197209

# Fractional statistical dynamics and fractional kinetics

José Luís da Silva  CCM, University of Madeira, Campus da Penteada, 9020-105
Funchal, Portugal<EMAIL_ADDRESS>Anatoly N. Kochubei  Institute of Mathematics,
National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka, Kyiv, 01601,
Ukraine<EMAIL_ADDRESS>Yuri Kondratiev  Department of Mathematics,
University of Bielefeld, D-33615 Bielefeld, Germany<EMAIL_ADDRESS>bielefeld.de

(Date: 17/03/2016; Revised 03/04/2016)

###### Abstract.

We apply the subordination principle to construct kinetic fractional
statistical dynamics in the continuum in terms of solutions to Vlasov-type
hierarchies. As a by-product we obtain the evolution of the density of
particles in the fractional kinetics in terms of a non-linear Vlasov-type
kinetic equation. As an application we study the intermittency of the
fractional mesoscopic dynamics.

###### Key words and phrases:

Configuration space, Caputo derivative, Vlasov-type kinetic equation,
correlation functions, Poisson flow.

###### 2000 Mathematics Subject Classification:

Primary 82C21; Secondary 34A08

##  1\. Introduction

A general scheme for the study of Markov dynamics for interacting particle
systems (IPS for short) in the continuum includes the following steps. We
start with a heuristic Markov generator $L$ defined on functions over a
configuration space of the system. Associated with this generator is a forward
Kolmogorov equation for states of the system (a.k.a. a Fokker–Planck equation
(FPE)). A solution to this equation gives the so-called statistical dynamics
of the model under consideration [11]. A constructive approach to the
existence and uniqueness problem for solution of the FPE and for the analysis
of its properties exploits the possibility of writing this equation as a
hierarchical chain of evolution equations for time dependent correlation
functions [11, 12]. This step corresponds to a microscopic description of the
system.

A mesoscopic level of the study is related with a Vlasov-type scaling limit
for the dynamics that leads to a kinetic or Vlasov hierarchy for correlation
functions. This scaling limit destroys the Markov property of the evolution of
the limiting Vlasov–Fokker–Planck equation (VFPE): for an initial probability
measure the solution, in general, is no longer a measure. Still, the resulting
dynamics has a conditional Markov property in the following sense. If we start
with a Poisson initial state then the solution of the VFPE will be given by a
flow of Poisson measures on the configuration space. In theoretical physics
this fact is known as the chaos propagation property.

The Poisson flow which appears in the Vlasov limit is completely characterized
by the density function $\rho_{t}(x)$, which corresponds to the Poisson
measure from the flow at time $t\geq 0$. A specific feature of the mesoscopic
limit is a non-linear Vlasov-type kinetic equation for this density. In most
cases this equation may be informally derived directly from the form of the
generator $L$. However, a rigorous realization of the above scheme is a non-
trivial task for each particular model [10, 11, 9]. The study of the resulting
kinetic equations for concrete Markov dynamics of interacting particle systems
in the continuum belongs to the general theory of non-local non-linear
evolution equations which has been under active development in recent years.

The aim of the present paper is to extend the concept of statistical dynamics
and related structures to the case of fractional time derivatives. From the
probabilistic point of view this means that we leave the Markov dynamical
framework by introducing a random time change in the corresponding Markov
process – see for example [28, 23]. In the language of functional analysis we
are no more in the arena of semigroup evolutions.

Below we discuss the concept of a fractional Fokker–Plank equation (FPE) and
the related fractional statistical dynamics, which is still an evolution in
the space of probability measures on the configuration space. The mesoscopic
scaling of the generator of this evolutions leads to the same result as for
the initial FPE. The latter leads us to the concept of a fractional VFPE. A
subordination principle provides for the representation of the solution to
this equation as a flow of measures that is a transformation of a Poisson flow
for the initial VFPE. Note that the density function for the fractal kinetics
is a subordination of the solution to the initial Vlasov equation. This
density characterizes the kinetic behavior of the fractional statistical
dynamics, but it is not the same as the solution to the Vlasov equation with a
fractional time derivative, as is typically assumed in theoretical physics.

In this paper we leave open the problem of rigorous realization of scaling
approach for particular models. Instead, our considerations are focused on
questions about the properties of subordinated flows. In particular, we
clarify the possibility of having time-dependent random point processes with
an asymptotic intermittency property as a result of subordination of Poisson
flows.

##  2\. Preliminaries

Let ${\mathcal{B}}({{\mathbb{R}}^{d}})$ be the family of all Borel sets in
${{\mathbb{R}}^{d}}$, $d\geq 1$ and let
${\mathcal{B}}_{b}({{\mathbb{R}}^{d}})$ denote the system of all bounded sets
in ${\mathcal{B}}({{\mathbb{R}}^{d}})$.

The space of $n$-point configurations in an arbitrary
$Y\in{\mathcal{B}}({{\mathbb{R}}^{d}})$ is defined by

$\Gamma_{Y}^{(n)}:=\big{\\{}\eta\subset Y\big{|}\;|\eta|=n\big{\\}},\quad
n\in{\mathbb{N}}.$

We also set $\Gamma_{Y}^{(0)}:=\\{\emptyset\\}$. As a set, $\Gamma_{Y}^{(n)}$
may be identified with the symmetrization of

$\widetilde{Y^{n}}=\big{\\{}(x_{1},\ldots,x_{n})\in Y^{n}\big{|}x_{k}\neq
x_{l}\text{ if }k\neq l\big{\\}}.$

The configuration space over the space ${{\mathbb{R}}^{d}}$ consists of all
locally finite subsets (configurations) of ${{\mathbb{R}}^{d}}$, namely,

