# SO(1,3) Yang–Mills solutions on Minkowski space via cosets

.
Kaushlendra<EMAIL_ADDRESS>
Institut für Theoretische Physik &
Leibniz Universität Hannover
Appelstraße 2, 30167 Hannover, Germany

( )

###### Abstract

We present a novel family of Yang–Mills solutions, with gauge group SO(1,3),
on Minkowski space that are geometrically distinguished into two classes, viz.
interior and exterior of the lightcone. We achieve this by foliating the
former with SO(1,3)/SO(3) cosets and the latter with SO(1,3)/SO(1,2) cosets
and analytically solving the Yang–Mills equation of an SO(1,3)-invariant gauge
field. The resulting fields and their stress-energy tensor, when translated to
the Minkowski space, diverge at the lightcone, but we demonstrate how this
stress-energy tensor could be regularised due to its unique algebraic
structure.

##  1 Introduction

Analytic solutions of Yang-Mills theory with compact gauge groups, such as
SU(2), and finite action are very few, like the ones presented in [2, 1, 3] or
more recently in [4, 5, 6]. Here we improve upon this situation by presenting
new solutions [7], albeit with a non-compact gauge group $G{=}\text{SO(1,3)}$,
i.e. the Lorentz group. The latter appears in a gauge theory formulation of
general relativity and could be relevant for emergent/modified theories of
gravity including supergravity and matrix models.

The construction of these solutions relies on the fact that, owing to the
natural action of the Lorentz group on Minkowski space $\mathds{R}^{1,3}$,
there exists foliations of $\mathds{R}^{1,3}$ into $G$-orbits that are
reductive and symmetric coset spaces $G/H$. Specifically, on the inside of the
lightcone we have $H{=}\text{SO(3)}$ and on the outside of the lightcone we
have $H{=}\text{SO(1,2)}$. The former is the Riemannian two-sheeted hyperbolic
space $H^{3}$, foliating the future and past of the lightcone with timelike
parameter $u$, while the latter is a pseudo-Riemmanian de Sitter space
$\text{dS}_{3}$, foliating the exterior of lightcone with spacelike parameter
$u$. The Yang–Mills dynamics on these spaces are separately studied by
considering a $G$-invariant gauge connection ${\cal A}$ and employing
dimensional reduction on $\mathds{R}\times G/H$ that yields a Newton’s
particle, parameterized by $u$ and subject to an inverted double-well
potential, which admits analytic solutions.

These field configurations are then pulled back to the respective domains of
the Minkowski space, yielding the color-electromagnetic fields—diverging at
the lightcone—that we then use to compute the stress-energy tensor. The latter
turns out to have the same form in both the domains; this form, curiously,
takes the shape of a pure improvement term. This fact can be used to
regularize the stress-energy tensor across the lightcone so that one can have
matching of the fields, defined on two domains of the spacetime, at the
lightcone.

##  2 Minkowskian geometry and its foliations

We can foliate the Minkowski space $\mathds{R}^{1,3}$, with metric
$(\eta_{\mu\nu})=(-,+,+,+)$ for $\mu,\nu=0,1,2,3$, in two parts as depicted in
Figure 1(a): (a) the lightcone-interior ${\cal T}$ with two-sheeted hyperbolic
space $H^{3}$ and (b) the lightcone-exterior ${\cal S}$ with single-sheeted de
Sitter space $\text{dS}_{3}$.

For the first case, the hyperbolic space $H^{3}$ is embedded in
$\mathds{R}^{1,3}$ algebraically as

$y{\cdot}y\equiv\eta_{\mu\nu}\,y^{\mu}\,y^{\nu}\ =\ -1\ ,$ (2.1)

and foliates—with a timelike parameter $u$ obeying
$\mathrm{e}^{u}=\sqrt{|x{\cdot}x|}$—the lightcone-interior ${\cal T}$ as222We
employ standard conventions $x^{0}{=}t,x^{1}{=}x,x^{2}{=}y,x^{3}{=}z$,
$\vec{x}:=(x^{1},x^{2},x^{3})$ and $r=\sqrt{\vec{x}{\cdot}\vec{x}}$ in this
article.

$\displaystyle\varphi_{{}_{\cal T}}$ $\displaystyle:\leavevmode\nobreak\
\mathds{R}\times H^{3}\rightarrow{\cal T}\ ,\quad(u,y^{\mu})\mapsto
x^{\mu}:=\mathrm{e}^{u}\,y^{\mu}\ ,$ (2.2) $\displaystyle\varphi_{{}_{\cal
T}}^{-1}$ $\displaystyle:\leavevmode\nobreak\ {\cal
T}\rightarrow\mathds{R}\times H^{3}\ ,\quad
x^{\mu}\mapsto(u,y^{\mu}):=\Bigl{(}\ln{\sqrt{|x{\cdot}x|}},\frac{x^{\mu}}{\sqrt{|x{\cdot}x|}}\Bigr{)}\
.$

With this, the metric on ${\cal T}$ becomes conformal to a Lorentzian cylinder
$\mathds{R}\times H^{3}$:

$\mathrm{d}{s}_{{}_{\cal T}}^{2}\ =\
\mathrm{e}^{2u}\left(-\mathrm{d}{u}^{2}+\mathrm{d}{s}_{H^{3}}^{2}\right)\ ,$
(2.3)

where $\mathrm{d}{s}_{H^{3}}^{2}$ is the flat metric on $H^{3}$ arising from
(2.1).

In the second case, we can embed $\text{dS}_{3}$ inside the Minkowski space
$\mathds{R}^{1,3}$ by

$y\cdot y\equiv\eta_{\mu\nu}\,y^{\mu}\,y^{\nu}\ =\ 1\ .$ (2.4)

The foliation of ${\cal S}$ follows analogous to the previous case, albeit
with a spacelike foliation parameter $u$ satisfying
$\mathrm{e}^{u}=\sqrt{|x{\cdot}x|}\equiv\sqrt{r^{2}-t^{2}}$:

$\displaystyle\varphi_{{}_{\cal S}}$ $\displaystyle:\leavevmode\nobreak\
\mathds{R}\times\text{dS}_{3}\rightarrow{\cal S}\ ,\quad(u,y^{\mu})\mapsto
x^{\mu}:=\mathrm{e}^{u}\,y^{\mu}\ ,$ (2.5) $\displaystyle\varphi_{{}_{\cal
S}}^{-1}$ $\displaystyle:\leavevmode\nobreak\ {\cal
S}\rightarrow\mathds{R}\times\text{dS}_{3}\ ,\quad
x^{\mu}\mapsto(u,y^{\mu}):=\Bigl{(}\ln{\sqrt{|x{\cdot}x|}},\frac{x^{\mu}}{\sqrt{|x{\cdot}x|}}\Bigr{)}\
,$

such that the metric on ${\cal S}$ becomes conformal to the metric on a
cylinder $\mathds{R}\times\text{dS}_{3}$,

$\mathrm{d}{s}_{{}_{\cal S}}^{2}\ =\
\mathrm{e}^{2u}\left(\mathrm{d}{u}^{2}+\mathrm{d}{s}_{\text{dS}_{3}}^{2}\right)\
,$ (2.6)

where the flat $\text{dS}_{3}$-metric $\mathrm{d}{s}_{\text{dS}_{3}}^{2}$ is
induced from (2.4).

(b) Every $H^{3}$ vector $V_{\alpha}$ is related to
$V_{0}\sim(1,0,0,0)^{\top}$ with a unique boost $\Lambda_{\alpha}$, yielding
its stability subgroup:
$\Lambda_{\alpha}\,\mathrm{SO}(3)\,\Lambda_{\alpha}^{\text{-}1}$. (c) The
above $H^{3}$ vectors $V_{\alpha}$ lies in one-to-one correspondence with
cosets $\Lambda_{\alpha}\,\mathrm{SO}(3)$ so that $H^{3}$, as a 3-dimensional
submanifold $\sigma(y)$ inside SO(1,3), arises through a choice of
representative $\sigma$ in each of these cosets.

(a) Foliation of interior (yellow region) and exterior of the lightcone with
$H^{3}$\- and $\text{dS}_{3}$-slices respectively. Every slicing has internal
coordinate $y$ and foliation parameter $u$.

Figure 1: Minkowski foliations and demonstration of
$H^{3}\cong\textrm{SO(1,3)/SO(3)}$.

##  3 Algebra/geometry of symmetric SO(1,3)-cosets

We consider here reductive coset spaces $G/H$ with 6-dimensional Lie group
$G=\textrm{SO(1,3)}$ that are also symmetric. This yields homogeneous spaces
$H^{3}$ and $\text{dS}_{3}$ with stablity subgroups $H{=}\textrm{SO(3)}$ and
$H{=}SO(1,2)$ respectively. The equivalence of SO(1,3)/SO(3) with $H^{3}$ is
geometrically illustrated through Figures 1(b) and 1(c).

The six generators $\\{I_{\scriptscriptstyle{A}}\\}$ of the Lie algebra
$\mathfrak{g}=\text{Lie}(G)$ are nothing but the canonical rotation ($J_{a}$)
and boost ($K_{a}$) generators of the Lorentz group:

$J_{1}{=}\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&{\text{-}1}\\\
0&0&1&0\end{pmatrix},J_{2}{=}\begin{pmatrix}0&0&0&0\\\ 0&0&0&1\\\ 0&0&0&0\\\
0&{\text{-}1}&0&0\end{pmatrix},J_{3}{=}\begin{pmatrix}0&0&0&0\\\
0&0&{\text{-}1}&0\\\ 0&1&0&0\\\
0&0&0&0\end{pmatrix},K_{1}{=}\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{pmatrix},K_{2}{=}\begin{pmatrix}0&0&1&0\\\ 0&0&0&0\\\ 1&0&0&0\\\
0&0&0&0\end{pmatrix},K_{3}{=}\begin{pmatrix}0&0&0&1\\\ 0&0&0&0\\\ 0&0&0&0\\\
1&0&0&0\end{pmatrix}\ .$ (3.1)

Moreover, for reductive cosets, $\mathfrak{g}$ splits into a Lie subalgebra
$\mathfrak{h}=\text{Lie}(H)$ and its orthogonal complement333The orthogonality
here is with respect to the Cartan–Killing metric, given by the trace of
adjoint representation of these generators $\\{I_{\scriptscriptstyle{A}}\\}$.
$\mathfrak{m}$; this is also reflected in the splitting of the generators
$\\{I_{\scriptscriptstyle{A}}\\}$ as follows

