text:
\section{Introduction}
\label{intro}

Heisenberg's celebrated uncertainty principle in its fundamental form does not take into account effects which are expected to arise when gravity is being accounted for. In fact, one of the low-energy implications associated with several candidate theories of quantum gravity predicts a correction to the aforementioned relation going by the name Generalized Uncertainty Principle (GUP) \cite{Amati87,Kempf95,Maggiore93,Adler01,GUPReview,AdlerDuality}. After seminal efforts have been devoted to this topic, the investigation revolving around the generalization of the fundamental quantum mechanical uncertainty has developed remarkably. As a matter of fact, the consequences entailed by the GUP deeply affects several aspects of black hole thermodynamics \cite{Sparsity,noncom,noncom2,noncom3,bis,ter,ter2,quar,ses,ong,set,ot,nov,dieci}, quantum field theory \cite{noui,frassino,qft,qft2,qft3,qft4,czech,czech2,pikovski,plenio,plenio2} and also quantum information \cite{dasprl,dasdecoherence,decoherence}.

As usually assumed, the GUP takes into account the gravitational uncertainty of position in connection with the existence of a minimal fundamental length scale in physics. However, it may not be the only gravitationally induced change. In particular, the curvature of space-time does exert an influence over quantum mechanical uncertainty relations. This is the regime of the Extended Uncertainty Principle (EUP) \cite{Mignemi2010,Anrade,Mureika,EUPThermod,Bambi2008}, whose purpose lies in the investigation of the uncertainty related to the background space-time.

Recently, a derivation of the EUP was performed in \cite{Schuermann2009,Schuermann2018}. It reflects the influence of spatial curvature on quantum-mechanics on 3-dimensional spacelike hypersurfaces of space-time. The method was applied to homogeneous and isotropic geometries of constant curvature $K$ and the corresponding EUP was calculated. This formalism was further developed in collaboration with one of the authors to accommodate for horizons \cite{RF2019} and arbitrary 3-dimensional manifolds, thus yielding an asymptotic extended uncertainty principle \cite{RF2020}. In short, the quantum mechanical wave function of interest is defined over a curved three-dimensional manifold and its domain constrained to a geodesic ball. Then, the radius of this ball serves as a diffeomorphism-invariant measure of position uncertainty. By computing the standard deviation of the momentum operator, one finds that there is a generic minimum which depends on the radius of the geodesic ball and on curvature invariants derived from the effective background. This represents a mathematically rigorous as well as physically motivated formulation of the EUP.

This paper goes further inasmuch it aims to provide the uncertainty relation experienced by a non-relativistic particle in a curved 4-dimensional background. To comply with this goal, we obtain the effective Lagrangian according to which the particle dynamics evolves. As is well-known in the literature \cite{gravitomagnetism}, the gravitational field splits up into a three-dimensional background metric, a gravitomagnetic vector field and a gravitoelectric scalar field. Due to the resulting gauge invariance which is not shared by the conjugated momentum, the physical momentum includes a contribution from the vector field. Adding these ingredients to the picture outlined above and taking into account subtleties of the quantization as first pointed out in \cite{DeWitt:1952js}, we finally achieve the desired result. This makes it possible to compare the impact of distinct theories of gravity (which describe different backgrounds) on the uncertainty relation.

The paper is organized as follows: in section \ref{sec: effham} we derive the effective Hamiltonian of a non-relativistic particle on a curved 4D background. This theory is quantized in section \ref{sec:QMtreat}, where we define the momentum uncertainty and the relevant operators whose expectation values appear therein and compute the curvature induced corrections to the uncertainty relation. The corresponding results are applied to several relevant space-times in section \ref{sec:app} to be subsequently discussed in section \ref{sec:disc}.

\section{Derivation of the effective Hamiltonian\label{sec: effham}}

In this section, we will derive the effective dynamics of a non-relativistic particle on a curved 4-dimensional background. To this aim, note that the action of a massive relativistic particle subjected to a curved geometry can be written as
\begin{align}
S	&=-m\int \D s\\
	&=-m\int \sqrt{-g_{\mu\nu}(x)\dot{x}^\mu\dot{x}^\nu} \D \tau\label{relaction}
\end{align}
with the background metric $g_{\mu\nu}$ (the Greek indices stand for space-time coordinates while Latin ones denote the spatial part), the four-velocity $\dot{x}^\mu=\D x^\mu/\D\tau$ ($\tau$ denotes the proper time along the curve) and the mass of the particle $m.$ Thus, the corresponding Lagrangian reads
\begin{equation}
L=-m\sqrt{-g_{\mu\nu}(x)\dot{x}^\mu \dot{x}^\nu}.\label{lagrangian2}
\end{equation}
Note that the action \eqref{relaction} is invariant under temporal reparametrisations $\tau'=f(\tau)$ for any sufficiently well-behaved function $f.$

After some algebra, the Lagrangian can be recast as
\begin{align}
L=  &-m\Big[\left(-g_{00}+g_{0i}g_{0j}h^{ij}\right)(\dot{x}^0)^2\nonumber\\
    &-\left(\dot{x}^0g_{0k}h^{ik}+\dot{x}^i\right)\left(\dot{x}^0g_{0l}h^{jl}+\dot{x}^j\right)g_{ij}\Big]^{1/2}
\end{align}
with $h^{ik}g_{kj}\equiv\delta^i_j.$ Thus, $h^{ij}$ is the inverse of the induced metric on hypersurfaces of constant $x^0.$ The Lagrangian can be further simplified by introducing the background field quantities
\begin{align}
    N=&\sqrt{-g_{00}+g_{0i}g_{0j}h^{ij}}\\
    N^i=&g_{0j}h^{ij}
\end{align}
which are readily identified as the lapse function and the shift vector in the 3+1 formalism \cite{ADM}. According to this approach to curved Lorentzian manifolds, any metric can be written as \cite{eric}
\begin{equation}
    \D s^2=-N^2\left(\D x^0\right)^2+h_{ij}(N^i\D x^0+\D x^i)(N^j\D x^0+\D x^j)
\end{equation}
with $h_{ij}\equiv g_{ij}.$ Thence, the Lagrangian reads
\begin{equation}
    L=-m\sqrt{N^2\left(\dot{x}^0\right)^2-\left(\dot{x}^0N^i+\dot{x}^i\right)\left(\dot{x}^0N^j+\dot{x}^j\right)h_{ij}}
\end{equation}
which under the assumption that $\dot{x}^0>0$ (\ie that coordinate time is moving in the same direction as the particle proper time) can be written as
\begin{align}
     L=&-mN\dot{x}^0\sqrt{1-\frac{\left(\dot{x}^0N^i+\dot{x}^i\right)\left(\dot{x}^0N^j+\dot{x}^j\right)h_{ij}}{N^2\left(\dot{x}^0\right)^2}}\\
      \equiv&-mN\dot{x}^0\sqrt{1-\epsilon}
\end{align}
where the last equality defines $\epsilon.$ In terms of the conjugate momenta $P_\mu=mg_{\mu\nu}\dot{x}^\nu/\sqrt{-g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu}$, we find $\epsilon/(1-\epsilon)=h^{ij}P_iP_j/m^2$, which means that the non-relativistic limit corresponds to $\epsilon\ll 1.$ Therefore, we can expand to obtain the effective non-relativistic Lagrangian
\begin{equation}
    L_{NR}=\frac{m}{2\dot{x}^0}\left(\dot{x}^0N^i+\dot{x}^i\right)\left(\dot{x}^0N^j+\dot{x}^j\right)G_{ij}-mN\dot{x}^0
\end{equation}
with the effective 3-metric $G_{ij}=h_{ij}/N.$ From this point onwards, this metric will be used to lower and raise indices and as the background for differential geometric quantities.

The effective non-relativistic action $S_{NR}=\int L_{NR}\D\tau$ still harbours the time reparametrization invariance alluded to above. For simplicity, we can fix the gauge by choosing $x^0=\tau$ to obtain the gauge-fixed non-relativistic Lagrangian
\begin{equation}
    L_{NR}=\frac{m}{2}\left(N^i+\dot{x}^i\right)\left(N^j+\dot{x}^j\right)G_{ij}-mN.
\end{equation}
A closer look at the Lagrangian tells us that it is of the form
\begin{equation}
    L_{NR}=\frac{m}{2}\dot{x}^i\dot{x}^jG_{ij}+m\dot{x}^iA_i-m\phi
\end{equation}
with $A_i=N^jG_{ij}$ and $\phi=N-N^iN^jG_{ij}/2.$ This is clearly reminiscent of the Lagrangian describing a charged non-relativistic particle minimally coupled to an electromagnetic gauge one-form $A_\mu=(\phi,A_i)$ where the mass $m$ plays the rôle of the charge.

On the other hand, the Lagrangian is additionally invariant under the gauge transformation $A\rightarrow A_i+\partial_i f,$ $\phi\rightarrow\phi-\dot{f},$ $G^{ij}\rightarrow G^{ij}$ for any scalar function $f(x^i,t)$ while the canonical momenta
\begin{equation}
    \pi_i=\frac{\partial L_{NR}}{\partial \dot{x}^i}=mG_{ij}\left(\dot{x}^j+N^j\right)
\end{equation}
are not, rendering them unobservable. Therefore, we define the gauge invariant physical momenta as
\begin{equation}
    p_i\equiv\pi_i-mN^jG_{ij}\label{phys}
\end{equation}
in terms of which the Hamiltonian reads
\begin{equation}
    H_{NR}=\frac{1}{2m}p_ip_jG^{ij}+m\phi.\label{effhamiltonian}
\end{equation}
Having found the effective Hamiltonian, we are now able to give a quantum mechanical description of non-relativistic particles in curved space-time.

\section{Quantum Mechanical treatment\label{sec:QMtreat}}

Aiming towards a generalization of the reasoning in \cite{RF2020} to four dimensional backgrounds, we have to find the quantum mechanical counterpart of the theory outlined in the previous section. For this purpose, we first construct an equivalent Hilbert space and introduce the basis in terms of which calculations will be performed. As we pointed out above, the effective Hamiltonian \eqref{effhamiltonian} entails a certain amount of gauge freedom which makes it necessary to define the gauge invariant (\ie physical) momenta $p_i.$ Moreover, we obtain the quantum operators representing their co- and contravariant versions and define the standard deviation accordingly. To conclude the section, we derive the flat space uncertainty relation and its higher-order corrections.

\subsection{Hilbert space and used basis}

In order to construct the Hilbert space, we have to find an appropriate measure defining the scalar product $\braket{\psi|\phi}=\int\D\mu\psi^*\phi.$ The problem at hand takes place in a 3-dimensional curved background defined by the canonical connection with respect to $G_{ij}$ and offers the gauge freedom mentioned above. These facts should be reflected by the measure making it gauge and diffeomorphism invariant. The only possible differential satisfying these constraints reads $\D\mu=\sqrt{G}\D^3x$ where $G=\text{det}G_{ij}.$

Furthermore, we confine the quantum states to a geodesic ball $B_\rho$ of radius $\rho$ by imposing Dirichlet boundary conditions as in \cite{RF2020}; hence, they reside in the Hilbert space $\mathcal{H}=L^2(B_{\rho}\subseteq {\rm I\!R}^3,\sqrt{G}\D^3 x).$ As the center of the geodesic ball $p_0$ is situated in the background manifold, it represents the expectation value of the position operator while $\rho$ can be interpreted as a measure of position uncertainty.

Since the negative Laplace-Beltrami operator $-\Delta$ is hermitian in $\mathcal{H},$ its eigenstates furnish an orthonormal basis of the Hilbert space. Due to the compactness of the domain of $\mathcal{H}$, the spectrum of said operator is discrete. Thus, there is a countably infinite number of base vectors satisfying the eigenvalue problem
\begin{equation}
(\Delta +\lambda_{nlm})\psi_{nlm}=0\hspace{1cm}\psi_{nlm}|_{\partial B_\rho}=0 \label{evprob}
\end{equation}
where $n,l,m$ stand for the three quantum numbers identifying the state.

In the case of a flat background, the solutions to the problem \eqref{evprob} can be obtained analytically and read in normalized form
\begin{equation}
    \psi_{nlm}=\sqrt{\frac{2}{\rho^3 j^2_{l+1}(j_{l,n})}}j_l\left(x_{nl}\frac{\sigma}{\rho}\right)Y^m_l(\chi,\gamma)\label{psinlm}
\end{equation}
with the spherical Bessel function $j_l,$ the spherical harmonics $Y^m_l$ and the pure number $j_{l,n}$ which is the nth solution of the equation $j_l(j_{l,n})=0$ or, in other words, the nth zero of the Bessel function $J_{l+1/2}.$ The corresponding eigenvalues are
\begin{equation}
    \lambda_{nl}=\frac{j_{l,n}^2}{\rho^2}.\label{flatev}
\end{equation}

\subsection{Physical momentum operator}

When promoting positions and momenta to quantum operators, we impose the canonical commutation relations $[\hat{x}^i,\hat{p}_j]=i\hbar\delta^i_j.$ In the position space representation, these are solved by the operators
\begin{equation}
\hat{x}^i\psi=x^i\psi\hspace{1cm}\hat{p}_i\psi=\left(-i\hbar\partial_i+F_i(x)\right)\psi
\end{equation}
where $\psi$ describes a general wave function. The arbitrary form $F_i(x)$ can be found imposing that the momentum operator be hermitian with respect to the Hilbert space measure $\D\mu$, \ie
\begin{align}
\braket{\psi|\hat{p}_i\phi}=\braket{\hat{p}_i\psi|\phi}
\end{align}
and aptly represents the physical momentum in the classical limit.

It can be shown that these criteria are uniquely satisfied by the momentum operator
\begin{align}
\hat{p}_i\psi	=&\left(\hat{\pi}_i-m\hat{N}^j\hat{G}_{ij}\right)\psi\\
					\equiv&-\left[i\hbar\left(\partial_i+\frac{1}{2}\Gamma^j_{ij}\right)+mN^jG_{ij}\right]\psi
\end{align}
where the second equality defines $\hat{\pi}_i$ and we introduced the Christoffel symbol
\begin{align}
\Gamma^k_{ij}=\frac{1}{2}G^{kl}\left(\partial_iG_{jk}+\partial_jG_{ik}-\partial_kG_{ij}\right).
\end{align}
Using this result, there is only one possible hermitian definition of the covariant momentum operator reading
\begin{align}
\hat{p}^i\psi	=&\left(\hat{\pi}^i-m\hat{N}^i\right)\psi\\
						\equiv&-\left[\frac{i\hbar}{2} \left(\left\{G^{ij},\partial_j\right\}+G^{ij}\Gamma^k_{kj}\right)+mN^i\right]\psi\\
						=&\frac{1}{2}\left\{\hat{G}^{ij},\hat{p}_j\right\}\psi
\end{align}
where $\hat{\pi}^i$ is defined according to the second equality.

Finally, the square of the momentum operator is of the form
\begin{equation}
\hat{p}^2=\hat{\pi}^2-2m\hat{p}_{mix}^2+m^2\hat{G}_{ij}\hat{N}^i\hat{N}^j
\end{equation}
where $\hat{\pi}^2\psi=-\hbar^2\Delta\psi$ and $\hat{p}^2_{mix}$ mixes canonical momenta and the shift.

