# Investigation of hydrogen incorporations in bulk infinite-layer nickelates

P. Puphal<EMAIL_ADDRESS>Max-Planck-Institute for Solid State Research,
Heisenbergstraße 1, 70569 Stuttgart, Germany  V. Pomjakushin  Laboratory for
Neutron Scattering and Imaging (LNS), Paul Scherrer Institute (PSI),
Forschungsstrasse 111, CH-5232 Villigen, Switzerland  R. A. Ortiz  Max-Planck-
Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart,
Germany  S. Hammoud  Max-Planck-Institute for Intelligent Systems,
Heisenbergstraße 3, 70569 Stuttgart, Germany  M. Isobe  Max-Planck-Institute
for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany  B.
Keimer  Max-Planck-Institute for Solid State Research, Heisenbergstraße 1,
70569 Stuttgart, Germany  M. Hepting  Max-Planck-Institute for Solid State
Research, Heisenbergstraße 1, 70569 Stuttgart, Germany

###### Abstract

Infinite-layer (IL) nickelates are an emerging class of superconductors, where
the Ni1+ valence state in a square planar NiO2 coordination can only be
reached via topotactic reduction of the perovskite phase. However, this
topotactic soft chemistry with hydrogenous reagents is still at a stage of
rapid development, and there is a number of open issues, especially
considering the possibility of hydrogen incorporation. Here we study the time
dependence of the topotactic transformation of LaNiO3 to LaNiO2 for powder
samples with x-ray diffraction and gas extraction techniques. While the
hydrogen content of the powder increases with time, neutron diffraction shows
no negative scattering of hydrogen in the LaNiO2 crystal lattice. The extra
hydrogen appears to be confined to grain boundaries or secondary-phase
precipitates. The average crystal structure, and possibly also the physical
properties, of the primary LaNiO2 phase are therefore not noticeably affected
by hydrogen residues created by the topotactic transformation.

##  I Introduction

Superconductivity exists in various hydrogen containing compounds, highlighted
by the recent discoveries of critical temperatures $T_{c}$ as high as room
temperature for hydride compounds formed under extreme pressures Shamp and
Zurek (2017); Snider  _et al._ (2020). Metal hydroxide-intercalated iron
chalcogenides, such as (Li0.8Fe0.2)OHFeSe Lu  _et al._ (2014), show
coexistence of antiferromagnetic order and superconductivity, a feature known
from some high-$T_{c}$ superconductors Kahsay and Singh (2016). Electron-doped
1111 iron pnictides as $R$FeAsO1-xHx Hanna  _et al._ (2011); Hiraishi  _et
al._ (2014), CeFeAsO1-xHx Matsuishi  _et al._ (2012), and the pnictogen-free
LaFeSiH Bernardini  _et al._ (2018) are a class, where the introduction of
charge carriers via doping with hydrogen drives the system from an
antiferromagnetically ordered state towards superconductivity. In layered
sodium cobalt oxyhydrate, NaxCoO${}_{2}\cdot$H2O Lynn  _et al._ (2003),
superconductivity with a similar hole/electron-doping behavior by chemical
substitution is observed as in the cuprate high-$T_{c}$ superconductors Keimer
_et al._ (2015).

For IL nickelates, a close relation and possible analogy to cuprate
superconductors was suggested already in 1999 Anisimov  _et al._ (1999) and
since the first discovery of superconductivity in the IL nickelate (Nd,Sr)NiO2
Li  _et al._ (2019), the observation of superconductivity has been confirmed
Zeng  _et al._ (2020); Li  _et al._ (2021); Gao  _et al._ (2021) and extended
to (Pr,Sr)NiO2 Osada  _et al._ (2020), (La,Sr)NiO2 Osada  _et al._ (2021),
(La,Ca)NiO2 Zeng  _et al._ (2021), and Nd6Ni5O12 Pan  _et al._ (2021).
Furthermore, a recent work reported superconductivity not only for films grown
on SrTiO3 substrates, but also on LSAT Ren  _et al._ (2021), which provides
enough evidence to consider thin film nickelates as a novel class of
superconductors. The possible presence of topotactic hydrogen in IL
nickelates, which depends on the rare-earth ion and/or epitaxial strain
Bernardini  _et al._ (2021); Alvarez  _et al._ (2021), was proposed in
theoretical studies Si  _et al._ (2020); Malyi  _et al._ (2021), and might
have substantial influence on the electronic and magnetic properties of the IL
nickelates, as LaNiO2H would realize a two-orbital Mott insulator Si  _et al._
(2020). A hint towards the possibility of hydrogen incorporation was provided
by an early study of topotactically reduced NdNiO3 films, which showed an
oxyhydride NdNiO3-xHy phase with a defect-fluorite structure in the surface
region Onozuka  _et al._ (2016). Furthermore, topotactic hydrogen can be found
in SrTiO3 thin films Kutsuzawa  _et al._ (2018), i.e. the material that is
commonly used as substrate and capping layer for IL nickelate films, which
again provides a possible route for inclusion of hydrogen in infinite-layer
nickelates thin films. While superconductivity has remained elusive in IL
nickelates in bulk form, with studies on powder samples reporting insulating
behavior Wang  _et al._ (2020); Li  _et al._ (2020), a first step towards
superconductivity was taken by our recent investigation of (La,Ca)NiO2 single
crystals Puphal  _et al._ (2021), where metallicity was observed.

In this work, we address the issue of possible hydrogen incorporations in IL
nickelates by examining bulk LaNiO3-xHy powder samples with a combination of
x-ray and neutron diffraction studies, gas extraction, as well as
complementary high pressure synthesis attempts of LaNiO2H.

##  II Methods

LaNiO3 powder samples with grain sizes of 0.5 $\mu$m were synthesized via the
citrate-nitrate method as described in Ref. Ortiz  _et al._ (2021). The method
is optimal as the relatively small grains enhance the surface to bulk ratio,
reducing the reduction times. Moreover, a higher purity can be reached than by
high pressure powder synthesis. The IL phase LaNiO2 can be obtained solely
through topotactic reduction of the perovskite LaNiO3 phase, which is here
achieved by using CaH2 as the reducing agent Wang  _et al._ (2020); Li  _et
al._ (2020). We reduced 50 mg of LaNiO3 powder wrapped in aluminum foil,
spacially separated from $250$ mg CaH2 powder at 280∘C as described in detail
in Ref. Ortiz  _et al._ (2021) for various times.

We measured the stoichiometry including the hydrogen content with a
combination of inductively coupled plasma mass spectroscopy (ICP-OES) and gas
extraction; the former with a Spectro Ciros CCD and the latter with an Eltra
ONH-2000 analyzer. For the determination of the oxygen and hydrogen content,
the powder samples were placed in a Ni crucible, clipped and heated, where the
carrier gas takes the oxygen and hydrogen out of the sample. The oxygen reacts
with carbon and CO2 is detected in a infrared-cell, while hydrogen is detected
by a thermal-conductivity-cell. Each measurement is repeated three times and
compared to a standard. The quoted error bars give the statistical error.

Powder x-ray diffraction (PXRD) data were collected using a Rigaku Miniflex
with a Cu Kα tube at room temperature. Neutron diffraction was performed at
the high-resolution powder neutron diffractometer HRPT Fischer  _et al._
(2000) at the spallation neutron source SINQ at the Paul Scherrer Institute in
Villigen. For the HRPT experiments, an amount of 340 mg of LaNiO2 was enclosed
into a vanadium can with an inner diameter of 6 mm, where the remaining space
of the vanadium can was shielded with Cd foil and the measurement was carried
out at 1.5 K in a 4He bath cryostat with a neutron wavelength of 1.15 Å.

##  III Results

Figure 1: Powder x-ray diffraction (PXRD) of reduced LaNiO3 after different
reduction times. Data were measured at $T=300$ K with Cu Kα radiation. Curves
are offset in vertical direction for clarity. The 0 h and 316 h data are
reproduced from Ref. Ortiz  _et al._ (2021).

As a first step, we checked the purity and stoichiometry of our starting
material in the perovskite phase. In PXRD we found no detectable impurites,
and ICP combined with measurements of gas extraction indicated a starting
stoichiometry of La0.99(1)Ni0.99(1)O2.99(3)H0.005(2). We studied the reduction
progress with time in detail by preparation of several ampules and extracting
samples after varying reduction times and analyzed all the products with PXRD.
We noticed that the progress of the reduction depends on the purity and sample
amount and grain size for a given time, as the reduction process is surface
dependent. Motivated by these observations, we carried out a standardized
reduction on the same batch of perovskite precursors. After one day of
reduction in the described process (see Methods) we found a full structural
transition to LaNiO2.5 (see Fig. 1). Our Rietveld refinement of the LaNiO2.5
crystal structure is in good agreement with previous reports on reduced powder
Crespin  _et al._ (1983); Alonso and Martínez-Lope (1995); Alonso  _et al._
(1997). Notably, reduction times of one day and longer are in stark contrast
to thin film samples, which can be fully reduced to the IL phase within hours
Li  _et al._ (2019); Zeng  _et al._ (2020); Li  _et al._ (2021); Gao  _et al._
(2021); Osada  _et al._ (2020, 2021); Zeng  _et al._ (2021); Ikeda  _et al._
(2016); Hepting  _et al._ (2020), presumably due to the enhanced thickness of
our grains, with sizes of 0.5 $\mu$m as extracted from scanning electron
microscopy (SEM) images. The reduction appears to be happening in domains, as
we found cluster spin glass behavior in a detailed magnetic characterization
Ortiz  _et al._ (2021) and thus after longer reudction times than one day, we
observe phase mixtures of LaNiO2.5 in $P2_{1}/n$ (#14) Alonso and Martínez-
Lope (1995), and LaNiO2 in $P4/mmm$ (#123) Hayward  _et al._ (1999), with a
slowly increasing fraction of the LaNiO2 phase as a function of time (see Fig.
2). Extracting the weight percentages from our refinement, we present the
reduction progress for the CaH2 reduction in analogy to a thermogravimetry
(TG) curve, which is shown in Fig. 2. The reduction process follows an
exponential decay, which we also observed in thermogravimetry studies in
hydrogen gas flow (not shown here). After a reduction time of approximately
316 h, the crystal structure can be refined assuming a single phase of IL
LaNiO2 (Fig. 2, and Ref. Ortiz  _et al._ (2021)). Notably, while the
structural transition does not progress further on an exponential time scale,
a small amount of apical oxygen remains in the crystal lattice, as will be
revealed by neutron diffraction below. However, for even longer reduction
times, the sample begins to decompose, forming Ni and La2O3, which becomes
clearly visible in PXRD after 400 h (Fig. 1 and Fig. 2).

In the inset of Fig. 2 we plot the hydrogen content in the resulting powder
samples versus the reduction time, obtained by a gas extraction method. While
the initial perovskite nickelate shows a hydrogen content of 0.005(2) wt%, we
find an increase of hydrogen with time (opposite to the oxygen content), with
0.065(3) wt% for 24 h, 0.094(3) wt% for 38 h, 0.015(5) wt% for 240 h and
finally 0.169(7) wt% after 320-376 h. Via gas extraction we find an oxygen
content of 14.1(2) wt% after 316 h and 14.0(2) wt% after 320-376 h of
reduction, where 14 wt% corresponding to a full reduction to the LaNiO2 IL
phase. If we assume that the amount of hydrogen would be incorporated into the
average crystal structure, the corresponding effective stoichiometry of the
sample investigated with neutrons would be La0.99(1)Ni0.99(1)O2.00(2)H0.39(2),
where La and Ni are determined via ICP. However, a first hint that this might
not be the case comes from the presence of small amounts of elemental Ni in
the PXRD of the 400 h sample (Fig. 1), which is known to trap hydrogen Louthan
_et al._ (1975). The Rietveld refinement of the neutron diffraction data below
also indicates a small amount of Ni. Furthermore, signatures of Ni
precipitates (below detection threshold of our PXRD) were also detected in
powders after shorter reduction times, since they give rise to ferromagnetic
contributions in the magnetic signal Ortiz  _et al._ (2021). The amount of Ni
likely increases with reduction time, which could explain the increasing
capability of incorporating hydrogen.

