#  12th Workshop on the CKM Unitarity Triangle
Santiago de Compostela, 18-22 September 2023
Search for CP violation in baryons with the LHCb detector

Chiara Mancuso1,2,3 on behalf of the LHCb collaboration
1Université Paris-Saclay  CNRS/IN2P3  IJCLab  Orsay  France
2INFN Sezione di Milano  Milano  Italy
3Università degli Studi di Milano  Milano  Italy

###### Abstract

The latest results from the LHCb collaboration for the search for CP violation
in $b$-baryon decays are reported here. The first article presented is the
search conducted in the $\Lambda_{b}^{0}\rightarrow p\pi^{-}\pi^{+}\pi^{-}$
decay, and the subsequent observation of P violation. The following paper
describes the search and the first observation of the
$\Lambda_{b}^{0}\rightarrow D^{0}pK^{-}$ decay, with $D^{0}\rightarrow K\pi$,
Cabibbo-favoured and doubly suppressed mode. This proceeding concludes with
the first amplitude analysis performed in $\Xi_{b}^{-}\rightarrow
pK^{-}K^{-}$, whose model allows for CP violation.

##  1 Introduction

Studying CP violation is fundamental to strengthen the Standard Model (SM) but
also to challenge its weaknesses. Within the framework of the SM, CP violation
is accommodated by the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1, 2], which
describes in weak interactions, the mixing between different quark flavours
and allows for a complex phase, thus providing a source for CP violation. The
CKM matrix offers a coherent explanation for the observed CP violating
effects, especially in the sector of mesons. CP violation in the $K$ [3], $B$
[4, 5] and $D$ [6] meson systems, is well-established experimentally and is
consistent with the predictions of the SM. Despite the successes of the CKM
matrix in explaining CP violation in mesons, the baryon sector remains a
largely uncharted territory. Indeed, the prediction of CP violation in baryons
has never been experimentally confirmed. In this context, the most recent
efforts of the LHCb collaboration are reported in the following.

##  2 Search for CP violation and observation of P violation in
$\Lambda_{b}^{0}\rightarrow p\pi^{-}\pi^{+}\pi^{-}$

The first study presented describes a detailed analysis of the decay
$\Lambda_{b}^{0}\rightarrow p\pi^{-}\pi^{+}\pi^{-}$ [7], using the data
obtained from proton-proton ($pp$) collisions, with an integrated luminosity
of 6.6 fb-1 collected over the period from 2011 to 2017 at energies of
$\sqrt{s}$ = 7, 8, and 13 TeV. The measurement employs two distinct and
independent techniques: the triple product asymmetries and the unbinned energy
test method.

The searches for CP violation are performed by separating P-odd and P-even
contributions. These investigations rely on the utilization of a rich control
sample, the Cabibbo-favoured
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}(\rightarrow pK^{-}\pi^{+})\pi^{-}$
decay, expected to exhibit no CP violation and is employed to evaluate
potential experimental biases and systematic effects.

The scalar triple products are built in the $\Lambda_{b}^{0}$ rest frame and
correspond to
$C_{\widehat{T}}\equiv\vec{p}_{p}\cdot\left(\vec{p}_{\pi_{\text{fast
}}^{-}}\times\vec{p}_{\pi^{+}}\right)$ and
$\bar{C}_{\widehat{T}}\equiv\vec{p}_{\bar{p}}\cdot\left(\vec{p}_{\pi_{\text{fast
}}^{+}}\times\vec{p}_{\pi^{-}}\right)$; by definition these are odd under
$\widehat{T}$ operator transformations, which reverses momentum and spin
three-vectors. CP and P violating effects appear as differences between the
triple product observables related by CP and P transformations. Indeed, the
asymmetries are defined as

$\displaystyle A_{\widehat{T}}$
$\displaystyle=\frac{N\left(C_{\widehat{T}}>0\right)-N\left(C_{\widehat{T}}<0\right)}{N\left(C_{\widehat{T}}>0\right)+N\left(C_{\widehat{T}}<0\right)}$
$\displaystyle\bar{A}_{\widehat{T}}$
$\displaystyle=\frac{\bar{N}\left(-\bar{C}_{\widehat{T}}>0\right)-\bar{N}\left(-\bar{C}_{\widehat{T}}<0\right)}{\bar{N}\left(-\bar{C}_{\widehat{T}}>0\right)+\bar{N}\left(-\bar{C}_{\widehat{T}}<0\right)},$

where $N$ and $\bar{N}$ are the number of events of $\Lambda_{b}^{0}$ and
$\bar{\Lambda_{b}^{0}}$. It follows that the CP and P violating asymmetries
are

$\displaystyle a_{CP}^{\widehat{T}\text{-odd }}$
$\displaystyle=\frac{1}{2}\left(A_{\widehat{T}}-\bar{A}_{\widehat{T}}\right)$
$\displaystyle a_{P}^{\widehat{T}\text{-odd }}$
$\displaystyle=\frac{1}{2}\left(A_{\widehat{T}}+\bar{A}_{\widehat{T}}\right).$

These quantities have been measured in the full phase space but also in
specific regions, as the decay investigated has a rich resonant structure
whose main contributions are given by
$N^{*+}\rightarrow\Delta^{++}(1234)\pi^{-}$, with
$\Delta^{++}(1234)\rightarrow p\pi^{+}$ and
$a_{1}^{-}(1260)\rightarrow\rho^{0}(770)\pi^{-}$, with
$\rho^{0}(770)\rightarrow\pi^{-}\pi^{+}$. These asymmetries are designed to be
substantially unaffected by asymmetries arising from particle-antiparticle
production and those induced by the detector. To enhance the sensitivity
towards localized CP violation effects, the phase space is segmented into bins
following two distinct binnings: the first one, labeled as A, divides the
dataset into 16 subsamples to explore the distribution of the polar and
azimuthal angles of the proton in the $\Delta^{++}$ rest frame; the second
one, labeled as B, divides the dataset into 10 subsamples, used to probe the
asymmetries as a function of $|\Phi|$, the angle between the planes defined by
the $p\pi^{-}_{fast}$ and $\pi^{+}\pi^{-}_{slow}$ systems in the
$\Lambda_{b}^{0}$ rest frame. The results for the full phase space are the
following

$\displaystyle a_{CP}^{\widehat{T}\text{-odd }}$ $\displaystyle=(-0.7\pm
0.7\pm 0.2)\%$ $\displaystyle a_{P}^{\widehat{T}\text{-odd }}$
$\displaystyle=(-4.0\pm 0.7\pm 0.2)\%.$

The value for obtained for $a_{CP}^{\widehat{T}\text{-odd }}$ is in agreement
with the CP conserving hypothesis, whereas the $a_{P}^{\widehat{T}\text{-odd
}}$ is clearly far from zero. Through the profile likelihood-ratio test, it is
found that the value has a significance level of 5.5 $\sigma$, suggesting a
violation of parity in this channel. Figure 1 shows the results for the binned
schemes $A$ and $B$ in two invariant mass regions: in $A_{1}$ and $B_{1}$, the
schemes are applied in the region where the resonance $a_{1}$ is dominant,
i.e. $m(p\pi^{+}\pi^{-}_{slow})>2.8$ $\text{GeV}/c^{2}$, while $A_{2}$ and
$B_{2}$ analyse the area in which the $N^{*+}$ resonance is dominant, i.e.
$m(p\pi^{+}\pi^{-}_{slow})<2.8$ $\text{GeV}/c^{2}$.

Figure 1: Measured asymmetries for the binning scheme (left) $A_{1}$ and
$A_{2}$ and (right) $B_{1}$ and $B_{2}$. The error bars represent the sum in
quadrature of the statistical and systematic uncertainties. The $\chi^{2}$ per
ndof is calculated with respect to the null hypothesis and includes
statistical and systematic uncertainties.

The energy test is a model-independent unbinned test sensitive to local
differences between two samples, which would be the case for CP violation. The
test is performed through the calculation of a test statistic

$T=\frac{1}{2n(n-1)}\sum_{i\neq
j}^{n}\psi_{ij}+\frac{1}{2\bar{n}(\bar{n}-1)}\sum_{i\neq
j}^{\bar{n}}\psi_{ij}-\frac{1}{n\bar{n}}\sum_{i=1}^{n}\sum_{j=1}^{\bar{n}}\psi_{ij},$
(2.1)

where there are $n$ ($\bar{n}$) candidates in the first (second) sample. The
first (second) term sums over pairs of candidates drawn from the first
(second) sample and the final term sums over pairs with one candidate drawn
from each sample. Each pair of candidates $ij$ is assigned a weight
$\psi_{ij}=e^{-d_{ij}^{2}/2\delta^{2}}$, where $d_{ij}$ is their Euclidean
distance in phase space, while the tunable parameter $\delta$ determines the
distance scale probed using the energy test. The phase space is defined using
the squared masses
$m^{2}(\pi^{+}),m^{2}(\pi^{+}\pi^{-}_{\text{slow}}),m^{2}(p\pi^{+}\pi^{-}_{\text{slow}}\pi^{-}_{\text{fast}})$,
$m^{2}(\pi^{-}\pi^{+}_{\text{slow}})$ and $m^{2}(p\pi^{-}_{\text{slow}})$. The
value of $T$ is large when there are significant localized differences between
samples and has an expectation of zero when there are no differences. The
distribution of $T$ under the hypothesis of no sample differences, and the
assignment of $p$-values, are determined using a permutation method. Results
for different configurations of the energy test are summarized in Table 1. All
CP violation searches using the energy test result in $p$-values with a
significance of 3$\sigma$ or smaller. The P violation test shows a
significance of 5.3 $\sigma$ at the two smaller scales probed.

Distance scale $\delta$ |  1.6 GeV${}^{2}/c^{4}$ |  2.7 GeV${}^{2}/c^{4}$ |  13 GeV${}^{2}/c^{4}$
---|---|---|---
$p$-value (CP conservation, P even) | $3.1\times 10^{-2}$ | $2.7\times 10^{-3}$ | $1.3\times 10^{-2}$
$p$-value (CP conservation, P odd) | $1.5\times 10^{-1}$ | $6.9\times 10^{-2}$ | $6.5\times 10^{-2}$
$p$-value (P conservation) | $1.3\times 10^{-7}$ | $4.0\times 10^{-7}$ | $1.6\times 10^{-1}$
Table 1: The $p$-values from the energy test for different distance scales and
test configurations.

##  3 Observation of the suppressed $\Lambda_{b}^{0}\rightarrow D^{0}pK^{-}$
decay with $D^{0}\rightarrow K^{-}\pi^{+}$ and measurement of its CP asymmetry

Another study the LHCb collaboration published is the search for the
$\Lambda_{b}^{0}\rightarrow DpK^{-}$, the Cabibbo-favoured and doubly Cabibbo-
suppressed mode $D\rightarrow K\pi$, where $D$ represents a superposition of
$D^{0}$ and $\bar{D^{0}}$ states [8]. In the same study, they performed a
measurement of the ratio of the branching fraction of the two decays and the
CP asymmetry of the suppressed mode. It is possible to perform an estimation
of the first quantity via the CKM matrix elements, obtaining

$R\approx\left|\frac{V_{cb}V^{*}_{us}}{V_{ub}V^{*}_{cs}}\right|^{2}=6.0.$
(3.1)

The $\Lambda_{b}^{0}\rightarrow[K^{-}\pi^{+}]_{D}pK^{-}$
($\Lambda_{b}^{0}\rightarrow[K^{+}\pi^{-}]_{D}pK^{-}$) decay with same
(opposite) sign kaons are referred to as the favoured (suppressed) decay
throughout this section. The suppressed decay object of this analysis is of
particular interest because its decay amplitude receives contributions almost
equally by both $b\rightarrow c$ and $b\rightarrow u$ transition amplitudes,
due to the comparable CKM suppression affecting the two $D$ decays. It is
expected that the interference of these two amplitudes, which is contingent on
the CKM angle $\gamma$, will be large. The data analysed comes from $pp$
collisions at an integrated luminosity of 9 fb-1 collected over the period
from 2011 to 2018 at center-of-mass energies of $\sqrt{s}$ = 7, 8, and 13 TeV.
The result of the selection is showed in Figure 2.

Figure 2: Distributions of the invariant mass for selected (left)
$\Lambda_{b}^{0}\rightarrow[K^{-}\pi^{+}]_{D}pK^{-}$ and (right)
$\Lambda_{b}^{0}\rightarrow[K^{+}\pi^{-}]_{D}pK^{-}$ candidates in the full
phase space (black points) corresponding to the favoured and suppressed
decays, respectively. The total fit model is indicated by the solid blue line,
and individual components are indicated

The CP measurement is performed both in the full phase space, but also in a
restricted one more sensitive to CP violation, where the $\Lambda^{*}$
resonance is predominant, $m^{2}(pK^{-})<5$ $\text{GeV}^{2}/c^{4}$. The CP
observable is the following

$A=\frac{\mathcal{B}(\Lambda_{b}^{0}\to[K^{+}\pi^{-}]_{D}pK^{-})-\mathcal{B}(\bar{\Lambda}_{b}^{0}\to[K^{-}\pi^{+}]_{D}\bar{p}K^{+})}{\mathcal{B}(\Lambda_{b}^{0}\to[K^{+}\pi^{-}]_{D}pK^{-})+\mathcal{B}(\bar{\Lambda}_{b}^{0}\to[K^{-}\pi^{+}]_{D}\bar{p}K^{+})},$
(3.2)

which has been computed taking into account the per-event efficiency of the
selection and the weights obtained using the sPlot technique. Also the ratio
between the suppressed and the favoured decays is expressed with the same
parametrization:

$\displaystyle A$
$\displaystyle=\frac{\sum_{i}w^{i}_{\text{SUP},\Lambda^{0}_{b}}/\epsilon^{i}-\sum_{i}w^{i}_{\text{SUP},\bar{\Lambda}^{0}_{b}}/\epsilon^{i}}{\sum_{i}w^{i}_{\text{SUP},\Lambda^{0}_{b}}/\epsilon^{i}+\sum_{i}w^{i}_{\text{SUP},\bar{\Lambda}^{0}_{b}}/\epsilon^{i}}$
$\displaystyle R$
$\displaystyle=\frac{\sum_{i}w^{i}_{\text{FAV}}/\epsilon^{i}}{\sum_{i}w^{i}_{\text{SUP}}/\epsilon^{i}}.$

The results are, in the full phase space and in the restricted one:

$\displaystyle A$ $\displaystyle=0.12\pm 0.09\text{
(stat.)}^{+0.02}_{-0.03}\text{ (syst.)}$ $\displaystyle A$
$\displaystyle=0.01\pm 0.16\text{ (stat.)}^{+0.03}_{-0.02}\text{ (syst.)}$
$\displaystyle R$ $\displaystyle=7.1\pm 0.8\text{ (stat.)}^{+0.4}_{-0.3}\text{
(syst.)}$ $\displaystyle R$ $\displaystyle=8.6\pm 1.5\text{
(stat.)}^{+0.4}_{-0.3}\text{ (syst.)}$

The analysis provided to the scientific community a first observation of the
suppressed $\Lambda_{b}^{0}\rightarrow[K^{+}\pi^{-}]_{D}pK^{-}$, and the ratio
of the branching fractions of the favoured and suppressed mode, which is,
within the uncertainties, compatible with the estimate computed using the
relevant CKM matrix elements. The asymmetry values are compatible with zero in
the regions studied.