Under the assumptions that it is hermitian and leads to the correct classical limit, this operator reads
\begin{align}
\hat{p}^2_{mix}=	&\frac{1}{2}\left\{\hat{\pi}_i,\hat{N}^i\right\}.
\end{align}
After some elementary algebra, it can be seen that it acts on wave functions as
\begin{equation}
\hat{p}^2_{mix}\psi=-\frac{i\hbar}{2}\left[\nabla_i\left(N^i\right)+2N^i\partial_i\right]\psi\label{pmix}
\end{equation}
with the covariant derivative $\nabla_i$ with respect to the canonical connection of $G_{ij}.$

Note that the first term in \eqref{pmix} is anti-hermitian, and since $\hat{p}^2_{mix}$ is hermitian it cancels out the anti-hermitian part of the second term so that we can rewrite $\hat{p}^2_{mix}$ as
\begin{align}
    \hat{p}^2_{mix}=\left(-i\hbar N^i\partial_i\right)_H\psi
\end{align}
where the subscript $H$ denotes the hermitian part.

Summing up the outcome of this subsection, the relevant operators act as
\begin{align}
\hat{p}_i\psi		=&\hbar\left[-i\left(\partial_i+\frac{1}{2}\Gamma^j_{ij}\right)-\frac{G_{ij}N^j}{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}_C}\right]\psi\label{conmom}\\
\hat{p}^i\psi	=&\hbar\left[-\frac{i}{2}\left(\left\{G^{ij},\partial_j\right\}+G^{ij}\Gamma^k_{jk}\right)-\frac{N^i}{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}_C}\right]\psi\label{covmom}\\
\hat{p}^2\psi	=&\hbar^2\left[-\Delta+\frac{2}{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}_C}\left(iN^i\partial_i\right)_H+\frac{N^iN^jG_{ij}}{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}_C^2}\right]\psi\label{squmom}
\end{align}
with the reduced Compton wavelength ${\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}_C=\hbar/m.$

Having figured out the position space representation of the contra-, covariant and squared momentum operators in \eqref{conmom}, \eqref{covmom} and \eqref{squmom} respectively, we can define the momentum uncertainty as
\begin{align}
\sigma_p\equiv\sqrt{\braket{\hat{p}^2}-\braket{\hat{p}^i}\braket{\hat{p}_i}}.\label{uncdef}
\end{align}
The remainder of this paper will be centered around the evaluation of this quantity.

\subsection{Perturbation around flat space}

Following the approach in \cite{RF2020}, we calculate the uncertainty relation perturbatively. In particular, we expand $G_{ij}$ in Riemann normal coordinates (RNC) $x^i$ to second order
\begin{align}
    G_{ij}&\simeq G^{(0)}_{ij}+G^{(2)}_{ij}\\
    &=\delta_{ij}-\frac{1}{3}R_{ikjl}|_{p_0}x^kx^l
\end{align}
with the Kronecker Delta $\delta_{ij}$ and the Riemann curvature tensor with respect to the metric $G_{ab}$ evaluated at $p_0.$ This leads to ensuing expansions in $\nabla_i=\nabla_i^{(0)}+\nabla_i^{(2)},\Delta=\Delta_{(0)}+\Delta_{(2)}$ due to
\begin{align}
    \Gamma^i_{jk}\simeq&\left(\Gamma^i_{ij}\right)^{(2)}\\
    =&\frac{1}{3}\left(R^i_{~jkm}+R^i_{~kjm}\right)|_{p_0}x^m.
\end{align}
Additionally, the measure in expanded form reads $\D\mu=\D\mu^{(0)}+\D\mu^{(2)}$, with $\D\mu^{(0)}=\D^3x$ and
\begin{equation}
    \D\mu^{(2)}=-\frac{1}{6}R_{ij}|_{p_0}x^ix^j\D^3x.
\end{equation}
Correspondingly, the scalar product is perturbed as $\braket{}\simeq\braket{}_0+\braket{}_2.$ How to treat quantum mechanical perturbation theory in this special case has been elaborated upon in \cite{RF2020}. Note that with $\braket{}^{(n)}$ we denote the nth correction to the whole amplitude including wave functions and operators inside, while $\braket{}_n$ signifies the nth correction to the scalar product.

Due to the spherical symmetry of the unperturbed problem, integrals appearing throughout the calculation will be solved in geodesic coordinates $(\sigma,\chi,\gamma)$ which were introduced in \cite{RF2020} and relate to RNC as spherical coordinates to Cartesian ones
\begin{equation}
x^i=\sigma(\sin\chi\cos\gamma,\sin\chi\sin\gamma,\cos\chi).
\end{equation}
As the metric has been expanded in RNC, the same should be done for the shift, which now reads
\begin{align}
    N^i\simeq&N^i_{(0)}+N^i_{(1)}+N^i_{(2)}\\
    =&N^i|_{p_0}+\nabla_jN^i|_{p_0}x^j+\nabla_j\nabla_kN^i|_{p_0}x^jx^k.
\end{align}
This means that, in principle, the shift could yield zeroth and first order corrections.

Finally, the momentum uncertainty is expanded as
\begin{align}
\sigma_p\simeq&\sqrt{\var{p}^{(0)}+\var{p}^{(1)}+\var{p}^{(2)}}\\
        =&\sigma_p^{(0)}\left(1+\frac{\var{p}^{(1)}}{2\var{p}^{(0)}}+\frac{\var{p}^{(2)}-\frac{\var{p}^{(1)}}{4\var{p}^{(0)}}}{2\var{p}^{(0)}}\right)
\end{align}
where we introduced the variance $\sigma_p^2.$ These contributions will be treated order by order.

\subsection{Unperturbed uncertainty relation}

To zeroth order the momentum uncertainty reads
\begin{equation}
    \sigma_p^{(0)}=\sqrt{\var{\pi}^{(0)}+\frac{2\hbar}{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}_C}N^i|_{p_0}\braket{i\hbar\partial_i+\pi_i}^{(0)}+\frac{\hbar^2}{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}^2_C}\var{N}^{(0)}}\label{0thorderunc}
\end{equation}
where we used that $i\partial_i$ is hermitian with respect to the unperturbed measure.

As $\pi_i^{(0)}=-i\hbar\partial_i,$ the term in the brackets vanishes. Moreover, it is a simple exercise to show that a similar cancellation occurs to the variance of the shift leaving us with
\begin{equation}
    \sigma_p^{(0)}=\sigma_\pi^{(0)}.
\end{equation}
Hence, the shift has no influence to zeroth order.

Note that these considerations hold independently of the state with respect to which the uncertainty is calculated. According to the formalism developed in \cite{Schuermann2018} and subsequently in \cite{RF2020}, we obtain an uncertainty relation by finding the state of minimum momentum uncertainty.

Yet, there it was falsely claimed that $\braket{\hat{\pi}^i}=\braket{\hat{\pi}_i}=0$ for arbitrary states in arbitrary backgrounds. This is indeed not the case. Nevertheless, as shown in the appendix, the state of minimal uncertainty is still the ground state for flat backgrounds. In geodesic coordinates, it reads
\begin{equation}
    \psi^{(0)}_{100}=\frac{1}{\sqrt{2\pi\rho}}\frac{\sin\left(\pi\frac{\sigma}{\rho}\right)}{\sigma}\label{groundstate}
\end{equation}
which is the leading-order contribution to $\psi_{100}.$ Consequently, the relation
\begin{equation}
    \sigma_p^{(0)}\left(\Psi^{(0)}\right)\geq\sigma_p^{(0)}\left(\psi_{100}^{(0)}\right)
\end{equation}
holds for the zeroth-order approximation $\Psi^{(0)}$ to every $\Psi\in\hil.$ As perturbations are assumed to be small, they cannot change this relation, which is why we deduce that
\begin{equation}
    \sigma_p(\Psi)\geq\sigma_p\left(\psi_{100}\right).
\end{equation}
As shown in the appendix, the ground state (as all eigenstates of the Laplace-Beltrami operator) satisfies
\begin{equation}
    \braket{\psi_{100}|\hat{\pi}^i\psi_{100}}=\braket{\psi_{100}|\hat{\pi}_i\psi_{100}}=0\label{expconjmomgeod}
\end{equation}
to all orders. Therefore, we obtain
\begin{equation}
    \sigma^{(0)}_p\geq\sqrt{-\hbar^2\braket{\psi_{100}|\Delta\psi_{100}}^{(0)}}=\hbar\pi/\rho.
\end{equation}
Now that the state of lowest momentum uncertainty is identified, it is time to evaluate the curvature induced corrections.

\subsection{Corrections}

First, according to \eqref{pmix}, we can generally write
\begin{equation}
\braket{\hat{p}_{mix}^2}=\hbar\text{Im}\braket{N^i\partial_i}.
\end{equation}
As the ground state is real, the integrand appearing in
\begin{align}
    \braket{\psi_{100}|N^i\partial_i\psi_{100}}=\int\D\mu \psi_{100} N^i\partial_i\psi_{100}
\end{align}
is purely real and so is the integral. Thus, in the ground state the expectation value of $\hat{p}^2_{mix}$ vanishes.

Furthermore, terms mixing expectation values of the shift and the momentum vanish identically due to \eqref{expconjmomgeod}. Then, the variance of the ground state equals
\begin{equation}
    \sigma_p^2\left(\psi_{100}\right)=\sigma_{\pi}^2\left(\psi_{100}\right)+\frac{\hbar^2}{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}_C^2}\sigma_{N}^2\left(\psi_{100}\right)
\end{equation}
non-perturbatively. Curvature corrections to $\var{\pi}$ appear to second order and the first-order correction to $\var{N}$ is subject to similar cancellations as in \eqref{0thorderunc}. Hence, the first-order contribution to the variance of the physical momentum operator vanishes
\begin{equation}
    \var{p}^{(1)}(\psi_{100})=0
\end{equation}
from which we deduce that the shift corrects the uncertainty relation at the same order as the background curvature.

Moreover, from \cite{RF2020} we already know that
\begin{equation}
    \var{\pi}^{(2)}(\psi_{100})=-\frac{1}{6}R|_{p_0}
\end{equation}
where $R|_{p_0}=G^{ij}G^{kl}R_{ikjl}|_{p_0}.$ At this point, we are left with the second-order correction to the variance, which after cancellations reads
\begin{align}
    \var{N}^{(2)}=\nabla_k N^i\nabla_l N^j G_{ij}|_{p_0}\left(\braket{x^k x^l}-\braket{x^k}\braket{x^j}\right)^{(0)}.
\end{align}
When evaluated with respect to the ground state, the second term in the bracket vanishes while the first yields
\begin{equation}
    \braket{\psi_{100}|x^jx^k\psi_{100}}^{(0)}=\frac{\rho^2}{18}\left(2-\frac{3}{\pi^2}\right)\delta^{jk}.
\end{equation}
Finally, lowering and raising indices with the effective metric $G_{ij},$ the second-order correction to the variance of the physical momentum operator equals
\begin{equation}
    \var{p}^{(2)}(\psi_{100})=-\frac{R|_{p_0}}{6}+\xi\frac{\rho^2}{2{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}_C^2}\nabla_jN_i\nabla^jN^i|_{p_0}
\end{equation}
where we introduced the mathematical constant $\xi=(2-3/\pi^2)/9.$

Observe that, though it seems to be of higher order at first glance due to the factor $\rho^2,$ the second term is actually of second order because the expansion done here is performed in terms of $\rho\sqrt{\mathcal{R}}$ where $\mathcal{R}$ denotes any curvature invariant with dimensions of squared inverse length. Thus, no assumptions were made concerning the factor $\rho^2/{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}^2_C.$

\subsection{Result}

Gathering all the results from the previous sections and introducing the Compton wavelength $\lambda_C\equiv2\pi{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}_C$, we obtain the uncertainty relation:
\begin{align}
    \sigma_p\rho\gtrsim &\pi\hbar\left[1-\frac{\rho^2R|_{p_0}}{12\pi^2}+\xi\frac{\rho^4}{\lambda_C^2}\nabla_jN_i\nabla^jN^i|_{p_0}\right].\label{finalunc}
\end{align}
In short, given a four metric $g_{\mu\nu}$ and an observer defining a foliation of space-time, the uncertainty relation \eqref{finalunc} can be computed by evaluating the Ricci scalar derived from $G_{ij}$ and the corresponding covariant derivative of the shift at $p_0.$

\section{Applications\label{sec:app}}

In what follows, we will employ the formalism developed up until now to present explicit results related to several relevant metrics. To achieve non-trivial solutions, we shall require to work with space-time metrics for which the shift vector $N^i$ in the $3+1$ decomposition is non-vanishing, otherwise the physical momentum \eqref{phys} coincides with the conjugate momentum $\pi_i.$ For the sake of continuity with Ref. \cite{RF2020}, we start our analysis with the G\"odel universe. Subsequently, we focus our attention on the first weak-field solution for a rotating source in the context of General Relativity (GR), namely the Lense-Thirring space-time. As the last two examples, we investigate space-times stemming from rotating compact objects in the framework of extended theories of gravity with the purpose of pinpointing the main differences with respect to the standard GR scenario.

\subsection{G\"odel universe}

The G\"odel solution \cite{godel} is a homogeneous and anisotropic space-time arising from Einstein's field equations for a perfect fluid with non-vanishing angular momentum. It essentially describes a rotating universe in which closed timelike curves are allowed, thus in principle permitting time travel. The line element associated with such a curved background written in cylindrical coordinates $(t,r^a)=(t,r,\phi,z)$ reads \cite{godel}
    \begin{align}\nonumber
\D s^2  =&-\D t^2-\frac{2r^2}{a\sqrt{2}}\D t \D\phi+\frac{\D r^2}{\left(1+\frac{r^2}{4a^2}\right)}\\
        &+r^2\left(1-\frac{r^2}{4a^2}\right)\D\phi^2+\D z^2\label{eqgodel}
\end{align}
with the constant parameter $a>0$ which has units of length and quantifies the angular momentum of matter. Have in mind that this slicing only covers the region $r< 2a.$

An observer orbiting circularly around the $z$-axis (\ie co-rotating with the Gödel universe) will experience the flow of proper time according to the time coordinate $t.$ The effective lapse, three-metric and shift from the point of view of this observer read
\begin{align}
N                                       =&\sqrt{1+\frac{2}{4\frac{a^2}{r^2}-1}}\\
G_{ab}\D r^a\D r^b                      =&\frac{1}{N}\left[\frac{\D r^2}{1+\frac{r^2}{4a^2}}+\left(1-\frac{r^2}{4a^2}\right)r^2\D\phi^2+\D z^2\right]\\
N^a\frac{\partial}{\partial r^a}        =&-\frac{a}{\sqrt{2}\left(a^2-\frac{r^2}{4}\right)}\partial_\phi.
\end{align}
As $r<2a$ in this slicing and the prefactors of terms containing higher powers of $r_0/2a$ (where $r_0=r|_{p_0}$) in the resulting uncertainty relation get ever smaller, we will only display the next-to-leading order for the sake of brevity.

