# Stochastic Processes and Mean Square Calculus on Fractal Curves

Alireza Khalili Golmankhaneh1111Corresponding author: Alireza Khalili
Golmankhaneh. Email addresses<EMAIL_ADDRESS>or
<EMAIL_ADDRESS>(Alireza Khalili Golmankhaneh ),
<EMAIL_ADDRESS>(Kerri Welch<EMAIL_ADDRESS>( Cristina Serpa ) ,
Kerri Welch2, Cristina Serpa3,4,5 Ivanka Stamova 6
1 Department of Physics, Urmia Branch, Islamic Azad University, Urmia 63896,
Iran
2Faculty at California Institute of Integral Studies, San Francisco, CA, USA
3 ISEL-Instituto Superior de Engenharia de Lisboa, Portugal
4 CMAFcIO-Centro de Matemática, Aplicações Fundamentais e Investigação
Operacional, Portugal
5 Instituto Piaget–Cooperativa para o Desenvolvimento Humano, Integral e
Ecológico, C.R.L., Portugal
6 Department of Mathematics, University of Texas at San Antonio, San Antonio,
TX 78249, U.S.A

###### Abstract

In this paper, random and stochastic processes are defined on fractal curves.
Fractal calculus is used to define cumulative distribution function,
probability density function, moments, variance, and correlation function of
stochastic process on fractal curves. A new framework which is a
generalization of mean square calculus is formulated. The sequence of random
variables on the fractal curve, fractal mean square continuity, mean square
$F^{\alpha}$-derivative, and fractal mean square integral. The mean square
solution of a fractal stochastic equation is derived and plotted in order to
show the details.

Keywords: $F^{\alpha}$-calculus; Fractal stochastic equation; Mean square
calculus; Fractal curve
MSC[2010]: 28A80, 60G12, 60G18, 60G20, 60H05

##  1 Introduction

Fractals are shapes that are seen in nature, such as clouds, mountains,
coastlines, blood vessels, heart rates, Romanesco broccoli, trees, frost
crystals, and so on [1]. Fractal geometry is suggested to characterize the
fractals and their properties. Fractals often have fractional dimensions and
are self-similar and their fractal dimension exceeds their topological
dimension [2]. Analysis on fractals was formulated by many researchers by
different methods such as measure theory, harmonic analysis, fractional space,
fractional calculus, stochastic process [3, 4, 5, 6, 7, 8, 9, 10, 11] The
Riemann-like method which is base of ordinary calculus, has been modified to
include functions with fractal support such as Cantor sets and curves. This
framework is simple, algorithmic with geometrical and physical meaning which
is called fractal calculus or $F^{\alpha}$-calculus [12, 13, 14, 15]. Sub-
diffusion and super diffusion were modeled by fractal local derivatives
without violating locality and central limit theorem [16]. Non-local fractal
derivatives were defined to model incompressible viscous fluid in fractal
media and processes with memory [17, 18]. Fractal calculus has been used to
present new model in physics in the fractal space and time [19]. These models
present power law and self-similar solutions and results [20]. Fractal
calculus was used to find the derivative and integral of the Weierstrass
function [21]. To do research in this direction we have generalized the mean
square calculus on fractal curves.
The outline of the paper is as follows:
Section 2 we give fractal calculus on curves. We define the stochastic and
random variables on fractal curves in Section 3. In Section 4 the fractal mean
square calculus is suggested on fractal curves and the corresponding
stochastic equation is solved. Finally, Section 5 is devoted to conclusion.

##  2 Preliminaries

In this section, we summarize the fractal calculus on fractal curves. [12, 13,
14, 20].

###  2.1 Fractal calculus on fractal curve

Let the image of the continuous function $f:R\rightarrow R^{n}$ be a fractal.

###### Definition 2.1

A fractal curve $F\subset R^{n}$ is called continuously parameterizable if
there exists a function such as

$w:[a_{1},b_{1}]\rightarrow F$ (1)

where $[a_{1},b_{1}]\subset R$, and $w$ is continuous, one-to-one and onto
$F$.

###### Definition 2.2

A subdivision $P_{[a,b]}$ of interval $[a,b]\subset[a_{1},b_{1}]$ is a finite
set of points such as

$P_{[a,b]}=\\{a=t_{0},t_{1},...,t_{n}=b\\}$ (2)

where $t_{i}<t_{i+1}$, and $[t_{i},t_{i+1}]$ is component of the $P_{[a,b]}$.
A subdivision $Q$ is called a refinement of $P$ if we have $P\subset Q$.

###### Definition 2.3

Let $F$ be a fractal curve, and a subdivision $P_{[a,b]}$ of $[a,b]$, then
$\sigma^{\alpha}[F,P]$ is defined by

$\sigma^{\alpha}[F,P]=\sum_{i=0}^{n-1}\frac{|w(t_{i+1})-w(t_{i})|^{\alpha}}{\Gamma(\alpha+1)}$
(3)

where $|*|$ denotes the Euclidean norm on $R^{n}$.

###### Definition 2.4

The coarse grained mass $\gamma_{\delta}^{\alpha}(F,a,b)$, for given $\delta$
is defined by

$\gamma_{\delta}^{\alpha}(F,a,b)=\inf_{P_{[a,b]}:|P|\leq\delta}\sigma^{\alpha}[F,P],$
(4)

where $|P|=\max_{0\leq i\leq n-1}(t_{i+1}-t_{i})$ for the subdivision $P$.

###### Definition 2.5

The mass function $\gamma^{\alpha}(F,a,b)$ is defined by

$\gamma^{\alpha}(F,a,b)=\lim_{\delta\rightarrow
0}\gamma_{\delta}^{\alpha}(F,a,b).$ (5)

The $\gamma^{\alpha}(F,a,b)$ is a monotonic function of $\delta$, and the
limit exists.