$\mathfrak{g}\ =\
\mathfrak{h}\oplus\mathfrak{m}\quad\implies\quad\\{I_{\scriptscriptstyle{A}}\\}\
=\ \\{I_{i}\\}\cup\\{I_{a}\\}\quad\quad\textrm{with}\quad
i=4,5,6\quad\textrm{and}\quad a=1,2,3\ ,$ (3.2)

where $\\{I_{i}\\}$ spans $\mathfrak{h}$ and $\\{I_{a}\\}$ spans
$\mathfrak{m}$. They satisfy following commutation relations

$[I_{i},I_{j}]\ =\ f_{ij}^{\ \ k}I_{k}\ ,\qquad[I_{i},I_{a}]\ =\ f_{ia}^{\ \
b}I_{b}\ ,\,\quad\textrm{and}\quad\,[I_{a},I_{b}]\ =\ f_{ab}^{\ \ i}I_{i}\ .$
(3.3)

Similarly, the left-invariant one-forms of SO(1,3) split into
$\\{e^{\scriptscriptstyle A}\\}=\\{e^{i}\\}\cup\\{e^{a}\\}$, such that the
following structure equations—with same structure coefficients as in (3.3)—are
satisfied

$\mathrm{d}e^{a}+f_{ib}^{\ \ a}\,e^{i}\wedge e^{b}\ =\
0\qquad\quad\textrm{and}\quad\qquad\mathrm{d}e^{i}+{\textstyle\frac{1}{2}}f_{jk}^{\
\ i}\,e^{j}\wedge e^{k}+{\textstyle\frac{1}{2}}f_{ab}^{\ \ i}\,e^{a}\wedge
e^{b}\ =\ 0\ .$ (3.4)

Here $e^{a}$ yields the metric on $G/H$ while $e^{i}=e^{i}_{a}\,e^{a}$ are
linearly dependent.

###  3.1 Case I: $H^{3}\cong\textrm{SO(1,3)/SO(3)}$

For SO(1,3)/SO(3) the splitting (3.2) and structure coefficients (3.3) are
given by

$I_{i}=J_{i-3}\ ,\ I_{a}=K_{a}\quad\implies\quad f_{ij}^{\ \ k}\ =\
\varepsilon_{i-3\;j-3\;k-3}\ ,\ f_{ia}^{\ \ b}\ =\ \varepsilon_{i-3\;a\,b}\ ,\
f_{ab}^{\ \ i}\ =\ -\varepsilon_{a\,b\,i-3}\ .$ (3.5)

The identification of coset space SO(1,3)/SO(3) with $H^{3}$ is seen from the
following maps

$\displaystyle\alpha_{{}_{\cal T}}$ $\displaystyle:\leavevmode\nobreak\
\mathrm{SO}(1,3)/\mathrm{SO}(3)\rightarrow H^{3}\ ,\quad[\Lambda_{\cal
T}]\mapsto y^{\mu}=(\Lambda_{{\cal T}})^{\mu}_{\;\;0}\ ,$ (3.6)
$\displaystyle\alpha_{{}_{\cal T}}^{-1}$ $\displaystyle:\leavevmode\nobreak\
H^{3}\rightarrow\mathrm{SO}(1,3)/\mathrm{SO}(3)\ ,\quad
y^{\mu}\mapsto[\Lambda_{\cal T}]\ ,$

where the representative $\Lambda_{\cal T}$ of the coset $[\Lambda_{\cal
T}]:=\\{\Lambda=\Lambda_{\cal T}h:h\in SO(3)\\}$ is given by

$\Lambda_{\cal T}\ =\
\begin{pmatrix}\gamma&\gamma\,\boldsymbol{\beta}^{\top}\\\
\gamma\,\boldsymbol{\beta}&\mathds{1}+(\gamma{-}1)\frac{\boldsymbol{\beta}\otimes\boldsymbol{\beta}}{\boldsymbol{\beta}^{2}}\end{pmatrix}\
,\quad\quad\textrm{with}\quad\beta^{a}=\frac{y^{a}}{y^{0}}\
,\quad\gamma=\frac{1}{\sqrt{1-\boldsymbol{\beta}^{2}}}=y^{0}\ ,$ (3.7)

and $\boldsymbol{\beta}^{2}=\delta_{ab}\,\beta^{a}\beta^{b}\geq 0$. It is
straightforward to verify that the above map $\alpha_{{}_{\cal T}}$ is well-
defined and, in fact, $\Lambda_{\cal T}$ is a generic boost obtained from
coset generators $I_{a}\in\mathfrak{m}$ as

$\Lambda_{\cal T}\ =\ \textrm{exp}(\eta^{a}\,I_{a})\
,\quad\quad\textrm{with}\quad\beta^{a}={\textstyle\frac{\eta^{a}}{\sqrt{\boldsymbol{\eta}^{2}}}}\tanh{\sqrt{\boldsymbol{\eta}^{2}}}\
,\quad\textrm{for}\quad\boldsymbol{\eta}^{2}=\delta_{ab}\,\eta^{a}\eta^{b}\ .$
(3.8)

We can now obtain the left-invariant one-forms by employing the Maurer–Cartan
prescription:

$\Lambda_{\cal T}^{-1}\,\mathrm{d}{\Lambda}_{\cal T}\ =\
e^{a}\,I_{a}+e^{i}\,I_{i}\ ;\quad
e^{a}=\Bigl{(}\delta^{ab}-\frac{y^{a}\,y^{b}}{y^{0}(1{+}y^{0})}\Bigr{)}\,\mathrm{d}y^{b}\quad\textrm{and}\quad
e^{i}=\varepsilon_{i-3\;a\,b}\,\frac{y^{a}}{1{+}y^{0}}\,\mathrm{d}y^{b}\ ,$
(3.9)

such that $e^{a}$ reproduces the metric on $H^{3}$ while $e^{i}$ become
linearly dependent as follows

$\mathrm{d}{s}_{H^{3}}^{2}\ =\ \delta_{ab}\,e^{a}\otimes
e^{b}\,\quad\textrm{and}\quad\,e^{i}=e^{i}_{a}\,e^{a}\
,\quad\textrm{with}\quad
e^{i}_{a}=\varepsilon_{a\,i-3\;b}\,\frac{y^{b}}{1{+}y^{0}}\ .$ (3.10)

###  3.2 Case II: $\text{dS}_{3}\cong$ SO(1,3)/SO(1,2)

For the coset space $\mathrm{SO}(1,3)/\mathrm{SO}(1,2)$ we chose the splitting
(3.2) as follows

$I_{i}\in\\{K_{1},K_{2},J_{3}\\}\,\quad\textrm{and}\quad\,I_{a}\in\\{J_{1},J_{2},K_{3}\\}\
,$ (3.11)

such that the structure coefficients (3.3) comes out to be

$f_{ij}^{\ \ k}\ =\ \varepsilon_{i-3\;j-3\;k-3}\,(1{-}2\,\delta_{k6})\ ,\qquad
f_{ia}^{\ \ b}\ =\
\varepsilon_{i-3\;a\,b}\,(1{-}2\,\delta_{a3})\,\quad\textrm{and}\quad\,f_{ab}^{\
\ i}\ =\ \varepsilon_{a\,b\,i-3}\ ,$ (3.12)

where no summation convention is used inside the brackets. As before, we
demonstrate the equivalence between $\text{dS}_{3}$ and
$\mathrm{SO}(1,3)/\mathrm{SO}(1,2)$ through following well-defined maps

$\displaystyle\alpha_{{}_{\cal S}}$ $\displaystyle:\leavevmode\nobreak\
\mathrm{SO}(1,3)/\mathrm{SO}(1,2)\rightarrow\mathrm{d}{\text{S}}_{3}\
,\quad[\Lambda_{\cal S}]\mapsto y^{\mu}:=(\Lambda_{\cal S})^{\mu}_{\;\;3}\ ,$
(3.13) $\displaystyle\alpha_{{}_{\cal S}}^{-1}$
$\displaystyle:\leavevmode\nobreak\
\mathrm{d}{\text{S}}_{3}\rightarrow\mathrm{SO}(1,3)/\mathrm{SO}(1,2)\ ,\quad
y^{\mu}\mapsto[\Lambda_{\cal S}]\ ,$

where the representative left-coset element $\Lambda_{\cal S}$ is again
obtained though exponentiation with coset generators
$\\{J_{1},J_{2},K_{3}\\}$. The resultant Maurer–Cartan one-forms look like

$\Lambda_{\cal S}^{-1}\,\mathrm{d}{\Lambda}_{\cal S}\ =\
e^{a}\,I_{a}+e^{i}\,I_{i}\ ;\quad
e^{a}=\mathrm{d}{y}^{3-a}-\frac{y^{3-a}}{1{+}y^{3}}\,\mathrm{d}{y}^{3}\quad\textrm{and}\quad
e^{i}=-\varepsilon_{i-3\;a\,b}\,\frac{y^{3-a}}{1{+}y^{3}}\,\mathrm{d}{y}^{3-b}\
.$ (3.14)

These one-forms behave as expected with $(\eta_{ab}){=}(-,+,+)$ due to the
stabilizer SO(1,2):

$\mathrm{d}{s}_{\mathrm{d}{\text{S}}_{3}}^{2}\ =\ \eta_{ab}\,e^{a}\otimes
e^{b}\,\quad\textrm{and}\quad\,e^{i}=e^{i}_{a}\,e^{a}\
,\quad\textrm{with}\quad
e^{i}_{a}=\varepsilon_{i-3\;a\,b}\,\frac{y^{3-b}}{1{+}y^{3}}\ .$ (3.15)

###  3.3 The lightcone exception

Before we move on, let us make a remark on the lightcone itself that sits here
as an exception in the following manner. It can be shown that both future and
past of the lightcone are individually isomorphic to SO(1,3)/ISO(2), where the
stability subgroup is a Euclidean group $\mathrm{E}(2){=}\mathrm{ISO}(2)$
generated by two translations and one rotation. However, this coset space is
non-reductive and, on top of that, this does not give rise to any foliation
here, making this case unsuitable to study Yang–Mills dynamics as discussed in
Section 4.

##  4 Yang–Mills fields from dimensional reduction

The study of Yang–Mills dynamics on $\mathds{R}\times G/H$ via dimensional
reduction is a well-known topic with excellent review in [8]. Given an
orthonormal frame $\\{e^{u}{:=}\mathrm{d}{u},e^{a}\\}$ on the cylinder
$\mathds{R}\times G/H$, we can write a generic connection one-form ${\cal A}$
in the “temporal” gauge ${\cal A}_{u}{=}0$ and its curvature two-form ${\cal
F}=\mathrm{d}{{\cal A}}+{\cal A}\wedge{\cal A}$ as follows

${\cal A}\ =\ {\cal A}_{a}\,e^{a}\qquad\implies\qquad{\cal F}={\cal
F}_{ua}\,e^{u}\wedge e^{a}+{\textstyle\frac{1}{2}}{\cal F}_{ab}\,e^{a}\wedge
e^{b}\ .$ (4.1)

Next, we expand the gauge field ${\cal A}_{a}$ in terms of full
SO(1,3)-generators (3.2) as ${\cal A}_{a}={\cal A}_{a}^{i}\,I_{i}+{\cal
A}_{a}^{b}\,I_{b}$ and impose $G$-invariance on this, yielding following two
conditions 444 We can write the second relation more succinctly as
$[I_{i},\widetilde{{\cal A}}_{a}]=f_{ia}^{\ \ b}\widetilde{{\cal A}}_{b}$ for
$\widetilde{{\cal A}}_{a}:={\cal A}_{b}^{a}\,I_{a}\in\mathfrak{m}$.