Figure 2: Oxygen and hydrogen content of reduced LaNiO3 after different
reduction times. Red symbols correspond to the oxygen content $x$ in LaNiO3-x,
extracted from Rietveld refinements of the PXRD data shown in Fig. 1. The blue
datapoint is extracted from Rietveld refinement of our neutron diffraction
data shown in Fig. 3. The solid red line is a guide to the eye. The black
symbols in the inset correspond to the hydrogen content $y$ in LaNiO3-xHy,
obtained from gas extraction. The solid black line is a guide to the eye.
Table 1:  Refined atomic coordinates of LaNiO2 in tetragonal space group
$P4/mmm$ with preferred orientation extracted from powder neutron diffraction
data at 1.5 K [Fig. 3]. LaNiO2 $|$ $a,b=3.9550(3)$ Å, $c=3.3588(3)$ Å
---
Atom | $x$ | $y$ | $z$ | $U[\text{\AA}^{2}]$ | Occ.
La (1d) | 0.5 | 0.5 | 0.5 | 0.0036(6) | 1
Ni (1a) | 0 | 0 | 0 | 0.0057(5) | 1
O (2f) | 0 | 0.5 | 0 | 0.0098(5) | 1
O (1b) | 0 | 0 | 0.5 | 0.0098(5) | 0.06(1)
Reliability factors | $\chi^{2}$ | $R_{B}$ | $R_{f}$ |  |
| 1.89 | 3.53 | 2.79 |  |

As a next step, we performed high resolution neutron diffraction experiments
on 340 mg of a sample prepared by mixing batches that had been subject to
reduction times of 320, 350 and 376 h, respectively, which are all in the
range just before the start of the clear decay shown in Fig. 2. The obtained
neutron diffraction pattern can be well refined (see Fig. 3) assuming LaNiO2
in the IL $P4/mmm$ structure (see Tab. 1), with a $c$-axis parameter of
3.3588(3) Å, which is lower than previously reported values Ortiz  _et al._
(2021); Hayward  _et al._ (1999); Puphal  _et al._ (2021), and 1.1(2) wt% of
elemental Ni was included as a secondary phase.

Figure 3: Neutron powder diffraction. The data is taken from mixed samples
reduced in the range of 320-376 h. The Rietfeld refinement includes the LaNiO2
phase and a Ni minority phase. Data were collected at 1.5 K with a neutron
wavelength of 1.15 Å.

Figure. 4 shows a Fourier map of the scattering density resulting from the
Rietveld refinement of the neutron data, where the range of the color scheme
is chosen to focus on the negative scattering lenght of hydrogen. While the
employed reduction temperatures (see Methods) are too low to enable
significant mobility of the La and Ni ions, the purpose of our topotactic
treatment is to reduce oxygen, which becomes mobile and is extracted by
reaction with hydrogen. We therefore focus on the scattering lengths of the
two elements O and H. According to the theoretical prediction of Ref. Si  _et
al._ (2020), intercalation of H in IL nickelates is most favorable on the
vacant apical oxygen sites with Wyckoff position (1b), or on interstitial
positions between the rare-earth ions (Wyckoff positions (2e) and (1c)).
Another theoretical work Malyi  _et al._ (2021) predicts that NdNiO2 is
unstable while NdNiO2H is not, and also proposes the apical oxygen site with
Wyckoff position (1b) for hydrogen. However, as shown in the Fourier maps of
Fig. 4, we find no significant negative scattering, which would appear as a
blue color in our plots. The slightly negative scattering realized as a ring
around a reflex, here in close proximity to La is an artifact from the Fourier
transformation, typically seen around larger ions Sano-Furukawa  _et al._
(2018), due to a signal cutoff from a finite q range. This is best seen in the
Fourier maps Fig. 4 E. We note, that the data can be refined with similar
reliability factors as in Tab. 1 by putting a mixture of O and H on Wyckoff
position (1b) and constraining their occupation. As the positive scattering
observed at this site could be a mixture of negative scattering H and positive
scattering O, which effectively clouds the negative scattering. Realized in a
substitution series with LaNiO3-xHx, which would yield occupations of O:
0.430(8) and H: 0.570(8) in refinements of our data, but this oxygen content
is way higher than what can be found in multiple gas extraction experiments
(La0.99(1)Ni0.99(1)O2.00(2)H0.39(2)). Most importantly the scenario of
LaNiO3-xHx would hint towards the possibility to synthesize the phase
directly, similar as observed in the iron pnictides. However, such synthesis
attempts have been carried out as described below and were not successful.
Furthermore, in the case of iron pnictides the hydrogen substitution leads to
no structural transitions Hanna  _et al._ (2011); Hiraishi  _et al._ (2014);
Matsuishi  _et al._ (2012) contrasting the evolution of the underlying
structural transitions in LaNiO3-x according to the PXRD data (see Fig.1) and
literature Hayward  _et al._ (1999). Another possibility to refine mixed
occupancy is by constraining an equal occupation of O and H as LaNiO2(OH)x,
which converges to 0.18(4). However, this would rather suggest an OH molecule
with typical distances around 0.84 Å, which would realize negative scattering
Sano-Furukawa  _et al._ (2018) and thus reduces the quality of the fit and
shifts the occupation again down to 0.06(1). Thus, we conclude that
intercalated hydrogen is unlikely in our LaNiO2+δ powder sample. Nevertheless,
we find the presence of a small amount of residual apical oxygen in Wyckoff
position (1b), which corresponds to a deviation $\delta=0.06(1)$ from the
ideal stoichiometry.

Figure 4: Fourier maps of the scattering-length density within the unit cell
of LaNiO2 extracted from neutron powder diffraction. A, B Refined unit cells
with cuts corresponding to the Fourier maps in C, D, with the intensity range
focused on the negative scattering-length (yellow to blue), with -3.74 fm
corresponding to hydrogen Sears (1992). The positive scattering (orange to
red) is capped at 1 (with 5.8 fm corresponding to oxygen). The black lines
mark the isosurface lines with zero scattering intensity. The theoretically
proposed hydrogen positions Si  _et al._ (2020); Malyi  _et al._ (2021) are
highlighted with blue dashed circles. E, F Fourier maps shown as a 3D
colormap.

Additionally, we attempted direct synthesis approaches of LaNiO3-xHx as e.g.
LaNiO2H via a Walker-type high pressure synthesis in NaCl crucibles. We mixed
La2O3 and NiO with NaH or CaH2 and heated it to $650^{\circ}$C -
$1000^{\circ}$C under a pressure of 5-7 GPa, but were unable to synthesize any
oxyhydride nickelate. Instead we obtained La(OH)3 and Ni, which is in contrast
to related cases, such as SmFeAsO1-xHx Hanna  _et al._ (2011), BaCrO2H Higashi
_et al._ (2021), BaScO2H Goto  _et al._ (2017), and SrVO2H Yamamoto  _et al._
(2017).

##  IV Conclusion

In summary, we have synthesized high quality LaNiO2+δ powders and found some
density of residual oxygen ($\delta=0.06(1)$) even at the final stage of the
topotactic reaction, just prior to decomposition of our samples. While our gas
extraction method reveals partial hydrogen inclusions in the powder samples,
high resolution neutron diffraction refinements show that there is no clear
topotactic hydrogen in LaNiO2, and direct attempts to synthesize LaNiO3-xHx
were unsuccessful. The hydrogen detected by gas extraction could be trapped in
Ni impurities that increase with increasing reduction time and/or ascribed to
phenomena at surfaces or grain boundaries such as water
adsorption/intercalation Baeumer  _et al._ (2021) on powder surfaces, or
partial formation of LaNiO3-xHx on the nm scale Onozuka  _et al._ (2016).
Notably, a relatively high density of grain boundaries/crystallographic
defects was reported for IL nickelate thin films Lee  _et al._ (2020), and
also surface effects can play a more decisive role in films. Thus, future
studies clarifying the presence and the impact of hydrogen in nickelate thin
film samples are highly desirable.

## Data availability statement

The original contributions presented in the study are included in the article;
further inquiries can be directed to the corresponding author.

## Author Contributions

PP, MI, BK, and MH conceived the project. RO carried out the topotactic
reductions. SH executed the ICP and gas extractions. PP and VP conducted the
neutron diffraction experiments. PP analyzed the data and prepared the
manuscript with input from all authors.

## Funding

We acknowledge financial support by the Center for Integrated Quantum Science
and Technology (IQST) and the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation): Projektnummer 107745057 - TRR 80. The Max Planck Society
is acknowledged for funding of the open access fee.

## Acknowledgements

We thank R. Merkle and A. Fuchs for the synthesis of LaNiO3 and H. Hoier for
preliminary PXRD characterizations. We acknowledge PXRD measurements by C.
Stefani from the X-ray Diffraction Scientific Facility at an early stage of
this work and S. Hammoud for carrying out the detailed ICP and gas extraction
measurements. The use of facilities of the Quantum Materials Department of H.
Takagi for the attempted high pressure synthesis of LaNiO2H is gratefully
acknowledged.

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Hence, applying Lemma 4.8, Proposition 2.10 and Corollary 2.12, we get

$\liminf_{\beta\downarrow
1}\,(\beta-1)^{2}\mathbb{E}_{\beta}\,\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\frac{S^{\prime}_{\beta}}{S_{\beta}}}\right)^{2}}\right]>1.$
(4.15)

Combining Equations (4.13), (4.14) and (4.15) yields

$0<\liminf_{\beta\downarrow
1}(\beta-1)^{2}\operatorname{Var}_{\beta}\mathopen{}\mathclose{{}\left(\frac{S^{\prime}_{\beta}}{S_{\beta}}}\right)\leq\limsup_{\beta\downarrow
1}(\beta-1)^{2}\operatorname{Var}_{\beta}\mathopen{}\mathclose{{}\left(\frac{S^{\prime}_{\beta}}{S_{\beta}}}\right)\leq
1.$ (4.16)

Finally, for the second term on the right-hand side of Equation (4.11), by
Cauchy–Schwarz inequality and then Equations (4.12) and (4.16), we get

$\mathopen{}\mathclose{{}\left\lvert\operatorname{Cov}_{\beta}\mathopen{}\mathclose{{}\left(\log
S_{\beta},\frac{S^{\prime}_{\beta}}{S_{\beta}}}\right)}\right\rvert\leq\mathopen{}\mathclose{{}\left(\operatorname{Var}_{\beta}\mathopen{}\mathclose{{}\left(\log
S_{\beta}}\right)\operatorname{Var}_{\beta}\mathopen{}\mathclose{{}\left(\frac{S^{\prime}_{\beta}}{S_{\beta}}}\right)}\right)^{1/2}=O\mathopen{}\mathclose{{}\left(\frac{\log\,\frac{1}{\beta-1}}{\beta-1}}\right),$

which proves that this term is negligible in Equation (4.11) and concludes the
proof. ∎

######  Lemma 4.8.

We have

$\liminf_{\beta\downarrow
1}\,(\beta-1)^{3}\,\mathbb{E}\mathopen{}\mathclose{{}\left[\frac{(S^{\prime}_{\beta})^{2}}{S_{\beta}}}\right]>C_{\star}.$

######  Remark 4.9.

A lower bound with a weak inequality could be easily obtained via
Cauchy–Schwarz inequality:

$\mathbb{E}\mathopen{}\mathclose{{}\left[\frac{(S^{\prime}_{\beta})^{2}}{S_{\beta}}}\right]\geq\frac{\mathbb{E}[S_{\beta}^{\prime}]^{2}}{\mathbb{E}\mathopen{}\mathclose{{}\left[S_{\beta}}\right]}\sim\frac{C_{\star}}{(\beta-1)^{3}},$

using Lemma 4.7 and Proposition 2.10. Equality at the first order in this
inequality would suggest that $S^{\prime}_{\beta}$ and $S_{\beta}$ are
colinear at first order (on the events that are significant for the first
moment). Therefore, the idea of the proof below is to find an event such that
the first moments of $S^{\prime}_{\beta}$ and $S_{\beta}$ given this event
have a different ratio than the one for the non-conditional first moments.