###### Definition 2.6

The $\gamma$-dimension of $F$ is defined by [13, 14, 20]

$dim_{\gamma}(F)=\inf\\{\alpha:\gamma^{\alpha}(F,a,b)=0\\}=\sup\\{\alpha:\gamma^{\alpha}(F,a,b)=\infty\\}.$
(6)

###### Definition 2.7

The staircase function $S_{F}^{\alpha}:[a_{1},b_{1}]\rightarrow R$ of order
$\alpha$ for a $F$ is defined by

$S_{F}^{\alpha}(t)=\left\\{\begin{array}[]{ll}\gamma^{\alpha}(F,a,b),&t\geq
p_{0}\\\ -\gamma^{\alpha}(F,a,b),&t<p_{0},\end{array}\right.$ (7)

where $p_{0}\in[a_{1},b_{1}]$, and $t\in[a_{1},b_{1}]$. Let consider $F$ for
which $S_{F}^{\alpha}(t)$ be strictly increasing, so it is invertible, and we
can write

$J(\theta)=S_{F}^{\alpha}(w^{-1}(\theta)),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \theta\in F.$
(8)

which is one-to-one.

###### Definition 2.8

Let $f:F\rightarrow R$. The limit of $f$ through points of $F$ is $l$, if for
given $\epsilon>0$ there exists $\delta>0$ such that

$\theta^{\prime}\in F,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ and,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
|\theta^{\prime}-\theta|<\delta\Rightarrow|f(\theta^{\prime})-l|<\varepsilon$
(9)

and is denoted by

$l=\underset{\theta^{\prime}\rightarrow\theta^{-}}{F_{-}lim}f(\theta^{\prime}).$
(10)

###### Definition 2.9

If $f:F\rightarrow R$, then the $F^{\alpha}$-derivative of function $f$ at
$\theta\in F$ is defined by

$D_{F}^{\alpha}f(\theta)=F-\lim_{\theta^{\prime}\rightarrow\theta}\frac{f(\theta^{\prime})-f(\theta)}{J(\theta^{\prime})-J(\theta)}$
(11)

if the limit exists.

###### Definition 2.10

Let us consider finite partition $P_{[a,b]}$ given in Definition 2.2. Then,
the $F^{\alpha}$-integral of $f:F\rightarrow R$ bounded function on $F$, is
defined by

$\int_{C(a,b)}f(\theta)d_{F}^{\alpha}\theta=\underset{\Delta_{n}\rightarrow
0}{F_{-}lim}\sum_{i=0}^{n}f(\theta_{i^{\prime}})[S_{F}^{\alpha}(w^{-1}(\theta_{i}))-S_{F}^{\alpha}(w^{-1}(\theta_{i-1}))]$
(12)

where $\theta=w(t)$, $\theta_{i^{\prime}}\in[\theta_{i-1},\theta_{i})$,
$\Delta_{n}=\max_{i}(\theta_{i}-\theta_{i-1})$ and segment $C(t_{1},t_{2})$ is
defined by

$C(t_{1},t_{2})=\\{w(t^{\prime}):t^{\prime}\in[t_{1},t_{2}]\\}.$ (13)

##  3 Random variable on fractal curve

In this section, we define random variable on fractal curve (RVFC) [22, 23,
24]. Let consider probability space $(\mathcal{S},\mathcal{F},\mathbb{P})$
where $S$ is the set of outcomes, $\mathcal{F}$ is the $\sigma$-algebra of
events, and $\mathbb{P}$ is the probability measure on sample space $S$ [25].

###### Definition 3.1

A random variable on the fractal curves $F$ (RVFC) is defined by

$X(\zeta):\mathcal{S}\rightarrow F$ (14)

where $\mathcal{S}$ is sample space.

###### Definition 3.2

The cumulative distribution function of RVRC is defined by

$F_{X}(\theta)=P(X(\zeta)\leq\theta)\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \theta\in F.$ (15)

###### Definition 3.3

The probability density function of RVRC is defined by

$f_{X}(\theta)=D_{F}^{\alpha}F_{X}(\theta).$ (16)

Then we can write

$F_{X}(\theta)=\int_{C(-\infty,t)}f_{X}(\theta)d_{F}^{\alpha}\theta$ (17)

where $\theta=w(t)$.

###### Definition 3.4

The mean of RVRC is defined by

$E(X)=\int_{-\infty}^{+\infty}\theta f_{X}(\theta)d_{F}^{\alpha}\theta$ (18)

###### Definition 3.5

The variance of RVRC is defined by

$Var(X)=\int_{-\infty}^{\infty}(\theta-E(X))^{2}f_{X}(\theta)d_{F}^{\alpha}\theta.$
(19)

###### Definition 3.6

The $m$-th moment of RVRC is defined by

$E(X^{m})=\int_{-\infty}^{+\infty}\theta^{m}f_{X}(\theta)d_{F}^{\alpha}\theta,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ m\in N$ (20)

###### Definition 3.7

A RVRC is called uniform if its probability density is given by

$f_{X}(\theta)=\left\\{\begin{array}[]{ll}\Gamma(\alpha+1),&\theta\in F\\\
0,&otherwise.\end{array}\right.$ (21)

and, its cumulative distribution function is

$F_{X}(\theta)=\int_{-\infty}^{\infty}\Gamma(\alpha+1)d_{F}^{\alpha}\theta=\left\\{\begin{array}[]{ll}J(\theta),&\theta\in
F\\\ 0,&otherwise.\end{array}\right.$ (22)

###### Definition 3.8

A RVRC is called memoryless if its probability density function is given by

$f_{X}(\theta)=\left\\{\begin{array}[]{ll}\lambda\exp(-\lambda\theta),&0<\theta,\\\
0,&\theta<0.\end{array}\right.$ (23)

Then its cumulative distribution function is

$F_{X}(\theta)=\left\\{\begin{array}[]{ll}1-\exp(-\lambda
J(\theta)),&\theta\geq 0\\\ 0,&\theta<0.\end{array}\right.$ (24)

where $\lambda>0$.

