# CPT and Lorentz symmetry tests with hydrogen using a novel in-beam hyperfine
spectroscopy method applicable to antihydrogen experiments

L. Nowak  C. Malbrunot111Permanent address: TRIUMF, 4004 Wesbrook Mall,
Vancouver, BC V6T 2A3, Canada and Physics Department, McGill University,
Montréal, Québec H3A 2T8, Canada and Department of Physics and Astronomy,
University of British Columbia, Vancouver BC, V6T 1Z1 Canada<EMAIL_ADDRESS>M. C. Simon  C. Amsler  S. Arguedas Cuendis222Present address: Consejo
Nacional de Rectores (CONARE), San José, Geroma, 10109, Costa Rica S.
Lahs333Present address: Université Paris-Saclay, CNRS, Laboratoire Aimé
Cotton, 91405, Orsay, France A. Lanz444Present address:University College
London, Gower St, London WC1E 6BT, United Kingdom  A. Nanda  M.
Wiesinger555Present address: Faculty of Physics, Ludwig-Maximilians-
Universität München, Am Coulombwall 1, 85748 Garching bei München, Germany T.
Wolz  E. Widmann

###### Abstract

We present a Rabi-type measurement of two ground-state hydrogen hyperfine
transitions performed in two opposite external magnetic field directions. This
puts first constraints at the level of
$2.3\text{\times}{10}^{-21}\text{\,}\mathrm{GeV}$ on a set of coefficients of
the Standard Model Extension, which were not measured by previous experiments.
Moreover, we introduce a novel method, applicable to antihydrogen hyperfine
spectroscopy in a beam, that determines the zero-field hyperfine transition
frequency from the two transitions measured at the same magnetic field. Our
value, $\nu_{0}=$1.420\,405\,751\,63(63)\text{\,}\mathrm{GHz}$$, is in
agreement with literature at a relative precision of 0.44 ppb. This is the
highest precision achieved on hydrogen in a beam, improving over previous
results by a factor of 6.

###### keywords:

ground-state hyperfine splitting , hydrogen , Standard Model Extension , CPT
and Lorentz violation , Rabi spectroscopy

††journal: Physics Letters B

[CERN]organization=CERN,addressline=1, Esplanade des Particules, city=Meyrin,
postcode=CH-1211, country=Switzerland [SMI] organization=Stefan Meyer
Institute for Subatomic Physics,addressline=Dominikanerbastei 16, city=Vienna,
postcode=A-1010, country=Austria [UniV]organization=University of Vienna,
Vienna Doctoral School in Physics,addressline=Universitaetsring 1,
city=Vienna, postcode=A-1010, country=Austria

##  1 Introduction

Precision spectroscopy of the hydrogen atom is receiving renewed interest, in
particular in the context of the proton radius puzzle [1], the determination
of the fine structure constant [2], and for comparison with its antimatter
counterpart, the antihydrogen ($\overline{\text{H}}$) atom, hitherto the only
synthesised stable atom made of antimatter. Techniques developed for precise
measurements of hydrogen atoms are being applied to antihydrogen and new
techniques are tested on hydrogen before being deployed in the more
challenging environment of antihydrogen experiments. One of the main
motivations for matter/antimatter comparisons is to test CPT symmetry (the
combination of the three discrete symmetries: Charge conjugation, Parity, and
Time reversal). Comparing the sensitivities of different tests of CPT symmetry
requires the use of a unified framework to parameterise deviations from this
symmetry. A framework that serves this purpose is the Standard Model Extension
(SME) [3, 4] which is an effective field theory that generalises the Standard
Model (SM) Lagrangian by adding terms violating CPT and Lorentz symmetry. The
models for testing Lorentz and CPT symmetry in atomic spectroscopy experiments
[5, 6] reveal that the sensitivity to coefficients that quantify a breaking of
CPT symmetry, known as SME coefficients, depends on the absolute precision of
the experiment. From this viewpoint the hyperfine splitting could provide the
most stringent test, as this transition has been determined with unrivalled
absolute uncertainties of a few mHz using hydrogen masers [7, 8, 9, 10, 11,
12]. Such precise maser measurements have provided tight constraints on SME
coefficients accessible through searches for sidereal variation [13, 14]
within the so-called minimal SME [5], restricted to Lorentz-violating operator
with mass dimension $d\leq 4$, expected to contain the dominant effects at low
energies. An extension of the framework was since developed, including
operators of arbitrary dimensions [15, 6]. Commonly appearing combinations of
SME coefficients are defined as effective SME coefficients. In this work we
consider the non-relativistic (NR) effective coefficients [15]. We denote the
NR coefficients in a laboratory frame on the Earth as
$\mathcal{K}^{\textrm{NR,lab}}$. The relation to the NR coefficients in the
Sun-centered frame, the frame commonly used to report the limits on the SME
coefficients, with the ones in the laboratory frame is given by the
transformation:

$\begin{split}\mathcal{K}^{\textrm{NR,lab}}_{w_{k10}}={\mathcal{K}^{\textrm{NR,Sun}}_{w_{k10}}\cos{\theta}}-\sqrt{2}\Re\mathcal{K}^{\textrm{NR,Sun}}_{w_{k11}}\sin{\theta}\cos{\omega_{\bigoplus}T_{\bigoplus}}\\\
+\sqrt{2}\Im\mathcal{K}^{\textrm{NR,Sun}}_{w_{k11}}\sin{\theta}\sin{\omega_{\bigoplus}T_{\bigoplus}}\end{split}$
(1)

where $w$ stands for electron or proton, $k$ represents the power of the
fermion’s momentum of the operator coupled to the coefficient, and
$T_{\bigoplus}$ represents the sidereal time, which is a location-dependent
time that is a convenient offset of the time coordinate of the Sun-centered
frame. Searches for variations of $\mathcal{K}^{\textrm{NR,lab}}_{w_{k10}}$ at
the sidereal frequency $\omega_{\bigoplus}$ using a hydrogen maser have
constrained the real and imaginary parts of
$\mathcal{K}^{\textrm{NR,Sun}}_{w_{k11}}$ while
$\mathcal{K}^{\textrm{NR,Sun}}_{w_{k10}}$ remained unconstrained.

This precise maser technique unfortunately cannot be applied to
$\overline{\textrm{H}}$ hyperfine spectroscopy as it relies on storage in a
bulb. One approach is to work with magnetically trapped
$\overline{\textrm{H}}$ as done successfully by the ALPHA collaboration [16].
First hyperfine spectroscopy results have been reported [17] although control
of systematic uncertainties in the highly inhomogeneous magnetic field
environment poses a challenge. An alternative approach for
$\overline{\textrm{H}}$ hyperfine spectroscopy has been proposed by the
ASACUSA collaboration [18, 19]. Producing a beam of $\overline{\textrm{H}}$
[20] enables Rabi-type measurements [21, 22], in which the hyperfine
interaction takes place in a region of well-controlled fields outside of the
$\overline{\textrm{H}}$ production trap. ASACUSA tested its
$\overline{\textrm{H}}$ Rabi-spectrometer [23] with hydrogen and reached an
absolute precision of $3.8\text{\,}\mathrm{Hz}$ (corresponding to a relative
precision of $2.7\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$) on the hyperfine
splitting frequency [24], thereby improving over previous in-beam measurements
[25, 26]. In that work, the $\sigma$ transition
($F,m_{F}$:1,0$\rightarrow$0,0, where $F$ represents the total angular
momentum quantum number and $m_{F}$ its $z$-axis projection, see Fig. 1) was
measured at various magnetic field values and extrapolated to zero field.
However, the sensitivity of the $\sigma$ transition to SME effects is strongly
suppressed at low fields and eventually vanishes at zero field. Similarly,
when the difference of the $\pi_{1}$ ($F,m_{F}$: 1,1 $\rightarrow$ 0,0) and
$\pi_{2}$ ($F,m_{F}$: 1,0 $\rightarrow$ 1,-1) transitions is used to extract
the zero-field hyperfine splitting [17] one becomes insensitive to SME
coefficients.

Here we employ quasi-simultaneous measurements of the $\sigma$ and $\pi_{1}$
transitions (hereafter referred to as $\pi$ transition) and use the results in
two ways: by measuring the $\pi$ transition at opposite magnetic field
orientation we constrain coefficients embedded in
$\mathcal{K}^{\textrm{NR,Sun}}_{w_{k10}}$. Using two transitions enables to
disentangle SME effects from shifts originating from magnetic field
uncertainties imposed by the change of field orientation. Furthermore, for
each resonance pair the zero-field hyperfine splitting can be calculated. This
novel method is applicable to antihydrogen as an accurate result is already
obtained from the measurement of a single ($\sigma$, $\pi$) transition pair,
while extrapolation to zero field requires multiple measurements.

Figure 1:  Breit-Rabi diagram: dependence of ground-state hydrogen hyperfine
energy states on an external static magnetic field ($B_{\text{stat}}$). The
energy levels that increase/decrease towards higher magnetic field amplitude
are labelled low/high field seeking (LFS/HFS) states. The two states with
$m_{F}=\pm 1$ experience a strong linear Zeeman shift. The mixing of the two
states with $m_{F}$=0 results in a weaker, second order Zeeman shift and also
in a suppressed sensitivity to potential CPT or Lorentz symmetry violations.
The inset illustrates the drastically different field dependence for the
$\sigma$ and $\pi_{1}$ transition within the region of magnetic fields probed
in this work.

##  2 Experimental setup

Figure 2:  Schematics of the experimental apparatus (total length
$\sim$5\text{\,}\mathrm{m}$$, not to scale). The hydrogen source region (1)
consists of a microwave-driven plasma dissociating H2 molecules, a cryostat
cooling hydrogen atoms passing through a PTFE tubing, a chopper modulating the
beam, and sextupole magnets for beam polarisation. In the interaction region
(2) a strip-line cavity provides the oscillating magnetic field
$B_{\text{osc}}$ at an angle of 45∘ with respect to the static external
magnetic field $B_{\text{stat}}$ produced by a set of McKeehan coils. The
assembly is enclosed in a three-layer cylindrical mu-metal shielding. The beam
is then spin-analysed in the detection region (3) by sextupole magnets before
it enters a quadrupole mass spectrometer that selectively counts protons.

The measurement principle follows Rabi’s magnetic resonance spectroscopy [21]:
polarised atoms (low field seekers - LFS) undergo a quantum transition in the
presence of an oscillating magnetic field. A magnetic field gradient then
separates the magnetic states and focuses LFS atoms on a detector. The count
rate at the detector is acquired as a function of the frequency of the
oscillating field to reveal a resonance pattern from which the transition
frequency can be determined.

The apparatus employed here is a modified version of the beamline designed to
characterise the ASACUSA antihydrogen hyperfine spectrometer [23], which
yielded the previous best in-beam measurement of the hydrogen hyperfine
structure [24]. Figure 2 shows a sketch of the experimental setup. It consists
of a source of cold, modulated, and polarised atomic hydrogen (1), a hyperfine
interaction region providing well-controlled static and oscillating magnetic
fields (2), and a detection system based on state selection and single ion
counting after electron impact ionisation (3).

Hydrogen molecules are dissociated in the microwave-driven plasma of an
Evenson cavity [27]. The hydrogen atoms lose energy through collisions in a
PTFE tubing sandwiched between two aluminum blocks connected to a cold finger
stabilised at $27\text{\,}\mathrm{K}$. The cooled hydrogen beam then passes
through an aperture to a second vacuum chamber housing a chopper which
modulates the beam at a frequency of $66\text{\,}\mathrm{Hz}$ for lock-in
amplification allowing for background suppression against residual hydrogen in
the detection region. At the exit of this vacuum chamber a ring aperture
($1^{\textrm{st}}$ ring aperture) blocks the central part of the divergent
beam before it enters the magnetic sextupole field produced by permanent
magnets arranged in a circular Halbach array [28]. At the typical radius of
the annular beam of $15\text{\,}\mathrm{mm}$ the sextupole field strength
reaches $190\text{\,}\mathrm{mT}$. At such high fields the two LFS states have
comparable magnetic moments, thus the trajectory restriction by the ring
apertures selects similar velocities for the two initial states.

The polarised beam then enters the radio-frequency (RF) cavity [23], which was
designed with a large geometrical acceptance (open diameter of
$100\text{\,}\mathrm{mm}$) as a compensation measure for the small yield of
$\overline{\textrm{H}}$ experiments. At the entrance and exit of the cavity,
the RF field is contained by two meshes with a combined transparency of 95%
and separated by $105.5\text{\,}\mathrm{mm}$, i.e. half a wavelength of the
hyperfine transition. A signal generator coupled to an amplifier provides the
RF waves, which are guided to an antenna in the cavity via a coupling-
maximising double stub tuner and a vacuum feedthrough. A second antenna picks
up the RF power for monitoring by a spectrum analyser. Both, the signal
generator and spectrum analyser are frequency-stabilised by a GPS-disciplined
rubidium clock with a long-term relative frequency stability on the order of
$\sim 1\times 10^{-12}$. A standing wave forms in the cavity leading to a
sinusoidal variation of the oscillating magnetic field, $B_{\text{osc}}$, in
the beam propagation direction, causing a double-dip resonance lineshape, as
illustrated in Fig. 3. For the derivation of the resonance shape see Ref.
[24]. The microwave cavity is tilted by 45∘ compared to the previous
arrangement. Therefore $B_{\text{osc}}$ has a component parallel and
perpendicular to the static magnetic field [29] as needed to drive the
$\sigma$ and $\pi$ transition, respectively.