$\Gamma=\Gamma_{{\mathbb{R}}^{d}}:=\big{\\{}\gamma\subset{{\mathbb{R}}^{d}}\big{|}|\gamma\cap\Lambda|<\infty,\text{
for all }\Lambda\in{\mathcal{B}}_{\mathrm{b}}({{\mathbb{R}}^{d}})\big{\\}}.$
(1)

The space $\Gamma$ is equipped with the vague topology, i.e., the minimal
topology for which all mappings
$\Gamma\ni\gamma\mapsto\sum_{x\in\gamma}f(x)\in{\mathbb{R}}$ are continuous
for any continuous function $f$ on ${{\mathbb{R}}^{d}}$ with compact support.
Note that the summation in $\sum_{x\in\gamma}f(x)$ is taken over only finitely
many points of $\gamma$ belonging to the support of $f$. It was shown in [18]
that with the vague topology $\Gamma$ may be metrizable and it becomes a
Polish space (i.e., a complete separable metric space). Corresponding to this
topology, the Borel $\sigma$-algebra ${\mathcal{B}}(\Gamma)$ is the smallest
$\sigma$-algebra for which all mappings

$\Gamma\ni\gamma\mapsto|\gamma_{\Lambda}|\in{\mathbb{N}}_{0}:={\mathbb{N}}\cup\\{0\\}$

are measurable for any $\Lambda\in{\mathcal{B}}_{b}({{\mathbb{R}}^{d}})$. Here
$\gamma_{\Lambda}:=\gamma\cap\Lambda$, and $|\cdot|$ the cardinality of a
finite set.

It follows that one can introduce the corresponding Borel $\sigma$-algebra,
which we denote by ${\mathcal{B}}(\Gamma_{Y}^{(n)})$. The space of finite
configurations in an arbitrary $Y\in{\mathcal{B}}({{\mathbb{R}}^{d}})$ is
defined by

$\Gamma_{0,Y}:=\bigsqcup_{n\in{\mathbb{N}}_{0}}\Gamma_{Y}^{(n)}.$

This space is equipped with the topology of disjoint unions. Therefore one can
introduce the corresponding Borel $\sigma$-algebra
${\mathcal{B}}(\Gamma_{0,Y})$. In the case of $Y={{\mathbb{R}}^{d}}$ we will
omit the index $Y$ in the notation, thus
$\Gamma_{0}:=\Gamma_{0,{{\mathbb{R}}^{d}}}$
$\Gamma^{(n)}:=\Gamma_{{{\mathbb{R}}^{d}}}^{(n)}$.

The restriction of the Lebesgue product measure $(dx)^{n}$ to
$\bigl{(}\Gamma^{(n)},{\mathcal{B}}(\Gamma^{(n)})\bigr{)}$ will be denoted by
$m^{(n)}$, and we set $m^{(0)}:=\delta_{\\{\emptyset\\}}$. The
Lebesgue–Poisson measure $\lambda$ on $\Gamma_{0}$ is defined by

$\lambda:=\sum_{n=0}^{\infty}\frac{1}{n!}m^{(n)}.$ (2)

For any $\Lambda\in{\mathcal{B}}_{b}({{\mathbb{R}}^{d}})$, the restriction of
$\lambda$ to $\Gamma_{\Lambda}:=\Gamma_{0,\Lambda}$ will be also denoted by
$\lambda$. The space $\bigl{(}\Gamma,{\mathcal{B}}(\Gamma)\bigr{)}$ is the
projective limit of the family of spaces
$\big{\\{}(\Gamma_{\Lambda},{\mathcal{B}}(\Gamma_{\Lambda}))\big{\\}}_{\Lambda\in{\mathcal{B}}_{b}({{\mathbb{R}}^{d}})}$.
The Poisson measure $\pi$ on $\bigl{(}\Gamma,{\mathcal{B}}(\Gamma)\bigr{)}$ is
given as the projective limit of the family of measures
$\\{\pi^{\Lambda}\\}_{\Lambda\in{\mathcal{B}}_{\mathrm{b}}({{\mathbb{R}}^{d}})}$,
where $\pi^{\Lambda}:=e^{-m(\Lambda)}\lambda$ is the probability measure on
$\bigl{(}\Gamma_{\Lambda},{\mathcal{B}}(\Gamma_{\Lambda})\bigr{)}$. Here
$m(\Lambda)$ is the Lebesgue measure of
$\Lambda\in{\mathcal{B}}_{b}({{\mathbb{R}}^{d}})$.

For any measurable function $f:{{\mathbb{R}}^{d}}\rightarrow{\mathbb{R}}$ we
define a _Lebesgue–Poisson exponent_

$e_{\lambda}(f,\eta):=\prod_{x\in\eta}f(x),\quad\eta\in\Gamma_{0};\quad
e_{\lambda}(f,\emptyset):=1.$ (3)

Then, by (2), for $f\in L^{1}({{\mathbb{R}}^{d}},dx)$ we obtain
$e_{\lambda}(f)\in L^{1}(\Gamma_{0},d\lambda)$ and

$\int_{\Gamma_{0}}e_{\lambda}(f,\eta)\,d\lambda(\eta)=\exp\left(\int_{\mathbb{R}^{d}}f(x)\,dx\right).$
(4)

A set $M\in{\mathcal{B}}(\Gamma_{0})$ is called bounded if there exists
$\Lambda\in{\mathcal{B}}_{b}({{\mathbb{R}}^{d}})$ and $N\in{\mathbb{N}}$ such
that $M\subset\bigsqcup_{n=0}^{N}\Gamma_{\Lambda}^{(n)}$. We will make use of
the following classes of functions on $\Gamma_{0}$: (i)
$L_{\mathrm{ls}}^{0}(\Gamma_{0})$ is the set of all measurable functions on
$\Gamma_{0}$ which have local support, i.e., $G\in L_{ls}^{0}(\Gamma_{0})$, if
there exists $\Lambda\in\mathcal{B}_{b}(\mathbb{R}^{d})$ such that
$G\upharpoonright_{\Gamma_{0}\setminus\Gamma_{\Lambda}}=0$, while (ii)
$B_{bs}(\Gamma_{0})$ is the set of bounded measurable functions with bounded
support, i.e., $G\upharpoonright_{\Gamma_{0}\setminus B}=0$ for some bounded
$B\in\mathcal{B}(\Gamma_{0})$.

