{S1}, we derive
\beq\label{RP1}\begin} \def\beq{\begin{equation}}  \def\F{\scr F{split} &\psi_t(x)- u_{T-t}(x)\\
&= \E\bigg[u_0(X_{t,T}^x)\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^TV_{T-r}(X_{t,r}^x)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}+\int_t^T g_{T-s}(X_{t,s}^x)\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^sV_{T-r}(X_{t,r}^x)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\bigg] - u_{T-t}(x)\\
&= \E\int_t^T \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^sV_{T-r}(X_{t,r}^x)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\big[\tilde}\def\[{\lfloor} \def\]{\rfloor f_s \cdot \nabla} \def\pp{\partial} \def\E{\mathbb E u_{T-s} \ \Big]\big(X_{t,s}^x\big)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\\
&= \int_t^T P_{t,s}^V \big[\tilde}\def\[{\lfloor} \def\]{\rfloor f_s  \cdot \nabla} \def\pp{\partial} \def\E{\mathbb E u_{T-s}  \big](x)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\ \ t\in [0,T],\end{split}\end{equation}
where $P_{t,s}^V= P_{t,s}^{\gg,V}$ for $\gg$ being the unique fixed point of $\Phi$.
Combining this with \eqref{B2}, \eqref{*88} and  $\|\nabla} \def\pp{\partial} \def\E{\mathbb E u\|_\infty<\infty$,
we find a constant $c_2>0$ such that
$$\|\psi_t-u_{T-t}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1} \le c_2 \bigg(\int_t^T\|\psi_s-u_{T-s}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}^{q_0}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\bigg)^{\ff 1 {q_0}},\ \ t\in [0,T].$$
Since  \eqref{*0} and \eqref{VV} imply that  $\|\psi_t-u_{T-t}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}$ is bounded in $t\in [0,T]$,
  we prove
  $\psi_t-u_{T-t}=0$ for all $t\in [0,T]$, so that the proof is finished.

(2)  Let $u$ be given in \eqref{SL}, and let $P_{t,s}^V=P_{t,s}^{\gg,V}$ for
the unique fixed point $\gg\in\tilde}\def\[{\lfloor} \def\]{\rfloor\GG$ of $\Phi$. We have
\beq\label{UT} u_t= \psi_{T-t,T}=P_{T-t,T}^V u_0+ \int_{T-t}^T P_{T-t,s}^Vg_{T-s}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\ \ t\in [0,T].\end{equation}
Let
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*}\tilde}\def\[{\lfloor} \def\]{\rfloor L_t = L_{T-t}+ F_{T-t}(\cdot, \psi_t,\nabla} \def\pp{\partial} \def\E{\mathbb E\psi_t)\cdot\nabla} \def\pp{\partial} \def\E{\mathbb E,  \ t\in [0,T].\end{align*}
 By Lemma \ref{N1'}(2),
$$u_t^{(1)}:= P_{T-t,T}^Vu_0,\ \ t\in [0,T]$$ satisfies
\beq\label{PD4} \begin} \def\beq{\begin{equation}}  \def\F{\scr F{split}&\pp_t u_t^{(1)}=  (\tilde}\def\[{\lfloor} \def\]{\rfloor L_{T-t} +V_t)u_t^{(1)},\ \ t\in [0,T],\\
&\| u^{(1)}\|_\infty+\|\nabla} \def\pp{\partial} \def\E{\mathbb E u^{(1)}\|_\infty+\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2u^{(1)}\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty.\end{split}\end{equation}