As shown in the inset of Fig. 1, the $\pi$ transition is much more sensitive
to the magnetic field and thus to magnetic field inhomogeneities. Hence, the
Helmholtz coils of the previous setup were replaced by a McKeehan
configuration [30], consisting of two pairs of coils of different radii [31],
powered by a precision bipolar current supply. The cuboidal two-layer magnetic
mu-metal shielding was upgraded to a three-layer cylindrical shielding.
Overall, the homogeneity of the static magnetic field, $B_{\text{stat}}$,
within the cavity volume was improved by a factor $\sim 20$, resulting in
$\sigma_{B_{\text{stat}}}/B_{\text{stat}}\sim$0.05\text{\,}\%$$ as established
from fluxgate measurements [31].

The atoms in LFS states entering the cavity undergo transitions when the RF
frequency and power are in the appropriate regime. Therefore the beam
downstream of the cavity will contain a fraction of the ($F=0$) high field
seeking (HFS) states again, that can be separated from the LFS states by
another magnetic field gradient. The large acceptance
($100\text{\,}\mathrm{mm}$ opening diameter) superconducting sextupole magnet
employed for this state analysis in [24], now in use in the ASACUSA
$\overline{\textrm{H}}$ experiment, was replaced by permanent sextupole
magnets equivalent to the ones providing the initial beam polarisation. For
the hydrogen experiment an open diameter of $40\text{\,}\mathrm{mm}$ is
sufficient, and the required field gradient can be achieved with a smaller
pole field strength. These analysing magnets remove all atoms that
transitioned to HFS states and focus the remaining LFS atoms through a pipe
aperture into a quadrupole mass spectrometer (QMS) located in the final
detection chamber. The QMS ionises the hydrogen atoms by electron impact.
Protons are selectively guided to a channeltron for counting at typical rates
of $\sim 5-$10\text{\,}\mathrm{kHz}$$ (see Fig. 3). The pressures in the five
differentially pumped vacuum chambers reduce from about
${10}^{-5}\text{\,}\mathrm{mbar}$ in the first chamber by roughly one order of
magnitude after each of the four apertures to about
${10}^{-10}\text{\,}\mathrm{mbar}$ in the detection chamber.

Figure 3:  Example of subsequently recorded $\sigma$ (left) and $\pi$ (right)
resonances. The effect of the magnetic field inhomogeneity on the resonance
shape is clearly visible on the $\pi$ resonance. In particular, the resonance
is asymmetric and at $\nu^{c}_{\pi}$ the rate is not returning to the baseline
as for the $\sigma$ resonance. This lineshape is the result of different field
inhomogeneities and is very well accounted for by the fitting algorithm as
shown from the residuals. The dotted line indicates the baseline $R_{0}$ which
is a fit parameter. The field inhomogeneities are also responsible for what
appears to be different beam rate drops between the $\sigma$ and $\pi$
transitions. In fact the two LFS states ($F=1,m_{F}=0,1$) are about equally
populated.

##  3 Measurement

Probing $\mathcal{K}^{\textrm{NR,Sun}}_{w_{k10}}$ requires a change of the
component of the static magnetic field, that projects onto the Earth’s
rotation axis. An efficient way to do this is by changing the direction of the
field. This corresponds to changing the polarity of the current in the coils,
therefore the two directions are labelled positive ($+$) and negative ($-$).

SME effects on the $\sigma$ transition are suppressed by roughly two orders of
magnitude with respect to the $\pi$ transition (see Eq. 3 and Eq. 4 and
associated discussion). We hence use the $\sigma$ transition to provide a
concomitant measurement of the magnetic field inside the interaction region
via the following method: based on the magnetic field dependence of the
transitions (deduced from the Breit-Rabi formulae [32], see Eq. 9 and Eq. 10
below) we calculate from the extracted centre frequency of the $\sigma$
transition ($\nu^{c}_{\sigma}$, see Fig. 3) the expected $\pi$ transition
frequency ($\nu^{c}_{\pi\leftarrow\sigma}$) at the same magnetic field and
compare it to the extracted centre frequency from our $\pi$ measurement
($\nu^{c}_{\pi}$). By comparing the obtained difference
$\Delta\nu_{\pi}=\nu^{c}_{\pi}-\nu^{c}_{\pi\leftarrow\sigma}$ for negative and
positive polarities most systematic effects cancel, while potential SME shifts
are enhanced. This double difference
($\Delta\nu_{\pi}^{+}\\!-\\!\Delta\nu_{\pi}^{-}$) is the final quantity used
to constrain SME effects. The impact of extracting
$\nu^{c}_{\pi\leftarrow\sigma}$ from $\nu^{c}_{\sigma}$ on the sensitivity to
different SME coefficients will be discussed in section 5.

It is worth noting that the second order Zeeman shift of the $\sigma$
transition has two consequences. On the one hand, a small frequency
uncertainty translates into a relatively large field uncertainty. The
acquisition of more data for $\sigma$ ($45\text{\,}\mathrm{min}$ per
resonance) than for $\pi$ ($15\text{\,}\mathrm{min}$) transitions mitigates
this effect. On the other hand, with the same absolute frequency uncertainty,
an increased precision on the magnetic field is reached at higher fields.
Therefore, measurements were taken at relatively high fields up to
$0.68\text{\,}\mathrm{mT}$, corresponding to a current
$I_{c}=$3\text{\,}\mathrm{A}$$ on the McKeehan coils.

A measurement series consisted of successive $\sigma$ and $\pi$ transitions
acquisitions at negative and positive polarities (a total of four transitions
were acquired within $2\text{\,}\mathrm{h}$). For each transition the
oscillating field frequency was varied in a random sequence of 51 points over
a range of $50\text{\,}\mathrm{kHz}$ (step size of $1\text{\,}\mathrm{kHz}$).
The absolute frequency was shifted by software before analysis, thus
effectively blinding the data. Figure 3 shows an example of the resonances
obtained at $-3\text{\,}\mathrm{A}$. Prior to acquiring rate spectra in the
frequency domain the RF input power was scanned to observe Rabi oscillations
at the frequency $\Omega_{\text{Rabi}}\propto B_{\text{osc}}$. The power in
the cavity was then set to drive the first state population inversion (called
$\pi$-pulse in conventional Rabi spectroscopy) as indicated in Fig. 4, where
$\nu^{c}$ is best resolved.

Over the course of two months a total of 576 resonance pairs were recorded at
six different coil current settings ($I_{c}=\pm$$2\text{\,}\mathrm{A}$,
$\pm$$2.5\text{\,}\mathrm{A}$, and $\pm$$3\text{\,}\mathrm{A}$) with 39 pairs
not passing data quality selection due to high rate instabilities.

In addition to the acquisition of ($\sigma$, $\pi$) resonance pairs, a series
of measurements of the $\pi$ transitions were taken at different oscillating
magnetic field strengths, resulting in two-dimensional maps, as exemplified in
Fig. 4. For these $\pi$ state-conversion data maps, recorded at the same six
different current settings as the main data, the oscillating field frequency
was stepped in a random sequence of 41 points over a range of
$80\text{\,}\mathrm{kHz}$ (step size of $2\text{\,}\mathrm{kHz}$) and the
oscillating magnetic field was scanned linearly up to values allowing to
observe the third state population inversion. These two-dimensional $\pi$ maps
provide an independent data set to assess the static magnetic field quality in
the cavity volume.

##  4 Analysis

The extraction of the centre frequencies $\nu^{\text{c}}$ from the data builds
upon the fit procedure developed for the experiment addressing only $\sigma$
transitions [24]. The rate fit function
$\mathcal{R}(\nu_{\text{RF}};R_{0},p,\nu_{c},\textrm{v}_{\textrm{H}},\Delta\nu^{x,y},B_{\text{osc}},\Delta
B^{z}_{1},\Delta B^{z}_{2})$ employed here has the frequency $\nu_{\text{RF}}$
as variable and eight parameters: the first two parameters, the baseline
$R_{0}$ and the polarisation $p$ scale the state conversion probability to the
observed rate. The next four parameters enter in the probability function
constructed from a spline interpolation on simulated two-dimensional maps (see
Fig. 4 left). The centre frequency $\nu^{c}$ is thus obtained as well as the
velocity of the atoms $\textrm{v}_{\textrm{H}}$, the frequency broadening
$\Delta\nu^{x,y}$ due to inhomogeneities of the static magnetic fields in the
(x,y)-plane, and the oscillating magnetic field strength $B_{\text{osc}}$.
$\textrm{v}_{\textrm{H}}$ and $\Delta\nu^{x,y}$ were extracted from all
resonances and then fixed in the final fit to an averaged value. Two different
velocity values, of $978\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ and
$1054\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$, were obtained for
$\sigma$ and $\pi$ transitions, respectively. The velocity difference
originates from the two involved initial states, which transmit sightly
differently through the magnetic sextupole fields. $\Delta\nu^{x,y}$
additionally depends on the current $I_{c}$ and was hence fixed for each
corresponding subset of data.

Finally, the parameters $\Delta B^{z}_{1}$ and $\Delta B^{z}_{2}$ encode the
first and second-order deviations from a uniform $B_{\textrm{stat}}$ in the
beam propagation direction (z) and modify the probability map underlying the
spline interpolations. Magnetic field measurements taken with fluxgates inside
the cavity, coils and shields assembly prior to their installation in the
hydrogen beamline revealed those field inhomogeneities. The magnitude of the
field, with the dominant component originating from the McKeehan coils aligned
along the $x$-direction, is well approximated by a second order polynomial:

$\centering B_{\textrm{stat}}(z)=B_{0}+\Delta B^{z}_{1}\ z+\Delta B^{z}_{2}(\
z^{2}-L_{\textrm{cav}}^{2}/12),\@add@centering$ (2)

where $z=0$ is the centre of the RF cavity of length $L_{\textrm{cav}}$, and
$B_{0}=\overline{B_{\text{stat}}}(z\in[-L_{\text{cav}}/2,L_{\text{cav}}/2])$
is the average magnetic field determining the centre frequency of the
transitions: $\nu^{c}_{\sigma,\pi}=\nu_{\sigma,\pi}(B_{0})$. The effects of
the $z$-dependence of $B_{\text{stat}}$ have to be taken into account by using
different probability maps. Sets for various values of $\Delta B^{z}_{1}$ and
$\Delta B^{z}_{2}$ have been produced by solving the von Neumann equation of
the four-level system, where the spatial dependence of $B_{\text{stat}}$
translates into a time dependence via the atoms velocity. The optimal values
were obtained by calculating the least-square deviations of such maps to the
measured $\pi$ probability maps (see Fig. 4 right). The quadratic coefficient
$\Delta B^{z}_{2}$ turned out to be crucial to explain asymmetries in the line
profiles. Those are strongest in the region of the second state population
inversion. Hence the $\pi$ data maps facilitate an accurate determination of
$\Delta B^{z}_{2}$. In contrast $\Delta B^{z}_{1}$ had much weaker effects and
an accurate determination was hampered by correlations with other fit
parameters. Therefore the linear parameter was set to zero and the systematic
effect resulting from this choice was assessed in the same way as for the
other two fixed fit parameters: the uncertainties were conservatively
estimated by varying their value around the central value by 1 or 3 standard
deviations. The resulting changes in $v^{c}$ were taken as the uncertainty.
Note that the uncertainty of the parameter $\Delta B^{z}_{1}$ is asymmetric as
positive and negative deviations from the chosen value of zero cause negative
shifts of $\nu^{\text{c}}$.

Figure 4:  Comparison of simulated and measured probabilities of conversion
from LFS to HFS states for the $\pi$ transition at $-3\text{\,}\mathrm{A}$ as
a function of the detuning ($\nu_{\text{RF}}-\nu_{\pi}$) and the amplitude of
the oscillating magnetic field $B_{osc}$. The simulated map results from
numerical solutions of the von Neumann equation for the hydrogen four-level
hyperfine system considering the oscillating and static magnetic fields
configuration of the experiment. The black dashed horizontal line indicates
the typical RF field driving strength used for the acquisition of ($\sigma$,
$\pi$) pairs. The effect of the non-homogeneous static magnetic field causing
an asymmetry in the resonance can clearly be seen and is particularly strong
in the region of the second state population inversion (labelled 2). This
general feature is very well reproduced in the simulation.

