M = \E_{\Upsilon} \XX = \frac{\E \Upsilon(\xi) \XX}{\E \Upsilon(\xi) },
\]
where $\Upsilon(\xi) = \OO_{\bfs \beta}\{ a \cdot \xi - a \cdot \infty \}$. Thus $M$ depends on the marked point $\xi$, the marked points in $\bfs \beta$, the nodes of $\XX$, and the modular parameters $z_k^l$, $z_k^r$ of the chordal standard domain. Denote by $\bfs z$ the modular parameters and $\bfs q$ the marked points of $\bfs \beta$. Define a function $m(\xi, \bfs z, \bfs q, t)$ by
\[
m(\xi, \bfs z, \bfs q, t) :=(R_{\xi, \bfs z, \bfs q} \, \| \, g_t^{-1}), \qquad R_{\xi, \bfs z, \bfs q} = \E_{\Upsilon} \XX.
\]
We need to prove that $M_t := m(\xi_t, \bfs z(t), \bfs q(t), t)$ is a martingale, where $\xi_t$ is the solution to the stochastic differential equation defined by \eqref{driving SDE} and \eqref{drift function}. By Ito's formula and \eqref{drift function} we have
\begin{align*}
d M_t &= (\pa_{\xi} m)(\xi_t, \bfs z(t), \bfs q(t), t) \, d \xi_t + \frac{1}{2} (\pa_{\xi}^2 m)(\xi_t, \bfs z(t), \bfs q(t), t) \, d \langle \xi \rangle_t + \frac{\pa m}{\pa t}(\xi_t, \bfs q(t),t) \, dt
\\
&\quad + \sum_k \left(\dot z_k(t) \pa_{z_k}m(\xi_t, \bfs z(t), \bfs q(t), t) + \overline{\dot z_k(t)} \bp_{z_k}m(\xi_t, \bfs z(t), \bfs q(t), t) \right) \, dt
\\
&\quad + \sum_k \dot q_k(t) \pa_{q_k} m(\xi_t, \bfs z(t), \bfs q(t), t) \\
&= \sqrt{\kappa} (\pa_{\xi} m) \, dB_t + \left( \Big(\kappa \, \pa_{\xi} \log \E \Upsilon(\xi) - r_0(\xi) \Big) \pa_{\xi} m + \frac{\kappa}{2} \pa_{\xi}^2 m + \frac{\pa m}{\pa t} -\nabla_{ \mathbb{H}_g, \bfs q } m \right) dt.
\end{align*}
The drift term vanishes by the BPZ-Cardy equations, we will now explain why. By definition of $m$ the time derivative $\pa m/\pa t$ is given by
\[
\frac{\pa m}{\pa t}(\xi, \bfs z, \bfs q, t) = \left. \frac{d}{ds} \right|_{s=0} \E \XX \, \| \, g_{t+s}^{-1}.
\]
Let $f_{s,t} = g_{t+s} \circ g_t^{-1}$. Then $\pa_s f_{s,t} = -v_{\xi_{t+s}} \circ f_{s,t}$ by the Loewner equation, with $f_{0,t} = \id$. We deduce that
\[
f_{s,t} = \id - s v_{\xi_t} + o(s), \quad s \downarrow 0.
\]
Since $g_{t+s} = f_{s,t} \circ g_t$, the definition of the Lie derivative gives
\[
\left. \frac{d}{ds} \right|_{s=0} \E \XX \, \| \, g_{t+s}^{-1}  = \left. \frac{d}{ds} \right|_{s=0} \E \XX \, \| \, g_{t}^{-1} \circ f_{s,t}^{-1}  = -\check{\LL}_{v_{\xi}} m \, \| \, g_t^{-1}.
\]
Taken together and invoking the BPZ-Cardy equations \eqref{BPZ-Cardy} (with $\xi$ on the real line), the last three calculations show that the drift term of $d M_t$ vanishes.
\end{proof}

We conclude with some example of martingale observables.
We first discuss the bosonic observable for these SLE systems, i.e. the one-point function of the modified Gaussian field $\Phi_{\bfs \beta}$. Here the modification comes in two different forms: one in the background charge $\bfs \beta$ and another through the OPE exponential $\Upsilon(\xi)$. This modification affects both the structure of the martingale observable and the associated SLE system. It is the analogue of the Schramm-Sheffield observable for chordal SLE$(\kappa, \bfs \rho)$ systems in simply connected domains.