In fact, any ${\mathcal{B}}(\Gamma_{0})$-measurable function $G$ on
$\Gamma_{0}$ is a sequence of functions
$\bigl{\\{}G^{(n)}\bigr{\\}}_{n\in{\mathbb{N}}_{0}}$, where $G^{(n)}$ is a
${\mathcal{B}}(\Gamma^{(n)})$-measurable function on $\Gamma^{(n)}$.

On $\Gamma$ we consider the set of cylinder functions
$\mathcal{F}_{cyl}(\Gamma)$. These functions are characterized by the relation
$F(\gamma)=F\upharpoonright_{\Gamma_{\Lambda}}(\gamma_{\Lambda})$.

The following mapping from $L_{ls}^{0}(\Gamma_{0})$ into
${{\mathcal{F}}_{cyl}}(\Gamma)$ which plays the key role in our further
considerations:

$KG(\gamma):=\sum_{\eta\Subset\gamma}G(\eta),\quad\gamma\in\Gamma,$ (5)

where $G\in L_{ls}^{0}(\Gamma_{0})$. (See, for example, [17], [21, 22]). The
summation in (5) is taken over all finite sub-configurations
$\eta\in\Gamma_{0}$ of the (infinite) configuration $\gamma\in\Gamma$; this
relationship is represented symbolically by $\eta\Subset\gamma$. The mapping
$K$ is linear, positivity preserving, and invertible, with

$K^{-1}F(\eta):=\sum_{\xi\subset\eta}(-1)^{|\eta\setminus\xi|}F(\xi),\quad\eta\in\Gamma_{0}.$
(6)

Here and in the sequel, inclusions like $\xi\subset\eta$ hold for
$\xi=\emptyset$ as well as for $\xi=\eta$. We denote the restriction of $K$
onto functions on $\Gamma_{0}$ by $K_{0}$.

A measure $\mu\in{\mathcal{M}}_{\mathrm{fm}}^{1}(\Gamma)$ is called locally
absolutely continuous with respect to (w.r.t.) a Poisson measure $\pi$ if for
any $\Lambda\in{\mathcal{B}}_{b}({{\mathbb{R}}^{d}})$ the projection of $\mu$
onto $\Gamma_{\Lambda}$ is absolutely continuous w.r.t. projection of $\pi$
onto $\Gamma_{\Lambda}$. By [17], there exists in this case a _correlation
functional_ $k_{\mu}:\Gamma_{0}\rightarrow{\mathbb{R}}_{+}$ such that the
following equality holds for any $G\in B_{bs}(\Gamma_{0})$:

$\int_{\Gamma}(KG)(\gamma)\,d\mu(\gamma)=\int_{\Gamma_{0}}G(\eta)k_{\mu}(\eta)\,d\lambda(\eta).$
(7)

Restrictions $k_{\mu}^{(n)}$ of this functional on $\Gamma_{0}^{(n)}$,
$n\in{\mathbb{N}}_{0}$, are called _correlation functions_ of the measure
$\mu$. Note that $k_{\mu}^{(0)}=1$.

##  3\. Mesoscopic statistical dynamics

In this section we introduce the general scheme of Vlasov scaling for the
Markov dynamics of IPS – interacting particle systems – on configuration
space. Thus we assume that the initial distribution (the state of particles)
in our system is a probability measure $\mu_{0}\in\mathcal{M}^{1}(\Gamma)$
with corresponding correlation function $k_{0}=(k_{0}^{(n)})_{n=0}^{\infty}$.
The distribution of particles at time $t>0$ is the measure
$\mu_{t}\in\mathcal{M}^{1}(\Gamma)$, and $k_{t}=(k_{t}^{(n)})_{n=0}^{\infty}$
its correlation function. If the evolution of states $(\mu_{t})_{t\geq 0}$ is
determined a priori by a heuristic Markov generator $L$, then $\mu_{t}$ is the
solution of the forward Kolmogorov equation (or Fokker–Plank equation (FPE)),

$\begin{cases}\frac{\partial\mu_{t}}{\partial t}&=L^{*}\mu_{t},\\\
\mu_{t}|t_{=0}&=\mu_{0},\end{cases}$ (8)

where $L^{*}$ is the adjoint operator. In terms of the time-dependent
correlation functions $(k_{t})_{t\geq 0}$ corresponding to $(\mu_{t})_{t\geq
0}$, the FPE may be rewritten as an infinite system of evolution equations

$\begin{cases}\frac{\partial k_{t}^{(n)}}{\partial
t}&=(L^{\triangle}k_{t})^{(n)},\\\ k_{t}^{(n)}|_{t=0}&=k_{0}^{(n)},\quad n\geq
0,\end{cases}$ (9)

where $L^{\triangle}$ is the image of $L^{*}$ in a Fock-type space of vector-
functions $k_{t}=(k_{t}^{(n)})_{n=0}^{\infty}$. In applications to concrete
models, the expression for the operator $L^{\triangle}$ is obtained from the
operator $L$ via combinatorial calculations (cf. [17]). The following diagram

---
$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{duality}}$$\scriptstyle{K}$$\textstyle{L^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{K^{*}}$$\textstyle{\hat{L}=K^{-1}LK}$$\textstyle{L^{\triangle}=\hat{L}^{*}=K^{*}L^{*}(K^{-1})^{*}}$

describes the relationships.