Figure 1: Graph of Eq. (24) choosing $\lambda=1$

In Figure 1, we have sketched Eq.(24).

###  3.1 Random processes on fractal curve

###### Definition 3.9

A fractal random process is a family of random variables denoted by

$X(\zeta,\tau),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\zeta\in\mathcal{S},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \tau\in F.$ (25)

If $\zeta=\zeta_{i}$ is constant, then $X(\zeta_{i},\tau)=X_{i}(\tau)$ is
called fractal sample function. For simplicity, let $X(\zeta,\tau)=X(\tau)$.

###### Definition 3.10

The correlation function of a fractal random process $X(\tau)$ is defined by

$R(\tau_{1},\tau_{2})=E[X(\tau_{1})X(\tau_{2})],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \tau_{1},\leavevmode\nobreak\
\tau_{2}\in F.$ (26)

If $\tau_{2}=\tau+\epsilon$, and $\tau_{1}=\tau$, then Eq.(26) turns to

$R(\tau)=E[X(\tau)X(\tau+\epsilon)],\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \tau\in F.$ (27)

##  4 Mean square $F^{\alpha}$-calculus on fractal curves

In this section, we give below some definitions to develop mean square (f.m.s)
calculus [25] which might be called fractal mean square calculus.

###### Definition 4.1

The inner product of two random variable on fractal curve $X_{1}$ and $X_{2}$
is defined by

$<X_{1},X_{2}>=E[X_{1}X_{2}].$ (28)

###### Definition 4.2

The distance between two random variable $X_{1}$ and $X_{2}$ is define by

$d(X_{1},X_{2})=||X_{1},X_{2}||$ (29)

###### Remark 1

The class of all second random variable $X_{1},X_{2},...,X_{n}$ on probability
space $(\mathcal{S},\mathcal{F},\mathbb{P})$ constitute a linear vector space
which is denoted by $L^{\alpha}_{2}$-space.

###### Definition 4.3

A random process on fractal curve $X(\tau)$ is called second order if [25]

$||X(\tau)||=(<X(\tau),X(\tau)>)^{1/2}=R(\tau,\tau)^{1/2}=(E[X(\tau)^{2}])^{1/2}<\infty,\leavevmode\nobreak\
\tau\in F.$ (30)

###### Definition 4.4

A stochastic process $X(\tau),\leavevmode\nobreak\ \tau\in F$ is a second
order stochastic process (s.p.) if, for every set
$\tau_{1},\tau_{2},...,\tau_{n}$ are elements of $L_{2}^{\alpha}$-space,
namely,

$||X(\tau)||^{2}=E[X(\tau)^{2}]<\infty,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \tau\in F.$ (31)

###### Definition 4.5

A sequence of random variable on fractal curve
$X(\tau_{1}),X(\tau_{2}),...,X(\tau_{n})$ is called mean square convergent to
$X(\tau)$ if we have

$\underset{n\rightarrow\infty}{F_{-}lim}\leavevmode\nobreak\
E[(X(\tau_{n})-X(\tau))^{2}]=0,$ (32)

or

$\underset{n\rightarrow\infty}{f.l.i.m}\leavevmode\nobreak\
X(\tau_{n})=X(\tau),$ (33)

where the symbol $f.l.i.m$ is denotes the fractal limit in mean square [25].

###### Theorem 4.1

Let $X_{n}$ and $X_{n^{\prime}}^{\prime}$ where $n,n^{\prime}\in
N\leavevmode\nobreak\ (=index\leavevmode\nobreak\ set)$ be two sequence of
second order f.r.v., if

$\underset{n\rightarrow n_{0}}{f.l.i.m}\leavevmode\nobreak\
X_{n}=X,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\textmd{and}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\underset{n^{\prime}\rightarrow n_{0}^{\prime}}{f.l.i.m}\leavevmode\nobreak\
X_{n^{\prime}}^{\prime}=X^{\prime}$ (34)

where $n_{0}$ and $n_{0}^{\prime}$ are limit points of $N$. Then

$\underset{n\rightarrow n_{0},n^{\prime}\rightarrow
n_{0}^{\prime}}{F_{-}lim}E\\{X_{n}X^{\prime}_{n^{\prime}}\\}=E\\{XX^{\prime}\\}.$
(35)

###### Proof 4.1

As $L_{2}^{\alpha}$-space is linear. Then

$\Delta X=X_{n}-X,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \Delta
X^{\prime}=X^{\prime}_{n^{\prime}}-X^{\prime},$ (36)

are second order f.r.v., and we have

$\displaystyle|E\\{X_{n}X^{\prime}_{n^{\prime}}-XX^{\prime}\\}|=|E\\{X\Delta
X^{\prime}\\}+E\\{X^{\prime}\Delta X\\}+E\\{\Delta X\Delta X^{\prime}\\}|$
$\displaystyle\leq||X||.||\Delta X^{\prime}||+||X^{\prime}||.||\Delta
X||+||\Delta X||.||\Delta X^{\prime}||,$ (37)

by using the hypothesis, namely,

$\underset{n\rightarrow n_{0}}{F_{-}lim}\leavevmode\nobreak\ ||\Delta
X||=0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\underset{n^{\prime}\rightarrow n_{0}^{\prime}}{F_{-}lim}\leavevmode\nobreak\
||\Delta X^{\prime}||=0$ (38)

we thus get

$\underset{n\rightarrow n_{0},n^{\prime}\rightarrow
n_{0}^{\prime}}{F_{-}lim}|E\\{X_{n}X^{\prime}_{n^{\prime}}-XX^{\prime}\\}|=0,$
(39)

and the proof is complete.