\begin{example}{Bosonic observable}
The simplest one-point observable is obtained by taking $X = \Phi_{\bfs \beta}(z)$, typically for $z$ in the interior of the chordal standard domain. Recall that $\bfs \beta = \sum_k \beta_k \cdot q_k$ for $q_k \in \R$ and the $\beta_k$ satisfy the neutrality condition $\sum \beta_k = b \chi$. Here, we have
\Upsilon(\xi) = \OO_{\bfs \beta}\{a \cdot \xi - a \cdot \infty, \bfs 0 \}.
Then an exercise in Wick's calculus shows that
\[
\E_{\Upsilon} \Phi_{\bfs \beta}(z) = \varphi_{\bfs \beta + a \cdot \xi - a \cdot \infty}(z).
\]
As a conformal field, the latter depends on the marked points $q_k$, $\xi$, and $z$. It is a $[0,0]$-differential in all of the $q$-variables and a PPS$(ib, -ib)$ form in the $z$ variable. By Theorem \ref{Thm_SLE MO}, it is a martingale observable for the SLE process with driving function
\[
d \xi_t = \sqrt{\kappa} \, d B_t + \Lambda_{D_t}(\xi_t) \, dt,
\]
with the drift function $\Lambda(\xi)$ given by
\[
\Lambda(\xi) := \kappa \pa_{\xi} \log \E \Upsilon(\xi) - r_0(\xi).
\]
Here $D_t = g_t(D_0 \backslash \gamma[0,t])$, where $\gamma$ is the SLE curve. Thus $D_t$ is yet another chordal standard domain, and the field $\Lambda_{D_t}$ is evaluated in this domain. The nature of the OPE exponential $\Upsilon(\xi)$ makes it difficult to give an exact expression for the term $\E \Upsilon(\xi)$, even in the standard coordinate chart. It only depends on the divisor $\bfs \beta$ and the marked point $\xi$. Similarly, the conformal field $\E \Upsilon \Phi_{\bfs \beta}(z)$ only depends on these divisors and $\xi$, along with the point $z \in D_0$. The coordinate chart $g_t^{-1}$ is well-defined around these marked points, so by evaluating the field $\E_{\Upsilon} \Phi_{\bfs \beta}(z)$ along the charts $g_t^{-1}$ the martingale in coordinates is
\[
t \mapsto -2b \arg g_t'(z) + 2 \sum_k \beta_k \Im \, G_D^+(g_t(q_k), g_t(z))
\]
by \eqref{Phi beta}. For $z$ on the real line (but distinct from the $q_k$ and $\xi$) it can be shown that $\E_{\Upsilon} \Phi_{\bfs \beta}(z)$ is piecewise constant.
\end{example}

In the multiply connected setting it is very difficult to provide any explicit formulas for the martingales, other than in terms of complicated special functions. Nonetheless we can see interesting functional relations arise between the correlation functions, even without explicit expressions available for them.

