# Gamma mixed fractional Lévy Ornstein - Uhlenbeck process

Héctor Araya1 Johanna Garzón2 Rolando Rubilar3
1 Instituto de Estadística, Facultad de Ciencias, Universidad de Valparaíso,
<EMAIL_ADDRESS>
2 Departamento de Matemática, Facultad de Ciencias, Universidad Nacional de
Colombia,
<EMAIL_ADDRESS>
3 Instituto de Estadística, Facultad de Ciencias, Universidad de Valparaíso,
<EMAIL_ADDRESS>

###### Abstract

In this article, we introduce a non Gaussian long memory process constructed
by the aggregation of independent copies of a fractional Lévy Ornstein-
Uhlenbeck process with random coefficients. Several properties and a limit
theorem are studied for this new process. Finally, some simulations of the
limit process are shown.

2010 AMS Classification Numbers: 60G10, 60G17, 60H05, 60H30. Key Words and
Phrases: non Gaussian, fractional Lévy process, Ornstein-Uhlenbeck, random
coefficients.

##  1 Introduction

An Ornstein-Uhlenbeck (OU) process is a diffusion process introduced by the
physicists Leonard Salomon Ornstein and George Eugene Uhlenbeck [28] to
describe the stochastic behavior of the velocity of a particle undergoing
Brownian motion. The OU process $X=\\{X(t),\ {t\geq 0}\\}$ is the solution of
the Langevin equation

$dX(t)=\alpha X(t)dt+\sigma dB(t),\ \ t\geq 0,$ (1)

where $X(0)=x\in\mathbb{R}$, $B=\\{B(t),t\geq 0\\}$ is a Brownian motion and
$\alpha,\sigma$ are constants. This process is stationary, Gaussian and
Markovian; in fact, it is the only stochastic process which has all these
three properties. This process is used for modelling in many different fields
such as physics, biology and finance among others (see [1, 6, 15, 28, 24, 29]
and references therein) and it has been widely generalized.

Different extensions of the Ornstein-Uhlenbeck processes have been obtained to
replace the Brownian motion in (1) by more general noise processes; for
example, Lévy OU [2], fractional OU [7], sub-fractional OU [23] or Hermite OU
processes [17]. These are introduced by solution of the Langevin equation with
driving noise given by a Lévy process, fractional Brownian motion, sub-
fractional Brownian motion, Hermite process respectively.

In the study of long-range dependence, Igoli and Terdik [14] defined a
generalization of OU process with this property. This process is called Gamma-
mixed Ornstein-Uhlenbeck processes and it is built via aggregation of a
sequence of random coefficient independent Ornstein-Uhlenbeck processes. Let
us be more precise. Given a sequence $(X_{k})_{k\in\mathbb{N}}$ of stochastic
processes such that for each $k\geq 1$, the process $X_{k}$ is the solution of
the Langevin equation

$dX_{k}(t)=\alpha_{k}X(t)dt+dB(t),$ (2)

where $B$ is a Brownian motion with time parameter $t\in\mathbb{R}$ and
$(-\alpha_{k})_{k\in\mathbb{N}}$ are independent random variables (also
independent of $B$) with Gamma distribution $\Gamma(1-h,\lambda)$ with
$h\in(0,1)$ and $\lambda>0$. The aggregated process is given by

$Y_{n}(t)=\frac{1}{n}\sum_{k=1}^{n}X_{k}(t),$

and it converges, as $n\to\infty$, to a stochastic process $Y$ which is a
stationary Gaussian process, semimartingale, asymptotically self-similar and
it has long-range dependence. This limit process is the so-called Gamma-mixed
Ornstein-Uhlenbeck. In a similar way, in [9] and [10] the authors studied the
case where $B$ in (2) is generalized to be a fractional Brownian motion and
Hermite process respectively; they define the fractional Ornstein-Uhlenbeck
process mixed with a Gamma distribution and the Hermite Ornstein-Uhlenbeck
process mixed with a Gamma distribution, both processes exhibit long range
dependence, the first one is a Gaussian process but the second one is not
Gaussian.

The aim of this paper is to define and study some properties of the Gamma
mixed fractional Lévy Ornstein-Uhlenbeck process obtained as the limit of the
aggregated OU process driven by fractional Lévy process and random coefficient
with Gamma distribution.

The fractional Lévy process (fLp) was defined by [19] as a generalization of
the moving average representation of fractional Brownian motion given by
Mandelbrot and Van Ness [18] when replacing the Brownian motion in this
integral representation by a Lévy process with zero mean, finite variance and
without Gaussian part. FLp is almost surely Hölderian, has stationary
increments and long range dependence, but unlike fractional Brownian motion,
this process is neither Gaussian nor self-similar process. In [11], authors
introduced the fractional Lévy Ornstein-Uhlenbeck process (fLOUp) as the
unique stationary pathwise solution of the Langevin equation driven by a fLp
and prove that its increments exhibit long range dependence. Recently, many
authors have studied fLp and the fractional Lévy Ornstein-Uhlenbeck process on
theoretical and applicable levels, see for example [3, 4, 13, 16, 25, 27, 30]
and the references therein.

This paper is organized as follows. In Section 2 we give a brief introduction
to the fLp and the stochastic calculus related to this. Fractional Lévy
Ornstein-Uhlenbeck process with random coefficient is introduced in Section 3.
In Section 4, we define the aggregated processwhich of fLOUp with random
coefficient and study its limit process, which we will call Gamma mixed
fractional Lévy Ornstein-Uhlenbeck process. Finally, in Section 5 we present
some simulations of the paths of the Gamma mixed fractional Lévy Ornstein-
Uhlenbeck process.

##  2 Preliminaries

In this section, we briefly recall some relevant aspects of the fractional
Lévy process (fLp), its main properties and stochastic integrals with respect
to this fLp. This process will be used in the remainder of the paper. We work
on a complete probability space $(\Omega_{L},{\cal F}_{L},\mathbb{P}_{L})$.

###  2.1 Fractional Lévy process

The fractional Lévy process $L^{d}=\\{L^{d}_{t},\ t\in\mathbb{R}\\}$, with
$d\in(0,1/2)$, is a non-Gaussian process defined as follows (see [19]):

$L_{t}^{d}=\int_{\mathbb{R}}f_{t}^{(d)}(s)dL(s),\quad t\in\mathbb{R},$ (3)

where the kernel function $f_{t}^{(d)}$ is given by

$f_{t}^{(d)}(s)={1\over\Gamma(d+1)}[(t-s)_{+}^{d}-(-s)_{+}^{d}],\ \ \
s\in\mathbb{R}$ (4)

and $L=\\{L(t),\ t\in\mathbb{R}\\}$ is a zero-mean two-sided Lévy process with
$\mathbb{E}(L_{1}^{2})<\infty$ and without Brownian component, i.e.

$L_{t}=L^{(1)}_{t}1_{\\{t\geq 0\\}}-L^{(2)}_{-t}1_{\\{t\leq 0\\}},$

where $L^{(1)}$ and $L^{(2)}$ are two independent copies of the same one-sided
Lévy process.

The following Lemma (see [19, 16]) establishes that the fLp is well defined in
the $L^{2}(\Omega)$-sense and gives its characteristic function.

######  Lemma 1.

Let $L=\\{L_{t}\\}_{t\in\mathbb{R}}$ be a two-sided Lévy process without
Brownian component such that $E[L(1)]=0$ and $E[L(1)^{2}]<\infty$. For
$t\in\mathbb{R}$, let $f_{t}\in L^{2}(\Omega)$. Then the integral
$S(t):=\int_{\mathbb{R}}f_{t}(u)dL(u)$ exists in the $L^{2}(\Omega)$ sense and
$E[S(t)]=0$. Furthermore, $S(t)$ satisfies the isometry

$E[(S(t))^{2}]=E[(L(1))^{2}]\|f_{t}(\cdot)\|_{L^{2}(\mathbb{R})},\ \ \
t\in\mathbb{R},$

the covariance function of process $S$ is given by

$\tilde{\Gamma}(s,t)=cov(S(s),S(t))=E[(L(1))^{2}]\int_{\mathbb{R}}f_{t}(u)f_{s}(u)du,\
\ s,t\in\mathbb{R}$

and the characteristic function of $S(t_{1}),\ldots,S(t_{m})$ for
$t_{1}<\ldots<t_{m}$ and $m\in\mathbb{N}$ is given by

$E_{L}\left[\exp\left(i\sum_{j=1}^{m}\theta_{j}S(t_{j})\right)\right]=\exp\left(\int_{\mathbb{R}}\psi\left(\sum_{j=1}^{m}\theta_{j}f_{t_{j}}(s)\right)ds\right),\
\ \ \text{for}\ \theta_{j}\in\mathbb{R},\ j=1,\ldots,m,$

where

$\psi(u)=\int_{\mathbb{R}}(e^{iux}-1-iux)\nu(dx),\quad u\in\mathbb{R}$

and $\nu$ is the Lévy measure of $L$.

From (3) we can see that the covariance function of $L^{d}$ is given by

$\mathbb{E}(L^{d}_{t}L^{d}_{s})=\frac{1}{2}V_{d}^{2}\left(|t|^{2d+1}+|s|^{2d+1}-|t-s|^{2d+1}\right),\quad
t,s\in\mathbb{R},$ (5)

where $V_{d}^{2}=\frac{\mathbb{E}(L_{1}^{2})}{2\Gamma(2d+2)\sin(\pi(d+1/2))}$.
Up to a scaling constant, this is the same covariance as fractional Brownian
motion.

The fractional Lévy process $L^{d}$ defined by (3) has the following
properties (see [19] for their proofs):

  * •

For any $\beta\in(0,d)$, the sample paths of $L^{d}$ are a.s $\beta$\- Hölder
continuous.

  * •

$L^{d}$ is a process with stationary increments and symmetric, i.e.
$\\{L^{d}_{-t}\\}_{t\in\mathbb{R}}\stackrel{{\scriptstyle(d)}}{{=}}\\{-L^{d}_{t}\\}_{t\in\mathbb{R}}$.

  * •

$L^{d}$ cannot be self-similar. However, $L^{d}$ is asymptotically self-
similar with parameter $0<d<0.5$, i.e

$\lim\limits_{c\rightarrow\infty}\left\\{\frac{L^{d}_{ct}}{c^{d}}\right\\}_{t\in\mathbb{R}}\stackrel{{\scriptstyle(d)}}{{=}}\left\\{B^{d}_{t}\right\\}_{t\in\mathbb{R}},$

where the equality is in the sense of finite dimensional distributions.

  * •

For $h>0$, the covariance between two increments $L^{d}_{t+h}-L^{d}_{t}$ and
$L^{d}_{s+h}-L^{d}_{s}$, where $s+h\leq t$ and $t-s=nh$ is

$\displaystyle\delta_{d}(n)$ $\displaystyle=$ $\displaystyle
V_{d}^{2}h^{2d+1}\left[(n+1)^{2d+1}+(n-1)^{2d+1}-2n^{2d+1}\right]$ (6)
$\displaystyle=$ $\displaystyle
V_{d}^{2}d(2d+1)h^{2d+1}n^{2d-1}+O(n^{2d-2}),\quad n\rightarrow\infty.$

  * •

The increments of $L^{d}$ exhibit long memory, in the sense that for $d>0$, we
have

$\sum_{n=1}^{\infty}\delta_{d}(n)=\infty.$

In the following, we recall two results from the reference [19] concerning to
stochastic integrals with respect to fractional Lévy process.

