##  Stability of measure solutions to a generalized Boltzmann
equation with collisions of a random number of particles ⋄

Henryk Gackia and £ukasz Stettnerb,⋆

aFaculty of Science and Technology, University of Silesia in Katowice, Bankowa
14,

40-007 Katowice, Poland

bInstitute of Mathematics Polish Academy of Sciences, Śniadeckich 8,

00-656 Warsaw, Poland

## Abstract

In the paper we study a measure version of the evolutionary nonlinear
Boltzmann-type equation in which we admit a random number of collisions of
particles. We consider first a stationary model and use two methods to find
its fixed points: the first based on Zolotarev seminorm and the second on
Kantorovich-Rubinstein maximum principle. Then a dynamic version of Boltzmann-
type equation is considered and its asymptotical stability is shown.

Keywords: Boltzmann equation, collisions of random particles, Zolotariev
seminorm, Kantorovich-Rubinstein maximum principle

2010 MSC: 82B31, 82B21.

##  1 Introduction

In the paper we consider a nonlinear evolutionary measure valued Boltzmann
type equation of the form

(1) $\frac{d\psi}{dt}+\psi=\mathbb{P}\,\psi\qquad\text{for}\qquad t\geq 0$

where the operator $\mathbb{P}$ maps $\mathcal{M}_{1}(\mathbb{R}_{+})$ the
space of probability measures on $\mathbb{R}_{+}=[0,\infty)$ into itself. We
are looking for
$\psi:\mathbb{R}_{+}\rightarrow\mathcal{M}_{sig}(\mathbb{R}_{+})$ with
$\psi_{0}\in\mathcal{M}_{1}(\mathbb{R}_{+})$, where
$\mathcal{M}_{sig}(\mathbb{R}_{+})$ (or shortly $\mathcal{M}_{sig}$) is the
space of all signed measures on $\mathbb{R}_{+}$.

Equation (1) is a generalized version of the equation considered in [27] (see
also section 8.9 in [17], or [6] for the motivation),

(2) $\frac{\partial u(t,x)}{\partial
t}+u(t,x)=\int\limits_{x}\limits^{\infty}\frac{dy}{y}\int\limits_{0}\limits^{y}u(t,y-z)u(t,z)dz:=P\,u(x)\qquad
t\geq 0,\qquad x\geq 0,$

which describes energy changes subject to the collision operator $P\,u(x)$ and
which was obtained

————————————————————————————————————————

⋄ Research supported by National Science Center, Poland by grant UMO-
$2016/23/B/ST1/00479$

⋆ Corresponding author

Email addresses<EMAIL_ADDRESS>(Henryk Gacki<EMAIL_ADDRESS>(£ukasz Stettner)

from Boltzmann equation corresponding to a spatially homogeneous gas with no
external forces using Abel transformation. To be more precise, in the theory
of dilute gases Boltzmann equation in the general form ${dF(t,x,v)\over
dt}=C(F(t,x,v))$ gives us an information about time, position and velocity of
particles of the dilute gas. This equation is a base for many mathematical
models of colliding particles. In particular, for a spatially homogeneous gas
we come to the equation (2) with additional conditions saying that its
solution $u$, for fixed $t$, is a density with first moment equal to $1$,
which in turn corresponds to the conservation law of mass and energy. The
operator $Pu$ is a density function of the random variable
$\eta(\xi_{1}+\xi_{2})$ where random variables $\eta$, $\xi_{1}$ and $\xi_{2}$
are independent and $\eta$ is uniformly distributed, while $\xi_{1}$,
$\xi_{2}$ have the same density function $u$. The assumption that $\eta$ has
uniform distribution on $[0,1]$ is quite restrictive and there are no physical
reasons to assume that the distribution of energy of particles can be
described only by its density (is absolutely continuous). Moreover collision
of two particles maybe replaced by collision of a random number of particles.
This is a reason that in what follows we shall consider a measure valued
version (1) of the equation (2).

Let

(3)
$D:=\big{\\{}\mu\in\mathcal{M}_{1}(\mathbb{R}_{+}):m_{1}(\mu)=1\big{\\}},\quad\text{with}\quad
m_{1}(\mu)=\int\limits_{0}\limits^{\infty}\,x\mu(dx),$

and denote by $\bar{D}$ a weak closure of $D$, which is of the form

(4) $\bar{D}:=\big{\\{}\mu\in\mathcal{M}_{1}(\mathbb{R}_{+}):m_{1}(\mu)\leq
1\big{\\}}.$

The operator $P$ defined in (2) describes collision of two particles. In what
follows we shall consider a general situation of collision of a random number
of particles. To describe the collision operator in this case we start from
recalling the convolution operator of order $n$ and the linear operator
$P_{\varphi}$, which is related to multiplication of random variables.

For every $n\in\mathbb{N}$ let
$P_{*n}:\mathcal{M}_{sig}\rightarrow\mathcal{M}_{sig}$, be given by the
formula

(5) $P_{*1}\mu:=\mu,\quad
P_{*(n+1)}\mu:=\mu*P_{*n}\mu\quad\text{for}\quad\mu\in\mathcal{M}_{sig}.$

It is easy to verify that
$P_{*n}(\mathcal{M}_{1}(\mathbb{R}_{+}))\subset\mathcal{M}_{1}(\mathbb{R}_{+})$
for every $n\in\mathbb{N}$. Moreover,
$P_{*n}|_{\mathcal{M}_{1}(\mathbb{R}_{+})}$ has a simple probabilistic
interpretation: if $\xi_{1},...,\xi_{n}$ are independent random variables with
the same probability distribution $\mu$, then $P_{*n}\mu$ is the probability
distribution of $\xi_{1}+...+\xi_{n}$.

The second class of operators we are going to study is related to
multiplication of random variables. Formal definition is as follows: given
$\mu,v\in\mathcal{M}_{sig}$, we define product $u\circ v$ by

(6) $(\mu\circ
v)(A):=\int_{\mathbb{R}_{+}}\int_{\mathbb{R}_{+}}\mathbf{1}_{A}(xy)\mu(dx)v(dy)\quad\text{for}\quad
A\in{\mathcal{B}_{\mathbb{R}_{+}}}.$

and

(7) $\big{\langle}f,\mu\circ
v\big{\rangle}=\int_{\mathbb{R}_{+}}\int_{\mathbb{R}_{+}}f(xy)\mu(dx)v(dy)$

for every Borel measurable $f:\mathbb{R}_{+}\rightarrow\mathbb{R}$ such that
$(x,y)\mapsto f(xy)$ is integrable with respect to the product of $|\mu|$ and
$|v|$. For fixed $\varphi\in\mathcal{M}_{1}$ define

(8) $P_{\varphi}\mu:=\phi\circ\mu\quad\text{for}\quad\mu\in\mathcal{M}_{sig}.$

Similarly as in the case of convolution it follows that
$P_{\varphi}(\mathcal{M}_{1})\subset\mathcal{M}_{1}$. For
$\mu\in\mathcal{M}_{1}$ the measure $P_{\varphi}\mu$ has an immediate
probabilistic interpretation: if $\varphi$ and $\mu$ are probability
distributions of random variables $\xi$ and $\eta$ respectively, then
$P_{\varphi}\mu$ is the probability distribution of the product $\xi\eta$.

We introduce now definition of more general version of $P$ allowing infinite
number of collisions:

(9)
$\mathbb{P}:=\sum_{i=1}^{\infty}\alpha_{i}P_{\varphi_{i}}P_{*_{i}}=\sum_{i=1}^{\infty}\alpha_{i}\mathbb{P}_{i},$

where we have $\mathbb{P}_{i}:=P_{\varphi_{i}}P_{*_{i}}$,
$\sum\limits_{i=1}\limits^{\infty}\alpha_{i}=1$, $\alpha_{i}\geq 0$,
$\varphi_{i}\in\mathcal{M}_{1}$ and $m_{1}(\varphi_{i})=1/i$ and the limit of
the series is considered in the weak topology sense that is,
$\sum\limits_{i=1}\limits^{n}\alpha_{i}P_{\varphi_{i}}P_{*_{i}}\Rightarrow\sum\limits_{i=1}\limits^{\infty}\alpha_{i}P_{\varphi_{i}}P_{*_{i}}$,
which means
$\sum\limits_{i=1}\limits^{n}\alpha_{i}P_{\varphi_{i}}P_{*_{i}}(f)\to\sum\limits_{i=1}\limits^{\infty}\alpha_{i}\mathbb{P}_{i}(f)$
as $n\to\infty$, for any continuous bounded function $f$ defined on
$\mathbb{R}$. From (9) it follows that
$\mathbb{P}\mathcal{M}_{1}\subset\mathcal{M}_{1}$. Using (5) and (8) it is
easy to verify that for $\mu\in D$,

(10) $m_{1}(P_{*i}\mu)=i\quad\text{and}\quad m_{1}(P_{\varphi_{i}}\mu)=1/i.$

Given $\mu\in\mathcal{M}_{1}$ the value of $\mathbb{P}\mu$ can be considered
as the probability distribution of a random variable $\zeta$ defined as

(11) $\zeta:=\eta_{\tau}(\sum_{j=1}^{\tau}\xi_{\tau j}),$

where we have sequences of independent random variables $\eta_{i}$,
$\xi_{ij}$, $j=1,2,\ldots,$ and $\tau$, such that $\xi_{ij}$ have the same
probability distribution $\mu$ for $j=1,2,\ldots,$, random variables
$\eta_{i}$ have the probability distribution $\varphi_{i}$ and random variable
$\tau$ takes values in the set $\left\\{1,2,\ldots\right\\}$ such that
$P\left\\{\tau=j\right\\}=\alpha_{j}$. Physically this means that the number
of colliding particles is random and energies of particles before a collision
are of independent quantities and that a particle after collision of $i$-th
particles takes the $\eta_{i}$ part of the sum of the energies of the
colliding particles.

We can also define $\tilde{\eta}_{i}:=i\eta_{i}$ and consider

(12) $\zeta:=\tilde{\eta}_{\tau}({1\over\tau}\sum_{j=1}^{\tau}\xi_{\tau j})$

and write

(13)
$\mathbb{P}=\sum_{i=1}^{\infty}\alpha_{i}\tilde{P}_{\varphi_{i}}\tilde{P}_{*_{i}}$

where $\tilde{P}_{\varphi_{i}}={P}_{\tilde{\varphi}_{i}}$ with
$\tilde{\varphi}_{i}$ being the probability distribution of $i\eta_{i}$, and
$\tilde{P}_{*_{i}}\mu$ is the probability distribution of
${\xi_{i1}+\ldots+\xi_{ii}\over i}$.