##  4 Search for CP violation in $\Xi_{b}^{-}\rightarrow pK^{-}K^{-}$ decays

The LHCb collaboration published the first amplitude analysis of a baryon
decay allowing for CP violation [9]. The analysis studied the data collected
by the LHCb detector from $pp$ collisions, with an integrated luminosity of 5
fb-1 collected over the period from 2011 to 2016 at collision energies of
$\sqrt{s}$ = 7, 8, and 13 TeV. The amplitude analysis started after the
selection performed on the dataset, which allowed to have $(63\pm 3)\%$ of
signal purity in Run I and $(70\pm 2)\%$ in Run II, considering only
candidates in the $m(pK^{-}K^{-})$ signal region of $\pm 40$ MeV around the
$\Xi_{b}^{-}$ mass. It is assumed that the $\Xi_{b}^{-}$ candidates are
produced in $pp$ collisions within the LHCb acceptance have negligible
polarisation, as observed for the $\Lambda_{b}^{0}$. With this assumption it
is possible to characterise the phase space of the $\Xi_{b}^{-}\rightarrow
pK^{-}_{1}K^{-}_{2}$ decay with only two independent kinematic variables
$m^{2}(pK^{-}_{1})$ and $m^{2}(pK^{-}_{2})$. Furthermore, to remove the Bose
symmetry coming from the exchange of the two $K$, the variables used for the
amplitude analysis are instead $m^{2}_{low}$ and $m^{2}_{high}$, denoting the
lower and the higher range of $m^{2}(pK^{-}_{1})$ and $m^{2}(pK^{-}_{2})$
respectively. The helicity formalism developed for the analysis relies on the
$\Sigma^{*}$ and $\Lambda^{*}$ resonances, in the channel
$\Xi_{b}^{-}\rightarrow(R\rightarrow pK^{-})K^{-}$, such that the differentiel
decay density can be expressed as

$\frac{d\mathit{\Gamma}^{Q}}{d\Omega}=\frac{1}{(8\pi
m_{\Xi_{b}})^{3}}\sum_{M_{\Xi_{b}},\lambda_{p}}\left|\sum_{R}A^{Q}_{R,M_{\Xi_{b}},\lambda_{p}}(\Omega)\right|^{2},$
(4.1)

where $Q=+1$ is for $\Xi_{b}^{-}$ and $Q=-1$ is for $\bar{\Xi}_{b}^{+}$, and
$A^{Q}_{R,M_{\Xi_{b}},\lambda_{p}}(\Omega)$ denotes the symmetrized decay
amplitude for a given intermediate state $R$, $\Xi_{b}$ spin component along a
chosen quantization axis $M_{\Xi_{b}}$ and proton helicity $\lambda_{b}$. To
formulate the model describing the signal, the resonance $\Lambda(1520)$ has
been taken as baseline, leaving the fit free to vary the helicity couplings of
all the other resonant and non-resonant components. The chosen model has been
obtained when adding further resonances was not resulting in a substantial
change in the negative log-likelihood, and is reported in Table 2.

State |  Mass (MeV/$c^{2}$) |  Width (MeV/$c^{2}$) | $J^{P}$ | $A_{CP}(10^{-2})$
---|---|---|---|---
$\Lambda(1405)$ | $1405.1\pm 1.3$ | $50.5\pm 2.0$ | $\frac{1}{2}^{-}$ | $-27\pm 34\,(\text{stat})\pm 73\,(\text{syst})$
$\Lambda(1520)$ |  $1518$ to $1520$ |  $15$ to $17$ | $\frac{3}{2}^{-}$ | $-1\pm 24\,(\text{stat})\pm 32\,(\text{syst})$
$\Lambda(1670)$ |  $1660$ to $1680$ |  $25$ to $50$ | $\frac{1}{2}^{-}$ | $-5\pm 9\,(\text{stat})\pm 8\,(\text{syst})$
$\Sigma(1385)$ | $1383.7\pm 1$ | $36\pm 5$ | $\frac{3}{2}^{+}$ | $3\pm 14\,(\text{stat})\pm 10\,(\text{syst})$
$\Sigma(1775)$ |  $1770$ to $1780$ |  $105$ to $135$ | $\frac{5}{2}^{-}$ | $-47\pm 26(\text{stat})\pm 14(\text{syst})$
$\Sigma(1915)$ |  $1900$ to $1935$ |  $80$ to $160$ | $\frac{5}{2}^{+}$ | $11\pm 26(\text{stat})\pm 22(\text{syst})$
Table 2: Components of the model, properties and results for the CP asymmetry
parameters.

The fit to data using the selected model is shown in Figure 3. There is no
indication of CP violation in the distributions, i.e., no significant
difference between $\Xi_{b}^{-}$ and $\bar{\Xi}_{b}^{+}$ decays.

Figure 3: Distributions of (top) $m_{low}$ and (bottom) $m_{high}$ for (left)
$\Xi_{b}^{-}$ and (right) $\bar{\Xi}_{b}^{+}$ candidates, with results of the
fits superimposed. The total fit result is shown as the blue solid curve, with
contributions from individual signal components and backgrounds.

## References

  * [1] Nicola Cabibbo.  Unitary symmetry and leptonic decays.  Phys. Rev. Lett., 10:531–533, Jun 1963.
  * [2] Makoto Kobayashi and Toshihide Maskawa.  $\mathit{CP}$-Violation in the Renormalizable Theory of Weak Interaction.  Progress of Theoretical Physics, 49(2):652–657, 02 1973.
  * [3] J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay.  Evidence for the $2\pi$ decay of the $\mathit{K}^{0}$ meson.  Phys. Rev. Lett., 13:138–140, Jul 1964.
  * [4] B. Aubert et al. Measurement of $\mathit{CP}$-violating asymmetries in $\mathit{B}^{0}$ decays to $\mathit{CP}$ eigenstates.  Phys. Rev. Lett., 86:2515–2522, Mar 2001.
  * [5] K. Abe et al. Observation of large $\mathit{CP}$ violation in the neutral $\mathit{B}$ meson system.  Phys. Rev. Lett., 87(9), August 2001.
  * [6] R. Aaij et al. Observation of $\mathit{CP}$ violation in charm decays.  Phys. Rev. Lett., 122:211803, May 2019.
  * [7] R. Aaij et al. Search for $\mathit{CP}$ violation and observation of $\mathit{P}$ violation in ${\mathrm{\Lambda}}_{b}^{0}\rightarrow p{\pi}^{-}{\pi}^{+}{\pi}^{-}$ decays.  Phys. Rev. D, 102:051101, Sep 2020.
  * [8] R. Aaij et al. Observation of the suppressed ${\Lambda}_{b}^{0}\rightarrow{D}^{0}p{K}^{-}$ decay with ${D}^{0}\rightarrow{K}^{-}\pi^{+}$ and measurement of its $\mathit{CP}$ asymmetry.  Physical Review D, 104(11), December 2021.
  * [9] R. Aaij et al. Search for $\mathit{CP}$ violation in ${\mathrm{\Xi}}_{b}^{-}\rightarrow p{K}^{-}{K}^{-}$ decays.  Phys. Rev. D, 104:052010, Sep 2021.


# On the principal eigenvalue for compound Poisson processes

Daesung Kim and Hyunchul Park Research supported in part by Research and
Creative Project Award from SUNY New Paltz.

###### Abstract

We investigate the explicit expression for the principal eigenvalue
$\lambda_{1}^{X}(D)$ for a large class of compound Poisson processes $X$ on a
bounded open set $D$ by examining its spectral heat content. When the jump
density of the compound Poisson process is radially symmetric and strictly
decreasing, we demonstrate that balls are the unique minimizers for
$\lambda_{1}^{X}(D)$ among all sets with equal Lebesgue measure. Furthermore,
we show that this uniqueness fails if the jump density is not strictly
decreasing.

##  1 Introduction

The principal eigenvalue of stochastic processes on domains is of fundamental
importance. One of the many practical implications of the principal eigenvalue
is its role as a large deviation parameter, determining the decay rate of
survival probability ([6, Equation 2.4]). An intriguing question in this
context is what shape of domains minimizes the principal eigenvalue, ensuring
the highest survival probability among all sets with equal Lebesgue measure.
The answer lies in the celebrated Faber-Krahn inequality, which asserts that
balls are the unique minimizers for the principal eigenvalue. However, except
in very few cases, the explicit expression for the principal eigenvalue is not
available, and one must rely on other objects such as torsion functions
(expected lifetime) or the inradius of the underlying domains to estimate the
principal eigenvalue (see [6]). Hence, calculating the precise value or
finding the concrete expression for the principal eigenvalue becomes an
important inquiry.

The primary objective of this paper is to study the principal eigenvalue of a
broad class of compound Poisson processes and derive its explicit expression.
Furthermore, when the jump density of the compound Poisson process is radially
symmetric and strictly decreasing, we demonstrate that the principal
eigenvalue is minimized on balls among all sets with equal Lebesgue measure
(see Corollary 2.2). On the other hand, we provide a counterexample that this
uniqueness of the minimal eigenvalue for balls breaks down in the absence of a
strictly decreasing jump density.

The main device to prove the main results, Theorems 2.1 and 2.3, is the
spectral heat content (SHC). For a Lévy process $Y=\\{Y_{t}\\}_{t\geq 0}$ and
a bounded open set $D$ in ${\mathbb{R}}^{d}$, we define the spectral heat
content on $D$ with respect to $Y$ by

$Q_{D}^{Y}(t)=\int_{D}{\mathbb{P}}_{x}(\tau_{D}^{Y}>t)\,dx,$

where $\tau_{D}^{Y}=\inf\\{t>0:Y_{t}\notin D\\}$ is the first exit time of $Y$
from $D$ and ${\mathbb{P}}_{x}$ is the law of $Y$ started at $x$. The spectral
heat content for jump processes has been studied intensively in the last
decade (see [7, 9, 10, 11] and references therein). When there exists a
discrete spectrum for the infinitesimal generator of the process, the spectral
heat content has the spectral representation (3.1), and one can obtain the
principal eigenvalue from this representation. For compound Poisson processes,
the spectral heat content also admits the jump decomposition (3.1), since
there are only a countable number of jumps of which the duration of the next
jump is strictly positive. This is a crucial property that holds only for
compound Poisson processes. For general Lévy processes whose Lévy measures are
infinite, the jump times are dense on $[0,\infty)$ and the jump decomposition
(3.1) of the spectral heat content does not hold. Extension to more general
Lévy processes with infinite Lévy measure is certainly an interesting and
natural question and this will be investigated in a future project.

The strategy to establish the main result is as follows. We first establish
two technical lemmas, Lemmas 4.4 and 4.5. In Lemma 4.4, we prove that the jump
chain, $S_{n}=J_{1}+\cdots+J_{n}$, conditioned on {$S_{n}\in D$} converges to
a uniform distribution on $D$ as $n\to\infty$. Intuitively speaking, if $D$ is
bounded, then it is difficult for the jump chain to stay inside a bounded open
set $D$ for large $n$, meaning that there should be no preference for $S_{n}$
to be in a certain region in $D$ than another. Hence, the distribution of
$S_{n}$ conditioned on $\\{S_{n}\in D\\}$ should converge to a uniform
distribution on $D$. Then, in Lemma 4.5, we compute the probability that the
jump chain $S_{n+1}$ started at $x\in D$ remains inside $D$ conditioned on it
has stayed inside $D$ up to $n$ and show that the conditional probability
converges to the probability that a single jump $J_{1}$ stays inside $D$
averaged on all starting points $x\in D$. From these two facts, together with
the comparison between the spectral decomposition (3.1) and the jump
decomposition (3.1), we derive the precise expression for the principal
eigenvalue in Theorem 2.1.