###### Theorem 4.2

Let $\\{X_{n}(\tau)\\},\leavevmode\nobreak\ \tau\in F$ be a sequence of second
order. Then it converges to second order process $X(\tau),\leavevmode\nobreak\
\tau\in F$ if, and only if,

$\underset{n,n^{\prime}\rightarrow
n_{0}}{F_{-}lim}R_{X_{n}X_{n^{\prime}}}(\tau,s)=\underset{n,n^{\prime}\rightarrow
n_{0}}{F_{-}lim}E\\{X_{n}(\tau),X_{n^{\prime}}(\tau)\\}=R_{XX}(\tau,s)=R(\tau,s),$
(40)

and $R_{XX}(\tau,s)$ is finite on $F\times F$.

###### Proof 4.2

To establish the if part, set $R(\tau,s)=R_{XX}(\tau,s)$, then

$||X_{n}-X_{n^{\prime}}||^{2}=E\\{X_{n}^{2}(\tau)\\}-2E\\{X_{n}(\tau)X_{n^{\prime}}(\tau)\\}+E\\{X_{n^{\prime}}^{2}(\tau)\\}$
(41)

It follows that

$\underset{n,n^{\prime}\rightarrow
n_{0}}{F_{-}lim}||X_{n}-X_{n^{\prime}}||^{2}=R(\tau,\tau)-2R(\tau,\tau)+R(\tau,\tau)=0,$
(42)

which implies

$\underset{n\rightarrow n_{0}}{f.l.i.m}\leavevmode\nobreak\
X_{n}(\tau)=X(\tau).$ (43)

To establish the only if, we make apply of Theorem 4.1. Substituting $X_{n}$
by $X_{n}(\tau)$, $X_{n}^{\prime}$ by $X_{n}(s)$, $X$ by $X(\tau)$, and
$X^{\prime}$ by $X(s)$, then Eq.(35) becomes

$\underset{n\rightarrow n_{0},n^{\prime}\rightarrow
n_{0}^{\prime}}{F_{-}lim}E[X_{n}(\tau)X_{n^{\prime}}(s)]=E[X(\tau)X(s)]=R(\tau,s)$
(44)

On setting $\tau=s$, we obtain

$\underset{n\rightarrow n_{0},n^{\prime}\rightarrow
n_{0}^{\prime}}{F_{-}lim}E[X_{n}(\tau)X_{n^{\prime}}(\tau)]=E[X(\tau)X(\theta)]=R(\tau,\tau)$
(45)

which proves only if part. Eq.(40) is established by setting $n=n^{\prime}$ in
Eq.(44).

###### Definition 4.6

A sequence of random variables on the fractal curve
$X(\tau_{1}),X(\tau_{2}),...,X(\tau_{n})$ is fundamental fractal in
probability if, for every $\epsilon>0$

$\underset{n,m\rightarrow\infty}{F_{-}lim}\leavevmode\nobreak\
P\big{\\{}|X(\tau_{m})-X(\tau_{n})|>\epsilon\big{\\}}=0,$ (46)

or

$\underset{n,m\rightarrow\infty}{f.l.i.m}\leavevmode\nobreak\
X(\tau_{m})=X(\tau_{n}).$ (47)

###### Definition 4.7

A sequence of random variables on the fractal curve
$X(\tau_{1}),X(\tau_{2}),...,X(\tau_{n})$ converges in probability to
$X(\tau)$ if we have

$\underset{n\rightarrow\infty}{F_{-}lim}\leavevmode\nobreak\
P\big{\\{}|X(\tau_{n})-X(\tau)|>\epsilon\big{\\}}=0,$ (48)

or

$\underset{n\rightarrow\infty}{f.l.i.m}\leavevmode\nobreak\
X(\tau_{n})=X(\tau).$ (49)

###### Definition 4.8

Let $X(\tau)$ be a random process on fractal curve. If

$\underset{\epsilon\rightarrow 0}{F_{-}lim}\leavevmode\nobreak\
E[(X(\tau+\epsilon)-X(\tau))^{2}]=0,$ (50)

or

$\underset{\epsilon\rightarrow 0}{f.l.i.m}\leavevmode\nobreak\
X(\tau+\epsilon)=X(\tau),$ (51)

where $\epsilon>0$ is infinitesimal [26], then $X(\tau)$ is called mean square
$F^{\alpha}$-continuous at $\tau$.

###### Definition 4.9

If a second order f.s.p $X(\tau),\leavevmode\nobreak\ \tau\in F$ is f.m.s
continuous at every point $\tau\in C(\tau_{1},\tau_{2})$, then $X(\tau)$ is
f.m.s. continuous on segment $C(\tau_{1},\tau_{2})$.