###### Proof.

Step 1: Working at finite $t$. Fix some $\beta>1$. Setting
$f_{\beta}(x)\coloneqq\mathrm{e}^{\beta\sqrt{2}x}$ and
$\partial_{\beta}f_{\beta}(x)\coloneqq\sqrt{2}x\mathrm{e}^{\beta\sqrt{2}x}$,
note that $S_{\beta}=\mathcal{C}(f_{\beta})$ and
$S_{\beta}^{\prime}=\mathcal{C}(\partial_{\beta}f_{\beta})$. Our first aim in
this step is to show

$\mathbb{E}\mathopen{}\mathclose{{}\left[\frac{(S^{\prime}_{\beta})^{2}}{S_{\beta}}}\right]=\lim_{t\to\infty}\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\frac{(\mathcal{C}_{t,r_{t}}^{*}(\partial_{\beta}f_{\beta}))^{2}}{\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})}}\right].$
(4.17)

For $K>0$, let $\chi_{K}$ denote a continuous function such that
$\mathbbm{1}_{[-K,K]}\leq\chi_{K}\leq\mathbbm{1}_{[-(K+1),K+1]}$. It follows
from the vague convergence stated in Equation (2.7) that

$\mathopen{}\mathclose{{}\left(\mathcal{C}_{t,r_{t}}^{*}(\chi_{K}f_{\beta}),\mathcal{C}_{t,r_{t}}^{*}(\chi_{K}\partial_{\beta}f_{\beta})}\right)\text{
under
}\widetilde{\mathbb{P}}_{t}\quad\xrightarrow[t\to\infty]{\text{(d)}}\quad\mathopen{}\mathclose{{}\left(\mathcal{C}(\chi_{K}f_{\beta}),\mathcal{C}(\chi_{K}\partial_{\beta}f_{\beta})}\right)\,\text{
under }\mathbb{P}.$

Moreover, recalling $\mathcal{C}_{t,r_{t}}^{*}$ is supported on $(-\infty,0]$
under $\widetilde{\mathbb{P}}_{t}$ we get

$\mathopen{}\mathclose{{}\left\lvert\frac{\mathcal{C}_{t,r_{t}}^{*}(\chi_{K}\partial_{\beta}f_{\beta})^{2}}{\mathcal{C}_{t,r_{t}}^{*}(\chi_{K}f_{\beta})}}\right\rvert\leq
2(K+1)^{2}\mathcal{C}_{t,r_{t}}^{*}(\chi_{K}f_{\beta})\leq
2(K+1)^{2}\mathcal{C}_{t,r_{t}}^{*}([-(K+1),0])\,,$

which is bounded in $L^{2}$ by Lemma 2.14. Hence,
$(\mathcal{C}_{t,r_{t}}^{*}(\chi_{K}\partial_{\beta}f_{\beta})^{2}/\mathcal{C}_{t,r_{t}}^{*}(\chi_{K}f_{\beta}))_{t}$
is uniformly integrable and we get

$\mathbb{E}\mathopen{}\mathclose{{}\left[\frac{\mathcal{C}(\chi_{K}\partial_{\beta}f_{\beta})^{2}}{\mathcal{C}(\chi_{K}f_{\beta})}}\right]=\lim_{t\to\infty}\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\frac{\mathcal{C}_{t,r_{t}}^{*}(\chi_{K}\partial_{\beta}f_{\beta})^{2}}{\mathcal{C}_{t,r_{t}}^{*}(\chi_{K}f_{\beta})}}\right].$
(4.18)

We now control the differences between the expectations in Equation (4.17) and
Equation (4.18) and show they are small when $K$ is large. We have

$\displaystyle\mathopen{}\mathclose{{}\left\lvert\frac{\mathcal{C}(\partial_{\beta}f_{\beta})^{2}}{\mathcal{C}(f_{\beta})}-\frac{\mathcal{C}(\chi_{K}\partial_{\beta}f_{\beta})^{2}}{\mathcal{C}(\chi_{K}f_{\beta})}}\right\rvert$
$\displaystyle\leq\frac{\mathopen{}\mathclose{{}\left\lvert\mathcal{C}(\partial_{\beta}f_{\beta})^{2}-\mathcal{C}(\chi_{K}\partial_{\beta}f_{\beta})^{2}}\right\rvert}{\mathcal{C}(f_{\beta})}+\mathcal{C}(\chi_{K}\partial_{\beta}f_{\beta})^{2}\mathopen{}\mathclose{{}\left\lvert\frac{1}{\mathcal{C}(f_{\beta})}-\frac{1}{\mathcal{C}(\chi_{K}f_{\beta})}}\right\rvert$
$\displaystyle\leq
2\,\mathcal{C}(\partial_{\beta}f_{\beta})\,\mathcal{C}\mathopen{}\mathclose{{}\left(\mathbbm{1}_{(-\infty,-K]}\,\partial_{\beta}f_{\beta}}\right)+(K+1)^{2}\mathcal{C}\mathopen{}\mathclose{{}\left(\mathbbm{1}_{(-\infty,-K]}\,\partial_{\beta}f_{\beta}}\right),$
(4.19)

where, for the first term, we used $\mathcal{C}(f_{\beta})\geq 1$ and the fact
that $\partial_{\beta}f_{\beta}$ is of constant sign on $(-\infty,0]$ which is
the support of $\mathcal{C}$ and, for the second term, we used
$\mathopen{}\mathclose{{}\left\lvert\mathcal{C}(\chi_{K}\partial_{\beta}f_{\beta})}\right\rvert/\mathcal{C}(f_{\beta})\leq\mathopen{}\mathclose{{}\left\lvert\mathcal{C}(\chi_{K}\partial_{\beta}f_{\beta})}\right\rvert/\mathcal{C}(\chi_{K}f_{\beta})\leq
K+1$. Then, writing
$\partial_{\beta}f_{\beta}(x)=\int_{-\infty}^{x}\sqrt{2}(\beta\sqrt{2}y+1)\mathrm{e}^{\beta\sqrt{2}y}\mathop{}\mathopen{}\mathrm{d}y$
and using Fubini’s theorem, we have

$\displaystyle\mathbb{E}\mathopen{}\mathclose{{}\left[\mathcal{C}(\partial_{\beta}f_{\beta})\mathcal{C}(\mathbbm{1}_{(-\infty,-K]}\partial_{\beta}f_{\beta})}\right]$
$\displaystyle=\int_{-\infty}^{0}\int_{-\infty}^{-K}\mathbb{E}\mathopen{}\mathclose{{}\left[\mathcal{C}([y,0])\,\mathcal{C}([z,-K])}\right]\,2(\beta\sqrt{2}z+1)\,\mathrm{e}^{\beta\sqrt{2}z}(\beta\sqrt{2}y+1)\mathrm{e}^{\beta\sqrt{2}y}\mathop{}\mathopen{}\mathrm{d}z\mathop{}\mathopen{}\mathrm{d}y$
$\displaystyle\leq C(\beta)(K+1)^{3/2}\mathrm{e}^{-(\beta-1)\sqrt{2}K},$

where we bounded the last expectation by $(\mathopen{}\mathclose{{}\left\lvert
y}\right\rvert+1)^{1/2}(\mathopen{}\mathclose{{}\left\lvert
z}\right\rvert+1)^{1/2}\mathrm{e}^{-\sqrt{2}(y+z)}$ using Cauchy–Schwarz
inequality and Equation (2.9), and where $C(\beta)$ denotes a constant
depending only on $\beta$ and which can change from line to line. Similarly,
we have
$\mathbb{E}[\mathcal{C}(\mathbbm{1}_{(-\infty,-K]}\partial_{\beta}f_{\beta})]\leq
C(\beta)(K+1)\mathrm{e}^{-(\beta-1)\sqrt{2}K}$ by using Equation (2.10).
Coming back to Equation (4.19), we get

$\mathopen{}\mathclose{{}\left\lvert\mathbb{E}\mathopen{}\mathclose{{}\left[\frac{\mathcal{C}(\partial_{\beta}f_{\beta})^{2}}{\mathcal{C}(f_{\beta})}}\right]-\mathbb{E}\mathopen{}\mathclose{{}\left[\frac{\mathcal{C}(\chi_{K}\partial_{\beta}f_{\beta})^{2}}{\mathcal{C}(\chi_{K}f_{\beta})}}\right]}\right\rvert\leq
C(\beta)(K+1)^{3}\mathrm{e}^{-(\beta-1)\sqrt{2}K},$

and the same holds true for $\mathcal{C}_{t,r_{t}}^{*}$ instead of
$\mathcal{C}$, uniformly in $t\geq 1$, by replacing Equations (2.9) and (2.10)
by Lemma 2.14 in the proof. Therefore, letting $K\to\infty$ in Equation (4.18)
yields Equation (4.17).

Step 2: Using Cauchy–Schwarz inequality conditionally on a well-chosen event.
We fix some parameters $a>0$ and $0<b<B$. For any $\beta>1$, letting
$r=r(\beta)\coloneqq a(\beta-1)^{-2}$, and for any $t>r$, we introduce the
event

$B_{r,t}=\mathopen{}\mathclose{{}\left\\{h_{t-r}(X_{t-r})-m_{t}+m_{r}\in[-B\sqrt{r},-b\sqrt{r}]}\right\\}.$
(4.20)

Then, we use Cauchy–Schwarz inequality given $B_{r,t}$ or given $B_{r,t}^{c}$
to get

$\displaystyle\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\frac{(\mathcal{C}_{t,r_{t}}^{*}(\partial_{\beta}f_{\beta}))^{2}}{\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})}}\right]$
$\displaystyle=\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\frac{(\mathcal{C}_{t,r_{t}}^{*}(\partial_{\beta}f_{\beta}))^{2}}{\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})}\mathrel{}\middle|\mathrel{}B_{r,t}}\right]\widetilde{\mathbb{P}}_{t}\mathopen{}\mathclose{{}\left(B_{r,t}}\right)+\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\frac{(\mathcal{C}_{t,r_{t}}^{*}(\partial_{\beta}f_{\beta}))^{2}}{\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})}\mathrel{}\middle|\mathrel{}B_{r,t}^{c}}\right]\widetilde{\mathbb{P}}_{t}\mathopen{}\mathclose{{}\left(B_{r,t}^{c}}\right)$
$\displaystyle\geq\frac{\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(\partial_{\beta}f_{\beta})\mathrel{}\middle|\mathrel{}B_{r,t}}\right]^{2}}{\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})\mathrel{}\middle|\mathrel{}B_{r,t}}\right]}\,\widetilde{\mathbb{P}}_{t}\mathopen{}\mathclose{{}\left(B_{r,t}}\right)+\frac{\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(\partial_{\beta}f_{\beta})\mathrel{}\middle|\mathrel{}B_{r,t}^{c}}\right]^{2}}{\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})\mathrel{}\middle|\mathrel{}B_{r,t}^{c}}\right]}\,\widetilde{\mathbb{P}}_{t}\mathopen{}\mathclose{{}\left(B_{r,t}^{c}}\right)$
$\displaystyle=\frac{\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(\partial_{\beta}f_{\beta})\mathbbm{1}_{B_{r,t}}}\right]^{2}}{\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})\mathbbm{1}_{B_{r,t}}}\right]}+\frac{\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(\partial_{\beta}f_{\beta})\mathbbm{1}_{B_{r,t}^{c}}}\right]^{2}}{\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})\mathbbm{1}_{B_{r,t}^{c}}}\right]},$
(4.21)

which yields a lower bound for the right-hand side of Equation (4.17), that we
now have to estimate.

Step 3: Estimating the expectations. Let $\varepsilon>0$. Our aim in this step
consists in proving that, for $\beta>1$ close enough to 1, there exists
$t_{0}>r$ such that, for any $t\geq t_{0}$,

$\displaystyle\mathopen{}\mathclose{{}\left\lvert(\beta-1)\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})\mathbbm{1}_{B_{r,t}}}\right]-C_{\star}\kappa}\right\rvert\leq\varepsilon,$
$\displaystyle\mathopen{}\mathclose{{}\left\lvert(\beta-1)^{2}\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(\partial_{\beta}f_{\beta})\mathbbm{1}_{B_{r,t}}}\right]+C_{\star}\kappa^{\prime}}\right\rvert\leq\varepsilon,$
(4.22)

where $\kappa,\kappa^{\prime}$ are constants depending only on the parameters
$a,b,B$ defined as follows

$\displaystyle\kappa$
$\displaystyle\coloneqq\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\varphi_{b,B}\mathopen{}\mathclose{{}\left(\frac{a\vee
w}{\mathopen{}\mathclose{{}\left\lvert
a-w}\right\rvert}}\right)\mathopen{}\mathclose{{}\left(\int_{0}^{\infty}u\mathrm{e}^{-u-u^{2}/(4w)}\mathop{}\mathopen{}\mathrm{d}u}\right)\frac{\mathop{}\mathopen{}\mathrm{d}w}{w^{3/2}}\,,$
$\displaystyle\kappa^{\prime}$
$\displaystyle\coloneqq\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\varphi_{b,B}\mathopen{}\mathclose{{}\left(\frac{a\vee
w}{\mathopen{}\mathclose{{}\left\lvert
a-w}\right\rvert}}\right)\mathopen{}\mathclose{{}\left(\int_{0}^{\infty}u^{2}\mathrm{e}^{-u-u^{2}/(4w)}\mathop{}\mathopen{}\mathrm{d}u}\right)\frac{\mathop{}\mathopen{}\mathrm{d}w}{w^{3/2}}\,,$
(4.23)

with $\varphi_{b,B}$ defined by

$\varphi_{b,B}(v)\coloneqq\int_{b\sqrt{v}}^{B\sqrt{v}}\sqrt{\frac{2}{\pi}}\,y^{2}\mathrm{e}^{-y^{2}/2}\mathop{}\mathopen{}\mathrm{d}y\,,\qquad
v\geq 0.$ (4.24)