Let $L^{1}(\mathbb{R})$ and $L^{2}(\mathbb{R})$ be the spaces of integrable
and square integrable real functions, respectively and $H$ the completion of
$L^{1}(\mathbb{R})\cap L^{2}(\mathbb{R})$ with respect to the norm
$\|g\|^{2}_{H}=E[L_{1}^{2}]\int_{\mathbb{R}}(I^{d}_{-}g)^{2}(u)du$, where
$(I^{d}_{-}{g})$ denotes the right-sided Riemann-Liouville fractional integral
defined by

$(I^{d}_{-}{g})(x)=\dfrac{1}{\Gamma(d)}\int_{x}^{\infty}g(t)(t-x)^{d-1}dt.$

######  Lemma 2.

If $g\in H$, then

$\int_{\mathbb{R}}g(s)dL^{d}_{s}=\int_{\mathbb{R}}(I^{d}_{-}{g})(s)dL_{s},$

where the equality is in $L^{2}$ sense.

The next second-order property of the stochastic integral respect to fLp will
be a key tool in this article.

######  Lemma 3.

If $|f|,|g|\in H$, then

$E\left[\int_{\mathbb{R}}f(s)dL^{d}_{s}\int_{\mathbb{R}}g(s)dL^{d}_{s}\right]=C_{d}\int_{\mathbb{R}}\int_{\mathbb{R}}f(t)g(s)|t-s|^{2d-1}dsdt,$

where

$C_{d}=\dfrac{\Gamma(1-2d)E[L_{1}^{2}]}{\Gamma(d)\Gamma(1-d)}.$ (7)

The following result from reference [11], established the solution of the
Langevin equation driven by a fractional Lévy process.

######  Theorem 1.

Let $L^{d}$ be an fLp, $d\in(0,1/2)$ and $\lambda>0$. Then the unique
stationary pathwise solution of the Langevin equation

$dX^{d,\lambda}(t)=\lambda X^{d,\lambda}(t)dt+dL^{d}_{t}$

is given a.s. by

$X^{d,\lambda}(t)=\int_{-\infty}^{t}e^{\lambda(t-u)}dL^{d}(u),\quad
t\in\mathbb{R},$ (8)

this process is called fractional Lévy-Ornstein-Uhlenbeck processes (fLOUp).

##  3 Ornstein-Uhlenbeck process with random coefficient

In this section, we study the fractional Lévy-Ornstein-Uhlenbeck processes
(fLOUp) with random coefficients. First, we establish the existence of the
solution, and then some properties of the process are shown.

We consider the fractional Lévy Ornstein-Uhlenbeck process
$V^{d}=(V^{d}(t),t\in\mathbb{R})$ given as the solution of

$dV^{d}(t)=\lambda V^{d}(t)dt+dL^{d}(t),\quad t\in\mathbb{R},$ (9)

where $L^{d}$ is a fLp with $d\in(0,1/2)$ and defined on $(\Omega_{L},{\cal
F}_{L},\mathbb{P}_{L})$; and the coefficient $\lambda$ is a random variable
defined on the probability space $(\Omega_{\lambda},{\cal
F}_{\lambda},\mathbb{P}_{\lambda})$ and independent by $L^{d}$. We assume that
$-\lambda$ follows a Gamma distribution with parameters $1-h$ and $\alpha$,
i.e. $-\lambda\sim\Gamma(1-h,\alpha)$ with $h\in(0,1)$ and $\alpha>0$.

From Theorem 1, the SDE given by (9) has the explicit solution

$V^{d}(t)=\int_{-\infty}^{t}e^{\lambda(t-u)}dL^{d}(u),\quad t\in\mathbb{R},$
(10)

where the initial condition is given by

$V^{d}(0)=\int_{-\infty}^{0}e^{-\lambda u}dL^{d}(u).$

Clearly, this process is stationary. Furthermore, by Lemma 2 we can get

$V^{d}(t)=\dfrac{1}{\Gamma(d)}\int_{\mathbb{R}}\int_{-\infty}^{t}e^{\lambda(t-v)}(v-u)_{+}^{d-1}dvdL(u),\quad
t\in\mathbb{R}.$

We will prove that for every $\omega_{\lambda}\in\Omega_{\lambda}$, the
process $V^{d}$ is well defined in $L^{2}(\Omega_{L})$. In fact, by Lemma 3 we
have that for $C_{d}=\dfrac{\Gamma(1-2d)}{\Gamma(d)\Gamma(1-d)}E[L_{1}^{2}]$

$\displaystyle E_{L}[(V^{d}(t))^{2}]$ $\displaystyle=$ $\displaystyle
C_{d}\int_{-\infty}^{t}\int_{-\infty}^{t}e^{\lambda(t-u)}e^{\lambda_{k}(t-v)}|u-v|^{2d-1}dudv$
(11) $\displaystyle=$ $\displaystyle
C_{d}\int_{0}^{\infty}\int_{0}^{\infty}e^{\lambda(u+v)}|u-v|^{2d-1}dudv$
$\displaystyle=$ $\displaystyle
2C_{d}\int_{0}^{\infty}\int_{0}^{u}e^{\lambda_{k}(u+v)}(u-v)^{2d-1}dvdu$
$\displaystyle=$ $\displaystyle 2C_{d}\int_{0}^{\infty}e^{2\lambda
u}\int_{0}^{u}e^{-\lambda v}v^{2d-1}dvdu$ $\displaystyle=$ $\displaystyle
2C_{d}\int_{0}^{\infty}e^{-\lambda v}v^{2d-1}\int_{v}^{\infty}e^{2\lambda
u}dudv$ $\displaystyle=$
$\displaystyle-\dfrac{C_{d}}{\lambda}\int_{0}^{\infty}e^{\lambda
v}v^{2d-1}dv=\dfrac{C_{d}}{(-\lambda)^{2d+1}}\Gamma(2d).$

######  Remark 1.

For $c\neq 0$, by (11) we have that
$E_{L}[(V^{d}(ct))^{2}]=\dfrac{C_{d}}{(-\lambda)^{2d+1}}\Gamma(2d)$. Hence the
processes $V^{d}$ is no self-similar.

######  Lemma 4.

The process $V^{d}$ (for fixed $\omega_{\lambda}\in\Omega_{\lambda}$) is
stationary, i.e. for $b>0$ and $t_{1}<\ldots<t_{n}$, with $n\in\mathbb{N}$

$(V^{d}(t_{1}+b),\ldots,V^{d}(t_{n}+b))\stackrel{{\scriptstyle(d)}}{{=}}(V^{d}(t_{1}),\ldots,V^{d}(t_{n})),$

where $\stackrel{{\scriptstyle(d)}}{{=}}$ means equality in the sense of
finite dimensional distributions.

Proof: For $b>0$, $u_{1},\ldots,u_{n}$ and $-\infty<t_{1}<\ldots<t_{n}$,
$n\in\mathbb{R}$, by the stationarity of the increments of $L^{d}$ we get:

$\displaystyle\sum_{i=1}^{n}u_{i}V^{d}(t_{i}+b)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}u_{i}\int_{-\infty}^{t_{i}+b}e^{\lambda(t_{i}+b-u)}dL^{d}(u)$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}$
$\displaystyle\sum_{i=1}^{n}u_{i}\int_{-\infty}^{t_{i}}e^{\lambda(t_{i}-u)}dL^{d}(u)$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{n}u_{i}V^{d}(t_{i}).$

∎ Using the results from [22, 20] already used in the reference [21] to
construct long memory process based on CARMA process driven by Lévy processes.
We will provide a spectral representation of the process $V^{d}$ given by
(10).

Let $L=\\{L(t)\\}_{t\in\mathbb{R}}$ be a two sided square integrable Lévy
process with $E[L(1)]=0$ and $E[L(1)^{2}]=\Sigma_{L}$, then there exists a
random orthogonal measure $\Phi_{L}$ with spectral measure $F_{L}$ such that
$E[\Phi_{L}(\Delta)]=0$ for any bounded Borel set $\Delta$,

$F_{L}(dt)=\dfrac{\Sigma_{L}}{2\pi}dt.$

Also, the random measure $\Phi_{L}$ is uniquely determined by

$\Phi_{L}([a,b))=\int_{\mathbb{R}}\dfrac{e^{-i\alpha a}-e^{-i\alpha b}}{2\pi
i\alpha}L(d\alpha),$ (12)

for all $-\infty<a<b<\infty$. Moreover,

$L(t)=\int_{\mathbb{R}}\dfrac{e^{i\alpha t}-1}{i\alpha}\Phi_{L}(d\alpha),\quad
t\in\mathbb{R}.$

Hence, for any function $f\in L^{2}(\mathbb{C})$,

$\int_{\mathbb{R}}f(\alpha)\Phi_{L}(d\alpha)=\dfrac{1}{2\pi}\int_{\mathbb{R}}\int_{\mathbb{R}}e^{-i\alpha
t}f(\alpha)d\alpha
L(dt)=\dfrac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\hat{f}(t)L(dt),$ (13)
$\int_{\mathbb{R}}\hat{f}(t)L(dt)=\int_{\mathbb{R}}\int_{\mathbb{R}}e^{i\alpha
t}\hat{f}(t)dt\Phi_{L}(d\alpha)=\sqrt{2\pi}\int_{\mathbb{R}}f(\alpha)\Phi_{L}(d\alpha),$
(14)

where

$\hat{f}(t)=\dfrac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-i\alpha
t}f(\alpha)d\alpha\quad\mbox{and}\quad
f(\alpha)=\dfrac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{i\alpha
t}\hat{f}(t)d\alpha.$

######  Lemma 5.

Let $V^{d}$ be a fLpOU with random coefficient given by (10). Then, the
spectral representation of $V^{d}$ is

$V^{d}(t)=\int_{\mathbb{R}}e^{it\alpha}\dfrac{1}{i\alpha-\lambda}\Phi_{M}(d\alpha),\quad
t\in\mathbb{R},$ (15)

where

$\Phi_{M}([a,b])=\int_{\mathbb{R}}\dfrac{e^{-ias}-e^{-ibs}}{2\pi
is}dL^{d}(s).$

Proof: By Theorem 2.8 in [20], we know that

$L^{d}(t)=\int_{\mathbb{R}}(e^{i\alpha
t}-1)(i\alpha)^{-(d+1)}\Phi_{L}(d\alpha),\quad t\in\mathbb{R}.$

Furthermore, following the proof of Theorem 2.8, Remark 2.9 and equality
(2.31) in the same reference, we can obtain

$\displaystyle V^{d}(t)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}}e^{\lambda(t-u)}1_{(-\infty,t)}(u)dL^{d}(u)$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}}\int_{\mathbb{R}}e^{i\alpha
u}e^{\lambda(t-u)}1_{(-\infty,t)}(u)du\Phi_{M}(d\alpha)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}}e^{it\alpha}\dfrac{1}{i\alpha-\lambda}\Phi_{M}(d\alpha).$

Then, the result is achieved. ∎

######  Corollary 1.

With almost no major effort, we can obtain, by Remark 2.9 in [20], that

$V^{d}(t)=\int_{\mathbb{R}}e^{it\alpha}\dfrac{1}{i\alpha-\lambda_{k}}(i\alpha)^{-d}\Phi_{L}(d\alpha),\quad
t\in\mathbb{R}.$

##  4 Gamma mixed fractional Lévy Ornstein - Uhlenbeck process

In this section, first, we will study a process defined by the aggregation of
independent fLOUp with random coefficient. Then we study its limit process,
which we call Gamma mixed fractional Lévy Ornstein - Uhlenbeck process. Some
properties of this limit process are given, namely, we give the characteristic
function of the finite dimensional distributions of the process, we
determinate its covariance, stationarity and long memory property; and
finally, we analyze the asymptotic behavior with respect to the parameter
$\alpha$.