Measure valued solutions to the Boltzmann-type equations with different
collision operator and even in multidimensional case were studied in a number
of papers see e.g. [22], [1] and [21]. In the paper [22] existence and
stability of measure solutions to the spatially homogeneous Boltzmann
equations that have polynomial and exponential moment production properties is
shown. In the paper [1] existence and uniqueness of measure solutions to one
dimensional Boltzmann dissipative equation and then their asymtotics is
considered. In this paper first stationary (steady state) equation for a
specific collision operator is studied and then dynamic fixed point theorem is
used. Asymptotics of solutions to the Boltzmann equation with infinite energy
to so called self similar solutions was studied in [7]. Asymptotic property of
self similar solutions to the Boltzmann Equation for Maxwell molecules was
then shown in [9]. Long time behaviour of the solutions to the nonlinear
Boltzmann equation for spacially uniform freely cooling inelastic Maxwell
molecules was studied in [5]. Stability of Boltzmann equation with external
potential was also considered in [28] and in the case of exterior problem in
[16]. Solutions of the Boltzmann equation with collision of $N$ particles and
their limit behaviour when $n\to\infty$ over finite time horizon were studied
in [2]. In [8], the N-particle model, which includes multi-particle
interactions was considered. It is shown that under certain natural
assumptions we obtain a class of equations which can be considered as the most
general Maxwell-type model.

In [26] an individual based model describing phenotypic evolution in
hermaphroditic populations which includes random and assortative mating of
individuals is introduced. By increasing the number of individuals to infinity
a nonlinear transport equation is obtained, which describes the evolution of
phenotypic probability distribution. The main result of the paper is a theorem
on asymptotic stability of trait (which concerns the model with more general
operator $P$) with respect to Fortet-Mourier metric.

Stability problems of the Boltzmann-type equation (1) with operator $P$
corresponding to collision of two particles was studied in the paper [18]. The
case with infinite number of particles was considered in [19] using Zolotariev
seminorm approach. Properties of stationary solutions corresponding two
collision of two particles were studied in [20].

In this paper we study stability of solutions to one dimensional Boltzmann-
type equation (1) with operator $P$ of the form $\mathbb{P}$ defined in (9).
We show that if this equation has a stationary solution $\mu^{*}$, such that
its support covers $\mathbb{R}_{+}$, then taking into account positivity of
solutions to (1)-(9) we have its asymptotical stability in Kantorovich -
Wasserstein metric to $\mu^{*}$. We consider first stationary equation and
look for fixed points of the operator $\mathbb{P}$. For this purpose we adopt
two methods to show the existence of fixed point of $\mathbb{P}$ with the
first moment equal to $1$. The first method is based on Zolotarev seminorm and
in some sense simplifies the method used in [19]. The second method is based
on Kantorovich-Rubinstein maximum principle and generalizes the results of
[14] and then also of [18] to the case of a random number of colliding
particles. We also show several characteristics of fixed points of
$\mathbb{P}$. Results on the stationary equation are then used to study
stability of the dynamic Boltzmann equation. The novelty of the paper is that
we consider the Boltzmann-type equation (1) with random unbounded number of
colliding particles and show its stability in Kantorovich - Wasserstein metric
using probabilistic methods, generalizing former results of [18], [19] and
[14]. To improve readability of the paper an appendix is added to the paper
where some important results, which are used in the paper, are formulated and
their proofs are sketched.

##  2 Properties of the operator $\mathbb{P}$

We study first several properties $\mathbb{P}$.

######  Proposition 2.1.

Operator $\mathbb{P}$ transforms the set $D$ or $\bar{D}$ into itself.It is
continuous with respect to weak topology in $\mathcal{M}_{1}(\mathbb{R}_{+})$
i.e. whenever $\mathcal{M}_{1}(\mathbb{R}_{+})\ni\mu_{n}\Rightarrow\mu$ we
have $\mathbb{P}\mu_{n}\Rightarrow\mathbb{P}\mu$ as $n\to\infty$. Furthermore
whenever
$m_{r}(\varphi_{i})=\int\limits_{\mathbb{R}_{+}}x^{r}\varphi_{i}(dx)<\infty$
and $m_{r}(\mu)=\int\limits_{\mathbb{R}_{+}}x^{r}\mu(dx)<\infty$ for $r\geq
1$, $i=1,2,\ldots$ then

(14)
$m_{r}(\mathbb{P}\mu)\leq\sum_{i=1}^{\infty}\alpha_{i}m_{r}(\varphi_{i})i^{r}m_{r}(\mu).$

###### Proof.

Note first that for each $\mu\in\mathcal{M}_{1}(\mathbb{R}_{+})$ we have that
$P_{\varphi_{i}}P_{*_{i}}\mu\in\mathcal{M}_{1}(\mathbb{R}_{+})$ for
$i=1,2,\ldots$ and therefore
$\mathbb{P}\mu\in\mathcal{M}_{1}(\mathbb{R}_{+})$. Moreover for $\mu\in D$

(15)
$m_{1}(P_{\varphi_{i}}P_{*_{i}}\mu)=\int_{0}^{\infty}\ldots\int_{0}^{\infty}x(y_{1}+y_{2}+\ldots+y_{i})\varphi_{i}(dx)\mu(dy_{1})\ldots\mu(dy_{i})={1\over
i}i=1$

so that $P_{\varphi_{i}}P_{*_{i}}\mu\in D$ and consequently
$\mathbb{P}:D\mapsto D$. Similarly $\mathbb{P}:\bar{D}\mapsto\bar{D}$. For a
given continuous bounded function $f:\mathbb{R}_{+}\to\mathbb{R}$ and
$\mathcal{M}_{1}(\mathbb{R}_{+})\ni\mu_{n}\Rightarrow\mu$, as $n\to\infty$ we
have

(16) $\displaystyle
P_{\varphi_{i}}P_{*_{i}}\mu_{n}(f)=\int_{0}^{\infty}\ldots\int_{0}^{\infty}f(x(y_{1}+y_{2}+\ldots+y_{i}))\varphi_{i}(dx)\mu_{n}(dy_{1})\ldots\mu_{n}(dy_{i})\to$
$\displaystyle\int_{0}^{\infty}\ldots\int_{0}^{\infty}f(x(y_{1}+y_{2}+\ldots+y_{i}))\varphi_{i}(dx)\mu(dy_{1})\ldots\mu(dy_{i})=P_{\varphi_{i}}P_{*_{i}}\mu(f)$

as $n\to\infty$. In fact, from continuity of

$(y_{1},y_{2},\ldots,y_{i})\to\int_{0}^{\infty}f(x(y_{1}+y_{2}+\ldots+y_{i}))\varphi_{i}(dx),$

and weak convergence of the measures
$\mu_{n}(dy_{1})\ldots\mu_{n}(dy_{i})\Rightarrow\mu(dy_{1})\ldots\mu(dy_{i})$,
using (16) we immediately obtain that
$\mathbb{P}\mu_{n}\Rightarrow\mathbb{P}\mu$ which is desired continuity
property. Now using $\zeta$ defined in (12), independency of random variables,
as well as convexity we obtain

(17) $\displaystyle m_{r}(\mathbb{P}\mu)$ $\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}_{+}}x^{r}\mathbb{P}\mu(dx)=E\left[\zeta^{r}\right]=\sum_{i=1}^{\infty}\alpha_{i}E\left[\eta^{r}_{i}\right]i^{r}E\left[\left({1\over
i}\sum_{j=1}^{i}\xi_{\tau j}\right)^{r}\right]$ $\displaystyle\leq$
$\displaystyle\sum_{i=1}^{\infty}\alpha_{i}m_{r}(\eta_{i})i^{r}m_{r}(\mu),$

which completes the proof. ∎

We comment below the formula (41)

######  Remark 2.1.

Note that since $m_{1}(\varphi_{i})={1\over i}$ we have
$\int\limits_{\mathbb{R}_{+}}x\varphi_{i}(dx)\leq\left(\int\limits_{\mathbb{R}_{+}}x^{r}\varphi_{i}(dx)\right)^{{1\over
r}}$ and consequently $m_{r}(\varphi_{i})\geq{1\over i^{r}}$, so that in
general we may not have that
$\sum\limits_{i=1}\limits^{\infty}\alpha_{i}m_{r}(\varphi_{i})i^{r}<\infty$.

This sum is finite however when for example we know that for a sufficiently
large $i$ we have that $\alpha_{i}m_{r}(\varphi_{i})\leq{1\over i^{\beta}}$
with $\beta>1+r$. Finiteness of the sum above shall play an important role in
the approach to study fixed points of $\mathbb{P}$ with the use of Zolotariev
seminorm (see section 3).

In what follows we are interested to find a fix point of the operator
$\mathbb{P}$ in the set $D$. It is clear clear $\mu=\delta_{0}$ is a fixed
point of $\mathbb{P}$ in $\bar{D}$. Typical way to find a fixed point is to
consider iterations $\mathbb{P}\mu$ for $\mu\in D$ and since the measures
$\left\\{\mathbb{P}\mu,\mathbb{P}^{2}\mu,\ldots\right\\}$ are tight, expect a
limit to be a fixed point. However even when such iteration converges in a
weak topology, its limit will be in $\bar{D}$ and it is not clear that such
limit will be a fixed point. Note furthermore that when we have more than one
particle, the operator $\mathbb{P}$ is nonlinear and we can not use several
techniques typical for linear operators.

######  Remark 2.2.

Notice that fixed point is not necessary in $D$ since the set $D$ is not
closed in the weak topology. In fact, when $\mu_{n}\left\\{{1\over
n}\right\\}={n\over n+1}$ and $\mu_{n}\left\\{{n}\right\\}={1\over n+1}$, then
$\mu_{n}\in D$ and $\mu_{n}\Rightarrow\mu:=\delta_{0}$ so that total mass of
$\mu$ is concentrated at $0$, and $\int_{0}^{\infty}x\mu(dx)=0$. Similary for
$0<\alpha<1$, letting $\mu_{n}(1-\alpha)=1-{1\over n}$,
$\mu_{n}(1)={1-\alpha\over n}$ and $\mu_{n}(n)={\alpha\over n}$ we clearly
have that $\mu_{n}\in D$ and $\mu_{n}\Rightarrow\delta_{1-\alpha}$, as
$n\to\infty$. One can show that the closure $\bar{D}$ of the set $D$ in the
weak topology consists of all probability measures
$\nu\in\mathcal{M}_{1}(\mathbb{R}_{+})$ such that $m_{1}(\nu)\leq 1$. Consider
the following example: $P\left\\{\varphi_{i}=0\right\\}={n\over n+1}$,
$P\left\\{\varphi_{i}=(n+1){1\over i}\right\\}={1\over n+1}$ and
$\mu([\Delta,\infty))=1$ with $m_{1}(\mu)=1$, $\Delta>0$ and fixed integer
$n\geq 1$. Then the support of $\mathbb{P}\mu$ consists of $0$ and the second
part contained in $[\Delta({n+1\over i}),\infty)$ and inductively the support
of $\mathbb{P}^{k}\mu$ contains $0$ and its second part is contained in
$[\Delta({(n+1)\over i})^{k},\infty)$, for positive integer $k$. Since
$\mathbb{P}^{k}\mu(0)={n\over n+1}$ it is clear that
$\mathbb{P}^{k}\mu\Rightarrow\delta_{0}$, as $k\to\infty$, so that limit of
$\mathbb{P}^{k}\mu$, which is also a fixed point of $\mathbb{P}$, is not in
$D$.