The evolution equation (9) is nothing but a hierarchical system of equations
to the Markov generator $L$. This system is the analogue of the BBGKY-
hierarchy of the Hamiltonian dynamics [4].

Our interest now turns to Vlasov-type scaling of stochastic dynamics for the
IPS in a continuum. This scaling leads to so-called kinetic description of the
considered model. In the language of theoretical physics we are dealing with a
mean-field type scaling which is adopted to preserve the spatial structure. In
addition, this scaling will lead to the limiting hierarchy, which possesses a
chaos propagation property. In other words, if the initial distribution is
Poisson (non-homogeneous) then the time evolution of states will maintain this
property. We refer to [10] for a general approach, concrete examples, and
additional references.

There exists a standard procedure for deriving Vlasov scaling
$L_{V}^{\triangle}$ from the generator $L^{\triangle}$ in (9). Heuristically,
$L_{V}^{\triangle}$ corresponds to a (non-Markov) generator $L_{V}$ on
observables which may be reconstructed form $L_{V}^{\triangle}$ just on the
level of combinatorial calculations. All together, it gives us the following
chain of transformed operators:

$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{V}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{V}^{\triangle}.}$

The specific type of scaling is dictated by the model in question. The process
leading from $L^{\triangle}$ to $L_{V}^{\triangle}$ produces a non-Markovian
generator $L_{V}$ since it lacks the positivity-preserving property. Therefore
instead of (8) we consider the following kinetic FPE,

$\begin{cases}\frac{\partial\mu_{t}}{\partial t}&=L_{V}^{*}\mu_{t},\\\
\mu_{t}|t_{=0}&=\mu_{0},\end{cases}$ (10)

and observe that if the initial distribution satisfies
$\mu_{0}=\pi_{\rho_{0}}$, then the solution is of the same type, i.e.,
$\mu_{t}=\pi_{\rho_{t}}$.

In terms of correlation functions, the kinetic FPE (10) gives rise to the
following Vlasov-type hierarchical chain (Vlasov hierarchy)

$\begin{cases}\frac{\partial k_{t}^{(n)}}{\partial
t}&=(L_{V}^{\triangle}k_{t})^{(n)},\\\ k_{t}^{(n)}|_{t=0}&=k_{0}^{(n)},\quad
n\geq 0.\end{cases}$ (11)

######  Remark 1.

  1. 1\. In applications it is important to consider the Lebesgue–Poisson exponents $k_{0}(\eta)=e_{\lambda}(\rho_{0},\eta)=\prod_{x\in\eta}\rho_{0}(x)$ as the initial condition. The scaling $L_{V}^{\triangle}$ should be such that the dynamics $k_{0}\mapsto k_{t}$ preserves this structure, or more precisely, $k_{t}$ should be of the same type

$k_{t}(\eta)=e_{\lambda}(\rho_{t},\eta)=\prod_{x\in\eta}\rho_{t}(x),\quad\eta\in\Gamma_{0}.$
(12)

  2. 2\. Relation (12) is known as the _chaos preservation property_ of the Vlasov hierarchy. It turns out that equation (12) implies, in general, a non-linear differential equation

$\frac{\partial\rho_{t}(x)}{\partial t}=\vartheta(\rho_{t})(x),\quad
x\in\mathbb{R}^{d},$ (13)

for $\rho_{t}$, which is called the _Vlasov-type kinetic equation_.

######  Remark 2.

In general, if one does not start with a Poisson measure, the solution will
leave the space $\mathcal{M}^{1}(\Gamma)$. To have a bigger class of initial
measures, we may consider the cone inside $\mathcal{M}^{1}(\Gamma)$ generated
by convex combinations of Poisson measures, denoted by $\mathbb{P}(\Gamma)$.

We would now like to generalize the above general scheme to obtain the analog
of _kinetic fractional statistical dynamics_ (or equivalently _mesoscopic
fractional statistical dynamics_). It would be tempting simply to replace the
usual time derivative in equation (13) by a time fractional derivative.
Because the equation (13) in general is non-linear, it is then much harder to
obtain a solution. But more essential is the question of the meaning of such
an equation. A naive use of the fractional derivative in the Vlasov equation
is not justified by the microscopic dynamics and its scaling. Our alternative
approach to realizing this generalization is described in the following
section.

##  4\. Fractional statistical dynamics

The procedure of Section 3 is suitable for describing non-Markov evolutions.
More precisely, in the FPE (8) we change the usual time derivative by the
Caputo–Djrbashian fractional time derivative $\mathbb{D}_{t}^{\alpha}$ (CDfd
for short) and then study the corresponding fractional dynamics.

In order to proceed, we first have to define the CDfd. Let
$f:\mathbb{R}_{+}\longrightarrow\mathbb{R}$ be given; then the CDfd of $f$ is
given in the Laplace transform domain by

$\big{(}\mathcal{L}\mathbb{D}_{t}^{\alpha}f\big{)}(s)=s^{\alpha}(\mathcal{L}f)(s)-s^{\alpha-1}f(0),\quad
s>0,\quad\alpha\in(0,1],$

where $\mathcal{L}f$ denotes the Laplace transform of $f$

$(\mathcal{L}f)(s)=\int_{0}^{\infty}e^{-st}f(t)\,dt.$

Another possible representation of the CDfd is

$\big{(}\mathbb{D}_{t}^{\alpha}f\big{)}(t)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int_{0}^{t}\frac{f(\tau)-f(0)}{(t-\tau)^{\alpha}}\,d\tau,\quad
0<\alpha<1.$

In case $f$ is absolutely continuous, we have

$\big{(}\mathbb{D}_{t}^{\alpha}f\big{)}(t)=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}\frac{f^{\prime}(\tau)}{(t-\tau)^{\alpha}}\,d\tau,\quad
0<\alpha<1.$

The definition of the CDfd has natural extensions to vector-valued or measure
valued functions on $\mathbb{R}_{+}$. We refer to the monographs [30] and [15]
for more details and references concerning the CDfd.