###  4.1 Aggregated fractional Lévy Ornstein-Uhlenbeck process

Consider a sequence of fLOUp with random coefficient
$V_{k}^{d}=(V_{k}^{d}(t),t\in\mathbb{R})$, $k\geq 1$ given by

$V_{k}^{d}(t)=\int_{-\infty}^{t}e^{\lambda_{k}(t-u)}dL^{d}(u),\quad
t\in\mathbb{R},\quad k\geq 1,$ (16)

where $L^{d}$ is a fLp with $d\in(0,1/2)$ defined on $(\Omega_{L},{\cal
F}_{L},\mathbb{P}_{L})$. The random variables $\lambda_{k}$ are assumed
independent and identically distributed defined on the probability space
$(\Omega_{\lambda},{\cal F}_{\lambda},\mathbb{P}_{\lambda})$, and for $k\geq
1$ we assume that $-\lambda_{k}$ follows a Gamma distribution with parameters
$1-h$ and $\alpha$, i.e. $-\lambda_{k}\sim\Gamma(1-h,\alpha)$ with $h\in(0,1)$
and $\alpha>0$. Furthermore, we also assume that the random variables
$(\lambda_{k})_{k\geq 1}$ are independent from $L^{d}$.

The aggregated fractional Lévy Ornstein-Uhlenbeck processs
$Z_{m}^{d}=(Z_{m}^{d}(t),t\in\mathbb{R})$ is defined by

$Z_{m}^{d}(t):=\dfrac{1}{m}\sum_{k=1}^{m}V_{k}^{d}(t),$ (17)

for $m\geq 1$ and $d\in(0,1/2)$.

######  Lemma 6.

For all $m\geq 1$, the aggregated fractional Lévy Ornstein-Uhlenbeck processs
$Z^{d}_{m}$ is stationary and

$E_{L}[(Z_{m}^{d}(t))^{2}]=\frac{C_{d}}{m^{2}}2\Gamma(2d)\sum_{k=1}^{m}\sum_{j=1}^{m}\frac{1}{-(\lambda_{k}+\lambda_{j})(-\lambda_{k})^{2d}},$

where $C_{d}$ is given by (7).

Proof: The similarity follows by Lemma 4.

From Lemma 3 and the change of variables $\tilde{u}=t-u$ and $\tilde{v}=t-v$,
we can see

$\displaystyle E_{L}[(Z_{m}^{d}(t))^{2}]$ $\displaystyle=$
$\displaystyle\frac{C_{d}}{m^{2}}\int_{-\infty}^{t}\int_{-\infty}^{t}\sum_{k=1}^{m}\sum_{j=1}^{m}e^{\lambda_{k}(t-u)}e^{\lambda_{j}(t-v)}|u-v|^{2d-1}dvdu$
$\displaystyle=$
$\displaystyle\frac{C_{d}}{m^{2}}\sum_{k=1}^{m}\sum_{j=1}^{m}\int_{0}^{\infty}\int_{0}^{\infty}e^{\lambda_{k}u+\lambda_{j}v}|u-v|^{2d-1}dvdu$
$\displaystyle=$
$\displaystyle\frac{C_{d}}{m^{2}}\sum_{k=1}^{m}\sum_{j=1}^{m}\left(\int_{0}^{\infty}\int_{0}^{u}e^{\lambda_{k}u+\lambda_{j}v}(u-v)^{2d-1}dvdu\right.+\left.\int_{0}^{\infty}\int_{u}^{\infty}e^{\lambda_{k}u+\lambda_{j}v}(v-u)^{2d-1}dvdu\right)$
$\displaystyle=$
$\displaystyle\frac{C_{d}}{m^{2}}\sum_{k=1}^{m}\sum_{j=1}^{m}\left(\int_{0}^{\infty}e^{(\lambda_{k}+\lambda_{j})u}\int_{0}^{u}e^{-\lambda_{j}v}v^{2d-1}dvdu\right.+\left.\int_{0}^{\infty}e^{(\lambda_{k}+\lambda_{j})u}\int_{0}^{\infty}e^{\lambda_{j}v}v^{2d-1}dvdu\right)$
$\displaystyle=$
$\displaystyle\frac{C_{d}}{m^{2}}\sum_{k=1}^{m}\sum_{j=1}^{m}\left(\int_{0}^{\infty}e^{\lambda_{j}v}v^{2d-1}\int_{v}^{\infty}e^{(\lambda_{k}+\lambda_{j})u}dudv\right.+\left.\int_{0}^{\infty}e^{\lambda_{j}v}v^{2d-1}\int_{0}^{\infty}e^{(\lambda_{k}+\lambda_{j})u}dudv\right)$
$\displaystyle=$
$\displaystyle\frac{C_{d}}{m^{2}}\sum_{k=1}^{m}\sum_{j=1}^{m}\frac{1}{-(\lambda_{k}+\lambda_{j})}\left(\int_{0}^{\infty}e^{\lambda_{k}v}v^{2d-1}dv+\int_{0}^{\infty}e^{\lambda_{j}v}v^{2d-1}dv\right)$
$\displaystyle=$
$\displaystyle\frac{C_{d}}{m^{2}}\Gamma(2d)\sum_{k=1}^{m}\sum_{j=1}^{m}\frac{1}{-(\lambda_{k}+\lambda_{j})}\left(\dfrac{1}{(-\lambda_{k})^{2d}}+\dfrac{1}{(-\lambda_{j})^{2d}}\right)$
$\displaystyle=$
$\displaystyle\frac{C_{d}}{m^{2}}2\Gamma(2d)\sum_{k=1}^{m}\sum_{j=1}^{m}\frac{1}{-(\lambda_{k}+\lambda_{j})(-\lambda_{k})^{2d}}.$

∎

###  4.2 Limit of aggregated fractional Lévy Ornstein-Uhlenbeck process

By (10) we can see that $Z_{m}^{d}$ can be written as

$\displaystyle Z_{m}^{d}(t)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{t}\dfrac{1}{m}\sum_{k=1}^{m}e^{\lambda_{k}(t-u)}dL^{d}(u)$
(18) $\displaystyle=$ $\displaystyle\int_{-\infty}^{t}f_{m}(t-u)dL^{d}(u).$

Moreover, by the law of large numbers we get

$\dfrac{1}{m}\sum_{k=1}^{m}e^{\lambda_{k}(t-u)}\rightarrow
E_{\lambda}\left[e^{\lambda_{1}(t-u)}\right]=\left(\dfrac{\alpha}{\alpha+t-u}\right)^{1-h},$
(19)

as $m\rightarrow\infty$, where the convergence is $P_{\lambda}$ almost surely.

Due to the previous result, a natural candidate to be the limit of the
aggregated fractional Lévy Ornstein-Uhlenbeck process is the process
$Z^{d}=(Z^{d}(t),t\in\mathbb{R})$ given by

$\displaystyle Z^{d}(t)$ $\displaystyle:=$
$\displaystyle\int_{-\infty}^{t}\left(\dfrac{\alpha}{\alpha+t-u}\right)^{1-h}dL^{d}(u)$
(20) $\displaystyle=$ $\displaystyle\int_{-\infty}^{t}g(t-u)dL^{d}(u),$

where $g(t)=\left(\dfrac{\alpha}{\alpha+t}\right)^{1-h}$.

We can see that the process $Z^{d}$ belongs to $L^{2}(\Omega_{L})$ if
$0<h<1/2-d$. Actually, by Lemma 3 and applying consecutively the change of
variables $\tilde{u}=t-u$, $\tilde{v}=t-v$, $x=u/\alpha$, $y=v/\alpha$,
$r=1/(1+x)$, $s=1/(1+y)$, and $\hat{u}=r/s$, we obtain

$\displaystyle E_{L}\left[(Z^{d}(t))^{2}\right]$ $\displaystyle=$
$\displaystyle
C_{d}\int_{-\infty}^{t}\int_{-\infty}^{t}g(t-u)g(t-v)|u-v|^{2d-1}dvdu$
$\displaystyle=$ $\displaystyle
C_{d}\int_{0}^{\infty}\int_{0}^{\infty}g(u)g(v)|u-v|^{2d-1}dvdu$
$\displaystyle=$ $\displaystyle
C_{d}\int_{0}^{\infty}\int_{0}^{\infty}\left(1+\dfrac{u}{\alpha}\right)^{h-1}\left(1+\dfrac{v}{\alpha}\right)^{h-1}|u-v|^{2d-1}dvdu$
$\displaystyle=$ $\displaystyle
C_{d}\alpha^{2d+1}\int_{0}^{\infty}\int_{0}^{\infty}\left(1+x\right)^{h-1}\left(1+y\right)^{h-1}|x-y|^{2d-1}dydx$
$\displaystyle=$ $\displaystyle
C_{d}\alpha^{2d+1}\int_{0}^{1}\int_{0}^{1}r^{1-h}s^{1-h}(rs)^{-2}\left|\dfrac{1}{r}-\dfrac{1}{s}\right|^{2d-1}drds$
$\displaystyle=$ $\displaystyle
2C_{d}\alpha^{2d+1}\int_{0}^{1}s^{-1-h}\int_{0}^{s}r^{-h-2d}\left(1-\dfrac{r}{s}\right)^{2d-1}drds$
$\displaystyle=$ $\displaystyle
2C_{d}\alpha^{2d+1}\int_{0}^{1}s^{-2(h+d)}\int_{0}^{1}\hat{u}^{-h-2d}(1-\hat{u})^{2d-1}d\hat{u}ds$
$\displaystyle=$
$\displaystyle\dfrac{2C_{d}\alpha^{2d+1}}{1-2(h+d)}B(1-h-2d,2d)=C_{\alpha,h,d}.$

Clearly, from the last line emerges the condition $0<h<1/2-d$.

Now, we present the main result of this section related to the limit of the
aggregated fractional Lévy Ornstein-Uhlenbeck process. This result is
analogous to that obtained in Theorem 3 in [10] for the aggregated fractional
Ornstein-Uhlenbeck processes.

######  Theorem 2.

Let $Z_{m}^{d}$ and $Z^{d}$ be defined by (17) and (20), respectively. Assume
that $0<h<1/2-d$. Then $P_{\lambda}$-a.s., for every $t\in\mathbb{R}$

$Z_{m}^{d}(t)\longrightarrow Z^{d}(t),\qquad\mbox{in}\quad L^{2}(\Omega_{L})$
(21)

and for $a,b\in\mathbb{R}$

$Z_{m}^{d}(t)\longrightarrow Z^{d}(t),\qquad\mbox{weakly \ in}\ \ C[a,b]\ \
\mbox{under}\quad P_{L},$ (22)

as $m\rightarrow\infty$.

Proof: By Lemma 3, (18) and (20), we have

$\displaystyle E_{L}[(Z^{d}_{m}(t)-Z^{d}(t))^{2}]$ $\displaystyle=$
$\displaystyle
E_{L}\left[\left(\int_{-\infty}^{t}\left[\dfrac{1}{m}\sum_{k=1}^{m}e^{\lambda_{k}(t-u)}-\left(\dfrac{\alpha}{\alpha+t-u}\right)^{h}\right]dL^{d}(u)\right)^{2}\right]$
$\displaystyle=$ $\displaystyle
E_{L}\left[\left(\int_{-\infty}^{t}\left[f_{m}(t-u)-g(t-u)\right]dL^{d}(u)\right)^{2}\right]$
$\displaystyle=$ $\displaystyle
C_{d}\int_{0}^{\infty}\int_{0}^{\infty}(f_{m}(u)-g(u))(f_{m}(v)-g(v)))|u-v|^{2d-1}dudv.$

Now, following the lines of the proof of Theorem 3 (first part) in [10] and
the fact that $0<h<1/2-d$, we can obtain that $P_{\lambda}$-a.s.

$\lim\limits_{m\rightarrow\infty}E_{L}[(Z^{d}_{m}(t)-Z^{d}(t))^{2}]=0.$

Hence we have the $P_{\lambda}$-a.s. convergence of the sequence
$(Z^{d}_{m})_{m\geq 1}$ in $L^{2}(\Omega_{L})$, which in turn implies the
$P_{\lambda}$-a.s. convergence of the finite dimensional distributions.

In order to prove the weak convergence, we only need to prove the tightness.
Due Theorem 12.3 in [5], it is sufficient to show that
$E_{L}[(Z_{m}^{d}(t)-Z_{m}^{d}(s))^{2}]\leq C(t-s)^{1+\rho}$, for $T>0$ fix
and $0\leq s<t\leq T$ where $\rho>0$ and C may depend uppon parameters.