On the other hand, when
$P\left\\{\varphi_{i}=\delta\right\\}={n\over(n+1)-i\delta}$ and
$P\left\\{\varphi_{i}=(n+1){1\over i}\right\\}={1-i\delta\over(n+1)-i\delta}$
with $i\delta<1$ and $\mu([\Delta,\infty))=1$ with $m_{1}(\mu)=1$, $\Delta>0$
we have that $\cup_{k}supp(\mathbb{P}^{k}\mu)$ is dense in $[0,\infty)$.

When support of $\varphi_{i}$ contains a sequence of positive real numbers
converging to $0$, then $\cup_{k}supp(\mathbb{P}^{k}\mu)$ is dense in
$[0,\infty)$ no matter what $\mu\in D$ is chosen. Assume additionally that
${1\over i}\in supp\varphi_{i}$ for each $i=1,2,\ldots$. Then
$supp(\mathbb{P}^{k}\mu)\subset supp(\mathbb{P}^{k+1}\mu)$, so that we have an
increasing sequence of closed sets $supp(\mathbb{P}^{k}\mu)$ which cover the
interval $(0,\infty)$.

Next two Propositions show properties of the fixed point of $\mathbb{P}$.

Let $\Lambda:=\left\\{i:\alpha_{i}>0\right\\}$

######  Proposition 2.2.

Let $\mu\in{D}$ be a fixed point of $\mathbb{P}$. When there are at least two
elements of $\Lambda$ and $\tilde{\varphi}_{i}=\delta_{1}$ for $i\in\Lambda$
then either $\mu=\delta_{1}$ or $m_{\beta}(\mu)=\infty$ for any $\beta>1$.
When $\mu=\delta_{1}$ then $\tilde{\varphi}_{i}=\delta_{1}$ for $i\in\Lambda$.

###### Proof.

Using (13) we have that the law of
$\tilde{\eta}_{\tau}({1\over\tau}\sum_{j=1}^{\tau}\xi_{\tau j})$, when the law
of $\xi_{ij}$ is $\mu$, is also $\mu$. Then for $\beta>1$ we have

(18)
$E\left\\{\tilde{\eta}_{\tau}^{\beta}({1\over\tau}\sum_{j=1}^{\tau}\xi_{\tau
j})^{\beta}\right\\}=\sum_{i=1}^{\infty}\alpha_{i}m_{\beta}(\tilde{\varphi}_{i})E\left\\{({1\over
i}\sum_{j=1}^{i}\xi_{ij})^{\beta}\right\\}=m_{\beta}(\mu).$

Notice now that by strict convexity we have for $\beta>1$ that $\left({1\over
i}\sum\limits_{j=1}^{i}\xi_{ij}\right)^{\beta}<{1\over
i}\sum\limits_{j=1}^{i}\xi_{ij}^{\beta}$ with equality only when
$\xi_{ij}=\xi_{ij^{\prime}}$ for $j^{\prime}\in\left\\{1,2,\ldots,i\right\\}$.

Therefore when there are at least one element in $\Lambda$ before
$i\in\Lambda$ and $\tilde{\varphi}_{j}=\delta_{1}$ for $j\in\Lambda$ we have

(19) $E\left\\{({1\over
i}\sum_{j=1}^{i}\tilde{\eta}_{i}\xi_{ij})^{\beta}\right\\}<\sum_{j=1}^{i}{1\over
i}E\left\\{\tilde{\eta}_{i}^{\beta}\xi_{ij}^{\beta}\right\\}=m_{\beta}(\tilde{\eta}_{i})m_{\beta}(\mu)=m_{\beta}(\mu)$

with strict inequality whenever $m_{\beta}(\mu)<\infty$. Since $\xi_{ij}$ and
$\xi_{ij^{\prime}}$ are independent for $j^{\prime}\neq j$, they should be
deterministic and because $m_{1}(\mu)=1$ we have that $\mu=\delta_{1}$. When
$\mu=\delta_{1}$ by (18) and (19) we have

(20)
$E\left\\{\tilde{\eta}_{\tau}^{\beta}({1\over\tau}\sum_{j=1}^{\tau}\xi_{\tau
j})^{\beta}\right\\}=m_{\beta}(\tilde{\varphi}_{i})m_{\beta}(\mu)=m_{\beta}(\mu)$

which implies that
$m_{\beta}(\tilde{\varphi}_{i})=1=m_{1}(\tilde{\varphi}_{i})$, which is again
true only when $\tilde{\varphi}_{i}=\delta_{1}$. ∎

In the next Proposition we adopt some arguments of Proposition 2.1 of [20]

######  Proposition 2.3.

Assume that for $1\neq k\in\Lambda$ there are $q_{1},q_{2}\in supp\varphi_{k}$
such that $q_{1}>{1\over k}$, $q_{2}<{1\over k}$. Under the above assumptions,
when $\mu^{*}$ is a fixed point of $\mathbb{P}$, its support is either
$\left\\{0\right\\}$ or $[0,\infty)$.

###### Proof.

Notice that the support of each $\varphi_{i}$ contains an element not greater
that ${1\over i}$. Therefore taking into account that $q_{2}<{1\over k}$ we
have that when $0\neq r\in supp\mu^{*}$ then the support of
$\mathbb{P}\mu^{*}$ contains an element not greater than

$\sum\limits_{i=1,i\neq
k}\limits^{\infty}\alpha_{i}r+\alpha_{k}q_{2}kr=\sum\limits_{i=1}\limits^{\infty}\alpha_{i}r+\alpha_{k}(q_{2}-{1\over
k})kr=r(1-(1-kq_{2})\alpha_{k})).$

Since $(1-(1-kq_{2})\alpha_{k})<1$ iterating the operator $\mathbb{P}$ we
obtain that $0\in supp\mu^{*}$. Therefore to show that $supp\mu^{*}$ in the
case when $supp\mu^{*}\neq\left\\{0\right\\}$ covers whole interval
$[0,\infty)$ it suffices to show that for any $r\in(0,\infty)\cap supp\mu^{*}$
we have that $A:=supp\left\\{\mathbb{P}_{k}^{i}\mu^{*},\ for\
i=1,2,\ldots\right\\}$ is dense in $[0,r]$. For a given $\varepsilon>0$ we can
find positive integer $m$ such that $\left({1\over
k}\right)^{m}\leq{\varepsilon\over rkq_{1}}$ and $(kq_{2})^{m}\leq 1$. Then we
can find positive integer $n$ such that
$(kq_{1})^{n-1}(kq_{2})^{m}\leq({1\over k})^{m}kq_{1}\leq 1$ and
$(kq_{1})^{n}(kq_{2})^{m}>1$. Consequently we have that
$q_{2}^{m}(kq_{1})^{n}\leq{\varepsilon\over r}$ and
$k^{m}q_{2}^{m}(kq_{1})^{n}\geq 1$. Therefore $iq_{2}^{m}(kq_{1})^{n}r\in A$
for $i=1,2,\ldots,k^{m}$, and $q_{2}^{m}(kq_{1})^{n}r\leq\varepsilon$ while
$k^{m}q_{2}^{m}(kq_{1})^{n}r\geq r$. Since $\varepsilon$ can be chosen
arbitrarily small and $A$ is closed we have that $[0,\infty)\subset A$, which
we obtain taking into account that together with $r$ the values $kq_{1}^{i}$
are in $supp\mu^{*}$. ∎

######  Remark 2.3.

In some cases there is only one fixed point of $\mathbb{P}$ which is
$\delta_{0}\notin D$. Given probability measure $\mu$ on $\mathbb{R}_{+}$
consider its characteristic function $\psi(t)=\int e^{itx}\mu(dx)$. If $\mu$
is a fixed point of $\mathbb{P}$ then in the case when
$\Lambda=\left\\{2\right\\}$, $\psi$ satisfies the following function equation

(21)
$\psi(t)=\int\limits_{\mathbb{R}_{+}}\left[\psi(tz)\right]^{2}\varphi_{2}(dz)$

Assume $\varphi_{2}={1\over 2}\delta_{0}+{1\over 2}\delta_{1}$. Then (21)
takes the form

(22) $\psi(t)={1\over 2}+{1\over 2}\left[\psi(t)\right]^{2}$

The solutions to this quadratic equation are constant functions $\psi(t)$ and
since $\psi(0)=1$ the only solution which is a characteristic function is
$\psi(t)\equiv 1$, which corresponds to $\mu=\delta_{0}$. Consequently we
don’t have fixed point of $\mathbb{P}$ in the set $D$.

In the space ${\mathcal{M}}_{1}(\mathbb{R}_{+})$ consider Kantorovich -
Wasserstein metric (see [15], Definition 4.3.1 or [10]) given by the formula

(23) $\|\mu_{1}-\mu_{2}\|_{\mathcal{K}}=\sup\\{|\mu_{1}(f)-\mu_{2}(f)|:\
f\in{\mathcal{K}}\\}\qquad\text{for}\qquad\mu_{1},\mu_{2}\in{\mathcal{M}}_{1}(\mathbb{R}_{+}),$

where ${\mathcal{K}}$ is the set of functions $f:\mathbb{R}_{+}\to R$ which
satisfy the condition

$\ |f(x)-f(y)|\leq|x-y|\quad\text{\rm for}\quad x,y\in\mathbb{R}_{+}.$

We recall now another, Forter-Mourier metric $\|\cdot\|_{\mathcal{F}}$ in
${\mathcal{M}}_{1}(\mathbb{R}_{+})$

(24) $\|\mu_{1}-\mu_{2}\|_{\mathcal{F}}=\sup\\{|\mu_{1}(f)-\mu_{2}(f)|:\
f\in{\mathcal{F}}\\}\qquad\text{for}\qquad\mu_{1},\mu_{2}\in{\mathcal{M}}_{1}(\mathbb{R}_{+}),$

where ${\mathcal{F}}$ consists of functions $f$ from ${\mathcal{K}}$ such that
$\|f\|_{sup}=\sup\limits_{x\in\mathbb{R}_{+}}|f(x)|\leq 1$.

Almost immediately we have that $\|f\|_{\mathcal{F}}\leq\|f\|_{\mathcal{K}}$
for $\mu\in{\mathcal{M}}_{1}(\mathbb{R}_{+})$ which means that Kantorovich -
Wasserstein metric is stronger than Fortet Mourier metric. Notice however that
the convergence of probability measures in Fortet Mourier metric is equivalent
to weak convergence (see [13] or [10]). We have

######  Lemma 2.1.

The operator $\mathbb{P}$ is nonexpansive in $\bar{D}$ in Kantorovich -
Wasserstein metric.