We will introduce the fractional statistical dynamics for a given Markov
generator $L$ by changing the time derivative in the FPE to the CDfd. The
resulting fractional Fokker–Planck dynamics (if it exists!) will act in the
space of states on $\Gamma$, i.e., it will preserve probability measures on
$\Gamma$. The fractional Fokker–Planck equation

$\begin{cases}\mathbb{D}_{t}^{\alpha}\mu_{t}^{\alpha}&=L^{*}\mu_{t}^{\alpha},\\\
\mu_{t}^{\alpha}|_{t=0}&=\mu_{0}^{\alpha}.\end{cases}$

describes a dynamical system with memory in the space of measures on $\Gamma$.
The corresponding evolution no longer has the semigroup property. However, if
the solution $\mu_{t}$ of equation (10) exists, then the subordination
principle (see [31], [1, 2] and references therein) gives the solution of the
equation FFPE, namely

$\mu_{t}^{\alpha}=\int_{0}^{\infty}\Phi_{\alpha}(\tau)\mu_{\tau
t^{\alpha}}\,d\tau.$ (13)

Here $\Phi_{\alpha}(z)$ is the Wright function

$\Phi_{\alpha}(z):=\sum_{n=0}^{\infty}\frac{(-z)^{n}}{n!\Gamma(-\alpha
n+1-\alpha)},$

a probability density function in $\mathbb{R}_{+}$). It is known (see, for
example, [13] and [26]) that

$\Phi_{\alpha}(t)\geq 0,\quad
t>0,\quad\int_{0}^{\infty}\Phi_{\alpha}(t)\,dt=1,$

and that the moments of $\Phi_{\alpha}$ are given by

$\int_{0}^{\infty}t^{\delta}\Phi_{\alpha}(t)\,dt=\frac{\Gamma(\delta+1)}{\Gamma(\alpha\delta+1)},\quad\delta>-1.$
(14)

Its Laplace transform is given by

$\int_{0}^{\infty}e^{-\tau t}\Phi_{\alpha}(\tau)\,d\tau=E_{\alpha}(-t),\quad
t>0,$

where $E_{\alpha}$ is the Mittag–Leffler function (see [15]):

$E_{\alpha}(z):=\sum_{n=0}^{\infty}\frac{z^{n}}{\Gamma(\alpha n+1)}.$

An application of the subordination principle may be justified in many
particular models where the evolution of correlation functions may be
constructed by means a $C_{0}$-semigroup in a proper Banach space. In general,
the subordination formula may be considered as a rule for the transformation
of Markov dynamics to fractional ones.

It is easy to see that $\mu_{t}^{\alpha}$ is a measure. Actually, positivity
follows from that fact that for any measurable set $A$ we have

$\mu_{t}^{\alpha}(A)=\int_{0}^{\infty}\Phi_{\alpha}(t,s)\mu_{s}(A)\,ds\geq 0,$

since $\mu_{s}$ is a measure and $\Phi_{\alpha}$ is a pdf. The
$\sigma$-additivity property may be verified using the standard procedure. The
FFPE equation may be written in terms of time-dependent correlation functions
as an infinite system of evolution equations, the so-called _hierarchical
chain_ :

$\begin{cases}\mathbb{D}_{t}^{\alpha}k_{\alpha,t}^{(n)}&=(L^{\triangle}k_{\alpha,t})^{(n)},\\\
k_{\alpha,t}^{(n)}|_{t=0}&=k_{\alpha,0}^{(n)},\quad n\geq 0.\end{cases}$

The evolution of the correlation functions should also be given by the
subordination principle. More precisely, if the solution $k_{t}$ of equation
(11) exists, then we have

$k_{\alpha,t}=\int_{0}^{\infty}\Phi_{\alpha}(\tau)k_{\tau t^{\alpha}}\,ds.$

##  5\.  Fractional kinetics and Poisson flows

As in the case of Markov statistical dynamics addressed above, we may consider
Vlasov-type scaling in the framework of the FFPE. We know that the kinetic
statistical dynamics for a Poisson initial state $\pi_{\rho_{0}}$ is given by
a flow of Poisson measures

$\mathbb{R}_{+}\ni t\mapsto\mu_{t}=\pi_{\rho_{t}}\in\mathcal{M}^{1}(\Gamma),$

where $\rho_{t}$ is the solution to the corresponding Vlasov kinetic equation.
Then the fractional kinetic dynamics of states may be defined as the
subordination of this flow (see comments above). Specifically, for
$0<\alpha<1$ we consider the subordinated flow

$\mu_{t}^{\alpha}:=\int_{0}^{\infty}\Phi_{\alpha}(\tau)\mu_{\tau
t^{\alpha}}\,d\tau=\int_{0}^{\infty}\Phi_{\alpha}(\tau)\pi_{\rho_{\tau
t^{\alpha}}}\,d\tau.$

The family of measures $\mu_{t}^{\alpha}$ is no longer a Poisson flow. We
would like to analyze the properties of these subordinated flows to
distinguish the effects of fractional evolution. Note first that the density
of the fractional kinetic state is given by the formula

$\rho_{t}^{\alpha}(x)=\int_{0}^{\infty}\Phi_{\alpha}(\tau)\rho_{\tau
t^{\alpha}}(x)\,d\tau.$

The latter is the subordination of the solution to the Vlasov equation and is
not related to a fractional Vlasov equation as it is expected in several
heuristic considerations in physics.