As an application of Theorem 2.1, we show that the principal eigenvalue for
symmetric compound Poisson processes becomes minimized on balls among all sets
with equal Lebesgue measure when the probability density function $j(x)$ of an
individual jump $J_{1}$ of the underlying compound Poisson process is radially
symmetric and strictly decreasing. The fact that balls are minimizers is
proved in [1, Corollary 5.5] for a more general class of Lévy processes, but
the ball being the unique minimizer for symmetric compound Poisson processes
has not been studied in the past to the best of the authors’ knowledge. A
natural question arises for the necessity of the condition that $j(x)$ being
radially symmetric or strictly decreasing. In Theorem 2.3, we show that the
uniqueness fails by considering a compound Poisson process whose jump
distribution is uniformly distributed over some set
$\mathcal{A}\subset{\mathbb{R}}^{d}$, whose Lévy density is obviously not
strictly decreasing. In fact, in Theorem 2.3, we characterize the maximal
spectral heat content under volume constraint and observe that this
characterization has two scenarios depending on the relative size of the
support of the jump and the underlying set $D$. When the Lebesgue measure
$|\mathcal{A}|$ of the support of the jump is relatively small compared to the
Lebesgue measure $|D|$, the spectral heat content becomes maximized when the
domain $D$ and the support of jump $\mathcal{A}$ are congruent ellipsoids. On
the other hand, when $|\mathcal{A}|$ is relatively large compared to $|D|$,
the spectral heat content becomes maximized as long as
$D-D\subset\mathcal{A}$. The intuition behind this fact is that when the
support of jump is much larger than the size of $D$, the probability that the
next jump stays inside $D$ does not depend on the location from which it
jumps, and each jump lands on $D$ with the same probability
$\frac{|D|}{|\mathcal{A}|}$. The rigorous proof of the two cases depends on
the characterization of the equality cases of Riesz’s inequality in [3]. When
there exist discrete and countable eigenvalues, the non-uniqueness of minimal
eigenvalues among all sets with equal Lebesgue measure follows from the non-
uniqueness of maximal spectral heat contents and (3.2).

The paper is organized as follows. In Section 2, we state the main results of
this paper. In Section 3, we recall necessary preliminaries such as Lévy
processes, their semigroups, and rearrangement inequalities. We provide the
proofs of the main results in Section 4.

The indicator function of a set $A$ is denoted by $1_{A}$. For a bounded set
$A$, its Lebesgue measure is denoted by $|A|$. Let
$A,B\subset{\mathbb{R}}^{d}$ be subsets of ${\mathbb{R}}^{d}$,
$a\in{\mathbb{R}}$ and $z\in{\mathbb{R}}^{d}$. We define $aA:=\\{ax:x\in
A\\}$, $z+A:=\\{z+x:x\in A\\}$, and $A+B:=\\{x+y:x\in A,y\in B\\}$. We use
${\mathbb{P}}_{x}$ and ${\mathbb{E}}_{x}$ to denote the probability measure
and expectation for processes starting at $x\in{\mathbb{R}}^{d}$. When $x=0$,
we use ${\mathbb{P}}$ and ${\mathbb{E}}$ instead of ${\mathbb{P}}_{0}$ and
${\mathbb{E}}_{0}$. We reserve notation $X=\\{X_{t}\\}_{t\geq 0}$ for compound
Poisson processes and $Y=\\{Y_{t}\\}_{t\geq 0}$ for generic Lévy processes.

##  2 Main Results

Let $D$ be a bounded open set in ${\mathbb{R}}^{d}$ and $\lambda_{1}^{X}(D)$
be the principal eigenvalue for the semigroup for a compound Poisson process
$X$ on $D$ and $j(x)$ be the probability density function for $J_{1}$ (see
Section 3 for definitions).

###### Theorem 2.1

Let $X=\\{X_{t}\\}_{t\geq 0}$ be a compound Poisson process with jump rate
$r>0$. Assume that $X$ satisfies Condition 4.1. For a bounded open set $D$,
its principal eigenvalue $\lambda_{1}^{X}(D)$ is given by

$\lambda_{1}^{X}(D)=\frac{r}{|D|}\int_{D}\int_{D^{c}}j(y-x)\,dydx.$

It follows from the Riesz’s rearrangement inequality (see [8, Theorem 3.9])

$\int_{D}\int_{D}j(y-x)\,dydx\leq\int_{D^{\ast}}\int_{D^{\ast}}j^{\ast}(y-x)\,dydx,$

where $j^{\ast}$ is the symmetric decreasing rearrangement of $j$ (see Section
3 for the definition). Then, Theorem 2.1 implies

$\lambda_{1}^{X}(D)\geq\frac{r}{|D|}\int_{D^{\ast}}\int_{(D^{\ast})^{c}}j^{\ast}(y-x)\,dydx=\lambda_{1}^{X^{\ast}}(D^{\ast}),$
(2.1)

where $D^{\ast}$ is a centered ball with $|D^{\ast}|=|D|$ and
$\lambda_{1}^{X}(D)$ and $\lambda_{1}^{X^{\ast}}(D^{\ast})$ are the principal
eigenvalues for the compound Poisson process $X$ in $D$ and the rotationally
symmetrized process $X^{\ast}$ in $D^{\ast}$ (see Subsection 4.2 for the
definition), respectively. Indeed, this was shown in [1] for more general Lévy
processes. The following result characterizes equality cases when the
probability density function $j(x)$ is radially symmetric and strictly
decreasing.

###### Corollary 2.2

Let $D$ be a bounded open set in ${\mathbb{R}}^{d}$ and $X^{\ast}$ be the
symmetrization of $X$. If the probability density function $j(x)$ of $J_{1}$
is radially symmetric and strictly decreasing, then the following Faber-Krahn
inequality holds;

$\lambda_{1}^{X}(D)\geq\lambda_{1}^{X^{\ast}}(D^{\ast}).$

Furthermore, the equality above holds if and only if $D$ is a ball.

Proof. The inequality follows from (2.1). The proof for the equality cases
follows from the equality case of strict rearrangement inequality [8, Theorem
3.9]. $\Box$

Corollary 2.2 asserts that the ball is the unique minimizer for the principal
eigenvalue for the compound Poisson process when $j(x)$ is radially symmetric
and strictly decreasing. In Theorem 2.3, we show that if the jump density is
not strictly decreasing, then the uniqueness of the minimizers for the
principal eigenvalue fails. In fact, we investigate cases that guarantee the
maximal spectral heat content for the compound Poisson process whose jump
distribution is uniformly distributed on some subset
$\mathcal{A}\subset{\mathbb{R}}^{d}$ without assuming Condition 4.1. When
there exist discrete eigenvalues, the non-uniqueness of minimal principal
eigenvalue follows from the non-uniqueness of maximal spectral heat content
and (3.2).

###### Theorem 2.3

Let $X=\\{X_{t}\\}_{t\geq 0}$ be a compound Poisson process, where the
sequence of individual jumps $\\{J_{i}\\}_{i=1}^{\infty}$ is uniformly
distributed on a bounded set $\mathcal{A}\subset{\mathbb{R}}^{d}$ and
$X^{\ast}$ be the symmetrization of $X$. If $|\mathcal{A}|<2^{d}|D|$, then the
followings are equivalent:

  1. 1.

$Q_{D}^{X}(t)=Q_{D^{\ast}}^{X^{\ast}}(t)$ for all $t>0$.

  2. 2.

$Q_{D}^{X}(t)=Q_{D^{\ast}}^{X^{\ast}}(t)$ for some $t>0$.

  3. 3.

$D=a+\alpha\mathcal{E}$ and $\mathcal{A}=\gamma\mathcal{E}$ for some
$a\in{\mathbb{R}}^{d}$ and some ellipsoid $\mathcal{E}\subset{\mathbb{R}}^{d}$
up to measure zero.

On the other hand, if $|\mathcal{A}|\geq 2^{d}|D|$, then the followings are
equivalent:

  1. 1.

$Q_{D}^{X}(t)=Q_{D^{\ast}}^{X^{\ast}}(t)$ for all $t>0$.

  2. 2.

$Q_{D}^{X}(t)=Q_{D^{\ast}}^{X^{\ast}}(t)$ for some $t>0$.

  3. 3.

$D-D\subset\mathcal{A}$.

  4. 4.

$Q_{D}^{X}(t)=|D|\exp\left(-rt\left(1-\frac{|D|}{|\mathcal{A}|}\right)\right)$
for all $t>0$.

Furthermore, assume that the spectral heat content for $X$ has the spectral
representation (3.1). If $|\mathcal{A}|<2^{d}|D|$, then $\lambda_{1}^{X}(D)$
is minimized if $D=a+\alpha\mathcal{E}$ and $\mathcal{A}=\gamma\mathcal{E}$
for some $a\in{\mathbb{R}}^{d}$ and some ellipsoid
$\mathcal{E}\subset{\mathbb{R}}^{d}$ up to measure zero. On the other hand, if
$|\mathcal{A}|\geq 2^{d}|D|$, then $\lambda_{1}^{X}(D)$ is minimized if
$D-D\subset\mathcal{A}$. Hence, the uniqueness of the minimal principal
eigenvalue for balls does not hold.

We close this section by introducing further questions related to our main
results.

  1. 1.

As a corollary of Theorem 2.1, we have shown that if the probability density
function $j(x)$ of $J_{1}$ is radially symmetric and strictly decreasing, then
$\lambda^{X}_{1}(D)=\lambda^{X}_{1}(D^{\ast})$ implies that $D$ must be a ball
(up to measure zero). A natural question is to ask the quantitative version or
stability estimate. That is, if $\lambda^{X}_{1}(D)$ is close to
$\lambda^{X}_{1}(D^{\ast})$, does it imply that $D$ is close to $D^{\ast}$ in
a certain sense?

  2. 2.

It is natural to ask if Theorem 2.3 holds for general Lévy processes other
than compound Poisson processes. For Brownian motions in ${\mathbb{R}}^{d}$,
it follows from [4, 12] that $Q^{BM}_{D}(t)=Q^{BM}_{D^{\ast}}(t)$ for some
$t>0$ if and only if $D$ is a ball, where $Q^{BM}_{D}(t)$ is the spectral heat
content for the Brownian motion. It is an interesting question to ask if this
could be extended for Lévy processes other than Brownian motions.

##  3 Preliminaries

###  3.1 Lévy Processes and Their Semigroups

Let $(\Omega,\mathcal{F},{\mathbb{P}})$ be a probability space. A
$d$-dimensional Lévy process $Y=\\{Y_{t}\\}_{t\geq 0}$ on
$(\Omega,\mathcal{F},{\mathbb{P}})$ is a stochastic process with càdlàg paths
and independent and stationary increments. The characteristic exponent
$\psi(\xi)$ of a Lévy process $Y=\\{Y_{t}\\}_{t\geq 0}$ is defined by
${\mathbb{E}}[e^{i\xi\cdot Y_{t}}]=e^{-t\psi(\xi)}$ for
$\xi\in{\mathbb{R}}^{d}$. It follows from the Lévy–Khintchine theorem that
$Y=\\{Y_{t}\\}_{t\geq 0}$ is a Lévy process with characteristic exponent
$\psi(\xi)$ if and only if there exists a triplet $(b,A,\nu)$ such that

$\displaystyle\psi(\xi)=ib\cdot\xi+\frac{1}{2}\xi\cdot
A\xi+\int_{{\mathbb{R}}^{d}}(1-e^{i\xi\cdot y}+(i\xi\cdot
y)1_{\\{x:|x|<1\\}}(y))\,\nu(dy),$

where $b\in{\mathbb{R}}^{d}$, $A$ is a positive semi-definite $d\times d$
matrix, and $\nu$ is a $\sigma$-finite Borel measure on
${\mathbb{R}}^{d}\setminus\\{0\\}$ satisfying
$\int_{{\mathbb{R}}^{d}\setminus\\{0\\}}(1\wedge|y|^{2})\,\nu(dy)<\infty$.

Let $N=\\{N_{t}\\}_{t\geq 0}$ be a Poisson process with rate $r$, that is
${\mathbb{P}}(N_{t}=k)=e^{-rt}\frac{(rt)^{k}}{k!}$, $k\geq 0$. Let
$S_{n}=J_{1}+\cdots+J_{n}$ be the jump chain, where
$\\{J_{i}\\}_{i=1}^{\infty}$ is a sequence of i.i.d. random variables
independent of $N$ whose characteristic exponent is given by
$\phi(\xi):={\mathbb{E}}[e^{i\xi J_{1}}]=\int_{{\mathbb{R}}^{d}}e^{i\xi
y}\mu(dy)$, where $\mu$ is a probability measure on ${\mathbb{R}}^{d}$ with
$\mu(\\{0\\})=0$. We define a compound Poisson process $X=\\{X_{t}\\}_{t\geq
0}$ with rate $r>0$ and a jump measure $\mu$ by $X_{t}:=S_{N_{t}}$. When each
$J_{i}$ is symmetric, that is $\mu(A)=\mu(-A)$ for all Borel sets $A$, then
$X$ is also symmetric and we will call $X$ a symmetric compound Poisson
process. Note that

$\displaystyle{\mathbb{E}}[e^{i\xi X_{t}}]$
$\displaystyle={\mathbb{E}}[e^{i\xi
S_{N_{t}}}]=\sum_{n=0}^{\infty}{\mathbb{E}}[e^{i\xi
S_{n}},N_{t}=n]=\sum_{n=0}^{\infty}e^{-rt}\frac{(rt)^{n}}{n!}\phi(\xi)^{n}=\exp\left(-rt\int_{{\mathbb{R}}^{d}}(1-e^{i\xi\cdot
y})\mu(dy)\right).$

By the Lévy-Khintchine formula, this shows that the Lévy measure of $X$ is
$r\mu$.