From Lemma 6 and (18)

$\displaystyle Z_{m}^{d}(t)-Z_{m}^{d}(s)$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}$ $\displaystyle
Z_{m}^{d}(t-s)-Z_{m}^{d}(0)$ $\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}$
$\displaystyle\dfrac{1}{m}\sum_{k=1}^{m}(e^{\lambda_{k}(t-s)}-1)\int_{-\infty}^{0}e^{-u\lambda_{k}}dL^{d}(u)+\int_{0}^{t-s}\dfrac{1}{m}\sum_{k=1}^{m}e^{\lambda_{k}(t-s-u)}dL^{d}(u)$
$\displaystyle=$ $\displaystyle I_{1}+I_{2}.$

Clearly, this implies

$E_{L}[(Z_{m}^{d}(t)-Z_{m}^{d}(s))^{2}]\leq
2E_{L}[I_{1}^{2}]+2E_{L}[I_{2}^{2}].$

We will estimate every term separately. In fact, by Lemma 3 we have

$\displaystyle E_{L}[I_{1}^{2}]$ $\displaystyle=$
$\displaystyle\dfrac{1}{m^{2}}\sum_{k_{1},k_{2}=1}^{m}(e^{\lambda_{k_{1}}(t-s)}-1)(e^{\lambda_{k_{2}}(t-s)}-1)E_{L}\left[\int_{-\infty}^{0}e^{-u\lambda_{k_{1}}}dL^{d}(u)\int_{-\infty}^{0}e^{-v\lambda_{k_{1}}}dL^{d}(v)\right]$
$\displaystyle=$
$\displaystyle\dfrac{1}{m^{2}}\sum_{k_{1},k_{2}=1}^{m}(e^{\lambda_{k_{1}}(t-s)}-1)(e^{\lambda_{k_{2}}(t-s)}-1)\int_{-\infty}^{0}\int_{-\infty}^{0}e^{-u\lambda_{k_{1}}}e^{-v\lambda_{k_{1}}}|u-v|^{2d-1}dudv.$

By similar arguments to those of [9], the law of large number and the fact
that $0<h<1/2-d$

$E_{L}[I_{1}^{2}]\leq C_{d,\lambda}(t-s)^{2}.$ (23)

With respect to $I_{2}$, by Lemma 3 and making the change of variable $z=u-v$,
we get

$\displaystyle E_{L}[I_{2}^{2}]$ $\displaystyle=$ $\displaystyle
E_{L}\left[\int_{0}^{t-s}\dfrac{1}{m}\sum_{k=1}^{m}e^{\lambda_{k}(t-s-u)}dL^{d}(u)\cdot\int_{0}^{t-s}\dfrac{1}{m}\sum_{k=1}^{m}e^{\lambda_{k}(t-s-v)}dL^{d}(v)\right]$
$\displaystyle=$ $\displaystyle
C_{d}\dfrac{1}{m^{2}}\sum_{k_{1},k_{2}=1}^{m}\int_{0}^{t-s}\int_{0}^{t-s}e^{\lambda_{k_{1}}(t-s-u)}e^{\lambda_{k_{2}}(t-s-v)}|u-v|^{2d-1}dudv$
$\displaystyle=$ $\displaystyle
C_{d}\dfrac{1}{m^{2}}\sum_{k_{1},k_{2}=1}^{m}e^{\lambda_{k_{1}}(t-s)}e^{\lambda_{k_{2}}(t-s)}\int_{0}^{t-s}\int_{0}^{t-s}e^{-\lambda_{k_{1}}u-\lambda_{k_{2}}v}|u-v|^{2d-1}dvdu$
$\displaystyle=$ $\displaystyle
2C_{d}\dfrac{1}{m^{2}}\sum_{k_{1},k_{2}=1}^{m}e^{\lambda_{k_{1}}(t-s)}e^{\lambda_{k_{2}}(t-s)}\int_{0}^{t-s}e^{-\lambda_{k_{1}}u}\int_{0}^{u}e^{-\lambda_{k_{2}}v}(u-v)^{2d-1}dvdu$
$\displaystyle=$ $\displaystyle
2C_{d}\dfrac{1}{m^{2}}\sum_{k_{1},k_{2}=1}^{m}e^{\lambda_{k_{1}}(t-s)}e^{\lambda_{k_{2}}(t-s)}\int_{0}^{t-s}e^{-u(\lambda_{k_{1}}+\lambda_{k_{2}})}\int_{0}^{u}e^{\lambda_{k_{2}}z}z^{2d-1}dvdu$
$\displaystyle\leq$ $\displaystyle
2C_{d}\dfrac{1}{m^{2}}\sum_{k_{1},k_{2}=1}^{m}e^{\lambda_{k_{1}}(t-s)}e^{\lambda_{k_{2}}(t-s)}\int_{0}^{t-s}e^{-\lambda_{k_{1}}u}u^{2d}du.$
$\displaystyle\leq$ $\displaystyle
2C_{d}(t-s)^{1+2d}\dfrac{1}{m}\sum_{k_{2}=1}^{m}e^{\lambda_{k_{2}}(t-s)}.$

Then, using the fact that $-\lambda_{k}\sim\Gamma(1-h,\alpha)$ with
$h\in(0,1)$ and $\alpha>0$, and $t>s$. We obtain

$E_{L}[I_{2}^{2}]\leq 2C_{d}(t-s)^{1+2d}.$ (24)

Finally, inequalities (23) and (24) imply

$E_{L}[(Z_{m}^{d}(t)-Z_{m}^{d}(s))^{2}]\leq
C_{T,d,\lambda,\alpha,h}(t-s)^{1+2d}.$

Since $d\in(0,1/2)$ the result is achieved. ∎

###  4.3 Properties of $\mathbf{Z^{d}}$

Now, we study some properties of the limit process $Z^{d}$.

######  Proposition 1.

Let $Z^{d}$ be given by (20) with $h\in(0,1/2-d)$. Then, the characteristic
function of $Z^{d}(t_{1}),Z^{d}(t_{2}),\ldots,Z^{d}(t_{m})$ with
$t_{1}<t_{2}<\ldots<t_{m}$ is given by

$E_{L}\left[\exp\left(i\sum_{j=1}^{m}\theta_{j}Z^{d}(t_{j})\right)\right]=\exp\left(\int_{\mathbb{R}}\psi\left[\sum_{j=1}^{m}\theta_{j}\tilde{f}_{t_{j},h,\alpha}(s)\right]ds\right),$

where

$\tilde{f}_{t_{j},h,\alpha}(s)=d\int_{-\infty}^{t_{j}}\left(\dfrac{\alpha}{\alpha+t_{j}-v}\right)^{1-h}(v-s)_{+}^{d-1}dv.$

Proof: The result follows by Lemma 1 in Section 2 (also see Proposition 3.3 in
[16]) and the fact that $Z^{d}$ belongs to $L^{2}(\Omega_{L})$ for
$h\in(0,1/2-d)$. ∎

######  Proposition 2.

Let $Z^{d}$ be given by (20) with $h\in(0,1/2-d)$. Then, $Z^{d}$ is a
stationary process with long memory.

Proof: By (20), the stationarity of the increments of $L^{d}$ and making the
change of variable $\tilde{u}=u-b$, we can get for every $b>0$

$\displaystyle Z^{d}(t+b)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{t+b}\left(\dfrac{\alpha}{\alpha+t+b-u}\right)^{1-h}dL^{d}(u)$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}$
$\displaystyle\int_{-\infty}^{t}\left(\dfrac{\alpha}{\alpha+t-u}\right)^{1-h}dL^{d}(u)=Z^{d}(t).$

With respect to the long memory property we will study the non summability of
the covariance. Since $Z^{d}$ is a stionary process, we have

$\displaystyle E_{L}[Z^{d}(a)Z^{d}(t+a)]$ $\displaystyle=$ $\displaystyle
E_{L}[Z^{d}(0)Z^{d}(t)].$

Then, Lemma 3 implies

$\displaystyle E_{L}[Z^{d}(0)Z^{d}(t)]$ $\displaystyle=$ $\displaystyle
C_{d}\int_{-\infty}^{0}\int_{-\infty}^{t}\left(\dfrac{\alpha}{\alpha+t-u}\right)^{1-h}\left(\dfrac{\alpha}{\alpha-v}\right)^{1-h}|u-v|^{2d-1}dudv$
$\displaystyle=$ $\displaystyle
C_{d}\alpha^{2-2h}\int_{-\infty}^{0}\int_{-\infty}^{t}\left(\alpha+t-u\right)^{h-1}\left(\alpha-v\right)^{h-1}|u-v|^{2d-1}dudv$
$\displaystyle=$ $\displaystyle
C_{d}\alpha^{2-2h}t^{2h+2d-1}\int_{-\infty}^{0}\int_{-\infty}^{1}\left(\dfrac{\alpha}{t}+1-y\right)^{h-1}\left(\dfrac{\alpha}{t}-x\right)^{h-1}|x-y|^{2d-1}dxdy$
$\displaystyle\sim$ $\displaystyle
C_{d}\alpha^{2-2h}t^{2h+2d-1}\int_{-\infty}^{0}\int_{-\infty}^{1}\left(1-y\right)^{h-1}\left(-x\right)^{h-1}|x-y|^{2d-1}dxdy$
$\displaystyle=$ $\displaystyle C_{d,\alpha,h}t^{2h+2d-1}.$

Then, the long memory property is obtain by just noticing that $h+d>0$. ∎

######  Remark 2.

Even if $d=0$, the long memory property is satisfied if $h>0$.

######  Remark 3.

We can see that the process $Z^{d}$ has the following property

$\dfrac{\rho(t)}{\rho(0)}\sim
C_{\alpha,h,d}t^{2h+2d-1},\mbox{with}\quad\rho(t)=E_{L}[Z^{d}(0)Z^{d}(t)],$

i.e., the process is almost self-similar (see [14] page 13 for details).

With respect to the behavior of the process $Z^{d}$ with respect to the
parameter $\alpha$ we have the following results

######  Proposition 3.

Let $t\geq 0$ and $d\in(0,1/2)$, then the random variable

$Z^{d}(t)-Z^{d}(0)$

converges en $L^{2}(\Omega_{L})$ as $\alpha\rightarrow\infty$ to the random
variable $L^{d}(t)$.

Proof: By (20) we obtain

$\displaystyle Z^{d}(t)-Z^{d}(0)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{t}\left(\dfrac{\alpha}{\alpha+t-u}\right)^{1-h}dL^{d}(u)-\int_{-\infty}^{0}\left(\dfrac{\alpha}{\alpha-u}\right)^{1-h}dL^{d}(u)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\left[\left(\dfrac{\alpha}{\alpha+t-u}\right)^{1-h}1_{(-\infty,t)}(u)-\left(\dfrac{\alpha}{\alpha-u}\right)^{1-h}1_{(-\infty,0)}(u)\right]dL^{d}(u)$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}}f_{t,d}(u)dL^{d}(u).$

Clearly, $f_{t,d}$ converges to $1_{(0,t)}$ as $\alpha\rightarrow\infty$.
Therefore, our candidate as a limit will be $L^{d}(t)$. In fact, by Lemma 3

$\displaystyle E_{L}[(Z^{d}(t)-Z^{d}(0)-L^{d}(t))^{2}]$ $\displaystyle=$
$\displaystyle
C_{d}\int_{\mathbb{R}}\int_{\mathbb{R}}\left[\left(\dfrac{\alpha}{\alpha+t-u}\right)^{1-h}1_{(-\infty,t)}(u)-\left(\dfrac{\alpha}{\alpha-u}\right)^{1-h}1_{(-\infty,0)}(u)-1_{(0,t)}(u)\right]$
$\displaystyle\times$
$\displaystyle\left[\left(\dfrac{\alpha}{\alpha+t-v}\right)^{1-h}1_{(-\infty,t)}(v)-\left(\dfrac{\alpha}{\alpha-v}\right)^{1-h}1_{(-\infty,0)}(v)-1_{(0,t)}(v)\right]|u-v|^{2d-1}dudv.$

Finally, the result is obtained by means of the dominated convergence theorem.
∎ If $\alpha\rightarrow 0$, then we get the following result.