It is reasonable to study the properties of subordinated flows from a more
general point of view when the evolution of densities $\rho_{t}(x)$ is not
necessarily related to a particular Vlasov-type kinetic equation. Similar
transformations of Poisson flows do appear due to completely different
motivations in several applications. See, for example, [8, 29] and, for the
related fractional Poisson process, [32, 20, 24, 25, 33, 27], and references
therein.

Below we will study certain properties of the resulting flows affected by
fractional dynamics.

###  5.1. Front propagation for the density

Let us consider a density evolution of the form

$\rho_{t}(x)={1\\!\\!1}_{[-1-vt,1+vt]}(x),\quad t\geq 0,\quad x\in\mathbb{R},$

where $v>0$ is the constant speed of the density front. The subordinated
density has the following representation

$\rho_{t}^{\alpha}(x)=\int_{0}^{\infty}\Phi_{\alpha}(\tau){1\\!\\!1}_{[-1-vt^{\alpha}\tau,1+vt^{\alpha}\tau]}(x)\,d\tau,$

and for $|x|>1$

$\rho_{t}^{\alpha}(x)=\int_{A(x,t)}^{\infty}\Phi_{\alpha}(\tau)\,d\tau,$

where

$A(x,\tau)=\frac{|x|-1}{vt^{\alpha}}.$

We have $\rho_{t}^{\alpha}(x)\to 1,t\to\infty,|x|>1$ and
$\rho_{t}^{\alpha}(x)\to 0,x\to\infty,t\geq 0$. Consider

$\Psi_{\alpha}(s)=\int_{s}^{\infty}\Phi_{\alpha}(\tau)\,d\tau.$

Due to monotonicity we may find a unique $s_{\alpha}$ s.t.
$\Psi_{\alpha}(s_{\alpha})=1/2$. Define the front of $\rho_{t}^{\alpha}$ for
given $t>0$ as $x\in\mathbb{R}$, for which $\rho_{t}^{\alpha}(x)=1/2$. The
motion of the front is then given by the formula

$|x|=1+s_{\alpha}vt^{\alpha}.$

The latter result means that in the subordinated dynamics the density will be
expanded sub-linearly and more slower for smaller $\alpha\in(0,1)$.

###  5.2. Intermittency for subordinated flows

Each measure from the flow $\mu_{t}^{\alpha}$ defines a generalized random
process on ${\mathbb{R}}^{d}$ given for $f\in C_{0}(\mathbb{R}^{d})$ by

$X_{f}(\gamma)=\sum_{x\in\gamma}f(x),\quad\gamma\in\Gamma.$

Let us consider the corresponding moments

$m_{t}^{p}(f)=\int X_{f}^{p}\,d\mu_{t}^{\alpha},\quad p\geq 1.$

The notion of asymptotic intermittency is well understood for regular random
fields; see for example [7, 6]. In the case of generalized random fields this
notion may be formulated as follows.

######  Definition 1.

(Intermittency via moments). The flow $\mu_{t}^{\alpha},t\geq 0$ has the
asymptotic intermittency property if for any $0\leq f\in
C_{0}(\mathbb{R}^{d})$ and for all $p_{1},\dots,p_{n}\in\mathbb{N}$ with
$p_{1}+\cdots+p_{n}=p$ one had

$\lim_{t\to\infty}\frac{m_{t}^{p}(f)}{m_{t}^{p_{1}}(f)\dots
m_{t}^{p_{n}}(f)}=\infty.$

This property means that moments of the random field grow in time
progressively with the order. In the case of random point processes the
leading growth of moments is defined in terms of correlation functions of the
corresponding orders.

This gives us the option of reformulating the definition of asymptotic
intermittency in terms more convenient for our purposes.

######  Definition 2.

(Intermittency via correlation functions). The flow $\mu_{t}^{\alpha},t\geq 0$
has the asymptotic intermittency property if for any $\eta\in\Gamma_{0}$ and
its decomposition $\eta=\eta_{1}\cup\dots\cup\eta_{n}$ in disjunct subsets for
the correlation function $k_{\mu_{t}^{\alpha}}$, one has

$\lim_{t\to\infty}\frac{k_{\mu_{t}^{\alpha}}(\eta)}{k_{\mu_{t}^{\alpha}}(\eta_{1})\dots
k_{\mu_{t}^{\alpha}}(\eta_{n})}=\infty.$

For a detailed discussion of relations between different versions of the
intermittency property for random point processes, see [16].

Let us consider the dynamics of the density given by $\rho_{t}(x)=e^{\beta
t^{\sigma}}$, $\beta,\sigma>0$. The flow of Poisson measures $\pi_{\rho_{t}}$
has, for each $t\geq 0$, correlation functions
$k_{\pi_{\rho_{t}}}^{(n)}(x_{1},\dots,x_{n})=e^{\beta nt^{\sigma}}$. Therefore
the intermittency is absent.

######  Theorem 1.

Let $0<\alpha<1$ be given. Consider the subordinated flow for the Poisson flow
introduced above,

$\mu_{t}^{\alpha}:=\int_{0}^{\infty}\Phi_{\alpha}(\tau)\pi_{\rho_{\tau
t^{\alpha}}}\,d\tau.$

Assume $\sigma(1-\alpha)<1$. Then the flow $\mu_{t}^{\alpha}$ has the
asymptotic intermittency property.