###### Definition 4.10

A random process $X(\tau)$ have mean square $F^{\alpha}$-derivative if we have

$\underset{\epsilon\rightarrow 0}{F_{-}lim}\leavevmode\nobreak\
E\bigg{[}\bigg{(}\frac{X(\tau+\epsilon)-X(\tau)}{J(\epsilon)}-D_{F}^{\alpha}X(\tau)\bigg{)}^{2}\bigg{]}=0,$
(52)

or

$\underset{\epsilon\rightarrow 0}{f.l.i.m}\leavevmode\nobreak\
\frac{X(\tau+\epsilon)-X(\tau)}{J(\epsilon)}=D_{F}^{\alpha}X(\tau).$ (53)

###### Definition 4.11

If a second order f.s.p $X(\tau),\leavevmode\nobreak\ \tau\in F$ is f.m.s
differentiable at every $\tau\in C(\tau_{1},\tau_{2})\subset F$ then $X(\tau)$
is f.m.s differentiable on $C(\tau_{1},\tau_{2})$.

###### Example 1

Consider the fractal stochastic process as

$X(\tau)=\cos(\tau+\Phi)$ (54)

where $\Phi$ is uniformly distributed on $F$. Its correlation function is

$R(\tau)=\frac{1}{2}\cos(\tau).$ (55)

As Eq.(55) is $F^{\alpha}$-differentiable infinitely hence $X(\tau)$ has f.m.s
derivative.

###### Theorem 4.3

If $X(\tau)$ and $Y(\tau)$ are f.m.s. differentiable at $\tau$, then we have

$D_{F}^{\alpha}[aX(\tau)+bY(\tau)]=aD_{F}^{\alpha}X(\tau)+bD_{F}^{\alpha}Y(\tau)$
(56)

where $a$ and $b$ are constant.

###### Proof 4.3

By the norm property, the proof is immediate as

$\displaystyle\bigg{\lVert}\frac{aX(\tau+\epsilon)+bY(\tau+\epsilon)-aX(\tau)-bX(\tau)}{\epsilon}-aD_{F}^{\alpha}X(\tau)+bD_{F}^{\alpha}Y(\tau)\bigg{\rVert}$
$\displaystyle\leq\bigg{\lVert}a\bigg{[}\frac{X(\tau+\epsilon)-X(\tau)}{\tau}-D_{F}^{\alpha}X(\tau)\bigg{]}\bigg{\rVert}+\bigg{\lVert}b\bigg{[}\frac{Y(\tau+\epsilon)-Y(\tau)}{\epsilon}-D_{F}^{\alpha}Y(\tau)\bigg{]}\bigg{\rVert},$
(57)

by hypothesis the last two terms tend to zero as $\epsilon\rightarrow 0$,
which completes the proof.

###### Theorem 4.4

If $f:F\rightarrow\mathbb{R}$ and $F^{\alpha}$-differentiable at $\tau\in F$,
and $X(\tau)$ is f.m.s differentiable at $\tau\in F$, then $f(\tau)$,
$X(\tau)$ is f.m.s differentiable at $\tau$ and we have

$D_{F}^{\alpha}[f(\tau)X(\tau)]=D_{F}^{\alpha}f(\tau)X(\tau)+f(\tau)D_{F}^{\alpha}X(\tau).$
(58)

###### Proof 4.4

To show this property, consider the norm property as

$\displaystyle\bigg{\lVert}\frac{f(\tau+\epsilon)X(\tau+\epsilon)-f(\tau)X(\tau)}{\tau}-D_{F}^{\alpha}f(\tau)X(\tau)-f(\tau)D_{F}^{\alpha}X(\tau)\bigg{\rVert}$
$\displaystyle\leq\bigg{\lVert}\frac{f(\tau+\epsilon)X(\tau+\epsilon)-f(\tau)X(\tau+\epsilon)}{\tau}-D_{F}^{\alpha}f(\tau)X(\tau)\bigg{\rVert}$
$\displaystyle+\bigg{\lVert}\frac{f(\tau)X(\tau+\epsilon)-f(\tau)X(\tau)}{\tau}-f(\tau)D_{F}^{\alpha}X(\tau)\bigg{\rVert}$
$\displaystyle\leq\bigg{\lVert}\bigg{[}\frac{f(\tau+\epsilon)-f(\tau)}{\epsilon}-D_{F}^{\alpha}f(\tau)\bigg{]}X(\tau+\epsilon)\bigg{\rVert}+\bigg{\lVert}D_{F}^{\alpha}f(\tau)\bigg{[}X(\tau+\epsilon)-X(\epsilon)\bigg{]}\bigg{\rVert}$
$\displaystyle+\bigg{\lVert}f(\tau)\bigg{[}\frac{X(\tau+\epsilon)-X(\tau)}{\epsilon}-D_{F}^{\alpha}f(\tau)]\bigg{\rVert}$
$\displaystyle\leq\bigg{|}\frac{f(\tau-\epsilon)-f(\tau)}{\epsilon}-D_{F}^{\alpha}f(\tau)\bigg{|}\bigg{\lVert}X(\tau+\epsilon)\bigg{\rVert}+\bigg{|}D_{F}^{\alpha}f(\tau)\bigg{|}\bigg{\lVert}X(\tau+\epsilon)-X(\tau)\bigg{\rVert}$
$\displaystyle+\bigg{|}f(t)\bigg{|}\bigg{\lVert}\frac{X(\tau+\epsilon)-X(\tau)}{\epsilon}-D_{F}^{\alpha}X(\tau)\bigg{\rVert}$
(59)

The Eq.(4.4) tends to zero as $\epsilon\rightarrow 0$. Thus the proof is
complete.