######  Proposition 4.

Let $t\geq 0$ and let us define $\tilde{Z}^{d}(t)$ by

$\tilde{Z}^{d}(t)=\alpha^{h-1}\int_{0}^{t}Z^{d}(s)ds,$

then, as $\alpha\rightarrow 0$ the random variable $\tilde{Z}^{d}(d)$
converges in $L^{2}(\Omega_{L})$ to the random variable $Y^{d}$ given by

$Y^{d}(t):=\dfrac{1}{h}\int_{\mathbb{R}}[(t-u)^{h}_{+}-(-u)_{+}^{h}]dL^{d}(u)$
(25)

Proof: By (20), we get

$\displaystyle\tilde{Z}^{d}(t)$ $\displaystyle=$
$\displaystyle\alpha^{h-1}\int_{0}^{t}\int_{-\infty}^{s}\left(\dfrac{\alpha}{\alpha+s-u}\right)^{1-h}dL^{d}(u)ds$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{-\infty}^{s}\left(\alpha+s-u\right)^{h-1}dL^{d}(u)ds$
$\displaystyle=$ $\displaystyle\int_{-\infty}^{t}\int_{v\vee
0}^{t}\left(\alpha+s-v\right)^{h-1}dsdL^{d}(v)$ $\displaystyle=$
$\displaystyle\dfrac{1}{h}\int_{-\infty}^{t}\left[\left(\alpha+t-v\right)^{h}-\left(\alpha+(v\vee
0)-v\right)^{h}\right]dL^{d}(v)$ $\displaystyle=$
$\displaystyle\dfrac{1}{h}\int_{-\infty}^{t}r_{\alpha,t,h}(v)dL^{d}(v).$

We can see that $r_{\alpha,t,h}(v)$ converges to $(t-v)^{h}_{+}-(-v)_{+}^{h}$
for every $v$ as $\alpha\rightarrow 0$. Therefore, by similar arguments to the
previous proposition the result is obtained. ∎

######  Proposition 5.

Let $Y^{d}=(Y^{d}(t))_{t\geq 0}$ with $Y^{d}(t)$ be given by (25), then
$Y^{d}$ is a stationary process with

$E_{L}[(Y^{d}(t))^{2}]=t^{2h+2d+1}E_{L}[(Y^{d}(1))^{2}].$

Proof: By (25) and taking $b>0$

$\displaystyle Y^{d}(t+b)-Y^{d}(b)$ $\displaystyle=$
$\displaystyle\dfrac{1}{h}\int_{\mathbb{R}}[(t+b-u)^{h}_{+}-(b-u)^{h}_{+}]dL^{d}(u)$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}$
$\displaystyle\dfrac{1}{h}\int_{\mathbb{R}}[(t-v)^{h}_{+}-(-v)^{h}_{+}]dL^{d}(v)=Y^{d}(t),$

here we have used the change of variable $v=u-b$ and the fact that $L^{d}$ is
a stationary increment process. With respect to the second part of the
statement, we have that

$\displaystyle E_{L}[(Y^{d}(t))^{2}]$ $\displaystyle=$
$\displaystyle\dfrac{1}{h^{2}}C_{d}\int_{\mathbb{R}}\int_{\mathbb{R}}((t-u)^{h}_{+}-(-u)_{+}^{h})((t-v)^{h}_{+}-(-v)_{+}^{h})|u-v|^{2d-1}dvdu$
$\displaystyle=$ $\displaystyle
t^{2h+2d+1}\dfrac{1}{h^{2}}C_{d}\int_{\mathbb{R}}\int_{\mathbb{R}}((1-u)^{h}_{+}-(-u)_{+}^{h})((1-v)^{h}_{+}-(-v)_{+}^{h})|u-v|^{2d-1}dvdu$
$\displaystyle=$ $\displaystyle t^{2h+2d+1}E_{L}[(Y^{d}(1))^{2}].$

∎

######  Remark 4.

Let us note that we can write

$\tilde{Y}^{d}(t)=\int_{\mathbb{R}}m_{t}(u)dL^{d}(u),$

where $m_{t}(u)=\dfrac{1}{h}[(t-u)^{h}_{+}-(-u)_{+}^{h}]$. Then, Proposition 5
and Theorem 3.1 in [16] imply

$Y^{d}(t)=\int_{\mathbb{R}}m_{t}(u)dL^{d}(u)=\int_{\mathbb{R}}(I^{g}_{-m})(u)dL(s),$

where (see Section 3 in the same reference for details)

$(I^{g}_{-m})(u):=\int_{u}^{\infty}m_{t}(v)g^{\prime}(v-u)dv=\int_{\mathbb{R}}m_{t}(v)g^{\prime}(v-u)dv$

and $g(u)=(u)^{d}_{+}$.

This implies that $Y^{d}$ can be seen as a type of Generalized fractional Lévy
process (see [16] for details).

##  5 Simulations

In this section we are interested in obtain some simulations related to the
process $Z^{d}$. First, let us recall how we can simulate a fractional Lévy
process.

In order to simulate sample paths from $L^{d}$ we use a Riemann-Stieltjes
approximation, that is, we approximate $L^{d}$ in the following way (see [19]
for details)

$\displaystyle L^{d}(t)$ $\displaystyle\approx$
$\displaystyle\frac{1}{\Gamma(d+1)}\left(\sum_{k=-n^{2}}^{0}\left[\left(t-\frac{k}{n}\right)^{d}-\left(-\frac{k}{n}\right)^{d}\right]\left(L_{(k+1)/n}-L_{k/n}\right)\right.$
$\displaystyle+\left.\sum_{k=0}^{\lfloor
nt\rfloor}\left(t-\frac{k}{n}\right)^{d}\left(L_{(k+1)/n}-L_{k/n}\right)\right).$

Figure 1: Sample paths of a fractional Lévy process for different values of
$d$. The approximation is made using Riemann-Stieltjes approximation with
driving Lévy process as a stationary Gamma process with $a=5$ and $b=15$
($a=1$ and $b=2$).

######  Remark 5.

An optimal form of simulating $L^{d}$ is shown in [26]. Here, we have used the
Riemann-Stieltjes approximation. This approximation can also be optimal if we
take $a_{n}=n^{2-d/1-d}$. An advantage of this procedure is that simulate
increments of a Lévy process is relatively easy (see [8] for details about the
simulation of increments of a Lévy process).

To simulate the process $Z^{d}$ we will use the following Riemann type
approximation

$Z^{d,(n)}(t)=\alpha^{1-h}\sum_{k=-a_{n}}^{\lfloor
nt\rfloor}\left(\alpha+t-\dfrac{k}{n}\right)^{h-1}\left(L^{d}\left(\dfrac{k+1}{n}\right)-L^{d}\left(\dfrac{k}{n}\right)\right),\quad
t\in\mathbb{R},$

as in [19] we take $a_{n}=n^{2}$.

Figure 2: Sample paths $Z^{d}$ of the limit process for different $\alpha$ and
$h=0.12$ $(a=1,b=2)$.

Figure 3: Sample paths $Z^{d}$ of the limit process for different $\alpha$ and
$h=0.12$ $(a=5,b=5)$.

## Acknowledgements

Héctor Araya was partially supported by Proyecto Fondecyt PostDoctorado, Chile
3190465, MEC 80190045, Math-Amsud 20-MATH-05 and Redes 190038. Johanna Garzón
was partially supported by HERMES project 52433.

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STABILITY ESTIMATES FOR THE COMPLEX MONGE-AMPÈRE AND HESSIAN EQUATIONS 111Work
supported in part by the National Science Foundation under grant DMS-1855947.

Bin Guo, Duong H. Phong, and Freid Tong

###### Abstract

A new proof for stability estimates for the complex Monge-Ampère and Hessian
equations is given, which does not require pluripotential theory. A major
advantage is that the resulting stability estimates are then uniform under
general degenerations of the background metric in the case of the Monge-Ampère
equation, and under degenerations to a big class in the case of Hessian
equations.

##  1 Introduction

Stability estimates for a non-linear partial differential equation are
estimates for how much the solution can vary, given the size of the variation
of the right hand side. Clearly, they are of great theoretical as well as
practical importance. Such estimates had been obtained by Kolodziej [12] for
the complex Monge-Ampère equation, by Dinew and Kolodziej [4] for complex
Hessian equations, and by Dinew and Zhang [6] for Monge-Ampère equations when
the background metric is not necessarily Kähler, but just big. In all cases,
the proofs made extensive use of pluripotential theory and the background
metric was fixed. It remained an open question whether these estimates can be
established without pluripotential theory, and whether they can be made
uniform under degenerations of the background metric, a situation which arises
frequently in geometric applications.

In [9], the authors developed a method for obtaining sharp $L^{\infty}$
estimates for the complex Monge-Ampère equation without pluripotential theory.
As explained in greater detail there, the method of [9] built on works of
Wang, Wang, Zhou [14] and of Chen and Cheng [3], particularly on the last two
authors’ idea of considering an associated complex Monge-Ampère equation. It
achieved the stated goal of giving an alternate PDE proof of $L^{\infty}$
estimates for the complex Monge-Ampère equation, but it also went considerably
beyond in, on one hand, applying to more general fully non-linear equations,
and on the other hand, allowing the background metrics to degenerate. It can
also give sharp gradient estimates [10], improving on the estimates in e.g.
[13, 2, 8, 7, 11].

The main goal of the present paper is to obtain stability estimates for the
complex Monge-Ampère and Hessian equations which are uniform under
degenerations. We use the method of [9]. We recover in the process the
stability estimates of [12, 4, 6], this time without pluripotential theory.
Our estimates are also uniform under general degenerations of the background
metric in the case of the Monge-Ampère equation, and under degenerations to a
big class in the case of Hessian equations. Thus we answer in the positive
both questions asked above.

##  2 Statement of the main results

Let $(X,\omega)$ be a compact Kähler manifold, $\chi$ a closed and non-
negative $(1,1)$-form, and set

$\displaystyle\omega_{t}=\chi+t\omega,\qquad t\in(0,1].$ (2.1)

Let $f,h\in C^{\infty}$ are smooth functions normalized by

$\int_{X}e^{f}\omega^{n}=\int_{X}e^{h}\omega^{n}=\int_{X}\omega^{n}=1,$

and consider the following complex Hessian equations

$(\omega_{t}+i\partial\bar{\partial}u_{t})^{k}\wedge\omega^{n-k}=c_{t}e^{f}\omega^{n},\quad(\omega_{t}+i\partial\bar{\partial}v_{t})^{k}\wedge\omega^{n-k}=c_{t}e^{h}\omega^{n}$
(2.2)

with the constants $c_{t}$ given by
$c_{t}=\int_{X}\omega_{t}^{k}\wedge\omega^{n-k}$. We normalize $u_{t}$ and
$v_{t}$ so that

$\max_{X}(u_{t}-v_{t})=\max_{X}(v_{t}-u_{t}).$

We will consider three cases:

$\displaystyle{\mathrm{\mathbf{I}}}:$ $\displaystyle k=n,\mbox{ and
}t\in(0,1]$ $\displaystyle{\mathrm{\mathbf{II}}}:$ $\displaystyle 1\leq
k<n,\mbox{ and }\chi=0,\,t=1.$ $\displaystyle{\mathrm{\mathbf{III}}}:$
$\displaystyle 1\leq k<n,\mbox{ and }t\in(0,1],\,\chi\mbox{ is big i.e.
$\int_{X}\chi^{n}>0$}$

which correspond respectively to the Monge-Ampère equations with
degenerations, the $k$-th Hessian equation with a fixed background metric, and
the $k$-th Hessian equation with degenerations. Different cases correspond to
different choices of test functions, and constants. So we will treat the cases
separately, when necessary.