###### Proof.

We are going to show that for each $i=1,2,\ldots$ the operators
$\mathbb{P}_{i}$ are nonexpansive in Kantorovich - Wasserstein metric. In
fact, for $f\in{\mathcal{K}}$ we and $\mu\in\bar{D}$ we have

(25)
$\mathbb{P}_{i}\mu(f)=\int\limits_{\mathbb{R}_{+}}\ldots\int\limits_{\mathbb{R}_{+}}f(r(x_{1}+x_{2}+\ldots+x_{i}))\varphi_{i}(dr)\mu(dx_{1})\mu(dx_{2})\ldots\mu(dx_{i}).$

Define for $\nu\in\bar{D}$

$\displaystyle\tilde{f}(x):=\int\limits_{\mathbb{R}_{+}}\ldots\int\limits_{\mathbb{R}_{+}}f(r(x+x_{2}+\ldots+x_{i})){\varphi}_{i}(dr)(\mu(dx_{2})\ldots\mu(dx_{i})+$
$\displaystyle\nu(dx_{2})\mu(dx_{3})\ldots\mu(dx_{i})+\nu(dx_{2})\nu(dx_{3})\mu(dx_{4})\ldots\mu(dx_{i})+\ldots+$
(26) $\displaystyle\nu(dx_{2})\nu(dx_{3})\ldots\nu(dx_{i})).$

Then $\tilde{f}\in{\mathcal{K}}$ (since $m_{1}({\varphi_{i}})={1\over i}$).
Furthermore

(27)
$\mathbb{P}_{i}\mu(f)-\mathbb{P}_{i}\nu(f)=\mu(\tilde{f})-\nu(\tilde{f}).$

Hence
$\|\mathbb{P}_{i}\mu-\mathbb{P}_{i}\nu\|_{\mathcal{K}}\leq\|\mu-\nu\|_{\mathcal{K}}$
and consequently

(28)
$\|\mathbb{P}\mu-\mathbb{P}\nu\|_{\mathcal{K}}\leq\sum_{i=1}^{\infty}\alpha^{i}\|\mathbb{P}_{i}\mu-\mathbb{P}_{i}\nu\|_{\mathcal{K}}\leq\|\mu-\nu\|_{\mathcal{K}},$

which completes the proof. ∎

######  Remark 2.4.

One can notice that operator $\mathbb{P}$ is not nonexpansive in Fortet
Mourier metric.

We also have

######  Corollary 2.1.

Operator $\mathbb{P}$ transforms $D$ into itself and is defined also as a
limit of $\sum\limits_{i=1}\limits^{n}\alpha_{i}\mathbb{P}_{i}$ in Kantorovich
- Wasserstein metric. Furthermore $D$ is convex and closed in Kantorovich -
Wasserstein metric.

###### Proof.

Notice that for $\mu\in D$ we have
$\left(\sum\limits_{i=1}\limits^{n}\alpha_{i}\right)^{-1}\sum\limits_{i=1}\limits^{n}\alpha_{i}\mathbb{P}_{i}\mu\Rightarrow\mathbb{P}\mu$
as $n\to\infty$ and

$m_{1}(\left(\sum_{i=1}^{n}\alpha_{i}\right)^{-1}\sum_{i=1}^{n}\alpha_{i}\mathbb{P}_{i}\mu)=1=m_{1}(\mathbb{P}\mu).$

Therefore by the second part of Theorem 6.2 we have that $\mathbb{P}\mu$ is a
limit of $\sum\limits_{i=1}\limits^{n}\alpha_{i}\mathbb{P}_{i}\mu$ in
Kantorovich - Wasserstein metric. Convexity of $D$ is obvious. Closedness of
$D$ in Kantorovich - Wasserstein metric follows almost immediately from the
following arguments: convergence in Kantorovich - Wasserstein metric implies
weak convergence and therefore the limit is a probability measure. Furthermore
convergence in Kantorovich - Wasserstein metric implies convergence of the
first moments. ∎

##  3 Fixed point of $\mathbb{P}$ using Zolotariev seminorm

In this section we shall introduce Zolotariev seminorm. Namely for
$\mu\in\mathcal{M}_{sig}(\mathbb{R}_{+})$ and $r\in(1,2)$ define

(29) $\|\mu\|_{r}:=\sup\left\\{\mu(f):f\in\mathcal{F}_{r}\right\\}$

where $\mathcal{F}_{r}$ consists of differentiable functions
$f:\mathbb{R}_{+}\to R$, which satisfy the condition

$\ |f^{\prime}(x)-f^{\prime}(y)|\leq|x-y|^{r-1}\quad\text{\rm for}\quad
x,y\in\mathbb{R}_{+}.$

One can notice that when $f\in\mathcal{F}_{r}$ then also function
$f(x)+\alpha+\beta x$ is in $\mathcal{F}_{r}$. Therefore $\|\mu\|_{r}=\infty$
whenever $\mu(\mathbb{R}_{+})\neq 0$ or $m_{1}(\mu)\neq 0$. The following
properties of Zolotariev seminorm will be used later on

######  Lemma 3.1.

Assume that

(30) $\sum_{i=1}^{\infty}\alpha_{i}m_{r}(\phi_{i})i^{r}<\infty$

then for $\mu,\nu\in D$, whenever $m_{r}(\mu)<\infty$ and $m_{r}(\nu)<\infty$
for $i=1,2,\ldots$, we have

(31) ${1\over r}|m_{r}(\mu)-m_{r}(\nu)|\leq\|\mu-\nu\|_{r}\leq{1\over
r}|m_{r}|(\mu-\nu):={1\over r}\int\limits_{\mathbb{R}_{+}}x^{r}|\mu-\nu|(dx),$
(32) $\|P_{*_{i}}\mu-P_{*_{i}}\nu\|_{r}\leq i\|\mu-\nu\|_{r},$ (33)
$\|P_{\varphi_{i}}P_{*_{i}}\mu-P_{\varphi_{i}}P_{*_{i}}\nu\|_{r}\leq
m_{r}(\varphi_{i})i\|\mu-\nu\|_{r},$ (34)
$\|\mathbb{P}\mu-\mathbb{P}\nu\|_{r}\leq\sum_{i=1}^{\infty}\alpha_{i}m_{r}(\phi_{i})i\|\mu-\nu\|_{r}.$

###### Proof.

When $f\in\mathcal{F}_{r}$ then $f(x)=f(0)+\int_{0}^{1}xf^{\prime}(tx)dt$.
Therefore

(35) $\displaystyle\mu(f)-\nu(f)$ $\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}_{+}}\int\limits_{0}\limits^{1}xf^{\prime}(tx)\mu(dx)-\int\limits_{\mathbb{R}_{+}}\int\limits_{0}\limits^{1}xf^{\prime}(tx)dt\nu(dx)$
$\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}_{+}}\int\limits_{0}\limits^{1}x(f^{\prime}(tx)-f^{\prime}(0))dt\mu(dx)-\int_{\mathbb{R}_{+}}\int_{0}^{1}x(f^{\prime}(tx)-f^{\prime}(0))dt\nu(dx)$
$\displaystyle\leq$
$\displaystyle\int\limits_{\mathbb{R}_{+}}\int\limits_{0}\limits^{1}x|f^{\prime}(tx)-f^{\prime}(0)|dt|\mu(dx)-\nu(dx)|$
$\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}_{+}}\int\limits_{0}\limits^{1}x|tx|^{r-1}dt|\mu(dx)-\nu(dx)|={1\over
r}|m_{r}|(\mu-\nu),$

which completes the proof of the second part of (31). For $f(x)={1\over
r}x^{r}$ we have that $f\in\mathcal{F}_{r}$ and then

(36) $\|\mu-\nu\|_{r}\geq{1\over
r}|\int\limits_{\mathbb{R}_{+}}x^{r}\mu(dx)-\int\limits_{\mathbb{R}_{+}}x^{r}\nu(dx)|\geq{1\over
r}|m_{r}(\mu)-m_{r}(\nu)|.$

Therefore we have (31).

To prove (32) notice that $P_{*_{i}}\mu-
P_{*_{i}}\nu=\sum\limits_{j=1}\limits^{i}\mu_{j}*\mu-\sum\limits_{j=1}\limits^{i}\mu_{j}*\nu$
where $\mu_{j}=(P_{*_{i-j}}\mu)*(P_{*_{j-1}}\nu)$ with
$P_{*_{0}}\mu=P_{*_{0}}\nu=\delta_{0}$. When $f\in\mathcal{F}_{r}$ then
$\bar{f}(y)=\int\limits_{\mathbb{R}_{+}}f(x+y)\mu_{j}(dx)$ is in
$\mathcal{F}_{r}$. Therefore

(37) $\displaystyle\|\mu_{j}*(\mu-\nu)\|_{r}$ $\displaystyle=$
$\displaystyle\sup_{f\in{\mathcal{F}_{r}}}|\int\limits_{\mathbb{R}_{+}}\int\limits_{\mathbb{R}_{+}}f((x+y)\mu_{j}(dx)(\mu-\nu)(dy)|$
$\displaystyle=$
$\displaystyle\sup_{f\in{\mathcal{F}_{r}}}|\int\limits_{\mathbb{R}_{+}}\bar{f}(y)(\mu-\nu)(dy)|\leq\|\mu-\nu\|_{r},$

from which (32) easily follows. Now

(38) $\displaystyle\|P_{\varphi_{i}}P_{*_{i}}\mu-
P_{\varphi_{i}}P_{*_{i}}\nu\|_{r}$ $\displaystyle=$
$\displaystyle\sup_{f\in{\mathcal{F}_{r}}}\left(\int\limits_{\mathbb{R}_{+}}\int\limits_{\mathbb{R}_{+}}f(zx)\varphi_{i}(dz)P_{*_{i}}\mu(dx)-\int\limits_{\mathbb{R}_{+}}\int\limits_{\mathbb{R}_{+}}f(zx)\varphi_{i}(dz)P_{*_{i}}\nu(dx)\right)$
$\displaystyle\leq$ $\displaystyle
m_{r}(\varphi_{i})\sup_{f\in{\mathcal{F}_{r}}}\left(\int\limits_{\mathbb{R}_{+}}\tilde{f}(x)P_{*_{i}}\mu(dx)-\int\limits_{\mathbb{R}_{+}}\tilde{f}(x)P_{*_{i}}\nu(dx)\right)$
$\displaystyle\leq$ $\displaystyle m_{r}(\varphi_{i})\|P_{*_{i}}\mu-
P_{*_{i}}\nu\|_{r},$

where $\tilde{f}(x)={1\over
m_{r}(\varphi_{i})}\int\limits_{\mathbb{R}_{+}}f(zx)\varphi_{i}(dx)$ is an
element of $\mathcal{F}_{r}$, provided that $f\in\mathcal{F}_{r}$. The
estimation (33) follows now directly from (32). From (33) we immediately
obtain (34). ∎

The main result of this section is the existence of a fixed point of
$\mathbb{P}$. We use the same assumptions as in the paper [19], where similar
result was obtained. Our proof is different and shows another properties of
the fixed point of $\mathbb{P}$. It is formulated as follows

######  Theorem 3.1.

Assume that for some $r\in(1,2)$ we have $m_{r}(\varphi_{i})<{1\over i}$ for
all $i\in\Lambda$, (30) is satisfied and there is $\mu\in D$ such that
$m_{r}(\mu)<\infty$. Then $\mathbb{P}^{n}\mu$ converges in Kantorovich -
Wasserstein metric to a unique $\mu^{*}\in D$, which is a fixed point of
$\mathbb{P}$ such that $m_{r}(\mu^{*})<\infty$.