Correspondingly, the observer defined above measures the uncertainty relation
\begin{equation}
    \sigma_p\rho\gtrsim \pi\hbar\left[1-\frac{\rho^2}{4\pi^2a^2}\left(1-\xi \frac{\rho^2}{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}_C^2}+\frac{17}{24}\frac{r_0^2}{a^2}\right)\right].
\end{equation}

\subsection{Lense-Thirring solution\label{subsec:lense}}

The phenomenon of frame-dragging was discovered only few years after the final settlement of GR. As a matter of fact, in 1918 Lense and Thirring found the weak-field limit for the space-time generated by a rotating body \cite{lense}. The main prediction of this solution is the existence of a precession of the orbits drawn by a test body, a feature that is completely absent in Newtonian mechanics. To get to this conclusion, they argued that in isotropic spherical coordinates $(t,r^e)=(t,r,\theta,\varphi)$ the metric tensor originating from a rotating source takes the form \cite{lense}
\begin{align}\label{eqlense}
\D s^2= &-(1+2\phi_{GR})\D t^2+(1-2\phi_{GR})\left(\D r^2+r^2\D\Omega^2\right)\nonumber\\
        &+4\phi_{GR} a_J\sin^2\theta \D\varphi \D t
\end{align}
where $\phi_{GR}=-GM/r$ is the usual Newtonian potential whereas $a_J=J/M$ is the rotational parameter with $J$ denoting the angular momentum of the source.

Note that the time coordinate $t$ used here corresponds to the time measured by a static observer at infinite distance from the gravitating body in the center. In turn, the uncertainty is calculated as it would be measured by this observer. In the non-relativistic limit, though, this provides a good approximation of the slicing carved out by the dynamical rest frame of the particle itself. Therefore, corrections to the results are expected to be of higher order. Analogous considerations apply to the extended models of gravity which are treated as corrections to \eqref{eqlense} below.

The static observer at infinity experiences the effective metric, shift and lapse
\begin{align}
N                                           \simeq&1-\phi_{GR}\\
N^e\frac{\partial}{\partial r^e}            \simeq&-2a_J\frac{\phi_{GR}}{r^2}\\
    G_{ef}\D r^e\D r^f                      \simeq&\left(1+3\phi_{GR}\right)\left(\D r^2+r^2\D\Omega^2\right).
\end{align}
Unfortunately, this reasoning leads to an uncertainty which is at least quadratic in the gravitational potential while the Lense-Thirring solution corresponds to a first-order expansion. In light of this, we generalize the discussion by approximating the Kerr metric in Boyer-Lindquist coordinates
\begin{align}
    \D s^2=&-\left(1+2\tilde{\phi}_{GR}\frac{\tilde{r}^2}{\Xi^2}\right)\D t^2+4\tilde{\phi}_{GR}a_J\sin^2\theta\frac{\tilde{r}^2}{\Xi^2}\D t\D \varphi\nonumber\\
    &+\frac{\Xi^2}{\Sigma}\D \tilde{r}^2+\Xi^2\D\theta^2+\tilde{r}^2\Big(1+\frac{a_J^2}{\tilde{r}^2}\nonumber\\
    &-2\tilde{\phi}_{GR}\sin^2\theta\frac{a_J^2}{\Xi^2}\Big)\sin^2\theta\D\varphi^2
\end{align}
with $\tilde{\phi}_{GR}=-GM/\tilde{r}$, $\Xi=\tilde{r}\sqrt{1+\cos^2\theta a_J^2/\tilde{r}^2}$ and $\Sigma=\tilde{r}^2(1+\tilde{\phi}_{GR}+a_J^2/\tilde{r}^2)$ to fourth order in $\tilde{\phi}_{GR}$ and $a_J/\tilde{r}$ simultaneously. Bear in mind that the Schwarzschild-like $\tilde{r}$ relates to the radial coordinate introduced with the Lense-Thirring metric as $\tilde{r}=r(1-\phi_{GR}/2)^2.$ As the resulting uncertainty relation is 3-diffeomorphism invariant, we will nevertheless provide it in terms of $r$ for the sake of future convenience.

Consequently, the static observer at infinity measures the uncertainty relation
\begin{align}
    \sigma_p\rho\geq&\pi\hbar\Bigg\{1+\phi_{GR}^2\frac{\rho^2}{48\pi^2r^2}\Bigg[10+30\phi_{GR}+55\phi_{GR}^2-\frac{a_J^2}{r^2}\nonumber\\
    &\times\Big(469-217\cos 2\theta-96\xi \frac{\rho^2}{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}_C^2}\left(7-3\cos 2\theta\right)\Big)\Bigg]\Bigg\}\Bigg|_{p_0}\\
    \equiv&\pi\hbar\left(1+\lambda_{LT}\right)
\end{align}
where the last line defines the (generalized) Lense-Thirring correction $\lambda_{LT}.$ We would like to remark that the dimensionless number $\rho^2/{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}_C^2$ is unconstrained; for this reason, in a suitable regime it can increase the effect of space-time rotation.

A sample orbit in the equatorial plane of the Kerr metric approximated as indicated above is given in figure \ref{fig:orb}, where the color of the curve changes with increasing proper time. The ensuing correction to the uncertainty relation is displayed in figure \ref{fig:uncorb} as a function of said proper time. The peaks correspond to the positions of smallest distance from the outer horizon along the trajectory. Their height is amplified by the ad-hoc choice of Compton wavelength of the particle $\lambda_C$ and the proximity of the orbit to the horizon which can only be realized surrounding black holes. A more conservative estimate of a measurement done on a geostationary satellite at $r|_{p_0}\sim 10^4\text{km}$ ($GM\sim 10^{-4}\text{km}$) using the largest quantum systems realized presently ($\rho\sim 10^{-15}\text{km}$) leads to a correction of the order $10^{-60}.$

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{"orbit.png"}
    \caption{The trajectory followed by a massive particle in the equatorial ($\tilde{x}$-$\tilde{y}$-) plane of the expanded Kerr metric as seen from above for $a_J/GM=0.5$. The starting point lies on the $x/GM$ axis at a distance $1000$ from the source with initial velocity (in coordinates $(t,r^e)$) $u(\tau =0)=(1.001,-0.010,0.000,0.000).$ The color ranging from violet to red indicates an increase in proper time $\tau$ whereas the black disk at the center symbolizes the outer horizon.}
    \label{fig:orb}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{"uncorb.png"}
    \caption{The corrections to the uncertainty relation experienced along the trajectory in figure \ref{fig:orb} by a particle of Compton wavelength $\lambda_C=10^{-4}GM$ as a function of proper time.}
    \label{fig:uncorb}
\end{figure}

\subsection{Fourth-order gravity}

Fourth-order gravity introduced by Stelle represents one of the first attempts to cure the quantization problems of the gravitational interaction. In particular, it was pointed out \cite{stelle} that the introduction of higher-derivative terms in the Einstein-Hilbert action can make the model renormalizable. To be more precise, according to the prescription in \cite{stelle}, the gravitational action from which to build up quantum gravity should be given by
\begin{equation}\label{actionstelle}
S=\frac{1}{16\pi^2G^2}\int d^4x\sqrt{-g}\left(R+\frac{\alpha}{2\hbar^2}R^2-\frac{\beta}{2\hbar^2}R_{\mu\nu}R^{\mu\nu}\right),
\end{equation}
where $\alpha$ and $\beta$ are dimensionful constants measured in units of inverse mass squared. However, the drawback of this model consists in the appearance of ghost-like degrees of freedom which undermine the unitarity of the underlying quantum field theory. Such a circumstance is a typical feature of local higher-order derivative gravity \cite{shapiro}.

For the current model, a Lense-Thirring-like solution has been recently obtained when analyzing the light bending due to quadratic theories of gravity \cite{luca}. In isotropic spherical coordinates, the aforementioned solution is
\begin{align}\label{geq}
\D s^2= &-(1+2\phi)\D t^2+(1-2\psi)\left(\D r^2+r^2\D\Omega^2\right)\nonumber\\
        &+2\xi\sin^2\theta \D\varphi \D t\,,
\end{align}
where the gravitational potentials are given by
\begin{align}
\phi=& \,\phi_{GR}\left(1+\frac{1}{3}e^{-m_0r/\hbar}-\frac{4}{3}e^{-m_2r/\hbar}\right)\label{eqstelle1}\\[2mm]
\psi=& \,\phi_{GR}\left(1-\frac{1}{3}e^{-m_0r/\hbar}-\frac{2}{3}e^{-m_2r/\hbar}\right)\label{eqstelle2}\\[2mm]
\xi=& \,2\phi_{GR}a_J\left[1-(1+m_2r/\hbar)e^{-m_2r/\hbar}\right]\label{eqstelle3}
\end{align}
with $m_0=2/\sqrt{12\alpha-\beta}$ and $m_2=\sqrt{2/\beta}$ being the masses of the spin-0 and spin-2 massive modes, respectively.

As the influence stemming from the higher-derivative terms ought to be small in comparison to the general relativistic effect, results should be given as corrections to the Lense-Thirring outcome. Written this way, the uncertainty relation reads for small gravitational potentials
\begin{align}
    \sigma_p\rho\geq&\pi\hbar\Bigg[1+\lambda_{LT}\nonumber\\
    &-\frac{\phi_{GR}|_{p_0}}{18\pi^2}\left(\frac{\rho^2}{\lambda_{m_0}^2}e^{-\frac{m_0 r_0}{\hbar}}+\frac{8\rho^2}{\lambda_{m_2}^2}e^{-\frac{m_2 r_0}{\hbar}}\right)\Bigg]
\end{align}
where $\lambda_{m_0}$ and $\lambda_{m_2}$ denote the Compton wavelengths corresponding to the respective massive gravitational modes.

As straightforwardly recognizable in the previous equation, for the current example we do not have to resort to a higher-order expansion of the Kerr-like solution in the context of the examined extended model of gravity, as the leading-order correction is already linear in $\phi_{GR}.$ This feature is shared by the upcoming analysis as well.

\subsection{Infinite-derivative gravity}

Starting from the above scenario and recalling that the reasoning in \cite{shapiro} prevents any local higher-order derivative gravity from being free from ghost fields, it is clear that one must give up on locality to arrive at a quantum gravitational model which is simultaneously renormalizable and unitary. Hovewer, non-locality should be manifest only in the currently unexplored UV regime, since all the available data acquired from gravitational experiments comply with the local behavior of gravity. Along this direction, it is possible to encounter the so-called infinite-derivative gravity theory, which precisely possesses the characteristics listed above. As the name suggests, the usual Einstein-Hilbert action is now accompanied by non-local functions of the curvature invariants; in the simplest form, the non-local gravitational action reads \cite{idg}
\begin{align}\nonumber
S=&\frac{1}{16\pi^2G^2}\int \D^4x\sqrt{-g}\left(R+R\frac{1-e^{-{\hbar^2\Box}/{\kappa^2}}}{4\Box}R\right.\\
&-\left.R_{\mu\nu}\frac{1-e^{-{\hbar^2\Box}/{\kappa^2}}}{2\Box}R^{\mu\nu}\right),\label{actionidg}
\end{align}
where $\kappa$ is the energy scale at which the non-local aspects of gravity are expected to be prominent.

As for the previous example, a Lense-Thirring-like solution can be analytically computed in this framework. Formally, the shape of the metric tensor is the same as the one exhibited in \eqref{geq}, with the difference that here the gravitational potentials are instead represented by
\begin{align}
\phi=\psi=& \,\phi_{GR}\mathrm{Erf}\left(\frac{\kappa\,r}{2\hbar}\right)\label{eqidg1}\\[2mm]
\xi=& \,2\phi_{GR}a_J\left[\mathrm{Erf}\left(\frac{\kappa\,r}{2\hbar}\right)-\frac{\kappa\,r}{\sqrt{\pi}\hbar}e^{-\kappa^2r^2/4\hbar^2}\right]\label{eqidg2}
\end{align}
where $\mathrm{Erf}(x)$ denotes the error function.

Again expressed as corrections to the Lense-Thirring result, the uncertainty relation for small gravitational potentials becomes in this case
\begin{equation}
    \sigma_p\rho\geq\pi\hbar\left(1+\lambda_{LT}-\frac{\phi_{GR}|_{p_0}}{4\pi^{\frac{5}{2}}}\rho^2r_0\frac{\kappa^3}{\hbar^3}e^{-\kappa^2 r_0^2/4\hbar}\right).
\end{equation}

\section{Discussion\label{sec:disc}}

Along the lines of the EUP prescription and by resorting to the dynamics of a non-relativistic particle in a curved background, we have shown how to generally derive a modification of the canonical uncertainty principle surged by the underlying gravitational field in which the motion takes place. To this aim, we have based our considerations on the notion of geodesic ball \cite{Schuermann2009,Schuermann2018} as a 3D-diffeomorphism invariant domain (whose radius yields a measure of position uncertainty). Moreover, we have properly defined a hermitian momentum operator that complies with the canonical commutation relations in the non-relativistic limit of the 3+1 formalism. Finally, perturbing around flat space-time and relying on a result for flat spaces obtained in an earlier work \cite{RF2020}, we have evaluated the gravitationally induced corrections to the uncertainty relation summarized by equation \eqref{finalunc}. To second order, it contains two new contributions, one proportional to the Ricci scalar of the effective three-dimensional metric and one to the squared covariant derivative of the shift vector, both of which are evaluated at the center of the geodesic ball (\ie the expectation value of the position operator).

Furthermore, we have explicitly computed the form of the above uncertainty relation for the G\"odel universe, the Lense-Thirring solution and its extension in the framework of fourth-order and infinite-derivative gravity. Remarkably, whilst the leading-order contribution goes like $\phi_{GR}^2$ in the Lense-Thirring scenario, for the extended models we observe a proportionality to $\phi_{GR}$. Therefore, there may be a regime in which the two terms are comparably important, thus leading to a simultaneous ``coexistence'' of the two quantities. A similar occurrence has also been addressed in different contexts, as for the case of the Casimir effect \cite{casimir}.