${\cal A}_{a}^{i}\ =\ e^{i}_{a}\,\quad\textrm{and}\quad\,{\cal
A}_{b}^{a}={\cal A}_{b}^{a}(u)\ ,\quad\textrm{with}\quad f_{ia}^{\ \ c}{\cal
A}_{b}^{a}\ =\ f_{ib}^{\ \ a}{\cal A}_{a}^{c}\ .$ (4.2)

Furthermore, for the symmetric spaces that concerns us, one finds that ${\cal
A}^{a}_{b}(u)=\phi(u)\,\delta^{a}_{b}$ such that our $G$-invariant gauge field
${\cal A}$ depends on a single real function $\phi$:

${\cal A}\ =\ I_{i}\,e^{i}+\phi(u)\,I_{a}\,e^{a}\ .$ (4.3)

The components of the field strength ${\cal F}$, using (4.2) and (3.4),
computes to

${\cal F}_{ua}\ =\ \dot{\phi}\,I_{a}\,\quad\textrm{and}\quad\,{\cal F}_{ab}\
=\ (\phi^{2}{-}1)f_{ab}^{\ \ i}\,I_{i}\
,\quad\quad\textrm{with}\quad\dot{\phi}:=\partial_{u}\phi\ ,$ (4.4)

yielding the the color-electric field ${\cal E}_{a}={\cal
F}_{au}\in\mathfrak{m}$ and -magnetic field ${\cal
B}_{a}={\textstyle\frac{1}{2}}\varepsilon_{abc}{\cal F}_{bc}\in\mathfrak{h}$
on the cylinder. Finally, to work out the dynamics of $\phi(u)$ we look at the
Yang–Mills action

$S_{\text{YM}}\ =\ -\frac{1}{4g^{2}}\int\mathrm{tr}_{\text{ad}}{\left({\cal
F}\wedge*{\cal F}\right)}\ ,$ (4.5)

which simplifies drastically in both cases, viz. interior of the lightcone
${\cal T}$ with $M^{3}:=H^{3}$ and exterior of the lightcone with
$M^{3}:=\text{dS}_{3}$, as follows

$S_{\text{YM}}\ =\ \frac{6}{g^{2}}\int_{\mathds{R}\times
M^{3}}\\!\\!\\!\mathrm{d}{\text{vol}}\
\bigl{(}\tfrac{1}{2}\dot{\phi}^{2}-V(\phi)\bigr{)}\ ;\qquad V(\phi)\ =\
-{\textstyle\frac{1}{2}}(\phi^{2}{-}1)^{2}$ (4.6)

Figure 2: Plot of $V(\phi)$

where
$\mathrm{d}{\text{vol}}={\textstyle\frac{1}{3!}}\varepsilon_{abc}\,\mathrm{d}{u}\wedge
e^{a}\wedge e^{b}\wedge e^{c}$ is the volume form. We immediately observe that
the above action represents a mechanical particle $\phi(u)$ in an inverted
double-well potential $V(\phi)$ depicted in Figure 2, which yields the
equation of motion for an anharmonic oscillator,

$\ddot{\phi}\ =\ -\frac{\partial V}{\partial\phi}\ =\ 2\,\phi\,(\phi^{2}{-}1)\
.$ (4.7)

This admits analytic solutions in terms of Jacobi elliptic functions. For
example, in the bounded case where the mechanical energy
${\textstyle\frac{1}{2}}\dot{\phi}^{2}+V(\phi)=:\epsilon\in[-\frac{1}{2},0]$
we have

$\phi_{\epsilon,u_{0}}(u)\ =\
f_{-}(\epsilon)\,\text{sn}\big{(}f_{+}(\epsilon)(u{-}u_{0}),k\big{)}\quad\textrm{with}\quad
f_{\pm}(\epsilon)=\sqrt{1\pm\sqrt{-2\epsilon}}\ ,\quad
k^{2}=\frac{f_{-}(\epsilon)}{f_{+}(\epsilon)}$ (4.8)

and a ‘time’-shift parameter $u_{0}$. This also include special cases, such as
a “kink”:

$\phi\ =\ \begin{cases}\
0\quad&\quad\textrm{for}\quad\epsilon=-{\textstyle\frac{1}{2}}\ ,\\\ \
\tanh{(u{-}u_{0})}\quad&\quad\textrm{for}\quad\epsilon=0\ ,\\\ \ \pm
1\quad&\quad\textrm{for}\quad\epsilon=0\ .\end{cases}$ (4.9)

Now in order to pull these solution back to ${\cal T}$ we transform the
orthonormal frame $\\{e^{u},e^{a}\\}$ on $\mathds{R}\times H^{3}$, using the
map $\varphi_{{}_{\cal T}}$ (2.2) and abbreviation $|x|:=\sqrt{|x{\cdot}x|}$,
as follows

$e^{u}:=\mathrm{d}{u}\ =\
\frac{t\,\mathrm{d}{t}-r\,\mathrm{d}{r}}{t^{2}-r^{2}}\qquad\quad\textrm{and}\quad\qquad
e^{a}\ =\
\frac{1}{|x|}\Bigl{(}\mathrm{d}{x}^{a}-\frac{x^{a}}{|x|}\,\mathrm{d}{t}+\frac{x^{a}}{|x|(|x|+t)}\,r\,\mathrm{d}{r}\Bigr{)}\
.$ (4.10)

The SO(1,3)-invariant gauge field ${\cal A}\equiv A$ (4.3) can be casted into
a Minkowski one-form

$A\ =\ \frac{1}{|x|}\left\\{\frac{\varepsilon_{ab}^{\ \
k-3}\,x^{a}}{|x|+t}\mathrm{d}{x}^{b}\,I_{k}+\phi(x)\Bigl{(}\mathrm{d}{x}^{a}-\frac{x^{a}}{|x|}\,\mathrm{d}{t}+\frac{x^{a}}{|x|(|x|+t)}\,r\,\mathrm{d}{r}\Bigr{)}I_{a}\right\\}\
,$ (4.11)

where $\phi(x){:=}\phi_{\epsilon,u_{0}}(u(x))$. We can then find the field
strength $F=F_{\mu\nu}\,\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu}$ on ${\cal
T}$ from its cylinder version ${\cal F}$ (4.4) using vierbein components
$e^{u}=e^{u}_{\mu}\,\mathrm{d}{x}^{\mu}$ and
$e^{a}=e^{a}_{\mu}\,\mathrm{d}{x}^{\mu}$ (4.10). The corresponding color-
electric $E_{i}:=F_{0i}$ and -magnetic
$B_{i}:=\frac{1}{2}\varepsilon_{ijk}\,F_{jk}$ fields read

$\displaystyle E_{a}$ $\displaystyle\ =\
\frac{1}{|x|^{3}}\left\\{\left(\phi^{2}{-}1\right)\varepsilon_{ab}^{\ \
i-3}\,x^{b}\,I_{i}-\dot{\phi}\Bigl{(}t\,\delta^{ab}-\frac{x^{a}\,x^{b}}{|x|+t}\Bigr{)}I_{b}\right\\}\
,$ (4.12) $\displaystyle B_{a}$ $\displaystyle\ =\
-\frac{1}{|x|^{3}}\left\\{\left(\phi^{2}{-}1\right)\Bigl{(}t\,\delta^{a\,i-3}-\frac{x^{a}\,x^{i-3}}{|x|+t}\Bigr{)}I_{i}+\dot{\phi}\,\varepsilon_{ab}^{\
\ c}\,x^{b}\,I_{c}\right\\}\ .$

An interesting feature of these fields is the presence of color-
electromagnetic duality, i.e. $E_{a}\rightarrow B_{a}$ and $B_{a}\rightarrow-
E_{a}$, which works when we simultaneously interchange
$\dot{\phi}\leftrightarrow(\phi^{2}{-}1)$ and and switch the generators as
follows: $I_{i}\rightarrow I_{a}$ and $I_{a}\rightarrow-I_{i}$ (plus some
obvious index adjustment). More importantly, we notice that the gauge field
${\cal A}$ (4.11) along with the electric $E_{i}$ and magnetic $B_{i}$ fields
(4.12) become singular at the lightcone $t{=}{\pm}r$. One can find out the
fields on ${\cal S}$ using (2.5) and following the same recipe as above. We
restrain from reproducing these results here owing to space constraint and
refer the reader to [7] for explicit form of such fields.

##  5 The stress-energy tensor

We can compute the stress-energy tensor, given by the expression

$T_{\mu\nu}\ =\
-\tfrac{1}{2g^{2}}\,\mathrm{tr}_{\text{ad}}{\left(F_{\mu\alpha}\,F_{\nu\beta}\,\eta^{\alpha\beta}-{\textstyle\frac{1}{4}}\eta_{\mu\nu}F^{2}\right)}\,\quad\textrm{with}\quad
F^{2}=F_{\mu\nu}\,F^{\mu\nu}\ ,$ (5.1)

for such Yang–Mills fields in a straightforward manner. Interestingly, we find
that the computation yields the same form of stress-energy tensor on both
sides of the lightcone that reads

$T\ =\
\frac{\epsilon}{g^{2}(r^{2}{-}t^{2})^{3}}\begin{pmatrix}3t^{2}{+}r^{2}&-4tx&-4ty&-4tz\\\
-4tx&t^{2}{+}4x^{2}{-}r^{2}&4xy&4xz\\\ -4ty&4xy&t^{2}{+}4y^{2}{-}r^{2}&4yz\\\
-4tz&4xz&4yz&t^{2}{+}4z^{2}{-}r^{2}\end{pmatrix}\ .$ (5.2)

It is worth emphasising here that the explicit form of $\phi$, like in (4.8),
is irrelevant here as $T$ only depends on the total mechanical energy
$\epsilon$. Moreover, this has a vanishing trace and presence of lightcone
singularity, as expected. Surprisingly, this admits a nice compact form that
can be recasted into a pure “improvement” term as follows,