##  5 Results

The unblinded results are summarised in Fig. 5. The centre frequency
$\nu_{c}^{\sigma}$ extracted for each $\sigma$ transition in a ($\sigma$,
$\pi$) data pair is used to compute, via the Breit-Rabi formula, the expected
$\nu^{c}_{\pi\leftarrow\sigma}$ (using the literature value
$\nu_{0}^{lit}=$1.420\,405\,751\,768(1)\text{\,}\mathrm{GHz}$$ [10]). This
value is then compared to $\nu^{c}_{\pi}$ extracted from the $\pi$ resonance,
see Fig. 5 (a). The difference $\Delta\nu_{\pi}^{+}\\!-\\!\Delta\nu_{\pi}^{-}$
between the two values at the two opposite magnetic field orientations
constitutes a test of the SME, see Fig. 5 (b). We note that the second-order
corrections to the Breit-Rabi formula computed in Ref. [33, 34] have, at the
low fields used in the experiment, negligible effect on this result as well as
on the extracted zero-field hyperfine splitting value, and are thus not
considered. The statistical uncertainty on
$\Delta\nu_{\pi}^{+}\\!-\\!\Delta\nu_{\pi}^{-}$ is primarily dominated by the
propagation of the $\nu^{c}_{\sigma}$ uncertainty due to the second order
dependency of the $\sigma$ transition on the magnetic field. Table 1 indicates
the scaling factor $\partial\nu_{\pi}/\partial\nu_{\sigma}$ for every current
which, for instance at $2\text{\,}\mathrm{A}$, leads to a
$\sim$$96\text{\,}\mathrm{Hz}$ uncertainty on $\nu_{\pi\leftarrow\sigma}$ from
an initial statistical uncertainty of $1.75\text{\,}\mathrm{Hz}$ on
$\nu_{\sigma}$. The correlation in the systematic uncertainties at different
currents was taken into account following Ref. [35, 36]. Systematic
uncertainties resulting from the choice of $\Delta B^{z}_{2}$ and $\Delta
B^{z}_{1}$ values and from the fixation of the parameters $v_{\text{H}}$ and
$\Delta\nu^{x,y}$ were added in quadrature and account for less than
$\sim$5\text{\,}\mathrm{Hz}$$, see Table 1. A $\sqrt{}(\chi_{\text{red}}^{2})$
inflation was adopted to take into account the quality of the individual
lineshape fits and contributed to the statistical uncertainty.

Figure 5:  Experimental results at the three McKeehan coil currents. a) Difference between the measured $\pi$ transition frequency ($\nu_{\pi}$) and the expected value ($\nu_{\pi\leftarrow\sigma}$) calculated from the Breit-Rabi formula based on the value of the magnetic field determined by the $\sigma$ transition. This difference is provided for positive ($\Delta\nu_{\pi}^{+}$) and negative polarities ($\Delta\nu_{\pi}^{-}$) and is consistent with expectation for both polarities. b) The consistency between the negative and positive polarity measurements provides a constraint on the magnitude of the coefficient of some SME terms (see text). The average of the difference between positive and negative polarities is consistent with zero at a precision of $51\text{\,}\mathrm{Hz}$ indicated by the grey band. c) The quasi-simultaneous measurements of $\pi$ and $\sigma$ transitions at the same magnetic field enable the determination of the zero-field hyperfine splitting $\Delta\nu_{0}$. The average is consistent with literature ($\nu_{0}^{\textrm{lit}}$) with an absolute precision of $0.63\text{\,}\mathrm{Hz}$.  $I_{\text{c}}$ | $2.0\text{\,}\mathrm{A}$ | $2.5\text{\,}\mathrm{A}$ | $3.0\text{\,}\mathrm{A}$
---|---|---|---
$B_{\text{stat}}$ | $0.46\text{\,}\mathrm{mT}$ | $0.57\text{\,}\mathrm{mT}$ | $0.68\text{\,}\mathrm{mT}$
$\partial\nu_{\pi}/\partial\nu_{\sigma}$ | 55.0 | 44.1 | 36.8
# pairs $+/-$ | 56 / 55 | 121 / 120 | 93 / 92
errors (Hz) | stat. | sys. | stat. | sys. | stat. | sys.
$\nu_{\sigma}^{+}$ | 1.75 | 0.14 | 1.26 | 0.20 | 1.40 | 0.22
$\nu_{\pi}^{+}$ | 2.37 | ${}^{+3.19}_{-4.25}$ | 1.69 | ${}^{+3.25}_{-4.30}$ | 2.11 | ${}^{+3.26}_{-4.31}$
$\nu_{\pi\leftarrow\sigma}^{+}$ | 96.3 | 7.70 | 55.7 | 8.82 | 51.7 | 8.10
$\Delta\nu_{\pi}^{+}$ | 96.4 | ${}^{+8.33}_{-8.79}$ | 55.8 | ${}^{+9.40}_{-9.81}$ | 51.8 | ${}^{+8.73}_{-9.18}$
$\nu_{\sigma}^{-}$ | 1.67 | 0.19 | 1.21 | 0.23 | 1.35 | 0.21
$\nu_{\pi}^{-}$ | 2.93 | ${}^{+3.18}_{-3.66}$ | 2.19 | ${}^{+3.34}_{-3.80}$ | 2.69 | ${}^{+3.38}_{-3.84}$
$\nu_{\pi\leftarrow\sigma}^{-}$ | 91.9 | 10.5 | 53.1 | 10.1 | 49.7 | 7.73
$\Delta\nu_{\pi}^{-}$ | 92.0 | ${}^{+10.9}_{-11.1}$ | 53.2 | ${}^{+10.7}_{-10.8}$ | 49.8 | ${}^{+8.44}_{-8.64}$
$\Delta\nu_{\pi}^{+}-\Delta\nu_{\pi}^{-}$ | 133 | ${}^{+13.7}_{-14.1}$ | 77.1 | ${}^{+14.2}_{-14.6}$ | 71.8 | ${}^{+12.2}_{-12.6}$
Table 1:  Error budget (in Hz). Systematic effects have been investigated for
parameters that were fixed during the final fit procedure, namely
$\textrm{v}_{\textrm{H}}$, $\Delta_{\nu}^{x,y}$, and both $\Delta B^{z}$
coefficients and added in quadrature. Asymmetric errors originate from the
one-sided error of $\Delta B^{z}_{1}$ (negligible for $\sigma$ transitions and
in the final result after rounding to two significant digits). The statistical
errors are slightly larger for $\pi$ measurements due to the three times
shorter acquisition time compared to $\sigma$ measurements and the altered
lineshape. In the case of $\nu_{\pi\leftarrow\sigma}$ the field-dependent
uncertainty propagation factor $\partial\nu_{\pi}/\partial\nu_{\sigma}$ (given
in the table header) has to be taken into account. The uncertainty on
$\Delta\nu_{\pi}$ is obtained by adding in quadrature those of $\nu_{\pi}$ and
$\nu_{\pi\leftarrow\sigma}$, and is clearly dominated by the latter. Finally,
the difference between $\Delta\nu_{\pi}$ for positive and negative polarity is
formed where the uncertainties are again added in quadrature. Sample sizes for
($\sigma$, $\pi$) pairs for each current and polarity are listed in the table
header.

The result, $\Delta\nu_{\pi}^{+}\\!-\\!\Delta\nu_{\pi}^{-}=$-19\pm
51\text{\,}\mathrm{Hz}$$ is consistent with the SM and allows to constrain the
SME Sun-centred frame coefficients $\mathcal{K}^{\textrm{NR,Sun}}_{w{k10}}$ in
Eq. (1). Those coefficients are decomposed into isotropic
($\mathcal{V}^{\textrm{NR,Sun}}_{w{k10}}$) and anisotropic coefficients
($\mathcal{T}^{\textrm{NR,Sun}}_{w{k10}}$). The isotropic terms shift all
ground-state hyperfine sub-levels by the same amount and are thus not
accessible by the measurement of the hyperfine splitting transitions. The
anisotropic terms would contribute to an energy shift as given by Eq. (48) of
Ref. [6]. Using the condensed notation
$\tilde{\mathcal{T}}_{wk10}^{\text{NR}}={\mathcal{T}}_{wk10}^{\text{NR,(0B)}}+2{\mathcal{T}}_{wk10}^{\text{NR,(1B)}}$
and including the magnetic field dependence yields the following expressions
for the $\sigma$ and $\pi$ transition:

$\displaystyle\nu_{\sigma}^{\textrm{SME}}=\nu_{\sigma}^{\text{SM}}+\frac{1}{2\pi\sqrt{3\pi}}\sum^{k=0,2,4}(\alpha
m_{r})^{k}(1+4\delta_{k4})\cdot$ (3) $\displaystyle x\
(1\\!+\\!x^{2})^{-\frac{1}{2}}\
\left(\tilde{\mathcal{T}}_{ek10}^{\text{NR,lab}}-\tilde{\mathcal{T}}_{pk10}^{\text{NR,lab}}\right),$
$\displaystyle\nu_{\pi}^{\textrm{SME}}=\nu_{\pi}^{\text{SM}}+\frac{1}{4\pi\sqrt{3\pi}}\sum^{k=0,2,4}(\alpha
m_{r})^{k}(1+4\delta_{k4})\cdot$ (4)
$\displaystyle\left[\tilde{\mathcal{T}}_{ek10}^{\text{NR,lab}}+\tilde{\mathcal{T}}_{pk10}^{\text{NR,lab}}+x\
(1\\!+\\!x^{2})^{-\frac{1}{2}}\left(\tilde{\mathcal{T}}_{ek10}^{\text{NR,lab}}-\tilde{\mathcal{T}}_{pk10}^{\text{NR,lab}}\right)\right],$

where $x=B_{\textrm{stat}}/B_{c}$ with $B_{c}=h\nu_{0}/(g_{+}\mu_{B})$ being a
characteristic magnetic field of $\sim$51\text{\,}\mathrm{mT}$$, hence $x\sim
10^{-2}$ for our magnetic field values. Here, we have introduced
$g_{\pm}=|g_{e}|\pm g_{p}m_{e}/m_{p}$, with
${m_{e}/m_{p}=5.44617021487(33)\times 10^{-4}}$ being the ratio of the
electron to proton mass, ${\mu_{B}=$5.7883818060(17)\times
10^{-5}\text{\,}\mathrm{eV}\text{\,}{\mathrm{T}}^{-1}$}$ is the Bohr magneton,
${|g_{e}|=2.00231930436256(35)}$ and ${g_{p}=5.5856946893(16)}$ are the
$g$-factors of the electron and proton, respectively [37].

In the limit $x\rightarrow 0$ the $\sigma$ transition becomes insensitive to
SME effects. However, through the procedure of evaluating
$\nu^{c}_{\pi\leftarrow\sigma}$, a suppressed SME effect on the $\sigma$
transition will be enhanced by the relative sensitivity of $\sigma$ and $\pi$
to the magnetic field (Zeeman shift) given by the factor

$\frac{\partial\nu_{\pi}}{\partial\nu_{\sigma}}=\frac{1}{2}\left(1+\frac{g_{-}}{g_{+}}\frac{\sqrt{1+x^{2}}}{x}\right).$
(5)

Since the Zeeman shift is dominated by the contribution of the electron
magnetic moment, this procedure reduces the sensitivity to the SME electron
coefficients by a factor $2g_{p}\mu_{N}/(g_{+}\mu_{B})\sim 0.003$, where
$\mu_{N}=$3.15245125844(96)\times
10^{-8}\text{\,}\mathrm{eV}\text{\,}{\mathrm{T}}^{-1}$$ is the nuclear
magneton [37], and enhances it by a factor $2|g_{e}|/g_{+}\sim 2$ for the
proton coefficients:

$\displaystyle|h\
(\Delta\nu_{\pi}^{+}-\Delta\nu_{\pi}^{-})|=\frac{\cos{\theta}}{\sqrt{3\pi}}\sum^{k=0,2,4}(\alpha
m_{r})^{k}(1+4\delta_{k4})\cdot$ (6)
$\displaystyle\left(\frac{2|g_{e}|}{g_{+}}\
\tilde{\mathcal{T}}_{pk10}^{\text{NR,Sun}}+\frac{2g_{p}\mu_{N}}{g_{+}\mu_{B}}\
\tilde{\mathcal{T}}_{ek10}^{\text{NR,Sun}}\right).$

The experimental apparatus was positioned at a latitude of
$\theta_{1}=$46.2\text{\,}\mathrm{\SIUnitSymbolDegree}$$ (CERN) and the beam
was oriented at an angle of
$\theta_{2}\approx$22.5(1.5)\text{\,}\mathrm{\SIUnitSymbolDegree}$$ relative
to the local north, lying tangential to the surface of the Earth. In this
configuration $\cos{\theta}=\sin{\theta_{1}}\sin{\theta_{2}}\approx
0.276(18)$. We hence obtain