###### Theorem 4.5

If a s.p. $X(\tau),\tau\in F$ is $n$ times f.m.s differentiable then we have

$E\\{D_{F}^{\alpha}X(\tau)\\}=D_{F}^{\alpha}E\\{X(\tau)\\}.$ (60)

###### Proof 4.5

To establish Eq.(60), let recall Definition (4.10), then we have

$\displaystyle
E\\{D_{F}^{\alpha}X(\tau)\\}=E\bigg{\\{}\underset{\epsilon\rightarrow
0}{f.l.i.m}\bigg{[}\frac{X(\tau+\epsilon)-X(\tau)}{\epsilon}\bigg{]}\bigg{\\}}$
$\displaystyle=\underset{\epsilon\rightarrow
0}{F_{-}lim}\bigg{\\{}\frac{E[X(\tau+\epsilon)]-E[X(\tau)]}{\epsilon}\bigg{\\}}$
$\displaystyle=D_{F}^{\alpha}E[X(\tau)],$ (61)

which completes the proof.

###### Definition 4.12

The second fractal generalized derivative of a second order s.p.
$X(\tau),\tau\in F$ is defined by

$\displaystyle\underset{\epsilon,\epsilon^{\prime}\rightarrow
0}{F_{-}lim}\bigg{(}\frac{1}{\epsilon\epsilon^{\prime}}\bigg{)}\Delta_{\epsilon}\Delta_{\epsilon^{\prime}}R(\tau,s)$
$\displaystyle=\underset{\epsilon,\epsilon^{\prime}\rightarrow
0}{F_{-}lim}\bigg{(}\frac{1}{\epsilon\epsilon^{\prime}}\bigg{)}\bigg{[}R(\tau+\epsilon,s+\epsilon^{\prime})-R(\tau+\epsilon,s)-R(\tau,s+\epsilon^{\prime})+R(\tau,s)\bigg{]}$
(62)

if it exists at $(\tau,\tau)$ and is finite.

###### Theorem 4.6

If a s.p. $X(\tau)$ is mean square $F^{\alpha}$-differentiable at point
$\tau$, then $X(\tau)$ is f.m.s continuous at $\tau$.

###### Proof 4.6

Since $\tau+\epsilon\in F$, then we write

$\displaystyle\underset{\epsilon\rightarrow 0}{F_{-}lim}\leavevmode\nobreak\
\bigg{\lVert}X(\tau+\epsilon)-X(\tau)\bigg{\rVert}^{2}=\underset{\epsilon\rightarrow
0}{F_{-}lim}\leavevmode\nobreak\
|\epsilon|^{2}\bigg{\lVert}\frac{[X(\tau+\epsilon)-X(\tau)]}{\epsilon}\bigg{\rVert}^{2}$
$\displaystyle=0\times\leavevmode\nobreak\ \underset{\epsilon\rightarrow
0}{F_{-}lim}\leavevmode\nobreak\
\bigg{(}\frac{1}{\epsilon^{2}}\bigg{)}\Delta_{\epsilon}\Delta_{\epsilon}R(\tau,\tau)=0.$
(63)

This is the desired conclusion.

###### Example 2

Consider a s.p. $X(\tau),\tau\in F$ as

$X(\tau)=A\tau,$ (64)

where $A$ is a second-order fractal r.v. with mean zero and variance
$\sigma^{2}$. Its correlation function is

$R(\tau,s)=\sigma^{2}\tau s.$ (65)

The second fractal generalized derivative of $X(\tau)$ is

$\displaystyle\underset{\epsilon,\epsilon^{\prime}\rightarrow
0}{F_{-}lim}\bigg{(}\frac{1}{\epsilon\epsilon^{\prime}}\bigg{)}\Delta_{\tau}\Delta_{\tau^{\prime}}R(\tau,s)$
$\displaystyle=\underset{\epsilon,\epsilon^{\prime}\rightarrow
0}{F_{-}lim}\bigg{(}\frac{\sigma^{2}}{\epsilon\epsilon^{\prime}}\bigg{)}\bigg{[}(\tau+\epsilon)(s+\epsilon^{\prime})-(\tau+\epsilon)s-\tau(s+\tau^{\prime})+\tau
s\bigg{]}$ $\displaystyle=\underset{\epsilon,\epsilon^{\prime}\rightarrow
0}{F_{-}lim}\bigg{(}\frac{\sigma^{2}}{\epsilon\epsilon^{\prime}}\bigg{)}[\epsilon\epsilon^{\prime}]=\sigma^{2}<\infty,$
(66)

which implies that $X(\tau)$ is f.m.s differentiable at every finite $\tau$.

###### Theorem 4.7

If the second fractal generalized derivative exist at $(\tau,\tau)$ for every
$\tau\in F$. Then the partial derivative

$D_{F,\tau}^{\alpha}R(\tau,s),\leavevmode\nobreak\
D_{F,s}^{\alpha}R(\tau,s),\leavevmode\nobreak\
D_{F,\tau}^{\alpha}D_{F,s}^{\alpha}R(\tau,s)$ (67)

exists and is finite on $F\times F$.

###### Proof 4.7

Since $X(\tau)$ is f.m.s differentiable, thus we can write

$\displaystyle
E\\{D_{F,\tau}^{\alpha}X(\tau)X(s)\\}=E\bigg{\\{}\underset{\epsilon\rightarrow
0}{f.l.i.m}\bigg{[}\frac{X(\tau+\epsilon)-X(\tau)}{\epsilon}\bigg{]}X(s)\bigg{\\}}$
$\displaystyle=\underset{\epsilon\rightarrow 0}{F_{-}lim}\leavevmode\nobreak\
\bigg{(}\frac{E\\{X(\tau+\epsilon)X(s)-X(\tau)X(s)\\}}{\epsilon}\bigg{)}$
$\displaystyle=\underset{\epsilon\rightarrow 0}{F_{-}lim}\leavevmode\nobreak\
[R(\tau+\epsilon,s)-R(\tau,s)]$ $\displaystyle=D_{F,\tau}^{\alpha}R(\tau,s).$
(68)