The derivation performed in this paper allows for a dual interpretation of curved energy-momentum spaces replacing geodesic balls in position space with their counterparts in momentum space and the momentum operator with the position operator. Hence, it provides a direct geometrical link between the resulting Generalized Uncertainty Principle and curvature in energy-momentum space as has been anticipated \eg in the geometrical description of Doubly Special Relativity as a theory of de Sitter momentum space. Thus, it paves the way towards momentum-space curvature corrected quantum mechanics on purely geometrical grounds.


text:
\section{Introduction}
In \cite{Ha}, Hariya obtained an integration by parts formula on a subset of the pinned path space on ${\mathbb R}^d$, which is a partial generalization of the work by Zambotti~\cite{Za}.
To state it more precisely, let $\Omega$ be a bounded domain in ${\mathbb R}^d$. We assume that the boundary of $\Omega$ is sufficiently smooth.
Take $a,b\in \Omega$ and define the path spaces as follows:
\begin{align*}
W_{a,b}&=\{w\in C([0,1]\to{\mathbb R}^d)\mid w(0)=a,\ w(1)=b\},\\
W_{a,b}^\Omega&=\{w\in C([0,1]\to \Omega)\mid w(0)=a,\ w(1)=b\},\\
H_0&=\left\{h\in C([0,1]\to{\mathbb R}^d)\left|\, \begin{array}{l}h(0)=h(1)=0,\ \text{$h$ is absolutely continuous}\\
\text{and }\int_0^1 |\dot h(s)|_{{\mathbb R}^d}^2\,ds<\infty\end{array}\right.\right\}.
\end{align*}
We regard $W_{a,b}^\Omega$ as a subset of $W_{a,b}$.
The topological boundary $\partial W_{a,b}^\Omega$ of $W_{a,b}^\Omega$ with respect to the uniform topology is given by
\[
  \partial W_{a,b}^\Omega=\left\{w\in W_{a,b}\left|
  \begin{array}{l}
  w(t)\in\overline{\Omega}\text{ for every $t\in[0,1]$ and}\\
  w(s)\in\partial \Omega\text{ for some $s\in(0,1)$}
  \end{array}
  \right.\right\},
\]
where $\overline{\Omega}$ and $\partial \Omega$ denote the closure and the boundary of $\Omega$ in ${\mathbb R}^d$, respectively.
We define a subset $\partial' W_{a,b}^\Omega$ of $\partial W_{a,b}^\Omega$ by
\[
\partial' W_{a,b}^\Omega=\{w\in \partial W_{a,b}^\Omega\mid{}\text{there exists a unique }s\in(0,1) \text{ such that } w(s)\in\partial\Omega\}.
\]
Let $\mu_{a,b}$ be the pinned Wiener measure on $W_{a,b}$.
For a smooth cylindrical function $F$ on $W_{a,b}$ and $h\in H_0$, Hariya~\cite{Ha} proved the identity
\begin{equation}\label{eq:ibp}
  \int_{W_{a,b}^\Omega} \partial_h F(w)\,\mu_{a,b}(dw)
  =\int_{W_{a,b}^\Omega}F(w)\langle h,w\rangle\,\mu_{a,b}(dw)
  + ({\rm BC}),
\end{equation}
where $\partial_h$ is the partial derivative along the direction of $h$, $\langle h,w\rangle$ denotes the Wiener integral $\int_0^1 h(s)\,dw(s)$, and (BC) is the ``boundary contribution,'' which is expressed as an integral over $\partial W_{a,b}^\Omega$.
The explicit expression of (BC) is provided in \cite{Ha}.
In this study, we provide only the following remarks on (BC).
\begin{enumerate}
\item[(a)] The mass of the measure on $\partial W_{a,b}^\Omega$ appearing in the integral representation of (BC) concentrates on $\partial' W_{a,b}^\Omega$.
\item[(b)] The integrand in (BC) contains the normal derivatives of the heat kernel density on $\Omega$ at $\partial \Omega$ with the Dirichlet boundary condition.
\end{enumerate}
The integration by parts formula \eqref{eq:ibp} implies that the indicator function $1_{W_{a,b}^\Omega}$ of $W_{a,b}^\Omega$ is a BV function, and we can construct the reflecting Ornstein-Uhlenbeck process on $W_{a,b}^\Omega$ with the Skorokhod representation (cf.\ Section~2 below).
On the other hand, property (b) above imposes on the strong regularity of $\Omega$ since we cannot expect the normal derivatives of the heat kernel density to exist at the boundary if $\partial \Omega$ is not very smooth.
If we are only interested in the probabilistic aspect, it is sufficient to prove that $1_{W_{a,b}^\Omega}$ is a BV function; in other words, even if we do not know the explicit expression of (BC), only proving the validity of the integration by parts is sufficient.
This is expected to be done under a milder assumption on $\Omega$ since such a claim can be proved only by a series of inequalities and not by equalities.
This is the objective of this paper.

In this paper, we introduce the concept of the {\em uniform exterior ball condition} for $\Omega$, which allows some singularity at $\partial\Omega$, and prove that $1_{W_{a,b}^\Omega}$ is a BV function under such a condition.
Based on this, we can construct the reflecting Ornstein-Uhlenbeck process on the closure of $W_{a,b}^\Omega$ and prove its Skorokhod representation.
Further, we prove that the mass of the measure on $\partial W_{a,b}^\Omega$ appearing in the Skorokhod representation concentrates on $\partial'W_{a,b}^\Omega$, which is consistent with  property~(a) above.
The proof is based on the quantitative estimates of Brownian motion on ${\mathbb R}^d$, and the method is different from that used in \cite{Ha}.
We expect that our method is sufficiently flexible to discuss more general situations.

This paper is organized as follows. In Section~2, we provide a framework and state the main theorem.
Some key estimates for the Brownian motion on ${\mathbb R}^d$ are proved in Section~3. These estimates are obtained by reducing them to a few detailed estimates of a one-dimensional Brownian motion with a constant drift.
The main theorem is proved in Section~4.
In the last section, we provide a few remarks.

\section{Framework and the main result}
First, we recall the concept of the BV functions on the Wiener space, according to \cite{FH}.
Let $(E,H,\mu)$ be an abstract Wiener space,
that is, $E$ is a separable Banach space, $H$ is a separable Hilbert
space densely and continuously embedded in $E$, and $\mu$ is a
Gaussian measure on $E$ that satisfies the condition
\[
  \int_E \exp\left(\sqrt{-1}\,l(z)\right)\mu(dz)=\exp\left(-|l|_H^2/2\right),
  \quad l\in E^*.
\]
Here, $^*$ denotes the topological dual and we use natural inclusions and an identification $E^*\subset H^*\cong H\subset E$.
When $M$ is a separable Hilbert space, $L^p(E\to M)$ denotes the $L^p$-space on $E$ with respect to $\mu$ which consists of $M$-valued functions.
When $M={\mathbb R}$, we omit $M$ from the notation.
Let $C_b^1({\mathbb R}^m)$ be the set of all bounded continuous functions $f$ on ${\mathbb R}^m$ such that all the first-order partial derivatives of $f$ are bounded and continuous.
Define
\begin{align*}
{\mathcal F} C_b^1&=\left\{ u\colon E\to {\mathbb R}\left|\,
\begin{array}{l}
u(z)=f(l_1(z),\ldots,l_m(z)),\
l_1,\ldots,l_m\in E^*,\\
f\in C_b^1({\mathbb R}^m)\text{ for some $m\in{\mathbb N}$}
\end{array}
\right.\right\},\\
({\mathcal F} C_b^1)_{E^*}&=\left\{ G\colon E\to E^*\left|\,
\begin{array}{l}
G(z)=\sum_{i=1}^m u_i(z)l_i,\
l_1,\ldots,l_m\in E^*,\\
u_1,\ldots,u_m\in {\mathcal F} C_b^1\text{ for some $m\in{\mathbb N}$}
\end{array}
\right.\right\}.
\end{align*}
For $u\in{\mathcal F} C_b^1$, an $H$-valued function $\nabla u$ on $E$ is given by the following identity:
\[
\langle \nabla u(z),l\rangle_H=\lim_{\varepsilon\to 0}(u(z+\varepsilon l)-u(z))/\varepsilon,
\quad l\in E^*\subset H,\ z\in E.
\]
Let $\nabla^*$ be a (formal) adjoint operator of $\nabla$, which is defined by the following relation:
\[
\langle \nabla^* G,u\rangle_{L^2(E)}=\langle G,\nabla u\rangle_{L^2(E\to H)},
\quad u\in {\mathcal F} C_b^1.
\]
We define the set of BV functions on $E$ as
\[
BV(E)=\left\{\rho\colon E\to {\mathbb R}\left|\,
\begin{array}{l}
\int_E |\rho|\max\{0,\log |\rho|\}^{1/2}\,d\mu<\infty \text{ and there exists}\\
C\ge0\text{ such that }
\left|\int_E (\nabla^* G)\rho\,d\mu\right|
\le C\| |G|_H\|_{L^\infty(E)}\\
\text{for all }G\in ({\mathcal F} C_b^1)_{E^*}
\end{array}
\right.\!\right\}.
\]
We shall now revisit several properties of BV functions on $E$.
\begin{thm}[({\cite[Theorems~3.7, 3.9]{FH}})] \label{th:1}
For $\rho\in L^1(E)$, the following are equivalent conditions.
\begin{enumerate}
\item $\rho\in BV(E)$.
\item There exists a sequence $\{\rho_n\}$ in ${\mathbb D}^{1,1}:=\overline{{\mathcal F} C_b^1}^{\|\nabla\cdot\|_{L^1(E\to H)}+\|\cdot\|_{L^1(E)}}$ such that $\rho_n$ converges to $\rho$ in $L^1(E)$ and $\|\nabla \rho_n\|_{L^1(E\to H)}$ is bounded in $n$.
\item (Integration by parts formula) There exist, a finite Borel measure $\nu$ and an $H$-valued function $\sigma$ on $E$ such that $|\sigma|_H=1$ $\nu$-a.e.\ and
\[
  \int_E (\nabla^* G)\rho\,d\mu=\int_E \langle G,\sigma\rangle_H\,d\nu,\quad
  G\in ({\mathcal F} C_b^1)_{E^*}.
\]
\end{enumerate}
\end{thm}
\begin{thm}[({\cite[Theorem~4.2]{FH}})] \label{th:2}
Let $\rho\in BV(E)$ and assume $\rho\ge0$ $\mu$-a.e.
Let $S$ be the support of the measure $\rho\,d\mu$.
Define a bilinear form on $L^2(S,\rho\,d\mu)$ by
\[
  {\mathcal E}(f,g)=\frac12\int_S \langle \nabla f,\nabla g\rangle_H \rho\,d\mu,\quad
  f,g\in{\mathcal F} C_b^1.
\]
Assume that $({\mathcal E},{\mathcal F} C_b^1)$ is closable on $L^2(S,\rho\,d\mu)$.
Then, its closure $({\mathcal E},{\mathcal F})$ is a quasi-regular, local, and conservative Dirichlet form on $L^2(S,\rho\,d\mu)$, and the following Skorokhod representation holds:
\begin{equation}\label{eq:skorohod}
X_t=X_0+B_t-\frac12\int_0^t X_s\,ds+\frac12\int_0^t \sigma(X_s)\,dA_s,\quad
t\ge0,\ P_w\text{-a.e.\ for q.e.\ $w$}.
\end{equation}
Here, $(X_t,P_w)$ is a diffusion process on $S$ associated with $({\mathcal E},{\mathcal F})$, $\{B_t\}$ is an $E$-valued Brownian motion starting at $0$, $\{A_t\}$ is an additive functional in Revuz correspondence with $\nu$, and $\nu$ and $\sigma$ are provided in Theorem~$\ref{th:1}$~$(3)$.
\end{thm}
Note that $\nu$ above is smooth with respect to the $({\mathcal E},{\mathcal F})$ from \cite[Theorem~3.9]{FH}, which justifies the consideration of the Revuz correspondence of $\nu$.
When $\rho$ is an indicator function, we term $\{X_t\}$ a reflecting Ornstein-Uhlenbeck process on $S$.
In such a case, the measure $\nu$ can be regarded as a surface measure of $S$.
\begin{thm}[({\cite[Theorem~3.15]{FH}})]\label{th:3}
Under the conditions described in Theorem~$\ref{th:2}$, if moreover $\rho$ is an indicator function of a set $U$, then the mass of $\nu$ concentrates on the topological boundary of $U$.
\end{thm}
We remark that the original assertion of Theorem~3.15 in \cite{FH} provides more detailed information on the support of $\nu$.

For $t>0$, $x,y\in{\mathbb R}^d$, we define
\[
p_t(x,y)=(2\pi t)^{-d/2}\exp\left(-\frac{|x-y|_{{\mathbb R}^d}^2}{2t}\right).
\]
Fix $a,b\in{\mathbb R}^d$, and let $W_{a,b}$ and $H_0$ as defined in Section~1.
The pinned Wiener measure $\mu_{a,b}$ on $W_{a,b}$ is a Borel probability measure such that for $0=t_0<t_1<\cdots<t_{N+1}=1$ and Borel sets $A_1,\ldots,A_N$ of ${\mathbb R}^d$,
\begin{align*}
&\mu_{a,b}[w\in W_{a,b}\mid  w_{t_i}-w_{t_{i-1}}\in A_i,\ i=1,\ldots,N]\\
&=p_1(a,b)^{-1}\int\!\cdots\!\int_{A_1\times \dots\times A_N}\prod_{i=1}^{N+1}p_{t_i-t_{i-1}}(x_{i-1},x_i)\,dx_1\cdots dx_N,
\end{align*}
where $x_0=a$ and $x_{N+1}=b$.
Then, $(W_{0,0},H_0,\mu_{0,0})$ is an abstract Wiener space.
When $(a,b)\ne (0,0)$, $W_{a,b}$ is not a linear space.
However, $W_{a,b}$ is isomorphic to $W_{0,0}$ as an  affine space according to the shift map \[\lambda_{a,b}\colon W_{a,b}\ni w\mapsto w-h_{a,b}\in W_{0,0},\] where $h_{a,b}(t)=a+(b-a)t$, $t\in[0,1]$, and $(W_{a,b},\mu_{a,b})$ is isomorphic to $(W_{0,0},\mu_{0,0})$ as a measure space according to the map $\lambda_{a,b}$.
Therefore, by pushing everything forward to $(W_{0,0},\mu_{0,0})$, we can define the concepts of ${\mathcal F} C_b^1$, $\nabla$, the BV space $BV(W_{a,b})$ etc., on $(W_{a,b},H_0,\mu_{a,b})$.
Furthermore, Theorems~\ref{th:1}, \ref{th:2}, and \ref{th:3} are valid on this space by appropriate modification.

Let $\Omega$ be a domain of ${\mathbb R}^d$.
We do not assume that $\Omega$ is bounded, but assume $\Omega\ne {\mathbb R}^d$.
For $x\in {\mathbb R}^d$ and $r>0$, $B(x,r)$ denotes the closed ball in ${\mathbb R}^d$ with center $x$ and radius $r$.
\begin{dfn}\label{def:uebc}
We state that $\Omega$ satisfies the {\em uniform exterior ball condition} if there exists $\delta>0$ such that for every $y\in\partial \Omega$, there exists $z\in{\mathbb R}^d\setminus \Omega$ satisfying ${B}(z,\delta)\cap \overline{\Omega}=\{y\}$.
\end{dfn}
For example, bounded domains with boundaries in the $C^2$-class and convex domains satisfy the uniform exterior ball conditions.
It may be said that this condition allows outward cusps, but not inward cusps.