On the other hand,   by Lemma \ref{N1'}(1),
$$ u_t^{(2)}:= \int_{T-t}^T P_{T-t,s}^V g_{T-s}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\ \ t\in [0,T]$$
satisfies
\beq\label{PD5} \begin} \def\beq{\begin{equation}}  \def\F{\scr F{split}&\pp_t u_t^{(2)}= (\tilde}\def\[{\lfloor} \def\]{\rfloor L_{T-t} +V_t)u_t^{(2)} + g_t,\ \ t\in [0,T],\\
&\| u^{(2)}\|_\infty+\|\nabla} \def\pp{\partial} \def\E{\mathbb E u^{(2)}\|_\infty+\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2u^{(2)}\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty,\end{split}\end{equation}
which together with \eqref{UT} and \eqref{PD4} finishes the proof.
\end{proof}

\section{Proof  of Theorem \ref{T1}   }

We first observe that by extending functions from $\T^d$ to $\R^d$ periodically, i.e. letting
$$f(x+k)=f(x),\ \ x\in\T^d,\ k\in \mathbb Z^d,$$
we extend the PDE   \eqref{E3} from $E=\T^d$ to $E=\R^d$ with assumptions $(H_{V,u_0})$, $(H_{a,b}), (H_{F,g})$ and $(H_{F,g}')$ invariant. So, the existence of the extended equation implies that of the original equation. On the other hand, if the original equation has two solutions with same  initial value, then their  periodical extensions solve
the extended equation, so that the uniqueness of the extended equation implies that of the original equation. Therefore,
it suffices to prove for $E=\R^d$.

In the following we set $E=\R^d$,  and prove Theorem \ref{T1}. We first prove under the following stronger assumption replacing  $(H_{F,g})$, then make extension by a truncation argument.

 \begin} \def\beq{\begin{equation}}  \def\F{\scr F{enumerate} \item[$(\tilde}\def\[{\lfloor} \def\]{\rfloor H_{F,g})$] $\|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+ \|\bar g\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty$, and there exist constants $K,\aa>0$ such that
 \begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &  |F_t(x,r_1,r_2)- F_t(x,\tilde}\def\[{\lfloor} \def\]{\rfloor r_1,\tilde}\def\[{\lfloor} \def\]{\rfloor r_2)| +   |g_t(x,r_1,r_2, r_3)- g_t(x,\tilde}\def\[{\lfloor} \def\]{\rfloor r_1,\tilde}\def\[{\lfloor} \def\]{\rfloor r_2,\tilde}\def\[{\lfloor} \def\]{\rfloor r_3)|  \\
&  \le K(|r_1-\tilde}\def\[{\lfloor} \def\]{\rfloor r_1|+|r_2-\tilde}\def\[{\lfloor} \def\]{\rfloor r_2|) +\aa |r_3-\tilde}\def\[{\lfloor} \def\]{\rfloor r_3|,\\
   &\qquad (t,x)\in [0,T]\times \R^d,\ r_1,\tilde}\def\[{\lfloor} \def\]{\rfloor r_1\in \R^m,\  r_2,\tilde}\def\[{\lfloor} \def\]{\rfloor r_2\in \R^{d\otimes m},\ r_3,\tilde}\def\[{\lfloor} \def\]{\rfloor r_3\in \R^{d\otimes d\otimes m}.\end{align*}
   \end{enumerate}

   \begin} \def\beq{\begin{equation}}  \def\F{\scr F{lem}\label{LW} Assume $(H_{a,b})$, $(H_{V,u_0})$ and $(\tilde}\def\[{\lfloor} \def\]{\rfloor F_{F,g})$ with small enough $\aa>0$ determined by $(H_{a,b})$ and $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+\|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}.$  The equation $\eqref{E3}$ has a unique solution in $\scr U(p_0,q_0)$ with  $T^*=T$ such that
   $$\sup_{t\in [0,T]} \|u_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1} +\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty.$$\end{lem}