Likewise, one can establish that

$E\\{X(\tau)D_{F,s}^{\alpha}X(s)\\}=D_{F,s}^{\alpha}R(\tau,s).$ (69)

Both Eqs. (4.7) and (69) exist and are finite on $F\times F$. Finally, we can
write

$\displaystyle
E\\{D_{F,\tau}^{\alpha}X(\tau)D_{F,s}^{\alpha}X(s)\\}=E\bigg{\\{}\underset{\epsilon,\epsilon^{\prime}\rightarrow
0}{f.l.i.m}\bigg{[}\frac{X(\tau+\epsilon)-X(\tau)}{\epsilon}\bigg{]}\bigg{[}\frac{X(s+\epsilon^{\prime})-X(s)}{\epsilon}\bigg{]}\bigg{\\}}$
$\displaystyle=\underset{\epsilon,\epsilon^{\prime}\rightarrow
0}{F_{-}lim}\leavevmode\nobreak\ \frac{1}{\epsilon}\leavevmode\nobreak\
E\bigg{\\{}\bigg{[}\frac{X(\tau+\epsilon)X(\tau+\epsilon^{\prime})-X(\tau+\epsilon)X(s)}{\epsilon^{\prime}}\bigg{]}-\bigg{[}\frac{X(\tau)X(\tau+\epsilon^{\prime})-X(\tau)X(s)}{\epsilon^{\prime}}\bigg{]}\bigg{\\}}$
$\displaystyle=\underset{\epsilon\rightarrow 0}{F_{-}lim}\leavevmode\nobreak\
\frac{1}{\epsilon}\leavevmode\nobreak\ \underset{\epsilon^{\prime}\rightarrow
0}{F_{-}lim}\leavevmode\nobreak\
\bigg{\\{}\frac{R(\tau+\epsilon,s+\epsilon^{\prime})-R(\tau+\epsilon,s)}{\epsilon^{\prime}}-\frac{R(\tau,s+\epsilon^{\prime})-R(\tau,s)}{\epsilon^{\prime}}\bigg{\\}}$
$\displaystyle=\underset{\epsilon\rightarrow 0}{F_{-}lim}\leavevmode\nobreak\
\frac{1}{\epsilon}\leavevmode\nobreak\
[D_{F,s}^{\alpha}R(\tau+\epsilon,s)-D_{F,s}^{\alpha}R(\tau,s)]=D_{F,\tau}^{\alpha}D_{F,s}^{\alpha}R(\tau,s),$
(70)

which exists and is finite on $F\times F$ and completes proof.

###### Definition 4.13

A second f.s.p $X(\tau)$ is f.m.s analytic on $F$, if it can be expanded in
the f.m.s. convergent Taylor series as

$X(\tau)=\sum_{n=0}^{\infty}\frac{(J(\tau)-J(\tau_{0}))^{n}}{n!}D_{F,\tau}^{n\alpha}X(\tau)\bigg{|}_{\tau=\tau_{0}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \tau,\tau_{0}\in F.$ (71)

###### Definition 4.14

Let $P_{[a,b]}$ be a finite subdivision which is given in Definition 2.10,
$X(\tau),\leavevmode\nobreak\ \tau\in F$ be a second order f.s.p. on
$[a,b]\subset F$, and $f(\tau,u)$ be a function on $\tau\in[a,b]$. For every
$u:F\rightarrow\mathbb{R}$ function, we form the random variable as

$Y_{n}(u)=\sum_{i=1}^{n}f(\tau_{i}^{\prime},u)X(\tau_{i}^{\prime})(J(\tau_{i})-J(\tau_{i-1}))$
(72)

Since $L_{2}^{\alpha}$-space is linear, so we have $Y_{n}(u)\in
L_{2}^{\alpha}$.

###### Definition 4.15

If for every $u$

$\underset{n\rightarrow\infty\Delta_{n}\rightarrow 0}{f.l.i.m}Y_{n}(u)=Y(u),$
(73)

exists for some sequence of subdivision $P$ defined in Definition 2.10, then
the $Y(u)$ is called the definite fractal mean square integral (f.m.s.i) of
$f(\tau,u)X(\tau)$ over the interval $[a,b]$ or mean square
$F^{\alpha}$-integral and it is denoted by

$Y(u)=\int_{C(a,b)}f(\tau,u)X(\tau)d_{F}^{\alpha}\tau.$ (74)

###### Theorem 4.8

The f.s.p $Y(u)$ defined by Eq.(74) exists if and only if the fractal double
integral

$\int_{C(a,b)}\int_{C(a,b)}f(\tau,u)f(s,u)R(\tau,s)d_{F}^{\alpha}\tau
d_{F}^{\alpha}s$ (75)

exists and is finite.

###### Proof 4.8

This theorem is an immediate result of the convergence in fractal mean square
Theorem 4.2. Namely, if we choose

$X_{n}(\tau)=\sum_{i=1}^{n}f(\tau_{i}^{\prime},\tau)X(\tau_{i}^{\prime})(J(\tau_{i})-J(\tau_{i-1})),\leavevmode\nobreak\
\leavevmode\nobreak\ \tau=u,\leavevmode\nobreak\ \leavevmode\nobreak\
n_{0}=\infty.$ (76)

###### Definition 4.16

An improper f.m.s integral is defined by

$\int_{C(a,\infty)}f(\tau,u)X(\tau)d_{F}^{\alpha}\tau=\underset{b\rightarrow\infty}{f.l.i.m}\int_{C(a,b)}f(\tau,u)X(\tau)d_{F}^{\alpha}\tau.$
(77)

It is easy to see that it exists if, and only if, the improper fractal double
integral

$\int_{C(a,\infty)}\int_{C(a,\infty)}f(\tau,u)f(s,u)R(\tau,s)d_{F}^{\alpha}\tau
d_{F}^{\alpha}s=\underset{b\rightarrow\infty}{F_{-}lim}\int_{C(a,b)}\int_{C(a,b)}f(\tau,u)f(s,u)R(\tau,s)d_{F}^{\alpha}\tau
d_{F}^{\alpha}s,$ (78)

exists and is finite.