For each case, we will make the following assumptions and choice of constants,

$\displaystyle{\mathrm{\mathbf{I}}}:$ $\displaystyle\|e^{h}\|_{L^{1}(\,{\rm
log}\,L)^{p_{1}}(\omega^{n})},\|e^{f}\|_{L^{1}(\,{\rm
log}\,L)^{p_{1}}(\omega^{n})}\leq K,\,p_{1}>n$
$\displaystyle{\mathrm{\mathbf{II}}}:$
$\displaystyle\|e^{h}\|_{L^{p_{2}}(\omega^{n})},\|e^{f}\|_{L^{p_{2}}(\omega^{n})}\leq
K,\,p_{2}>\frac{n}{k},\mbox{ and }q_{2}=\frac{p_{2}-1}{p_{2}-n/k}$
$\displaystyle{\mathrm{\mathbf{III}}}:$
$\displaystyle\|e^{h}\|_{L^{p_{3}}(\omega^{n})},\|e^{f}\|_{L^{p_{3}}(\omega^{n})}\leq
K,\,p_{3}>\frac{n^{2}}{k},\mbox{ and }q_{3}=\frac{p_{3}-1}{p_{3}-n/k}$

for a fixed constant $K>0$.

The equations (2.2) admit smooth solutions by [15, 5]. By [9] (also [12, 4]),
under the above assumptions, the oscillations ${\mathrm{osc}}u_{t}$ and
${\mathrm{osc}}v_{t}$ of the solutions are uniformly bounded independently of
$t$ in all three cases ${\mathrm{\mathbf{I}}},{\mathrm{\mathbf{I}I}}$, and
${\mathrm{\mathbf{I}II}}$. Let $\beta_{0}>1$ denote such an upper bound
depending only on $n,k,\omega,\chi,p_{a}$ and $10K$.

To start with, we will define a positive function $\gamma_{a}(r)$ with
$\gamma_{a}(r)\to 0$ as $r\to 0$. Each case has different choice of such a
function. We define $\gamma_{a}$ case by case:

$\displaystyle{\mathrm{\mathbf{I}}}:$
$\displaystyle\gamma_{1}(r)=r^{\frac{1}{\delta_{1}}-n+1},\mbox{ where
}\delta_{1}=\frac{p_{1}-n}{p_{1}n}<\frac{1}{n}$
$\displaystyle{\mathrm{\mathbf{II}}}:$
$\displaystyle\gamma_{2}(r)=r^{(n+1)q_{2}-n}$
$\displaystyle{\mathrm{\mathbf{III}}}:$
$\displaystyle\gamma_{3}(r)=r^{(n+1)q_{3}-n}$

###### Theorem 1

Let the assumptions and notations be as above. Then we have in all three cases
listed in (2)

$\sup_{X}|u_{t}-v_{t}|\leq C\|e^{f}-e^{h}\|_{L^{1}}^{1/(n+3+\sigma_{a})},$
(2.3)

where in each case, $\sigma_{a}>0$ is the power of $r$ in $\gamma_{a}(r)$,
i.e. $\gamma_{a}(r)=r^{\sigma_{a}}$, and $C$ is a constant depending only on
$n,k,\omega,\chi$, $K>0$ and $p_{a}$.

We observe that this theorem improves on all results known so far. More
specifically in case I, we get uniform stability for a degenerating family,
and it does not even matter whether $\chi$ is big or not. Kolodziej [12]
proved this case for a fixed Kähler metric, and Dinew and Zhang [6] proved it
for a fixed big class. In case II, we slightly sharpen the known stability
result in [4], where the RHS in the inequality is
$\|e^{f}-e^{h}\|_{L^{q^{\prime}}}$ for some $q^{\prime}>1$, while we are able
to prove the inequality for $q^{\prime}=1$. In case III, we obtain a uniform
stability theorem for Hessian equations when the class remains big. This is
completely new, and relies in particular on the uniform $L^{\infty}$ estimate
in [9] for solutions with degenerating big classes.

We note that the exponent $1/(n+3+\sigma_{a})$ is not sharp in general. The
sharp exponent can be obtained by replacing Lemma 1 below by a result from
[6]. We leave the details to the interested readers.

##  3 Proof of Theorem 1

For the proof, it is convenient to restate the theorem as follows. Under the
above assumptions, if in each case we have

$\|e^{f}-e^{h}\|_{L^{1}(\omega^{n})}\leq\gamma_{a}(r)r^{n+3},$ (3.1)

then there is a small $r_{0}>0$ such that for all $0<r\leq r_{0}$

$\sup_{X}|u_{t}-v_{t}|\leq Cr,$

for all $t$ listed in each case in (2) and $C$ depends only on
$n,k,\omega,\chi$, $K>0$ and the corresponding $p_{a}$ in each case. This is
the version which we shall prove.

We begin with a lemma due to Kolodziej [12]. The proof is almost identical to
that in [12], but since we would like to avoid the use of pluripotential
theory, some additional smoothing is needed in the proof, and we provide a
full proof of this lemma.

Choose a small $\bar{r}_{0}\in(0,\frac{1}{10})$ such that
$\gamma_{a}(\bar{r}_{0})\bar{r}_{0}^{n}\leq\frac{1}{5}$. We then fix an
$0<r<\bar{r}_{0}$. We remark that all relevant constants are independent of
$r$. Later on we will choose an even smaller $r_{0}>0$.

By switching the roles of $u_{t}$ and $v_{t}$ if necessary, we may assume

$\int_{\\{v_{t}\leq u_{t}\\}}(e^{f}+e^{h})\omega^{n}\leq 1.$

Denote $E_{j}:=\\{v_{t}\leq u_{t}-j\beta_{0}r\\}$. The next lemma states that
over the set $E_{2}$, the integral of $e^{h}$ is small.

###### Lemma 1

In each case $a=$ I, II, III, we have

$\int_{E_{2}}e^{h}\omega^{n}\leq C_{0}\gamma_{a}(r)r^{n},$

for some constant $C_{0}=1+\frac{2}{(\frac{3}{2})^{1/k}-1}$.

Proof. We calculate

$\int_{E_{0}}e^{h}\omega^{n}=\frac{1}{2}\int_{E_{0}}(e^{f}+e^{h})+(e^{h}-e^{f})\omega^{n}\leq\frac{1}{2}(1+\frac{1}{5})=\frac{3}{5}.$
(3.2)

Take a sequence of positive smooth functions $\tau_{j}$ that converge
uniformly to $\chi_{E_{0}}$ such that $\tau_{j}\equiv 1$ on $E_{0}$. Consider
a sequence of smooth positive functions

$e^{h_{j}}=\frac{3}{2}\tau_{j}e^{h}+c_{j}(1-\tau_{j})e^{h}$

where $c_{j}>0$ are chosen so that $\int_{X}e^{h_{j}}\omega^{n}=1$. It is not
hard to see from (3.2) that for $j>>1$, $\frac{1}{20}\leq c_{j}\leq 3.$ Hence
when $j>>1$

$\bullet$ in case I, we have $\|e^{h_{j}}\|_{L^{1}(\,{\rm
log}\,L)^{p_{1}}}\leq 5K$,

$\bullet$ in case II, we have $\|e^{h_{j}}\|_{L^{p_{2}}}\leq 5K$,

$\bullet$ in case III, we have $\|e^{h_{j}}\|_{L^{p_{3}}}\leq 5K$.

We solve the following Hessian equations which admit known to admit unique
smooth solutions [15, 5],

$(\omega_{t}+i\partial\bar{\partial}\rho_{j})^{k}\wedge\omega^{n-k}=c_{t}e^{h_{j}}\omega^{n},\quad\sup_{X}\rho_{j}=0\mbox{
and }\omega_{t}+i\partial\bar{\partial}\rho_{j}\in\Gamma_{k},$

where $\Gamma_{k}$ is the usual open convex cone in $k$-th Hessian equations.
By the choice of $\beta_{0}$, we have $-\beta_{0}\leq\rho_{j}\leq 0$ (see
[9]). The following Newton inequality holds pointwise for any $1\leq l\leq k$

$\omega_{t,u_{t}}^{l}\wedge\omega_{t,\rho_{j}}^{k-l}\wedge\omega^{n-k}\geq\Big{(}\frac{\omega_{t,u_{t}}^{k}\wedge\omega^{n-k}}{\omega^{n}}\Big{)}^{l/k}\Big{(}\frac{\omega^{k}_{t,\rho_{j}}\wedge\omega^{n-k}}{\omega^{n}}\Big{)}^{(k-l)/k}\omega^{n}.$

Then on the set $E_{0}\backslash G$ where $G=\\{e^{f}\leq(1-r^{2})e^{h}\\}$,
we have

$\omega_{t,u_{t}}^{l}\wedge\omega_{t,\rho_{j}}^{k-l}\wedge\omega^{n-k}\geq
c_{t}(1-r^{2})^{l/k}(\frac{3}{2})^{(k-l)/k}e^{h}\omega^{n}.$

It follows that on $E_{0}\backslash G$

$\displaystyle\omega_{t,r\rho_{j}+(1-r)u_{t}}^{k}\wedge\omega^{n-k}$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{k}\frac{k!}{l!(k-l)!}r^{k-l}(1-r)^{l}\omega_{t,u_{t}}^{l}\wedge\omega_{t,\rho_{j}}^{k-l}\wedge\omega^{n-k}$
(3.3) $\displaystyle\geq$ $\displaystyle
c_{t}\sum_{l=0}^{k}\frac{k!}{l!(k-l)!}r^{k-l}(1-r)^{l}(1-r^{2})^{l/k}(\frac{3}{2})^{(k-l)/k}e^{h}\omega^{n}$
$\displaystyle=$ $\displaystyle
c_{t}\big{(}(1-r)(1-r^{2})^{1/k}+r(\frac{3}{2})^{1/k}\big{)}^{k}e^{h}\omega^{n}\geq
c_{t}(1+b_{0}r)e^{h}\omega^{n}$

where $b_{0}=\frac{1}{2}((\frac{3}{2})^{1/k}-1)>0$, since $r$ is chosen to be
small.

Note that by (3.1)

$r^{2}\int_{G}e^{h}\omega^{n}\leq\int_{G}(e^{h}-e^{f})\omega^{n}\leq\gamma_{a}(r)r^{n+3},$

which implies

$\int_{G}e^{h}\omega^{n}\leq\gamma_{a}(r)r^{n+1}.$ (3.4)

Adding the same constant to $u_{t}$ and $v_{t}$, we may assume without loss of
generality $-\beta_{0}\leq u_{t}\leq 0$. The following inclusion relation
holds from the definition

$E_{2}\subset E:=\\{v_{t}\leq r\rho_{j}+(1-r)u_{t}-\beta_{0}r\\}\subset
E_{0}.$

All functions involved are smooth so by the comparison principle and (3.3),

$\displaystyle c_{t}(1+b_{0}r)\int_{E\backslash G}e^{h}\omega^{n}$
$\displaystyle\leq$
$\displaystyle\int_{E}\omega_{t,r\rho_{j}+(1-r)u_{t}}^{k}\wedge\omega^{n-k}$
$\displaystyle\leq$
$\displaystyle\int_{E}\omega_{v_{t}}^{k}\wedge\omega^{n-k}=c_{t}\int_{E\backslash
G}e^{h}\omega^{n}+c_{t}\int_{G}e^{h}\omega^{n}$

Combined with (3.4) this implies

$\int_{E\backslash G}e^{h}\omega^{n}\leq\frac{1}{b_{0}}\gamma_{a}(r)r^{n}$

It follows that

$\int_{E_{2}}e^{h}\omega^{n}\leq\int_{E\backslash
G}e^{h}\omega^{n}+\int_{G}e^{h}\omega^{n}\leq(1+\frac{1}{b_{0}})\gamma_{a}(r)r^{n}.$

The Lemma is proved.