###### Proof.

Under (30) using Proposition 2.1 we have that
$m_{r}(\mathbb{P}^{n}\mu)<\infty$ for $n=1,2,\ldots$. By (34) we have that
$\|\mathbb{P}^{j+1}\mu-\mathbb{P}^{j}\mu\|_{r}\leq\lambda^{j}|\mathbb{P}\mu-\mu\|_{r}$,
where $\lambda=\sum\limits_{i=1}\limits^{\infty}\alpha_{i}m_{r}(\phi_{i})i$.
Therefore

(39)
$\|\mathbb{P}^{n}\mu-\mu\|_{r}=\|\sum\limits_{j=1}\limits^{n}\mathbb{P}^{j}\mu-\mathbb{P}^{j-1}\mu\|_{r}\leq\sum\limits_{j=1}\limits^{n}\lambda^{j}\|\mathbb{P}\mu-\mu\|_{r},$

where $\mathbb{P}^{0}\mu=\mu$. Since by (31)

(40) $\|\mathbb{P}^{n}\mu-\mu\|_{r}\geq{1\over
r}|m_{r}(\mathbb{P}^{n}\mu)-m_{r}(\mu)|,$

we therefore have that for $n=1,2,\ldots$

(41) $m_{r}(\mathbb{P}^{n}\mu)\leq{r\over
1-\lambda}\|\mathbb{P}\mu-\mu\|_{r}+m_{r}(\mu):=\kappa<\infty.$

The sequence of probability measures
$\left\\{\mu,\mathbb{P}\mu,\mathbb{P}^{2}\mu,\ldots\right\\}$ is tight and
therefore there is $\mu^{*}$ and subsequence $(n_{k})$ such that
$\mathbb{P}^{n_{k}}\mu\Rightarrow\mu^{*}$ as $k\to\infty$. By (41) and
Corollary 6.1 we have that $\|\mathbb{P}^{n_{k}}\mu-\mu^{*}\|_{\mathcal{K}}\to
0$ as $k\to\infty$. Consequently $\mu^{*}\in D$. Since for $K>0$

(42) $\int\limits_{\mathbb{R}_{+}}(x^{r}\wedge K)\mathbb{P}^{n_{k}}\mu(dx)\leq
m_{r}(\mathbb{P}^{n_{k}}\mu)\leq\kappa$

and therefore letting $k\to\infty$ we have that
$\int_{\mathbb{R}_{+}}(x^{r}\wedge K)\mu^{*}(dx)\leq\kappa$, which by Fatou
lemma gives that $m_{r}(\mu^{*})\leq\kappa$. Now

(43)
$\|\mathbb{P}^{n_{k}}\mathbb{P}\mu-\mathbb{P}^{n_{k}}\mu\|_{\mathcal{K}}=\|\mathbb{P}\mathbb{P}^{n_{k}}\mu-\mathbb{P}^{n_{k}}\mu\|_{\mathcal{K}}\to\|\mathbb{P}\mu^{*}-\mu^{*}\|_{\mathcal{K}}:=\beta$

and

$\displaystyle\|\mathbb{P}\mathbb{P}\mu^{*}-\mathbb{P}\mu^{*}\|_{\mathcal{K}}=\lim_{k\to\infty}\|\mathbb{P}\mathbb{P}\mathbb{P}^{n_{k}}\mu-\mathbb{P}\mathbb{P}^{n_{k}}\mu\|_{\mathcal{K}}\geq$
(44)
$\displaystyle\lim_{k\to\infty}\|\mathbb{P}\mathbb{P}^{n_{k+1}}\mu-\mathbb{P}^{n_{k+1}}\mu\|_{\mathcal{K}}=\|\mathbb{P}\mu^{*}-\mu^{*}\|_{\mathcal{K}}=\beta,$

so that taking into account that
$\|\mathbb{P}\mathbb{P}\mu^{*}-\mathbb{P}\mu^{*}\|_{\mathcal{K}}\leq\|\mathbb{P}\mu^{*}-\mu^{*}\|_{\mathcal{K}}$
we obtain that
$\|\mathbb{P}\mathbb{P}\mu^{*}-\mathbb{P}\mu^{*}\|_{\mathcal{K}}=\|\mathbb{P}\mu^{*}-\mu^{*}\|_{\mathcal{K}}$.
By small modification of (43) and (3) we obtain that for any $n=0,1,\ldots$

(45)
$\|\mathbb{P}^{n+1}\mu^{*}-\mathbb{P}^{n}\mu^{*}\|_{\mathcal{K}}=\|\mathbb{P}\mu^{*}-\mu^{*}\|_{\mathcal{K}}.$

On the other hand by (34) we have that
$\|\mathbb{P}^{n+1}\mu^{*}-\mathbb{P}^{n}\mu^{*}\|_{r}\leq\lambda^{n}\|\mathbb{P}\mu^{*}-\mu^{*}\|_{r}$.
Therefore
$\lim\limits_{n\to\infty}\|\mathbb{P}^{n+1}\mu^{*}-\mathbb{P}^{n}\mu^{*}\|_{r}=0$.
By Theorem 6.3 we also have that
$\lim\limits_{n\to\infty}\|\mathbb{P}^{n+1}\mu^{*}-\mathbb{P}^{n}\mu^{*}\|_{\mathcal{K}}=0$.
Therefore $\|\mathbb{P}\mu^{*}-\mu^{*}\|_{\mathcal{K}}=0$ and $\mu^{*}$ is a
fixed point of $\mathbb{P}$. If there is another weak limit $\nu^{*}$ of
subsequence of $\mathbb{P}^{n}\mu$, then
$\|\mathbb{P}^{n}\mu^{*}-\mathbb{P}^{n}\nu^{*}\|_{r}\to 0$, as $n\to\infty$
and by Theorem 6.3 again we have that
$\|\mu^{*}-\nu^{*}\|_{\mathcal{K}}=\lim\limits_{n\to\infty}\|\mathbb{P}^{n+1}\mu^{*}-\mathbb{P}^{n}\mu^{*}\|_{\mathcal{K}}=0$.
Consequently any weak limit of a subsequence of $\mathbb{P}^{n}\mu$ is equal
to $\mu^{*}$, which means that $\mathbb{P}^{n}\mu\Rightarrow\mu^{*}$ and the
convergence also holds in Kantorovich - Wasserstein metric. ∎

######  Remark 3.1.

We may not exclude the case in which we have another fixed point $\nu^{*}\in
D$ of $\mathbb{P}$ such that $m_{r}(\nu^{*})=\infty$ for each $r>1$. Notice
furthermore that in the case when $\varphi_{i}$ is uniformly distributed over
the interval $[0,{1\over i}]$ then we have $m_{r}(\varphi_{i})={1\over
r+1}\left({2\over i}\right)^{r}<{1\over i}$ for all $r\in(1,2)$ and
$i\in\Lambda\setminus\left\\{1\right\\}$, so that if (30) is satisfied and
$1\notin\Lambda$, by the above theorem we have existence of a unique fixed
point of $\mathbb{P}$ in $D$ with finite $r$-th moment.

##  4 Contraction property of the operator $\mathbb{P}$

We have the following generalization of Theorem 5.2.2 of [14]

######  Theorem 4.1.

Assume for some $i\in\Lambda$ we have that $0$ is an accumulation point of
$\varphi_{i}$, where $m_{1}(\varphi_{i})={1\over i}$. Then for $\mu,\nu\in{D}$
such that $\mu\neq\nu$ and

(46) $supp(P_{*(i-1)}(\mu+\nu))=\mathbb{R}_{+}$

we have

(47)
$\|\mathbb{P}_{i}\mu-\mathbb{P}_{i}\nu\|_{\mathcal{K}}<\|\mu-\nu\|_{{\mathcal{K}}}$

and consequently

(48)
$\|\mathbb{P}\mu-\mathbb{P}\nu\|_{\mathcal{K}}<\|\mu-\nu\|_{\mathcal{K}}.$

###### Proof.

Recall that $\mathbb{P}_{i}={P}_{\varphi_{i}}{P}_{*_{i}}$ and assume that
$\|\mathbb{P}_{i}\mu-\mathbb{P}_{i}\nu\|_{\mathcal{K}}=\|\mu-\nu\|_{\mathcal{K}}$.
By Theorem 6.4 there is $f_{0}\in\mathcal{K}$ such that

(49) $\|\mathbb{P}_{i}\mu-\mathbb{P}_{i}\nu\|_{\mathcal{K}}=\langle
f_{0},\mathbb{P}_{i}\mu-\mathbb{P}_{i}\nu\rangle.$

Then

$\displaystyle\|\mu-\nu\|_{{\mathcal{K}}}=\int_{{\mathbb{R}_{+}}^{i+1}}f_{0}((x_{1}+x_{2}+\ldots+x_{i})r){\varphi}_{i}(dr)$
(50)
$\displaystyle\left[\mu(dx_{1})\mu(dx_{2})\ldots\mu(dx_{i})-\nu(dx_{1})\nu(dx_{2})\ldots\nu(dx_{i})\right]=\langle
f_{1},\mu-\nu\rangle,$

where

$\displaystyle
f_{1}(x)=\int_{{\mathbb{R}_{+}}^{i}}f_{0}((x+x_{2}+\ldots+x_{i})r){\varphi}_{i}(dr)\left[\mu(dx_{2})\ldots\mu(dx_{i})+\right.$
$\displaystyle\nu(dx_{2})\mu(dx_{3})\ldots\mu(dx_{i})+\nu(dx_{2})\nu(dx_{3})\mu(dx_{4})\ldots\mu(dx_{i})+\ldots+$
(51)
$\displaystyle\left.\nu(dx_{2})\nu(dx_{3})\nu(dx_{4})\ldots\mu(dx_{i})+\nu(dx_{2})\nu(dx_{3})\nu(dx_{4})\ldots\nu(dx_{i})\right].$

Clearly, using again the fact that $m_{1}(\varphi_{i})={1\over i}$ we have
that ${f}_{1}\in\mathcal{K}$.