###### Proof.

The $n$-th correlation function of $\mu_{t}^{\alpha}$ is given by

$\displaystyle(\rho_{t}^{\alpha})^{(n)}(x_{1},\ldots,x_{n})=\int_{0}^{\infty}\Phi_{\alpha}(\tau)(\rho_{\tau
t^{\alpha}}(x))^{n}\,d\tau$
$\displaystyle\qquad=\int_{0}^{\infty}\Phi_{\alpha}(\tau)e^{n\beta
t^{\sigma\alpha}\tau^{\sigma}}\,d\tau=\sum_{k=0}^{\infty}\frac{(n\beta
t^{\sigma\alpha})^{k}}{k!}\int_{0}^{\infty}\Phi_{\alpha}(\tau)\tau^{\sigma
k}\,d\tau$ $\displaystyle\qquad=\sum_{k=0}^{\infty}\frac{(n\beta
t^{\sigma\alpha})^{k}}{k!}\frac{\Gamma(\sigma k+1)}{\Gamma(\sigma
k\alpha+1)}=\sum_{k=0}^{\infty}\frac{n^{k}z^{k}}{k!}\frac{\Gamma(\sigma
k+1)}{\Gamma(\sigma k\alpha+1)},$

where $z:=\beta t^{\sigma\alpha}$. It is known [5] that the series converges
for all values of $z$ if and only if $\sigma<1/(1-\alpha)$, and that

$(\rho_{t}^{\alpha})^{(n)}(x_{1},\ldots,x_{n})\sim
C(nz)^{1/(2\mu)}\exp\big{(}c(nz)^{1/\mu}\big{)},$

where $C,c>0$, $\mu=1+\sigma(\alpha-1)$. Since $0<\mu<1$, this asymptotic
behavior implies intermittency. In fact, due to Definition 2, we need to
consider the limiting behavior of the ratio

$\frac{\exp(c(nz)^{\frac{1}{\mu}}}{\exp\sum_{k=1}^{m}\big{(}c(n_{k}z)^{\frac{1}{\mu}}\big{)}},\quad
z=\beta t^{\sigma\alpha}$

for $t\to\infty$ under the assumption $\sum_{k=1}^{m}n_{k}=n$. This limit is
equal $+\infty$ due to the inequality

$\left(\sum_{k=1}^{m}n_{k}\right)^{{\frac{1}{\mu}}}>\sum_{k=1}^{m}(n_{k})^{{\frac{1}{\mu}}}$

for $1/\mu>1$ (see [3], Chapter 1, § 16). ∎

###  5.3. Polynomially growing density

Let us now consider the case of a polynomial density $\rho_{t}(x)=(1+t)^{p}$,
$p\in\mathbb{N}$. For any $n\in\mathbb{N}$, the $n$th correlation function is
given by

$\displaystyle(\rho_{t}^{\alpha})^{(n)}(x_{1},\ldots,x_{n})=\int_{0}^{\infty}\Phi_{\alpha}(\tau)\rho_{\tau
t^{\alpha}}(x_{1})\ldots\rho_{\tau t^{\alpha}}(x_{n})\,d\tau$
$\displaystyle\qquad=\int_{0}^{\infty}\Phi_{\alpha}(\tau)(1+\tau
t^{\alpha})^{pn}\,d\tau=\sum_{j=0}^{np}\binom{np}{j}t^{\alpha
j}\int_{0}^{\infty}\tau^{j}\Phi_{\alpha}(\tau)\,d\tau$
$\displaystyle\qquad=\sum_{j=0}^{np}\binom{np}{j}t^{\alpha
j}\frac{\Gamma(j+1)}{\Gamma(j\alpha+1)}=\sum_{j=0}^{np}\frac{(np)!}{(np-j)!}t^{\alpha
j}\frac{1}{\Gamma(\alpha j+1)}$
$\displaystyle\qquad=\frac{(np)!}{\Gamma(\alpha np+1)}t^{\alpha
np}+o(t^{\alpha np}).$

In particular, for $n=1$, the 1st correlation function is equal to

$(\rho_{t}^{\alpha})^{(1)}=\rho_{t}^{\alpha}=\frac{p!}{\Gamma(\alpha
p+1)}t^{\alpha p}+o(t^{\alpha p}).$

Therefore we obtain

$\displaystyle\frac{(\rho_{t}^{\alpha})^{(n)}}{(\rho_{t}^{\alpha})^{n}}$
$\displaystyle=$ $\displaystyle\frac{(np)!}{\Gamma(\alpha np+1)}t^{\alpha
np}\times\left(\frac{\Gamma(\alpha p+1)}{p!}\right)^{n}\frac{1}{t^{\alpha
pn}}+o(1)$ $\displaystyle=$ $\displaystyle\frac{(np)!}{\Gamma(\alpha
np+1)}\times\left(\frac{\Gamma(\alpha p+1)}{p!}\right)^{n}+o(1),$

which is constant as $t$ goes to infinity. In conclusion, the power growth of
the 1st correlation function is not sufficient to organize intermittency in
the subordinated flow. Summarizing the above considerations, we conclude that
subordinating a flow (which corresponds to the dynamics of a system without
intermittency) is a way to _organize intermittency_.

##  6\. Examples

In this section we apply the general scheme of the fractional statistical
dynamics developed here to concrete models, namely the _contact model_ and the
_pure birth model_ , also known as the _Surgailis pure birth model_.