We now come to the proof of Theorem 1 proper. We normalize $u_{t}$ as in the
proof of Lemma 1. For $s\geq 0$, we set
$\Omega_{s}=\\{v_{t}\leq(1-r)u_{t}-3\beta_{0}r-s\\}$. Note that
$\Omega_{s}\subset E_{2}$ for any $s\geq 0$.

We follow the same strategy as in [9]. We choose a sequence of smooth positive
functions $\eta_{j}:{{\bf R}}\to{{{\bf R}}}_{+}$ such that

$\eta_{j}(x)=x+\frac{1}{j},\quad\mbox{ when }x\geq 0,$

and

$\eta_{j}(x)=\frac{1}{2j},\quad\mbox{ when }x\leq-\frac{1}{j},$

and $\eta_{j}(x)$ lies between $1/2j$ and $1/j$ for $x\in[-1/j,0]$. Clearly
$\eta_{j}\to\eta_{\infty}(x)=x\cdot\chi_{{\bf R}_{+}}(x)$ pointwise as
$j\to\infty$.

We solve the complex Monge-Ampère equations

$(\omega_{t}+i\partial\bar{\partial}\psi_{j})^{n}=c_{t}^{{n/}{k}}\frac{\eta_{j}(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)}{A_{s,j}}e^{\frac{n}{k}h}\omega^{n},\quad\sup\psi_{j}=0.$

As $j\to\infty$ we have by the dominated convergence theorem

$A_{s,j}=\frac{c_{t}^{\frac{n}{k}}}{V_{t}}\int_{X}\big{(}\eta_{j}(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)\big{)}e^{\frac{n}{k}h}\omega^{n}\to
A_{s}$

where the constant $A_{s}$ is defined by

$A_{s}:=\frac{c_{t}^{\frac{n}{k}}}{V_{t}}\int_{\Omega_{s}}(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)e^{\frac{n}{k}h}\omega^{n}.$

Consider
$\Phi=-\varepsilon\big{(}-\psi_{j}+\Lambda\big{)}^{\frac{n}{n+1}}+\big{(}-v_{t}+(1-r)u_{t}-3\beta_{0}r-s\big{)}$
where

$\varepsilon=\Big{(}\frac{k(n+1)}{n^{2}}\Big{)}^{\frac{n}{n+1}}c(n,k)^{-\frac{1}{n+1}}A_{s,j}^{\frac{1}{n+1}},\quad\mbox{where
$c(n,k)$ is the one in (\ref{eqn:str})}$
$\Lambda=\Big{(}\frac{n}{n+1}\frac{\varepsilon}{r}\Big{)}^{n+1}=C(n,k)\frac{A_{s,j}}{r^{n+1}}$

Suppose $\sup\Phi=\Phi(x_{0})$ for some point $x_{0}$ in $X$. If
$x_{0}\not\in\Omega_{s}^{\circ}$, then by definition $\Phi(x_{0})<0$.
Otherwise $x_{0}\in\Omega_{s}^{\circ}$. We calculate as in [9]. First note
that for
$G^{i\bar{j}}=\frac{\partial}{\partial(\omega_{t,v_{t}})_{i\bar{j}}}\,{\rm
log}\,\sigma_{k}(\omega_{t,v_{t}})$

${\rm det}G^{i\bar{j}}\geq c(n,k)c_{t}^{-\frac{n}{k}}e^{-\frac{n}{k}h}$ (3.5)

for some computable constant $c(n,k)>0$. It follows that at $x_{0}$

$\displaystyle 0$ $\displaystyle\geq$ $\displaystyle
G^{i\bar{j}}(\Phi)_{i\bar{j}}(x_{0})$ $\displaystyle\geq$
$\displaystyle\frac{n^{2}\varepsilon}{n+\delta}(-\psi_{j}+\Lambda)^{-\frac{1}{n+1}}\Big{(}{\rm
det}G\cdot{\rm
det}\omega_{t,\psi_{j}}\Big{)}^{1/n}-k+\Big{(}r-\frac{n\varepsilon}{n+1}\Lambda^{-\frac{1}{n+1}}\Big{)}G^{i\bar{j}}(\omega_{t})_{i\bar{j}}$
$\displaystyle\geq$
$\displaystyle\frac{n^{2}\varepsilon}{n+1}(-\psi_{j}+\Lambda)^{-\frac{1}{n+1}}c(n,k)^{1/n}\Big{(}\frac{-v_{t}+(1-r)u_{t}-3\beta_{0}r-s}{A_{s,j}}\Big{)}^{1/n}-k.$

By the choice of $\varepsilon$ and $\Lambda$, we deduce that $\Phi(x_{0})\leq
0$. Thus $\Phi\leq 0$ on $X$ and this implies

$\displaystyle\int_{\Omega_{s}}\,{\rm
exp}\,\Big{\\{}c_{0}\Big{(}\frac{-v_{t}+(1-r)u_{t}-3\beta_{0}r-s}{A_{s,j}^{1/(n+1)}}\Big{)}^{\frac{n+1}{n}}\Big{\\}}\omega^{n}$
$\displaystyle\leq$ $\displaystyle\int_{\Omega_{s}}\,{\rm
exp}\,\Big{(}-\alpha_{0}\psi_{j}+\alpha_{0}\Lambda\Big{)}\omega^{n}$ (3.6)
$\displaystyle\leq$ $\displaystyle C\,{\rm
exp}\,\Big{\\{}\frac{A_{s,j}}{r^{n+1}}\Big{\\}}$

for some small $c_{0}=c_{0}(n,k,\omega,\chi)>0$, $C=C(n,k,\omega,\chi)>0$, and
$\alpha_{0}$ a fixed number satisfying $0<\alpha_{0}<\alpha(X,\omega)$, where
$\alpha(X,\omega)$ is the $\alpha$-invariant of $(X,\omega)$. Letting
$j\to\infty$ gives

$\int_{\Omega_{s}}\,{\rm
exp}\,\Big{\\{}c_{0}\Big{(}\frac{(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)}{A_{s}^{1/(n+1)}}\Big{)}^{\frac{n+1}{n}}\Big{\\}}\omega^{n}\leq
C\,{\rm exp}\,\Big{(}\frac{A_{s}}{r^{n+1}}\Big{)}.$ (3.7)

To estimate $\frac{A_{s}}{r^{n+1}}$ in (3.7), we need to consider separately
the three cases I, II and III.

$\bullet$ Case I. In this case $k=n$ and $c_{t}=V_{t}$. By Lemma 1 we deduce
that

$\displaystyle\frac{A_{s}}{r^{n+1}}$ $\displaystyle=$
$\displaystyle\frac{c_{t}^{n/k}}{V_{t}}\frac{1}{r^{n+1}}\int_{\Omega_{s}}(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)e^{h}\omega^{n}$
$\displaystyle\leq$
$\displaystyle\frac{1}{r^{n+1}}\int_{\Omega_{s}}(-v_{t}+(1-r)u_{t}-3\beta_{0}r)e^{h}\omega^{n}$
$\displaystyle\leq$
$\displaystyle\frac{C(n,\beta_{0})}{r^{n+1}}\int_{E_{2}}e^{h}\omega^{n}\leq
C_{0}C(n,\beta_{0})\gamma_{1}(r)r^{-1}$ $\displaystyle\leq$ $\displaystyle
C(n,\beta_{0})\quad\mbox{ by the choice of $\gamma_{1}(r)$.}$

$\bullet$ Case II. In this case, under our normalization, $V_{1}=c_{1}=1$. As
in Case I, we have

$\displaystyle\frac{A_{s}}{r^{n+1}}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{n+1}}\int_{\Omega_{s}}(-v_{1}+(1-r)u_{1}-3\beta_{0}r-s)e^{\frac{n}{k}h}\omega^{n}$
$\displaystyle\leq$
$\displaystyle\frac{1}{r^{n+1}}\int_{\Omega_{s}}(-v_{1}+(1-r)u_{1}-3\beta_{0}r)e^{\frac{n}{k}h}\omega^{n}$
$\displaystyle\leq$
$\displaystyle\frac{C(n,\beta_{0})}{r^{n+1}}\int_{E_{2}}e^{\frac{n}{k}h}\omega^{n}=\frac{C(n,\beta_{0})}{r^{n+1}}\int_{E_{2}}e^{(\frac{n}{k}-1)h}e^{h}\omega^{n}$
$\displaystyle\leq$
$\displaystyle\frac{C(n,\beta_{0})}{r^{n+1}}\Big{(}\int_{E_{2}}e^{q_{2}^{*}(\frac{n}{k}-1)h}e^{h}\omega^{n}\Big{)}^{1/q_{2}^{*}}\Big{(}\int_{E_{2}}e^{h}\Big{)}^{1/q_{2}}$
$\displaystyle\leq$
$\displaystyle\frac{C(n,\beta_{0})}{r^{n+1}}\Big{(}\int_{E_{2}}e^{p_{2}h}\omega^{n}\Big{)}^{1/q_{2}^{*}}\Big{(}\int_{E_{2}}e^{h}\Big{)}^{1/q_{2}}$
$\displaystyle\leq$ $\displaystyle
C(n,k,\beta_{0},K)\gamma_{2}(r)^{\frac{1}{q_{2}}}r^{\frac{n}{q_{2}}-n-1}=C(n,k,\beta_{0},K),$

where $\frac{1}{q_{2}}+\frac{1}{q_{2}^{*}}=1$ and in the last equation we use
the choice of the function $\gamma_{2}(r)$.

$\bullet$ Case III. We note that since $[\chi]$ is big,
$V_{t}\geq\int_{X}\chi^{n}>0$, hence $\frac{c_{t}^{n/k}}{V_{t}}\leq
C_{\omega,\chi}$ for a uniform $C_{\omega,\chi}=C_{\omega,\chi}(n,k)>0$ which
we will fix throughout the proof below. Then we have

$\displaystyle\frac{A_{s}}{r^{n+1}}$ $\displaystyle=$
$\displaystyle\frac{c_{t}^{n/k}}{V_{t}}\frac{1}{r^{n+1}}\int_{\Omega_{s}}(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)e^{\frac{n}{k}h}\omega^{n}$
$\displaystyle\leq$
$\displaystyle\frac{C(n,\omega,\chi,k,\beta_{0})}{r^{n+1}}\Big{(}\int_{E_{2}}e^{p_{3}h}\omega^{n}\Big{)}^{1/q_{3}^{*}}\Big{(}\int_{E_{2}}e^{h}\Big{)}^{1/q_{3}}$
$\displaystyle\leq$ $\displaystyle
C(n,k,\omega,\chi,K)\gamma_{3}(r)^{\frac{1}{q_{3}}}r^{\frac{n}{q_{3}}-n-1}=C(n,k,\omega,\chi,K),$

where $\frac{1}{q_{3}}+\frac{1}{q_{3}^{*}}=1$ and in the last identity we use
the choice of the function $\gamma_{3}(r)$.

So for all cases I, II and III, we get from (3.7) that

$\int_{\Omega_{s}}\,{\rm
exp}\,\Big{\\{}c_{0}\Big{(}\frac{-v_{t}+(1-r)u_{t}-3\beta_{0}r-s}{A_{s}^{1/(n+1)}}\Big{)}^{\frac{n+1}{n}}\Big{\\}}\omega^{n}\leq
C,$ (3.8)

for some constant $C>0$ depending on $n,k,\chi,\omega,K$ and the exponents
$p_{1},p_{2},p_{3}$ in each case, respectively. In particular this $C$ is
independent of the choice of $r\in(0,\bar{r}_{0})$.

We choose $p>n$ as $p=p_{1}$ in case I, and arbitrary and large $p>n$ in cases
II and III.