$T_{\mu\nu}\ =\
\mbox{$\partial$}^{\rho}S_{\rho\mu\nu}\qquad\quad\textrm{with}\quad\qquad
S_{\rho\mu\nu}\ =\
\frac{\epsilon}{g^{2}}\,\frac{x_{\rho}\eta_{\mu\nu}-x_{\mu}\eta_{\rho\nu}}{(x{\cdot}x)^{2}}\
,$ (5.3)

where the term $S_{\rho\mu\nu}$ can be expressed using abbreviation
$(\tilde{S}_{\rho})_{\mu\nu}:={\textstyle\frac{g^{2}(x{\cdot}x)^{2}}{\epsilon}}S_{\rho\mu\nu}$
as

$\tilde{S}_{0}\ =\ \begin{pmatrix}0&0&0&0\\\ x&-t&0&0\\\ y&0&-t&0\\\
z&0&0&-t\end{pmatrix}\ ,\quad\tilde{S}_{1}\ =\ \begin{pmatrix}-x&t&0&0\\\
0&0&0&0\\\ 0&-y&x&0\\\ 0&-z&0&x\end{pmatrix}\ ,\quad\tilde{S}_{2}\ =\
\begin{pmatrix}-y&0&t&0\\\ 0&y&-x&0\\\ 0&0&0&0\\\ 0&0&-z&y\end{pmatrix}\
,\quad\tilde{S}_{3}\ =\ \begin{pmatrix}-z&0&0&t\\\ 0&z&0&-x\\\ 0&0&z&-y\\\
0&0&0&0\end{pmatrix}\ .$ (5.4)

Naturally, one hopes to glue the two expressions for stress-energy tensors to
find a single expression valid across the Minkowski spacetime. The price to
pay here is the singularity at the lightcone, which can be remedied through
the following regularization procedure

$S^{\textrm{reg}}_{\rho\mu\nu}\ =\
\frac{\epsilon}{g^{2}}\,\frac{x_{\rho}\eta_{\mu\nu}-x_{\mu}\eta_{\rho\nu}}{(x{\cdot}x+\delta)^{2}}\qquad\Rightarrow\qquad
T^{\textrm{reg}}_{\mu\nu}\ =\
\frac{\epsilon}{g^{2}}\,\frac{4\,x_{\mu}x_{\nu}-\eta_{\mu\nu}x{\cdot}x+3\,\delta\,\eta_{\mu\nu}}{(x{\cdot}x+\delta)^{3}}\
.$ (5.5)

This nonsingular improvement term (with a finite regularization parameter
$\delta$) yields vanishing energy and momenta as their fall-off behaviour at
spatial infinity is fast enough.

An alternate route to regularization could be to directly shift only the
denominator of $T_{\mu\nu}$ in (5.3) via $x{\cdot}x\mapsto x{\cdot}x+\delta$.
We can then improve the resultant stress-energy tensor up to the term in (5.5)
so as to obtain the following energy-momentum tensor candidate that is regular
and that also vanishes as $\delta\to 0$,

$T^{\delta}_{\mu\nu}\ =\
\frac{\epsilon}{g^{2}}\,\frac{4\,x_{\mu}x_{\nu}-\eta_{\mu\nu}x{\cdot}x}{(x{\cdot}x+\delta)^{3}}\
\sim\
\frac{\epsilon}{g^{2}}\,\frac{-3\,\delta\,\eta_{\mu\nu}}{(x{\cdot}x+\delta)^{3}}\
.$ (5.6)

##  6 Conclusion

Starting from the geometry of the Minkowski foliations with $H^{3}$\-
(interior of the lightcone) and $\text{dS}_{3}$-slices (exterior of the
lightcone) and exploring the origin of these symmetric spaces through cosets
of the gauge group SO(1,3), we have obtained analytic solutions of Yang–Mills
equation on Minkowski space that, however, diverge at the lightcone. We
achieved this by first solving a SO(1,3)-invariant configuration on the
cylinder $\mathds{R}\times\text{SO(3)}/H$, with $H{=}\textrm{SO(3)}$ on the
interior and $H{=}\textrm{SO(1,2)}$ on the exterior of the lightcone, using
dimensional reduction technique of gauge theory and then translating these
solutions to two different domains of Minkowski spacetime, seperated by the
lightcone, with their respective foliation maps. We then computed the stress-
energy tensor in both cases and found out that they have the same form. Not
only this, when written compactly, we were able to cast it into a pure
improvement term, a fact that helped us in finding a regularized candidate for
the stress-energy tensor, defined throughout the spacetime; how this modified
stress-energy tensor arise from a source term remains an open question though.

## Acknowledgements

This work was supported by the Deutscher Akademischer Austauschdienst (DAAD)
with grant number 57381412. The author thanks Olaf Lechtenfeld, Gabriel
Picanço Costa and Jona Röhrig for contribution in [7]. The author is also
grateful to OL for a careful reading of the draft and suggesting improvements.

## References

  * [1] V. De Alfaro, S. Fubini and G. Furlan, A new classical solution of the Yang-Mills field equations, Phys. Lett. 65B, 163 (1976), doi:10.1016/0370-2693(76)90022-8.
  * [2] M. Lüscher, SO(4)-symmetric solutions of Minkowskian Yang-Mills field equations, Phys. Lett. 70B, 321 (1977), doi:10.1016/0370-2693(77)90668-2.
  * [3] B.M. Schechter, Yang-Mills theory on the hypertorus, Phys. Rev. D 16, 3015 (1977), doi:10.1103/PhysRevD.16.3015.
  * [4] T.A. Ivanova, O. Lechtenfeld and A.D. Popov, Solutions to Yang-Mills Equations on Four-Dimensional de Sitter Space, Phys. Rev. Lett. 119, 061601 (2017), doi:10.1103/PhysRevLett.119.061601.
  * [5] T.A. Ivanova, O. Lechtenfeld and A.D. Popov, Finite-action solutions of Yang-Mills equations on de Sitter $\text{dS}_{4}$ and anti-de Sitter $\text{AdS}_{4}$ spaces, JHEP 11, 017 (2017), doi:10.1007/JHEP11(2017)017.
  * [6] T.A. Ivanova and O. Lechtenfeld, Yang–Mills instantons and dyons on group manifolds, Phys. Lett. B 670, 91 (2008), doi:10.1016/j.physletb.2008.10.027.
  * [7] K. Kumar, O. Lechtenfeld, G. Picanço Costa and J. Röhrig, Yang–Mills solutions on Minkowski space via non-compact coset spaces, Phys. Lett. B 835, 137564 (2022), doi:10.1016/j.physletb.2022.137564.
  * [8] D. Kapetanakis and G. Zoupanos, Coset space dimensional reduction of gauge theories, Phys. Rept. 219, 4 (1992), doi:10.1016/0370-1573(92)90101-5.


# Non trivial optimal sampling rate for estimating a Lipschitz-continuous
function in presence of mean-reverting Ornstein-Uhlenbeck noise

Enrico Bernardi 111Dipartimento di Scienze Statistiche Paolo Fortunati,
Università di Bologna. Bologna, Italy. 222e-mail<EMAIL_ADDRESS>Alberto Lanconelli 11footnotemark: 1 333e-mail<EMAIL_ADDRESS>Christopher S. A. Lauria 11footnotemark: 1 444e-mail:
<EMAIL_ADDRESS>Berk Tan Perçin 11footnotemark: 1 555e-mail:
<EMAIL_ADDRESS>
###### Abstract

We examine a mean-reverting Ornstein-Uhlenbeck process that perturbs an
unknown Lipschitz-continuous drift and aim to estimate the drift’s value at a
predetermined time horizon by sampling the path of the process. Due to the
time varying nature of the drift we propose an estimation procedure that
involves an online, time-varying optimization scheme implemented using a
stochastic gradient ascent algorithm to maximize the log-likelihood of our
observations. The objective of the paper is to investigate the optimal sample
size/rate for achieving the minimum mean square distance between our estimator
and the true value of the drift. In this setting we uncover a trade-off
between the correlation of the observations, which increases with the sample
size, and the dynamic nature of the unknown drift, which is weakened by
increasing the frequency of observation. The mean square error is shown to be
non monotonic in the sample size, attaining a global minimum whose precise
description depends on the parameters that govern the model. In the static
case, i.e. when the unknown drift is constant, our method outperforms the
arithmetic mean of the observations in highly correlated regimes, despite the
latter being a natural candidate estimator. We then compare our online
estimator with the global maximum likelihood estimator.

Key words and phrases: Ornstein-Uhlenbeck process, positive correlation, log-
likelihood function, stochastic gradient ascent.

AMS 2000 classification: 60H10, 62F10, 65C20.

##  1 Introduction and statement of the main result

High frequency samples of a noisy continuous time signal may produce strongly
correlated observations which have a negative impact on the performance of
statistical estimation methods. This phenomenon has been observed in a number
of different contexts: [11] and [15] explore the complex nature of sampling
effects in tracking the movement of individual cells or animals, [19] proposes
a statistical counterpart of the sampling rate’s lower bound from ordinary
spectral analysis of time series related to aliasing, [1] shows that in
presence of market microstructure noise the optimal sampling rate for
estimating the variance of log returns through their sum of squares is finite,
[17] proves that the spacing between observations affects the estimation of
the variance of a Fraction Brownian Motion.

The aim of the present paper is to analyse this phenomenon in a particular
setting. Our framework is very specific but nevertheless pretty common in
several applied settings (see for instance [2]).
Specifically, let $\\{X_{t}\\}_{t\in[0,1]}$ be our _observation process_ which
is assumed to be of the form

$\displaystyle X_{t}:=m^{\star}(t)+Y_{t},\quad t\in[0,1],$ (1.1)

where

  * •

$m^{\star}:[0,1]\to\mathbb{R}$ is an _unknown_ Lipschitz-continuous function,
i.e. there exists $K>0$ such that

$\displaystyle|m^{\star}(t)-m^{\star}(s)|\leq K|t-s|\quad\mbox{ for all
}t,s\in[0,1];$ (1.2)

  * •

$\\{Y_{t}\\}_{t\geq 0}$ is a mean-reverting Ornstein-Uhlenbeck process, i.e.

$\displaystyle\begin{cases}dY_{t}=-\theta Y_{t}dt+\sigma dB_{t},\quad t>0;\\\
Y_{0}\thicksim\mathtt{N}\left(0,\frac{\sigma^{2}}{2\theta}\right).\end{cases}$
(1.3)

Here, the initial condition $Y_{0}$ is independent of the Brownian Motion
$\\{B_{t}\\}_{t\geq 0}$ while $\theta$ and $\sigma$ are positive real
parameters. The parameter $\theta$ models the _strength of mean-reversion_
towards zero and $\sigma$ the _amplitude of fluctuation_ of the process.