We consider $W_{a,b}^\Omega$, $\partial W_{a,b}^\Omega$, and $\partial'W_{a,b}^\Omega$ as defined in Section~1.
Let $\overline{W_{a,b}^\Omega}=W_{a,b}^\Omega\cup\partial W_{a,b}^\Omega$.
The main theorem in this paper is as follows.
\begin{thm}\label{th:main}
Assume that $\Omega$ satisfies the uniform exterior ball condition.
Then, $1_{\overline{W_{a,b}^\Omega}}\in BV(W_{a,b})$.
Further, the bilinear form $({\mathcal E}',{\mathcal F} C_b^1)$ on $L^2(\overline{W_{a,b}^\Omega},\mu_{a,b}|_{\overline{W_{a,b}^\Omega}})$ defined by
\[
{\mathcal E}'(f,g)=\frac12\int_{\overline{W_{a,b}^\Omega}}\langle \nabla f,\nabla g\rangle_{H_0}\,d\mu_{a,b},
\quad f,g\in{\mathcal F} C_b^1
\]
 is closable, and its closure $({\mathcal E}',{\mathcal F}')$ is a quasi-regular, local, and conservative Dirichlet form.
Moreover, when $(X'_t,P'_w)$ denotes the diffusion process on $\overline{W_{a,b}^\Omega}$ associated with $({\mathcal E}',{\mathcal F}')$,
$(X_t,P_w):=(\lambda_{a,b}(X'_t),P'_{\lambda_{a,b}^{-1}(w)}\circ\lambda_{a,b}^{-1})$ satisfies the Skorokhod representation~\eqref{eq:skorohod} with $(E,H,\mu)=(W_{0,0},H_0,\mu_{0,0})$ and $\rho=1_{\lambda_{a,b}\left(\overline{W_{a,b}^\Omega}\right)}$.
Furthermore, $\partial W_{a,b}^\Omega\setminus\partial' W_{a,b}^\Omega$ has a null capacity that is associated with $({\mathcal E}',{\mathcal F}')$.
In particular, the mass of the measure $\nu$ that  corresponds to $\rho$ in Theorem~$\ref{th:1}$~$(3)$ concentrates on $\lambda_{a,b}(\partial' W_{a,b}^\Omega)$.
\end{thm}
\section{Some estimates for (pinned) Brownian motion}
Subsequently, $C_i$ denotes an insignificant positive constant and a domain $\Omega$ in ${\mathbb R}^d$ is assumed to satisfy the uniform exterior ball condition.

We define a Lipschitz function $q$ on ${\mathbb R}^d$ by
\[
q(x)=\inf_{y\in{\mathbb R}^d\setminus\Omega}|x-y|_{{\mathbb R}^d}-\inf_{y\in\Omega}|x-y|_{{\mathbb R}^d}.
\]
For $r\ge0$, set $\Omega_r=\{x\in{\mathbb R}^d\mid q(x)>r\}$.
Note that $\Omega_0=\Omega$ and $\{q(x)\ge0\}=\overline{\Omega}$.

Let $W=C([0,\infty)\to{\mathbb R}^d)$.
Let $\{\hat P_x\}_{x\in{\mathbb R}^d}$ be the probability measures on $W$ such that the coordinate process $\{\omega_t\}_{t\ge0}$ is a $d$-dimensional Brownian motion starting at $x$ under $\hat P_x$ for each $x\in{\mathbb R}^d$.
For $t\ge0$, let $\hat{\mathcal F}_t$ be a $\sigma$-field generated by $\{\{\omega_s\in D\}; s\in[0,t],\ D \text{ is a Borel set of }{\mathbb R}^d\}$.
Then, $\{\hat{\mathcal F}_t\}$ is a minimal filtration to which $\{\omega_t\}$ is adapted on the canonical measurable space $(W,\hat{\mathcal F}_\infty)$.
For an $\{\hat{\mathcal F}_t\}$-stopping time $\tau$, define
$\hat{\mathcal F}_\tau=\{A\in\hat{\mathcal F}_\infty\mid A\cap\{\tau\le t\}\in\hat{\mathcal F}_t \text{ for all }t\ge0\}$.
We denote the integral with respect to $\hat P_x$ by $\hat E_x$.
The shift operator $\theta_s\colon W\to W$ is defined by $(\theta_s \omega)_t=\omega_{s+t}$, $t\ge0$.
\begin{lmm}\label{lem:ito}
Let $x\in\overline\Omega$.
Choose $y\in\partial\Omega$ and $z\in{\mathbb R}^d\setminus\Omega$ such that $q(x)=|x-y|_{{\mathbb R}^d}$ and $B(z,\delta)\cap\overline\Omega=\{y\}$, where $\delta$ is provided in Definition~$\ref{def:uebc}$.
Let $K=(d-1)/(2\delta)$ and $R_t=|\omega_t-z|_{{\mathbb R}^d}$ for $\omega=\{\omega_t\}\in W$.
Then, for each $u>0$,
\[
\{R_t\ge\delta \text{ for all }t\in[0,u]\}\subset\{R_t\le q(x)+\delta+Kt+S_t \text{ for all }t\in[0,u]\}
\]
up to a $\hat P_x$-null set.
Here, $S_t$ is a one-dimensional Brownian motion under $\hat P_x$ starting at $0$ that is defined by
\[
   S_t(\omega)=\sum_{i=1}^d\int_0^t\frac{\omega_s^{(i)}-z^{(i)}}{R_s}\,d\omega^{(i)}_s,\quad
   \omega_s=(\omega^{(1)}_s,\ldots,\omega^{(d)}_s),\
   z=(z^{(1)},\ldots,z^{(d)}).
\]
\end{lmm}
\begin{proof}
Define an $\{\hat{\mathcal F}_t\}$-stopping time $\sigma$ by $\sigma=\inf\{t\ge0\mid R_t=0\}$.
Note that $R_0=|x-z|_{{\mathbb R}^d}=q(x)+\delta$ $\hat P_x$-a.e.
By virtue of It\^o's formula,
\[
  R_t=q(x)+\delta+\int_0^t\frac{d-1}{2R_s}\,ds+S_t
  \quad\text{on }\{t<\sigma\}\quad
  \hat P_x\text{-a.e.}
\]
Therefore, the assertion holds.
\end{proof}
\begin{prp}\label{prop:1}
There exists $C_1>0$ such that for every $u>0$ and $x\in\overline\Omega$,
\[
  \hat P_x\left[\inf_{t\in[0, u]}q(\omega_t)\ge0\right]\le C_1(1+u^{-1/2})q(x).
\]
\end{prp}
\begin{proof}
We retain the notations in Lemma~\ref{lem:ito}, from which
\[
\hat P_x\left[ R_t\ge\delta\text{ for all }t\in[0,u]\right]
\le \hat P_x\left[q(x)+\delta+K t+S_t\ge\delta\text{ for all }t\in[0,u]\right].
\]
Let $r>q(x)$ and define $\eta=\inf\{t\ge0\mid Kt+S_t\le -r\}$.
The law of $\eta$ under $\hat P_x$ is given by
\[
\hat P_x[\eta\in dt]=1_{(0,\infty)}(t)\frac{r}{\sqrt{2\pi t^3}}\exp\left(-\frac{(r+Kt)^2}{2t}\right)\,dt+(1-\exp(-2Kr))\delta_{\infty}(dt),
\]
where $\delta_\infty$ is a delta measure at $\infty$. (See, e.g., \cite[p.~295]{BS}.)
Then, we have
\begin{align*}
\hat P_x\left[\inf_{t\in[0, u]}q(\omega_t)\ge0\right]
&\le \hat P_x\left[ R_t\ge\delta\text{ for all }t\in[0,u]\right]
\le \hat P_x[\eta>u]\\
&=\int_u^\infty \frac{r}{\sqrt{2\pi t^3}}\exp\left(-\frac{(r+Kt)^2}{2t}\right)\,dt
+1-\exp(-2Kr)\\
&\le \int_u^\infty \frac{r}{\sqrt{2\pi t^3}}\,dt
+2Kr
=\sqrt{\frac2\pi}\frac{r}{\sqrt u}+2Kr.
\end{align*}
Letting $r\to q(x)$, we obtain the desired inequality.
\end{proof}
For $r>0$, define an $\{\hat{\mathcal F}_t\}$-stopping time $\tau_r$ by $\tau_r=\inf\{t\ge0\mid \omega_t\not\in\Omega_r\}$.
Let $\hat P_x^r$ be the law of $\tau_r$ under $\hat P_x$.
\begin{lmm}\label{lem:differentiable}
$\hat P_x^r([0,t])$ is differentiable in $t$ on $(0,\infty)$ and there exists a constant $C_2>0$ such that
$
\frac{d}{dt}\hat P_x^r([0,t])\le C_2 t^{-1}
$.
The constant $C_2$ is taken independently of $x$, $r$ and $t$.
\end{lmm}
\begin{proof}
  It is sufficient to consider the case that $x\in\Omega_r$.
  Let $p^r_t(\cdot,\cdot)$ be the transition density of the Brownian motion of $\Omega_r$ killed at $\partial \Omega_r$.
  Then,
  \begin{align*}
    \hat P_x^r([0,t])&=\hat P_x[\tau_r\le t]
    =1-\int_{\Omega_r}p^r_t(x,z)\,dz\\
    &=1-\iint_{\Omega_r\times\Omega_r}p^r_s(x,y)p^r_{t-s}(y,z)\,dy\,dz
  \end{align*}
  for $0<s<t$.
  From \cite[Theorem~6.17]{Ou}, $p^r_{t-s}(y,z)$ is differentiable in $t$ on $(s,\infty)$ for a.e.\ $(y,z)$ and the following estimate holds:
  \begin{align*}
  \frac{d}{dt}\hat P_x^r([0,t])
  &\le \iint_{\Omega_r\times\Omega_r} p^r_s(x,y)\left|\frac{\partial}{\partial t}p^r_{t-s}(y,z)\right|\,dy\,dz\\
  &\le C_3\iint_{\Omega_r\times\Omega_r} p^r_s(x,y)(t-s)^{-d/2-1}\exp\left(-\frac{C_4|y-z|_{{\mathbb R}^d}^2}{t-s}\right)\,dy\,dz\\
  &\le C_5 (t-s)^{-1}\int_{\Omega_r} p^r_s(x,y)\,dy
  \le C_5 (t-s)^{-1},
  \end{align*}
  where $C_3$, $C_4$ and $C_5$ are taken independently of $x$, $r$, $s$ and $t$. By letting $s=t/2$, we complete the proof.
\end{proof}
\begin{prp}\label{prop:path}
There exists $C_6>0$ such that for all $u>0$, $r>0$, and  $x\in \Omega$,
\[
  \hat P_x\left[0\le\inf_{t\in[0,u]}q(\omega_t)\le r\right]\le C_6(1+u^{-1/2})r.
\]
\end{prp}
\begin{proof}
First, let $x\in\Omega\setminus \Omega_r$.
From Proposition~\ref{prop:1} and the fact that $0<q(x)\le r$,
\[
\hat P_x\left[0\le\inf_{t\in[0,u]}q(\omega_t)\le r\right]
\le C_1(1+u^{-1/2})q(x)
\le C_1(1+u^{-1/2})r.
\]
Next, let $x\in\Omega_r$. Then,
{\allowdisplaybreaks
\begin{align*}
&\hat P_x\left[0\le\inf_{t\in[0,u]}q(\omega_t)\le r\right]
= \hat P_x\left[\tau_r\le u,\ 0\le\inf_{t\in[\tau_r,u]}q(\omega_t)\right]\\
&= \hat P_x\left[\tau_r\le u,\ 0\le\inf_{t\in[0,u-\tau_r]}q((\theta_{\tau_r}\omega)_t)\right]\\
&= \sum_{k=1}^\infty \hat P_x\left[2^{-k}u<u-\tau_r\le 2^{-k+1}u,\ 0\le\inf_{t\in[0,u-\tau_r]}q((\theta_{\tau_r}\omega)_t)\right]\\
&\le \sum_{k=1}^\infty \hat P_x\left[2^{-k}u<u-\tau_r\le 2^{-k+1}u,\ 0\le\inf_{t\in[0,2^{-k}u]}q((\theta_{\tau_r}\omega)_t)\right].
\end{align*}%
}%
Here, we used $\hat P_x[\tau_r=u]=0$ in the third line, which follows from Lemma~\ref{lem:differentiable}.
From the strong Markov property and Proposition~\ref{prop:1},
\begin{align*}
&\hat P_x\left[\left.2^{-k}u<u-\tau_r\le 2^{-k+1}u,\ 0\le\inf_{t\in[0,2^{-k}u]}q((\theta_{\tau_r}\omega)_t)\right|\hat{\mathcal F}_{\tau_r}\right]\\
&=1_{\{2^{-k}u<u-\tau_r\le 2^{-k+1}u\}}\cdot \hat P_{\omega_{\tau_r}}\!\left[0\le\inf_{t\in[0,2^{-k}u]}q(\omega_t)\right]\\
&\le C_1 1_{\{2^{-k}u<u-\tau_r\le 2^{-k+1}u\}}\cdot (1+(2^{-k}u)^{-1/2})r\\
&\le C_1 1_{\{2^{-k}u<u-\tau_r\le 2^{-k+1}u\}}\cdot (1+((u-\tau_r)/2)^{-1/2})r.
\end{align*}
Therefore,
{\allowdisplaybreaks
\begin{align*}
&\hat P_x\left[0\le\inf_{t\in[0,u]}q(\omega_t)\le r\right]\\
&\le C_1 r\hat E_x[1+((u-\tau_r)/2)^{-1/2};\ \tau_r\le u]\\
&\le C_1 r\int_0^u \left(1+\left(\frac{u-s}2\right)^{-1/2}\right)\,\hat P_x^r(ds)\\
&\le C_1 r(1+(u/4)^{-1/2})\hat P_x[\tau_r\le u/2]
+C_1 r\int_{u/2}^u \left(1+\left(\frac{u-s}2\right)^{-1/2}\right)C_2s^{-1}\,ds\\
&\le C_1 r(1+(u/4)^{-1/2})+\frac{2C_1C_2 r}{u}\int_{u/2}^u\left(1+\left(\frac{u-s}2\right)^{-1/2}\right)\,ds\\
&\le C_6(1+u^{-1/2})r.
\end{align*}%
}%
Here, we used Lemma~\ref{lem:differentiable} in the third line.
This completes the proof.
\end{proof}
Let $\hat P_{a,b}$ be a probability measure on $W$ such that $\{\omega_t\}_{t\in[0,1]}$ is a pinned Brownian motion under $\hat P_{a,b}$ with $\omega_0=a$ and $\omega_1=b$.
The following lemma is proved by the definition of $\hat P_{a,b}$ and the monotone class theorem.
\begin{lmm}\label{lem:A}
For $t\in [0,1)$, $A\in \hat{\mathcal F}_t$, and a Borel set $D$ of ${\mathbb R}^d$,
\[
  \hat P_{a,b}[A\cap \{\omega_t\in D\}]
  \le \hat P_{a}[A\cap \{\omega_t\in D\}]
  \cdot \sup_{y\in D}\frac{p_{1-t}(y,b)}{p_1(a,b)}.
\]
\end{lmm}
\begin{lmm}\label{lem:B}
Let $\tau$ be an $\{\hat{\mathcal F}_t\}$-stopping time and $A\in \hat{\mathcal F}_\tau$.
Let $D$ be an open set of ${\mathbb R}^d$.
Then,
\[
\hat P_{a,b}[\{\tau<1\}\cap A\cap\{\omega_\tau\in D\}]
\le \hat P_a[\{\tau<1\}\cap A\cap\{\omega_\tau\in \overline{D}\}]
\cdot \sup_{t\in(0,1],\,y\in D}\frac{p_t(y,b)}{p_1(a,b)}.
\]
Here, $\overline{D}$ is a closure of $D$ in ${\mathbb R}^d$.
\end{lmm}
\begin{proof}
Consider a sequence of $\{\hat{\mathcal F}_t\}$-stopping times $\{\tau_n\}$ such that each $\tau_n$ takes only finite values of $\{t_n^{(k)}\}_{k\in\Lambda_n}$ and $\tau_n\downarrow\tau$.
Here, $\Lambda_n$ is an index set consisting of finite elements.
Then,
\begin{align*}
\{\tau<1\}\cap A\cap\{\omega_{\tau}\in D\}
&\subset\liminf_{n\to\infty}
(\{\tau_n<1\}\cap A\cap\{\omega_{\tau_n}\in D\})\\
&\subset\limsup_{n\to\infty}
(\{\tau_n<1\}\cap A\cap\{\omega_{\tau_n}\in D\})\\
&\subset \{\tau<1\}\cap A\cap\{\omega_{\tau}\in \overline{D}\}.
\end{align*}
For $t\in[0,1)$, $\{\tau_n=t\}\cap A\cap \{\omega_{\tau_n}\in D\}\in\hat {\mathcal F}_t$.
Therefore, from Lemma~\ref{lem:A},
\begin{align*}
&\hat P_{a,b}[\{\tau_n<1\}\cap A\cap \{\omega_{\tau_n}\in D\}]\\
&=\sum_{k\in \Lambda_n,\,t_n^{(k)}<1}\hat P_{a,b}[\{\tau_n=t_n^{(k)}\}\cap A\cap\{\omega_{\tau_n}\in D\}]\\
&\le\sum_{k\in \Lambda_n,\,t_n^{(k)}<1}\hat P_{a}[\{\tau_n=t_n^{(k)}\}\cap A\cap\{\omega_{\tau_n}\in D\}]
\cdot\sup_{y\in D}\frac{p_{1-t_n^{(k)}}(y,b)}{p_1(a,b)}\\
&\le\sum_{k\in \Lambda_n,\,t_n^{(k)}<1}\hat P_{a}[\{\tau_n=t_n^{(k)}\}\cap A\cap\{\omega_{\tau_n}\in D\}]
\cdot\sup_{t\in(0,1],\,y\in D}\frac{p_t(y,b)}{p_1(a,b)}\\
&=\hat P_a[\{\tau_n<1\}\cap A\cap\{\omega_{\tau_n}\in D\}]
\cdot\sup_{t\in(0,1],\,y\in D}\frac{p_t(y,b)}{p_1(a,b)}.
\end{align*}
By letting $n\to\infty$, we complete the proof from Fatou's lemma.
\end{proof}
Denote the Borel $\sigma$-field on $[0,\infty)$ by ${\cal B}([0,\infty))$.
\begin{lmm}\label{lem:C}
Let $\tau$ be an $\{\hat{\mathcal F}_t\}$-stopping time such that $\tau\le1$
and $\Gamma\subset [0,\infty)\times W$ an element in ${\cal B}([0,\infty))\otimes\hat{\mathcal F}_\infty$.
Assume that
\[
\{\tau<1\}\cap\{((1-\tau)/2,\theta_\tau\omega)\in\Gamma\}\in \hat{\mathcal F}_{(1+\tau)/2}.
\]
Then, for a Borel set $D$ of ${\mathbb R}^d$,
\begin{align*}
&\hat P_{a,b}[\{\tau<1\}\cap\{\omega_\tau\in D\}\cap
\{((1-\tau)/2,\theta_\tau\omega)\in\Gamma\}]\\
&\le \sup_{x\in D}\hat E_x\left[\sup_{s\in(0,1/2]}1_\Gamma(s,\omega)\cdot s^{-d/2}\right]\exp(|a-b|_{{\mathbb R}^d}^2/2).
\end{align*}
\end{lmm}
\begin{proof}
Let $c\in(0,1)$. Then,
\begin{align*}
&\hat P_{a,b}[\{\tau<1\}\cap\{\omega_\tau\in D\}\cap
\{((1-\tau)/2,\theta_\tau\omega)\in\Gamma\}]\\
&=\sum_{k=0}^\infty\hat P_{a,b}[\{c^{k+1}<1-\tau\le c^k\}\cap\{\omega_\tau\in D\}\cap
\{((1-\tau)/2,\theta_\tau\omega)\in\Gamma\}].
\end{align*}
Since $c^{k+1}<1-\tau\le c^k$ implies that $\tau< 1-c^{k+1}$ and $(1+\tau)/2<1-c^{k+1}/2$, by combining the assumption,
the set in $\hat P_{a,b}[{}\cdots{}]$ belongs to $\hat{\mathcal F}_{1-c^{k+1}/2}$.
From Lemma~\ref{lem:A} and the strong Markov property, the above equation is dominated by
\begin{align*}
&\sum_{k=0}^\infty\hat P_a[\{c^{k+1}<1-\tau\le c^k\}\cap\{\omega_\tau\in D\}\cap
\{((1-\tau)/2,\theta_\tau\omega)\in\Gamma\}]\\
&\phantom{\sum_{k=0}^\infty}\cdot p_1(a,b)^{-1}(\pi c^{k+1})^{-d/2}\\
&\le \hat E_a[1_{\{\tau<1\}\cap\{\omega_\tau\in D\}\cap
\{((1-\tau)/2,\theta_\tau\omega)\in\Gamma\}}\cdot (\pi(1-\tau)c)^{-d/2}]\cdot p_1(a,b)^{-1}\\
&\le \hat E_a\!\left[1_{\{\tau<1\}\cap\{\omega_\tau\in D\}}
 \sup_{s\in(0,1/2]}\left(1_\Gamma(s,\theta_\tau\omega)\cdot (2\pi sc)^{-d/2}\right)\right]\cdot p_1(a,b)^{-1}\\
&= \hat E_a\!\left[1_{\{\tau<1\}\cap\{\omega_\tau\in D\}}
\hat E_{\omega_\tau}\!\left[\sup_{s\in(0,1/2]}1_\Gamma(s,\omega)\cdot (2\pi sc)^{-d/2}\right]\right]\cdot p_1(a,b)^{-1}\\
&\le \sup_{x\in D}
\hat E_x\!\left[\sup_{s\in(0,1/2]}1_\Gamma(s,\omega)\cdot s^{-d/2}\right]\cdot c^{-d/2}\exp(|a-b|_{{\mathbb R}^d}^2/2).
\end{align*}
By letting $c\to1$, we reach the conclusion.
\end{proof}
\begin{prp}\label{prop:2}
  There exists $C_7>0$ such that for every $r>0$,
  \begin{equation}\label{eq:r}
  \mu_{a,b}\left[w\in W_{a,b}\left|\;0\le\inf_{t\in[0,1]}q(w(t))\le r\right]\right.
  \le C_7 r.
  \end{equation}
\end{prp}
\begin{proof}
  Let $\alpha=\min\{q(a),q(b)\}/2$.
  It is sufficient to prove that there exists $C_7>0$ such that \eqref{eq:r} holds for all $r\in(0,\alpha/3)$.
  Choose $r\in(0,\alpha/3)$ and let $V=B(b,\alpha)$ and $V'=B(b,\alpha/2)$. Then,
  \begin{align*}
  &\mu_{a,b}\left[w\in W_{a,b}\left|\ 0\le\inf_{t\in[0,1]}q(w(t))\le r\right.\right]\\
  &\le \hat P_{a,b}[\tau_r<1\text{ and }\omega_t\in\overline{\Omega}\setminus V \text{ for all }t\in[\tau_r,(\tau_r+1)/2]]\\
  &\phantom{{}\le{}}{}+ \hat P_{a,b}\left[\begin{array}{l}\tau_r<1,\ \omega_t\in V \text{ for some }t\in[\tau_r,(\tau_r+1)/2],\\
  \text{and }
  \omega_t\in\overline{\Omega} \text{ for all }t\in[\tau_r,(\tau_r+1)/2]
  \end{array}
  \right]\\
  &=:I_1+I_2.
\end{align*}
For $I_1$, Lemma~\ref{lem:B} with $\tau=\min\{(\tau_r+1)/2, 1\}$ implies
\begin{align*}
  I_1&= \hat P_{a,b}[\tau_r<1,\ \omega_t\in\overline{\Omega}\setminus V \text{ for all }t\in[\tau_r,\tau],\ \text{and }\omega_\tau\in{\mathbb R}^d\setminus V']\\
  &\le\hat P_a[\tau_r<1,\ \omega_t\in\overline{\Omega}\setminus V \text{ for all }t\in[\tau_r,\tau]]\cdot \sup_{t\in(0,1],\ y\in{\mathbb R}^d\setminus V'}\frac{p_t(y,b)}{p_1(a,b)}.
\end{align*}
Now,
  \[
  \sup_{t\in(0,1],\ y\in{\mathbb R}^d\setminus V'}\frac{p_t(y,b)}{p_1(a,b)}
  \le p_1(a,b)^{-1}\sup_{t\in(0,1]}(2\pi t)^{-d/2}\exp\left(-\frac{(\alpha/2)^2}{2t}\right)
  \le C_8
  \]
  and
  \begin{align*}
  &\hat P_a[\tau_r<1,\ \omega_t\in\overline{\Omega}\setminus V \text{ for all }t\in[\tau_r,\tau]]\\
  &\le \hat P_a[\tau_r<1,\ (\theta_{\tau_r} \omega)_t\in\overline{\Omega} \text{ for all }t\in[0,(1-\tau_r)/2]]\\
  &\le \sum_{k=1}^\infty \hat P_a[2^{-k}<1-\tau_r\le 2^{-k+1},\ (\theta_{\tau_r} \omega)_t\in\overline{\Omega} \text{ for all }t\in[0,2^{-k-1}]].
  \end{align*}
  Since
  \begin{align*}
  &\hat P_a[2^{-k}<1-\tau_r\le 2^{-k+1},\ (\theta_{\tau_r} \omega)_t\in\overline{\Omega} \text{ for all }t\in[0,2^{-k-1}]\mid \hat{\mathcal F}_{\tau_r}]\\
  &=1_{\{2^{-k}<1-\tau_r\le 2^{-k+1}\}}\cdot\hat P_{\omega_{\tau_r}}[\omega_t\in\overline{\Omega} \text{ for all }t\in[0,2^{-k-1}]]\\
  &\le 1_{\{2^{-k}<1-\tau_r\le 2^{-k+1}\}}\cdot C_1(1+2^{(k+1)/2})r\\
  &\le 1_{\{2^{-k}<1-\tau_r\le 2^{-k+1}\}}\cdot C_1(1+2(1-\tau_r)^{-1/2})r
  \end{align*}
from the strong Markov property and Proposition~\ref{prop:1},
  \begin{align*}
  I_1&\le C_8C_1r \hat E_a[1_{\{\tau_r<1\}}\cdot(1+2(1-\tau_r)^{-1/2})]\\
  &\le C_8C_1 r(1+2\sqrt2)\hat P_a[\tau_r\le 1/2]
  +C_8C_1 r \int_{1/2}^1 (1+2(1-s)^{-1/2})\cdot C_2 s^{-1}\,ds\\
  &\le C_9 r,
  \end{align*}
  by virtue of Lemma~\ref{lem:differentiable}.