\begin{example}{A functional relation for a sequence of correlation functions derived from $T_{\bfs \beta}$}
In the introduction we argued that CFT can be used to derive functional equations between correlation functions. Via the Virasoro field $T_{\bfs \beta}$ we can use the Eguchi-Ooguri form of Ward's equation to derive a recursive sequence of correlation functions. For the setup, let $D$ be a chordal standard domain and fix a sequence of distinct points $z_1, z_2, \ldots \in D \cup \R$. Fix a background charge $\bfs \beta = \sum_k \beta_k \cdot q_k$ where one has $q_k \in \R$ and the $\beta_k$ satisfy the neutrality condition $\sum \beta_k = b \chi$. Assume that the $q_k$ are all distinct from the $z_j$. We will consider the conformal fields
\[
V_n = \E \Upsilon(\xi) \XX_n, \qquad \XX_n = T_{\bfs \beta}(z_1) \ldots T_{\bfs \beta}(z_n),
\]
where in this case $\Upsilon(\xi) = \OO_{\bfs \beta}\{a \cdot \xi - a \cdot \infty, \bfs 0 \}$ and we recall that $a$ is a solution to $2a(a+b) = 1$. Here $\xi$ is distinct from the $q_k$ and $z_j$. As a (non-random) conformal field, $V_n$ is
\begin{itemize}
\item a $[0,0]$-differential at the $q_k$, \smallskip
\item a Schwarzian form of order $c/12 = 1/12 - b^2$ at the $z_j$, $j=1, \ldots, n$, \smallskip
\item and an $[h_{1,2}, 0]$-differential at $\xi$, where $h_{1,2} = a^2/2 - ab$.
\end{itemize}
It will also transform as a differential at infinity, but in these calculations we only consider conformal maps that fix infinity and have the hydrodynamic normalization there, so the transformation rule at infinity does not enter in.
More precisely, the Loewner vector field has a triple zero at infinity and the Lie derivative at infinity in Ward's equation does not appear in the identity chart of a chordal standard multiply connected domain.
Note the sequence $\XX_{n+1}$ is built up by successive multiplications by the stress tensor $T_{\bfs \beta}$, i.e. there is the recursive relation $\XX_{n+1} = T_{\bfs \beta}(z_{n+1}) \XX_{n}$. At the same time, the Ward equation relates multiplication by the stress tensor to the Lie derivative. By the Ward equation \eqref{eq: Ward_no_integral}, and recalling that $A_{\bfs \beta} = T_{\bfs \beta} - \E\,T_{\bfs \beta}$, we obtain the recursive equation
\begin{align}\label{eq: example_V_recursion}
2 V_{n+1} = 2 V_n \E\,T_{\bfs \beta}(z_{n+1}) + (\LL_{v_{z_{n+1} }}^+ + \nabla_{ \mathbb{H}_g, \bfs q }) V_n.
\end{align}
In a simply connected domain, this recursion formula coincides with that for an avoiding probability of the SLE$(8/3)$ curve (in the $c=0$ case), see \cite[Section 14.5]{KM13} and [25]. This is closely related to the restriction property and we refer to [1] for a related recent work in a multiply connected domain.
In principle, \eqref{eq: Virasoro_recursion} can be solved given the one-point function $\E\,T_{\bfs \beta}(z)$ and an initial value $V_0 = \E \Upsilon(\xi) $.
Although $V_n$ is not a martingale observable, Theorem \ref{Thm_SLE MO} states that $M_n := V_n/\E \Upsilon(\xi)$ is a martingale observable for an SLE process, when evaluated on the random charts $g_t^{-1}$. By dividing through \eqref{eq: example_V_recursion} by $\E \Upsilon(\xi)$ and using the Leibniz rule we obtain
\begin{align}
\begin{split} \label{eq: example_M_recursion}
2 M_{n+1} &= 2 M_n \E \, T_{\bfs \beta}(z_{n+1}) + (\LL_{v_{z_{n+1} }}^+ + \nabla_{\mathbb{H}_{g}, \bfs q }) M_n + \frac{M_n}{\E \Upsilon(\xi)} (\LL_{v_{z_{n+1} }}^+  + \nabla_{\mathbb{H}_{g},\bfs q}) \E \Upsilon(\xi).
\end{split}
\end{align}
In the higher genus case that we consider both will be expressed in terms of complicated special functions, which makes it difficult to find a closed form solution to the recursive equation. The transformation rules for $V_n$ allow us to compute the Lie derivative term as
\begin{align*}
\LL_{v_{z_{n+1}}}^+ V_n & = v_{ z_{n+1} }(\xi) \pa_{\xi} V_n + h_{1,2} v_{ z_{n+1} }'(\xi) V_n %+ \sum_k v_{ z_{n+1} }(q_k) \pa_{q_k} V_n
+ \sum_{j=1}^n \left[ v_{ z_{n+1} }(z_j) \pa_{z_j} + 2 v_{ z_{n+1} }'(z_j) \right] V_n + \frac{c}{12} \sum_{j=1}^n v_{ z_{n+1} }'''(z_j) V_{n,j},
\end{align*}
where $V_{n,j} = \E \Upsilon(\xi) \XX_{n,j}$ and $\XX_{n,j} = T_{\bfs \beta}(z_1) \ldots \hat{T}_{\bfs \beta}(z_j) \ldots T_{\bfs \beta}(z_n)$. Here, the hat indicates that term is left out of the product.
Combining the above, it follows that
\begin{align} \label{eq: Virasoro_recursion}
\begin{split}
2 V_{n+1} & = \bigg[  2  \E\,T_{\bfs \beta}(z_{n+1})  +   \sum_j \Big( v_{ z_{n+1} }(z_j) \pa_{z_j} + 2 v_{ z_{n+1} }'(z_j) \Big)  + \mathfrak{D}_{n}[\xi,\bfs q]  \bigg] V_n
+ \frac{c}{12} \sum_j v_{ z_{n+1} }'''(z_j) V_{n,j},
\end{split}
\end{align}
\begin{align*}
\mathfrak{D}_{n}[\xi,\bfs q] &:=    h_{1,2} v_{ z_{n+1} }'(\xi) +  v_{ z_{n+1} }(\xi) \pa_{\xi}    + \nabla_{ \mathbb{H}_g, \bfs q }.
\end{align*}
After normalizing \eqref{eq: Virasoro_recursion} the sequence $M_n$ satisfies
\begin{align}
\begin{split}
2 M_{n+1} &  = \bigg[  2  \E\,T_{\bfs \beta}(z_{n+1})   +   \sum_j \Big( v_{ z_{n+1} }(z_j) \pa_{z_j} + 2 v_{ z_{n+1} }'(z_j) \Big)   + \mathfrak{D}_{n}[\xi,\bfs q]  \bigg] M_n \\
&\quad + \frac{c}{12} \sum_j v_{ z_{n+1} }'''(z_j) M_{n,j} +  \frac{\mathfrak{D}_{n}[\xi,\bfs q] \E \Upsilon(\xi) }{ \E \Upsilon(\xi) }  M_n,
\end{split}
\end{align}
where $M_{n,j} = V_{n,j} / \E \Upsilon(\xi)$. This formula agrees with \eqref{eq: example_M_recursion} since each $M_n$ is a $[0,0]$-differential at the $q_k$ and at $\xi$, and a Schwarzian form of order $c/12$ at $z_1, \ldots, z_n$, whereas $\E \Upsilon(\xi)$ is an $[h_{1,2},0]$-differential at $\xi.$
\end{example}