Our problem is to estimate $m^{\star}(1)$ with a finite sample of observations
of the process $\\{X_{t}\\}_{t\in[0,1]}$, sampled at equally spaced instants
of time.
Let $N\in\mathbb{N}$ and assume we observe the process $\\{X_{t}\\}_{t\geq 0}$
in the time interval $[0,1]$ at the discrete times
$\left\\{\frac{1}{N},\frac{2}{N},...,1\right\\}$. Then, according to our model
specification we have

$\displaystyle
X_{\frac{t}{N}}\thicksim\mathtt{N}\left(m_{t}^{\star},\frac{\sigma^{2}}{2\theta}\right)\quad\mbox{
and
}\quad\mathtt{cov}\left(X_{\frac{t}{N}},X_{\frac{s}{N}}\right)=\frac{\sigma^{2}}{2\theta}e^{-\frac{\theta}{N}|t-s|}\quad\mbox{
for all $t,s\in\\{1,...,N\\}$},$ (1.4)

where for notational convenience we set

$\displaystyle m^{\star}_{t}:=m^{\star}\left(\frac{t}{N}\right),\quad
t\in\\{1,...,N\\}.$

We remark that each of the observations
$X_{\frac{1}{N}},X_{\frac{2}{N}}...,X_{1}$ has a different unknown mean by
virtue of the dynamic nature of the drift $t\mapsto m^{\star}(t)$,
nevertheless, assumption (1.2) entails that

$\displaystyle|m^{\star}_{t}-m^{\star}_{s}|\leq\frac{K}{N}|t-s|,\quad\mbox{
for all $t,s\in\\{1,...,N\\}$}.$ (1.5)

Thus large values of $N$ dampen the time-varying nature of the sequence
$t\mapsto m^{\star}_{t}$. However, large values of $N$ increase -according to
equation (1.4)- the correlation between observations. Such contrasting effects
caused by increases of $N$ are an important element of the setting, we will
showcase the existence of a trade-off between them.

We stress that we are facing a non classical statistical problem: our aim is
to estimate $m^{\star}(1)$, or in the previously introduced notation
$m^{\star}_{N}$, which is the mean of $X_{1}$, and to do so we can only
utilize (in addition to the value of $X_{1}$) the observations
$X_{\frac{k}{N}}$, $k\in\\{1,...,N-1\\}$, which are realisations of random
variables with means $m^{\star}_{1},...,m^{\star}_{N-1}$, different from
$m^{\star}_{N}$. The Lipschitz-continuity assumption that entails (1.5)
provides information on the difference between unknown means and hence will be
crucial in deriving an estimator for $m^{\star}(1)$.
To estimate $m^{\star}_{N}$ we set

$\displaystyle\hat{m}_{1}:=X_{\frac{1}{N}}\quad\mbox{ and
}\quad\hat{m}_{t}:=(1-\alpha)\hat{m}_{t-1}+\alpha X_{\frac{t}{N}},\quad
t\in\\{2,...,N\\},$ (1.6)

where the parameter $\alpha\in[0,1]$ is called _learning rate_. Our suggested
estimator for $m^{\star}_{N}$ is $\hat{m}_{N}$, defined as solution of the
recursive equation (1.6). Notice that $\hat{m}_{N}$ can be written explicitly
as

$\displaystyle\hat{m}_{N}=\beta^{N-1}X_{\frac{1}{N}}+\alpha\sum_{j=2}^{N}\beta^{N-j}X_{\frac{j}{N}}.$
(1.7)

Scheme (1.6) is known in the econometric literature as _simple exponential
smoothing_ : it was originally defined in [6], and then further developed in
[7] and [8]. Simple exponential smoothing was originally employed as a way to
generate forecasts for time series data by giving more weight to recent
observations while exponentially diminishing the influence of older data, but
it has also been used to estimate the trend of a time series, see for instance
[5].
In our setting it is possible to individuate a statistical justification for
the estimation rule (1.6). In fact, if we rewrite equation (1.6) for
$t\in\\{2,...,N\\}$ as

$\displaystyle\hat{m}_{t}=$
$\displaystyle\hat{m}_{t-1}+\alpha(X_{\frac{t}{N}}-\hat{m}_{t-1})$
$\displaystyle=$
$\displaystyle\hat{m}_{t-1}+\tilde{\alpha}\frac{d}{d\hat{m}}\ln\left(\frac{1}{\sqrt{\pi\sigma^{2}/\theta}}e^{-\frac{\left(X_{\frac{t}{N}}-\hat{m}\right)^{2}}{\sigma^{2}/\theta}}\right)\Bigg{|}_{\hat{m}=\hat{m}_{t-1}}$

with $\tilde{\alpha}:=\frac{2\alpha\theta}{\sigma^{2}}$, one can identify
(1.6) with a stochastic gradient ascent algorithm where the function to be
optimized is the log-likelihood of the $t$-th observation, embedding
$\hat{m}_{N}$ in the framework of maximum likelihood estimation. There is
however a crucial difference from the usual optimization framework: the
likelihood function one wishes to optimize changes at each time $t$, since the
observations have different unknown means, and hence the maximum of those
functions is time-varying and not static.
Tracking a sequence of optima through time is a problem that has already been
considered in a variety of settings, see for instance [13, 14, 18, 9, 10].
Using the nomenclature of [16], these types of problems have been called
_time-varying_ optimization problems since their aim is to find the optimum of
an objective function that varies with time.
The aforementioned lift of the maximum likelihood principle to a dynamic
context has been explored, with varying degrees of generality, in recent
papers [12], [3] and [4], which show how the scheme defined by equation (1.6)
(or a suitable variation) is able to track the evolution of a time-varying
parameter.
We are now ready to state our main result whose proof is given in Section 4.1.

######  Theorem 1.1.

Let $\hat{m}_{N}$ be defined by the recursive equation (1.6). Then,
$\mathbb{E}[|\hat{m}_{1}-m^{\star}_{1}|^{2}]=\frac{\sigma^{2}}{2\theta}$ and
for $N\geq 2$ we have

$\displaystyle\mathbb{E}[|\hat{m}_{N}-m^{\star}_{N}|^{2}]\leq$
$\displaystyle\beta^{2(N-1)}\frac{\sigma^{2}}{2\theta}+\left(\beta^{2}\frac{1+\beta}{1-\beta}\frac{K^{2}}{N^{2}}+\alpha^{2}\frac{\sigma^{2}}{2\theta}\frac{e^{\frac{\theta}{N}}+\beta}{e^{\frac{\theta}{N}}-\beta}\right)\frac{1-\beta^{2(N-1)}}{1-\beta^{2}}$
$\displaystyle-2\frac{K^{2}}{N^{2}}\frac{\beta^{2N}-\beta^{N+1}}{\alpha^{2}}+\alpha\frac{\sigma^{2}}{\theta}\frac{e^{\frac{\theta}{N}}-1}{1-\beta
e^{-\frac{\theta}{N}}}\frac{\beta^{2N-1}e^{-\frac{\theta}{N}}-\beta^{N}e^{-\theta}}{\beta
e^{\frac{\theta}{N}}-1};$ (1.8)

here, to ease the notation, we have set $\beta:=1-\alpha$. Moreover, the
previous inequality is sharp: if the unknown drift $t\mapsto m^{\star}(t)$ is
linear, then it becomes an equality.

######  Remark 1.2.

  * •

The last ratio in (1.1) is well defined when $\beta\neq
e^{-\frac{\theta}{N}}$. This constraint stems from the requirement of the
geometric sum formula, which necessitates this condition for its application.
Nevertheless, the result remains valid without imposing this constraint,
provided that one eschews the geometric sum formula and retains the complete
summation term.

  * •

A simple verification shows that the upper bound (1.1) converges, as $N$ tends
to infinity, to $\frac{\sigma^{2}}{2\theta}$ which coincides with
$\mathbb{E}[|\hat{m}_{1}-m^{\star}_{1}|^{2}]$. This fact yields a non
monotonic behavior of the function
$N\mapsto\mathbb{E}[|\hat{m}_{N}-m^{\star}_{N}|^{2}]$.

  * •

If in (1.1) we Taylor-expand $e^{\frac{\theta}{N}}$ in a neighborhood of zero,
and neglect terms of order $O\left(\frac{1}{N^{2}}\right)$, we obtain

$\displaystyle\mathbb{E}[|\hat{m}_{N}-m^{\star}_{N}|^{2}]\leq$
$\displaystyle\frac{1+\frac{\theta}{N(1+\beta)}}{1+\frac{\theta}{N(1-\beta)}}\frac{\sigma^{2}}{2\theta}.$

Since the ratio in front of $\frac{\sigma^{2}}{2\theta}$ is smaller than one,
and converges to one as $N$ tends to infinity, we see that the upper bound
(1.1) converges, as $N$ tends to infinity, to $\frac{\sigma^{2}}{2\theta}$
from below. Thus the non monotonic function
$N\mapsto\mathbb{E}[|\hat{m}_{N}-m^{\star}_{N}|^{2}]$ possesses a minimum.

The rest of the paper is organized as follows: in Section 2 we propose several
plots of the upper bound (1.1) as a function of $N$ with the aim of
illustrating the role of the various parameters of the model on the
performance of our estimator. In Section 3, we focus on the case of a constant
drift $m^{\star}$ and compare our method with two classical estimators
suitable for investigating static problems: the arithmetic mean and the global
maximum likelihood estimator. In the final section, we present the proof of
our main theorem along with the derivation of two formulas used in the
analysis of the static case.

##  2 Numerical illustration

In this section we visually examine the impact of the various parameters in
our model (1.1)-(1.2)-(1.3) on the upper bound (1.1). In particular, we aim to
shed light on the relationship between $K$ and the learning rate $\alpha$, as
well as the role of $\theta$, in changing the graph of (1.1) as a function of
$N$.

In Figure 1 we plotted the upper bound (1.1) versus $N$ in the static case,
i.e. when $K=0$, for two different sets of values for $\theta$ and $\sigma$.
We explored four different learning rates, and an inspection of the graphs
indicates better performance for lower values of $\alpha$.

Figure 1: Plots of the sharp upper bound (1.1) as a function of $N$ with $K=0$
and different learning rates $\alpha$. In the left figure we set $\theta=1$
and $\sigma=1$ while in the right figure we chose $\theta=2$ and $\sigma=0.5$.
The least mean square error is attained in correspondence of small values of
the learning rate, as expected in the case of a static drift $m^{\star}$ (i.e.
when $K=0$), and for relatively small sample size.

In Figure 2 the Lipschitz constant $K$ is set equal to $0.5$, thus the dynamic
nature of the drift $m^{\star}$ assumes importance. In this case the smallest
value of the learning rate $\alpha$ does not always result in the best
performance.