$\mid
h(\Delta\nu_{\pi}^{+}-\Delta\nu_{\pi}^{-})\mid\frac{\sqrt{3\pi}}{\cos{\theta}}=$0.9(2.3)e-21\text{\,}\mathrm{GeV}$$
(7)

which is consistent with zero. The sensitivity of the measurement is used to
extract the following limit on the sum of the involved spherical anisotropic
SME terms, which decompose into CPT-odd ($g_{wk10}^{\textrm{NR,Sun}}$) and
CPT-even ($H_{wk10}^{\textrm{NR,Sun}}$) coefficients:

$\displaystyle\Bigl{|}\sum^{k=0,2,4}(\alpha
m_{r})^{k}(1+4\delta_{k4})\bigg{(}\frac{2|g_{e}|}{g_{+}}\Big{(}g_{pk10}^{\textrm{NR(0B),Sun}}-H_{pk10}^{\textrm{NR(0B),Sun}}$
(8)
$\displaystyle+2g_{pk10}^{\textrm{NR(1B),Sun}}\\!-\\!2H_{pk10}^{\textrm{NR(1B),Sun}}\Big{)}\\!+\\!\frac{2g_{p}\mu_{N}}{g_{+}\mu_{B}}\Big{(}g_{ek10}^{\textrm{NR(0B),Sun}}\\!-\\!H_{ek10}^{\textrm{NR(0B),Sun}}$
$\displaystyle+2g_{ek10}^{\textrm{NR(1B),Sun}}-2H_{ek10}^{\textrm{NR(1B),Sun}}\Big{)}\bigg{)}\Bigr{|}<2.3\times
10^{-21}\,\text{GeV}.$

This constitutes the first limit on this subset of non-relativistic spherical
coefficients. Assuming only one coefficient is non-zero at a time, we can
extract the limits provided in Table 2 on the individual coefficients.

Coefficient $\mathcal{K}$ | Constraint on $|\mathcal{K}|$
---|---
proton
$H_{p010}^{\textrm{NR(0B)},\textrm{Sun}}$, $g_{p010}^{\textrm{NR(0B)},\textrm{Sun}}$ |  $<1.2\times 10^{-21}$ GeV
$H_{p010}^{\textrm{NR(1B)},\textrm{Sun}}$, $g_{p010}^{\textrm{NR(1B)},\textrm{Sun}}$ |  $<5.8\times 10^{-22}$ GeV
$H_{p210}^{\textrm{NR(0B)},\textrm{Sun}}$, $g_{p210}^{\textrm{NR(0B)},\textrm{Sun}}$ |  $<8.4\times 10^{-11}$ GeV-1
$H_{p210}^{\textrm{NR(1B)},\textrm{Sun}}$, $g_{p210}^{\textrm{NR(1B)},\textrm{Sun}}$ |  $<4.2\times 10^{-11}$ GeV-1
$H_{p410}^{\textrm{NR(0B)},\textrm{Sun}}$, $g_{p410}^{\textrm{NR(0B)},\textrm{Sun}}$ |  $<1.2$ GeV-3
$H_{p410}^{\textrm{NR(1B)},\textrm{Sun}}$, $g_{p410}^{\textrm{NR(1B)},\textrm{Sun}}$ |  $<0.6$ GeV-3
electron
$H_{e010}^{\textrm{NR(0B)},\textrm{Sun}}$, $g_{e010}^{\textrm{NR(0B)},\textrm{Sun}}$ |  $<7.7\times 10^{-19}$ GeV
$H_{e010}^{\textrm{NR(1B)},\textrm{Sun}}$, $g_{e010}^{\textrm{NR(1B)},\textrm{Sun}}$ |  $<3.8\times 10^{-19}$ GeV
$H_{e210}^{\textrm{NR(0B)},\textrm{Sun}}$, $g_{e210}^{\textrm{NR(0B)},\textrm{Sun}}$ |  $<5.5\times 10^{-8}$ GeV-1
$H_{e210}^{\textrm{NR(1B)},\textrm{Sun}}$, $g_{e210}^{\textrm{NR(1B)},\textrm{Sun}}$ |  $<2.8\times 10^{-8}$ GeV-1
$H_{e410}^{\textrm{NR(0B)},\textrm{Sun}}$, $g_{e410}^{\textrm{NR(0B)},\textrm{Sun}}$ |  $<8.0\times 10^{2}$ GeV-3
$H_{e410}^{\textrm{NR(1B)},\textrm{Sun}}$, $g_{e410}^{\textrm{NR(1B)},\textrm{Sun}}$ |  $<4.0\times 10^{2}$ GeV-3
Table 2:  Constraints on the individual proton and electron non-relativistic
spherical coefficients $\mathcal{K}^{\textrm{NR,Sun}}_{w{k10}}$ for $k\leq 4$
derived from Eq.8 under the assumption that one coefficient is non-zero at a
time.

In the absence of an SME signal the ($\sigma,\pi$) pairs were used to
calculate the zero-field hyperfine splitting $\nu_{0}^{\textrm{meas}}$ from
the Breit-Rabi equations [32] by elimination of $B_{\text{stat}}$ in the
equations of the transition frequencies:

$\nu_{\sigma}(B_{\text{stat}})=\sqrt{\nu_{0}^{2}+\left(\frac{\mu_{B}g_{+}B_{\text{stat}}}{h}\right)^{2}},$
(9)
$\nu_{\pi}(B_{\text{stat}})=\frac{1}{2}\left(\nu_{0}+\frac{\mu_{B}g_{-}B_{\text{stat}}}{h}+\sqrt{\nu_{0}^{2}+\left(\frac{\mu_{B}g_{+}B_{\text{stat}}}{h}\right)^{2}}\right),$
(10)
$\nu_{0}^{\text{meas}}=\frac{g_{+}^{2}(2\nu_{\pi}^{\text{c}}-\nu_{\sigma}^{\text{c}})+g_{-}\sqrt{g_{-}^{2}(\nu_{\sigma}^{\text{c}})^{2}-4g_{+}^{2}(\nu_{\pi}^{\text{c}})^{2}+4g_{+}^{2}\nu_{\pi}^{\text{c}}\nu_{\sigma}^{\text{c}}}}{g_{+}^{2}+g_{-}^{2}}.$
(11)

The weighted average of $\nu_{0}^{\textrm{meas}}$ for all pairs within a
current and polarity set are shown with respect to the literature value in
Fig. 5 (c). Analogously to the SME uncertainty treatment, correlated
uncertainties have been taken into account following Ref. [35, 38]. The
combined data set of 537 ($\sigma,\pi$) pairs yields
$\nu^{\textrm{meas}}_{0}=$1.420\,405\,751\,63(63)\text{\,}\mathrm{GHz}$$. The
result is consistent with the literature value:
$\nu_{0}^{\textrm{lit}}-\nu_{0}^{\textrm{meas}}=$0.14\pm 0.59(stat)\pm
0.23(syst)\text{\,}\mathrm{Hz}$$ with a relative precision of 4.4$\times
10^{-10}$ and improves the previous determination of this quantity in a beam
by a factor 6.

##  6 Outlook

The ASACUSA hydrogen programme has been developed to characterise the
apparatus needed for the spectroscopy of the hyperfine structure of
antihydrogen in a beam, the main goal pursued by the ASACUSA-Cusp
collaboration. The work presented here has demonstrated the sub-Hz potential
of the apparatus on the determination of the zero-field hyperfine transition
and highlighted the intricate effects of field inhomogeneities on the
resonance lineshape, which will be relevant in the antihydrogen case as well.
Measurements at opposite static magnetic field orientations lead to first
constraints at the level of $2.3\text{\times}{10}^{-21}\text{\,}\mathrm{GeV}$
on CPT and Lorentz-violation coefficients of the SME. Our method provides an
enhanced sensitivity to proton coefficients with respect to electron
coefficients by a factor equal to the ratio of the electron to proton magnetic
moments. This result is statistically limited and dominated by the
determination of the magnetic field by the $\sigma$ transition. An improvement
could thus be obtained by operating at higher fields or by using advanced
magnetometry methods within the cavity volume, which would remove the need for
a concomitant measurement of the $\sigma$ transition. For example, the Zeeman
shift for the $\pi$ transition at the magnetic fields considered here is of
the order of $14\text{\,}\mathrm{Hz}\text{\,}{\mathrm{nT}}^{-1}$. A co-
magnetometry with absolute precision of $350\text{\,}\mathrm{pT}$ would reduce
the combined systematic and statistical uncertainties below
$5\text{\,}\mathrm{Hz}$, therefore improving by a factor 5 on the constraints
to SME proton coefficients and more than 3 orders of magnitude on the electron
coefficients. Further improvements could be obtained with a more homogeneous
static magnetic field.

Precise magnetometry is also of interest in the context of antihydrogen where
higher, as well as time dependent, magnetic stray fields are present in the
experiment. In the future, direct comparison between hydrogen and antihydrogen
will provide constraints specifically on the CPT-odd operators.

## Acknowledgements

We wish to thank the numerous undergraduate students, CERN and SMI summer
students who assisted in various ways throughout the development of this
experiment. We want to express our gratitude to the staff of the SMI Advanced
Instrumentation group for hardware support. We are indebted to the CERN
cryolab team for their help, in particular to Torsten Koettig. We would like
to thank Fritz Caspers for advising on RF aspects and the TE-MSC-LMF group, in
particular Attilio Milanese, Dmitry Chechenev, and Carlos Lopez for the design
assistance and construction of the McKeehan coils. We warmly thank Arnaldo
Vargas for the many fruitful discussions on the Standard Model Extension. This
work was supported by the CERN-Austrian Doctoral Student Program, the Austrian
Science Fund (FWF) [W1252-N27] (Doktoratskolleg Particles and Interactions),
and the National Research Council of Canada (NRC).

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# Sub-exponential Rate of Convergence to Equilibrium for Processes on the
Half-Line

Andrey Sarantsev  Department of Mathematics and Statistics, University of
Nevada, Reno<EMAIL_ADDRESS>This article is dedicated to Professor Anna
Panorska
on the occasion of her 20 years at the University of Nevada, Reno.

###### Abstract.

A powerful tool for studying long-term convergence of a Markov process to its
stationary distribution is a Lyapunov function. In some sense, this is a
substitute for eigenfunctions. For a stochastically ordered Markov process on
the half-line, Lyapunov functions can be used to easily find explicit rates of
convergence. Our earlier research focused on exponential rate of convergence.
This note extends these results to slower rates, including power rates, thus
improving results of (Douc, Fort, Guillin, 2009).

##  1\. Introduction

###  1.1. Lyapunov functions and long-term convergence

Take a continuous-time Markov process $X=(X(t),\,t\geqslant 0)$ with
transition function $P^{t}(x,A):=\mathbb{P}(X(t)\in A\mid X(0)=x)$, and
semigroup $P^{t}f(x)=\mathbb{E}\left[f(X(t))\mid X(0)=x\right]$. Suppose $X$
has a unique stationary distribution $\pi$, a probability measure on its state
space $S$ such that if the process starts from $X(0)\sim\pi$, then for every
$t\geqslant 0$ we have: $X(t)\sim\pi$. Does the measure $P^{t}(x,\cdot)$
converge to $\pi$ for all $x$ as $t\to\infty$, in which distance, and how
fast?

For continuous-time Markov chains on a finite state space, the answer depends
on the generating matrix $\mathcal{L}$ of transition intensities: Loosely
speaking, the eigenvalue closest to $0$ gives us the exponential convergence
rate. For general state spaces $S$, for example in $\mathbb{R}^{d}$, this
description of eigenvalues is difficult or impossible. This is because
$\mathcal{L}$ is no longer a matrix: It is an operator defined by

$\mathcal{L}f(x)=\lim\limits_{t\downarrow
0}\frac{1}{t}\left[P^{t}f(x)-f(x)\right]$

on a certain space of functions $f:S\to\mathbb{R}$. It is often hard to find
eigenvalues and eigenfunctions of $\mathcal{L}$, but there is a useful
substitute: Lyapunov functions. There are various versions of this concept,
but the basic property is (loosely speaking) that this is a function
$V:S\to[1,\infty)$ such that for some compact subset $K\subseteq S$,

(1) $\mathcal{L}V(x)\leqslant-c<0,\,x\in S\setminus K.$

Then under some additional technical conditions the process $X$ is ergodic:
That is, $P^{t}(x,\cdot)\to\pi$ as $t\to\infty$ for every $x\in S$ in the
total variation norm:

(2) $\sup\limits_{A\subseteq S}|P^{t}(x,A)-\pi(A)|\to 0,\,t\to\infty.$

Let us explain the intuition. The function $V$ measures the “altitude” reaches
by the process $X$. When the process is outside of the “valley” which is the
compact set $K$, it is compelled by the condition (1) to “decrease altitude”.
This random process has negative “drift” until it gets back to the “valley”.
This “gravity force” implies (after additional technical work) its ergodicity.
See articles [27, 28] for rigorous exposition, and the classic book [30] for
similar concepts on discrete-time Markov chains. Under a stronger than (1)
condition: For a constant $k>0$ and a compact $K\subseteq S$,

(3) $\mathcal{L}V(x)\leqslant-kV(x),\,x\in S\setminus K,$

we get exponential ergodicity: $P^{t}(x,\cdot)\to\pi$ as $t\to\infty$ for all
$x\in S$ as fast as $Ce^{-\varkappa t}$ for some $C,\varkappa>0$. See rigorous
statements in [13]. However, finding or estimating $\varkappa$ proved to be
very hard (compared to finding or estimating $k$ from (3), which is often easy
in practice); see for example [29, 31]. Among other articles on this topic,
let us mention [5, 9]; and applications to Markov Chain Monte Carlo
techniques, [33].