  We will estimate a value for $I_2$.
  From Lemma~\ref{lem:C} with $\tau=\min\{\tau_r,1\}$, $D=\partial\Omega_r$, and
  \[
  \Gamma=\left\{(s,\omega)\in[0,\infty)\times W\left|\;
  \begin{array}{l}
  \omega_t\in V \text{ for some }t\in[0,s]\text{ and}\\
  \omega_t\in\overline{\Omega} \text{ for all }t\in[0,s]
  \end{array}\right.\right\},
  \]
  we obtain
  \[
  I_2\le C_{10} \sup_{x\in \partial\Omega_r}\hat E_x\left[\sup_{s\in(0,1/2]}1_\Gamma(s,\omega)\cdot s^{-d/2}\right].
  \]
  By letting $f(\omega)=\sup_{s\in(0,1/2]}1_\Gamma(s,\omega)\cdot s^{-d/2}$, we have
  \begin{equation}\label{eq:I2}
  I_2
  \le C_{10}\sup_{x\in\partial\Omega_r}\hat E_x[f]
  =C_{10} \sup_{x\in\partial\Omega_r}\int_0^\infty \hat P_x[f> u]\,du.
  \end{equation}
  Let $x\in\partial\Omega_r$, and define $y$, $z$, $K$, $R_t$, and $S_t$ as in Lemma~\ref{lem:ito}.
  We have $|x-z|_{{\mathbb R}^d}=\delta+r\in(\delta,\delta+\alpha)$ and $|b-z|_{{\mathbb R}^d}\ge\delta+q(b)\ge\delta+2\alpha$.
  Define the stopping times with respect to the canonical augmentation of $\{\hat{\mathcal F}_t\}$ by $\{\hat P_x\}_{x\in{\mathbb R}^d}$ as follows:
  \begin{align*}
  \rho&=\inf\{t\ge0\mid R_t\notin [\delta, \delta+\alpha)\},\\
  \rho'&=\inf\{t\ge0\mid r+\delta+Kt+S_t\notin [\delta,\delta+\alpha)\}.
  \end{align*}
  Since $\omega_t\in V$ implies $R_t\ge|b-z|_{{\mathbb R}^d}-\alpha\ge\delta+\alpha$,
  \begin{align*}
  &\hat P_x\left[f> u\right]\\
  &\le\hat P_x[(s,\omega)\in\Gamma \text{ for some }s< u^{-2/d}]\\
  &\le \hat P_x[\rho< u^{-2/d}\text{ and }R_\rho=\delta+\alpha]\\
  &\le \hat P_x[\rho'< u^{-2/d}\text{ and }r+\delta+K\rho'+S_{\rho'}=\delta+\alpha]
  \quad\text{(from Lemma~\ref{lem:ito})}\\
  &\le \hat E_x[\exp(1-u^{2/d}\rho');\, r+K\rho'+S_{\rho'}=\alpha]\\
  &=e^{1+K(\alpha-r)}{\sinh\left(r\sqrt{2u^{2/d}+K^2}\right)}\left/{\sinh\left(\alpha\sqrt{2u^{2/d}+K^2}\right)}\right..
  \quad \text{(cf.\ \cite[p.~309]{BS})}
  \end{align*}
  Since $(v/4)e^{v/2}\le \sinh v\le v e^v$ for $v\ge0$, the above term is dominated by
  \begin{align*}
  &\frac{e^{1+K(\alpha-r)}r\sqrt{2u^{2/d}+K^2}\exp\left(r\sqrt{2u^{2/d}+K^2}\right)}{(\alpha/4)\sqrt{2u^{2/d}+K^2}\exp\left(\alpha\sqrt{2u^{2/d}+K^2}/2\right)}\\
  &=4\alpha^{-1}e^{1+K(\alpha-r)}r\exp\left((r-\alpha/2)\sqrt{2u^{2/d}+K^2}\right)\\
  &\le 4\alpha^{-1}e^{1+K\alpha}r\exp(-\sqrt{2}\alpha u^{1/d}/6).
  \end{align*}
Substituting this estimate into \eqref{eq:I2},
we obtain $I_2\le C_{11} r$.
  This completes the proof.
\end{proof}
\section{Proof of Theorem~\ref{th:main}}
In this section, we prove Theorem~\ref{th:main}.
We retain the notations in the previous sections.
We will utilize the following theorem.
\begin{thm}[(\cite{ES})]\label{th:ES}
Let $F$ be a measurable function on $W_{a,b}$ and $H_0$-Lipschitz; in other words, there exists $C>0$ such that
\[
|F(w+h)-F(w)|\le C|h|_{H_0},\quad
w\in W_{a,b},\ h\in H_0.
\]
Then, if $\int_{W_{a,b}} F^2\,d\mu_{a,b}<\infty$, $F$ belongs to ${\mathbb D}^{1,2}$.
Here, ${\mathbb D}^{1,2}$ is a first order $L^2$-Sobolev space on $W_{a,b}$ that is defined as the completion of ${\mathcal F} C_b^1$ with respect to the norm $(\|\nabla\cdot\|_{L^2(W_{a,b}\to H_0,\mu_{a,b})}^2+\|\cdot\|_{L^2(W_{a,b},\mu_{a,b})}^2)^{1/2}$.
Moreover, $|\nabla F(\omega)|_{H_0}\le C$ $\mu_{a,b}$-a.e.
\end{thm}