Let $D$ be a bounded open set in ${\mathbb{R}}^{d}$ and $Y=\\{Y_{t}\\}_{t\geq
0}$ a symmetric Lévy process. Let
$P^{D}_{t}f(x)={\mathbb{E}}_{x}[f(Y_{t});t<\tau_{D}]$ be a semigroup
associated with the killed process $Y^{D}$, where $Y^{D}_{t}=Y_{t}$ if
$t<\tau_{D}^{Y}:=\inf\\{t>0:Y_{t}\notin D\\}$ and $Y^{D}_{t}=\partial$ for
$t\geq\tau_{D}^{Y}$, where $\partial$ is a cemetery state. The infinitesimal
generator $\mathcal{L}$ for the semigroup $(P^{D}_{t})_{t\geq 0}$ is given by

$\displaystyle\mathcal{L}f(x)=\lim_{t\downarrow
0}\frac{P^{D}_{t}f(x)-f(x)}{t},$

whenever the limit exists, where the limit is taken in the supremum norm. The
generator of the strongly continuous semigroup is non-positive definite and
self-adjoint. We assume that the generator $\mathcal{L}$ and the semigroup
$P^{D}_{t}$ admit discrete spectra $\\{\lambda_{n}^{Y}(D)\\}_{n=1}^{\infty}$
and $\\{e^{-t\lambda_{n}^{Y}(D)}\\}_{n=1}^{\infty}$ such that
$P^{D}_{t}\varphi_{n}=e^{-t\lambda_{n}^{Y}(D)}\varphi^{Y}_{n}$ and
$\mathcal{L}\varphi^{Y}_{n}=\lambda_{n}^{Y}(D)\varphi^{Y}_{n}$, where
$\\{\varphi^{Y}_{n}\\}$ is an orthonormal basis of eigenfunctions in
$L^{2}(D)$.

Assume that $Y$ admits a transition density $p_{D}^{Y}(t,x,y)$ on $D$. It is
well-known that $p_{D}^{Y}(t,x,y)$ has the following representation

$p_{D}^{Y}(t,x,y)=\sum_{n=1}^{\infty}e^{-t\lambda^{Y}_{n}}\varphi^{Y}_{n}(x)\varphi^{Y}_{n}(y).$

From this representation, the spectral heat content can be expressed as

$Q_{D}^{Y}(t)=\int_{D}{\mathbb{P}}^{x}(\tau^{Y}_{D}>t)\,dx=\int_{D}\int_{D}p_{D}^{Y}(t,x,y)dydx=\sum_{n=1}^{\infty}e^{-t\lambda^{Y}_{n}}\left(\int_{D}\varphi_{n}^{Y}(y)dy\right)^{2}.$
(3.1)

We call the above expression the spectral representation of the spectral heat
content. This spectral representation will play an important role in proving
Theorem 2.1. In particular, from (3.1) one can recover the principal
eigenvalue $\lambda_{1}^{Y}(D)$ by the following expression

$\lambda^{Y}_{1}(D)=-\lim_{t\to\infty}\frac{\ln Q_{D}^{Y}(t)}{t}.$ (3.2)

Besides the spectral decomposition, which holds for a large class of Lévy
processes, the spectral heat content for a compound Poisson process has the
following jump decomposition. Note that $X_{t}=S_{N_{t}}$, where
$S_{n}=J_{1}+\cdots+J_{n}$, $N_{t}$ is an independent Poisson process with
rate $r$ and $\\{J_{i}\\}_{i=1}^{\infty}$ is a sequence of i.i.d. random
variables. Since $\\{J_{i}\\}_{i=1}^{\infty}$ and $\\{N_{t}\\}_{t>0}$ are
independent,

$\displaystyle Q_{D}^{X}(t)=$
$\displaystyle\int_{D}{\mathbb{P}}_{x}(\tau^{X}_{D}>t)dx$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\int_{D}{\mathbb{P}}_{x}(\tau^{X}_{D}>t,N_{t}=n)dx$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\int_{D}{\mathbb{P}}_{x}(\bigcap_{k=1}^{n}\\{S_{k}\in
D\\},N_{t}=n)dx$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}e^{-rt}\frac{(rt)^{n}}{n!}\int_{D}A(x,n,D)dx,$
(3.3)

where $A(x,n,D)={\mathbb{P}}_{x}(\bigcap_{k=1}^{n}\\{S_{k}\in D\\})$.

###  3.2 Rearragement Inequalities

For a subset $D\subset{\mathbb{R}}^{d}$ with $|D|<\infty$, its symmetric
rearrangement $D^{\ast}$ is a centered ball with the same volume as $D$. That
is, $D^{\ast}=B(0,r)$, where $\omega_{d}r^{d}=|D|$ and
$\omega_{d}=\frac{\pi^{d/2}}{\Gamma(1+d/2)}$ is the volume of a unit ball in
${\mathbb{R}}^{d}$. For a nonnegative function $f$ with
$|\\{x\in{\mathbb{R}}^{d}:f(x)>t\\}|<\infty$ for all $t>0$, the symmetric
decreasing rearrangement $f^{\ast}$ is defined by

$\displaystyle f^{\ast}(x)=\int_{0}^{\infty}1_{\\{f(x)>t\\}^{\ast}}\,dt.$

Riesz’s rearrangement inequality [8, Theorems 3.7 and 3.9] states that

$\int_{{\mathbb{R}}^{d}}\int_{{\mathbb{R}}^{d}}f(x)g(x-y)h(y)\,dxdy\leq\int_{{\mathbb{R}}^{d}}\int_{{\mathbb{R}}^{d}}f^{\ast}(x)g^{\ast}(x-y)h^{\ast}(y)\,dxdy.$

More generally, if $f_{1},f_{2},\cdots,f_{m}$ are nonnegative functions with
$|\\{x\in{\mathbb{R}}^{d}:f_{i}(x)>t\\}|<\infty$ for all $t>0$ and for
$i=1,2,\cdots,m$, and $B=(b_{ij})$ is a $k\times m$ matrix for $k\leq m$, then

$\displaystyle\int\cdots\int\prod_{i=1}^{m}f_{i}\left(\sum_{j=1}^{k}b_{ij}x_{j}\right)\,dx_{1}\cdots
dx_{k}\leq\int\cdots\int\prod_{i=1}^{m}f^{\ast}_{i}\left(\sum_{j=1}^{k}b_{ij}x_{j}\right)\,dx_{1}\cdots
dx_{k}.$ (3.4)

This inequality is called the Brascamp–Lieb–Luttinger inequality [2, Theorem
1.2]. We use the following notations

$\mathcal{J}(f,g,h):=\int(f*g)(x)h(x)dx,\quad\mathcal{J}(A,B,C):=\mathcal{J}(1_{A},1_{B},1_{C}).$

A natural question arises for the characterization of equality cases in
Riesz’s rearrangement inequality. It follows from [8, Theorem 3.9] that if
$g=g^{\ast}$ and strictly decreasing, then
$\mathcal{J}(f,g,h)=\mathcal{J}(f^{\ast},g^{\ast},h^{\ast})$ implies that $f$
and $h$ are radially symmetric up to the same translation, that is, there
exists $y\in{\mathbb{R}}^{d}$ such that $f(x)=f^{\ast}(x-y)$ and
$h(x)=h^{\ast}(x-y)$. In [3], the author characterized the equality cases for
Riesz’s rearrangement inequality when $f,g$, and $h$ are indicator functions.

###### Theorem 3.1

[3, Theorem 1] Let $A$, $B$ and $C$ be measurable sets of finite positive
measure in ${\mathbb{R}}^{d}$. Denote by $A^{*}$, $B^{*}$, $C^{*}$ the
centered balls of the same measure as $A$, $B$, and $C$, respectively, and let
$\alpha$, $\beta$, and $\gamma$ be the radii of these balls. If
$|\alpha-\beta|<\gamma<\alpha+\beta$, then there is equality in

$\mathcal{J}(A,B,C)\leq\mathcal{J}(A^{*},B^{*},C^{*})$ (3.5)

if and only if, up to sets of measure zero,

$A=a+\alpha\mathcal{E},\quad B=b+\beta\mathcal{E},\quad
C=c+\gamma\mathcal{E},$

where $\mathcal{E}$ is a centered ellipsoid in ${\mathbb{R}}^{d}$, and $a,b$
and $c=a+b$ are vectors in ${\mathbb{R}}^{d}$. Otherwise, permute the three
sets so that $\gamma\geq\alpha+\beta$. Then, the equality in (3.5) holds if
and only if $A+B\subset C$ up to a set of measure zero.

##  4 Proofs of Main Results

###  4.1 Proof of Theorem 2.1

In this section, we prove Theorem 2.1. Let $X=\\{X_{t}\\}_{t\geq 0}$ be a
compound Poisson process with rate $r>0$ and a jump measure $\mu$. Recall that
$\phi(\xi)={\mathbb{E}}[e^{i\xi J_{1}}]=\int_{{\mathbb{R}}^{d}}e^{i\xi
y}\mu(dy)$, where $J_{i}$ are jumps of the compound Poisson process. In this
section, we assume the following conditions.

###### Condition 4.1

  1. 1.

$J_{1}$ has a symmetric probability density function $j(x)$.

  2. 2.

${\mathbb{E}}[J_{1}]=0$ and $\text{Var}(J_{1})<\infty$.

  3. 3.

There exists a countable set of discrete eigenvalues
$\\{\lambda_{n}^{X}(D)\\}_{n=1}^{\infty}$ for the infinitesimal generator of
$X$ killed upon leaving $D$.

###### Remark 4.2

  1. 1.

Under the condition $(2)$ in (4.1), $X_{t}$ has a symmetric probability
density function. Indeed, for any Borel set $A$

$\displaystyle{\mathbb{P}}(X_{t}\in A)={\mathbb{P}}(\sum_{i=1}^{N_{t}}J_{i}\in
A)=\sum_{n=0}^{\infty}{\mathbb{P}}(\sum_{i=1}^{n}J_{i}\in A,N_{t}=n)$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}e^{-rt}\frac{(rt)^{n}}{n!}\int_{A}j^{\otimes
n}(x)dx.$

Hence, $p(t,x):=\sum_{n=0}^{\infty}e^{-rt}\frac{(rt)^{n}}{n!}j^{\otimes n}(x)$
is the probability density function for $X_{t}$.

  2. 2.

Furthermore, if $j(x)$ is bounded, then it is easy to see that $p(t,x)$ is
also bounded. This implies that the operator
$T_{t}f(x)={\mathbb{E}}_{x}[f(X_{t}^{D})]=\int_{D}p^{D}(t,x,y)f(y)dy$, where
$p^{D}(t,x,y)=p(t,x,y)-r^{D}(t,x,y)$ with
$r^{D}(t,x,y)={\mathbb{E}}_{x}[\tau_{D}\leq t,p(t-\tau_{D},X_{\tau_{D}},y)]$
and $p(t,x,y):=p(t,y-x)$, is self-adjoint and Hilbert-Schmidt operator on
$L^{2}(D)$ for any bounded open set $D$, hence the condition $(3)$ of
Condition 4.1 is satisfied for a bounded set $D$. Hence, any symmetric
compound Poisson processes with finite mean and variance with a bounded
probability density function $j(x)$ for $J_{1}$ satisfy all conditions in
Condition 4.1. This confirms that Condition 4.1 covers a large class of
symmetric compound Poisson processes.

  3. 3.

Note that $j\in L^{2}({\mathbb{R}}^{d})$ from the condition
$\text{Var}(J_{i})<\infty$. Hence, it follows from the Plancherel’s formula
$\frac{1}{(2\pi)^{d}}\int_{{\mathbb{R}}^{d}}|\phi(\xi)|^{2}d\xi=\int_{{\mathbb{R}}^{d}}|j(x)|^{2}dx<\infty$
that $|\phi(\xi)|^{2}$ is integrable on ${\mathbb{R}}^{d}$. This condition is
necessary in Lemma 4.3 to guarantee a uniform convergence.

We recall the following version of the central limit theorem. The proof is
given in [5, XV.5 Theorem 2] and the comment below it.

######  Lemma 4.3 ([5, XV.5 Theorem 2])

Let $\\{J_{i}\\}_{i=1}^{\infty}$ be a sequence of i.i.d. random variables with
probability density function $j(x)$. Suppose that $\text{Var}(J_{1})<\infty$
and there exists a positive integer $k$ such that $|\phi(\xi)|^{k}$ is
integrable on ${\mathbb{R}}^{d}$, where $\phi(\xi)={\mathbb{E}}[e^{i\xi
J_{1}}]$ is the characteristic function of $J_{1}$. Let $j_{n}(x)$ be the
density of $\frac{\sum_{i=1}^{n}J_{i}}{\sqrt{n\text{Var}(J_{1})}}$. Then,
uniformly on $x\in{\mathbb{R}}^{d}$,

$\lim_{n\to\infty}j_{n}(x)=(2\pi)^{-d/2}e^{-\frac{|x|^{2}}{2}}.$

Recall that $S_{n}=\sum_{i=1}^{n}J_{i}$ is the jump chain of the underlying
compound Poisson process. We show that conditional on $S_{n}\in D$, the
distribution of $S_{n}$ converges weakly to the uniform distribution on $D$.

###### Lemma 4.4

Let $D$ be a bounded open set. Suppose the Condition 4.1 holds true. For any
$x\in{\mathbb{R}}^{d}$,

$\lim_{n\to\infty}{\mathbb{E}}_{x}[e^{i\xi\cdot S_{n}}|S_{n}\in
D]=\frac{1}{|D|}\int_{{\mathbb{R}}^{d}}e^{i\xi\cdot
w}1_{D}(w)\,dw,\quad\xi\in{\mathbb{R}}^{d}.$

Furthermore, the convergence is uniform for all $x\in E$.