There is one special case when we simply conclude that $\varkappa=k$ from (3):
The state space $S=\mathbb{R}_{+}:=[0,\infty)$, the exceptional compact set is
$K=\\{0\\}$, and the process $X$ is stochastically ordered. The latter means
that for any $x_{2}\geqslant x_{1}\geqslant 0$, we can couple two copies
$X_{1}$ and $X_{2}$ of this process starting from $X_{i}(0)=x_{i}$, on the
same probability space, so that $X_{1}(t)\leqslant X_{2}(t)$ for all
$t\geqslant 0$. This statement was shown in [23] and improved in [34] using
the coupling method. We used this method in risk theory [17] and for more
general processes, so-called Walsh diffusions, which include as a particular
case diffusions on the real line, [20].

###  1.2. Our results

One can also have a condition stronger than (1), but weaker than (3): For some
increasing concave function $\varphi:[1,\infty)\to[0,\infty)$ and some compact
set $K\subseteq S$,

(4) $\mathcal{L}V(x)\leqslant-\varphi(V(x)),\,x\in S\setminus K.$

It is shown in [11] that this implies (under some additional technical
conditions) a subexponential, or subgeometric, convergence rate
$P^{t}(x,\cdot)\to\pi$ to $t\to\infty$. That is, the total variation distance
converges to zero, but at a rate slower than exponential, given by
$Ce^{-\varkappa t}$. In that article and earlier ones [16, 38], they were able
to find explicit rates, similar to $ct^{-\alpha}$ for some $C,\alpha>0$, or
$C\exp\left[-\lambda t^{\beta}\right]$ for $C,\lambda>0$ and $\beta\in(0,1)$.

In this note, as opposed to [11, 16], we assume that the compact set
$K=\\{0\\}$ on the state space $S=[0,\infty)$, and that the process $X$ is
stochastically ordered. We improve upon the results of these articles cited
above. In other words, we combine ideas from [11, 23, 34] in the context of
subgeometric ergodicity. This leads to simple and elegant proofs. We explore
how our new results fit into exponential ergoicity framework from [34].

We give examples of reflected diffusions, jump-diffusions, and Lévy processes
on $\mathbb{R}_{+}$. These processes behave as, correspondingly, diffusions,
jump-diffusions, and Lévy proceses, as long as they do not hit zero. When this
happens, they are reflected back inside the positive half-line. In our
examples, they have power rates of convergence: $h(t):=t^{-\beta}$ for
$\beta>0$. These processes have been extensively studied, from the original
article [37] on stochastic differential equations with reflection on the half-
line, to recent articles [2] on multidimensional reflected jump-diffusions and
[3, 4] on reflected Lévy processes. These processes have applications in
queueing theory (heavy traffic approximation), see [7, 21] and references
therein; and, more recently, in financial mathematics, see [14, 19] and
references therein.

We prove convergence not only in total variation norm, but in stronger norms.
Such convergence implies the convergenc of moments up to a certain order. As
in [11], there is a tradeoff between the norm and the rate of convergence.

###  1.3. Overview of related research

Similar questions were studied with relation to functional inequalities: log-
Sobolev, Poincare, Talagrand. The latter inequalities are related to
probabilities of large deviations from the median. We cite the books [39,
Chapter 1] and [22]. However, they deal with exponential rate of convergence.
This is the case of spectral gap: when the generator has eigenvalues with
negative real parts, separated from zero. Here, we deal with subexponential
rate of convergence, when there is no spectral gap. There is some literature
on this case: see an old article [32] and very recent articles [6, 18] and
citations therein. But they deal with convergence in $L^{2}$-norm: If $\pi$ is
the stationary distribution, then for $u:S\to\mathbb{R}$ with
$\int_{S}u^{2}(x)pi(\mathrm{d}x)<\infty$,

$\int_{S}\left[(P^{t}u)(x)-\int_{S}(P^{t}u)(y)\,\pi(\mathrm{d}y)\right]^{2}\,\pi(\mathrm{d}x)\leqslant
C(u)f(t),\,f(t)\to 0,\,t\to\infty,$

Our estimates are in the total variation norm as in (2) or stronger related
norms, which take into account the starting point $x$, instead of integrating
over $x$ with respect to $\pi$.

One can also study convergence in Wasserstein distance. This is another family
of distance metrics which is different than total variation and other distance
measures from this article. Wasserstein distance takes into account the metric
structure of the state space, which is related to weak convergence. In fact,
convergence of measures in Wasserstein distance is equivalent to weak
convergence plus convergence of moments of the given order. The author studied
this question for exponential convergence rate for jump-diffusion processes in
[36]. For subgeometric convergence, this was studied in [8].

###  1.4. Organization of the article

In Section 2, we state all notation and definitions, present main results, and
discuss their relationship with existing research. In Section 3, we present
examples, and Section 4 is devoted to proofs of main results.

###  1.5. Acknowledgements

We thank Professor Mark M. Meerschaert for invitation to give a talk in
November 2016 at the Colloquium at the Department of Statistics & Probability,
Michigan State University in East Lansing, and for useful discussion there. We
thank the Department of Mathematics & Statistics, University of Nevada in
Reno, for a supportive atmosphere for research and professional development.

##  2\. Notation, Definitions, and Main Results

###  2.1. Notation and definitions

Let $\mathbb{R}_{+}:=[0,\infty)$. For two probability measures $\rho_{1}$ and
$\rho_{2}$ on $\mathbb{R}_{+}$, we can define their stochastic maximum
$\rho_{0}:=\rho_{1}\vee\rho_{2}$ is defined by
$\rho_{0}((x,\infty)):=\rho_{1}((x,\infty))\vee\rho_{2}((x,\infty))$,
$x\geqslant 0$. For a signed measure $\nu$ on $\mathbb{R}_{+}$ and a function
$f:\mathbb{R}_{+}\to\mathbb{R}$, we define
$(\nu,f):=\int_{0}^{\infty}f(x)\nu(\mathrm{d}x)$. For a function
$V:\mathbb{R}_{+}\to[1,\infty)$, the $V$-norm of a signed measure $\nu$ on
$\mathbb{R}_{+}$ is

$\lVert\nu\rVert_{V}:=\sup\limits_{|f|\leqslant V}|(\nu,f)|,$

where the sup is taken over all functions $f:\mathbb{R}_{+}\to\mathbb{R}$ such
that $|f(x)|\leqslant V(x)$. If $V\equiv 1$, then this norm is denoted by
$\lVert\cdot\rVert_{\mathrm{TV}}$ and is called the total variation norm. A
family of Borel measures $(Q_{x})_{x\geqslant 0}$ on $\mathbb{R}_{+}$ is
called stochastically ordered if $Q_{x}([z,\infty))\leqslant
Q_{y}([z,\infty))$ for $0\leqslant x\leqslant y$ and $z\geqslant 0$.

We operate on a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geqslant 0},\mathbb{P})$. Consider a
Markov process $X=(X(t),\,t\geqslant 0)$ on $\mathbb{R}_{+}$ with transition
kernel $P^{t}(x,\cdot)$. Formally, $P^{t}(x,A)=\mathbb{P}(X(t)\in A\mid
X(0)=x)$. This process $X$ has positivity property if $P^{t}(x,B)>0$ for any
$t>0$, $x\geqslant 0$, and a Borel subset $B\subseteq\mathbb{R}_{+}$ of
positive Lebesgue measure. For any initial value $X(0)=x$, we can construct a
copy of this Markov process. If the initial distribution is $X(0)\sim\rho$,
then the distribution of $X(t)$ is written as

$\rho P^{t},\quad\mbox{where}\quad\rho
P^{t}(B):=\int_{0}^{\infty}P^{t}(x,B)\rho(\mathrm{d}x)\quad\mbox{for}\quad
B\subseteq\mathbb{R}_{+}.$

The transition semigroup $(P^{t})_{t\geqslant 0}$ is defined as

$P^{t}f(x)=\mathbb{E}\left[f(X(t))\mid
X(0)=x\right]=\int_{0}^{\infty}P^{t}(x,\mathrm{d}y)f(y).$

This process has generator $\mathcal{L}$ with domain
$\mathcal{D}(\mathcal{L})$:

$\mathcal{L}f(x)=\lim\limits_{t\downarrow
0}\frac{1}{t}\left(P^{t}f(x)-f(x)\right),\quad f\in\mathcal{D}(\mathcal{L}).$

This Markov process is called stochastically ordered if for every $t\geqslant
0$, the family of measures $(P^{t}(x,\cdot))_{x\geqslant 0}$ is stochastically
ordered. A probability measure $\pi$ on $\mathbb{R}_{+}$ is called a
stationary distribution for this Markov process if $X(0)\sim\pi$ implies
$X(t)\sim\pi$ for all $t>0$:

$\int_{0}^{\infty}P^{t}(x,A)\pi(\mathrm{d}x)=\pi(A)\quad\forall
A\subseteq\mathbb{R}_{+}.$

Take a strictly increasing concave $\varphi:[1,\infty)\to\mathbb{R}_{+}$.
Define

(5) $\Phi(s):=\int_{1}^{s}\frac{\mathrm{d}u}{\varphi(u)}.$

Assuming this function is finite for all $s>1$, it is strictly increasing
$\Phi:[1,\infty)\to\mathbb{R}_{+}$, thus it has a well-defined inverse
$\Psi:=\Phi^{-1}$ on $[0,\Phi(\infty))$, with $\Phi(\Psi(v))\equiv v$ for
$v\geqslant 0$. Note that $\Psi(\infty)=\infty$: If $\Psi(\infty)=a<\infty$,
then $\Phi(a)=\infty$, which contradicts our assumptions.

A $\varphi$-Lyapunov function for this process $X$ is a function
$V:\mathbb{R}_{+}\to(0,\infty)$ such that the following process is a local
$(\mathcal{F}_{t})_{t\geqslant 0}$-supermartingale:

(6)
$\varphi(V(X(t\wedge\tau_{0})))+\int_{0}^{t\wedge\tau_{0}}\mathcal{L}V(X(s))\,\mathrm{d}s,\quad
t\geqslant 0,$

where $\tau_{0}:=\inf\\{t\geqslant 0\mid X(t)=0\\}$ is the hitting time of
$0$.

######  Remark 1.

If $V\in\mathcal{D}(\mathcal{L})$, this condition (6) is equivalent to
$\mathcal{L}V(x)\leqslant-\varphi(V(x))$ for $x>0$. But it is more convenient
for us to write this condition in the form (6). We consider functions with
$V^{\prime}(0)\neq 0$, but reflected processes have generator domain
$\mathcal{D}(\mathcal{L})$ restricted by $V^{\prime}(0)=0$.

######  Example 1.

For $\varphi(s)=ks,\,k>0$, this becomes modified Lyapunov function from [34].

###  2.2. Main results

Take a function $G(t,u):=\Psi(\Phi(u)+t)$, $u\geqslant 1$, $t\geqslant 0$.