By Theorem 6.4 there are two points $x_{1},x_{2}\in\mathbb{R}_{+}$ such that
$x_{1}<x_{2}$ and $|{f}_{1}(x_{2})-{f}_{1}(x_{1})|=x_{2}-x_{1}$. Since
${f}_{1}$ is nonexpansive (Lipschitz with constant less or equal to $1$) we
have that ${f}_{1}(x)=\theta x+\sigma$, for $x\in(x_{1},x_{2})$ with
$|\theta|=1$. Therefore
$|{f}_{1}(x_{1}+\varepsilon)-{f}_{1}(x_{1})|=\varepsilon$ for
$\varepsilon\in(0,x_{2}-x_{1})$. Replacing $f_{0}$ by $-f_{0}$ we may assume
that ${f}_{1}(x_{1}+\varepsilon)-{f}_{1}(x_{1})=\varepsilon$ for
$\varepsilon\in(0,x_{2}-x_{1})$. We are going now to show that for
$x\in\mathbb{R}_{+}$

(52) $f_{0}(x)=x+c$

with a constant $c\in\mathbb{R}$. Consider now $u_{1},u_{2}\in\mathbb{R}_{+}$
such that $u_{1}<u_{2}$. We want to show that then

(53) $f_{0}(u_{2})-f_{0}(u_{1})\geq u_{2}-u_{1},$

which by nonexpansiveness of $f_{0}$ implies that
$f_{0}(u_{2})-f_{0}(u_{1})=u_{2}-u_{1}$ and therefore $f_{0}$ is of the form
(52). Assume conversely that $f_{0}(u_{2})-f_{0}(u_{1})<u_{2}-u_{1}$. Since
$f_{0}$ as a Lipschitzian mapping is almost everywhere differentiable there is
$\bar{u}\in(u_{1},u_{2})$ such that $f_{0}^{\prime}(\bar{u})<1$ and for
$\delta\in(0,\delta_{0})$ we have

(54) ${f_{0}(\bar{u}+\delta)-f_{0}(\bar{u})\over\delta}<1.$

Define

(55)
$h(y_{2},\ldots,y_{i},r,\varepsilon)={f_{0}((x_{1}+\varepsilon+y_{2}+\ldots+y_{i})r)-f_{0}((x_{1}+y_{2}+\ldots+y_{i})r)\over\varepsilon
r}.$

By definition of $f_{1}$ for $\varepsilon\in(0,x_{2}-x_{1})$ we have

$\displaystyle
1={{f}_{1}(x_{1}+\varepsilon)-{f}_{1}(x_{1})\over\varepsilon}=\int_{(\mathbb{R}_{+})^{i-1}}\int_{\mathbb{R}_{+}}h(y_{2},\ldots,y_{i},r,\varepsilon)r{\phi}_{i}(dr)\left[\mu(dy_{2})\right.$
$\displaystyle\ldots\mu(dy_{i})+\nu(dy_{2})\mu(dy_{3})\ldots\mu(dy_{i})+\nu(dy_{2})\nu(dy_{3})\mu(dy_{4})\ldots\mu(dy_{i})+$
$\displaystyle\left.\ldots+\nu(dy_{2})\nu(dy_{3})\nu(dy_{4})\ldots\mu(dy_{i})+\nu(dy_{2})\nu(dy_{3})\nu(dy_{4})\ldots\nu(dy_{i})\right]:=$
(56)
$\displaystyle\int_{(\mathbb{R}_{+})^{i}}h(y_{2},\ldots,y_{i},r,\varepsilon)q(dy_{2},\ldots,dy_{i},dr),$

where we define implicitly probability measure $q$. Since $0$ is an
accumulation point of $supp\phi$ there is $\bar{r}\in supp{\phi}$ such that
$x_{1}\bar{r}<\bar{u}$. Then there is
$(\bar{y}_{2},\bar{y}_{3},\ldots,\bar{y}_{i})\in supp(P_{*(i-1)}(\mu+\nu))$
such that

(57)
$\bar{u}-x_{1}\bar{r}=(\bar{y}_{2}+\bar{y}_{3}+\ldots,\bar{y}_{i})\bar{r}.$

Consequently for every $\bar{\varepsilon}\in(0,x_{2}-x_{1})$ such that
$\bar{\varepsilon}\bar{r}<\delta_{0}$ we have

(58)
$h(\bar{y}_{2}+\bar{y}_{3}+\ldots,\bar{y}_{i},\bar{r},\bar{\varepsilon})={f_{0}(\bar{u}+\bar{\varepsilon}\bar{r})-f_{0}(\bar{u})\over\bar{\varepsilon}\bar{r}}<1.$

Since $h\leq 1$ by continuity of $h$ and full support of $P_{*(i-1)}(\mu+\nu)$
we have that

(59)
$\int_{(\mathbb{R}_{+})^{i}}h(y_{2},\ldots,y_{i},r,\varepsilon)q(dy_{2},\ldots,dy_{i},dr)<1,$

a contradiction to (4). Therefore we have equality in (53). Consequently
$f_{0}(x)=x+c$ for a constant $c$. Since $\mathbb{P}_{i}\mu$ and
$\mathbb{P}_{i}\nu\in D$ we therefore have $\langle
f_{0},\mathbb{P}_{i}\mu-\mathbb{P}_{i}\nu\rangle=m_{1}(\mathbb{P}_{i}\mu)-m_{1}(\mathbb{P}_{i}\nu)=0$
and by (49)

$\|\mathbb{P}_{i}\mu-\mathbb{P}_{i}\nu\|_{\mathcal{K}}=\|\mu-\nu\|_{\mathcal{K}}=0,$

which contradicts the fact that $\mu\neq\nu$.

∎

######  Remark 4.1.

Condition $supp(P_{*(i-1)}(\mu+\nu))=\mathbb{R}_{+}$ is not very restrictive.
It holds in particular when $supp(\mu+\nu)=\mathbb{R}_{+}$ or when
$supp\mu=\mathbb{R}_{+}$.

We have the following consequences of Theorem 4.1

######  Corollary 4.1.

If for some $i\in\Lambda$ we have that $0$ is an accumulation point of
$\phi_{i}$ and $\mu^{*}$ is a weak accumulation point of $\mathbb{P}^{n}\mu$
for $\mu\in{D}$ i.e. there is a sequence $(n_{k})$ such that
$\mathbb{P}^{n_{k}}\mu\Rightarrow\mu^{*}$, as $k\to\infty$ and
$supp\mu^{*}=\mathbb{R}_{+}$, $\mu^{*}\in D$ then $\mu^{*}$ is a fixed point
of $\mathbb{P}$.

###### Proof.

Notice that $m_{1}(\mathbb{P}^{n_{k}}\mu)=m_{1}(\mu^{*})$ so that by Theorem
6.2 we have that $\|\mathbb{P}^{n_{k}}\mu-\mu^{*}\|_{\mathcal{K}}\to 0$ as
$k\to\infty$. Therefore

(60)
$\|\mathbb{P}^{n_{k+1}}\mathbb{P}\mu-\mathbb{P}^{n_{k+1}}\mu\|_{\mathcal{K}}\leq\|\mathbb{P}^{n_{k}+1}\mathbb{P}\mu-\mathbb{P}^{n_{k}+1}\mu\|_{\mathcal{K}}\leq\|\mathbb{P}^{n_{k}}\mathbb{P}\mu-\mathbb{P}^{n_{k}}\mu\|_{\mathcal{K}}\leq\|\mathbb{P}\mu-\mu\|_{\mathcal{K}},$

so that there is a limit
$\lim_{k\to\infty}\|\mathbb{P}^{n_{k}}\mathbb{P}\mu-\mathbb{P}^{n_{k}}\mu\|_{\mathcal{F}_{0}}:=\beta$
and by continuity

(61)
$\|\mathbb{P}^{n_{k}}\mathbb{P}\mu-\mathbb{P}^{n_{k}}\mu\|_{\mathcal{K}}=\|\mathbb{P}\mathbb{P}^{n_{k}}\mu-\mathbb{P}^{n_{k}}\mu\|_{\mathcal{K}}\to\|\mathbb{P}\mu^{*}-\mu^{*}\|_{\mathcal{K}}=\beta.$

If $\mathbb{P}\mu^{*}\neq\mu^{*}$ and assumptions of Corollary are satisfied
then by Theorem 4.1 we have that
$\|\mathbb{P}\mathbb{P}\mu^{*}-\mathbb{P}\mu^{*}\|_{\mathcal{K}}<\|\mathbb{P}\mu^{*}-\mu^{*}\|_{\mathcal{K}}=\beta$,
while

$\displaystyle\|\mathbb{P}\mathbb{P}\mu^{*}-\mathbb{P}\mu^{*}\|_{\mathcal{K}}=\lim_{k\to\infty}\|\mathbb{P}\mathbb{P}\mathbb{P}^{n_{k}}\mu-\mathbb{P}\mathbb{P}^{n_{k}}\mu\|_{\mathcal{K}}\geq$
(62)
$\displaystyle\lim_{k\to\infty}\|\mathbb{P}\mathbb{P}^{n_{k+1}}\mu-\mathbb{P}^{n_{k+1}}\mu\|_{\mathcal{K}}=\|\mathbb{P}\mu^{*}-\mu^{*}\|_{\mathcal{K}}=\beta$

and we have a contradiction. Therefore $\mathbb{P}\mu^{*}=\mu^{*}$. ∎

######  Corollary 4.2.

If for some $i\in\Lambda$ we have that $0$ is an accumulation point of
$\phi_{i}$ and $\mu^{*}\in{D}$ is a fixed point of $\mathbb{P}$, then there
are no other fixed point $\nu^{*}\in{D}$. Consequently for $\nu\in D$ we have
that any weakly convergent subsequence $\mathbb{P}^{n_{k}}\nu$ either
converges to $\mu^{*}$, as $n\to\infty$, or to a measure
$\tilde{\mu}\in\bar{D}\setminus D$.

###### Proof.

By Proposition 2.3 since $\mu^{*}\in{D}$ we have that
$supp\mu^{*}=\mathbb{R}_{+}$. Then we use again Theorem 4.1. Namely when
$\nu^{*}\in{D}$ is another fixed point of $\mathbb{P}$ we have

(63)
$\|\mu^{*}-\nu^{*}\|_{\mathcal{K}}=\|\mathbb{P}\mu^{*}-\mathbb{P}\nu^{*}\|_{\mathcal{K}}<\|\mu^{*}-\nu^{*}\|_{\mathcal{K}},$

which is a contradiction. Any sequence $(\mathbb{P}^{n}\nu)$ is compact in
Fortet Mourier metric and its subsequence converges to a measure
$\tilde{\mu}$, and the convergence is also in Kantorovich Wasserstein metric
whenever $m_{1}(\mu)=1$. In the last case we have $\tilde{\mu}=\mu^{*}$ by
uniqueness of fixed point of $\mathbb{P}$ in $D$. ∎

Using Theorem 6.1 we can now strengthen the last Corollary.