Figure 2: Plots of the sharp upper bound (1.1) as a function of $N$ with
$K=0.5$ and different learning rates $\alpha$. In the left figure we set
$\theta=1$ and $\sigma=1$ while in the right figure we chose $\theta=2$ and
$\sigma=0.5$. The increase of $K$ now results in large mean square errors for
small learning rates.

The plots of Figure 3 consider $K=1.2$ (left) and $K=2$ (right). Here, the
deterministic behaviour of the drift dominates the uncertainty associated with
the observations: the red line in the last plot shows the almost irrelevant
contribution of frequent observations in reducing the mean square error.
Further increase of $K$ imply large mean square errors in correspondence to
small learning rates.

Figure 3: Plots of the sharp upper bound (1.1) as a function of $N$ with
$K=1.2$ (left figure), $K=2$ (right figure), $\theta=1$, $\sigma=1$ and
different learning rates $\alpha$.

##  3 The static case $K=0$

The case $K=0$, i.e. $m^{\star}(t)=m^{\star}$ for all $t\in[0,1]$, gives us
observations $X_{\frac{1}{N}}$, $X_{\frac{2}{N}}$,…, $X_{1}$ that are Gaussian
with common unknown mean $m^{\star}$ and common variance
$\frac{\sigma^{2}}{2\theta}$. One may therefore wonder whether an application
of standard statistical tools can provide lower mean square errors compared to
those derived with scheme (1.6). In this section we compare our method with
the arithmetic mean of the observations and the (global) maximum likelihood
estimator. Here, we emphasize the term ’global’ to highlight that our scheme
(1.6) is also based on the maximum likelihood principle, embedded in the
stochastic gradient ascent algorithm, but employed in an online fashion -one
observation at a time. While this feature is unnecessary in the current static
framework, it becomes crucial in the broader context presented earlier, where
the Lipschitz condition (1.2) (and thus the relationship between the unknown
parameters $m^{\star}_{t}$ for $t\in\\{1,...,N\\}$) implies that the order of
observations is significant.

###  3.1 Comparison with the arithmetic mean

The mean square error of the arithmetic mean of our observations
$X_{\frac{1}{N}}$, $X_{\frac{2}{N}}$,…, $X_{1}$ is given by

$\displaystyle\mathbb{E}\left[\left(\frac{1}{N}\sum_{i=1}^{N}X_{\frac{i}{N}}-m^{\star}\right)^{2}\right]=\frac{1}{N^{2}}\frac{\sigma^{2}}{2\theta}\frac{Ne^{-\frac{\theta}{N}}-Ne^{\frac{\theta}{N}}+2-2e^{-\theta}}{(1-e^{\frac{\theta}{N}})(1-e^{-\frac{\theta}{N}})},\quad
N\geq 1.$ (3.1)

A proof of this identity can be found in Section 4.2 below. We remark in
passing the marginal role of $\sigma^{2}$ in (3.1) where it trivially acts as
a multiplicative constant.
In Figure 4 we compare the performance of (1.1) with the one of (3.1); the
plots describe their behaviour as functions of $N$. It is interesting to note
that, if one uses the time-varying learning rate $\frac{1}{t}$ in (1.6), the
solution to the corresponding recursive equation coincides with the arithmetic
mean $\frac{1}{N}\sum_{i=1}^{N}X_{\frac{i}{N}}$. Hence, we are here comparing
two different instances of the same general method, one with constant learning
rate $\alpha$ and one with vanishing learning rate $\frac{1}{t}$. We remark
that, when the drift $m^{\star}$ is non constant, a vanishing learning rate
proves inadequate for effectively tracking the dynamic nature of the problem.
In Figure 4 we set $\alpha=0.005$, $\sigma=1$ and analyse the sensitivity of
the graph to an increase of $\theta$ from $0.1$ up to $15$, i.e., a decrease
of the correlation between the observations. When correlation is high, our
methods performs better than the arithmetic mean for a specific range of $N$
values; low correlation regimes are instead slightly favorable to the
arithmetic mean and provide an almost vanishing mean square error.

Figure 4: Comparison between the sharp upper bound (1.1) with $K=0$ and
$\alpha=0.005$ (orange line) and the right hand side of (3.1) (blue line). The
parameter $\sigma$ is set equal to one while $\theta=0.1$ (top left figure),
$\theta=1$ (top right figure), $\theta=5$ (bottom left figure) and $\theta=15$
(bottom right figure). The mean square error of our estimator performs better
than the arithmetic mean for a suitable range of $N$ values. As $\theta$
increases, that is, when the correlation decreases, the performance disparity
between the two statistical procedures diminishes, ultimately resulting in the
arithmetic mean surpassing our method. We observe the monotonic behavior of
the blue line, in contrast to the orange line, which attains a global minimum
at non-trivial values of $N$.

###  3.2 Comparison with the maximum likelihood estimator

The (global) maximum likelihood estimator $\hat{M}$ of $m^{\star}$ is defined
as

$\displaystyle\hat{M}:=\operatorname*{arg\,max}_{m\in\mathbb{R}}\frac{1}{(2\pi|C|)^{\frac{N}{2}}}\exp\left\\{-\frac{1}{2}\sum_{i,j=1}^{N}a_{ij}\left(X_{\frac{i}{N}}-m\right)\left(X_{\frac{j}{N}}-m\right)\right\\}$

where $C$ stands for the covariance matrix of the vector
$\left(X_{\frac{1}{N}},X_{\frac{2}{N}},...,X_{1}\right)$, i.e.,
$c_{ij}=\frac{\sigma^{2}}{2\theta}e^{-\frac{\theta}{N}|i-j|}$ for
$i,j\in\\{1,...,N\\}$, $|C|$ denotes the determinant of $C$ while
$A=\\{a_{ij}\\}_{i,j=1,..,N}$ stands for the inverse of $C$. To compute the
$\operatorname*{arg\,max}$ we take the logarithm of the likelihood function
and differentiate with respect to $m$, this yields

$\displaystyle\hat{M}=\frac{\sum\limits_{i,j=1}^{N}a_{ij}X_{\frac{j}{N}}}{\sum\limits_{i,j=1}^{N}a_{ij}},$

and its mean square error is given by

$\displaystyle\mathbb{E}\left[\left(\hat{M}-m^{\star}\right)^{2}\right]=\frac{\sigma^{2}(1-e^{-2\frac{\theta}{N}})}{2\theta\left(N+(N-2)e^{-2\frac{\theta}{N}}-2(N-1)e^{-\frac{\theta}{N}}\right)}.$
(3.2)

A proof of this formula can be found in Section 4.3.
In Figure 5 we compare the performance of the upper bound of our estimator
(1.1) with the maximum likelihood one (3.2), the plots describe their
behaviour as functions of $N$. Both methods employ the maximum likelihood
principle; however, the method associated with scheme (1.6) processes one
observation at a time, whereas the global maximum likelihood estimator
utilizes the entire dataset simultaneously. Consequently, we cannot expect our
estimator $\hat{m}_{N}$ to outperform $\hat{M}$ in the static setting.
Nonetheless, the subsequent plots will demonstrate that in low-correlation
regimes, the two methods yield nearly equivalent results. As was the case for
the arithmetic mean, in the dynamic setting with the Lipschitz Assumption 1.2,
the global maximum likelihood estimator loses its efficacy and adapting it to
this new context is not straightforward.
In Figure 5 we set $\alpha=0.005$, $\sigma=1$ and analyse the sensitivity of
the maximum likelihood estimator and our proposed estimator to an increase of
$\theta$ from $0.1$ up to $15$, i.e., a decrease of the correlation between
the observations.

Figure 5: A comparison between the sharp upper bound (1.1) with $K=0$ and
$\alpha=0.005$ (orange line) and the right hand side of (3.1) (blue line). The
parameter $\sigma$ is set to one while $\theta=0.1$ (top left figure),
$\theta=1$ (top right figure), $\theta=5$ (bottom left figure) and $\theta=15$
(bottom right figure). The (global) maximum likelihood estimator performs
better than our method. However, as $\theta$ increases, i.e., as the
correlation becomes weaker, the gap between the performances of the two
statistical procedures reduces. We again notice the monotonic behavior of the
blue line in contrast to the orange one which achieves a global minimum for
non trivial values of $N$.

##  4 Proofs

This section is devoted to the proofs of Theorem (1.1), formula (3.1) and
formula (3.2).

###  4.1 Proof of Theorem 1.1

The proof given here is inspired by the one contained in [4]. First of all we
recall that

$\displaystyle\hat{m}_{t}=\beta^{t-1}X_{\frac{1}{N}}+\alpha\sum_{j=2}^{t}\beta^{t-j}X_{\frac{j}{N}}.$
(4.1)

Now,

$\displaystyle\hat{m}_{t}-m^{\star}_{t}=\beta(\hat{m}_{t-1}-m^{\star}_{t})+\alpha\left(X_{\frac{t}{N}}-m^{\star}_{t}\right),$
(4.2)

and taking squares we obtain

$\displaystyle(\hat{m}_{t}-m^{\star}_{t})^{2}=\beta^{2}(\hat{m}_{t-1}-m^{\star}_{t})^{2}+\alpha^{2}\left(X_{\frac{t}{N}}-m^{\star}_{t}\right)^{2}+2\alpha\beta(\hat{m}_{t-1}-m^{\star}_{t})\left(X_{\frac{t}{N}}-m^{\star}_{t}\right).$
(4.3)

Considering the last term we see that

$\displaystyle(\hat{m}_{t-1}-m^{\star}_{t})\left(X_{\frac{t}{N}}-m^{\star}_{t}\right)=$
$\displaystyle\left(\beta^{t-2}X_{\frac{1}{N}}+\alpha\sum_{j=2}^{t-1}\beta^{t-1-j}X_{\frac{j}{N}}-m^{\star}_{t}\right)\left(X_{\frac{t}{N}}-m^{\star}_{t}\right)$
$\displaystyle=$
$\displaystyle\beta^{t-2}X_{\frac{1}{N}}\left(X_{\frac{t}{N}}-m^{\star}_{t}\right)+\alpha\sum_{j=2}^{t-1}\beta^{t-1-j}X_{\frac{j}{N}}\left(X_{\frac{t}{N}}-m^{\star}_{t}\right)$
$\displaystyle-m^{\star}_{t}\left(X_{\frac{t}{N}}-m^{\star}_{t}\right)$
$\displaystyle=$
$\displaystyle\beta^{t-2}\left(X_{\frac{1}{N}}-m^{\star}_{1}\right)\left(X_{\frac{t}{N}}-m^{\star}_{t}\right)$
$\displaystyle+\alpha\sum_{j=2}^{t-1}\beta^{t-1-j}\left(X_{\frac{j}{N}}-m^{\star}_{j}\right)\left(X_{\frac{t}{N}}-m^{\star}_{t}\right)$
$\displaystyle+\quad\mbox{zero mean terms}.$