   \begin} \def\beq{\begin{equation}}  \def\F{\scr F{proof} We will make use of the fixed point theorem for a map $\Psi$ induced by \eqref{E3}. Let ${\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T$ be the Banach space of maps
$$h: [0,T]\times \R^d\rightarrow} \def\gg{\gamma \R^m$$
satisfying
$$\|h\|_{\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T:= \sup_{t\in [0,T]} \|h_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 h\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty.$$
Given $h\in{\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T$, consider the PDE
\beq\label{PD} \pp_t u_t^h = (L_t+V_t) u_t^h + F_t (\cdot, u_t^h,\nabla} \def\pp{\partial} \def\E{\mathbb E u_t^h)\cdot \nabla} \def\pp{\partial} \def\E{\mathbb E u_t^h + g_t(\cdot, h_t,\nabla} \def\pp{\partial} \def\E{\mathbb E h_t,\nabla} \def\pp{\partial} \def\E{\mathbb E^2 h_t),\ \ t\in [0,T].\end{equation}
By $(\tilde}\def\[{\lfloor} \def\]{\rfloor H_{F,g})$, we have $\|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty$ and
\beq\label{SB}\begin} \def\beq{\begin{equation}}  \def\F{\scr F{split} &\|g(\cdot, h,\nabla} \def\pp{\partial} \def\E{\mathbb E h,\nabla} \def\pp{\partial} \def\E{\mathbb E^2 h)\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)}\le \|\bar g\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)} \\
&+ K\bigg(\int_0^t \|h_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}^{q_0}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\bigg)^{\ff 1 {q_0}}+ \aa\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 h\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)}<\infty,\ \ t\in [0,T],\end{split}\end{equation}
so that $(H_{F,g}'')$ holds for $g(\cdot, h,\nabla} \def\pp{\partial} \def\E{\mathbb E h,\nabla} \def\pp{\partial} \def\E{\mathbb E^2 h)$ replacing $g$.
According to Theorem \ref{T1'} and Theorem \ref{CC}, this together with $(H_{V, u_0})$ and  $(H_{a,b})$   implies that \eqref{PD} has a unique solution in the class $\scr U(p_0,q_0)$ up to time
$T$, such that
\beq\label{XX} \|u^h\|_\infty\le k(u_0,V,g),\end{equation}
and for some constant $c >0$ determined by $(H_{a,b})$ and $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+\|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$,
\beq\label{XX0}\begin} \def\beq{\begin{equation}}  \def\F{\scr F{split}& \|u_t^h\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+ \|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u^h\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)}\le c\big(\|u_0\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+ \|g(\cdot, h,\nabla} \def\pp{\partial} \def\E{\mathbb E h,\nabla} \def\pp{\partial} \def\E{\mathbb E^2 h)\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)}\big)\\
&\le c \big(\|u_0\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+   \|\bar g\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)}\big) + cK \bigg(\int_0^t \|h_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}^{q_0}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\bigg)^{\ff 1 {q_0}}+ c\aa\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 h\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)},\\
&\qquad  \ \ h\in{\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T, t\in [0,T].\end{split} \end{equation}
So, it suffices to prove that the map
$$\Psi: {\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T\ni h\mapsto u^h\in {\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T$$ has a unique fixed point.

To this end, we let $\aa\in (0,\ff 1 {2c})$ where $c>0$ is determined by $(H_{a,b})$ and $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+\|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$, and denote
  $c_0:= c \big(\|u_0\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+   \|\bar g\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)}\big)$, and let
$${\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T_N:=\Big\{h\in {\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T:\ \|h\|_{N}:=\sup_{t\in [0,T]} \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-Nt} \big(\|h_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 h\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)}\big)\le N c_0\Big\},\ \ N\ge 2.$$
We will find a constant $N_0\ge 2$ such that for any $N\ge N_0$, $\Psi:{\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T_N\rightarrow} \def\gg{\gamma {\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T_N$ has a unique fixed point.