Define $\eta:{{\bf R}}_{+}\to{{\bf R}}_{+}$ by $\eta(x)=(\,{\rm
log}\,(1+x))^{p}$. Note that $\eta$ is a strictly increasing function with
$\eta(0)=0$, and let $\eta^{-1}$ be its inverse function. If we let

$\displaystyle\Psi:=\frac{c_{0}}{2}\Big{(}\frac{-v_{t}+(1-r)u_{t}-3\beta_{0}r-s}{A_{s}^{1/(n+1)}}\Big{)}^{\frac{n+1}{n}}$
(3.9)

then we have for any $z\in\Omega_{s}$, by the generalized Young’s inequality
with respect to $\eta$,

$\displaystyle\Psi(z)^{p}e^{\frac{n}{k}h(z)}$ $\displaystyle\leq$
$\displaystyle\int_{0}^{\,{\rm
exp}\,({\frac{n}{k}h(z)})}\eta(x)dx+\int_{0}^{\Psi(z)^{p}}\eta^{-1}(y)dy$
$\displaystyle\leq$ $\displaystyle
e^{\frac{n}{k}h(z)}(1+|h(z)|)^{p}+C(p)e^{2\Psi(z)}$

We integrate both sides in the inequality above over $z\in\Omega_{s}$, and get
by (3.8) that

$\displaystyle\int_{\Omega_{s}}\Psi(z)^{p}e^{\frac{n}{k}h(z)}\omega^{n}\leq\|e^{h}\|_{L^{n/k}(\,{\rm
log}\,L)^{p}}+C,$

where the constant $C>0$ depends only on $n,k,\omega,\chi,p,K$. In view of the
definition of $\Psi$, this implies

$\int_{\Omega_{s}}(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)^{\frac{(n+1)p}{n}}e^{\frac{n}{k}h}\omega^{n}\leq
CA_{s}^{\frac{p}{n}}\big{(}\|e^{h}\|_{L^{n/k}(\,{\rm log}\,L)^{p}}+1\big{)}.$
(3.10)

It follows from the Hölder inequality that

$\displaystyle A_{s}$ $\displaystyle=$
$\displaystyle\frac{c_{t}^{n/k}}{V_{t}}\int_{\Omega_{s}}(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)e^{\frac{n}{k}h}\omega_{X}^{n}$
$\displaystyle\leq$
$\displaystyle\Big{(}\frac{c_{t}^{n/k}}{V_{t}}\int_{\Omega_{s}}(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)^{\frac{(n+1)p}{n}}e^{\frac{n}{k}h}\omega^{n}\Big{)}^{\frac{n}{(n+1)p}}\cdot\Big{(}\frac{c_{t}^{n/k}}{V_{t}}\int_{\Omega_{s}}e^{\frac{n}{k}h}\omega^{n}\Big{)}^{1/q}$
$\displaystyle\leq$ $\displaystyle
CA_{s}^{\frac{1}{n+1}}\Big{(}\|e^{h}\|_{L^{n/k}(\,{\rm
log}\,L)^{p}}+1)\Big{)}^{\frac{n}{(n+1)p}}\cdot\Big{(}\frac{c_{t}^{n/k}}{V_{t}}\int_{\Omega_{s}}e^{\frac{n}{k}h}\omega^{n}\Big{)}^{1/q}$

where $q>1$ satisfies $\frac{n}{p(n+1)}+\frac{1}{q}=1$, i.e.
$q=\frac{p(n+1)}{p(n+1)-n}$. The inequality above yields

$A_{s}\leq C\Big{(}\|e^{h}\|_{L^{n/k}(\,{\rm
log}\,L)^{p}}+1\Big{)}^{1/p}\cdot\Big{(}\frac{c_{t}^{n/k}}{V_{t}}\int_{\Omega_{s}}e^{\frac{n}{k}h}\omega^{n}\Big{)}^{\frac{1+n}{qn}}=B_{0}\Big{(}\frac{c_{t}^{n/k}}{V_{t}}\int_{\Omega_{s}}e^{\frac{n}{k}h}\omega^{n}\Big{)}^{1+\delta_{0}}.$
(3.11)

Observe that the exponent of the integral on the right hand of (3.11)
satisfies

$\frac{1+n}{qn}=\frac{pn+p-n}{pn}=1+\delta_{0}>1,\quad\mbox{for
$\delta_{0}:=\frac{p-n}{pn}>0$.}$

We remark that $\delta_{0}$ can be chosen to be close to $1/n$ in cases II and
III by picking $p$ large enough. Furthermore, we note that

$B_{0}:=C\Big{(}\|e^{h}\|_{L^{n/k}(\,{\rm log}\,L)^{p}}+1\Big{)}^{1/p}$ (3.12)

is a constant depending only on $n,k,\omega,\chi,K$, and the exponents
$p_{1},p_{2},p_{3}$ in each case, respectively, and in particular, it is
independent of $r$ with $r\in(0,\bar{r}_{0})$.

If we define

$\phi(s)=\frac{c_{t}^{n/k}}{V_{t}}\int_{\Omega_{s}}e^{\frac{n}{k}h}\omega^{n},$

then (3.11) shows that if $\Omega_{s+s^{\prime}}\not=\emptyset$ then

$s^{\prime}\phi(s+s^{\prime})\leq B_{0}\phi(s)^{1+\delta_{0}},\quad\mbox{for
all }s^{\prime}\geq 0\mbox{ and }s\geq 0.$ (3.13)

We now choose $r_{0}<\bar{r}_{0}$ small in each case as follows.

Case I. We choose $r_{0}>0$ small so that for $r\in(0,r_{0})$

$\displaystyle B_{0}\phi(0)^{\delta_{0}}\leq
B_{0}\Big{(}\int_{E_{2}}e^{h}\omega^{n}\Big{)}^{\delta_{0}}\leq
B_{0}C_{0}^{\delta_{0}}(\gamma_{1}(r)r^{n})^{\delta_{0}}\leq
B_{0}C_{0}^{\delta_{0}}(\gamma_{1}(r_{0})r_{0}^{n})^{\delta_{0}}\leq\frac{1}{2}$
(3.14)

and $\phi(0)\leq C_{0}\gamma_{1}(r)r^{n}<\bar{C}r^{1/\delta_{0}}$ by Lemma 1
for some uniform $\bar{C}$.

Case II. We choose $r_{0}>0$ small so that for all $r\in(0,r_{0})$

$\displaystyle B_{0}\phi(0)^{\delta_{0}}$ $\displaystyle\leq$ $\displaystyle
B_{0}\Big{(}\int_{E_{2}}e^{\frac{n}{k}h}\omega^{n}\Big{)}^{\delta_{0}}$
$\displaystyle\leq$ $\displaystyle
B_{0}\Big{(}\int_{E_{2}}e^{p_{2}h}\omega^{n}\Big{)}^{\delta_{0}/q_{2}^{*}}\Big{(}\int_{E_{2}}e^{h}\omega^{n}\Big{)}^{\delta_{0}/q_{2}}$
$\displaystyle\leq$ $\displaystyle
B_{0}C_{0}^{\delta_{0}/q_{2}}\Big{(}\int_{E_{2}}e^{p_{2}h}\omega^{n}\Big{)}^{\delta_{0}/q_{2}^{*}}(\gamma_{2}(r)r^{n})^{\delta_{0}/q_{2}}$
$\displaystyle\leq$ $\displaystyle
B_{0}C_{0}^{\delta_{0}/q_{2}}\Big{(}\int_{E_{2}}e^{p_{2}h}\omega^{n}\Big{)}^{\delta_{0}/q_{2}^{*}}(\gamma_{2}(r_{0})r_{0}^{n})^{\delta_{0}/q_{2}}\leq\frac{1}{2}$

where $\frac{1}{q}_{2}+\frac{1}{q^{*}_{2}}=1$ and we also have

$\phi(0)\leq
C_{0}^{1/q_{2}}\Big{(}\int_{E_{2}}e^{p_{2}h}\omega^{n}\Big{)}^{1/q_{2}^{*}}(\gamma_{2}(r)r^{n})^{1/q_{2}}<\bar{C}r^{1/\delta_{0}}$

for some uniform $\bar{C}>0$ by the definition of $\gamma_{2}(r)$.

Case III. We choose $r_{0}>0$ small so that for $r\in(0,r_{0})$

$\displaystyle B_{0}\phi(0)^{\delta_{0}}$ $\displaystyle\leq$ $\displaystyle
B_{0}C_{\omega,\chi}^{\delta_{0}}\Big{(}\int_{E_{2}}e^{\frac{n}{k}h}\omega^{n}\Big{)}^{\delta_{0}}$
$\displaystyle\leq$ $\displaystyle
B_{0}C_{\omega,\chi}^{\delta_{0}}\Big{(}\int_{E_{2}}e^{p_{3}h}\omega^{n}\Big{)}^{\delta_{0}/q_{3}^{*}}\Big{(}\int_{E_{2}}e^{h}\omega^{n}\Big{)}^{\delta_{0}/q_{3}}$
$\displaystyle\leq$ $\displaystyle
B_{0}C_{\omega,\chi}^{\delta_{0}}C_{0}^{\delta_{0}/q_{3}}\Big{(}\int_{E_{2}}e^{p_{3}h}\omega^{n}\Big{)}^{\delta_{0}/q_{3}^{*}}(\gamma_{3}(r)r^{n})^{\delta_{0}/q_{3}}$
$\displaystyle\leq$ $\displaystyle
B_{0}C_{\omega,\chi}^{\delta_{0}}C_{0}^{\delta_{0}/q_{3}}\Big{(}\int_{E_{2}}e^{p_{3}h}\omega^{n}\Big{)}^{\delta_{0}/q_{3}^{*}}(\gamma_{3}(r_{0})r_{0}^{n})^{\delta_{0}/q_{3}}\leq\frac{1}{2}$

where $\frac{1}{q}_{3}+\frac{1}{q^{*}_{3}}=1$ and we also have

$\phi(0)\leq
C_{\omega,\chi}C_{0}^{1/q_{3}}\Big{(}\int_{E_{2}}e^{p_{3}h}\omega^{n}\Big{)}^{1/q_{3}^{*}}(\gamma_{3}(r)r^{n})^{1/q_{3}}<\bar{C}r^{1/\delta_{0}}$

by the choice of $\gamma_{3}(r)$.

It is clear that in all cases, $r_{0}$ and $\bar{C}$ depend only on the given
data, namely, $n,k,\omega,\chi,K$ and $p_{a}$, and we have
$B_{0}\phi(0)^{\delta_{0}}\leq\frac{1}{2}$ and
$\phi(0)\leq\bar{C}r^{1/\delta_{0}}$.

Define a sequence of increasing real numbers $(s_{j})$ inductively such that
$s_{0}=0$ and

$s_{j+1}=\sup\\{s>s_{j}|\phi(s)>\frac{1}{2}\phi(s_{j})\\}.$

Then we can show that (see [9]) $\phi(s_{j})\leq 2^{-j}\phi(s_{0})$ and

$s_{j+1}-s_{j}\leq 2B_{0}2^{-j\delta_{0}}\phi(0)^{\delta_{0}}.$

Thus the limit $S_{\infty}=\lim_{j\to\infty}s_{j}$ satisfies

$S_{\infty}\leq\frac{2B_{0}}{1-2^{-\delta_{0}}}\phi(0)^{\delta_{0}}\leq\frac{2B_{0}\bar{C}^{\delta_{0}}}{1-2^{-\delta_{0}}}r=\hat{C}r.$

Hence the set $\Omega_{\hat{C}r}=\emptyset$, and we conclude that

$v_{t}\geq(1-r)u_{t}-3\beta_{0}r-\hat{C}r,\quad\mbox{or equivalently }\quad
v_{t}-u_{t}\geq-Cr,$

for some uniform constant $C>0$ depending only on the given data. By the
normalization $\max(u_{t}-v_{t})=\max(v_{t}-u_{t})$, it is clear that
$v_{t}-u_{t}\leq Cr$. The proof of Theorem 1 is complete.

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Department of Mathematics & Computer Science, Rutgers University, Newark, NJ
07102 USA

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Department of Mathematics, Columbia University, New York, NY 10027 USA

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