Proof. Let $j_{n}(u)$ be the transition density of
$\frac{S_{n}}{\sigma\sqrt{n}}$ where $\text{Var}(J_{1})=\sigma^{2}$. By the
change of variable $\sigma\sqrt{n}u=v$

$\displaystyle{\mathbb{E}}_{x}\left[e^{i\xi\cdot S_{n}}\mid S_{n}\in D\right]$
$\displaystyle=\frac{{\mathbb{E}}_{x}\left[e^{i\xi\cdot S_{n}};S_{n}\in
D\right]}{{\mathbb{P}}_{x}(S_{n}\in
D)}=\frac{{\mathbb{E}}\left[e^{i\xi\cdot(S_{n}+x)};S_{n}\in
D\right]}{{\mathbb{P}}(S_{n}+x\in D)}$
$\displaystyle=\frac{{\mathbb{E}}\left[e^{i\xi\cdot
x}e^{i\xi\cdot\sigma\sqrt{n}\frac{S_{n}}{\sigma\sqrt{n}}};\frac{S_{n}}{\sigma\sqrt{n}}\in\frac{D-x}{\sigma\sqrt{n}}\right]}{{\mathbb{P}}\left(\frac{S_{n}}{\sigma\sqrt{n}}\in\frac{D-x}{\sigma\sqrt{n}}\right)}$
$\displaystyle=\frac{\int_{{\mathbb{R}}^{d}}e^{i\xi\cdot
x}e^{i\sigma\sqrt{n}\xi\cdot
u}1_{\frac{D-x}{\sigma\sqrt{n}}}(u)j_{n}(u)\,du}{\int_{{\mathbb{R}}^{d}}1_{\frac{D-x}{\sigma\sqrt{n}}}(u)j_{n}(u)\,du}$
$\displaystyle=\frac{\int_{{\mathbb{R}}^{d}}e^{i\xi\cdot x}e^{i\xi\cdot
v}1_{D-x}(v)j_{n}(\frac{v}{\sigma\sqrt{n}})(\sigma\sqrt{n})^{-d}\,dv}{\int_{{\mathbb{R}}^{d}}1_{D-x}(v)j_{n}(\frac{v}{\sigma\sqrt{n}})(\sigma\sqrt{n})^{-d}\,dv}$
$\displaystyle=\frac{\int_{{\mathbb{R}}^{d}}e^{i\xi\cdot(x+v)}1_{D}(x+v)j_{n}(\frac{v}{\sigma\sqrt{n}})\,dv}{\int_{{\mathbb{R}}^{d}}1_{D}(x+v)j_{n}(\frac{v}{\sigma\sqrt{n}})\,dv}.$

Let $\gamma(u)=(2\pi)^{-d/2}e^{-\frac{|u|^{2}}{2}}$. By the change of variable
$w=v+x$, we obtain

$\displaystyle\left|\int_{{\mathbb{R}}^{d}}e^{i\xi\cdot(x+v)}1_{D}(x+v)j_{n}\left(\frac{v}{\sigma\sqrt{n}}\right)\,dv-\int_{{\mathbb{R}}^{d}}e^{i\xi\cdot
w}1_{D}(w)\gamma(0)\,dw\right|$
$\displaystyle\qquad\leq\int_{{\mathbb{R}}^{d}}e^{i\xi\cdot(x+v)}1_{D}(x+v)\left|j_{n}\left(\frac{v}{\sigma\sqrt{n}}\right)-\gamma(0)\right|\,dv$
$\displaystyle\qquad\leq|D|\sup_{v\in
D-x}\left|j_{n}\left(\frac{v}{\sigma\sqrt{n}}\right)-\gamma(0)\right|$
$\displaystyle\qquad\leq|D|\sup_{v\in
D-D}\left|j_{n}\left(\frac{v}{\sigma\sqrt{n}}\right)-\gamma(0)\right|,$

where $D-D=\\{x-y:x,y\in D\\}$. It follows from Lemma 4.3 that the last
expression converges to 0 as $n\to\infty$. Using the same argument, uniformly
for all $x\in E$, we have

$\lim_{n\to\infty}\int_{{\mathbb{R}}^{d}}1_{D}(x+v)j_{n}\left(\frac{v}{\sigma\sqrt{n}}\right)\,dv=|D|\gamma(0).$

Hence, uniformly for all $x\in E$,

$\lim_{n\to\infty}\frac{\int_{{\mathbb{R}}^{d}}e^{i\xi(x+v)}1_{D}(x+v)j_{n}\left(\frac{v}{\sigma\sqrt{n}}\right)\,dv}{\int_{{\mathbb{R}}^{d}}1_{D}(x+v)j_{n}\left(\frac{v}{\sigma\sqrt{n}}\right)\,dv}=\frac{1}{|D|}\int_{{\mathbb{R}}^{d}}e^{i\xi\cdot
w}1_{D}(w)\,dv.$

$\Box$

###### Lemma 4.5

Let $D$ be a bounded open set. Suppose that the Condition 4.1 holds true. For
any $x\in{\mathbb{R}}^{d}$ we have

$\lim_{n\to\infty}{\mathbb{P}}_{x}\left(S_{n+1}\in
D\Big{|}\bigcap_{k=1}^{n}\\{S_{k}\in
D\\}\right)=\frac{1}{|D|}\int_{D}{\mathbb{P}}_{x}(J_{1}\in
D)\,dx=\frac{1}{|D|}\int_{D}\int_{D}j(x+u)\,dudx.$

Furthermore, the convergence is uniform for all $x\in E$.

Proof. By the strong Markov property, we have

$\displaystyle{\mathbb{P}}_{x}\left(S_{n+1}\in
D\Big{|}\bigcap_{k=1}^{n}\\{S_{k}\in D\\}\right)$
$\displaystyle=\frac{{\mathbb{P}}_{x}\left(S_{n+1}\in
D,\bigcap_{k=1}^{n}\\{S_{k}\in
D\\}\right)}{{\mathbb{P}}_{x}\left(\bigcap_{k=1}^{n}\\{S_{k}\in D\\}\right)}$
$\displaystyle=\frac{{\mathbb{E}}_{x}\left[{\mathbb{P}}_{S_{n}}(J_{n+1}\in
D);\bigcap_{k=1}^{n}\\{S_{k}\in
D\\}\right]}{{\mathbb{P}}_{x}\left(\bigcap_{k=1}^{n}\\{S_{k}\in D\\}\right)}.$

From Lemma 4.4 the distribution of $S_{n}$, conditioned on $S_{n}\in D$,
converges weakly to the uniform distribution on $E$. Hence, the independence
and the weak convergence yield that

$\lim_{n\to\infty}{\mathbb{P}}_{S_{n}}(J_{n+1}\in
D)=\frac{1}{|D|}\int_{D}{\mathbb{P}}_{z}(J_{1}\in D)\,dz,$

as it is conditioned on the event $S_{n}\in D$ in the numerator. Now, the
conclusion follows from the bounded convergence theorem and cancellation
between the same expressions
${\mathbb{P}}_{x}\left(\bigcap_{k=1}^{n}\\{S_{k}\in D\\}\right)$ in the
numerator and denominator. $\Box$

We are now ready to prove Theorem 2.1.

Proof of Theorem 2.1. Recall from (3.1) that the spectral heat content
$Q_{D}^{X}(t)$ has the following jump decomposition

$\displaystyle Q_{D}^{X}(t)$
$\displaystyle=\int_{D}{\mathbb{P}}_{x}(\tau^{X}_{D}>t)dx=\sum_{n=0}^{\infty}e^{-rt}\frac{(rt)^{n}}{n!}\int_{D}A(x,n,D)dx,$

where $A(x,n,D)={\mathbb{P}}_{x}(\bigcap_{k=1}^{n}\\{S_{k}\in D\\})$. Let
$\alpha=\frac{1}{|D|}\int_{D}{\mathbb{P}}_{z}(J_{1}\in D)\,dz$ for simplicity.
From (3.2), it suffices to show that

$\lim_{t\to\infty}\frac{\log Q^{X}_{D}(t)}{t}=-r(1-\alpha).$

Given $\varepsilon>0$ with $\varepsilon<\alpha$, it follows from Lemma 4.5
that there exists $N=N(\varepsilon)$ such that

$\alpha-\varepsilon<{\mathbb{P}}_{x}\left(S_{n}\in
D\Big{|}\bigcap_{k=1}^{n-1}\\{S_{k}\in D\\}\right)<\alpha+\varepsilon$ (4.1)

for all $n\geq N$ and $x\in D$. Hence, for all $n\geq N$ and $x\in D$

$\displaystyle A(x,n,D)$
$\displaystyle={\mathbb{P}}_{x}\left(\bigcap_{k=1}^{N-1}\\{S_{k}\in
D\\}\right)\prod_{k=N}^{n}{\mathbb{P}}_{x}\left(S_{k}\in
D\Big{|}\bigcap_{j=1}^{k-1}\\{S_{j}\in D\\}\right)$
$\displaystyle\leq\prod_{k=N}^{n}{\mathbb{P}}_{x}\left(S_{k}\in
D\Big{|}\bigcap_{j=1}^{k-1}\\{S_{j}\in D\\}\right)$
$\displaystyle\leq(\alpha+\varepsilon)^{n-N+1}.$

Using a trivial upper bound $A(x,n,D)\leq 1$,

$\displaystyle Q_{D}^{X}(t)$
$\displaystyle=\sum_{n=0}^{\infty}e^{-rt}\frac{(rt)^{n}}{n!}\int_{D}A(x,n,D)\,dx$
$\displaystyle\leq|D|\left(\sum_{n=0}^{N-1}e^{-rt}\frac{(rt)^{n}}{n!}+\sum_{n=N}^{\infty}e^{-rt}\frac{(rt)^{n}}{n!}(\alpha+\varepsilon)^{n-N+1}\right)$
$\displaystyle\leq|D|\left(\sum_{n=0}^{N-1}e^{-rt}\frac{(rt)^{n}}{n!}+(\alpha+\varepsilon)^{-N+1}e^{rt(\alpha+\varepsilon-1)}\right).$

Taking logarithm and letting $t\to\infty$

$\limsup_{t\to\infty}\frac{\log
Q^{X}_{D}(t)}{t}\leq-r(1-\alpha)+r\varepsilon.$

Since $\varepsilon>0$ is arbitrary we conclude that

$\displaystyle\limsup_{t\to\infty}\frac{\log
Q^{X}_{D}(t)}{t}\leq-r(1-\alpha).$

For the lower bound, it follows from (4.1) that for all $n\geq N$ and $x\in
D$,

$\displaystyle A(x,n,D)$
$\displaystyle={\mathbb{P}}_{x}\left(\bigcap_{k=1}^{N-1}\\{S_{k}\in
D\\}\right)\prod_{k=N}^{n}{\mathbb{P}}_{x}\left(S_{k}\in
D\Big{|}\bigcap_{j=1}^{k-1}\\{S_{j}\in D\\}\right)$
$\displaystyle\geq{\mathbb{P}}_{x}\left(\bigcap_{k=1}^{N-1}\\{S_{k}\in
D\\}\right)(\alpha-\varepsilon)^{n-N+1}.$

Hence,

$\displaystyle Q^{X}_{D}(t)$
$\displaystyle\geq\sum_{n=N}^{\infty}e^{-rt}\frac{(rt)^{n}}{n!}\int_{D}{\mathbb{P}}_{x}\left(\bigcap_{k=1}^{N-1}\\{S_{k}\in
D\\}\right)(\alpha-\varepsilon)^{n-N+1}\,dx$ $\displaystyle\geq
e^{-rt}(\alpha-\varepsilon)^{-N+1}\left(e^{rt(\alpha-\varepsilon)}-\sum_{n=0}^{N-1}\frac{(rt)^{n}}{n!}(\alpha-\varepsilon)^{n}\right)\int_{D}{\mathbb{P}}_{x}\left(\bigcap_{k=1}^{N-1}\\{S_{k}\in
D\\}\right)\,dx.$

Hence, for any constant
$c<(\alpha-\varepsilon)^{-N+1}\int_{D}{\mathbb{P}}_{x}\left(\bigcap_{k=1}^{N-1}\\{S_{k}\in
D\\}\right)\,dx$, we have

$Q^{X}_{D}(t)\geq
ce^{-rt}\left(e^{rt(\alpha-\varepsilon)}-\sum_{n=0}^{N-1}\frac{(rt)^{n}}{n!}(\alpha-\varepsilon)^{n}\right).$

Furthermore, since $\alpha>\varepsilon$ and $N$ is fixed, we may decrease the
value of $c$, if necessary, so that

$Q^{X}_{D}(t)\geq ce^{-rt(1-\alpha+\varepsilon)},$

for all sufficiently large $t>0$. Taking logarithm and letting $t\to\infty$
yield

$\liminf_{t\to\infty}\frac{\log
Q^{X}_{D}(t)}{t}\geq-r(1-\alpha)-r\varepsilon.$

Since $\varepsilon>0$ is arbitrary we conclude that

$\displaystyle\liminf_{t\to\infty}\frac{\log
Q^{X}_{D}(t)}{t}\geq-r(1-\alpha).$

$\Box$

###  4.2 Proof of Theorem 2.3

In this section, we prove Theorem 2.3 when the distribution of $J_{1}$ of the
underlying compound Poisson process is uniformly distributed on a set
$\mathcal{A}\subset{\mathbb{R}}^{d}$. Note that we do not impose the Condition
4.1 for the current section, which were used to derive Lemmas 4.4 and 4.5. For
a compound Poisson process $X_{t}=S_{N_{t}}$, we define the symmetrized
compound Poisson process $X^{\ast}_{t}=S^{\ast}_{N_{t}}$ where
$\\{J_{i}^{\ast}\\}_{i=1}^{\infty}$ is a sequence of i.i.d. random variables
with probability density function $j^{\ast}$ and
$S^{\ast}_{n}=\sum_{i=1}^{n}J^{\ast}_{i}$. We call $X^{\ast}$ the
symmetrization of $X$. Then, we define

$A(x,n,D)={\mathbb{P}}_{x}(\bigcap_{k=1}^{n}\\{S_{k}\in D\\}),\quad
A^{\ast}(x,n,D)={\mathbb{P}}_{x}(\bigcap_{k=1}^{n}\\{S^{\ast}_{k}\in D\\}).$

Note that

$\displaystyle\int_{D}A(x,n,D)\,dx$
$\displaystyle=\int_{{\mathbb{R}}}1_{D}(x)\int_{({\mathbb{R}})^{n}}1_{D}(x+x_{1})\cdots
1_{D}(x+x_{1}+\cdots+x_{n})\prod_{i=1}^{n}j(x_{i})\,dx_{1}\cdots dx_{n}dx$
$\displaystyle=\int_{{\mathbb{R}}^{n+1}}\prod_{l=0}^{n}1_{D}\left(x+\sum_{k=1}^{l}x_{k}\right)\prod_{i=1}^{n}j(x_{i})\,dx_{1}\cdots
dx_{n}dx.$

