\section*{A signed-measure problem for reflected Brownian motion}

Let
\[
\mathbb{R}_+^d=[0,\infty)^d,
\qquad
F_i=\{x\in\mathbb{R}_+^d:x_i=0\},
\qquad i=1,\ldots,d.
\]
Fix a drift vector \(\mu\in\mathbb{R}^d\), a positive-definite covariance matrix
\(\Sigma\in\mathbb{R}^{d\times d}\), and a reflection matrix
\(R\in\mathbb{R}^{d\times d}\). Denote the \(i\)th column of \(R\) by \(R_i\).

A matrix \(R\) is called an \emph{\(S\)-matrix} if there exists
\(w\in\mathbb{R}_+^d\) such that \(Rw>0\). It is called \emph{completely-\(S\)}
if each of its principal submatrices is an \(S\)-matrix. Throughout, assume
that \(R\) is completely-\(S\).

An \((\Sigma,\mu,R)\)-semimartingale reflecting Brownian motion (SRBM) in
\(\mathbb{R}_+^d\) is a process \(Z\) admitting the representation
\[
Z(t)=X(t)+RY(t),\qquad t\geq 0,
\]
where \(X\) is a Brownian motion with drift \(\mu\) and covariance matrix
\(\Sigma\), and \(Y=(Y_1,\ldots,Y_d)\) is a continuous, coordinatewise
nondecreasing process satisfying
\[
Y(0)=0,
\qquad
\int_0^\infty \mathbf 1_{\{Z_i(t)>0\}}\,dY_i(t)=0,
\qquad i=1,\ldots,d,
\]
and
\[
Z(t)\in\mathbb{R}_+^d,\qquad t\geq 0.
\]
Thus \(Y_i\) can increase only when \(Z\) lies on the face \(F_i\), and the
corresponding reflection direction is \(R_i\).

Assume that \(Z\) has a stationary probability distribution, denoted by
\(\pi_*\). For sufficiently smooth \(f\), define
\[
Lf(x)
=
\frac12
\sum_{j,k=1}^d
\Sigma_{jk}
\frac{\partial^2 f}{\partial x_j\partial x_k}(x)
+
\sum_{j=1}^d
\mu_j
\frac{\partial f}{\partial x_j}(x),
\]
and
\[
D_i f(x)
=
R_i\cdot\nabla f(x)
=
\sum_{j=1}^d
R_{ji}
\frac{\partial f}{\partial x_j}(x),
\qquad i=1,\ldots,d.
\]

A tuple of finite measures
\[
(\pi,\nu_1,\ldots,\nu_d)
\]
on \(\mathbb{R}_+^d\), with \(\nu_i\) supported on \(F_i\), is said to satisfy
the \emph{basic adjoint relationship} (BAR) if
\[
\int_{\mathbb{R}_+^d} Lf(x)\,\pi(dx)
+
\sum_{i=1}^d
\int_{F_i} D_i f(x)\,\nu_i(dx)
=0
\]
for every \(f\in C_c^\infty(\mathbb{R}^d)\).

By Ito formula, the stationary distribution \(\pi_*\) of \(Z(t)\), together
with the boundary occupation measures
\[
\nu_i^*(A)
=
\mathbb E_{\pi_*}
\left[
\int_0^1
\mathbf 1_{\{Z(t)\in A\}}\,dY_i(t)
\right],
\qquad i=1,\ldots,d,
\]
satisfies the BAR.

\begin{proposition}[Known BAR characterization]
Let \(\pi,\nu_1,\ldots,\nu_d\) be finite nonnegative measures on
\(\mathbb{R}_+^d\), and suppose that \(\nu_i\) is supported on \(F_i\) for each
\(i=1,\ldots,d\). If \((\pi,\nu_1,\ldots,\nu_d)\) satisfies the BAR, then
\[
\pi=c\,\pi_*
\]
for some constant \(c\geq 0\).
\end{proposition}

\begin{problem}[Signed BAR characterization]
Let \(\pi,\nu_1,\ldots,\nu_d\) be finite signed measures on
\(\mathbb{R}_+^d\), with \(\nu_i\) supported on \(F_i\) for each
\(i=1,\ldots,d\). Suppose that
\[
\int_{\mathbb{R}_+^d} Lf(x)\,\pi(dx)
+
\sum_{i=1}^d
\int_{F_i} D_i f(x)\,\nu_i(dx)
=0
\]
for every \(f\in C_c^\infty(\mathbb{R}^d)\). Prove that there exists a
constant \(c\in\mathbb{R}\) such that
\[
\pi=c\,\pi_*.
\]
\end{problem}