\begin{comment}

Finally, we conclude with a martingale observable coming from the OPE exponentials $\Upsilon(\xi)$.

\begin{example}{An OPE exponential as a martingale observable}
Fix $b \in \R$ and $x, y \in \R$ and consider the divisor
\[
\bfs \beta = \tfrac{1}{2} b \chi \cdot x + \tfrac{1}{2} b \chi \cdot y
\]
Also choose distinct points $\xi_1, \xi_2  \in \R$ and let
\[
\Upsilon_1(\xi_1) = \OO_{\bfs \beta}[a \cdot \xi_1 - a \cdot \infty, \bfs 0], \quad \Upsilon_2(\xi_2) = \OO_{\bfs \beta}[a \cdot \xi_2 - a \cdot \infty, \bfs 0].
\]
Since $\Upsilon_i \in \FF_{\bfs \beta}$ for $i=1,2$, Theorem \ref{Thm_SLE MO} implies that the conformal field
\[
\E_{\Upsilon_1} \Upsilon_2
\]
is a martingale observable for the SLE process with driving function
\[
d \xi_{1,t} = \sqrt{\kappa} \, d B_{1,t} + \Lambda_{1, D_t}(\xi_t) \, dt
\]
As usual $\kappa$ is determined from $b$ via $b = \sqrt{\kappa/8} - \sqrt{2/\kappa}$, The drift function $\Lambda_1(\xi_1)$ is the conformal field given by
\[
\Lambda_1(\xi_1) := \kappa \partial_{\xi_1} \log \E \Upsilon_1(\xi_1) - r_0(\xi_1)
\]
Again $D_t = g_t(D_0 \backslash \gamma[0,t])$ where $\gamma$ is the SLE curve. In this case we can given an explicit formula for the martingale observable and for the drift function, which we will do shortly. Before we do, however, observe that Theorem \ref{Thm_SLE MO} \textit{also} implies that the conformal field
\[
\E_{\Upsilon_2} \Upsilon_1
\]
is a martingale observable for an SLE process, but one in which all the ``1'' indices are replaced by ``2''. This is because both $\Upsilon_1(\xi_1)$ and $\Upsilon_2(\xi_2)$ are valid OPE exponentials in the statement of Theorem \ref{Thm_SLE MO}, and both are in the OPE family $\FF_{\bfs \beta}$.
\end{example}

\end{comment}

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