Firstly, we take $N_0\ge 2$ such that
$$cK\bigg(\int_0^T \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-q_0N_0s}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\bigg)^{\ff 1 {q_0}}\le \ff 1 2.$$
Combining this with \eqref{XX0} and $c\aa\le \ff 1 2$, we obtain
$$\|\Psi(h)\|_N=\|u^h\|_N\le c_1 +\ff 1 2 \|h\|_N\le c_0 +\ff N 2 c_0\le N c_0,\ \ h\in {\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T_N, N\ge N_0.$$
So, $\psi:   {\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T_N\rightarrow} \def\gg{\gamma {\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T_N$  for $N\ge N_0$.  Thus,
\beq\label{X*}\sup_{t\in [0,T]} \|u_t^h\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\le   \|\Psi(h)\|_{\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T\le c(N):= c_0 N \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{NT},\ \ h\in {\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T_N, N\ge N_0.\end{equation}
It remains to prove  that $\Psi$ has a unique fixed point in ${\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T_N$ for any $N\ge N_0$.

For any $h,\tilde}\def\[{\lfloor} \def\]{\rfloor h\in {\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T_N,$ let
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} f_t:=&\,\big\{F_{T-t}(\cdot, u_{T-t}^{\tilde}\def\[{\lfloor} \def\]{\rfloor h}, \nabla} \def\pp{\partial} \def\E{\mathbb E u_{T-t}^{\tilde}\def\[{\lfloor} \def\]{\rfloor h})- F_{T-t}(\cdot, u_{T-t}^{h}, \nabla} \def\pp{\partial} \def\E{\mathbb E u_{T-t}^{h})\big\}\cdot \nabla} \def\pp{\partial} \def\E{\mathbb E u_t^{\tilde}\def\[{\lfloor} \def\]{\rfloor h}\\
&+ g_{T-t}(\cdot, \tilde}\def\[{\lfloor} \def\]{\rfloor h_{T-t}, \nabla} \def\pp{\partial} \def\E{\mathbb E  \tilde}\def\[{\lfloor} \def\]{\rfloor h_{T-t}, \nabla} \def\pp{\partial} \def\E{\mathbb E^2  \tilde}\def\[{\lfloor} \def\]{\rfloor h_{T-t})- g_{T-t}(\cdot, h_{T-t}, \nabla} \def\pp{\partial} \def\E{\mathbb E h_{T-t}, \nabla} \def\pp{\partial} \def\E{\mathbb E^2 h_{T-t}),\ \ t\in [0,T]\end{align*}
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} L_t^h := L_{T-t}+ F_{T-t}(\cdot, u_{T-t}^{h}, \nabla} \def\pp{\partial} \def\E{\mathbb E u_{T-t}^{h})\cdot\nabla} \def\pp{\partial} \def\E{\mathbb E,\ \ t\in [0,T].\end{align*}