Note that $\kappa,\kappa^{\prime}\in(0,1)$ and they tend to $1$ as $b\to 0$
and $B\to\infty$ (which means $\mathbbm{1}_{B_{r,t}}\to 1$). To prove the
first inequality in Equation (4.22), we write

$\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})\mathbbm{1}_{B_{r,t}}}\right]=\int_{0}^{\infty}\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}([-x,0])\mathbbm{1}_{B_{r,t}}}\right]\beta\sqrt{2}\,\mathrm{e}^{-\beta\sqrt{2}x}\mathop{}\mathopen{}\mathrm{d}x.$

Then, we apply Lemma 4.10 for some $\theta\in(0,1)$ and with $\varepsilon>0$
introduced earlier to estimate the part
$x\in[\theta\sqrt{r},\theta^{-1}\sqrt{r}]$ of the integral and use Lemma 2.14
to bound the remaining part. Note also that $\varphi_{b,B}\leq 1$ and
$\int_{0}^{\infty}\frac{x\mathrm{e}^{-x^{2}/(2s)}}{\sqrt{2\pi}s^{3/2}}\mathop{}\mathopen{}\mathrm{d}s=1$.
Therefore, for $\beta>1$ close enough to 1 (equivalently $r$ large enough),
there exists $t_{0}>r$ such that, for any $t\geq t_{0}$,

$\displaystyle\mathopen{}\mathclose{{}\left\lvert\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})\mathbbm{1}_{B_{r,t}}}\right]-C_{\star}\int_{0}^{\infty}\mathopen{}\mathclose{{}\left(\int_{0}^{\infty}\varphi_{b,B}\mathopen{}\mathclose{{}\left(\frac{r\vee
s}{\mathopen{}\mathclose{{}\left\lvert
r-s}\right\rvert}}\right)\frac{x\mathrm{e}^{-x^{2}/(2s)}}{\sqrt{2\pi}s^{3/2}}\mathop{}\mathopen{}\mathrm{d}s}\right)\beta\sqrt{2}\,\mathrm{e}^{-(\beta-1)\sqrt{2}x}\mathop{}\mathopen{}\mathrm{d}x}\right\rvert$
$\displaystyle\leq\varepsilon\int_{0}^{\infty}\beta\sqrt{2}\mathrm{e}^{-(\beta-1)\sqrt{2}x}\mathop{}\mathopen{}\mathrm{d}x+C\int_{[\theta\sqrt{r},\theta^{-1}\sqrt{r}]^{c}}\beta\sqrt{2}\,\mathrm{e}^{-(\beta-1)\sqrt{2}x}\mathop{}\mathopen{}\mathrm{d}x$
$\displaystyle=\frac{\beta}{\beta-1}\mathopen{}\mathclose{{}\left(\varepsilon+C\mathopen{}\mathclose{{}\left(1-\mathrm{e}^{-\theta\sqrt{a}}+\mathrm{e}^{-\theta^{-1}\sqrt{a}}}\right)}\right),$
(4.25)

recalling that $r=a(\beta-1)^{-2}$. Choosing $\theta$ small enough and
considering $\beta<2$, the right-hand side of Equation (4.25) is smaller that
$3\varepsilon/(\beta-1)$. Then, we rewrite the double integral on the left-
hand side of Equation (4.25) by using Fubini’s theorem and changing variables
with $u=(\beta-1)\sqrt{2}x$ and $w=s(\beta-1)^{2}$, which shows that this
double integral equals $\beta\kappa/(\beta-1)$. This proves the first
inequality in Equation (4.22) (with $4\varepsilon$ instead of $\varepsilon$).
The second inequality is proved by writing

$\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})\mathbbm{1}_{B_{r,t}}}\right]=-\int_{0}^{\infty}\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}([-x,0])\mathbbm{1}_{B_{r,t}}}\right]\sqrt{2}(\beta\sqrt{2}x-1)\mathrm{e}^{-\beta\sqrt{2}x}\mathop{}\mathopen{}\mathrm{d}x$

and then proceeding similarly (note that $\beta\sqrt{2}x-1$ can be replaced by
$\beta\sqrt{2}x$ up to a negligible error as $\beta\downarrow 1$).

Step 4: Conclusion. We first claim that for $\theta\in(0,1)$ and $\epsilon>0$,
there exists $r_{0}>0$ such that, for any $r\geq r_{0}$, there exists
$t_{0}>0$ such that, for any $t\geq t_{0}$ and any
$v\in[\theta\sqrt{r},\theta^{-1}\sqrt{r}]$,

$\mathopen{}\mathclose{{}\left\lvert\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}([-v,0])}\right]-C_{\star}\mathrm{e}^{\sqrt{2}v}}\right\rvert\leq\varepsilon\,\mathrm{e}^{\sqrt{2}v}.$

This is a slightly stronger version of [21, Lemma 5.2] where no uniformity for
$v\in[\theta\sqrt{r},\theta^{-1}\sqrt{r}]$ is stated: however, the
aforementioned claim follows from their proof (more precisely, it follows from
[21, Lemma 5.6] in the same way as Lemma 4.10 below follows from Lemma A.1).
Then, we deduce from this claim that, for $\beta>1$ close enough to 1, up to a
modification of $t_{0}$, we also have, for any $t\geq t_{0}$,

$\displaystyle\mathopen{}\mathclose{{}\left\lvert(\beta-1)\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})}\right]-C_{\star}}\right\rvert\leq\varepsilon,$
$\displaystyle\mathopen{}\mathclose{{}\left\lvert(\beta-1)^{2}\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(\partial_{\beta}f_{\beta})}\right]+C_{\star}}\right\rvert\leq\varepsilon,$

where these inequalities are obtained in the same way as Equations (4.22) have
been obtained from Lemma 4.10 in Step 3. Combining this with Equations (4.17),
(4.21) and (4.22), we get

$\mathbb{E}\mathopen{}\mathclose{{}\left[\frac{(S^{\prime}_{\beta})^{2}}{S_{\beta}}}\right]\geq\frac{1}{(\beta-1)^{3}}\mathopen{}\mathclose{{}\left(\frac{(C_{\star}\kappa^{\prime}-\varepsilon)^{2}}{C_{\star}\kappa+\varepsilon}+\frac{(C_{\star}(1-\kappa^{\prime})-\varepsilon)^{2}}{C_{\star}(1-\kappa)+\varepsilon}}\right).$

Letting $\varepsilon\to 0$, this proves

$\liminf_{\beta\downarrow
1}(\beta-1)^{3}\,\mathbb{E}\mathopen{}\mathclose{{}\left[\frac{(S^{\prime}_{\beta})^{2}}{S_{\beta}}}\right]\geq
C_{\star}\mathopen{}\mathclose{{}\left(\frac{(\kappa^{\prime})^{2}}{\kappa}+\frac{(1-\kappa^{\prime})^{2}}{1-\kappa}}\right).$
(4.26)

By Lemma 4.11, we can choose $a,b,B$ such that $\kappa\neq\kappa^{\prime}$
and, together with the fact that $\kappa,\kappa^{\prime}\in(0,1)$, this
implies that the right-hand side of Equation (4.26) is larger than
$C_{\star}$. ∎

The proof of the following lemma is postponed to Subsection A.2.

######  Lemma 4.10.

Let $\theta\in(0,1)$, $\varepsilon,a>0$ and $0<b<B$. There exists $r_{0}>0$
such that, for any $r\geq r_{0}$, there exists $t_{0}>0$ such that, for any
$t\geq t_{0}$ and any $v\in[\theta\sqrt{r},\theta^{-1}\sqrt{r}]$,

$\mathopen{}\mathclose{{}\left\lvert\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}([-v,0])\mathbbm{1}_{B_{r,t}}}\right]-C_{\star}\mathrm{e}^{\sqrt{2}v}\int_{0}^{\infty}\varphi_{b,B}\mathopen{}\mathclose{{}\left(\frac{r\vee
s}{\mathopen{}\mathclose{{}\left\lvert
r-s}\right\rvert}}\right)\frac{v\mathrm{e}^{-v^{2}/(2s)}}{\sqrt{2\pi}s^{3/2}}\mathop{}\mathopen{}\mathrm{d}s}\right\rvert\leq\varepsilon\mathrm{e}^{\sqrt{2}v}.$

######  Lemma 4.11.

Recall the definition of $\kappa$ and $\kappa^{\prime}$ in Equation (4.23).
There exist $a>0$ and $0<b<B$ such that $\kappa\neq\kappa^{\prime}$.

###### Proof.

We proceed by contradiction: assume that, for any $a>0$ and $0<b<B$,
$\kappa=\kappa^{\prime}$. Then, differentiating the relation
$\kappa=\kappa^{\prime}$ w.r.t. $B$, we get, for any $a,B>0$, the following
quantity is the same for $k=1$ and $k=2$:

$\displaystyle\int_{0}^{\infty}\frac{a\vee
w}{\mathopen{}\mathclose{{}\left\lvert
a-w}\right\rvert}\exp\mathopen{}\mathclose{{}\left(-\frac{B^{2}}{2}\frac{a\vee
w}{\mathopen{}\mathclose{{}\left\lvert
a-w}\right\rvert}}\right)\mathopen{}\mathclose{{}\left(\int_{0}^{\infty}u^{k}\mathrm{e}^{-u-u^{2}/(4w)}\mathop{}\mathopen{}\mathrm{d}u}\right)\,\frac{\mathop{}\mathopen{}\mathrm{d}w}{w^{3/2}}.$

On the other hand, a direct calculation shows that, for $k\in\\{1,2\\}$,

$\displaystyle\int_{0}^{\infty}\mathopen{}\mathclose{{}\left(\int_{0}^{\infty}u^{k}\mathrm{e}^{-u-u^{2}/(4w)}\mathop{}\mathopen{}\mathrm{d}u}\right)\,\frac{\mathop{}\mathopen{}\mathrm{d}w}{w^{3/2}}=2\sqrt{\pi}$

and therefore is the same for $k=1$ and $k=2$. Therefore, we deduce that the
following quantity is the same for $k=1$ and $k=2$:

$\int_{0}^{\infty}\mathopen{}\mathclose{{}\left(\exp\mathopen{}\mathclose{{}\left(-\frac{B^{2}}{2}\frac{a\vee
w}{\mathopen{}\mathclose{{}\left\lvert
a-w}\right\rvert}}\right)-\mathrm{e}^{-B^{2}/2}}\right)\mathopen{}\mathclose{{}\left(\int_{0}^{\infty}u^{k}\mathrm{e}^{-u-u^{2}/(4w)}\mathop{}\mathopen{}\mathrm{d}u}\right)\,\frac{\mathop{}\mathopen{}\mathrm{d}w}{w^{3/2}}.$
(4.27)

Our goal is now to study the behavior as $a\to 0$ of this quantity, to find a
contradiction. We decompose the main integral in Equation (4.27) into a part
$w\in[0,a)$ in which we change $w$ to $x=a/(a-w)$ and $u$ to
$v=u(x/(a(x-1)))^{1/2}$, and a part $w\in(a,\infty)$ in which we change $w$ to
$x=w/(w-a)$ and $u$ to $v=u((x-1)/(ax))^{1/2}$. This yields that the following
quantity is the same for $k=1$ and $k=2$:

$\begin{split}&a^{k/2}\int_{1}^{\infty}\mathopen{}\mathclose{{}\left(x\mathrm{e}^{-xB^{2}/2}-\mathrm{e}^{-B^{2}/2}}\right)\Biggl{(}\frac{(x-1)^{k/2-1}}{x^{k/2+1}}\int_{0}^{\infty}v^{k}\mathrm{e}^{-v^{2}/4-v(\frac{a(x-1)}{x})^{1/2}}\mathop{}\mathopen{}\mathrm{d}v\\\
&\hskip
156.49014pt{}+\frac{x^{k/2-1}}{(x-1)^{k/2+1}}\int_{0}^{\infty}v^{k}\mathrm{e}^{-v^{2}/4-v(\frac{ax}{x-1})^{1/2}}\mathop{}\mathopen{}\mathrm{d}v\Biggr{)}\mathop{}\mathopen{}\mathrm{d}x.\end{split}$
(4.28)

For $k=1$, both integrals w.r.t. $v$ in (4.28) converge as $a\to 0$ towards
$\int_{0}^{\infty}v\mathrm{e}^{-v^{2}/4}\mathop{}\mathopen{}\mathrm{d}v=2$ by
dominated convergence. Hence, by dominated convergence again but in the
integral w.r.t. $x$, (4.28) equals