From Proposition~\ref{prop:2}, for any $r>0$,
\[
\mu_{a,b}\left[\overline{W_{a,b}^\Omega}\setminus W_{a,b}^\Omega\right]
=\mu_{a,b}\left[ \inf_{s\in[0,1]}
q(w(s))=0\right]
\le C_7 r.
\]
Therefore, $\mu_{a,b}\left[\overline{W_{a,b}^\Omega}\setminus W_{a,b}^\Omega\right]=0$.
By combining this with the remark in \cite[p.~230]{Fu},
the bilinear form $({\mathcal E}',{\mathcal F} C_b^1)$ is closable on $L^2(\overline{W_{a,b}^\Omega},\mu_{a,b}|_{\overline{W_{a,b}^\Omega}})$, and the closure $({\mathcal E}',{\mathcal F}')$ is a quasi-regular, local, and conservative Dirichlet form.
In particular, we obtain the diffusion process $(X'_t,P'_x)$ on $\overline{W_{a,b}^\Omega}$ associated with $({\mathcal E}',{\mathcal F}')$.

Next, we prove that $1_{\overline{W_{a,b}^\Omega}} \in BV(W_{a,b})$.
Define $F(w)=\inf_{t\in[0,1]}q(w(t))$ for $w\in W_{a,b}$.
For $n\in{\mathbb N}$, we define $f_n(s)=\min\{\max\{0, ns\},1\}$ for $s\in{\mathbb R}$ and $\rho_n(w)=f_n\left(F(w) \right)$ for $w\in W_{a,b}$.
Then, since ${W_{a,b}^\Omega}=\{F(w)>0\}$, we obtain $\lim_{n\to \infty}\rho_n =1_{{W_{a,b}^\Omega}}$ $\mu_{a,b}$-a.e.\ and in $L^1(W_{a,b},\mu_{a,b})$.
Therefore, from Theorem~\ref{th:1}, it is sufficient to prove
$\sup_n \|\nabla\rho_n\|_{L^1(W_{a,b}\to H_0,\mu_{a,b})} <\infty$.
We note that $q(x)$ is a Lipschitz function on ${\mathbb R}^d$ with Lipschitz constant 1; thus, we obtain the following estimate for $w\in W_{a,b}$ and $h\in H_0$
\begin{align*}
	|F(w+h)-F(w)| &= \left|\inf_{t\in[0,1]}q(w(t)+h(t))-\inf_{t\in[0,1]}q(w(t))\right|\\
	&\le \sup_{t\in[0,1]}|q(w(t)+h(t))-q(w(t))|\\
	&\le \sup_{t\in[0,1]}|h(t)|
	\le |h|_{H_0}.
\end{align*}
Thus, $F$ is $H_0$-Lipschitz continuous.
From Theorem~\ref{th:ES}, we deduce that $F\in {\mathbb D}^{1,2}$ and $|\nabla F|_{H_0}\le 1$ $\mu_{a,b}$-a.e.

Now, we use the chain rule of $H_0$-derivative to obtain
\begin{align*}
	\|\nabla \rho_n\|_{L^1(W_{a,b}\to H_0,\mu_{a,b})} &\le \left\|n 1_{\{0\le F \le 1/n\}}|\nabla F|_{H_0}\right\|_{L^1(W_{a,b},\mu_{a,b})}\\
	&\le n\mu_{a,b}\left[0\le \inf_{t\in[0,1]}q(w(t)) \le \frac1n\right].
\end{align*}
According to Proposition~\ref{prop:2}, $\sup_n \|\nabla\rho_n\|_{L^1(W_{a,b}\to H_0,\mu_{a,b})} <\infty$.

By virtue of Theorem~\ref{th:2}, the process $X_t:=\lambda_{a,b}(X'_t)$ satisfies the Skorokhod equation \eqref{eq:skorohod}.

Next, we will prove $\mathop{\rm Cap}\nolimits\left(\partial W_{a,b}^\Omega\setminus\partial' W_{a,b}^\Omega\right)=0$, where $\mathop{\rm Cap}\nolimits$ denotes the capacity associated with $({\mathcal E}',{\mathcal F}')$.
When $w\in \partial W_{a,b}^\Omega\setminus\partial' W_{a,b}^\Omega$, there exist at least two points $t\in(0,1)$ such that $w(t)\in \partial\Omega$.
Therefore,
\begin{align}\label{eq:cap2}
&\partial W_{a,b}^\Omega\setminus\partial' W_{a,b}^\Omega\\ &\subset \bigcup_{\begin{subarray}{c}0<s_1<s_2<1\\s_1,s_2\in{{\mathbb Q}}\end{subarray}}
\left\{ w\in W_{a,b}\left|\;\inf_{t\in[0,s_1]}q(w(t))=0,\ \inf_{t\in[s_1,s_2]}q(w(t))=0
\right\}\right.. \nonumber
\end{align}
For $\alpha,\beta \in {\mathbb R}$ and $s_1,s_2\in (0,1)$ with $s_1<s_2$, we define
\[A_{s_1,s_2,(\alpha ,\beta)} =\left\{w\in W_{a,b}\left|\;\inf_{t\in[0,s_1]}q(w(t))=\alpha,\ \inf_{t\in[s_1,s_2]}q(w(t))=\beta\right\}\right..\]
The right-hand side of \eqref{eq:cap2} is rewritten as $\bigcup_{0<s_1<s_2<1,\,s_1,s_2\in{{\mathbb Q}}} A_{s_1,s_2,(0,0)}$.
For a subset $G$ of ${\mathbb R}^2$, we denote $\bigcup_{(\alpha , \beta)\in G}A_{s_1,s_2,(\alpha ,\beta)}$ by $A_{s_1,s_2,G}$.

Fix $s_1,s_2\in (0,1)$ with $s_1<s_2$.
We define a map $f\colon W_{a,b}\to {\mathbb R}^2$ by
\[
f(w)=\Bigl(\inf_{t\in[0,s_1]}q(w(t)),\ \inf_{t\in[s_1,s_2]}q(w(t))\Bigr).
\]
We denote an open ball in ${\mathbb R}^2$ with its center at $0$ and radius $r$ by $B(r)$.
By the continuity of $f$, $A_{s_1,s_2,B(r)}$ is an open neighborhood of $A_{s_1,s_2,(0,0)}$ in $W_{a,b}$.

Take $\varepsilon >0$ and $\lambda \in (0,1)$.
We choose a smooth function $g$ on $[0,\infty)$ such that
\[
g(t)=\left\{\begin{array}{lcl}
1 &&\text{for } t \in [0,\lambda \varepsilon),\\
\dfrac{3\log (t/{\varepsilon})}{\log \lambda}-1 &&\text{for } t\in (\lambda^{5/9}\varepsilon , \lambda^{4/9}\varepsilon),\\
0 &&\text{for }t\in (\varepsilon, \infty),
\end{array}\right.
\]
and $3(t\log \lambda)^{-1}\le g'(t) \le 0$ for all $t\in(0,\infty)$.
We define a function $\zeta\colon {{\mathbb R}}^2\to{\mathbb R}$ by $\zeta(x,y)=\sqrt{x^2+y^2}$ and set
$\iota = g\circ \zeta $.
Since $\iota\circ f$ is a bounded $H_0$-Lipschitz continuous function, it belongs to ${{\mathbb D}}^{1,2}$ --- to ${{\mathcal F}'}$ in particular --- by virtue of Theorem~\ref{th:ES}.
Moreover, $\iota\circ f=1$ on $A_{s_1,s_2,B(\lambda \varepsilon)}$.
Denoting the gradient operator on ${\mathbb R}^2$ by $\nabla_{{\mathbb R}^2}$, we have
{\allowdisplaybreaks
\begin{align*}
{{\mathcal E}'}(\iota\circ f,\iota\circ f)
&=\frac12\int_{\overline{W_{a,b}^\Omega}} |\nabla (\iota\circ f)|^2_{H_0} \, d\mu_{a,b}\\
&=\frac12\int_{\overline{W_{a,b}^\Omega}} |\langle (\nabla f)(w),(\nabla_{{{\mathbb R}}^2} \,\iota)(f(w)) \rangle_{{{\mathbb R}}^2}|^2_{H_0} \, \mu_{a,b}(dw)\\
&\le C_{12}\int_{\overline{W_{a,b}^\Omega}}  |(\nabla_{{\mathbb R}^2} \,\iota)(f(w))|_{{\mathbb R}^2}^2\, \mu_{a,b}(dw)\\
&=C_{12}\int_{\{(x,y)\in {\mathbb R}^2 \mid x\geq 0 ,y\geq 0\}}|\nabla_{{\mathbb R}^2} \,\iota|_{{\mathbb R}^2}^2\, d(f_*(\mu_{a,b}|_{\overline{W_{a,b}^\Omega}}))\\
&=:I_3.
\end{align*}}%
Here, $f_*(\mu_{a,b}|_{\overline{W_{a,b}^\Omega}})$ denotes the image measure of $\mu_{a,b}|_{\overline{W_{a,b}^\Omega}}$ by $f$.
In the second line, $\langle \cdot,\cdot \rangle_{{{\mathbb R}}^2}$ denotes a pairing between the elements in $H_0\otimes {\mathbb R}^2$ and in ${\mathbb R}^2$ and has values in $H_0$.
The inequality from the second line to the third follows from the fact that $f$ is $H_0$-Lipschitz continuous.
Now,
\begin{align*}
|\nabla_{{\mathbb R}^2} \,\iota|_{{\mathbb R}^2}^2
&=\left({\partial \iota}/{\partial x}\right)^2+\left({\partial \iota}/{\partial y}\right)^2	\\
&=\left(g'\circ \zeta(x,y)\right)^2\frac{x^2}{x^2+y^2}+\left(g'\circ \zeta(x,y)\right)^2\frac{y^2}{x^2+y^2}\\
&=\left(g'\circ \zeta(x,y)\right)^2.
\end{align*}
By considering $\xi=(\zeta\circ f)_*(\mu_{a,b}|_{\overline{W_{a,b}^\Omega}})$, we obtain
\[
	I_3=C_{12}\int_{0}^{\infty}g'(r)^2\,\xi(dr) \le 9C_{12}\int_{\lambda\varepsilon}^{\varepsilon}(\log \lambda)^{-2}r^{-2} \,\xi(dr)=:I_4.
\]
From Lemma~\ref{lem:A}, the Markov property of $(\omega_t,\hat P_x)$ and Proposition~\ref{prop:path},
{\allowdisplaybreaks
\begin{align*}
\Xi(r)&:=\xi([0,r])
= (f_*(\mu_{a,b}|_{\overline{W_{a,b}^\Omega}}))(\zeta^{-1}([0,r]))\\
&=\mu_{a,b}\left[ w\in \overline{W_{a,b}^\Omega} \left|\ \inf_{t\in[0,s_1]}q(w(t))^2+\inf_{t\in[s_1,s_2]}q(w(t))^2 \le r^2\right. \right]\\
&\le\mu_{a,b}\left[ w\in \overline{W_{a,b}^\Omega} \left|\ 0\le\inf_{t\in[0,s_1]}q(w(t))\le r,\ 0\le\inf_{t\in[s_1,s_2]}q(w(t)) \le r\right. \right]\\
&\le \hat P_a \left[\ 0\le \inf_{t\in[0,s_1]}q(\omega_t)\le r,\  0\le \inf_{t\in[s_1,s_2]}q(\omega_t)\le r \right]
\cdot \frac{p_{1-s_2}(b,b)}{p_1(a,b)}\\
&=C_{13}\hat E_a\left[1_{\{0\le \inf_{t\in[0,s_1]}q(\omega_t)\le r\}}\hat P_{\omega_{s_1}}\left[ 0\le \inf_{t\in[0,s_2-s_1]}q(\omega_t)\le r  \right]\right]\\
&\le C_{14}(1+(s_2-s_1)^{-1/2})r \hat P_a\left[ 0 \le \inf_{t\in[0,s_1]}q(\omega_t)\le r\right]\\
&\le  C_{15}(1+(s_2-s_1)^{-1/2})(1+s_1^{-1/2})r^2
=C_{16}r^2.
\end{align*}%
}%
Thus, we obtain
\begin{align*}
I_4&=9C_{12}\int_{\lambda\varepsilon}^{\varepsilon}(\log{\lambda})^{-2} \frac{1}{r^2} \,d\Xi(r)\\
&=9C_{12}(\log{\lambda})^{-2}  \left\{\left[\frac{\Xi(r)}{r^2}\right]_{\lambda\varepsilon}^{\varepsilon}+\int_{\lambda\varepsilon}^{\varepsilon}\frac{2\Xi(r)}{r^3} \,dr\right\}\\
&\le C_{17}(\log{\lambda})^{-2}  \Bigl(1+\int_{\lambda\varepsilon}^{\varepsilon}\frac{1}{r}  \,dr\Bigr)\\
&=C_{17}(\log{\lambda})^{-2} (1-\log {\lambda}).
\end{align*}
Therefore,
\begin{align*}
\mathop{\rm Cap}\nolimits(A_{s_1,s_2,(0,0)})
&\le \mathop{\rm Cap}\nolimits(A_{s_1,s_2,B(\lambda \varepsilon)})\\
&\le {\mathcal E}'(\iota\circ f,\iota\circ f) + \|\iota\circ f\|^2_{L^2({\overline{W_{a,b}^\Omega}},\mu|_{\overline{W_{a,b}^\Omega}})}\\
&\le {\mathcal E}'(\iota\circ f,\iota\circ f) +\Xi(\varepsilon)\\
&\le  C_{17}(\log{\lambda})^{-2}  (1-\log {\lambda})+C_{16}\varepsilon^2.
\end{align*}
By letting $\varepsilon \to 0$ and $\lambda \to 0$, we obtain $\mathop{\rm Cap}\nolimits(A_{s_1,s_2,(0,0)})=0$.
Therefore,
\[
\mathop{\rm Cap}\nolimits(\partial W_{a,b}^\Omega\setminus \partial' W_{a,b}^\Omega)
\le\sum_{0<s_1<s_2<1,\,s_1,s_2\in{{\mathbb Q}}}\mathop{\rm Cap}\nolimits(A_{s_1,s_2,(0,0)})=0.
\]
The last claim follows from the above result and Theorem~\ref{th:3}, and the fact that $\nu$ is a smooth measure.
This completes the proof of Theorem~\ref{th:main}.