It follows from Brascamp–Lieb–Luttinger inequality (3.4) that

$\displaystyle\int_{D}A(x,n,D)\,dx$
$\displaystyle\leq\int_{{\mathbb{R}}^{n+1}}\prod_{l=0}^{n}1_{D^{\ast}}\left(x+\sum_{k=1}^{l}x_{k}\right)\prod_{i=1}^{n}j^{\ast}(x_{i})\,dx_{1}\cdots
dx_{n}dx$ $\displaystyle=\int_{D^{\ast}}A^{\ast}(x,n,D^{\ast})\,dx.$

We observe that the spectral heat content is invariant under linear
isomorphisms. Let $T:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ be a linear
isomorphism with $|\det(T)|=1$ and $D,\mathcal{A}\subset{\mathbb{R}}^{d}$ be
open sets. Let $\\{J_{i}\\}_{i=1}^{\infty}$,
$\\{\tilde{J}_{i}\\}_{i=1}^{\infty}$ be sequences of random variables with
densities $j$ and $\tilde{j}=T\circ j$ respectively. Let $N_{t}$ be a Poisson
process with rate $r>0$, $S_{n}=\sum_{i=1}^{n}J_{i}$, and
$\tilde{S}_{n}=\sum_{i=1}^{n}\tilde{J}_{i}$. Define $X_{t}=S_{N_{t}}$ and
$\tilde{X}_{t}=\tilde{S}_{N_{t}}$. Then, we have
$Q_{D}^{X}(t)=Q_{T(D)}^{\tilde{X}}(t)$ for all $t>0$. Indeed, the result
follows from

$Q_{D}^{X}(t)=\sum_{n=0}^{\infty}e^{-rt}\frac{(rt)^{n}}{n!}\int_{D}A(x,n,D)\,dx$

where $A(x,n,D)={\mathbb{P}}_{x}(\bigcap_{k=1}^{n}\\{S_{k}\in D\\})$ and

$\int_{D}A(x,n,D)\,dx=\int_{T(D)}\tilde{A}(x,n,T(D))\,dx,$

by the change of variables.

###### Proposition 4.6

Let $X_{t}=\sum_{i=1}^{N_{t}}J_{i}$ be a compound Poisson process, where
$J_{i}$ is uniformly distributed on a set $\mathcal{A}\subset{\mathbb{R}}^{d}$
and $X^{\ast}$ be the symmetrization of $X$. Assume that
$|\mathcal{A}|<2^{d}|E|$. Then, the followings are equivalent:

  1. 1.

$Q_{D}^{X}(t)=Q^{X^{\ast}}_{D^{*}}(t)$ for all $t>0$.

  2. 2.

$Q_{D}^{X}(t)=Q^{X^{\ast}}_{D^{*}}(t)$ for some $t>0$.

  3. 3.

$D=a+b\mathcal{E}$ and $\mathcal{A}=c\mathcal{E}$, where
$a\in{\mathbb{R}}^{d}$, $b,c\in{\mathbb{R}}$, and $\mathcal{E}$ is a centered
ellipsoid in ${\mathbb{R}}^{d}$.

Proof. Clearly, the first condition implies the second.

Secondly, assume that $Q_{E}^{X}(t)=Q^{X^{\ast}}_{E^{\ast}}(t)$ for some
$t>0$. From [1, Corollary 5.5] $Q_{E}^{X}(t)\leq Q^{X^{\ast}}_{E^{\ast}}(t)$
and from the jump decomposition (3.1) of the spectral heat content, we must
have

$\int_{D}A(x,n,D)\,dx=\int_{D^{*}}A^{*}(x,n,D^{*})\,dx$

for all $n\in{\mathbb{N}}$. By the change of variable $y=-x$

$\displaystyle\int_{D}A(x,1,D)\,dx$
$\displaystyle=\int_{D}{\mathbb{P}}_{x}(J_{1}\in
D)\,dx=\int_{-D}{\mathbb{P}}_{-y}(J_{1}\in D)\,dy$
$\displaystyle=\int_{{\mathbb{R}}^{d}}1_{-D}(y)\int_{{\mathbb{R}}^{d}}1_{D}(-y+z)\frac{1_{\mathcal{A}}(z)}{|\mathcal{A}|}\,dzdy$
$\displaystyle=\int_{{\mathbb{R}}^{d}}\frac{1_{\mathcal{A}}(z)}{|\mathcal{A}|}\int_{{\mathbb{R}}^{d}}1_{-D}(y)1_{D}(-y+z)\,dydz$
$\displaystyle=\int_{{\mathbb{R}}^{d}}\frac{1_{\mathcal{A}}(z)}{|\mathcal{A}|}(1_{-D}\ast
1_{D})(z)dz$ $\displaystyle=\mathcal{J}(1_{-D},1_{D},1_{\mathcal{A}}).$

Similarly,
$\int_{D^{*}}A^{*}(x,1,D^{*})\,dx=\mathcal{J}(1_{(-D)^{*}},1_{D^{*}},1_{\mathcal{A}^{*}})$,
which in turn implies
$\mathcal{J}(1_{-D},1_{D},1_{\mathcal{A}})=\mathcal{J}(1_{(-D)^{*}},1_{D^{*}},1_{\mathcal{A}^{*}})$.
Recall that in Theorem 3.1 $\alpha,\beta$, and $\gamma$ represent the radii of
underlying three sets arranged in an increasing order. In our setting,
$\alpha$ and $\beta$ are the radius of $D^{*}$ and $\gamma$ is the radius of
$\mathcal{A}^{*}$. Hence, the condition $|\alpha-\beta|<\gamma<\alpha+\beta$
in Theorem 3.1 is equivalent to $|\mathcal{A}|<|2D|=2^{d}|D|$. It follows from
Theorem 3.1 $D=a+b\mathcal{E}$ and $\mathcal{A}=c\mathcal{E}$ for some
centered ellipsoid $\mathcal{E}$.

Finally, suppose that $D=a+b\mathcal{E}$ and $\mathcal{A}=c\mathcal{E}$, where
$a\in{\mathbb{R}}^{d}$ and $\mathcal{E}$ is a centered ellipsoid in
${\mathbb{R}}^{d}$. Since the spectral heat content is translation invariant,
we may assume that $a=0$. Then, there exists a linear isomorphism $T$ such
that $T(D)=D^{\ast}$ and $T(\mathcal{A})=\mathcal{A}^{\ast}$. By the change of
variables, for any $t>0$ we have

$\displaystyle Q_{D}^{X}(t)$
$\displaystyle=\sum_{n=0}e^{-rt}\frac{(rt)^{n}}{n!}\int_{D}A(x,n,D)\,dx$
$\displaystyle=\sum_{n=0}e^{-rt}\frac{(rt)^{n}}{n!}\int_{T(D)}\tilde{A}(x,n,T(D))\,dx=Q^{X^{\ast}}_{D^{\ast}}(t).$

$\Box$

Now we study the case $|\mathcal{A}|\geq 2^{d}|E|$.

###### Proposition 4.7

Let $X_{t}=S_{N_{t}}$ be a compound Poisson process whose jump distribution is
uniform on a set $\mathcal{A}\subset{\mathbb{R}}^{d}$ and $X^{\ast}$ be the
symmetrization of $X$. Assume that $|\mathcal{A}|\geq 2^{d}|D|$. Then, the
followings are equivalent:

  1. 1.

$Q_{D}^{X}(t)=Q^{X^{\ast}}_{D^{*}}(t)$ for all $t>0$.

  2. 2.

$Q_{D}^{X}(t)=Q^{X^{\ast}}_{D^{*}}(t)$ for some $t>0$.

  3. 3.

$D-D\subset\mathcal{A}$ up to measure zero, where $D-D:=\\{x-y:x,y\in D\\}$.

  4. 4.

$Q_{D}^{X}(t)=|D|\exp\left(-rt\left(1-\frac{|D|}{|\mathcal{A}|}\right)\right)$
for all $t>0$.

Proof. Clearly, the first condition implies the second.

Secondly, suppose that $Q_{D}^{X}(t)=Q_{D^{\ast}}^{X^{\ast}}(t)$ for some
$t>0$. It follows from the proof of Proposition 4.6 we have
$\mathcal{J}(1_{-D},1_{D},1_{\mathcal{A}})=\mathcal{J}(1_{D^{*}},1_{D^{*}},1_{\mathcal{A}^{*}})$.
The condition $|\mathcal{A}|\geq 2^{d}|D|$ is equivalent to
$\gamma\geq\alpha+\beta$ in Theorem 3.1 and from Theorem 3.1 we conclude
$D-D\subset\mathcal{A}$.

Thirdly, assume that $D-D\subset\mathcal{A}$. Without lose of generality, we
may assume $x=0$ and $0\in D$. Since $D=D-0\subset D+(-D)\subset\mathcal{A}$

$\displaystyle
A(0,1,D)=\int_{{\mathbb{R}}^{d}}1_{D}(y)\frac{1}{|\mathcal{A}|}1_{\mathcal{A}}(y)\,dy=\frac{1}{|\mathcal{A}|}\int_{{\mathbb{R}}^{d}}1_{D\cap\mathcal{A}}(y)\,dy=\frac{1}{|\mathcal{A}|}\int_{{\mathbb{R}}^{d}}1_{D}(y)\,dy=\frac{|D|}{|\mathcal{A}|}.$

Similarly, by the chain rule for conditional probability, we have

$A(0,n,D)={\mathbb{P}}\left(\bigcap_{k=1}^{n}\\{S_{k}\in
D\\}\right)=\prod_{k=1}^{n}{\mathbb{P}}\left(S_{k}\in
D\Big{|}\bigcap_{j=1}^{k-1}\\{S_{j}\in D\\}\right).$

We will show that ${\mathbb{P}}\left(S_{k}\in
D\Big{|}\bigcap_{j=1}^{k-1}\\{S_{j}\in D\\}\right)=\frac{|D|}{|\mathcal{A}|}$
for each $1\leq k\leq n$. Let $j_{k-1}(y_{1},\cdots,y_{k-1})$ be the joint
density for $(S_{1},\cdots,S_{k-1})$. Since $\\{S_{j}\\}_{j=1}^{k-1}$ and
$J_{k}$ are independent, their joint density is
$j_{k-1}(y_{1},\cdots,y_{k-1})\cdot\frac{1}{|\mathcal{A}|}1_{\mathcal{A}}(z)$.
Hence,

$\displaystyle{\mathbb{P}}\left(S_{k}\in D|\bigcap_{j=1}^{k-1}\\{S_{j}\in
D\\}\right)=\frac{{\mathbb{P}}\left(\bigcap_{j=1}^{k-1}\\{S_{j}\in
D\\},S_{k-1}+J_{k}\in
D\right)}{{\mathbb{P}}\left(\bigcap_{j=1}^{k-1}\\{S_{j}\in D\\}\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{{\mathbb{P}}\left(\bigcap_{j=1}^{k-1}\\{S_{j}\in
D\\}\right)}\int_{({\mathbb{R}}^{d})^{k-1}}\int_{{\mathbb{R}}^{d}}1_{D^{k-1}}(y_{1},\cdots,y_{k-1})1_{D-y_{k-1}}(z)j_{k-1}(y_{1},\cdots,y_{k-1})\frac{1_{\mathcal{A}}(z)}{|\mathcal{A}|}dzdy_{1}\cdots
dy_{k-1}$ $\displaystyle=$
$\displaystyle\frac{1}{{\mathbb{P}}\left(\bigcap_{j=1}^{k-1}\\{S_{j}\in
D\\}\right)}\int_{({\mathbb{R}}^{d})^{k-1}}1_{D^{k-1}}(y_{1},\cdots,y_{k-1})j_{k-1}(y_{1},\cdots,y_{k-1})\int_{{\mathbb{R}}^{d}}1_{D-y_{k-1}}(z)\frac{1_{\mathcal{A}}(z)}{|\mathcal{A}|}dzdy_{1}\cdots
dy_{k-1}$ $\displaystyle=$
$\displaystyle\frac{1}{{\mathbb{P}}\left(\bigcap_{j=1}^{k-1}\\{S_{j}\in
D\\}\right)}\int_{({\mathbb{R}}^{d})^{k-1}}1_{D^{k-1}}(y_{1},\cdots,y_{k-1})j_{k-1}(y_{1},\cdots,y_{k-1})\frac{|(D-y_{k-1})\cap\mathcal{A}|}{|\mathcal{A}|}dy_{1}\cdots
dy_{k-1}$ $\displaystyle=$
$\displaystyle\frac{|D-y_{k-1}|}{|\mathcal{A}|}=\frac{|D|}{|\mathcal{A}|},$

where we used the fact $D-y_{k-1}\subset D-D\subset\mathcal{A}$. This
establishes ${\mathbb{P}}\left(S_{k}\in D\Big{|}\bigcap_{j=1}^{k-1}\\{S_{j}\in
D\\}\right)=\frac{|D|}{|\mathcal{A}|}$ for each $1\leq k\leq n$ and
$A(0,n,D)=\left(\frac{|D|}{|\mathcal{A}|}\right)^{n}$ for all $n\geq 1$. Then,

$\displaystyle
Q_{D}^{X}(t)=\sum_{n=0}^{\infty}e^{-rt}\frac{(rt)^{n}}{n!}\int_{D}A(x,n,E)dx=\sum_{n=0}^{\infty}|D|\left(\frac{|D|}{|\mathcal{A}|}\right)^{n}\cdot
e^{-rt}\frac{(rt)^{n}}{n!}=|D|\exp\left(-rt\left(1-\frac{|D|}{|\mathcal{A}|}\right)\right).$

Finally, assume the spectral heat content is given by
$Q_{D}^{X}(t)=|D|\exp\left(-rt\left(1-\frac{|D|}{|\mathcal{A}|}\right)\right)$.
By repeating the same argument for $X^{\ast}$ and $D^{\ast}$, we know that
$Q_{D^{\ast}}^{X^{\ast}}(t)=|D|\exp\left(-rt\left(1-\frac{|D|}{|\mathcal{A}|}\right)\right)$
and this implies $Q_{D}^{X}(t)=Q_{D^{\ast}}^{X^{\ast}}(t)$ for all $t>0$.
$\Box$

We are now ready to prove Theorem 2.3.