$2a^{1/2}\int_{1}^{\infty}\mathopen{}\mathclose{{}\left(x\mathrm{e}^{-xB^{2}/2}-\mathrm{e}^{-B^{2}/2}}\right)\Biggl{(}\frac{(x-1)^{-1/2}}{x^{3/2}}+\frac{x^{-1/2}}{(x-1)^{3/2}}\Biggr{)}\mathop{}\mathopen{}\mathrm{d}x+o(a^{1/2}),$

as $a\to 0$. For $k=2$, we cannot use the same argument: the second domination
cannot be justified. Instead, we bound the first integral w.r.t. $v$ by a
constant and, for the second one, for some $\varepsilon>0$, we write

$\int_{0}^{\infty}v^{2}\mathrm{e}^{-v^{2}/4-v(\frac{ax}{x-1})^{1/2}}\mathop{}\mathopen{}\mathrm{d}v=\mathopen{}\mathclose{{}\left(\frac{x-1}{ax}}\right)^{3\varepsilon}\int_{0}^{\infty}u^{2}\mathrm{e}^{-u^{2}(\frac{x-1}{ax})^{2\varepsilon}/4-u(\frac{ax}{x-1})^{1/2-\varepsilon}}\mathop{}\mathopen{}\mathrm{d}u\leq
C\mathopen{}\mathclose{{}\left(\frac{x-1}{ax}}\right)^{3\varepsilon},$

by bounding the last integral by
$\int_{0}^{\infty}u^{2}\mathrm{e}^{-u^{2}/4}\mathop{}\mathopen{}\mathrm{d}u$
if $(x-1)/ax\geq 1$ and by
$\int_{0}^{\infty}u^{2}\mathrm{e}^{-u}\mathop{}\mathopen{}\mathrm{d}u$
otherwise. With $\varepsilon<1/6$, this proves that (4.28) is a
$o\mathopen{}\mathclose{{}\left(a^{1/2}}\right)$ for $k=2$. Since (4.28) is
the same for $k=1$ and $k=2$, this implies

$\int_{1}^{\infty}\mathopen{}\mathclose{{}\left(x\mathrm{e}^{-xB^{2}/2}-\mathrm{e}^{-B^{2}/2}}\right)\Biggl{(}\frac{(x-1)^{-1/2}}{x^{3/2}}+\frac{x^{-1/2}}{(x-1)^{3/2}}\Biggr{)}\mathop{}\mathopen{}\mathrm{d}x=0,$
(4.29)

for any $B>0$. But the left-hand side of Equation (4.29) tends to infinity as
$B\to\infty$, so this is a contradiction and concludes the proof. ∎

##  Appendix A Proof of technical results on the decoration of the BBM

This section is dedicated to the proof of several technical results concerning
the decoration of the branching Brownian motion, mostly based on ideas
introduced in [21]. Therefore, we explain how to adapt their argument and use
their notation without introducing it.

###  A.1 Uniform bounds for moments of level sets

###### Proof of Lemma 2.14.

The bound on the second moment is a direct consequence of Lemma A.6, so we
focus here on the first moment. However, it could also be deduced from the
proof of [21, Lemma 5.3], in a similar way as what is done below for the first
moment.

To bound uniformly
$\widetilde{\mathbb{E}}_{t}[\mathcal{C}_{t,r_{t}}^{*}([-v,0])]$, we follow the
proof of [21, Lemma 5.2], which establishes an asymptotic equivalent for the
first moment of $\mathcal{C}_{t,r_{t}}^{*}([-v,0])$ as $t\to\infty$ and then
$v\to\infty$. The proof begins by writing

$\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}([-v,0])}\right]=\frac{\widetilde{\mathbb{E}}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}([-v,0])\mathbbm{1}_{\\{\max_{x\in
L_{t}}h_{t}(x)\leq
m_{t}\\}}\mathrel{}\middle|\mathrel{}h_{t}(X_{t})=m_{t}}\right]}{\widetilde{\mathbb{P}}\mathopen{}\mathclose{{}\left(\,\max_{x\in
L_{t}}h_{t}(x)\leq
m_{t}\mathrel{}\middle|\mathrel{}h_{t}(X_{t})=m_{t}}\right)},$ (A.1)

where the denominator satisfies, for some constant $C_{1}>0$, as $t\to\infty$,

$\widetilde{\mathbb{P}}\mathopen{}\mathclose{{}\left(\max_{x\in
L_{t}}h_{t}(x)\leq
m_{t}\mathrel{}\middle|\mathrel{}h_{t}(X_{t})=m_{t}}\right)\sim\frac{C_{1}}{t},$
(A.2)

by [21, Lemmata 3.1 and 3.4] (the constant $C_{1}$ equals $2f^{(0)}(0)g(0)$
with notation of [21, Lemma 3.4]). In particular, there exists a constant
$c>0$ such that the probability in Equation (A.2) is at least $c/t$ for any
$t\geq 1$. Therefore, it remains to prove that there exist $C>0$ and
$t_{0}\geq 0$ such that, for any $t\geq t_{0}$ and $v\geq 0$,

$\widetilde{\mathbb{E}}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}([-v,0])\mathbbm{1}_{\\{\max_{x\in
L_{t}}h_{t}(x)\leq
m_{t}\\}}\mathrel{}\middle|\mathrel{}h_{t}(X_{t})=m_{t}}\right]\leq\frac{C}{t}\mathrm{e}^{\sqrt{2}v}.$
(A.3)

Indeed, for $t\leq t_{0}$, bounding the indicator function by 1 and
$\mathcal{C}_{t,r_{t}}^{*}([-v,0])\leq\\#L_{t_{0}}$, the left-hand side of
Equation (A.3) is at most

$\widetilde{\mathbb{E}}\mathopen{}\mathclose{{}\left[\\#L_{t_{0}}\mathrel{}\middle|\mathrel{}h_{t}(X_{t})=m_{t}}\right]=\widetilde{\mathbb{E}}\mathopen{}\mathclose{{}\left[\\#L_{t_{0}}}\right]=\int_{0}^{t_{0}}\mathrm{e}^{t_{0}-s}\cdot
2\mathop{}\mathopen{}\mathrm{d}s=C(t_{0}),$ (A.4)

where the first equality follows from the fact that displacement of the spine
and branching of the BBM are independent and the second equality uses the
facts that the spine branches at rate 2, giving birth to standard BBMs and
that a standard BBM has in mean $\mathrm{e}^{r}$ particles at time $r$.

We now focus on proving Equation (A.3). Applying successively [21, Lemmata 5.4
and 5.5] (note that $j_{t,v}(s)=j_{t,v}^{\geq 0}(s)$), we get for any $t$ such
that $r_{t}\leq t/2$ and any $v\geq 0$,

$\displaystyle\widetilde{\mathbb{E}}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}([-v,0])\mathbbm{1}_{\\{\max_{x\in
L_{t}}h_{t}(x)\leq
m_{t}\\}}\mathrel{}\middle|\mathrel{}h_{t}(X_{t})=m_{t}}\right]$
$\displaystyle=2\int_{0}^{r_{t}}j_{t,v}(s)\mathop{}\mathopen{}\mathrm{d}s$
$\displaystyle\leq\frac{C}{t}(v+1)\,\mathrm{e}^{\sqrt{2}v}\int_{0}^{\infty}\frac{\mathrm{e}^{-v^{2}/(16s)}+\mathrm{e}^{-v/2}}{(s+1)\sqrt{s}}\mathop{}\mathopen{}\mathrm{d}s.$

This last integral is smaller than $C((v+1)^{-1}+\mathrm{e}^{-v/2})$ for any
$v\geq 0$ (see [21, Equations (5.44)-(5.45)] for details) and so we get
Equation (A.3). ∎

###  A.2 First moment of level sets on a particular event

We prove in this subsection Lemma 4.10. This is a (non-trivial) refinement of
the proof of [21, Lemma 5.2].

###### Proof of Lemma 4.10.

Recalling the definition of $\widetilde{\mathbb{P}}_{t}$ in Equation (2.6), we
first have

$\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}([-v,0])\mathbbm{1}_{B_{r,t}}}\right]=\frac{\widetilde{\mathbb{E}}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}([-v,0])\mathbbm{1}_{B_{r,t}}\mathbbm{1}_{\\{\max_{x\in
L_{t}}h_{t}(x)\leq
m_{t}\\}}\mathrel{}\middle|\mathrel{}h_{t}(X_{t})=m_{t}}\right]}{\widetilde{\mathbb{P}}\mathopen{}\mathclose{{}\left(\max_{x\in
L_{t}}h_{t}(x)\leq
m_{t}\mathrel{}\middle|\mathrel{}h_{t}(X_{t})=m_{t}}\right)}.$ (A.5)

Recall from Equation (A.2) that the denominator in Equation (A.5) is
asymptotically equivalent to $C_{1}/t$, so we now focus on the numerator.

We introduce the event

$\mathcal{B}_{r,t}\coloneqq\mathopen{}\mathclose{{}\left\\{\widehat{W}_{t,r}\in[-B\sqrt{r},-b\sqrt{r}]}\right\\},$

which is analog to $B_{r,t}$ (see Equation (4.20)) but for the process
$(\widehat{W}_{t,s})_{s\in[0,t]}$ defined in [21, Equation (3.1)] as

$\widehat{W}_{t,s}\coloneqq W_{s}-\gamma_{t,s},\quad\text{with
}\gamma_{t,s}\coloneqq\frac{3}{2\sqrt{2}}\mathopen{}\mathclose{{}\left(\log_{+}s-\frac{s}{t}\log_{+}t}\right),$
(A.6)

for any $0\leq s\leq t$, with $W$ a standard Brownian motion under
$\mathbb{P}$. Defining, for $v\geq 0$ and $0\leq s,r\leq t$,

$j_{t,v,r}(s)\coloneqq\mathbb{E}\mathopen{}\mathclose{{}\left[J_{t,v}(s)\mathbbm{1}_{\mathcal{B}_{r,t}}\middle|\widehat{W}_{t,0}=\widehat{W}_{t,t}=0}\right],$

where $J_{t,v}(s)$ is introduced in [21, Equation (5.9)], it follows from the
proof of [21, Lemma 5.4], that

$\widetilde{\mathbb{E}}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}([-v,0])\mathbbm{1}_{B_{r,t}}\mathbbm{1}_{\\{\max_{x\in
L_{t}}h_{t}(x)\leq
m_{t}\\}}\mathrel{}\middle|\mathrel{}h_{t}(X_{t})=m_{t}}\right]=2\int_{0}^{r_{t}}j_{t,v,r}(s)\mathop{}\mathopen{}\mathrm{d}s.$
(A.7)

As in [21, Equation (5.17)], for any $M\geq 0$, we split $j_{t,v,r}(s)$ into

$\displaystyle j_{t,v,r}^{<M}(s)$
$\displaystyle\coloneqq\mathbb{E}\mathopen{}\mathclose{{}\left[J_{t,v}(s)\mathbbm{1}_{\mathcal{B}_{r,t}}\mathbbm{1}_{\\{\lvert\widehat{W}_{t,s}\rvert<M\\}}\middle|\widehat{W}_{t,0}=\widehat{W}_{t,t}=0}\right],$
$\displaystyle j_{t,v,r}^{\geq M}(s)$
$\displaystyle\coloneqq\mathbb{E}\mathopen{}\mathclose{{}\left[J_{t,v}(s)\mathbbm{1}_{\mathcal{B}_{r,t}}\mathbbm{1}_{\\{\lvert\widehat{W}_{t,s}\rvert\geq
M\\}}\middle|\widehat{W}_{t,0}=\widehat{W}_{t,t}=0}\right].$

We postpone the estimate of $j_{t,v,r}^{<M}(s)$ to Lemma A.1 (which replace
Lemma 5.6 in [21]). With this lemma in hand, we can conclude the proof. Let
$\delta\in(0,\theta\wedge\frac{1}{2})$ and $M>0$. Considering $t$ large enough
such that $\delta^{-1}r\leq r_{t}$, we decompose the right-hand side of
Equation (A.7) as

$\displaystyle 2\int_{[\delta
r,\delta^{-1}r]}j_{t,v,r}^{<M}(s)\mathop{}\mathopen{}\mathrm{d}s+2\int_{[\delta
r,\delta^{-1}r]}j_{t,v,r}^{\geq
M}(s)\mathop{}\mathopen{}\mathrm{d}s+2\int_{[\delta
r,\delta^{-1}r]^{c}}j_{t,v,r}(s)\mathop{}\mathopen{}\mathrm{d}s,$

where the two last terms are negligible. Indeed, it is proved in [21, Equation
(5.41)] (and the paragraph around) that