Therefore, taking the expectation in (4.3) gives

$\displaystyle\mathbb{E}[(\hat{m}_{t}-m^{\star}_{t})^{2}]=$
$\displaystyle\beta^{2}\mathbb{E}[(\hat{m}_{t-1}-m^{\star}_{t})^{2}]+\alpha^{2}\frac{\sigma^{2}}{2\theta}+2\alpha\beta\left(\beta^{t-2}\gamma(t-1)+\alpha\sum_{j=2}^{t-1}\beta^{t-1-j}\gamma(t-j)\right)$
$\displaystyle=$
$\displaystyle\beta^{2}\mathbb{E}[(\hat{m}_{t-1}-m^{\star}_{t})^{2}]+\alpha^{2}\frac{\sigma^{2}}{2\theta}+2\alpha\beta\left(\beta^{t-1}\gamma(t-1)+\alpha\sum_{j=1}^{t-1}\beta^{t-1-j}\gamma(t-j)\right)$
$\displaystyle=$
$\displaystyle\beta^{2}\mathbb{E}[(\hat{m}_{t-1}-m^{\star}_{t})^{2}]+\alpha^{2}\frac{\sigma^{2}}{2\theta}+2\alpha\beta^{t}\gamma(t-1)+2\alpha^{2}\sum_{j=1}^{t-1}\beta^{t-j}\gamma(t-j)$
$\displaystyle=$
$\displaystyle\beta^{2}\mathbb{E}[(\hat{m}_{t-1}-m^{\star}_{t})^{2}]+\alpha^{2}\frac{\sigma^{2}}{2\theta}+2\alpha\beta^{t}\gamma(t-1)+2\alpha^{2}\sum_{k=1}^{t-1}\beta^{k}\gamma(k).$

Now,

$\displaystyle\mathbb{E}[(\hat{m}_{t-1}-m^{\star}_{t})^{2}]=$
$\displaystyle\mathbb{E}[(\hat{m}_{t-1}-m^{\star}_{t-1})^{2}]+K_{t}^{2}-2K_{t}\mathbb{E}[\hat{m}_{t-1}-m^{\star}_{t-1}],$

with $K_{t}:=m_{t}^{\star}-m_{t-1}^{\star}$. On the other hand, taking the
expectation in (4.2) we get

$\displaystyle\mathbb{E}[\hat{m}_{t}-m^{\star}_{t}]=\beta\mathbb{E}[\hat{m}_{t-1}-m^{\star}_{t-1}]-\beta
K_{t},$

with
$\mathbb{E}[\hat{m}_{1}-m^{\star}_{1}]=\mathbb{E}\left[X_{\frac{1}{N}}-m^{\star}_{1}\right]=0$;
this means

$\displaystyle\mathbb{E}[\hat{m}_{t}-m^{\star}_{t}]=-\beta\sum_{j=2}^{t}\beta^{t-j}K_{j}.$

Therefore, setting $D_{t}:=\mathbb{E}[(\hat{m}_{t}-m^{\star}_{t})^{2}]$ we
obtain

$\displaystyle D_{t}=$
$\displaystyle\beta^{2}\left(D_{t-1}+K_{t}^{2}+2K_{t}\sum_{j=2}^{t-1}\beta^{t-j}K_{j}\right)+\alpha^{2}\frac{\sigma^{2}}{2\theta}+2\alpha\beta^{t}\gamma(t-1)+2\alpha^{2}\sum_{k=1}^{t-1}\beta^{k}\gamma(k).$
(4.4)

At this point we upper bound each $K_{j}$ in (4.4) with $\frac{K}{N}$ to get

$\displaystyle D_{t}\leq$
$\displaystyle\beta^{2}D_{t-1}+\beta^{2}\frac{K^{2}}{N^{2}}\left(1+2\sum_{j=2}^{t-1}\beta^{t-j}\right)+\alpha^{2}\frac{\sigma^{2}}{2\theta}+2\alpha\beta^{t}\gamma(t-1)+2\alpha^{2}\sum_{k=1}^{t-1}\beta^{k}\gamma(k)$
(4.5) $\displaystyle=$
$\displaystyle\beta^{2}D_{t-1}+\beta^{2}\frac{1+\beta-2\beta^{t-1}}{1-\beta}\frac{K^{2}}{N^{2}}+\alpha^{2}\frac{\sigma^{2}}{2\theta}+2\alpha\beta^{t}\gamma(t-1)+2\alpha^{2}\sum_{k=1}^{t-1}\beta^{k}\gamma(k).$

We now replace $\gamma(k)$ with
$\frac{\sigma^{2}}{2\theta}e^{-\frac{\theta}{N}k}$; this gives

$\displaystyle\sum_{k=1}^{t-1}\beta^{k}\gamma(k)=\frac{\sigma^{2}}{2\theta}\sum_{k=1}^{t-1}\beta^{k}e^{-\frac{\theta}{N}k}=\frac{\sigma^{2}}{2\theta}\sum_{k=1}^{t-1}(\beta
e^{-\frac{\theta}{N}})^{k}=\frac{\sigma^{2}}{2\theta}\frac{\beta
e^{-\frac{\theta}{N}}-(\beta e^{-\frac{\theta}{N}})^{t}}{1-\beta
e^{-\frac{\theta}{N}}},$

and hence

$\displaystyle D_{t}\leq$
$\displaystyle\beta^{2}D_{t-1}+\beta^{2}\frac{1+\beta-2\beta^{t-1}}{1-\beta}\frac{K^{2}}{N^{2}}+\alpha^{2}\frac{\sigma^{2}}{2\theta}+\alpha\frac{\sigma^{2}}{\theta}\beta^{t}e^{-\frac{\theta}{N}(t-1)}+\alpha^{2}\frac{\sigma^{2}}{\theta}\frac{\beta
e^{-\frac{\theta}{N}}-(\beta e^{-\frac{\theta}{N}})^{t}}{1-\beta
e^{-\frac{\theta}{N}}}$ $\displaystyle=$
$\displaystyle\beta^{2}D_{t-1}+\beta^{2}\frac{1+\beta-2\beta^{t-1}}{1-\beta}\frac{K^{2}}{N^{2}}+\alpha^{2}\frac{\sigma^{2}}{2\theta}+\alpha\frac{\sigma^{2}}{\theta}e^{\frac{\theta}{N}}(\beta
e^{-\frac{\theta}{N}})^{t}+\alpha^{2}\frac{\sigma^{2}}{\theta}\frac{\beta
e^{-\frac{\theta}{N}}-(\beta e^{-\frac{\theta}{N}})^{t}}{1-\beta
e^{-\frac{\theta}{N}}}.$

Simple algebraic manipulations on the last three terms above give

$\displaystyle\alpha^{2}\frac{\sigma^{2}}{2\theta}+\alpha\frac{\sigma^{2}}{\theta}e^{\frac{\theta}{N}}(\beta
e^{-\frac{\theta}{N}})^{t}+\alpha^{2}\frac{\sigma^{2}}{\theta}\frac{\beta
e^{-\frac{\theta}{N}}-(\beta e^{-\frac{\theta}{N}})^{t}}{1-\beta
e^{-\frac{\theta}{N}}}$
$\displaystyle\quad\quad=\alpha\frac{\sigma^{2}}{\theta}\frac{e^{\frac{\theta}{N}}-1}{1-\beta
e^{-\frac{\theta}{N}}}(\beta
e^{-\frac{\theta}{N}})^{t}+\alpha^{2}\frac{\sigma^{2}}{2\theta}\frac{e^{\frac{\theta}{N}}+\beta}{e^{\frac{\theta}{N}}-\beta};$

this implies

$\displaystyle D_{t}\leq$
$\displaystyle\beta^{2}D_{t-1}+\beta^{2}\frac{1+\beta-2\beta^{t-1}}{1-\beta}\frac{K^{2}}{N^{2}}+\alpha\frac{\sigma^{2}}{\theta}\frac{e^{\frac{\theta}{N}}-1}{1-\beta
e^{-\frac{\theta}{N}}}(\beta
e^{-\frac{\theta}{N}})^{t}+\alpha^{2}\frac{\sigma^{2}}{2\theta}\frac{e^{\frac{\theta}{N}}+\beta}{e^{\frac{\theta}{N}}-\beta}$
$\displaystyle=$
$\displaystyle\beta^{2}D_{t-1}+\beta^{2}\frac{1+\beta}{1-\beta}\frac{K^{2}}{N^{2}}+\alpha^{2}\frac{\sigma^{2}}{2\theta}\frac{e^{\frac{\theta}{N}}+\beta}{e^{\frac{\theta}{N}}-\beta}-\frac{2}{1-\beta}\frac{K^{2}}{N^{2}}\beta^{t+1}$
$\displaystyle+\alpha\frac{\sigma^{2}}{\theta}\frac{e^{\frac{\theta}{N}}-1}{1-\beta
e^{-\frac{\theta}{N}}}(\beta e^{-\frac{\theta}{N}})^{t}$ $\displaystyle=$
$\displaystyle\beta^{2}D_{t-1}+C_{1}-C_{2}\beta^{t+1}+C_{3}(\beta
e^{-\frac{\theta}{N}})^{t},$

with

$\displaystyle
C_{1}:=\beta^{2}\frac{1+\beta}{1-\beta}\frac{K^{2}}{N^{2}}+\alpha^{2}\frac{\sigma^{2}}{2\theta}\frac{e^{\frac{\theta}{N}}+\beta}{e^{\frac{\theta}{N}}-\beta},\quad
C_{2}:=\frac{2}{1-\beta}\frac{K^{2}}{N^{2}},\quad
C_{3}:=\alpha\frac{\sigma^{2}}{\theta}\frac{e^{\frac{\theta}{N}}-1}{1-\beta
e^{-\frac{\theta}{N}}}.$ (4.6)