######  Theorem 1.

Take a concave increasing function $\varphi:[1,\infty)\to[0,\infty)$ with
finite $\Phi$ from (5). Assume $X$ is a semimartingale and a stochastically
ordered Markov process on $\mathbb{R}_{+}$. Take a $\varphi$-Lyapunov function
$V$ such that for some nondecreasing functions
$U,r:\mathbb{R}_{+}\to\mathbb{R}_{+}$,

(7) $h(t)U(x)\leqslant G(t,V(x)),\quad t,x\geqslant 0.$

Then we have the following results:

  1. (a)

Take two copies $X_{1}$ and $X_{2}$ of $X$ starting from $x_{1}$ and $x_{2}$,
with $0\leqslant x_{1}\leqslant x_{2}$. Then

$\lVert P^{t}(x_{1},\cdot)-P^{t}(x_{2},\cdot)\rVert_{U}\leqslant
2h^{-1}(t)V(x_{2}),\quad t\geqslant 0.$

  2. (b)

Take two copies $X_{1}$ and $X_{2}$ of $X$ starting from distributions
$\rho_{1}$ and $\rho_{2}$ on $\mathbb{R}_{+}$. Then we can write
$X_{i}(t)\sim\rho_{i}P^{t}$, $i=1,2$. Then

$\lVert\rho_{1}P^{t}-\rho_{2}P^{t}\rVert_{U}\leqslant
2h^{-1}(t)(\rho_{1}\vee\rho_{2},V),\quad t\geqslant 0.$

  3. (c)

Take a copy $X$ starting from $X(0)\sim\rho$ with $(\rho,V)<\infty$. Assume
$X$ has a unique stationary distribution $\pi$ with $(\pi,V)<\infty$, then

$\lVert\rho P^{t}-\pi\rVert_{U}\leqslant 2h^{-1}(t)(\pi\vee\rho,V).$

######  Remark 2.

In Theorem 1, we can let $U:=1$ and $h:=\Psi$. Then we get convergence in the
total variation norm. The rate of this convergence is given by $1/\Psi(t)$.
This is stronger than in [11], where the rate of convergence for the total
variation norm is $1/\varphi(\Psi(t))$. Indeed, in the above example
$\varphi(u):=\sqrt{u}$, and so $\varphi(\Psi(t))=o(\Psi(t))$ as $t\to\infty$.

######  Remark 3.

Let us discuss these results in exponential ergodicity case, when
$\varphi(s)=ks,\,k>0$. Then $\Phi(s)=k^{-1}\ln(s)$, and $\Psi(v):=e^{kv}$.
Thus $G(t,u)=ue^{kt}$, and we can take $U:=V$ and $r(t):=e^{kt}$. Here, we
have perfect decomposition of $G$ into a product form.

######  Example 2.

Even if $\varphi$ is bounded, we can get nontrivial results. For example,
$\varphi(x)=k>0$ implies $\Phi(x)=k^{-1}(x-1)$ and $\Psi(x)=kx+1$. Thus
$G(t,x)=\Psi(\Phi(x)+t)=x+kt$, and we can let $h(t):=2kt^{1/2}$,
$U(x):=V^{1/2}(x)$, or use more general techniques from the Appendix.

######  Remark 4.

The existence and uniqueness of the stationary distribution $\pi$ and the
property $(\pi,V)<\infty$ can be obtained from the Lyapunov conditions in
their classic form, as in [11, 27, 28]. Often we can find (another version of)
a Lyapunov function $\hat{V}:\mathbb{R}_{+}\to[1,\infty)$ in
$\mathcal{D}(\mathcal{L})$ such that for constants $x_{0},b>0$:

(8) $\mathcal{L}\hat{V}(x)\leqslant-\varphi(\hat{V}(x))+b1_{[0,x_{0}]}(x).$

We also need the following positivity property: $P^{t}(x,B)>0$ for every
$t>0$, $x\geqslant 0$, and every Borel set $B\subseteq\mathbb{R}_{+}$ of
positive Lebesgue measure. Then the process has a unique stationary
distribution $\pi$, and $(\pi,\hat{V})<\infty$. This follows from [35, Lemma
2.3, Theorem 2.6]; see also [27, 28]. In practice, for processes on
$\mathbb{R}_{+}$, this function $\hat{V}$ can sometimes be constructed as
$\hat{V}(x):=V(\psi(x))$, where $\psi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a
nondecreasing function with $\psi(x)=0$ for $x\leqslant x_{1}$ and $\psi(x)=x$
for $x\geqslant x_{2}$, where $0<x_{1}<x_{2}$; and, finally, $\psi(x)\leqslant
x$ for all $x\geqslant 0$. Then $\hat{V}(x)\leqslant V(x)\leqslant
C\hat{V}(x)$ for some constant $C>1$, and $(\pi,\hat{V})<\infty$ implies
$(\pi,V)<\infty$.

##  3\. Examples

###  3.1. Reflected diffusions

Take functions $g,\sigma:\mathbb{R}_{+}\to\mathbb{R}$. For a one-dimensional
$(\mathcal{F}_{t})_{t\geqslant 0}$-Brownian motion $W=(W(t),\,t\geqslant 0)$,
consider a stochastic differential equation (SDE):

(9) $\mathrm{d}Z(t)=g(Z(t))\,\mathrm{d}t+\sigma(Z(t))\,\mathrm{d}W(t).$

######  Definition 1.

A solution to the SDE (9) with reflection on $\mathbb{R}_{+}$ starting from
$x\geqslant 0$ is defined as an adapted process $Z=(Z(t),\,t\geqslant 0)$ with
a.s. continuous trajectories, such that

$Z(t)=x+\int_{0}^{t}g(Z(s))\,\mathrm{d}s+\int_{0}^{t}\sigma(Z(s))\,\mathrm{d}W(s)+\ell(t),\,t\geqslant
0,$

where $W$ is an $(\mathcal{F}_{t})_{t\geqslant 0}$-Brownian motion, and
$\ell=(\ell(t),\,t\geqslant 0)$ is a continuous nondecreasing process with
$\ell(0)=0$, increasing only when $Z(t)=0$. It is also called a reflected
diffusion on $\mathbb{R}_{+}$ with drift $g$ and diffusion $\sigma^{2}$.

This process is well-defined for continuous $g,\sigma$, stochastically ordered
and has generator

$\mathcal{L}f(x)=g(x)f^{\prime}(x)+\frac{1}{2}\sigma^{2}(x)f^{\prime\prime}(x),\
f\in C^{2}(\mathbb{R}_{+}),\ f^{\prime}(0)=0.$

Thus for all functions $f\in C^{2}(\mathbb{R}_{+})$, even when
$f^{\prime}(0)\neq 0$ and $f$ is not in the domain of the generator, the
following process is an $(\mathcal{F}_{t})_{t\geqslant 0}$-supermartingale:

$f(Z(t\wedge\tau_{0}))-\int_{0}^{t\wedge\tau_{0}}\mathcal{L}f(Z(s))\,\mathrm{d}s,\quad
t\geqslant 0.$

We can prove that $\mathcal{L}V(x)\leqslant-\varphi(V(x))$ for $x>0$, even
when $V^{\prime}(0)\neq 0$, and this would prove that $V$ is a
$\varphi$-Lyapunov function. Assume for some constants $a,c>0$ and
$\alpha\in(0,1)$,

(10) $g(x)\leqslant-a(1+cx)^{\alpha-1},\quad x>0.$

######  Example 3.

Take a function $V(x)=1+cx$. We get: $V^{\prime}(x)=c$,
$V^{\prime\prime}(x)=0$, and thus

$\displaystyle\mathcal{L}V(x)\leqslant-
ac(1+cx)^{\alpha-1}=-\varphi(V(x)),\quad\varphi(s):=acs^{1-\alpha}.$

The function $\varphi$ is increasing and concave. We can compute

(11)
$\displaystyle\begin{split}\Phi(s):=\int_{1}^{s}&\frac{\mathrm{d}y}{\varphi(y)}=\frac{1}{ac\alpha}\left[s^{\alpha}-1\right],\quad\Psi(v):=\left[ac\alpha
v+1\right]^{1/\alpha};\\\ G(t,x)&:=\Psi(\Phi(x)+t)=\left[ac\alpha
t+x^{\alpha}\right]^{1/\alpha}.\end{split}$

Thus we can deduce (7) from (36) from the Appendix:

$\displaystyle\Psi(t,V(x))$ $\displaystyle=\left[ac\alpha\cdot
t+a^{\alpha}(1+cx)^{\alpha}\right]^{1/\alpha}\geqslant h(t)U(x),$
$\displaystyle h(t)$ $\displaystyle:=\left[H^{-1}(ac\alpha\cdot
t)\right]^{1/\alpha},\quad
U(x):=\left[K^{-1}(a^{\alpha}(1+cx)^{\alpha})\right]^{1/\alpha}.$

For the example in (35), we get: $H^{-1}(x)=p^{1/p}x^{1/p}$ and
$K^{-1}(y):=q^{1/q}y^{1/q}$, thus

$h(t):=(acp\alpha)^{1/\alpha p}t^{1/(\alpha p)},\quad
U(x):=q^{1/(q\alpha)}a^{1/\alpha}(1+cx)^{1/q}.$

If $p$ is larger, then $q$ is smaller. This illustrates the general principle:
Taking a stronger $\lVert\cdot\rVert_{U}$, for a larger function $U$, leads to
slower convergence rate $h$, and vice versa.

######  Example 4.

Now assume the condition (10) holds, and moreover
$\sigma(x)\equiv\sigma=\mathrm{const}$. Take a function $V(x):=(1+\lambda
x)^{\beta}$ for $\beta>1$ and $\lambda\geqslant c$ to be determined later.
Then we have:

$\displaystyle V^{\prime}(x)$ $\displaystyle=\beta\lambda\cdot(1+\lambda
x)^{\beta-1},\quad V^{\prime\prime}(x)=\lambda^{2}\beta(\beta-1)(1+\lambda
x)^{\beta-2},$ $\displaystyle\mathcal{L}V(x)\leqslant-a\beta$
$\displaystyle\lambda(1+cx)^{\alpha-1}(1+\lambda
x)^{\beta-1}+\frac{1}{2}\lambda^{2}\sigma^{2}\beta(\beta-1)(1+\lambda
x)^{\beta-2}\leqslant-A(\lambda)(1+\lambda x)^{\beta-\alpha},$ $\displaystyle
A(\lambda):=a\beta\lambda-\frac{1}{2}\sigma^{2}\beta(\beta-1)\lambda^{2},\quad
x>0.$

Thus we can take $\varphi(s):=A(\lambda)s^{1-\alpha/\beta}$. Since
$\alpha<1<\beta$, then $0<1-\alpha/\beta<1$ and this is a concave increasing
function. This is only true if we can make sure that $A(\lambda)>0$. We need

(12) $\lambda<\frac{2a}{\sigma^{2}(\beta-1)}.$

But on the other hand, we need $\lambda\geqslant c$. We got a prerequisite
inequality:

(13)
$\frac{2a}{\sigma^{2}(\beta-1)}>c\,\Leftrightarrow\,\beta<1+\frac{\sigma^{2}}{2ac}.$

If we satisfy conditions (12) and (13), then we can constuct this Lyapunov
function. It gives us better rates of convergence and stronger $U$-norm
$\lVert\cdot\rVert_{U}$. But we did this under (10) combined with constant
diffusion coefficient. If we have similar estimates for a non-constant
diffusion coefficient, we need to modify our Lyapunov function.

######  Remark 5.

Let us follow Remark 8 to show for both examples that the stationary
distribution $\pi$ exists and is unique, and $(\pi,V)<\infty$. For each case,
taking $\hat{V}(x):=V(\psi(x))$ as at the end of Section 2, we get a function
$\hat{V}$ such that $\hat{V}^{\prime}(0)=0$ and the condition (8) holds. Since
the process $X$ has the positivity property $P^{t}(x,B)>0$, it has a unique
stationary distribution $\pi$ which satisfies $(\pi,\hat{V})<\infty$. Since
$\hat{V}(x)=V(x)$ for large enough $x$, we can automatically conclude that
$(\pi,V)<\infty$.

###  3.2. Reflected jump-diffusions

In addiition to the above notation, take a family $(\nu_{x})_{x\geqslant 0}$
of finite Borel measures on $\mathbb{R}_{+}$ with
$\nu_{x}(\mathbb{R}_{+})=M<\infty$ for all $x\geqslant 0$. Then we can augment
the above reflected diffusion with these jumps: They occur with intensity $M$,
and the destination of a jump from $x\geqslant 0$ is distributed as
$M^{-1}\nu_{x}(\cdot)$. As long as the family $(\nu_{x})_{x\geqslant 0}$ is
weakly continuous: $\nu_{y}\to\nu_{x}$ weakly as $y\to x$, and $g,\sigma$
satisfy the same assumptions as in the previous subsection, this process can
be constructed by piecing out, see [34], and it exists in the weak sense and
is unique in law. This is a Markov process with the following generator, for
$f\in C^{2}(\mathbb{R}_{+})$ with $f^{\prime}(0)=0$:

(14)
$\mathcal{L}f(x)=g(x)f^{\prime}(x)+\frac{1}{2}\sigma^{2}(x)f^{\prime\prime}(x)+\int_{0}^{\infty}[f(y)-f(x)]\,\nu_{x}(\mathrm{d}y).$

It is shown in [34] that if this family of measures is stochastically ordered,
then the process $Z$ is also stochastically ordered. Take a Lyapunov function
$V(x)=1+cx$ for $c>0$. Applying the generator from (14) even though
$V^{\prime}(0)\neq 0$, we get:

$\mathcal{V}(x)=cm(x),\quad
m(x):=g(x)+\int_{0}^{\infty}(y-x)\,\nu_{x}(\mathrm{d}y).$

One can think of $m(x)$ as “average drift” at the point $x\geqslant 0$ which
is the usual drift $g(x)$ combined with the “implied drift” created by jumps.
This is the “velocity” with which the process “wants to move” to the right
while at location $x$. If $m(x)$ instead of $g(x)$ satisfies (10), then we can
make the same conclusions as for the reflected diffusion above. We need to
show that the stationary distribution $\pi$ exists, is unique, and satisfies
$(\pi,V)<\infty$, as in Remark 8.