######  Example 3.

(Surgailis pure birth model). This is an example in which the kinetic
fractional statistical dynamics is a mixture of Poisson measures. The
Surgailis pure birth model (zero mortality) has generator given by

$(LF)(\gamma)=z\int_{\mathbb{R}^{d}}[F(\gamma\cup x)-F(\gamma)]\,dx.$

(cf. [19]). Starting from from the Poisson initial distribution
$\mu_{0}=\pi_{\rho_{0}}$, the solution of the FPE

$\begin{cases}\frac{\partial\mu_{t}}{\partial t}&=L^{*}\mu_{t},\\\
\mu_{t}|_{t=0}&=\pi_{\rho_{0}}\end{cases}$

is of the same type

$\mu_{t}=\pi_{zt+\rho_{0}}.$

The solution of the fractional FPE

$\begin{cases}\mathbb{D}_{t}^{\alpha}\nu_{t}^{\alpha}&=L^{*}\nu_{t}^{\alpha},\\\
\nu_{t}^{\alpha}|_{t=0}&=\pi_{\rho_{0}}\end{cases}$

is then given by the subordination principle as

$\nu_{t}^{\alpha}=\int_{0}^{\infty}\Phi_{\alpha}(s)\pi_{zt^{\alpha}s+\rho_{0}}\,ds.$

Hence the solution $\nu_{t}^{\alpha}$, $t>0$ is a mixture of Poisson measures.
The correlation function of the Poisson measure $\pi_{zt^{\alpha}s+\rho_{0}}$
is $((zt^{\alpha}s+\rho_{0})^{n})_{n=0}^{\infty}$, and therefore the
correlation function of the mixture $\nu_{t}^{\alpha}$ is, for $n\geq 0$,

$\displaystyle r_{t,\alpha}^{(n)}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\Phi_{\alpha}(s)(zt^{\alpha}s+\rho_{0})^{n}\,ds$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{n}\binom{n}{j}(zt^{\alpha})^{j}\rho_{0}^{n-j}\int_{0}^{\infty}\Phi_{\alpha}(s)s^{j}\,ds.$

The absolute moments of $\Phi_{\alpha}$ (cf. eq. (14)) satisfy

$\int_{0}^{\infty}s^{j}\Phi_{\alpha}(s)\,ds=\frac{\Gamma(j+1)}{\Gamma(\alpha
j+1)},\quad j>-1.$

Accordingly, the $n$th order correlation function of the measure
$\nu_{t}^{\alpha}$ reduces to

$r_{t,\alpha}^{(n)}=\sum_{j=0}^{n}\binom{n}{j}\rho_{0}^{n-j}(zt^{\alpha})^{j}\frac{j!}{\Gamma(\alpha
j+1)}=(zt^{\alpha})^{n}\frac{n!}{\Gamma(\alpha
n+1)}+o\big{(}(zt^{\alpha})^{n}\big{)}.$

In particular

$\big{(}r_{t,\alpha}^{(1)}\big{)}^{n}=\left(\frac{zt^{\alpha}}{\Gamma(\alpha+1)}\right)^{n}+o\big{(}(zt^{\alpha})^{n}\big{)},$

and thus

$\displaystyle\frac{r_{t,\alpha}^{(n)}}{\big{(}r_{t,\alpha}^{(1)}\big{)}^{n}}$
$\displaystyle=$
$\displaystyle(zt^{\alpha})^{n}\frac{n!}{\Gamma(n\alpha+1)}\times\left(\frac{\Gamma(\alpha+1)}{zt^{\alpha}}\right)^{n}+o((zt^{\alpha})^{n})$
$\displaystyle=$
$\displaystyle\frac{(n-1)!}{\alpha\Gamma(n\alpha)}(\Gamma(\alpha+1))^{n}+o((zt^{\alpha})^{n}).$

From this we see that as $t\rightarrow\infty$ the above coefficient does not
explode, which tell us that this model has no asymptotic intermittency. In
other words, the power growth of the correlation function corresponding to the
FPE is not sufficient to realize asymptotic intermittency of the kinetic
fractional statistical dynamics. In the next example we show that under strong
growth on the $n$th order correlation function of the FPE (exponential
growth), the kinetic fractional statistical dynamics does exhibit asymptotic
intermittency.

######  Example 4.

(Contact model). The contact model is one of the simplest models in the theory
of IPS. Nevertheless, it has interesting properties, e.g., its asymptotic
behavior and the structure of its equilibrium measures. We refer to [10] for
more details.

The generator $L$ of the stochastic dynamics is given informally by

$(LF)(\gamma)=\sum_{x\in\gamma}m\big{(}F(\gamma\backslash
x)-F(\gamma)\big{)}+\int_{\mathbb{R}^{d}}b(x,\gamma)\big{(}F(\gamma\cup
x)-F(\gamma)\big{)}\,dx.$

Here $m>0$ is a constant mortality rate and the birth rate is

$b(x,\gamma)=\sum_{y\in\gamma}a(x-y),$

where $0\leq a\in L^{1}(\mathbb{R}^{d})$ is even.

In the kinetic limit, the correlation functions of the contact model in the
super critical regime are given by

$r_{t}^{(n)}(x_{1},\ldots,x_{n})=C^{n}e^{\beta nt}$

for certain $C,\beta>0$ [19], [10]. The correlation functions of the solution
for the fractional kinetic dynamics are then given by

$r_{t,\alpha}^{(n)}=C^{n}\int_{0}^{\infty}\Phi_{\alpha}(s)e^{\beta
nt^{\alpha}s}\,ds=C^{n}E_{\alpha}(\beta nt^{\alpha}),\quad n\in\mathbb{N}.$

Using the asymptotic behavior of the Mittag-Leffler function $E_{\alpha}$ as
$t\rightarrow\infty$ (see eq. (6.4) in [14]), we can conclude that the kinetic
fractional statistical dynamics in the contact model does exhibit asymptotic
intermittency. Of course, this statement is a particular case of Theorem 1 for
$\sigma=1$.

Acknowledgments. We would like to thank the financial support of the DFG
through SFB 701, Bielefeld University and the project I&D: UID/MAT/04674/2013.

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