######  Corollary 4.3.

Assume there is $i\in\Lambda$ such that $0$ be an accumulation point of
$\varphi_{i}$. Assume that $\mu^{*}\in{D}$ is a fix point of $\mathbb{P}$.
Then for $\nu\in{D}$ for any limit of weakly convergent subsequence
$\mathbb{P}^{n_{k}}\nu$ which belongs to $D$ is also in Kantorovich
Wasserstein metric. Furthermore if the sequence $n\to\mathbb{P}^{n}\nu$ is
uniformly integrable we have

(64) $\lim_{n\to\infty}\|\mathbb{P}^{n}\nu-\mu^{*}\|_{\mathcal{K}}=0.$

###### Proof.

As above using Proposition 2.3, since $\mu^{*}\in{D}$, we have that
$supp\mu^{*}=\mathbb{R}_{+}$. The first part easily follows from Corollary
4.2. It remains to notice only that uniform integrability of
$n\to\mathbb{P}^{n}\nu$ together with weak compactness implies compactness of
$n\to\mathbb{P}^{n}\nu$ in Kantorovich Wasserstein metric. Since any
subsequence converges to the same limit in Kantorovich Wasserstein metric we
have (64). ∎

##  5 Asymptotic stability of the nonlinear Boltzmann-type equation

Consider now the following equation in the space of signed measures

(65) $\frac{d\psi}{dt}+\psi=\mathbb{P}\,\psi\qquad\text{for}\qquad t\geq 0$

with initial condition

(66) $\psi\,(0)=\psi_{0},$

where $\psi_{0}\in\mathcal{M}_{1}(\mathbb{R}_{+})$ and
$\psi:\mathbb{R}_{+}\rightarrow\mathcal{M}_{sig}(\mathbb{R}_{+})$. In this
section we show that the equation (1) may by considered in a convex closed
subset $D$ of a vector space of signed measures
$\mathcal{M}_{sig}(\mathbb{R}_{+})$. This approach seems to be quite natural
and it is related to the classical results concerning the semigroups and
differential equations on convex subsets of Banach spaces (see [11], [19]).
For details see Appendix.

We finish the paper with sufficient conditions for the asymptotic stability of
solutions of the equation (65) with respect to Kantorovich–Wasserstein metric.

Equation (65) together with the initial condition (66) may be considered in a
convex subset ${D}$ of the vector space of finite signed measures
$\mathcal{M}_{sig}$.

######  Corollary 5.1.

Assume $\varphi_{i}\in\mathcal{M}_{1},i\in\\{1,2,...,\\}$ is such that
$m_{1}(\varphi_{i})={1\over i}$, $\mathbb{P}$ is given by (9) with
$\sum\limits_{i=1}\limits^{\infty}\alpha_{i}=1$, $\alpha_{i}\geq 0$. Then for
every $\psi_{0}\in D$ there exists a unique solution $\psi$ of problem (65),
(66) taking values in ${D}$.

The solutions of (65), (66) generate a semigroup of Markov operators
$(P^{t})_{t\geq 0}$ on D given by

(67) ${P}^{t}\,u_{0}=u(t)\qquad\text{for}\qquad t\in{\mathbb{R}}_{+},\qquad
u_{0}\in{D}.$

Now using Theorem 4.1 we can easily derive the following result

######  Theorem 5.1.

Let $\mathbb{P}$ be operator given by (9). Moreover, let
$(\varphi_{1},\varphi_{2},...)$ be a sequence of probability measures such
that $m_{1}(\varphi)={1\over i}$ and $\alpha_{i}\geq 0$ be a sequence of
nonnegative numbers such that $\sum\limits_{i=1}\limits^{\infty}\alpha_{i}=1$.
Assume that $0$ is an accumulation point of $supp\varphi_{i}$ for some
$i\in\Lambda$. If $\mathbb{P}$ has a fixed point $\mu^{*}\in D$ then

(68) $\lim_{t\to\infty}||\psi(t)-\mu^{*}||_{\mathcal{K}}=0$

for every sequentially compact (in Kantorovich Wasserstein metric) solution
$\psi$ of (65), (66).

###### Proof.

First we have to prove that $(P^{t})_{t\geq 0}$ is nonexpansive on $D$ with
respect to Kantorovich-Wasserstein metric. For this purpose let
$\eta_{0},\vartheta_{0}\in D$ be given. For $t\in{\mathbb{R}}_{+}$ define

(69) $\upsilon(t)=P^{t}\,\eta_{0}-P^{t}\,\vartheta_{0}.$

Using (95), Corollary 2.1 and (96) it is easy to see that

(70)
$v(t)=e^{-t}v_{0}+\int\limits_{0}\limits^{t}e^{-(t-s)}(\mathbb{P}(P^{s}\eta_{0})-\mathbb{P}(P^{s}\vartheta_{0}))ds\quad\textnormal{for}\quad
t\in\mathbb{R}_{+}.$

From Corollary 5.1 it follows immediately that

(71) $||v(t)||_{\mathcal{K}}\leq
e^{-t}||v(0)||_{\mathcal{K}}+\int\limits_{0}\limits^{t}e^{-(t-s)}||v(s)||_{\mathcal{K}}ds\quad\textnormal{for}\quad
t\in\mathbb{R}_{+}.$

Consequently

(72)
$f(t)\leq||v(0)||_{\mathcal{K}}+\int\limits_{0}\limits^{t}f(s)ds\quad\textnormal{for}\quad
t\in\mathbb{R}_{+},$

where $f(t)=e^{t}||v(t)||_{\mathcal{K}}$. From the Gronwall inequality it
follows that

(73) $f(t)\leq e^{t}||v(0)||_{\mathcal{K}}$

This is equivalent to the fact that $(P^{t})_{\geq t}$ is nonexpansive on $D$
with respect to Kantorovich - Wasserstein metric. Notice that $\mu^{*}$ as a
fixed point of $\mathbb{P}$ is a stationary solution to the equation (65) i.e.
$P^{t}\mu^{*}=\mu^{*}$. To complete the proof it is sufficient to verify
condition (84) of Theorem 6.1.

From (95) and Proposition 2.3 and Theorem 4.1 it follows immediately that for
$\psi_{0}\in D$ and $t>0$

$\displaystyle\|P^{t}\psi_{0}-\mu^{*}\|_{\mathcal{K}}\leq
e^{-t}\,\|\,\psi_{0}-\mu^{*}\|_{\mathcal{K}}+\int\limits_{0}\limits^{t}\,e^{-(t-s)}\,\|\mathbb{P}P^{s}\psi_{0}-\mathbb{P}\mu^{*}\|_{\mathcal{K}}\,ds<$
(74) $\displaystyle
e^{-t}\,\|\,\psi_{0}-\mu^{*}\|_{\mathcal{K}}+(1-e^{-t})\,\|P^{s}\psi_{0}-\mu^{*}\|_{\mathcal{K}}\leq\|\,\psi_{0}-\mu^{*}\|_{\mathcal{K}}.$

By Theorem 6.1 we immediately obtain (68). ∎

We shall now study nonlinear Boltzmann equation (65) using Zolotariev seminorm
following the results of [19]. Consider time discretized version of (65) with
discretization step $h\in(0,1)$

(75)
$\frac{d\psi_{h}}{dt}(d_{h}(t))+\psi_{h}(d_{h}(t))=\mathbb{P}\,\psi_{h}(d_{h}(t))\qquad\text{for}\qquad
t\geq 0$

with initial condition

(76) $\psi_{h}\,(0)=\psi_{0},$

where $\psi_{0}\in\mathcal{M}_{1}(\mathbb{R}_{+})$ and $d_{h}(t)=nh$ for
$t\in[nh,(n+1)h)$. Then

(77)
$\psi_{h}((n+1)h)=(1-h)\psi_{h}(nh)+h\mathbb{P}\psi_{h}(nh):=\mathbb{P}_{h}\psi_{h}(nh).$

Notice that fixed point of the operator $\mathbb{P}$ is also a fixed point of
$\mathbb{P}_{h}$ and vice versa. We have

######  Lemma 5.1.

Under assumptions of Theorem 3.1 we have that

  * (i)

when $m_{r}(\mu)<\infty$ then $m_{r}(\mathbb{P}_{h}^{n}\mu)<\infty$ for any
positive integer $n$ and $\mu\in\mathcal{M}_{1}(\mathbb{R}_{+})$,

  * (ii)

$\|\mathbb{P}_{h}\mu-\mathbb{P}_{h}\nu\|_{r}\leq\lambda_{h}\|mu-\nu\|_{r}$ for
$\mu,\nu\in\mathcal{M}_{1}(\mathbb{R}_{+})$ such that $m_{r}(\mu)<\infty$,
$m_{r}(\nu)<\infty$ with $\lambda_{h}=1-h(1-\lambda)$, where
$\lambda=\sum\limits_{i=1}\limits^{\infty}\alpha_{i}m_{r}(\varphi_{i})i$,

  * (iii)

$\|\mathbb{P}_{h}^{n}\mu-\mu^{*}\|_{\mathcal{K}}\leq 2^{1+{1\over
r}}\lambda_{h}^{{n\over r}}\left(\|\mu-\mu^{*}\|_{r}\right)^{1\over r}\leq K$
with $\mu^{*}\in D$ being the unique fixed point of $\mathbb{P}$ and
$K={1\over r}2^{1+{1\over r}}(m_{r}(\mu)+m_{r}(\mu^{*}))$.

###### Proof.

Under (30) using Proposition 2.1 we have that $m_{r}(\mathbb{P}\mu)<\infty$
and therefore also $m_{r}(\mathbb{P}_{h}\mu)<\infty$. Then (i) follows by
induction. (ii) follows directly from the definition of $\mathbb{P}$ and (34).
For fixed point $\mu^{*}$ of $\mathbb{P}$, which is also a fixed point of
$\mathbb{P}_{h}$ and $\mu\in\mathcal{M}_{1}(\mathbb{R}_{+})$ such that
$m_{r}(\mu)<\infty$ we have

(78) $\|\mathbb{P}_{h}^{n}\mu\|_{r}\leq\lambda_{h}^{n}\|\mu-\nu\|_{r}.$

Therefore using Theorem 6.3 and then (31) we obtain

(79) $\|\mathbb{P}_{h}^{n}\mu-\mu^{*}\|_{\mathcal{K}}\leq 2^{1+{1\over
r}}\lambda_{h}^{n\over r}(\|\mu-\mu^{*}\|_{\mathcal{K}})^{1\over r}\leq K$

with $K={1\over r}2^{1+{1\over r}}(m_{r}(\mu)+m_{r}(\mu^{*}))$. ∎

We recall now Lemma 3 of [19].