Assume that there exist $0<c_{1}<c_{2}$ such that for $x\geqslant c_{2}$,
$\nu_{x}([0,c_{1}])=0$. Take $\psi(x)=0$ for $x\leqslant c_{1}/2$ and
$\psi(x)=x$ for $x\geqslant c_{1}$. Then for $x\geqslant c_{2}$ we have:

$\displaystyle\int_{0}^{\infty}$
$\displaystyle(\hat{V}(y)-\hat{V}(x))\,\nu_{x}(\mathrm{d}y)=\int_{0}^{\infty}(V(\psi(y))-V(\psi(x)))\,\nu_{x}(\mathrm{d}y)$
$\displaystyle=\int_{c_{1}}^{\infty}(V(\psi(y))-V(\psi(x)))\,\nu_{x}(\mathrm{d}y)=\int_{c_{1}}^{\infty}(V(y)-V(x))\,\nu_{x}(\mathrm{d}y)$
$\displaystyle=\int_{0}^{\infty}(V(y)-V(x))\,\nu_{x}(\mathrm{d}y);$
$\displaystyle g(x)$
$\displaystyle\hat{V}^{\prime}(x)+\frac{1}{2}\sigma^{2}(x)\hat{V}^{\prime\prime}(x)=g(x)V^{\prime}(x)+\frac{1}{2}\sigma^{2}(x)V^{\prime\prime}(x);$

and thus the generator formally applied to $V$ will be the same as if applied
to $\hat{V}\in\mathcal{D}(\mathcal{L})$:

$\mathcal{L}\hat{V}(x)=\mathcal{L}V(x),\quad x\geqslant c_{2}.$

If $\mathcal{L}\hat{V}(x)$ is bounded on $[0,c_{2}]$ (it is easy to check in
practice) then we have the condition (8). Together with positivity property,
proved in [34], this implies existence and uniqueness of the stationary
distribution $\pi$ and $(\pi,V)<\infty$.

######  Example 5.

Try the following process: $g(x)=-3(x+1)^{-0.5}$, $M=2$ and $\nu_{x}$ is
defined as

$\nu_{x}(\mathrm{d}y)=\lambda(x)\exp\left[\lambda(x)(x-y)\right]1_{\\{y>x\\}}\,\mathrm{d}y,\quad\lambda(x):=(x+1)^{0.5}.$

In other words, the process jumps with constant intensity $2$ to the right,
and the displacement is distributed as an exponential random variable with
rate $\lambda(x)$ (and therefore mean $\lambda^{-1}(x)$). Then the average
rate of displacement from jumps is $2\lambda^{-1}(x)$. Thus

$m(x)=-3(x+1)^{-0.5}+2(x+1)^{-0.5}=-(x+1)^{-0.5},\quad x\geqslant 0,$

This gives us subexponential convergence, as in Example 3 above.

We could also weaken the assumption of finite first moment for $\nu_{x}$
Instead of stating a general result, we consider a particular case. Assume
that there exists a finite measure $\mu$ on the positive half-line
$\mathbb{R}_{+}$ such that for all $B\subseteq\mathbb{R}_{+}$,

$\nu_{x}(x+B)=\mu(B),\,\nu_{x}[0,x)=0,\,x\in\mathbb{R}_{+}.$

We fix a $\beta\in(0,1)$ and assume

$m_{\beta}:=\int_{0}^{\infty}x^{\beta}\mu(\mathrm{d}x)<\infty.$

In addition, assume there exist constants $C,x_{0}>0$ such that
$g(x)\leqslant-Cx^{1-\beta}$ for $x\geqslant x_{0}$. Then we can take a
Lyapunov function $V(x)=1+x^{\beta}$. We get: For $x\geqslant x_{0}$,

$\displaystyle\mathcal{L}V(x)$
$\displaystyle=g(x)V^{\prime}(x)+\frac{\sigma^{2}(x)}{2}V^{\prime\prime}(x)+\int_{0}^{\infty}[V(x+y)-V(x)]\mu(\mathrm{d}y)$
$\displaystyle=g(x)\beta
x^{\beta-1}+\frac{\sigma^{2}(x)}{2}\beta(\beta-1)x^{\beta-2}+\int_{0}^{\infty}\left[(x+y)^{\beta}-x^{\beta}\right]\mu(\mathrm{d}y)$
$\displaystyle\leqslant-C+\int_{0}^{\infty}y^{\beta}\mu(\mathrm{d}y)=-C+m_{\beta}.$

The last inequality follows from the estimate $(x+y)^{\beta}\leqslant
x^{\beta}+y^{\beta}$ for $x,y\geqslant 0$. If $m_{\beta}<C$, then Theorem 1
holds with $\varphi$ as in Remark 2.

###  3.3. Reflected Lévy processes

Any Lévy process on the real line can be decomposed into the sum of two
components: a Brownian motion with constant drift and diffusion (the
continuous component), and a pure jump Lévy process $\mathcal{J}$. In this
article, we assume this pure jump process is nondecreasing. Its jumps are
governed by a spectral measure $\mu$. If the spectral measure is finite, then
there are a.s. finitely many jumps (a Poisson number) during any time interval
$[0,t]$, and $\mathcal{J}$ is, in fact, a compound Poisson process. However,
if $\mu(\mathbb{R}_{+})=\infty$, then this process $\mathcal{J}$ makes
infinitely many jumps during any time interval $[0,t]$. This $\mu$ is a
$\sigma$-finite Borel measure on $\mathbb{R}_{+}$. This measure must satisfy

(15) $\int_{0}^{\infty}(1\wedge x)\,\mu(\mathrm{d}x)<\infty.$

We take a reflected version $X$ of $L$ on the half-line using the Skorohod
reflection mapping

$X(t):=L(t)+\sup\limits_{0\leqslant s\leqslant t}\left[\max(-L(s),0)\right].$

The resulting process will have values in $\mathbb{R}_{+}$ and behave like a
Lévy process $L$ as long as it is strictly inside $\mathbb{R}_{+}$. When it
hits $0$, it is reflected back. If $\mu$ is finite, we are back in the case of
reflected jump-diffusions, discussed in the previous subsection. In this
subsection, we are mostly interested in the case when
$\mu(\mathbb{R}_{+})=\infty$. This process is defined by a triple
$(g,\sigma,\mu)$ where $g\in\mathbb{R}_{+}$ is a drift, $\sigma>0$ is a
diffusion coefficient, and it can be decomposed as

(16) $L(t)=L(0)+gt+\sigma W(t)+\mathcal{J}(t),\quad t\geqslant 0,$

where $W$ is a Brownian motion. The generator of $L$ is given by

(17)
$\mathcal{L}f(x)=gf^{\prime}(x)+\frac{1}{2}\sigma^{2}f^{\prime\prime}(x)+\int_{0}^{\infty}\left[f(x+z)-f(x)\right]\,\mu(\mathrm{d}z).$

Impose a condition which is a bit stronger than (15), that the first moment is
finite:

(18) $m_{1}:=\int_{0}^{\infty}z\,\mu(\mathrm{d}z)<\infty.$

The reflected version has the same generator (17), but for functions $f\in
C^{2}(\mathbb{R}_{+})$ with $f^{\prime}(0)=0$. Formally apply the generator to
the function $V(x):=1+x$, although $V^{\prime}(0)\neq 0$. Then
$\mathcal{L}V(x)=g+m_{1}$. If $g<-m_{1}$, then this process is ergodic. We
prove this by taking $\hat{V}(x):=V(\psi(x))$ as in Remark 8. Then we can
apply Theorem 1 using Remark 2. Similarly to the previous subsection, we get
Moreover, assume another condition stronger than (18):

(19) $\int_{1}^{\infty}e^{\lambda_{0}x}\,\mu(\mathrm{d}x)<\infty.$

Take $V(x):=e^{\lambda x}$ for $\lambda\in(0,\lambda_{0})$. Then

$\displaystyle\mathcal{L}V(x)$ $\displaystyle=V(x)k(\lambda),\quad
k(\lambda):=\lambda
g+\frac{\sigma^{2}\lambda^{2}}{2}+\int_{0}^{\infty}\left[e^{\lambda
z}-1\right]\,\mu(\mathrm{d}z).$

And the integral in the right-hand side is finite because of (18) and (19).
Similarly to [34], we prove in Lemma 1 that there exists a $\lambda>0$ such
that $k(\lambda)<0$. Reflected Lévy process also has positivity property, by
Lemma 2. Thus by using Remark 8 we get that there exists a unique stationary
distribution $\pi$, which satisfies $(\pi,V)<\infty$, and for any such
$\lambda$ we get:

$\lVert
P^{t}(x,\cdot)-\pi(\cdot)\rVert_{V}\leqslant(V(x)+(\pi,V))e^{k(\lambda)t},\quad
x,t\geqslant 0.$

This is the same exponential rate of convergence as in [34] for reflected Lévy
processes with finite spectral measure (which is a compound Poisson process
plus a Brownian motion).

######  Lemma 1.

Under conditions (18) and (19), if $g<-m_{1}$, then $k(\lambda)<0$ for some
$\lambda>0$.

###### Proof.

This follows from differentiability under the integral with respect to
$\lambda$ at $\lambda=0$, which we shall prove now. For all
$\lambda\in(0,\lambda_{0})$, there exist constants $C_{1},C_{2}>0$ such that
for all $z\geqslant 0$, we get: $ze^{\lambda z}\leqslant
C_{1}z+C_{2}e^{\lambda_{0}z}$. They can be taken independent of $\lambda$.
From this elementary estimate combined with (18) and (19), we get:

$\int_{0}^{\infty}\left[ze^{\lambda z}\right]\,\mu(\mathrm{d}z)<\infty.$

From here, we can deduce by Lebesgue dominated convergence theorem that

$\frac{\mathrm{d}}{\mathrm{d}\lambda}\int_{0}^{\infty}\left[e^{\lambda
z}-1\right]\,\mu(\mathrm{d}z)=\int_{0}^{\infty}\frac{\partial}{\partial\lambda}\left[e^{\lambda
z}-1\right]\,\mu(\mathrm{d}z)=\int_{0}^{\infty}\left[ze^{\lambda
z}\right]\,\mu(\mathrm{d}z).$

Thus, taking derivative of $k(\lambda)$ and letting $\lambda=0$, we get:

$k^{\prime}(0)=g+\int_{0}^{\infty}z\,\mu(\mathrm{d}z)=g+m_{1}<0.$

This, together with previous computations, completes the proof. ∎

######  Lemma 2.

For any $t>0$, $x\geqslant 0$, and a set $B\subseteq\mathbb{R}_{+}$ of
positive Lebesgue measure, the transition kernel of the reflected process
satisfies $P^{t}(x,B)>0$.