Solving for $t\in\\{2,...,N\\}$ the following recursive inequality

$\displaystyle D_{t}\leq$
$\displaystyle\beta^{2}D_{t-1}+C_{1}-C_{2}\beta^{t+1}+C_{3}(\beta
e^{-\frac{\theta}{N}})^{t},$

with $D_{1}=\frac{\sigma^{2}}{2\theta}$, yields

$\displaystyle D_{N}\leq$
$\displaystyle\beta^{2(N-1)}\frac{\sigma^{2}}{2\theta}+\sum_{t=2}^{N}\beta^{2(N-t)}(C_{1}-C_{2}\beta^{t+1}+C_{3}(\beta
e^{-\frac{\theta}{N}})^{t})$ $\displaystyle=$
$\displaystyle\beta^{2(N-1)}\frac{\sigma^{2}}{2\theta}+C_{1}\frac{1-\beta^{2(N-1)}}{1-\beta^{2}}-C_{2}\beta^{2N}\sum_{t=2}^{N}(\beta^{-1})^{t-1}+C_{3}\beta^{2N}\sum_{t=2}^{N}(\beta^{-1}e^{-\frac{\theta}{N}})^{t}$
$\displaystyle=$
$\displaystyle\beta^{2(N-1)}\frac{\sigma^{2}}{2\theta}+C_{1}\frac{1-\beta^{2(N-1)}}{1-\beta^{2}}-C_{2}\frac{\beta^{2N}-\beta^{N+1}}{\beta-1}+C_{3}\frac{\beta^{2N-1}e^{-\frac{\theta}{N}}-\beta^{N}e^{-\theta}}{\beta
e^{\frac{\theta}{N}}-1}.$

In conclusion, recalling notation (4.6), we obtain, for all $N\geq 2$, the
bound

$\displaystyle D_{N}\leq$
$\displaystyle\beta^{2(N-1)}\frac{\sigma^{2}}{2\theta}+\left(\beta^{2}\frac{1+\beta}{1-\beta}\frac{K^{2}}{N^{2}}+\alpha^{2}\frac{\sigma^{2}}{2\theta}\frac{e^{\frac{\theta}{N}}+\beta}{e^{\frac{\theta}{N}}-\beta}\right)\frac{1-\beta^{2(N-1)}}{1-\beta^{2}}$
$\displaystyle-\frac{2}{1-\beta}\frac{K^{2}}{N^{2}}\frac{\beta^{2N}-\beta^{N+1}}{\beta-1}+\alpha\frac{\sigma^{2}}{\theta}\frac{e^{\frac{\theta}{N}}-1}{1-\beta
e^{-\frac{\theta}{N}}}\frac{\beta^{2N-1}e^{-\frac{\theta}{N}}-\beta^{N}e^{-\theta}}{\beta
e^{\frac{\theta}{N}}-1},$

with $D_{1}=\frac{\sigma^{2}}{2\theta}$: this proves formula (1.1). The
statement regarding the sharpness of the inequality is immediately obtained by
observing that the use of an inequality bound throughout the proof occurs only
at (4.5), where we upper bound each $K_{j}$ with $\frac{K}{N}$. Recalling that
$K_{j}:=m^{\star}_{j}-m^{\star}_{j-1}$ and that
$m^{\star}_{j}=m^{\star}\left(\frac{j}{N}\right)$, we see that in the case of
a linear drift $m^{\star}:[0,1]\to\mathbb{R}$ we can write

$\displaystyle
K_{j}=m^{\star}_{j}-m^{\star}_{j-1}=m^{\star}\left(\frac{j}{N}\right)-m^{\star}\left(\frac{j-1}{N}\right)=\frac{K}{N},$

obtaining the sharpness.

###  4.2 Proof of formula (3.1)

We have

$\displaystyle\mathbb{E}\left[\left(\frac{1}{N}\sum_{i=1}^{N}X_{\frac{i}{N}}-m^{\star}\right)^{2}\right]=$
$\displaystyle\mathbb{E}\left[\left(\frac{1}{N}\sum_{i=1}^{N}\left(X_{\frac{i}{N}}-m^{\star}\right)\right)^{2}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{N^{2}}\sum_{i,j=1}^{N}\mathbb{E}\left[\left(X_{\frac{i}{N}}-m^{\star}\right)\left(X_{\frac{j}{N}}-m^{\star}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{N^{2}}\sum_{i,j=1}^{N}\mathtt{cov}\left(X_{\frac{i}{N}},X_{\frac{j}{N}}\right).$

We now utilize (1.4) to obtain

$\displaystyle\mathbb{E}\left[\left(\frac{1}{N}\sum_{i=1}^{N}X_{\frac{i}{N}}-m^{\star}\right)^{2}\right]=$
$\displaystyle\frac{1}{N^{2}}\sum_{i,j=1}^{N}\mathtt{cov}\left(X_{\frac{i}{N}},X_{\frac{j}{N}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{N^{2}}\sum_{i,j=1}^{N}\frac{\sigma^{2}}{2\theta}e^{-\frac{\theta}{N}|i-j|}$
$\displaystyle=$
$\displaystyle\frac{1}{N^{2}}\frac{\sigma^{2}}{2\theta}\left(N+2\sum_{i>j}e^{-\frac{\theta}{N}(i-j)}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{N^{2}}\frac{\sigma^{2}}{2\theta}\left(N+2\sum_{i=2}^{N}\sum_{j=1}^{i-1}e^{-\frac{\theta}{N}(i-j)}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{N^{2}}\frac{\sigma^{2}}{2\theta}\left(N+2\sum_{i=2}^{N}e^{-\frac{\theta}{N}i}\sum_{j=1}^{i-1}e^{\frac{\theta}{N}j}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{N^{2}}\frac{\sigma^{2}}{2\theta}\left(N+2\sum_{i=2}^{N}e^{-\frac{\theta}{N}i}\frac{e^{\frac{\theta}{N}}-e^{\frac{\theta}{N}i}}{1-e^{\frac{\theta}{N}}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{N^{2}}\frac{\sigma^{2}}{2\theta}\left(N+\frac{2}{1-e^{\frac{\theta}{N}}}\sum_{i=2}^{N}\left(e^{-\frac{\theta}{N}(i-1)}-1\right)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{N^{2}}\frac{\sigma^{2}}{2\theta}\left(N+\frac{2}{1-e^{\frac{\theta}{N}}}\left(\frac{e^{-\frac{\theta}{N}}-e^{-\theta}}{1-e^{-\frac{\theta}{N}}}-(N-1)\right)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{N^{2}}\frac{\sigma^{2}}{2\theta}\frac{Ne^{-\frac{\theta}{N}}-Ne^{\frac{\theta}{N}}+2-2e^{-\theta}}{(1-e^{\frac{\theta}{N}})(1-e^{-\frac{\theta}{N}})}.$

Comparing the first and last members above we get the desired formula (3.1).

###  4.3 Proof of formula (3.2)

Adding and subtracting $m^{\star}$ from $X_{\frac{j}{N}}$ in

$\displaystyle\hat{M}=\frac{\sum\limits_{i,j=1}^{N}a_{ij}X_{\frac{j}{N}}}{\sum\limits_{i,j=1}^{N}a_{ij}}$

gives

$\displaystyle\hat{M}=\frac{\sum\limits_{i,j=1}^{N}a_{ij}\left(X_{\frac{j}{N}}-m^{\star}\right)+\sum\limits_{i,j=1}^{N}a_{ij}m^{\star}}{\sum\limits_{i,j=1}^{N}a_{ij}}=\frac{\sum\limits_{i,j=1}^{N}a_{ij}\left(X_{\frac{j}{N}}-m^{\star}\right)}{\sum\limits_{i,j=1}^{N}a_{ij}}+m^{\star},$

and hence

$\displaystyle\hat{M}-m^{\star}=\frac{\sum\limits_{i,j=1}^{N}a_{ij}\left(X_{\frac{j}{N}}-m^{\star}\right)}{\sum\limits_{i,j=1}^{N}a_{ij}}.$

We can therefore compute the mean square error of the (global) maximum
likelihood estimator as

$\displaystyle\mathbb{E}[(\hat{M}-m^{\star})^{2}]=\frac{\sum\limits_{i,j=1}^{N}\sum\limits_{k,l=1}^{N}a_{ij}a_{kl}\mathbb{E}\left[\left(X_{\frac{j}{N}}-m^{\star}\right)\left(X_{\frac{l}{N}}-m^{\star}\right)\right]}{\left(\sum\limits_{i,j=1}^{N}a_{ij}\right)^{2}}$
$\displaystyle\quad\quad=\frac{\sum\limits_{i,j=1}^{N}\sum\limits_{k,l=1}^{N}a_{ij}a_{kl}c_{jl}}{\left(\sum\limits_{i,j=1}^{N}a_{ij}\right)^{2}}=\frac{\sum\limits_{i=1}^{N}\sum\limits_{k,l=1}^{N}a_{kl}\sum\limits_{j=1}^{N}a_{ij}c_{jl}}{\left(\sum\limits_{i,j=1}^{N}a_{ij}\right)^{2}}$
$\displaystyle\quad\quad=\frac{\sum\limits_{i=1}^{N}\sum\limits_{k,l=1}^{N}a_{kl}\delta_{il}}{\left(\sum\limits_{i,j=1}^{N}a_{ij}\right)^{2}}=\frac{\sum\limits_{i=1}^{N}\sum\limits_{k=1}^{N}a_{ki}}{\left(\sum\limits_{i,j=1}^{N}a_{ij}\right)^{2}}$
$\displaystyle\quad\quad=\frac{1}{\sum\limits_{i,j=1}^{N}a_{ij}}.$

Here, $\\{\delta_{il}\\}_{i,l=1,...,N}$ stands for the Kronecker symbol and we
have exploited the fact that $A$ is the inverse of $C$. On the other hand, a
direct verification shows that

$\displaystyle
A=\frac{2\theta}{\sigma^{2}(1-e^{-2\frac{\theta}{N}})}\begin{bmatrix}1&-e^{-\frac{\theta}{N}}&0&0&\cdot&0\\\
-e^{-\frac{\theta}{N}}&1+e^{-2\frac{\theta}{N}}&-e^{-\frac{\theta}{N}}&0&\cdot&0\\\
0&-e^{-\frac{\theta}{N}}&1+e^{-2\frac{\theta}{N}}&-e^{-\frac{\theta}{N}}&\cdot&0\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
0&\cdot&0&-e^{-\frac{\theta}{N}}&1+e^{-2\frac{\theta}{N}}&-e^{-\frac{\theta}{N}}\\\
0&0&\cdot&0&-e^{-\frac{\theta}{N}}&1\end{bmatrix},$

and hence

$\displaystyle\sum\limits_{i,j=1}^{N}a_{ij}=\frac{2\theta}{\sigma^{2}(1-e^{-2\frac{\theta}{N}})}\left(N+(N-2)e^{-2\frac{\theta}{N}}-2(N-1)e^{-\frac{\theta}{N}}\right).$

To conclude,

$\displaystyle\mathbb{E}[(\hat{M}-m^{\star})^{2}]=\frac{\sigma^{2}(1-e^{-2\frac{\theta}{N}})}{2\theta\left(N+(N-2)e^{-2\frac{\theta}{N}}-2(N-1)e^{-\frac{\theta}{N}}\right)}.$

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