######  Lemma 5.2.

Under the assumptions of Theorem 3.1, when $\mu=\psi_{0}$ is such that
$m_{r}(\mu)<\infty$, we have

(80) $\|\psi_{h}(t)-\psi(t)\|_{\mathcal{K}}\leq 4Kh(e^{2t}-1).$

We can now formulate

######  Theorem 5.2.

Under assumptions of Theorem 3.1 when $\mu=\psi_{0}$ is such that
$m_{r}(\mu)<\infty$ we have that

(81) $\|\psi(t)-\mu^{*}\|_{\mathcal{K}}\leq Ke^{-{t\over r}(1-\lambda)}.$

###### Proof.

We follow the arguments of the proof of Theorem 1 of [19]. Fix $t>0$ and for a
given $\varepsilon>0$ find positive integer $n$ such that ${t\over
n}4K(e^{2t}-1)\leq\varepsilon$ Then by Lemma 5.2

(82)
$\|\psi(t)-\mu^{*}\|_{\mathcal{K}}\leq\|\psi(t)-\psi_{h}(t)\|_{\mathcal{K}}+\|\psi_{h}(t)-\mu^{*}\|_{\mathcal{K}}\leq\varepsilon+K(1-{t\over
n}(1-\lambda))^{n\over r}.$

Since $1-x\leq e^{-x}$ for $x\geq 0$ we have that $(1-{t\over
n}(1-\lambda))^{n\over r}\leq e^{-{t\over r}(1-\lambda)}$ and the claim
follows from (82) taking into account that $\varepsilon$ could be chosen
arbitrarily small. ∎

##  6 Appendix

On a given complete metric space $(E,\rho)$ consider a continuous operator $T$
or continuous semigroup $(T_{t})$, for $t\geq 0$ transforming $(E,\rho)$ into
itself. Denote by $\omega(x)$ the set of all limiting points of the trajectory
$n\to T^{n}x$ or $t\to T_{t}x$ respectively. We say that $n\to T^{n}x$ or
$t\to T_{t}x$ is sequentially compact if from every sequence $T^{n_{k}}x$,
$T_{t_{n_{k}}}x$ respectively, one could choose a convergent subsequence. Let
${\mathcal{Z}}$ be the set of all $x$ such that the trajectory $t\to T_{t}x$
($n\to T^{n}x$) is sequentially compact. We shall assume that ${\mathcal{Z}}$
is a nonempty set and let
$\Omega=\bigcup\limits_{\mu\in{\mathcal{Z}}}\omega(\mu)$. We have the
following result formulated for semigroup $T_{t}$, which naturally holds for
continuous operator $T$

######  Theorem 6.1.

(see Theorem 5.1.2 of [14]) Assume that $T_{t}$ is nonexpansive i.e.

(83) $\rho(T_{t}x,T_{t}y)\leq\rho(x,y)$

for $t\geq 0$ and there is $x^{*}\in\Omega$ such that for every $x\in\Omega$,
$x\neq x^{*}$ there is $t(x)$ such that

(84) $\rho(T_{t(x)}x,T_{t(x)}x^{*})<\rho(x,x^{*}).$

Then for $z\in{\mathcal{Z}}$ we have

(85) $\lim_{t\to\infty}\rho(T_{t}z,x^{*})=0.$

######  Definition 6.1.

Sequence of probability measures $\mu_{n}$ defined on $\mathbb{R}_{+}$ is
uniformly integrable when

(86) $\sup_{n}\int\limits_{M}\limits^{\infty}x\mu_{n}(dx)\to 0,$

whenever $M\to\infty$.

######  Theorem 6.2.

Assume that for sequence of probability measures $\mu_{n}$ defined on
$\mathbb{R}_{+}$ we have that $m_{1}(\mu_{n})<\infty$ and
$\mu_{n}\Rightarrow\mu$, as $n\to\infty$. Then $m_{1}(\mu_{n})\to m_{1}(\mu)$
if and only if measures $\mu_{n}$ are uniformly integrable. Furthermore for
sequence of probability measures $\mu_{n}$ defined on $\mathbb{R}_{+}$ such
that $m_{1}(\mu_{n})<\infty$ and we have that convergence
$\mu_{n}\Rightarrow\mu$, as $n\to\infty$, together with convergence of
$m_{1}(\mu_{n})\to m_{1}(\mu)$ is equivalent to convergence
$\|\mu_{n}-\mu\|_{\mathcal{K}}\to 0$.

###### Proof.

By Skorokhod theorem (25.6 of [3]) there is a probability space $(\Omega,F,P)$
and nonnegative random variables $X_{n}$, $X$ with laws $\mu_{n}$ and $\mu$
respectively such that $X_{n}(\omega)\to X(\omega)$ for each
$\omega\in\Omega$. Uniform integrability of $\mu_{n}$ is equivalent to uniform
integrability of $X_{n}$. By Theorem II T21 of [23] uniform integrability of
$X_{n}$ is equivalent to the convergence $m_{1}(\mu_{n})\to m_{1}(\mu)$. To
prove the last statement of Theorem notice that when
$\|\mu_{n}-\mu\|_{\mathcal{K}}\to 0$ we have also
$\|\mu_{n}-\mu\|_{\mathcal{F}}\to 0$, so that $\mu_{n}\Rightarrow\mu$ and
$m_{1}(\mu_{n})\to m_{1}(\mu)$. Assume now that $\mu_{n}\Rightarrow\mu$, as
$n\to\infty$ and $m_{1}(\mu_{n})\to m_{1}(\mu)$. Then $X_{n}$ defined above
converges to $X$ in $L^{1}(P)$ norm. In particular for any function $f$ with
Lipschitz constant not greater than $1$ we have

(87) $|\mu_{n}(f)-\mu(f)|=|E\left[f(X_{n})\right]-E\left[f(X)\right]|\leq
E\left[|f(X_{n})-f(X)|\right]\leq E|X_{n}-X|\to 0$

as $n\to\infty$, which means that we have also convergence in
$\|\cdot\|_{\mathcal{K}}$ norm, which completes the proof. ∎

######  Remark 6.1.

The result above in not unexpected. For given measure $\mu\in D$ define
$\bar{\mu}(A):=\int\limits_{A}x\mu(dx)$ for Borel measurable set $A$. Then
compactness of the closure of the sequence $\left\\{\bar{\mu}_{n}\in
D\right\\}$ is by Theorem 6.2 of [4] equivalent to the tightness of measures
$\left\\{\bar{\mu}_{n}\right\\}$, which is equivalent to (86).

######  Corollary 6.1.

Whenever $D\ni\mu_{n}\Rightarrow\mu$ and $\sup_{n}m_{\beta}(\mu_{n})<\infty$
for some $\beta>1$ we have $\|\mu_{n}-\mu\|_{\mathcal{K}}\to 0$ as
$n\to\infty$.

###### Proof.

It is clear that

(88)
$\sup_{n}\int\limits_{M}\limits^{\infty}x\mu_{n}(dx)\leq\sup_{n}\Big{(}\int\limits_{0}\limits^{\infty}x^{\beta}\mu_{n}(dx)\Big{)}^{1\over\beta}\Big{(}\int\limits_{0}\limits^{\infty}1_{x\geq
M}\mu_{n}(dx)\Big{)}^{\beta-1\over\beta}.$

Now $\int\limits_{0}\limits^{\infty}1_{x\geq M}\mu_{n}(dx)\leq{1\over M}$, so
that $\mu_{n}$ is uniformly integrable and it remains to use Theorem 6.2.∎

Before we formulate next theorem we define metric $d_{r}$ in the space of
probability measures defined on $\mathbb{R}_{+}$ with finite $r$-th moments,
where $r\in[1,2)$. Namely for probability measures $\mu$ and $\nu$ such that
$m_{r}(\mu)<\infty$ and $m_{r}(\nu)<\infty$ let

(89) $d_{r}(\mu,\nu):=\inf\left\\{\left(E(|X-Y|^{r})\right)^{1\over
r}\right\\},$

where infimum taken over probability measures $P$ on $\mathbb{R}_{+}^{2}$ such
that their marginals are $\mu$ and $\nu$ respectively. We have

######  Theorem 6.3.

For $\mu,\nu\in D$ such that $m_{r}(\mu)<\infty$ and $m_{r}(\nu)<\infty$ with
$r\in(1,2)$ we have

(90) $\|\mu-\nu\|_{\mathcal{K}}\leq 2(2\|\mu-\nu\|_{r})^{1\over r}.$

###### Proof.

By [25] we have that $d_{r}(\mu,\nu)\leq 2(2\|\mu-\nu\|_{r})^{1\over r}$.
Clearly $d_{1}(\mu,\nu)\leq d_{r}(\mu,\nu)$. Since by Theorem 20.1 of [12]
$d_{1}(\mu,\nu)=\|\mu-\nu\|_{\mathcal{K}}$ we obtain (90). ∎

We recall now Kantorovich-Rubinstein maximum principle for our metric
$\|\cdot\|_{\mathcal{K}}$, see Corollary 6.2 of [24]

######  Theorem 6.4.

For probability measures $\mu,\nu$ defined on $\mathbb{R}_{+}$ there exists
$f_{0}\in{\mathcal{K}}$ such that

(91) $\|\mu-\nu\|_{\mathcal{K}}=\langle f_{0},\mu-\nu\rangle.$

Moreover when $f_{0}\in{\mathcal{K}}$ satisfies (91) for measures $\mu\neq\nu$
defined on $\mathbb{R}_{+}$ then there are two different points
$x_{1},x_{2}\in\mathbb{R}_{+}$ such that
$|f_{0}(x_{1})-f_{0}(x_{2})|=|x_{1}-x_{2}|$.

Finally we recall now some known results related with ordinary differential
equations in Banach spaces. For details see [11].

Let $(E,\|\cdot\|)$ be a Banach space and let $\tilde{D}$ be a closed, convex,
nonempty subset of $E$. In the space $E$ we consider an evolutionary
differential equation

(92) $\frac{du}{dt}=-u+\tilde{P}\,u\qquad\text{for}\qquad
t\in{\mathbb{R}}_{+}$

with the initial condition

(93) $u(0)=u_{0},\qquad u_{0}\in\tilde{D},$

where $\tilde{P}:\tilde{D}\to\tilde{D}$ is a given operator.

Function $u:{\mathbb{R}}_{+}\to E$ is called a solution to the problem (92),
(93) if it is strongly differentiable on ${\mathbb{R}}_{+}$,
$u(t)\in\tilde{D}$ for all $t\in{\mathbb{R}}_{+}$ and $u$ satisfies relations
(92), (93).

We have

######  Theorem 6.5.

Assume that the operator $\tilde{P}:\tilde{D}\to\tilde{D}$ satisfies Lipschitz
condition

(94) $\|\tilde{P}\,v-\tilde{P}\,w\|\leq l\,\|v-w\|\qquad\textnormal{for}\qquad
u,w\in\tilde{D},$

where $l$ is a nonnegative constant. Then for every $u_{0}\in\tilde{D}$ there
exists a unique solution $u$ to the problem (92), (93).

The standard proof of Theorem 6.5 is based on the fact, that function
$u:\mathbb{R}_{+}\to\tilde{D}$ is a solution to (92), (93) if and only if it
is continuous and satisfies the integral equation

(95)
$u(t)=e^{-t}\,u_{0}+\int\limits_{0}\limits^{t}\,e^{-(t-s)}\,\tilde{P}\,u(s)\,ds\qquad\text{for}\qquad
t\in{\mathbb{R}}_{+}.$

Due to completeness of $\tilde{D}$, the integral on the right hand side is
well defined and equation (95) may be solved by the method of successive
approximations. Observe that, thanks to the properties of $\tilde{D}$, for
every $u_{0}\in\tilde{D}$ and for every continuous function
$u:{\mathbb{R}}_{+}\to\tilde{D}$ the right hand side of (95) is also a
function with values in $\tilde{D}$. The solutions of (95) generate a
semigroup of operators $(\tilde{P}^{\ t})_{t\geq 0}$ on $\tilde{D}$ given by
the formula

(96) $\tilde{P}^{\ t}\,u_{0}=u(t)\qquad\text{for}\qquad
t\in{\mathbb{R}}_{+},\qquad u_{0}\in\tilde{D}.$

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