Then
\beq\label{WR} w_t:= u_{T-t}^h - u_{T-t}^{\tilde}\def\[{\lfloor} \def\]{\rfloor h}=\Psi_{T-t}(h)- \Psi_{T-t}(\tilde}\def\[{\lfloor} \def\]{\rfloor h)\end{equation}   satisfies $w\in \tilde}\def\[{\lfloor} \def\]{\rfloor H_{q_0}^{2,p_0}(0,T)$ and solves the PDE
$$(\pp_t + L_t^h +V_{T-t}) w_t = f_t,\ \ t\in [0,T],\ w_T=0.$$
By Lemma \ref{LN3}, \eqref{XX},    \eqref{X*},   $(H_{V, u_0}),$  $(H_{a,b})$ and $(\tilde}\def\[{\lfloor} \def\]{\rfloor H_{F,g})$ imply   that for some constant  $c_1 >0$ determined by $(H_{a,b})$ and $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+ \|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$,
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} & \|w_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 w\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)} \le c_1 \|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)}\\
&\le c_1  K\bigg(\int_t^T \big\{c(N)\|w_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+ \|h_s-\tilde}\def\[{\lfloor} \def\]{\rfloor h_{T-s}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\big\}^{q_0}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\bigg)^{\ff 1 {q_0}} \\
&\quad + c_1 \aa    \|\nabla} \def\pp{\partial} \def\E{\mathbb E^2(h_{T-\cdot}-\tilde}\def\[{\lfloor} \def\]{\rfloor h_{T-\cdot})\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(t,T)},\ \ t\in [0,T].\end{align*}
Then  for any $\ll>0$, this and \eqref{WR} yield
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} \| \Psi(h)-\Psi(\tilde}\def\[{\lfloor} \def\]{\rfloor h)\|_\ll  \le \vv(\ll) \big[\|h-\tilde}\def\[{\lfloor} \def\]{\rfloor h\|_\ll  +  \|\Psi(h)-\Psi(\tilde}\def\[{\lfloor} \def\]{\rfloor h)\|_\ll\big],\ \ h,\tilde}\def\[{\lfloor} \def\]{\rfloor h \in{\bf H}}\def\gg{\gamma} \def\n{{\mathbf n}}\def\GG{\Gamma}\def\T{\mathbb T_N \end{align*}
for $\vv(\ll):=   \max\big\{c_1\aa,\ (c(N)\lor 1)c_1K\sup_{t\in [0,T]}\big(\int_t^T \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{-\ll q_0 (s-t)}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s \big)^{\ff 1 {q_0}}\big\}.$
Taking  $\ll>0$   large enough and $\aa \in (0, \ff 1 {2c_1})$ such that
$\vv(\ll)<\ff 1 2$, we see that    $\Psi$ is $\|\cdot\|_\ll$-contractive in $H_N$,  and hence has a unique fixed point.