###### Proof.

Assume $x>0$. Without loss of generality, we can assume
$B\subseteq[\delta,\infty)$ for some $\delta>0$. Then with probability $p>0$,
this process never hits $0$ until time $t$. In this case, it behaves as a non-
reflected Lévy process, which has positivity property $Q^{t}(x,B)>0$. Thus
$P^{t}(x,B)\geqslant pQ^{t}(x,B)>0$. Next, if $x=0$, then
$P^{t/2}(0,(0,\infty))>0$, and thus
$P^{t}(x,B)\geqslant\int_{0}^{\infty}P^{t/2}(y,B)P^{t/2}(0,\mathrm{d}y)>0$
(the integral of a positive function over a positive measure is positive).
Thus we reduced the case $x=0$ to the case $x>0$. ∎

##  4\. Proof of Theorem 1

###  4.1. Overview of the proof

The main idea is similar to [34, Theorem 4.1, Theorem 5.2]. It is known from
[25, Theorem 5] that if $X$ has trajectories which are a.s. right-continuous
with left limits, then being stochastically ordered is equivalent to the
following statement: For every $0\leqslant x_{1}\leqslant x_{2}$, there exists
a coupling of two versions $X_{1}$ and $X_{2}$ of $X$, starting from
$X_{1}(0)=x_{1}$ and $X_{2}(0)=x_{2}$: A probability space and copies $X_{1}$
and $X_{2}$ defined on this space such that $X_{1}(t)\leqslant X_{2}(t)$ for
all $t\geqslant 0$ a.s. Let us define a coupling time
$\tau_{0}:=\inf\\{t\geqslant 0\mid X_{1}(t)=X_{2}(t)\\}$. We can assume
$X_{1}(t)=X_{2}(t)$ for $t>\tau_{0}$, so after this coupling time, the
processes coincide. For any function $g:\mathbb{R}_{+}\to\mathbb{R}$ with
$|g|\leqslant U$, we get (the last inequality from nondecreasing $U$):

(20)
$\displaystyle\begin{split}h(t)&\left|\mathbb{E}[g(X_{1}(t))]-\mathbb{E}[g(X_{2}(t))]\right|=h(t)\left|\mathbb{E}[g(X_{1}(t))]1_{\\{\tau_{0}>t\\}}-\mathbb{E}[g(X_{2}(t))1_{\\{\tau_{0}>t\\}}]\right|\\\
&\leqslant
h(t)\left(\mathbb{E}\left[U(X_{1}(t))1_{\\{\tau_{0}>t\\}}\right]+\mathbb{E}\left[U(X_{2}(t))1_{\\{\tau_{0}>t\\}}\right]\right)\leqslant
2\mathbb{E}\left[h(t)U(X_{2}(t))1_{\\{\tau_{0}>t\\}}\right].\end{split}$

Now we define $\tau:=\inf\\{t\geqslant 0\mid X_{2}(t)=0\\}$. Then
$X_{1}(\tau)\leqslant X_{2}(\tau)=0$ and thus $X_{1}(\tau)=0$. Therefore,
$\tau\geqslant\tau_{0}$ a.s. Combining this observation with (20), we get:

(21)
$h(t)\left|\mathbb{E}[g(X_{1}(t))]-\mathbb{E}[g(X_{2}(t))]\right|\leqslant
2\mathbb{E}\left[h(t)U(X_{2}(t))1_{\\{\tau>t\\}}\right].$

######  Lemma 3.

The function $G$ satisfies the boundary conditions

$G(t,1)=\Psi(t),\,t\geqslant 0;\quad G(0,u)=u,\,u\geqslant 1,$

and for $t\geqslant 0,\,u\geqslant 1$ the following equation and inequalities
hold:

(22) $\frac{\partial G}{\partial t}(t,u)=\varphi(u)\frac{\partial G}{\partial
u}(t,u),\quad\frac{\partial G}{\partial u}(t,u)\geqslant
0,\quad\frac{\partial^{2}G}{\partial u^{2}}(t,u)\leqslant 0.$

This technical lemma was partially proved in [11], but we want to collect
these results and give (a straightforward) proof for the sake of completeness.
The key part of the proof of Theorem 1 is the following lemma. Take a copy of
$X$ starting from $X(0)=x_{0}\geqslant 0$.

######  Lemma 4.

The following process $K=(K(t),\,t\geqslant 0)$ is a local
$(\mathcal{F}_{t})_{t\geqslant 0}$-supermartingale:

$K(t):=G(t\wedge\tau,\,V(X(t\wedge\tau))),\,t\geqslant 0.$

Assume Lemma 4 is already proved. Let us show that $\tau<\infty$ a.s. Since
$G$ is nondecreasing with respect to $u$, we get: $G(t,u)\geqslant
G(t,1)=\Psi(t)$. Letting $n=1,2,\ldots$ we get:

(23)
$G(0,V(X(0)))=\mathbb{E}K(0)\geqslant\mathbb{E}K(n)\geqslant\mathbb{E}\left[G(n\wedge\tau,V(X(n\wedge\tau)))\right]\geqslant\mathbb{E}\Psi(n\wedge\tau).$

Assume the converse: $\tau=\infty$ with positive probability. Then

(24) $\mathbb{E}\Psi(n\wedge\tau)\geqslant\Psi(n)\mathbb{P}(\tau=\infty).$

But $\Psi(\infty)=\infty$, as stated in subsection 2.1. Letting $n\to\infty$,
we compare (23) and (24) and arrive at a contradiction. This proves that
$\tau<\infty$ a.s. Since $K(t)\geqslant 0$ for all $t\geqslant 0$, then by
Fatou’s lemma we can remove the word “local” from Lemma 4. From (7) we get:

(25)
$\displaystyle\begin{split}h(t)&\mathbb{E}\left[U(X_{2}(t))1_{\\{\tau>t\\}}\right]=\mathbb{E}\left[h(t\wedge\tau)U(X_{2}(t\wedge\tau))1_{\\{\tau>t\\}}\right]\\\
&\leqslant\mathbb{E}\left[G(t\wedge\tau,V(X_{2}(t\wedge\tau)))1_{\\{\tau>t\\}}\right]\leqslant\mathbb{E}\left[G(t\wedge\tau,V(X_{2}(t\wedge\tau)))\right]\\\
&=\mathbb{E}K(t)\leqslant\mathbb{E}K(0)=G(0,V(x_{2}))=V(x_{2}).\end{split}$

Combining (20) with (25), we get:

$h(t)\left|\mathbb{E}[g(X_{1}(t))]-\mathbb{E}[g(X_{2}(t))]\right|\leqslant
2V(x_{2}).$

Dividing by $h(t)$ and taking the supremum over all
$g:\mathbb{R}_{+}\to\mathbb{R}$ with $|g|\leqslant U$, we complete the proof
of (a). The proof of (b) is done similarly, and (c) follows from (b) and the
observation that $\pi P^{t}=\pi$ for a stationary distribution $\pi$ and any
$t\geqslant 0$.

###  4.2. Proof of Lemma 22

The boundary conditions are easy to show: $G(0,u)=\Psi(\Phi(u))=u$ for
$u\geqslant 1$, and $G(t,1)=\Psi(\Phi(1)+t)=\Psi(t)$ for $t\geqslant 0$. Next,
the function $\varphi:[1,\infty)\to\mathbb{R}_{+}$ is increasing. Thus the
function $\Phi:[1,\infty)\to\mathbb{R}_{+}$ satisfies $\Phi(1)=0$, increasing,
and concave. Next, $\Psi:\mathbb{R}_{+}\to[1,\infty)$ satisfies $\Psi(0)=1$,
increasing, and convex. By the chain rule,
$\Psi^{\prime}(v)=\left[\Phi^{\prime}(\Psi(v))\right]^{-1}=\varphi(\Psi(v))$.
Let us do computation of derivatives of $G$:

(26) $\displaystyle\begin{split}\frac{\partial G}{\partial
t}(t,u)&=\Psi^{\prime}(\Phi(u)+t)=\varphi(\Psi(\Phi(u)+t))=\varphi(G(t,u));\\\
\frac{\partial G}{\partial
u}(t,u)&=\Psi^{\prime}(\Phi(u)+t)\cdot\Phi^{\prime}(u)=\frac{\varphi(\Psi(\Phi(u)+t))}{\varphi(u)}=\frac{\varphi(G(t,u))}{\varphi(u)}\geqslant
0.\end{split}$

These two equations from (26) prove the equality and the first inequality in
(22). Finally, we need to prove the second inequality in (22), that is,
concavity of $G$ with respect to $u$:

(27) $\frac{\partial^{2}G}{\partial u^{2}}(t,u)=\frac{\partial}{\partial
u}\left(\frac{\varphi(G(t,u))}{\varphi(u)}\right)=\frac{1}{\varphi^{2}(u)}\left[\varphi^{\prime}(G(t,u))\frac{\partial
G}{\partial u}(t,u)\varphi(u)-\varphi(G(t,u))\varphi^{\prime}(u)\right].$

Next, using the second equation in (26), we get:

(28) $\varphi^{\prime}(G(t,u))\frac{\partial G}{\partial
u}(t,u)\varphi(u)=\varphi^{\prime}(G(t,u))\varphi(G(t,u)).$

Finally, $\varphi^{\prime}$ is nonincreasing. Since $\Psi$ is nondecreasing,
$G(t,u)=\Psi(\Phi(u)+t)\geqslant\Psi(\Phi(u))=u$ for all $u\geqslant 1$ and
$t\geqslant 0$. Thus $\varphi^{\prime}(G(t,u))\leqslant\varphi^{\prime}(u)$.
Multiplying this inequality by $\varphi(G(t,u))\geqslant 0$ and using (27) and
(28), we complete the proof.

###  4.3. Proof of Lemma 4

We know that the following process is an $(\mathcal{F}_{t})_{t\geqslant
0}$-supermartingale:

(29)
$N(t)=V(X(t\wedge\tau))+\int_{0}^{t\wedge\tau}\varphi(V(X(s)))\,\mathrm{d}s,\,t\geqslant
0.$

When we write in differential notation, we let $df(t)\leqslant 0$ if $f$ is a
nonincreasing function, and $\mathrm{d}f(t)\leqslant\mathrm{d}g(t)$ if $g-f$
is a nondecreasing function. For $t<\tau$, since $V$ is smooth and $X$ is a
semimartingale, then $V(X(\cdot))$ is also a semimartingale. Thus we can apply
Itô’s formula:

(30) $\displaystyle\begin{split}&\mathrm{d}K(t)=\mathrm{d}G(t,V(X(t)))\\\
&=\frac{\partial G}{\partial t}(t,V(X(t)))\,\mathrm{d}t+\frac{\partial
G}{\partial
u}(t,V(X(t)))\,\mathrm{d}V(X(t))+\frac{1}{2}\frac{\partial^{2}G}{\partial
u^{2}}(t,V(X(t)))\,\mathrm{d}\langle V(X)\rangle_{t}.\end{split}$

From concavity of $G$ with respect to $u$ found in Lemma 22, and from (30), we
get:

(31) $\mathrm{d}K(t)\leqslant\frac{\partial G}{\partial
t}(t,V(X(t)))\,\mathrm{d}t+\frac{\partial G}{\partial
u}(t,V(X(t)))\,\mathrm{d}V(X(t)).$

On the other hand, from (29) we get:
$\mathrm{d}N(t)\leqslant\mathrm{d}V(X(t))+\varphi(V(X(t)))\,\mathrm{d}t+\mathrm{d}M(t)$
for some local $(\mathcal{F}_{t})_{t\geqslant 0}$-martingale
$M=(M(t),\,t\geqslant 0)$. Thus for $t<\tau$ we have:
$\mathrm{d}V(X(t))=\mathrm{d}N(t)-\mathrm{d}M(t)-\varphi(V(X(t)))\,\mathrm{d}t$.
Next, from above calculations in (31) and (29), we get:

(32) $\displaystyle\begin{split}\mathrm{d}K(t)&\leqslant\frac{\partial
G}{\partial t}(t,V(X(t)))\,\mathrm{d}t+\frac{\partial G}{\partial
u}(t,V(X(t)))\,\mathrm{d}N(t)\\\ &-\frac{\partial G}{\partial
u}(t,V(X(t)))\,\mathrm{d}M(t)-\frac{\partial G}{\partial
u}(t,V(X(t)))\,\varphi(V(X(t)))\,\mathrm{d}t.\end{split}$

From computations of derivatives of $G$ we get:

(33) $\frac{\partial G}{\partial t}(t,u)=\varphi(u)\frac{\partial G}{\partial
u}(t,u).$

Plugging (33) into (32), we get: $\mathrm{d}K(t)\leqslant\frac{\partial
G}{\partial u}(t,V(X(t)))\,(\mathrm{d}N(t)-\mathrm{d}M(t))$. But $N$ is an
$(\mathcal{F}_{t})_{t\geqslant 0}$-supermartingale, and $M$ is a local
$(\mathcal{F}_{t})_{t\geqslant 0}$-martingale. Thus $N-M$ is also an
$(\mathcal{F}_{t})_{t\geqslant 0}$-supermartingale, and the same can be said
about $K$, since $\frac{\partial G}{\partial u}\geqslant 0$. This completes
the proof.

##  5\. Appendix

Below is a method to find $h$ and $U$ such that (7) holds. Take any Young pair
$(H,K)$ of strictly increasing functions $\mathbb{R}_{+}\to\mathbb{R}_{+}$
such that

(34) $H(0)=K(0)=0,\quad H(\infty)=K(\infty)=\infty,\quad xy\leqslant
H(x)+K(y),\quad x,y\geqslant 0.$

The following example follows from Young’s inequality:

(35) $H(x)=\frac{x^{p}}{p},\quad K(y):=\frac{y^{q}}{q},\quad
p,\,q>0,\quad\frac{1}{p}+\frac{1}{q}=1.$

Assume (34) holds. Since $H$ and $K$ are strictiy increasing, define their
inverses $H^{-1},K^{-1}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ such that
$H(H^{-1}(x))=x$ and $K(K^{-1}(y))=y$ for all $x,y\geqslant 0$. Then

(36) $H^{-1}(x)K^{-1}(y)\leqslant x+y,\,x,y\geqslant 0.$

For example, inverses of (35) are given by $H^{-1}(x):=(px)^{1/p}$,
$K^{-1}(y):=(qy)^{1/q}$.

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