Proof of Theorem 2.3. The first part on the equivalent conditions follows from
Propositions 4.6 and 4.7.

Assume that the spectral heat content for the process $X$ has the spectral
representation (3.1). From [1, Corollary 5.5], $Q_{D}^{X}(t)\leq
Q_{D^{\ast}}^{X^{\ast}}(t)$ for all $t>0$, and the principal eigenvalue
$\lambda_{1}^{X}$ can be obtained using (3.2). Hence, any equivalent condition
on the maximality of the spectral heat content guarantees the minimality of
the principal eigenvalue. $\Box$

Acknowledgment: The second-named author is grateful to Professor Younghwan Son
(POSTECH, South Korea) for the discussion and helpful comments on Lemmas 4.4
and 4.5.

## References

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Daesung Kim

School of Mathematics, Georgia Institute of Technology, GA 30332, USA

E-mail<EMAIL_ADDRESS>
Hyunchul Park

Department of Mathematics, State University of New York at New Paltz, NY
12561,USA

School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea

E-mail<EMAIL_ADDRESS>


# On finite categories of algebraic varieties

Junho Peter Whang  Department of Mathematical Sciences and RIM, Seoul National
University<EMAIL_ADDRESS>
###### Abstract.

We prove that the finiteness of a finitely generated category of irreducible
algebraic varieties over a field of characteristic zero is decidable. We also
obtain a Burnside finiteness criterion for such a category, with applications
to algebraic dynamical systems of several maps.

##  1\. Introduction

### 1.1.

Jacob [4] showed that the finiteness of a finitely generated monoid of
matrices over a field is decidable. This paper provides a nonlinear
generalization of this result for finitely generated categories of irreducible
algebraic varieties, over fields of characteristic zero. In this paper, by an
algebraic variety we mean a reduced separated scheme of finite type over a
field. Let us define a _system_ in a category $C$ to be a quiver (i.e.,
directed multigraph) whose vertices are objects in $C$ and whose arrows are
morphisms in $C$ between the vertices. In other words, a system in $C$
specifies the generators of a subcategory of $C$.

######  Theorem 1.1.

Let $k$ be a field of characteristic zero. There exists an algorithm to
determine, given an explicit finite system of irreducible algebraic varieties
over $k$, whether or not the category it generates is finite.

We make precise the notion of an explicitly given system of varieties in
Section 2.2. We prove Theorem 1.1 by induction on the complexity of the
system, using two ingredients: an effective form of nonlinear Selberg’s lemma
due to Bass-Lubotzky [1], and the observation that dominant endormophisms of
finite order on integral schemes are automorphisms (Lemma 3.1). A similar
argument yields a solution to the strong Burnside problem for categories of
varieties in characteristic zero, generalizing (in characteristic zero) the
work of McNaughton-Zalcstein [6] on matrix monoids. Let us say that a category
is _torsion_ if every endomorphism of every object generates a cyclic monoid
of finite order under composition.

######  Theorem 1.2.

Let $k$ be a field of characteristic zero. Let $C$ be a finitely generated
subcategory of the category of irreducible algebraic varieties over $k$. Then
$C$ is finite if and only if it is torsion.

Over general fields, one can deduce a weaker finiteness criterion for
categories of varieties, establishing the analogue of the bounded Burnside
problem. This relies on Zelmanov’s resolution of the restricted Burnside
problem [11, 12]. One source of motivation for our work is the study of finite
orbits in dynamics of several maps on algebraic varieties. For instance, we
prove the following.

######  Corollary 1.3.

Let $k$ be a field. Let $M$ be a finitely generated monoid acting on an
algebraic variety $V/k$. Let $x\in V(k)$. Then $M\cdot x$ is finite if and
only if

$\sup_{N}|N\cdot x|<\infty$

where $N$ runs over all $2$-generated submonoids of $M$.

Given a set $X$ and a monoid $M$ of endomorphisms of $X$, let us say that a
point $x\in X$ is _$M$ -periodic_ if the $M$-orbit $M\cdot x$ is finite and
$M$ permutes the elements of $M\cdot x$. In the case where $k$ is a field of
characteristic zero and $M$ is a finitely generated monoid acting on a variety
$V/k$, one can combine a refined form of Corollary 1.3 with [10, Theorem 1.2]
to obtain the following.

######  Theorem 1.4.

Let $k$ be a field of characteristic zero. Let $M$ be a finitely generated
monoid acting on an algebraic variety $V/k$. Let $x\in V(k)$. Then the
following are equivalent:

  1. (a)

$x$ is $M$-periodic.

  2. (b)

$x$ is $N$-periodic for every $2$-generated submonoid $N\leq M$.

If moreover $M$ is a group, then the above are equivalent to:

  1. (c)

$x$ is $\langle f\rangle$-periodic for every $f\in M$.

This paper is organized as follows. Section 2 collects the necessary
background, including results of Bass-Lubotzky [1] and Zelmanov [11, 12]. In
Section 3, we prove Theorems 1.1 and 1.2. In Section 4, we consider dynamical
corollaries of our main results, and in particular prove Corollary 1.3 and
Theorem 1.4.

###  1.2. Acknowledgments

I thank Abhishek Oswal for helpful conversations. In particular, the proof of
Theorem 1.1 was inspired by an unpublished collaborative work on finite matrix
monoids. This work was supported by the Samsung Science and Technology
Foundation under Project Number SSTF-BA2201-03.

##  2\. Background

###  2.1. Notations

Let us set up the notations and terminology that will be used throughout the
paper. A quiver is a directed multigraph. Given a quiver $Q$, we write $Q_{0}$
for the class of its vertices and $Q_{1}$ for the class of its arrows. Let
$s,t:Q_{1}\rightrightarrows Q_{0}$ denote the source and target maps of $Q$.
We shall say that a quiver is _small_ if $Q_{0}$ and $Q_{1}$ are sets. We
shall view a category as a quiver along with a composition law on its arrows
that satisfies the usual axioms.

Let $C$ be a category. If $X,Y\in C_{0}$, we write
$\operatorname{Hom}_{C}(X,Y)$ for the set of morphisms in $C$ from $X$ to $Y$.
We write $\operatorname{End}_{C}(X)=\operatorname{Hom}_{C}(X,X)$. Given $X\in
C_{0}$, we write $C_{X}$ for the full subcategory of $C$ with a single object
$X$. Thus, $(C_{X})_{1}=\operatorname{End}_{C}(X)$. In this paper, a _monoid_
is a small category whose object set is a singleton. A monoid is _cyclic_ if
it is generated by a single endomorpism. A _groupoid_ is a small category
whose morphisms are all invertible.

######  Definition 2.1.

Let $C$ be a small category.

  1. (1)

The _order_ of $C$ is $|C|=|C_{1}|$.

  2. (2)

We say $C$ is _finite_ if it has finite order.

  3. (3)

We say $C$ is _torsion_ if every cyclic submonoid of $C$ is finite.

  4. (4)

We say $C$ is _$n$ -torsion_ if every cyclic submonoid of $C$ has order $\leq
n$.

Forgetting the composition law on a category $C$, we obtain a quiver (i.e.
directed multigraph) whose collection of vertices is $C_{0}$, whose collection
of arrows is $C_{1}$, and whose source and target maps are $s$ and $t$.

######  Definition 2.2.

A _system_ in a category $C$ to be a small subquiver of the quiver underlying
$C$. Given a system $S$ in $C$, let $\langle S\rangle$ denote the subcategory
of $C$ generated by $S$, i.e. smallest subcategory $C^{\prime}$ of $C$ such
that $S_{0}=C^{\prime}_{0}$ and $S_{1}\subseteq C^{\prime}_{1}$. We say that a
category $C$ is _finitely generated_ if there is a finite quiver $S$ in $C$
(i.e. with $|S_{0}|$ and $|S_{1}|$ finite) such that $C=\langle S\rangle$.

######  Definition 2.3.

Let $S$ be a quiver. A _path_ in $S$ from $v\in S_{0}$ to $w\in S_{0}$ is a
sequence $f_{1},\dots,f_{k}\in S_{1}$ of arrows such that $s(f_{1})=v$,
$t(f_{k})=w$, and $t(f_{i})=s(f_{i+1})$ for $i=1,\dots,k-1$. Two vertices
$v,w\in S_{0}$ are _path-equivalent_ if there is a path in $S$ from $v$ to $w$
and there is a path in $S$ from $w$ to $v$. We denote by $S^{\circ}$ the
quiver obtained from $S$ by deleting the arrows between vertices that are not
path-equivalent. A system $S$ is _path-connected_ if $S=S^{\circ}$. A _path-
component_ of $S$ is a maximal subquiver of $S$ that is path-connected.

######  Lemma 2.4.

A finitely generated category $C$ is finite if and only if $C^{\circ}$ is
finite.

###### Proof.

If $C$ is finite, then obviously $C^{\circ}$ is finite. Suppose conversely
that $C^{\circ}$ is finite. Let $N$ be the number of path-components of $C$.
Let $S$ be a finite system of generators for $C$. Every morphism of $f$ can be
written in the form

$f=g_{k}h_{k}g_{k-1}h_{k-1}\dots g_{2}h_{1}g_{0}$

for some $k\leq N$, where $g_{i}\in c(C)_{1}$ and $h_{i}\in S_{1}\setminus
c(C)_{1}$ for each $i=0,\dots,k$. It follows that $C_{1}$ is finite, so $C$ is
finite. ∎

###  2.2. Bass-Lubotzky

We recall a theorem of Bass-Lubotzky [1, Corollary (1.2)].

######  Theorem 2.5.

Let $k$ be an arbitrary ring. Let $G$ be a finitely generated group of
automorphisms of a scheme $V$ of finite presentation over $k$.

  1. (1)

$G$ is residually finite.

  2. (2)

If $V$ is flat over ${\mathbb{Z}}$, then $G$ is virtually torsionfree.

We shall also need an effective form of the second part of Theorem 2.5, whose
formulation we recall as follows. Let $k$ be a finitely generated subring of
$\bar{\mathbb{Q}}$. Let $Z$ be a scheme flat of finite type over $k$. Given a
finite set $X$ of closed points of $Z$, let

$A_{X}=\prod_{x\in X}{\mathcal{O}}_{Z,x}\quad\text{and}\quad J_{X}=\prod_{x\in
X}{\mathfrak{m}}_{Z,x}$

where ${\mathcal{O}}_{Z,x}$ denotes the local ring of $Z$ at $x$ and
${\mathfrak{m}}_{Z,x}$ is the maximal ideal of ${\mathcal{O}}_{Z,x}$, with
residue field $\kappa(x)={\mathcal{O}}_{Z,x}/{\mathfrak{m}}_{Z,x}$. We shall
say that $X$ has _residue characteristic $p$_ if $\kappa(x)=p$ for every $x\in
X$. Following [1], we shall say that $X$ is _effective_ if there is a finite
affine open covering $(U_{i})_{i\in I}$ of $Z$ such that the natural morphism
${\mathcal{O}}_{Z}(U_{i})\to\prod_{x\in X\cap U_{i}}{\mathcal{O}}_{Z,x}$ is
injective for every $i\in I$. The following holds.

######  Proposition 2.6.

Suppose that $X$ is a finite effective set of closed points in $Z$ with
residue characteristic $p$. Then the order of any torsion element in

$\Gamma_{X}=\ker(\operatorname{Aut}(Z/k)\to\operatorname{Aut}Z(A_{X}/J_{X}^{2}))$

is a power of $p$. In particular, if $X$ and $X^{\prime}$ are finite effective
sets in $Z$ having distinct residue characteristics $p\neq p^{\prime}$, then
$\Gamma_{X}\cap\Gamma_{X^{\prime}}$ is a normal torsionfree subgroup of finite
index in $\operatorname{Aut}(Z/k)$.

###### Proof.

The proof is given in [1, pp.4-5]. See also [10, Section 2.1] for a summary. ∎

###  2.3. Burnside problem

We refer to [8] for a summary of the history of the Burnside problem. Here, we
recall their formal statements of its variants.

######  Problem 2.7 (Strong Burnside’s problem).

Let $G$ be a finitely generated group all of whose elements are torsion. Is
$G$ finite?

######  Problem 2.8 (Bounded Burnside’s problem).

Let $G$ be a finitely generated group all of finite exponent. Is $G$ finite?

######  Problem 2.9 (Restricted Burnside’s problem).

Are there only finitely many finite groups with given number of generators and
given exponent?

While both the strong Burnside’s problem and bounded Burnside’s problem have
negative answers in general (in the strong case by work of Golod-Shafarevich
[3], and in the bounded case by work of Adian and Novikov [7]), for linear
groups they admit affirmative answers (due to Schur [9] and Burnside [2],
respectively). The restricted Burnside problem was solved affirmatively by
Zelmanov [11, 12]. An immediate corollary of his work is the following
characterization of finite groups.

######  Theorem 2.10.

A group $G$ is finite if and only if it is finitely generated, residually
finite, and of finite exponent.

In the meanwhile, analogues of Burnside’s problems for other algebraic
structures such as semigroups have been studied. In [6], McNaughton-Zalcstein
[6] established the analogue of the strong Burnside’s problem for semigroups
(or monoids) of matrices over arbitrary fields. Theorem 1.2 serves to
generalize this result, in characteristic zero, to categories of algebraic
varieties.