\end{proof}

\begin} \def\beq{\begin{equation}}  \def\F{\scr F{proof}[Proof of Theorem \ref{T1}]
(1)  For any $n\in\mathbb N$, let  $$\varphi_n(r):= r1_{\{|r|\le n\}}+ \ff{nr}{|r|}1_{\{|r|>n\}},\ \ r\in \R^m\cup\R^{d\otimes m}.$$
We take the following truncation of $F_t$:
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*}  &F_t^{(n)}(x,r_1,r_2)= F_t(x,\varphi_n(r_1), \varphi_n(r_2)),\ \ g_t^{(n)}(x,r_1,r_2, r_3)= g_t(x,\varphi_n(r_1), \varphi_n(r_2), r_3),\\
  & \ t\ge 0, x\in\R^d, (r_1,r_2, r_3)\in \R^m\times \R^{d\otimes m}\times \R^{d\otimes d\otimes d\otimes m}.\end{align*}
Then  $(H_{F,g})$ implies $(\tilde}\def\[{\lfloor} \def\]{\rfloor H_{F,g})$ for $(F^{(n)}, g^{(n)})$ replacing $(F,g)$. So, by Lemma \ref{LW},   the equation
\beq\label{**} \begin} \def\beq{\begin{equation}}  \def\F{\scr F{split} & \pp_t u_t^{(n)}= (L_t +V_t)u_t^{(n)}+ F_t^{(n)}(\cdot, u_t^{(n)}, \nabla} \def\pp{\partial} \def\E{\mathbb E u_t^{(n)})\cdot\nabla} \def\pp{\partial} \def\E{\mathbb E u_t^{(n)}+  g_t^{(n)}(\cdot, u_t^{(n)}, \nabla} \def\pp{\partial} \def\E{\mathbb E u_t^{(n)}, \nabla} \def\pp{\partial} \def\E{\mathbb E^2 u_t^{(n)}),\\
&\qquad t\in [0,T],\  u_0^{(n)}=u_0\end{split} \end{equation}
has a unique solution in $\scr U(p_0,q_0)$ with  $T^*=T,$ and
\beq\label{C*} \sup_{t\in [0,T]} \|u_t^{(n)}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1} +\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u^{(n)}\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)} <\infty,\ \ n\ge 1.\end{equation}
Let
\beq\label{TN} T_n:=T\land \inf\{t\ge 0: \|u_t^{(n)}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1} \ge n\}.\end{equation}
Then $u_t^{(n)}$ solves \eqref{E3} up to time $T_n$. Letting $T_0=0$ and
$$T^*:=\lim_{n\rightarrow} \def\gg{\gamma\infty} T_n,$$
   the uniqueness of \eqref{**} for every $n\ge 1$ implies
   $$u_t^{(n)}= u_t^{(m)},\ \ t\in [0, T_{n}\land T_m],\ \ n,m\in \mathbb N,$$ so that  $\eqref{E3}$ has a unique solution
$$u_t:=\sum_{n=1}^\infty u_t^{(n)}1_{[T_{n-1}, T_n)}(t),\ \ t\in [0,T^*)$$ in the class $\scr U(p_0,q_0)$,   provided   $T^*>0$.