$\limsup_{\eta\to
0}\limsup_{M\to\infty}\limsup_{v\to\infty}\limsup_{t\to\infty}\frac{t}{\mathrm{e}^{\sqrt{2}v}}\cdot\mathopen{}\mathclose{{}\left(2\int_{[\eta
v^{2},\eta^{-1}v^{2}]}j_{t,v}^{\geq
M}(s)\mathop{}\mathopen{}\mathrm{d}s+2\int_{[\eta
v^{2},\eta^{-1}v^{2}]^{c}}j_{t,v}(s)\mathop{}\mathopen{}\mathrm{d}s}\right)=0,$

and therefore

$\limsup_{\delta\to
0}\limsup_{M\to\infty}\limsup_{r\to\infty}\limsup_{t\to\infty}\sup_{v\in[\theta\sqrt{r},\theta^{-1}\sqrt{r}]}\mathopen{}\mathclose{{}\left(2\int_{[\delta
r,\delta^{-1}r]}j_{t,v,r}^{\geq
M}(s)\mathop{}\mathopen{}\mathrm{d}s+2\int_{[\delta
r,\delta^{-1}r]^{c}}j_{t,v,r}(s)\mathop{}\mathopen{}\mathrm{d}s}\right)=0.$
(A.8)

Now, setting $S_{\delta}\coloneqq[\delta
r,(1-\delta)r]\cup[(1+\delta)r,\delta^{-1}r]$, we have

$2\int_{[\delta
r,\delta^{-1}r]}j_{t,v,r}^{<M}(s)\mathop{}\mathopen{}\mathrm{d}s=2\int_{S_{\delta}}j_{t,v,r}^{<M}(s)\mathop{}\mathopen{}\mathrm{d}s+2\int_{(1-\delta)r}^{(1+\delta)r}j_{t,v,r}^{<M}(s)\mathop{}\mathopen{}\mathrm{d}s.$

This last integral is also negligible for $\delta$ small enough (in the same
sense as in Equation (A.8)), because

$j_{t,v,r}^{<M}(s)\leq j_{t,v}(s)=j_{t,v}^{\geq 0}(s)\leq
C\,\frac{(v+1)\mathrm{e}^{\sqrt{2}v}}{s^{3/2}t}\leq
C(\theta)\,\frac{\mathrm{e}^{\sqrt{2}v}}{rt}\,,$

for any $M>0$, $0\leq r\leq t/4$, $s\in[(1-\delta)r,(1+\delta)r]$ and
$v\in[\theta\sqrt{r},\theta^{-1}\sqrt{r}]$ by [21, Lemma 5.5]. Finally, using
Lemma A.1, we have, as $t\to\infty$, then $r\to\infty$ and then $M\to\infty$,
uniformly in $v\in[\theta\sqrt{r},\theta^{-1}\sqrt{r}]$,

$\displaystyle
2\int_{S_{\delta}}j_{t,v,r}^{<M}(s)\mathop{}\mathopen{}\mathrm{d}s\sim\frac{2C_{2}}{t}\,v\mathrm{e}^{\sqrt{2}v}\int_{S_{\delta}}\frac{\mathrm{e}^{-v^{2}/(2s)}}{s^{3/2}}\,\varphi_{b,B}\mathopen{}\mathclose{{}\left(\frac{r\vee
s}{\mathopen{}\mathclose{{}\left\lvert
r-s}\right\rvert}}\right)\mathop{}\mathopen{}\mathrm{d}s.$

Coming back to Equation (A.5) and letting $\delta\to 0$, this proves that
there exists $r_{0}>0$ such that, for any $r\geq r_{0}$, there exists
$t_{0}>0$ such that, for any $t\geq t_{0}$ and any
$v\in[\theta\sqrt{r},\theta^{-1}\sqrt{r}]$,

$\mathopen{}\mathclose{{}\left\lvert\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}([-v,0])\mathbbm{1}_{B_{r,t}}}\right]-\frac{2C_{2}}{C_{1}}v\mathrm{e}^{\sqrt{2}v}\int_{0}^{\infty}\varphi_{b,B}\mathopen{}\mathclose{{}\left(\frac{r\vee
s}{\mathopen{}\mathclose{{}\left\lvert
r-s}\right\rvert}}\right)\frac{\mathrm{e}^{-v^{2}/(2s)}}{s^{3/2}}\mathop{}\mathopen{}\mathrm{d}s}\right\rvert\leq\varepsilon\,\mathrm{e}^{\sqrt{2}v}.$
(A.9)

Using the relation666 The authors of [21] do not keep precisely track of
constants but this relation can be deduced from a careful reading of their
paper. Alternatively letting $b\to 0$ and $B\to\infty$, we have
$\mathbbm{1}_{B_{r,t}}\to 1$ and $\varphi_{b,B}\to 1$ and Equation (A.9) gives
an alternative proof of their [21, Lemma 5.2], showing that the constant $C$
appearing there equals $2\sqrt{2\pi}C_{2}/C_{1}$. Then, a quick look at the
proof of [21, Proposition 1.5] ensures that this constant $C$ is actually
$C_{\star}$.  $C_{\star}=2\sqrt{2\pi}\,C_{2}/C_{1}$ gives the desired result.
∎

######  Lemma A.1.

Let $a>0$, $0<b<B$ and $\delta\in(0,1/2)$. As $t\to\infty$, then $r\to\infty$
and then $M\to\infty$, we have

$j_{t,v,r}^{<M}(s)\sim\frac{C_{2}}{ts^{3/2}}\,v\mathrm{e}^{\sqrt{2}v-v^{2}/(2s)}\,\varphi_{b,B}\mathopen{}\mathclose{{}\left(\frac{r\vee
s}{\mathopen{}\mathclose{{}\left\lvert r-s}\right\rvert}}\right),$

uniformly $s\in[\delta r,(1-\delta)r]\cup[(1+\delta)r,\delta^{-1}r]$ and
$v\in[\delta\sqrt{r},\delta^{-1}\sqrt{r}]$, with

$C_{2}\coloneqq\frac{2\sqrt{2}}{\pi}f^{(0)}(0)g(0)\int_{\mathbb{R}}f(z)g(z)^{2}\mathrm{e}^{\sqrt{2}z}\mathop{}\mathopen{}\mathrm{d}z\in(0,\infty),$

where $f^{(0)},f,g$ are positive functions introduced in [21, Lemma 3.4].

###### Proof.

We follow ideas from the proof of [21, Lemma 5.6], but instead of only
distinguishing according to the value of $\widehat{W}_{t,s}$, we also
distinguish according to the value of $\widehat{W}_{t,r}$. For comparison, the
constant $C$ appearing in the statement of [21, Lemma 5.6] equals $C_{2}$
introduced here.

We start with the case $r\leq s$, that is $s\in[(1+\delta)r,\delta^{-1}r]$.
Then, we have, with the notations from [21],

$\displaystyle
j_{t,v,r}^{<M}(s)=\int_{-B\sqrt{r}}^{-b\sqrt{r}}\int_{-M}^{M}p_{t}((r,y){;}(s,z))q_{t}((0,0){;}(r,y))q_{t}((r,y){;}(s,z))q_{t}((s,z){;}(t,0))e_{s,v}(z)\mathop{}\mathopen{}\mathrm{d}z\mathop{}\mathopen{}\mathrm{d}y.$

The function $(y,z)\mapsto p_{t}((r,y){;}(s,z))$ is the density of
$(\widehat{W}_{t,r},\widehat{W}_{t,s})$ given
$\widehat{W}_{t,0}=\widehat{W}_{t,t}=0$. It is explicitly given by (recall the
definition of $\gamma_{t,s}$ in Equation (A.6))

$\displaystyle p_{t}((r,y){;}(s,z))$
$\displaystyle=\frac{1}{2\pi}\sqrt{\frac{t}{r(s-r)(t-s)}}\,\exp\mathopen{}\mathclose{{}\left(-\frac{(s(y+\gamma_{t,r})-r(z+\gamma_{t,s}))^{2}}{2rs(s-r)}-\frac{t(z+\gamma_{t,s})^{2}}{2s(t-s)}}\right)$
$\displaystyle\sim\frac{1}{2\pi\sqrt{r(s-r)}}\,\exp\mathopen{}\mathclose{{}\left(-\frac{sy^{2}}{2r(s-r)}}\right),$

as $t\to\infty$ and then $r\to\infty$, uniformly in
$s\in[(1+\delta)r,\delta^{-1}r]$, $y\in[-B\sqrt{r},-b\sqrt{r}]$ and
$z\in[-M,M]$. Furthermore, it follows from [21, Lemma 3.4] that

$\displaystyle q_{t}((0,0){;}(r,y))$
$\displaystyle=q_{r}((0,0){;}(r,y))\sim\frac{2(-y)f^{(0)}(0)}{r},$
$\displaystyle q_{t}((r,y){;}(s,z))$
$\displaystyle=q_{s}((r,y){;}(s,z))\sim\frac{2(-y)g(z)}{s-r},$ $\displaystyle
q_{t}((s,z){;}(t,0))$
$\displaystyle\sim\frac{2f^{(s)}(z)g(0)}{t-s}\sim\frac{2f(z)g(0)}{t},$

as $t\to\infty$ and then $r\to\infty$, uniformly in $s$, $y$ and $z$ as
before. Finally, it follows from [21, Lemma 4.2] (see also [21, Equation
(5.37)]) that

$e_{s,v}(z)\sim
v\mathrm{e}^{\sqrt{2}v-v^{2}/(2s)}\,\frac{g(z)}{\sqrt{\pi}}\,\mathrm{e}^{\sqrt{2}z},$

as $r\to\infty$, uniformly in $v\in[\delta\sqrt{r},\delta^{-1}\sqrt{r}]$ and
$s,z$ as before. Altogether, this proves

$\displaystyle
j_{t,v,r}^{<M}(s)\sim\frac{4}{\pi^{3/2}}\,\frac{f^{(0)}(0)g(0)}{tr^{3/2}(s-r)^{3/2}}\,v\mathrm{e}^{\sqrt{2}v-v^{2}/(2s)}\int_{-B\sqrt{r}}^{-b\sqrt{r}}y^{2}\mathrm{e}^{-sy^{2}/(2r(s-r))}\mathop{}\mathopen{}\mathrm{d}y\int_{-M}^{M}f(z)g(z)^{2}\mathrm{e}^{\sqrt{2}z}\mathop{}\mathopen{}\mathrm{d}z,$

as $t\to\infty$ and then $r\to\infty$, uniformly in $s$ and $v$ as before. A
change of variable in the first integral and letting $M\to\infty$ in the
second integral (the fact that this integral converges to a finite limit is
justified at the end of the proof of [21, Lemma 5.6]) yields the result.

The case $s\leq r$, that is $s\in[\delta r,(1-\delta)r]$ is similar: we write

$\displaystyle
j_{t,v,r}^{<M}(s)=\int_{-B\sqrt{r}}^{-b\sqrt{r}}\int_{-M}^{M}p_{t}((s,z){;}(r,y))q_{t}((0,0){;}(s,z))q_{t}((s,z){;}(r,y))q_{t}((r,y){;}(t,0))e_{s,v}(z)\mathop{}\mathopen{}\mathrm{d}z\mathop{}\mathopen{}\mathrm{d}y$

and use the same asymptotics as before, the main difference being

$\displaystyle p_{t}((s,z){;}(r,y))$
$\displaystyle=\frac{1}{2\pi}\,\sqrt{\frac{t}{s(r-s)(t-r)}}\,\exp\mathopen{}\mathclose{{}\left(-\frac{(r(z+\gamma_{t,s})-s(y+\gamma_{t,r}))^{2}}{2rs(r-s)}-\frac{t(y+\gamma_{t,r})^{2}}{2r(t-r)}}\right)$
$\displaystyle\sim\frac{1}{2\pi\sqrt{s(r-s)}}\,\exp\mathopen{}\mathclose{{}\left(-\frac{y^{2}}{2(s-r)}}\right).$

This yields the result in that case. ∎

###  A.3 Cross-moments of level sets

Our aim in this section is to prove Proposition 2.13. For this, we follow the
proof of [21, Proposition 1.5], which bounds
$\mathbb{E}[\mathcal{C}([-v,0])^{2}]$. This proof is based on a series of five
lemmas, that we re-state here in a new version tuned for our purpose of
dealing with two level sets of different levels $v$ and $v^{\prime}$.