\section{Concluding remarks}
\begin{enumerate}
\item  In a similar and simpler way, we can prove the counterpart of Theorem~\ref{th:main} that concerns the one-sided pinned path space on $\Omega$; this theorem was proved in \cite{U}.
  More precisely, we define the path spaces as follows:
\begin{align*}
W_{a}&=\{w\in C([0,1]\to{\mathbb R}^d)\mid w(0)=a\},\\
W_{a}^\Omega&=\{w\in C([0,1]\to \Omega)\mid w(0)=a\},\\
H&=\left\{h\in C([0,1]\to{\mathbb R}^d)\left|\,
\begin{array}{l}
h(0)=0,\ \text{$h$ is abslutely continuous}\\
\text{and } \int_0^1 |\dot h(s)|_{{\mathbb R}^d}^2\,ds<\infty
\end{array}\right.\right\},\\
  \partial W_{a}^\Omega&=\left\{w\in W_{a}\left|\,
  \begin{array}{l}
  w(t)\in\overline{\Omega}\text{ for every $t\in[0,1]$ and}\\
  w(s)\in\partial \Omega\text{ for some $s\in(0,1]$}
  \end{array}
  \right.\right\},\\
  \partial' W_{a}^\Omega&=\left\{w\in \partial W_{a}^\Omega\left|\,
  \begin{array}{l}
  \text{there exists a unique }s\in(0,1]\\
   \text{such that } w(s)\in\partial\Omega
   \end{array}\right.\right\},\\
  \overline{W_{a}^\Omega}&=W_{a}^\Omega\cup \partial W_{a}^\Omega.
\end{align*}
Let $\mu_a$ be the probability measure on $W_a$ that is the law of the $d$-\hspace{0pt}dimensional Brownian motion starting at $a$.
Then, we can prove the claim that is modified by replacing $W_{a,b}$, $W_{a,b}^\Omega$, $\mu_{a,b}$, $W_{0,0}$, $\mu_{0,0}$, $H_0$, and $\lambda_{a,b}$ in Theorem~\ref{th:main} by $W_a$, $W_a^\Omega$, $\mu_a$, $W_0$, $\mu_0$, $H$, and $\lambda_a\colon W_a\ni w\mapsto w-a\in W_0$, respectively.
Also, from \cite[Theorem~4.4]{Hi}, $BV(W_0)\cap \bigcap_{q>1}L^q(W_0,\mu_0)\subset {\mathbb D}^{\alpha,p}$ if $p>1$ and $\alpha p<1$, where ${\mathbb D}^{\alpha,p}$ is a Sobolev space on $W_0$ with differentiability index $\alpha$ and integrability index $p$ according to the Malliavin calculus.
Therefore, we obtain the following theorem, which is a generalization of a part of the results of \cite{AZ}.
\begin{thm}
Assume that $0\in\Omega$ and $\Omega$ satisfies the uniform exterior ball condition.
Then, $\mu_0(\partial W_0^\Omega)=0$ and $1_{W_0^\Omega}\in BV(W_0)$.
Moreover, for any real numbers $\alpha$ and $p$ such that $p>1$ and $\alpha p<1$,
the function $1_{W_0^\Omega}$ belongs to ${\mathbb D}^{\alpha,p}$.
\end{thm}
\item Precisely speaking, the diffusion process associated with $({\mathcal E}',{\mathcal F}')$ should be called the {\em modified} reflecting Ornstein-Uhlenbeck process as in \cite{Fu,FH}, since ${\mathcal F}'$ is defined as the completion of ${\mathcal F} C_b^1$ and may be strictly smaller than the ``canonical'' first order $L^2$-Sobolev space $H^1(W_{a,b}^\Omega)$.
When ${\mathcal F}'$ is equal to $H^1(W_{a,b}^\Omega)$ remains an open problem;
a partial answer is provided in \cite{Hi0}.
\end{enumerate}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
The perturbative exploring of quark-gluon plasma (QGP) has some
difficulties in describing the physics of QGP and phase
transitions. However, it was realized 30 years ago that
nonperturbative (np) vacuum fields are strong (\cite{1}) and later
it was predicted (\cite{2}) and confirmed on the lattice
(\cite{3}) that the magnetic part of gluon condensate does not
decrease at $T>T_c$ and even grows as $T^4$ at large $T$ (\cite{3a}).

Therefore it is natural to apply the np approach, the Field
Correlator Method (FCM) (\cite{4}) to the problem of QGP and phase
transitions, which was done in a series of papers
(\cite{5}-\cite{9}). As a result one obtains np equation of state
(EoS) of QGP and the full picture of phase transition, including
an unbiased prediction for the critical temperature $T_c(\mu)$ for
different number of flavors $n_f$.

\section{Nonperturbative EoS of QGP} We split the gluonic
field $A_\mu$ into the background field $B_\mu$ and the (valence
gluon) quantum field $a_\mu$: $A_\mu=B_\mu+a_\mu$ both satisfying
the periodic boundary conditions.

The partition function averaged both in perturbative and np fields
is \begin{equation} Z(V,T)=\langle Z(B+a)\rangle_{B,a}\label{1}\end{equation}

Exploring free energy $F(T,\mu)=-T \ln \langle Z(B)\rangle_B$ that
contains perturbative and np interactions of quarks and gluons
(which also includes creation and dissociation of bound states) we
follow so-called Single Line Approximation (SLA). Namely, we
assume that quark-gluon system for $T>T_c$ stays gauge invariant,
as it was for $T<T_c$, and neglect all perturbative interactions
in the first approximation. Nevertheless in SLA already exist a
strong interaction of gluons (and quarks) with np vacuum fields.
This interaction consists of colorelectric (CE) and colormagnetic
(CM) parts. The CE part in deconfinement phase creates np
self-energy contribution for every quark and gluon embedded in
corresponding Polyakov line. An important point is that Polyakov
line is computed from the gauge invariant $q\bar q$ (gg) Wilson
loop, which for np $D_1^E$ interaction splits into individual
quark (gluon) contributions. As for CM part - its consideration is
beyond the SLA, because as has been recently shown in paper
(\cite{10})
 strong CM fields are responsible for creation of
bound states of white combinations of quarks and gluons.

To proceed with FCM we apply the nonabelian Stokes theorem and the
Gaussian approximation to compute the Polyakov line in terms of np field correlators

\begin{multline*}
L_{fund} = \frac{1}{N_c} tr\ P\exp\left(ig \int_0^\beta B_4(z)dz^4\right)= \\
\frac{1}{N_c} tr\
\exp\left(-\frac{g^2}{2}\int_{S_n}\int_{S_n}d\sigma_{\mu\nu}(u)
d\sigma_{\lambda\sigma}(v)D_{\mu\nu,\lambda\sigma}\right)
\end{multline*}

with \begin{equation} D_{\mu\nu,\lambda\sigma}\equiv g^2\langle
F_{\mu\nu}(u)\Phi(u,v)F_{\lambda\sigma}(v)\Phi(v,u)\rangle
\label{2} \end{equation}

$D_1^E$ and $D^E$ arise from CE field strengths: \begin{equation} \frac{1}{N_c}
D_{0i,0k}= \delta_{ik} \left[D^E+D_1^E+u_4^2 \frac{\partial D_1^E}
{\partial u_4^2} \right]+u_i u_k \frac{\partial D_1^E}{\partial
\vec{u}^2}\label{3}\end{equation}

As a result the Polyakov loop can be expressed in terms of
"potentials" $V_1$ and $V_D$
\begin{equation}
L_{fund}=\exp\left(-\frac{V_1(T)+2V_D}{2T}\right),
L_{adj}=\left(L_{fund}\right)^{9/4}, \label{4} \end{equation}

with $V_1(T)\equiv V_1(\infty,T)$, $V_D\equiv V_D(r^*,T)$
(\cite{5}) \begin{equation} V_1(r,T)=\int_0^\infty d\nu (1-\nu T) \int_0^r
d\xi\ \xi D_1^E(\sqrt{\xi^2+\nu^2})\label{5}\end{equation}

\begin{equation} V_D(r,T)=2\int_0^\infty d\nu (1-\nu T)\int_0^r d\xi\ (r-\xi)
D^E(\sqrt{\xi^2+\nu^2})\label{6}\end{equation}

In what follows we use the Polyakov line fit (\cite{8,9}) \begin{equation}
L_{fund}\left(x=\frac{T}{T_c},T\right)=\exp \left(
-\frac{.175\mbox{Gev}}{(1.35x-1)2T}    \right) \label{7} \end{equation}

The free energy $F(T)$ of quarks and gluons in SLA can be
expressed as a sum over all Matsubara winding numbers $n$
 with coefficients $L_{fund}^n$ and $L_{adj}^n$ for quarks and
 gluons respectively. For nonzero chemical potential $\mu$ one can
  keep $L_{fund,adj}$ independent of $\mu$, treating np fields
 as strong and unchanged by $\mu$ in the first approximation.

The final formulas for pressure of qgp are (\cite{7,9})
 \begin{equation} p_q\equiv
\frac{P^{SLA}_q}{T^4}=\frac{n_f}{\pi^2}\left[\Phi_\nu\left(\frac{\mu-\frac{V_1}{2}}{T}\right)+
\Phi_\nu\left(-\frac{\mu+\frac{V_1}{2}}{T}\right)\right]\label{8}\end{equation} where
$\nu=m_q/T$ and
\begin{equation}\Phi_\nu(a)=\int_0^\infty\frac{z^4}{\sqrt{z^2+\nu^2}}\frac{1}{(e^{\sqrt{z^2+\nu^2}}+1)}\label{9}\end{equation}
\begin{equation} p_{gl}=\frac{P^{SLA}_{gl}}{T^4}=\frac{8}{3\pi^2}\int_0^\infty
\frac{z^3dz}{e^{z+9/4 V_1}-1}\label{10}\end{equation}

The energy density is $\varepsilon=T^2\frac{\partial}{\partial
T}\left(\frac{P}{T}\right)_V$ and the speed of sound in plasma is
$c_s^2=\frac{\partial P}{\partial \varepsilon}$. In Fig.5 $c_s^2$
is shown calculated with the use of (\ref{8}), (\ref{10}) and
compared to lattice data for $\mu=0$ from (\cite{11}). No lattice
calculations has yet been done for sound speed at nonzero baryon
density, though our theory allows to do that and as is shown in
(\cite{12}) the result for $\mu>0$ does not differ much from the
case $\mu=0$.

\begin{figure}[t]
  \begin{minipage}{5cm}
     \centering
     \includegraphics[scale=0.55]{0312015L.eps}
     \caption{Fit (\ref{7}) of Polyakov line for $n_f=0$ and
     $n_f=2$)(black curves) to the lattice data (\cite{11}).}
     \label{}
  \end{minipage}
\hfill
  \begin{minipage}{5cm}
     \centering
     \includegraphics[scale=0.45]{0608003p.eps}
     \caption{Analytic (\ref{8}), (\ref{10}) and lattice (\cite{11}) curves for pressure of QGP with $n_f=0,2+1,3$ from (\cite{9}).}
     \label{}
  \end{minipage}
\end{figure}

\begin{figure}[h]
  \begin{minipage}{5cm}
     \centering
     \includegraphics[scale=0.5]{0608003e.eps}
     \caption{Analytic and lattice (\cite{11}) curves for energy density of QGP with $n_f=2+1$ and $n_f=3$ from (\cite{9}).}
     \label{}
  \end{minipage}
\hfill
  \begin{minipage}{5cm}
     \centering
     \includegraphics[scale=0.35]{0703228p.eps}
     \caption{The phase transition curve $T_c(\mu)$ (in GeV) from
     (\ref{9}) as function of quark chemical potential $\mu$ (in GeV)
      for $n_f=2$ (upper curve) and $n_f=3$ (lower curve) and
      $\Delta G_2=0.0034\ \mbox{GeV}^4$ from (\cite{8}).}
     \label{}
  \end{minipage}
\end{figure}

\begin{figure}[!h]
\centering
\includegraphics[height=8cm]{0608003c.eps}
\caption{Sound speed for $\mu=0$ and $n_f=3$ (blue dashed curve)
compared to lattice data from (\cite{11}).}
\end{figure}

\section{Phase transition} To obtain the curve of phase transition
one needs to define pressure $P_I$ in the confined phase and
$P_{II}$ in the deconfined phase, taking into account that vacuum
energy density contributes to the free energy, and hence to the
pressure: \begin{equation} P_I=|\varepsilon_{vac}|+P_{hardon},\ \ \
P_{II}=|\varepsilon_{vac}^{dec}|+(p_q+p_{gl})T^4. \label{11}\end{equation}
Having formulas for pressure (which contain parameter of
$L_{fund}(x)$ (\ref{7})) we may write for the phase
transition curve $T_c(\mu)$: \begin{equation}
T_c(\mu)=\left(\frac{(11-\frac{2}{3}n_f)\Delta G_2}{32
(p_q+p_{gl})}\right)^{1/4}\label{12} \end{equation} here $\Delta
\varepsilon_{vac}=|\varepsilon_{vac}-\varepsilon_{vac}^{dec}|=(11-\frac{2}{3}n_f)/32\Delta
G_2$. In particular, for the expected value of $\Delta G_2/G_2(stand)\approx 0.4$ one obtains
 $T_c=0.27 \mbox{ GeV } (n_f=0),\ 0.19 \mbox{ GeV } (n_f=2),\ 0.17 \mbox{ GeV } (n_f=3)$ in
 good agreement with lattice data.

\section{Summary} The EoS of QGP is written, where the
only np input is the Polyakov line. It should be stressed, that
only the modulus of the Polyakov line enters in EoS due to gauge
invariance. The phase transition curve $T_c(\mu)$ and speed of
sound $c_s^2(T)$ are obtained and agree well with lattice data. An
important point of the work is that the only parameter used to
receive the final physical quantities from the initial QCD
Lagrangian is the Polyakov line taken from lattice data, and is in
agreement with analytic estimate for $T=T_c$ (\cite{5}).

\section{Acknowledgments} The financial support of RFFI grant
06-02-17012 is acknowledged.
\section*{References}