To prove $T^*>0$, it suffices to show that $T_n>0$ for  large    $n>0.$
For fixed $T\in (0,\infty)$, let $P_{t,s}^{(n),V}$ be the Feynman-Kac semigroup generated by
$$L_{T-t}+V_{T-t}+ F_{T-t}^{(n)}(\cdot, u_{T-t}^{(n)},\nabla} \def\pp{\partial} \def\E{\mathbb E u_{T-t}^{(n)})\cdot\nabla} \def\pp{\partial} \def\E{\mathbb E,\ \ t\in [0,T];$$
that is
\beq\label{ASS}   P_{t,s}^{(n),V}f(x):= \E\big[f(X_{t,s}^{(n),x}) \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_t^s V_{T-r} (X_{t,r}^{(n),x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\big],\ \ 0\le t\le s\le T, f\in\B_b(\R^d)\end{equation}
for $X_{t,s}^{(n),x}$ solving the SDE
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D  X_{t,s}^{(n),x}= \sigma} \def\ess{\text{\rm{ess}}}\def\s{{\bf s}_s(X_{t,s}^{(n),x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D W_s+ \big\{b_{T-s}+F^{(n)}_{T-s}(\cdot, u_{T-s}^{(n)},\nabla} \def\pp{\partial} \def\E{\mathbb E u_{T-s}^{(n)})\big\} (X_{t,s}^{(n),x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\\
&\qquad\qquad  s\in [t,T],\ X_{t,t}^{(n),x}=x.\end{align*}
Since $(H_{F,g})$ implies   $\|F^{(n)}\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty,$  by $(H_{a,b})$ and \cite[Theorem 2.1]{W21e}, this SDE is well-posed.
By \eqref{**} and It\^o's formula, we find a martingale $M_s$ such that
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \Big\{u_{T-s}^{(n)}(X_{T-t,s}^{(n),x}) \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{\int_{T-t}^sV_{T-r}(X_{t,r}(X_{t,r}^{(n),x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\Big\}\\
&= \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D M_s-g_{T-s}^{(n)}(\cdot, u_{T-s}^{(n)}, \nabla} \def\pp{\partial} \def\E{\mathbb E u_{T-s}^{(n)},\nabla} \def\pp{\partial} \def\E{\mathbb E^2u_{T-s}^{(n)})(X_{T-t,s}^{(n),x})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\ \ s\in [T-t,T].\end{align*}  So,
\beq\label{AS} u_t^{(n)}= P_{T-t,T}^{(n),V} u_0 +\int_{T-t}^T P_{T-t,s}^{(n),V} g_{T-s}^{(n)}(\cdot, u_{T-s}^{(n)}, \nabla} \def\pp{\partial} \def\E{\mathbb E u_{T-s}^{(n)},\nabla} \def\pp{\partial} \def\E{\mathbb E^2u_{T-s}^{(n)})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\ \ t\in [0,T],\end{equation}
which implies
$\|u^{(n)}\|_\infty \le k(u_0,g,V),$  so that
$$\big|F^{(n)}_{T-s}(\cdot, u_{T-s}^{(n)},\nabla} \def\pp{\partial} \def\E{\mathbb E u_{T-s}^{(n)})\big|\le \bar F_{T-s},\ \ s\in [0,T].$$
Combining this with $(H_{a,b})$ and $\|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty$,
by Lemma \ref{N1'}, we find a constant $c>0$ determined by $(H_{a,b})$ and $\|\bar F\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+ \|V\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}$, such that for any $n\ge 1$ and any $t\in [0,T],$
\beq\label{AS1}   \|P_{T-t,T}^{(n),V} u_0\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\le c,\ \
    \bigg\|\int_{T-t}^T  P_{T-t,s}^{(n),V} f_s\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\bigg\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\le c  \|f\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(T-t,T)},\ \ f\in \tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T),
  \end{equation}
 \beq\label{AS'}  \|\nabla} \def\pp{\partial} \def\E{\mathbb E P_{t,s}^{(n),V} f\|_\infty\le c (s-t)^{-\ff 1 2}\|f\|_\infty,\ \ f\in \B_b(\R^d),\ \ s\in (t,T].\end{equation}
 Combining   \eqref{AS} with \eqref{AS1}, and noting that $(H_{F,g})$ implies
$$\big\|g^{(n)} (\cdot, u^{(n)},\nabla} \def\pp{\partial} \def\E{\mathbb E u^{(n)}, \nabla} \def\pp{\partial} \def\E{\mathbb E^2 u^{(n)})\big\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}\le 2 nK_n + \|\bar g\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}+\aa\|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u^{(n)}\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,T)}<\infty,$$ we obtain
 $$\lim_{t\downarrow 0} \|u_t^{(n)}\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}\le \lim_{t\downarrow 0}\big\{\|P_{T-t,T}^{(n),V}u_0\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}
 + c\|g^{(n)}(\cdot, u^{(n)}, \nabla} \def\pp{\partial} \def\E{\mathbb E u^{(n)}, \nabla} \def\pp{\partial} \def\E{\mathbb E^2 u^{(n)})\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)}\big\}\le c,\ \ n\ge 1.$$
By \eqref{TN},  this implies $T_n>0$ for $n>c. $