In the following lemma, as in [21, Lemma 4.3], we work with a 2-spine BBM
defined under the probability measure $\widetilde{\mathbb{P}}^{(2)}$ as
follows. It starts with one particle at 0 at time 0 which is part of the
spines 1 and 2. Particles belonging to $m$ spines branch at rate $2^{m}$ and
move according to a standard Brownian motion. At a branching point, for each
$k\in\\{1,2\\}$, if the parent was part of spine $k$, then one of both
children is chosen uniformly at random to be part of spine $k$. We denote
$X_{t}(k)$ the particle at time $t$ which is part of the spine $k$.y For
$x,y\in L_{t}$, we write $d(x,y)=r$ if the most recent common ancestor of $x$
and $y$ died at time $t-r$.

######  Lemma A.2.

There exists $C>0$ such that, for any $0\leq r\leq t$ and $v\leq
v^{\prime}\leq u$,

$\displaystyle\widetilde{\mathbb{P}}^{(2)}\mathopen{}\mathclose{{}\left(\widehat{h}_{t}(X_{t}(1))\geq
v\,,\,\widehat{h}_{t}(X_{t}(2))\geq v^{\prime}\,,\,\widehat{h}_{t}^{*}\leq
u\mathrel{}\middle|\mathrel{}d(X_{t}(1),X_{t}(2))=r}\right)$
$\displaystyle\leq\frac{C\mathrm{e}^{-t-r}}{1+(r\wedge(t-r))^{3/2}}(u_{+}+1)\mathrm{e}^{\sqrt{2}u}(u-v+1)(u-v^{\prime}+1)\mathrm{e}^{-\sqrt{2}(v+v^{\prime})}\mathopen{}\mathclose{{}\left(\mathrm{e}^{-(u-v)^{2}/(4t)}+\mathrm{e}^{-(u-v)/2}}\right).$

###### Proof.

This is a new version of [21, Lemma 4.3] and we explain how to adapt its
proof. Similarly as [21, Equation (4.15)], the probability in the statement
equals

$\displaystyle\int_{\mathbb{R}}\widetilde{\mathbb{P}}\mathopen{}\mathclose{{}\left(\widehat{h}_{r}(X_{r})\geq
v-z,\widehat{h}^{*}_{r}\leq
u-z}\right)\widetilde{\mathbb{P}}\mathopen{}\mathclose{{}\left(\widehat{h}_{r}(X_{r})\geq
v-z,\widehat{h}^{*}_{r}\leq u-z}\right)$
$\displaystyle\qquad{}\times\widetilde{\mathbb{P}}\mathopen{}\mathclose{{}\left(\widehat{h}_{t-r}(X_{t-r})-m_{t,r}\in\mathop{}\mathopen{}\mathrm{d}z,\widehat{h}^{*}_{t}(\mathrm{B}(X_{t})^{c})\leq
u}\right).$ (A.10)

Following [21], we split the integral according to $z\leq u$ and $z>u$.

For $z>u$, using [21, Equation (4.2)] for the first one and [21, Equation
(4.3)] for the second one, we bound the product of the two first probabilities
in Equation (A.10) by

$C\mathrm{e}^{-2r}(u-v+1)(u-v^{\prime}+1)\mathrm{e}^{-\sqrt{2}(v+v^{\prime})}\mathrm{e}^{2\sqrt{2}z-\frac{3}{2}(z-u)}\mathopen{}\mathclose{{}\left(\mathrm{e}^{-(v-z)^{2}/(4t)}+\mathrm{e}^{(v-z)/2}}\right).$

This is exactly the same as [21, Equation (4.20)] up to the factor
$(u-v^{\prime}+1)\mathrm{e}^{-\sqrt{2}v^{\prime}}$ where primes have been
added. Therefore, this part of the integral is dealt with exactly the same way
as in [21].

For $z\leq u$, we use [21, Equation (4.2)] for both first probabilities in
Equation (A.10) and note that, because $v\leq v^{\prime}$,

$\displaystyle\mathopen{}\mathclose{{}\left(\mathrm{e}^{-(v-z)^{2}/(4t)}+\mathrm{e}^{(v-z)/2}}\right)\mathopen{}\mathclose{{}\left(\mathrm{e}^{-(v^{\prime}-z)^{2}/(4t)}+\mathrm{e}^{(v^{\prime}-z)/2}}\right)$
$\displaystyle\leq\mathopen{}\mathclose{{}\left(\mathrm{e}^{-(v-z)^{2}/(4t)}+\mathrm{e}^{(v-z)/2}}\right)\mathopen{}\mathclose{{}\left(1+\mathrm{e}^{(v-z)/2}}\right)$
$\displaystyle\leq\mathopen{}\mathclose{{}\left(\mathrm{e}^{-(v-z)^{2}/(4t)}+\mathrm{e}^{v-z}+2\mathrm{e}^{(v-z)/2}}\right).$

This shows the product of the two first probabilities in Equation (A.10) is at
most

$C\mathrm{e}^{-2r}(u-v+1)(u-v^{\prime}+1)\mathrm{e}^{-\sqrt{2}(v+v^{\prime})}\mathrm{e}^{2\sqrt{2}z}(u-z+1)^{2}\mathopen{}\mathclose{{}\left(\mathrm{e}^{-(v-z)^{2}/(4t)}+\mathrm{e}^{(v-z)/2}+\mathrm{e}^{(v-z)/2}}\right).$

There are two differences with [21, Equation (4.19)]: the factor
$(u-v^{\prime}+1)\mathrm{e}^{-\sqrt{2}v^{\prime}}$ where primes have been
added (but this adds no new difficulty), and the additional term
$\mathrm{e}^{(v-z)/2}$ in the last parentheses. This latter gives rise to the
following new term, which should be added to the integral in [21, Equation
(4.21)],

$\int_{-\infty}^{u}\mathrm{e}^{\frac{v}{2}+(\sqrt{2}-\frac{1}{2})z)}(u-z+1)^{3}\mathop{}\mathopen{}\mathrm{d}z\leq
C\mathrm{e}^{\sqrt{2}u}\mathrm{e}^{-(u-v)/2},$

which can be included in the upper bound of the statement of the lemma after
taking care of the other factors (see [21, Equation (4.22)]: this is exactly
how another part of the integral in [21, Equation (4.21)] is bounded). ∎

######  Lemma A.3.

There exists $C>0$ such that, for any $t\geq 0$ and $v\leq v^{\prime}\leq u$,

$\displaystyle\mathbb{E}\mathopen{}\mathclose{{}\left[\mathcal{E}_{t}([v,\infty))\mathcal{E}_{t}([v^{\prime},\infty));\widehat{h}_{t}^{*}\leq
u}\right]$ $\displaystyle\leq
C(u_{+}+1)\mathrm{e}^{\sqrt{2}u}(u-v+1)(u-v^{\prime}+1)\mathrm{e}^{-\sqrt{2}(v+v^{\prime})}\mathopen{}\mathclose{{}\left(\mathrm{e}^{-(u-v)^{2}/(4t)}+\mathrm{e}^{-(u-v)/2}}\right).$

###### Proof.

This follows from Lemma A.2 in the same way as [21, Lemma 4.4] follows from
[21, Lemma 4.3]. ∎

######  Lemma A.4.

For any $v,v^{\prime}\geq 0$,

$\displaystyle\widetilde{\mathbb{E}}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}([-v,0])\,\mathcal{C}_{t,r_{t}}^{*}([-v^{\prime},0])\,;\,\widehat{h}_{t}^{*}\leq
0\mathrel{}\middle|\mathrel{}\widehat{h}_{t}(X_{t})=0}\right]$
$\displaystyle=4\int_{0}^{r_{t}}\int_{0}^{r_{t}}j_{t,v,v^{\prime}}(s,s^{\prime})\mathop{}\mathopen{}\mathrm{d}s^{\prime}\mathop{}\mathopen{}\mathrm{d}s+2\int_{0}^{r_{t}}j_{t,v,v^{\prime}}(s,s)\mathop{}\mathopen{}\mathrm{d}s,$

where
$j_{t,v,v^{\prime}}(s,s^{\prime})\coloneqq\widehat{\mathbb{E}}_{t}[J_{t,v}(s)J_{t,v^{\prime}}(s^{\prime})]$.

###### Proof.

This follows directly from the proof of [21, Lemma 5.4]. ∎

######  Lemma A.5.

There exists $C>0$ such that, for any $t\geq 0$, $s,s^{\prime}\in[0,t/2]$ and
$v\geq v^{\prime}\geq 0$,

$\displaystyle j_{t,v,v^{\prime}}(s,s^{\prime})$
$\displaystyle\leq\frac{C(v+1)(v^{\prime}+1)\mathrm{e}^{\sqrt{2}(v+v^{\prime})}}{t(s\wedge
s^{\prime}+1)\sqrt{s\wedge s^{\prime}}(\mathopen{}\mathclose{{}\left\lvert
s^{\prime}-s}\right\rvert+1)\sqrt{\mathopen{}\mathclose{{}\left\lvert
s^{\prime}-s}\right\rvert+\mathbbm{1}_{s=s^{\prime}}}}\mathopen{}\mathclose{{}\left(\mathrm{e}^{-v^{2}/(16s)}+\mathrm{e}^{-v/4}}\right).$

###### Proof.

This is a new version of [21, Lemma 5.5].

For the case $s\neq s^{\prime}$, we first assume that $s<s^{\prime}$ but do
not assume $v\geq v^{\prime}$. Then it follows directly from the proof in [21]
that

$\displaystyle j_{t,v,v^{\prime}}(s,s^{\prime})$
$\displaystyle\leq\frac{C(v+1)(v^{\prime}+1)\mathrm{e}^{\sqrt{2}(v+v^{\prime})}}{t(s+1)\sqrt{s}(s^{\prime}-s+1)\sqrt{s^{\prime}-s+\mathbbm{1}_{s=s^{\prime}}}}\mathopen{}\mathclose{{}\left(\mathrm{e}^{-v^{2}/(16s)}+\mathrm{e}^{-v/4}}\right)\mathopen{}\mathclose{{}\left(\mathrm{e}^{-(v^{\prime})^{2}/(16s^{\prime})}+\mathrm{e}^{-v^{\prime}/4}}\right).$

Then, we use
$j_{t,v,v^{\prime}}(s,s^{\prime})=j_{t,v^{\prime},v}(s^{\prime},s)$ to cover
the case $s>s^{\prime}$ (this is fine because we removed the assumption $v\geq
v^{\prime}$). Finally, for $v\geq v^{\prime}$, we bound
$\mathrm{e}^{-(v^{\prime})^{2}/(16s^{\prime})}+\mathrm{e}^{-v^{\prime}/4}\leq
2$. This yields the desired result.

The case $s=s^{\prime}$ is also identical to the proof of [21, Lemma 5.5],
applying here Lemma A.3 instead of [21, Lemma 4.4]. ∎

######  Lemma A.6.

There exists $C>0$ such that, for any $t\geq 1$ and $v\geq v^{\prime}\geq 0$,

$\displaystyle\widetilde{\mathbb{E}}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}([-v,0])\mathcal{C}_{t,r_{t}}^{*}([-v^{\prime},0])\mathrel{}\middle|\mathrel{}\widehat{h}_{t}^{*}=\widehat{h}_{t}(X_{t})=0}\right]$
$\displaystyle\leq C(v^{\prime}+1)\mathrm{e}^{-\sqrt{2}(v+v^{\prime})}.$

###### Proof.

Proceeding as in Equations (A.1), (A.2), (A.3) and (A.4)777For this step note
that we end up here with $\mathbb{E}[(\\#L_{t_{0}})^{2}]$, which is also a
finite constant depending on $t_{0}$ using similar arguments, including the
fact that the second moment of the number of particles at time $r$ in a
standard BBM is finite (more precisely it equals
$2\mathrm{e}^{2r}-\mathrm{e}^{r}$)., it is enough to prove that there exist
$C>0$ and $t_{0}\geq 0$ such that, for any $t\geq t_{0}$ and $v\geq 0$,

$\widetilde{\mathbb{E}}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}([-v,0])\mathcal{C}_{t,r_{t}}^{*}([-v^{\prime},0])\mathbbm{1}_{\widehat{h}_{t}^{*}\leq
0}\mathrel{}\middle|\mathrel{}\widehat{h}_{t}(X_{t})=0}\right]\leq\frac{C}{t}(v^{\prime}+1)\mathrm{e}^{\sqrt{2}v}.$
(A.11)

We choose $t_{0}$ such that, for any $t\geq t_{0}$, $r_{t}\leq t/2$. Using
Lemmas A.4 and A.5, the left-hand side of Equation (A.11) is at most