##  3\. Proofs of Theorems 1.1 and 1.2

###  3.1. Explicitly given systems of varieties

Here, we make precise our notion of an explicitly given finite systems of
algebraic varieties, generally following the spirit of the paragraph after
[10, Theorem 1.2].

  1. (1)

A finitely presented ring $k$ is explicitly given if it is given as a quotient
of a polynomial ring with coefficients in ${\mathbb{Z}}$, and a finite set of
generators for the kernel of the quotient map is specified. A ring
homomorphism between explicitly given finitely presented rings is explicitly
given if the images of the generators (given by the explicit finite
presentation) of the domain ring are specified. In what follows, let $k$ be an
explicitly given finitely presented ring.

  2. (2)

An affine scheme of finite presentation over $k$ is explicitly given if it is
the spectrum of an explicitly given $k$-algebra. A morphism between two
explicitly given affine schemes of finite presentation over $k$ is explicitly
given if it is induced by an explicitly given $k$-algebra homomorphism between
their coordinate rings.

  3. (3)

A scheme $V$ separated of finite presentation over $k$ is explicitly given if
it is written as an explicit finite union of explicitly given open affine
schemes $U_{i}$ with affine overlaps $U_{i}\cap U_{j}$, such that the gluing
morphisms $U_{i}\cap U_{j}\to U_{i}$ are explicitly given. We shall call
$(U_{i})$ an effective presentation of $V$.

  4. (4)

Let $V$ and $W$ be explicitly given schemes of finite presentation over $k$,
with effective presentations $(U_{i})_{i\in I}$ and $(T_{j})_{j\in J}$
respectively. A morphism $f:V\to W$ over $k$ is explicitly given if there is
another effective presentation $(U_{i}^{\prime})_{i\in I^{\prime}}$ of $V$ and
an explicit function $j:I^{\prime}\to J$ such that $f(U_{i}^{\prime})\subseteq
T_{j(i)}$ for every index $i$, and the following hold:

    1. (a)

$f|_{U_{i}^{\prime}}:U_{i}^{\prime}\to T_{j(i)}$ is an explicitly given
morphism for all $i\in I$, and

    2. (b)

the inclusions of $U_{i}^{\prime}\cap U_{k}$ into $U_{i}^{\prime}$ and
$U_{k}^{\prime}$ are explicitly given morphisms for all $(i,k)\in
I^{\prime}\times I$.

Finally, a finite system $S$ of algebraic varieties over a field $K$ is
explicitly given if there is an explicitly given finitely presented domain
$k\subset K$ and a finite system $S_{k}$ of schemes over $k$, whose vertices
are explicitly given separated schemes of finite type over $k$ and whose
arrows are explicitly given morphisms between those schemes, such that $S$ is
obtained from $S_{k}$ by base change.

###  3.2. Decidability of finiteness

Here, we shall prove Theorem 1.1. Let us begin with a lemma.

######  Lemma 3.1.

Let $X$ be an integral separated scheme. If $f$ is a dominant endomorphism of
$X$ of finite order, then $f$ is an automorphism.

###### Proof.

Let $k(X)$ be the function field of $X$. Since $f$ is dominant, it induces an
inclusion $f^{*}:k(X)\to k(X)$. Since there exist $0\leq m<n$ such that
$f^{m}=f^{n}$, in fact $f^{*}$ must induce an automorphism of $k(X)$, and
$(f^{n-m})^{*}$ is the identity on $k(X)$. This implies that $f^{n-m}$ is the
identity on a dense open subscheme of $X$. Since $X$ is separated, this
implies that $f^{n-m}={\textup{id}}_{X}$. ∎

######  Proposition 3.2.

Let $k$ be a finitely presented domain of characteristic zero. Let $S$ be a
finite path-connected system of dominant morphisms between integral separated
schemes of finite type over $k$. If $\langle S\rangle$ is finite, then
$|\langle S\rangle|\leq C(S_{0})$ where $C(S_{0})$ is a constant that only
depends on $S_{0}$.

###### Proof.

Let $S$ be a finite system as in the statement of the proposition. By Lemma
3.1 and our assumptions on $S$, every endomorphism of an object in $\langle
S\rangle$ is an automorphism of finite order. It follows that $\langle
S\rangle$ is a groupoid. Fix $Z\in S_{0}$. For every $W\in S_{0}$, the set
$\operatorname{Hom}_{\langle S\rangle}(Z,W)$ is a torsor under the group
$\langle S\rangle_{Z}$, so we have $|\langle S\rangle|=|\langle
S\rangle_{Z}|^{|S_{0}|}$. Now, $\langle S\rangle_{Z}$ is a finite subgroup of
$\operatorname{Aut}(Z/k)$. Fix two closed points $x$ and $x^{\prime}$ of the
integral scheme $Z$ such that the characteristics of the residue fields
$\kappa(x)$ and $\kappa(x^{\prime})$ are coprime. Setting $X=\\{x\\}$ and
$X^{\prime}=\\{x^{\prime}\\}$, we see by Proposition 2.6 that the composition
of group homomorphisms

$\langle
S\rangle_{Z}\to\operatorname{Aut}(Z/k)\to\operatorname{Aut}Z(A_{X}/J_{X}^{2})\times\operatorname{Aut}(A_{X}/J_{X}^{2})$

is injective. Since the right hand side only depends on $Z$, we are done. ∎

######  Theorem 1.1.

Let $k$ be a field of characteristic zero. There exists an algorithm to
determine, given an explicit finite system of irreducible algebraic varieties
over $k$, whether or not the category it generates is finite.

###### Proof.

Let $S$ be a finite system of irreducible algebraic varieties over
$\bar{\mathbb{Q}}$. By spreading out, we can explicitly determine a finitely
generated subring of $\bar{\mathbb{Q}}$ such that $S$ descends to a finite
system of integral separated schemes flat of finite type over $k$. We shall
proceed by induction on $|S_{1}|$ and $\max_{V\in S_{0}}\dim V$. By Lemma 2.4,
we may assume that $S$ is path-connected. If every $f\in S_{1}$ is dominant,
then by Proposition 3.2 it is decidable whether or not $\langle S\rangle$ is
finite. So assume that some $f\in S_{1}$ is not dominant, and write $A,B\in
S_{0}$ for the source and target of $f$ respectively. Let $Z$ be the Zariski
closure of the image of $f$ in $B$, equipped with the reduced closed subscheme
structure. Note that $Z$ is integral, separated, and flat of finite type over
$k$. Consider the finite systems $S^{\prime}$ and $S^{\prime\prime}$ where:

  * (a)

$S_{0}^{\prime}=S_{0}$ and $S_{1}^{\prime}=S_{1}\setminus\\{f\\}$, and

  * (b)

$S_{0}^{\prime\prime}=\\{Z\\}$ and $S_{1}^{\prime\prime}=\\{(f\circ
g)|_{Z}:g\in\operatorname{Hom}_{\langle S^{\prime}\rangle}(B,A)\\}$.

Since $|S_{1}^{\prime}|<|S_{1}|$, by inductive hypothesis it is decidable
whether or not $\langle S^{\prime}\rangle$ is finite, so we may assume it is.
This implies that $|S_{1}^{\prime\prime}|$ is finite, and since $Z$ has
smaller dimension than $B$, by inductive hypothesis it is decidable whether or
not $\langle S^{\prime\prime}\rangle$ is finite. We may assume $\langle
S^{\prime\prime}\rangle$ is finite. Then for every $V,W\in S_{0}$, if
$g\in\operatorname{Hom}_{\langle S\rangle}(V,W)$ then either
$g\in\operatorname{Hom}_{\langle S^{\prime}\rangle}(V,W)$ or

$g=h_{2}\circ e\circ f\circ h_{1}$

for some $h_{1}\in\operatorname{Hom}_{\langle S^{\prime}\rangle}(V,A)$,
$e\in\langle S^{\prime\prime}\rangle_{1}$, and
$h_{2}\in\operatorname{Hom}_{\langle S^{\prime}\rangle}(B,W)$. It follows that
$\langle S\rangle$ is finite. This completes the induction and the proof. ∎

###  3.3. Burnside finiteness criterion

We now prove the following extended version of Theorem 1.2, by adapting our
proof of Theorem 1.1.

######  Theorem 3.3.

Let $k$ be a field. Let $C$ be a finitely generated category of irreducible
algebraic varieties over $k$. Then $C$ is finite if and only if:

  1. (1)

$C$ is $n$-torsion for some $n\geq 1$, or

  2. (2)

$C$ is torsion and $k$ has characteristic zero.

###### Proof.

The “only if” direction is clear, so we shall focus on the “if” direction. Let
$S$ be a finite system generating $C$. As in the proof of Theorem 1.1, we
proceed by induction on $|S_{1}|$ and $\max_{V\in S_{0}}\dim V$. By Lemma 2.4,
we may assume that $S$ is path-connected. First, suppose that every $f\in
S_{1}$ is dominant. Since $C$ is torsion, arguing as in the proof of
Proposition 3.2, we see that $C$ is a finitely generated groupoid, and hence
its finiteness is equivalent to the finiteness of the finitely generated group
$C_{V}$ for some (or any) $V\in C_{0}$. Now, we have the following.

  1. (1)

If $C$ is $n$-torsion for some $n\geq 1$, then since the group $C_{V}$ is
residually finite by Theorem 2.5(1), it is finite by the resolution of the
restricted Burnside problem (Theorem 2.10).

  2. (2)

If $C$ is torsion and $k$ has characteristic zero, then since $C_{V}$ is
virtually torsionfree by Theorem 2.5 it is finite.

This proves the theorem in the case where every $f\in S_{1}$ is dominant. If
some $f\in S_{1}$ is not dominant, then we argue as in the proof of Theorem
1.1 to construct systems $S^{\prime}$ and $S^{\prime\prime}$ with smaller
complexity, such that finiteness of $C^{\prime}=\langle S^{\prime}\rangle$ and
$C^{\prime\prime}=\langle S^{\prime\prime}\rangle$ imply the finiteness of
$C$. If $C$ satisfies condition (1) or (2) of the theorem, then clearly
$C^{\prime}$ and $C^{\prime\prime}$ also satisfy the same condition. Thus,
$C^{\prime}$ and $C^{\prime\prime}$ are finite by inductive hypothesis, and
hence $C$ is finite. ∎

##  4\. Dynamical corollaries

We record some dynamical corollaries of Theorem 3.3, by transferring
properties of monoids to their orbits. The following is a refined version of
Corollary 1.3.

######  Corollary 4.1.

Let $k$ be a field. Let $M$ be a finitely generated monoid acting on an
algebraic variety $V/k$. Let $x\in V(k)$. The following are equivalent.

  1. (1)

$M\cdot x$ is finite.

  2. (2)

There exists $n\geq 1$ with $|N\cdot x|\leq n$ for every $2$-generated
submonoid $N\leq M$.

  3. (3)

There exists $n\geq 1$ with $|\langle f\rangle\cdot g(x)|\leq n$ for every
$f,g\in M$.

###### Proof.

It is clear that $(1)\implies(2)\implies(3)$, so it remains to show
$(3)\implies(1)$. Suppose $(3)$ holds. Let $Z$ be the Zariski closure of
$M\cdot x$ in $V$, equipped with the reduced closed subscheme structure. Let
us now construct a finite system $S$ of irreducible varieties, where the
elements of $S_{0}$ are the irreducible components of $Z$ and the elements of
$S_{1}$ are the restrictions of the elements of a finite generating set of $M$
to the irreducible components of $Z$. Let $C=\langle S\rangle$. If $f\in
C_{1}$ is any endomorphism of an object $A$ in $C$, we claim that $f$ has
finite order $\leq n$, i.e., $C$ is $n$-torsion. Indeed, there exists some
$0\leq m<n$ such that $A(k)$ contains a Zariski dense set $T$ of points
satisfying $f^{n}(y)=f^{m}(y)$ for all $y\in T$. Since $A$ is integral and
separated, it follows that $f^{n}=f^{m}$ on $A$. Thus $C$ is $n$-torsion, and
by Theorem 3.3 it follows that $C$ is finite. But then the orbit of $x$ must
be finite. ∎

######  Theorem 1.4.

Let $k$ be a field of characteristic zero. Let $M$ be a finitely generated
monoid acting on an algebraic variety $V/k$. Let $x\in V(k)$. Then the
following are equivalent:

  1. (a)

$x$ is $M$-periodic.

  2. (b)

$x$ is $N$-periodic for every $2$-generated submonoid $N\leq M$.

If moreover $M$ is a group, then the above are equivalent to:

  1. (c)

$x$ is $\langle f\rangle$-periodic for every $f\in M$.

###### Proof.

The implication (a)$\implies$(b) is clear, so let us show (b)$\implies$(a).
Assume (b). By [10, Theorem 1.2], there is a constant $n\geq 0$, depending
only on $V$ and the finitely generated ring over which $V$, $x$, and $M$ are
defined, such that $|N\cdot x|\leq n$ for every $2$-generated submonoid $N\leq
M$ by $N$-periodicity of $x$. This implies (a) by Corollary 4.1. Suppose now
that $M$ is a group. Since (a)$\implies$(c) is clear, we shall show
(c)$\implies$(a). Let us assume (c). In light of Corollary 4.1, it suffices to
show that there is a constant $n\geq 0$ such that $|\langle f\rangle\cdot
g(x)|\leq n$ for every $f,g\in M$, or (since $M$ is a group) equivalently
$|\langle h^{-1}fh\rangle\cdot x|\leq n$ for every $f,h\in M$, or equivalently
$|\langle g\rangle\cdot x|\leq n$ for every $g\in M$. Now, since $\langle
g\rangle\cdot x$ is finite for every $g\in M$ by hypothesis, the desired
result follows again by Theorem [10, Theorem 1.2]. ∎

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