(2) Assume $(H_{F,g}')$, we intend to prove $T^*=\infty$. For $n\ge 1$ let $ P_{t,s}^{(n),V}$ be in \eqref{ASS}. Since $u_t=u_t^{(n)}$ for $t \in [0,T_n]$,  we have
$$g_s^{(n)}(\cdot, u_s^{(n)}, \nabla} \def\pp{\partial} \def\E{\mathbb E u_s^{(n)}, \nabla} \def\pp{\partial} \def\E{\mathbb E^2 u_s^{(n)})= g_s (\cdot, u_s, \nabla} \def\pp{\partial} \def\E{\mathbb E u_s, \nabla} \def\pp{\partial} \def\E{\mathbb E^2 u_s),\ \ s\in [0,T_n].$$
Then with the integral transform $s\mapsto T-s$,
\eqref{AS} implies
$$ u_t= P_{T-t,T}^{(n),V} u_0 +\int_{0}^t P_{T-t,T-s}^{(n),V} g_{s}(\cdot, u_{s}, \nabla} \def\pp{\partial} \def\E{\mathbb E u_{s},\nabla} \def\pp{\partial} \def\E{\mathbb E^2u_{s})\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\ \ t\in [0, T_n].$$
Combining this with \eqref{AS1}, \eqref{AS'}, we find a constant $c_1>0$ such that
\begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &\|u_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+ \|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)}
 \le c\|u_0\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}  + c  \|g(\cdot, 0,0,\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u)\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)}\\
 &\qquad +  c\int_0^t (t-s)^{-\ff 1 2} \|g_{s}(\cdot, u_{s}, \nabla} \def\pp{\partial} \def\E{\mathbb E u_{s},\nabla} \def\pp{\partial} \def\E{\mathbb E^2u_{s})-g_s(\cdot,0,0,\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u_s)\|_\infty \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s.\end{align*}
 Since   $(H_{F,g}')$ implies
 \begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &  \|g(\cdot, 0,0,\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u)\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)}\le \|\bar g  \|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)}  + \aa \|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)},\\
 & \|g_{s}(\cdot, u_{s}, \nabla} \def\pp{\partial} \def\E{\mathbb E u_{s},\nabla} \def\pp{\partial} \def\E{\mathbb E^2u_{s})-g_s(\cdot,0,0,\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u_s)\|_\infty \le C  (\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+\|u_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}) \{\log(\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+ \|u_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1})\}^\theta,\end{align*}
 we find a constant $c_1>0$ such that
\beq\label{*PP} \begin} \def\beq{\begin{equation}}  \def\F{\scr F{split}& \|u_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+ \|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)}
 \le c_1   +c\aa \|\nabla} \def\pp{\partial} \def\E{\mathbb E^2 u\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)} \\
 &\quad + c_1  \int_0^t (t-s)^{-\ff 1 2} (\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+\|u_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}) \{\log(\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+ \|u_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1})\}^\theta \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\ \ t\in [0,T_n].\end{split} \end{equation}
Since  H\"older's inequality implies
 \begin} \def\beq{\begin{equation}}  \def\F{\scr F{align*} &\int_0^t (t-s)^{-\ff 1 2}  (\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+\|u_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}) \{\log(\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+ \|u_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1})\}^\theta \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s \\
&\le
\bigg(\int_0^t (t-s)^{-\ff 1 {2(1-\theta)}}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\bigg)^{ 1-\theta}\bigg(\int_0^t  (\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+\|u_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1})^{\theta^{-1}}\log  (\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+ \|u_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1})    \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\bigg)^\theta,\end{align*}
by  $\theta\in (0,\ff 1 2)$, when $\aa\in (0,\ff 1 c)$  we find a constant $c_2>0$ such that \eqref{*PP} implies
\beq\label{*PP0}\begin} \def\beq{\begin{equation}}  \def\F{\scr F{split} &(\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+\|u_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1}+\|u\|_{\tilde}\def\[{\lfloor} \def\]{\rfloor L_{q_0}^{p_0}(0,t)})^{\theta^{-1}} \\
&\le c_2+ c_2 \int_0^t (\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+\|u_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1})^{\theta^{-1}}   \log(\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+ \|u_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1})   \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r,\ \ t\in [0, T_n], n\ge 1.\end{split}\end{equation}
By Gronwall's inequality, we obtain
$$(\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+\|u_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1})^{\theta^{-1}} \le c_2\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{c_2\int_0^t \log(\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+ \|u_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1})   \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r},\ \ t\in [0, T_n], n\ge 1.$$
Thus, there exists a constant $c_3>0$ such that
$$\log (\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+ \|u_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1})\le c_3+ c_3\int_0^t \log(\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+ \|u_s\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1})   \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r,\ \ t\in [0,T_n], n\ge 1.$$
Using Gronwall's inequality again, we derive
$$\log (\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega+ \|u_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1})\le c_3\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega}  \def\oo{\omega^{c_3  T},\ \ t\in [0,  T_n], n\ge 1.$$
Therefore, there exists a constant $c_4>0$ such that
\beq\label{AD} \sup_{t\in [0,  T_n]}  \|u_t\|_{\scr C}      \def\aaa{\mathbf{r}}     \def\r{r_b^1} \le c_4,\ \ n\ge 1.\end{equation}
By \eqref{TN},  if $T_n<T$ then this implies
$n\le c_4$. Thus, $T_n\ge T$ for $n> c_4$. Hence  $T^*=T_n=T$  for large $n\ge 1$, so that \eqref{NN} follows from \eqref{*PP0} and \eqref{AD}.

\end{proof}

{\footnotesize

redpajama_set_name: RedPajamaArXiv