$\displaystyle\frac{C}{t}(v+1)(v^{\prime}+1)\mathrm{e}^{\sqrt{2}(v+v^{\prime})}\Biggl{(}\int_{0}^{r_{t}}\int_{0}^{r_{t}}\frac{(\mathrm{e}^{-v^{2}/(16s)}+\mathrm{e}^{-v/4})}{(s\wedge
s^{\prime}+1)\sqrt{s\wedge s^{\prime}}(\mathopen{}\mathclose{{}\left\lvert
s^{\prime}-s}\right\rvert+1)\sqrt{\mathopen{}\mathclose{{}\left\lvert
s^{\prime}-s}\right\rvert}}\mathop{}\mathopen{}\mathrm{d}s^{\prime}\mathop{}\mathopen{}\mathrm{d}s$
$\displaystyle\hskip
128.0374pt{}+\int_{0}^{r_{t}}\frac{(\mathrm{e}^{-v^{2}/(16s)}+\mathrm{e}^{-v/4})}{(s+1)\sqrt{s}}\mathop{}\mathopen{}\mathrm{d}s\Biggr{)}$
$\displaystyle\leq\frac{C}{t}(v+1)(v^{\prime}+1)\mathrm{e}^{\sqrt{2}(v+v^{\prime})}\int_{0}^{r_{t}}\frac{(\mathrm{e}^{-v^{2}/(16s)}+\mathrm{e}^{-v/4})}{(s+1)\sqrt{s}}\mathop{}\mathopen{}\mathrm{d}s,$

by integrating w.r.t. $s^{\prime}$ first. Then, proceeding as in the proof of
[21, Lemma 5.3], this last integral is at most $C/(v+1)$ and this yields the
result. ∎

###### Proof of Proposition 2.13.

This follows from Lemma A.6 in the same way as [21, Proposition 1.5] follows
from [21, Lemma 5.3]. ∎

###  A.4 Small moments of $S_{\beta}$

######  Lemma A.7.

For any $\gamma\in(0,1)$, there exists $C=C(\gamma)>0$ and $t_{0}\geq 0$, such
that, for any $t\geq t_{0}$ and any $\beta\in(1,2]$,

$\widetilde{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})^{\gamma}}\right]\leq\begin{cases}C(\beta-1)^{1-2\gamma}&\text{if
}\gamma\in(1/2,1),\\\ C\log\frac{1}{\beta-1}&\text{if }\gamma=1/2,\\\
C&\text{if }\gamma\in(0,1/2),\end{cases}$

where $f_{\beta}\colon x\in\mathbb{R}\mapsto\mathrm{e}^{\beta\sqrt{2}x}$.

###### Proof.

Proceeding as in Equations (A.1) and (A.2), it is enough to prove that there
exist $C>0$ such that, for any $t\geq t_{0}$ and $\beta\in(1,2]$,

$\widetilde{\mathbb{E}}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})^{\gamma}\mathbbm{1}_{\\{\widehat{h}_{t}^{*}\leq
0\\}}\mathrel{}\middle|\mathrel{}\widehat{h}_{t}(X_{t})=0}\right]\leq\frac{C}{t}\times\begin{cases}(\beta-1)^{1-2\gamma}&\text{if
}\gamma\in(1/2,1),\\\ \log\frac{1}{\beta-1}&\text{if }\gamma=1/2,\\\
1&\text{if }\gamma\in(0,1/2),\end{cases}$ (A.12)

where $t_{0}$ is chosen such that, for any $t\geq t_{0}$, $r_{t}\leq t/2$.

We first decompose $\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})$ along the spine as
it is done for level sets in [21, Lemma 5.4]. Setting (this replaces
$J_{t,v}(s)$ defined in [21, Equation (5.9)])

$K_{t,\beta}(s)\coloneqq\mathcal{E}_{s}^{s}(f_{\beta}(\,\cdot\,+\widehat{W}_{t,s}))\times\mathbbm{1}_{\\{\widehat{h}_{s}^{s*}\leq-\widehat{W}_{t,s}\\}}\times\mathbbm{1}_{\mathcal{A}_{t}},$

and
$\widehat{\mathbb{P}}_{t}\coloneqq\mathbb{P}(\,\cdot\,|\widehat{W}_{t,0}=\widehat{W}_{t,t}=0)$,
we have

$\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})\mathbbm{1}_{\\{\widehat{h}_{t}^{*}\leq
0\\}}\text{ under
}\widetilde{\mathbb{P}}(\,\cdot\,|\widehat{h}_{t}(X_{t})=0)\overset{(\mathrm{d})}{=}\int_{0}^{r_{t}}K_{t,\beta}(s)\mathcal{N}(\mathop{}\mathopen{}\mathrm{d}s)\text{
under }\widehat{\mathbb{P}}_{t},$

where $\mathcal{N}$ is a Poisson point process on $\mathbb{R}_{+}$ with
intensity $2\mathop{}\mathopen{}\mathrm{d}s$. Using subadditivity of $x\mapsto
x^{\gamma}$ (note that the integral above is actually a finite sum) and then
proceeding as in the proof of [21, Lemma 5.4], we get

$\widetilde{\mathbb{E}}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})^{\gamma}\mathbbm{1}_{\\{\widehat{h}_{t}^{*}\leq
0\\}}\mathrel{}\middle|\mathrel{}\widehat{h}_{t}(X_{t})=0}\right]\leq\widehat{\mathbb{E}}_{t}\mathopen{}\mathclose{{}\left[\int_{0}^{r_{t}}K_{t,\beta}(s)^{\gamma}\mathcal{N}(\mathop{}\mathopen{}\mathrm{d}s)}\right]=2\int_{0}^{r_{t}}k_{t,\beta,\gamma}(s)\mathop{}\mathopen{}\mathrm{d}s,$
(A.13)

with
$k_{t,\beta,\gamma}(s)\coloneqq\widehat{\mathbb{E}}_{t}[K_{t,v}(s)^{\gamma}]$.

We now aim at proving that there exists $C>0$ such that, for any $t\geq 0$,
$s\in[0,t/2]$ and $\beta\in(1,2]$,

$k_{t,\beta,\gamma}(s)\leq\frac{C}{t(s+1)\sqrt{s}}\mathopen{}\mathclose{{}\left((s+1)\wedge(\beta-1)^{-2}}\right)^{\gamma}.$
(A.14)

For this, we follow the ideas from the proof of [21, Lemma 5.5] (in the case
$M=0$). Conditioning on $\widehat{W}_{t,s}$, we get, similarly as [21,
Equation (5.20)],

$k_{t,\beta,\gamma}(s)=\int_{\mathbb{R}}q_{t}((0,0){;}(s,z))\mathbb{E}\mathopen{}\mathclose{{}\left[\mathcal{E}_{s}(f_{\beta}(\,\cdot\,+z))^{\gamma}\mathbbm{1}_{\\{\widehat{h}_{s}^{*}\leq-z\\}}}\right]q_{t}((s,z){;}(t,0))p_{t}(s,z)\mathop{}\mathopen{}\mathrm{d}z.$
(A.15)

The single difference with [21] is inside the expectation. Using Jensen’s
inequality and then writing
$f_{\beta}(x+z)=\int_{0}^{\infty}\beta\sqrt{2}\mathrm{e}^{-\beta\sqrt{2}v}\mathbbm{1}_{\\{x\geq-
v-z\\}}\mathop{}\mathopen{}\mathrm{d}v$ for $x\leq-z$, we get

$\displaystyle\mathbb{E}\mathopen{}\mathclose{{}\left[\mathcal{E}_{s}(f_{\beta}(\,\cdot\,+z))^{\gamma}\mathbbm{1}_{\\{\widehat{h}_{s}^{*}\leq-z\\}}}\right]^{1/\gamma}$
$\displaystyle\leq\mathbb{E}\mathopen{}\mathclose{{}\left[\mathcal{E}_{s}(f_{\beta}(\,\cdot\,+z))\mathbbm{1}_{\\{\widehat{h}_{s}^{*}\leq-z\\}}}\right]$
$\displaystyle=\int_{0}^{\infty}\beta\sqrt{2}\mathrm{e}^{-\beta\sqrt{2}v}\mathbb{E}\mathopen{}\mathclose{{}\left[\mathcal{E}_{s}([-v,0]-z)\mathbbm{1}_{\\{\widehat{h}_{s}^{*}\leq-z\\}}}\right]\mathop{}\mathopen{}\mathrm{d}v.$
(A.16)

Using [21, Lemma 4.2], the right-hand side of Equation (A.16) is at most

$\displaystyle
C(z_{-}+1)\mathrm{e}^{\sqrt{2}z}\int_{0}^{\infty}(v+1)\mathrm{e}^{-(\beta-1)\sqrt{2}v}\mathopen{}\mathclose{{}\left(\mathrm{e}^{-(v+z)^{2}/4s}+\mathrm{e}^{-(v+z)/2}}\right)\mathop{}\mathopen{}\mathrm{d}v.$

The part of the last integral due to the term $\mathrm{e}^{-(v+z)/2}$ is
bounded by $C\mathrm{e}^{-z/2}$. For the other part, it can be bounded by
$\int_{0}^{\infty}(v+1)\mathrm{e}^{-(\beta-1)\sqrt{2}v}\mathop{}\mathopen{}\mathrm{d}v\leq
C(\beta-1)^{-2}$ or by
$\int_{\mathbb{R}}(v+1)\mathrm{e}^{-(v+z)^{2}/4s}\mathop{}\mathopen{}\mathrm{d}v\leq
C(\mathopen{}\mathclose{{}\left\lvert z}\right\rvert+1)(s+1)$. Therefore, we
proved

$\mathbb{E}\mathopen{}\mathclose{{}\left[\mathcal{E}_{s}(f_{\beta}(\,\cdot\,+z))^{\gamma}\mathbbm{1}_{\\{\widehat{h}_{s}^{*}\leq-z\\}}}\right]\leq
C(\mathopen{}\mathclose{{}\left\lvert
z}\right\rvert+1)^{2\gamma}\mathrm{e}^{\gamma\sqrt{2}z}\mathopen{}\mathclose{{}\left(\mathrm{e}^{-z/2}+\mathopen{}\mathclose{{}\left((s+1)\wedge(\beta-1)^{-2}}\right)}\right)^{\gamma}.$

Coming back to Equation (A.15) and bounding the other factors in the integrand
as in [21, Equations (5.23) and (5.24)], we get

$k_{t,\beta,\gamma}(s)\leq\frac{C}{t(s+1)\sqrt{s}}\int_{\mathbb{R}}\mathopen{}\mathclose{{}\left(z_{-}+\mathrm{e}^{-\frac{3}{2}z_{+}}}\right)(\mathopen{}\mathclose{{}\left\lvert
z}\right\rvert+1)^{2\gamma+1}\mathrm{e}^{\gamma\sqrt{2}z}\mathopen{}\mathclose{{}\left(\mathrm{e}^{-\frac{\gamma}{2}z}+\mathopen{}\mathclose{{}\left((s+1)\wedge(\beta-1)^{-2}}\right)^{\gamma}}\right)\mathop{}\mathopen{}\mathrm{d}z,$

and Equation (A.14) follows.

Finally, we come back to Equation (A.13) and apply Equation (A.14) (we use
here $r_{t}\leq t/2$) to get

$\displaystyle\widetilde{\mathbb{E}}\mathopen{}\mathclose{{}\left[\mathcal{C}_{t,r_{t}}^{*}(f_{\beta})^{\gamma}\mathbbm{1}_{\widehat{h}_{t}^{*}\leq
0}\mathrel{}\middle|\mathrel{}\widehat{h}_{t}(X_{t})=0}\right]$
$\displaystyle\leq\frac{C}{t}\int_{0}^{\infty}\frac{1}{(s+1)\sqrt{s}}\mathopen{}\mathclose{{}\left((s+1)\wedge(\beta-1)^{-2}}\right)^{\gamma}\mathop{}\mathopen{}\mathrm{d}s$
$\displaystyle\leq\frac{C}{t}\mathopen{}\mathclose{{}\left(\int_{0}^{(\beta-1)^{-2}}\frac{\mathop{}\mathopen{}\mathrm{d}s}{(s+1)^{1-\gamma}\sqrt{s}}+(\beta-1)^{-2\gamma}\int_{(\beta-1)^{-2}}^{\infty}\frac{\mathop{}\mathopen{}\mathrm{d}s}{(s+1)\sqrt{s}}}\right)$

and Equation (A.12) follows. ∎

## Acknowledgements

The authors warmly thank Bernard Derrida for very stimulating discussions and
the GdR Branchement for financial support. Michel Pain also acknowledges
support from the PEPS JCJC grant n°246644.

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