###### Remark 2

The mean square integral of f.s.p $X(\tau),\leavevmode\nobreak\ \tau\in F$ is
defined by using Eq.(74) and setting $f(\tau,u)=1$ as follows:

$\displaystyle Z(\tau)$
$\displaystyle=\int_{C(t_{0},\tau)}X(\theta)d_{F}^{\alpha}\theta$ (79)
$\displaystyle=\underset{\Delta^{\alpha}J(\theta_{i}),\Delta^{\alpha}J(\theta_{k})\rightarrow
0}{F_{-}lim}\leavevmode\nobreak\
E\bigg{\\{}\bigg{[}\sum_{i}X(\theta_{i})\Delta^{\alpha}J(\theta_{i})-\sum_{k}X(\theta_{k})\Delta^{\alpha}J(\theta_{k})\bigg{]}^{2}\bigg{\\}},$
(80)

where $\theta_{0}<\theta_{1}<...<\theta$ , $\theta_{i}=w(t_{i})$,
$\Delta^{\alpha}J(\theta_{k})=J(\theta_{k})-J(\theta_{k-1})$ and
$\Delta^{\alpha}J(\theta_{i})=J(\theta_{i})-J(\theta_{i-1})$.

###### Theorem 4.9

If $X(\tau)$ is f.m.s continuous on $[a,\tau]\subset F$, then

$Y(\tau)=\int_{C(a,\tau)}X(s)d_{F}^{\alpha}s$ (81)

is f.m.s continuous on $[a,\tau]$. It is also f.m.s differentiable on
$[a,\tau]$ with

$D_{F,\tau}^{\alpha}Y(\tau)=X(\tau).$ (82)

###### Proof 4.9

We only prove the second part on $F^{\alpha}$-differentiability, since the
first part is evident using Theorem 4.6. Let us consider

$\displaystyle\bigg{\lVert}\frac{1}{\epsilon}\bigg{[}\int_{C(a,\tau+\epsilon)}X(s)d_{F}^{\alpha}s-\int_{C(a,\tau)}X(s)d_{F}^{\alpha}s\bigg{]}-X(\tau)\bigg{\rVert}$
$\displaystyle=\bigg{\lVert}\frac{1}{\epsilon}\int_{C(a,\tau+\epsilon)}[X(s)-X(\tau)]d_{F}^{\alpha}s\bigg{\rVert}$
$\displaystyle\leq|\frac{1}{\epsilon}|\int_{C(a,\tau+\epsilon)}\bigg{\lVert}[X(s)-X(\tau)]\bigg{\rVert}d_{F}^{\alpha}s$
$\displaystyle\leq\max_{s\in[\tau,\tau+\epsilon]}\bigg{\lVert}X(s)-X(\tau)\bigg{\rVert}$
(83)

as $\epsilon\rightarrow 0$ from the hypothesis, we arrive at the result.

###### Corollary 4.9.1

If $X(\tau)$ is f.m.s integrable on $F$ and $f(\tau,s)$ if $F$-continuous on
$F\times F$ with a finite first partial derivative
$D_{F,\tau}^{\alpha}f(\tau,s)$, then the f.m.s derivative of

$Y(\tau)=\int_{C(a,\tau)}f(\tau,s)X(s)d_{F}^{\alpha}s$ (84)

exists at all $\tau\in F$, and

$D_{F,\tau}^{\alpha}Y(\tau)=\int_{C(a,\tau)}D_{F,\tau}^{\alpha}f(\tau,s)X(s)d_{F}^{\alpha}s+f(\tau,\tau)X(\tau).$
(85)

The proof is obvious. This result may called the fractal mean square
counterpart of the Leibniz rule in ordinary calculus.

###### Theorem 4.10

If $X(\tau)$ be m.s $F^{\alpha}$-differentiable on $F$, and let
$f:F\rightarrow\mathbb{R}$ be $F$-continuous on $F\times F$, whose partial
derivative $D_{F,s}^{\alpha}f(\tau,s)$ exists. If

$Y(\tau)=\int_{C(a,\tau)}f(\tau,s)D_{F,s}^{\alpha}X(s)d_{F}^{\alpha}s,$ (86)

then

$Y(\tau)=f(\tau,s)X(s)\bigg{|}_{a}^{\tau}-\int_{C(a,\tau)}D_{F,s}^{\alpha}f(\tau,s)X(s)d_{F}^{\alpha}s.$
(87)

The proof is straightforward. This property is fractal mean square counter
part of fundamental theorem of ordinary calculus and may called the
fundamental theorem of fractal mean square calculus.

###### Example 3

Let us consider stochastic second order mean square on fractal curve as

$(D_{F}^{\alpha})^{2}X(\tau)+A^{2}X(\tau)=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
X(\tau)|_{\tau=0}=X_{0},\leavevmode\nobreak\ \leavevmode\nobreak\
\textmd{and}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
D_{F}^{\alpha}X(\tau)|_{\tau=0}=X_{1}.$ (88)

where $A^{2}$ is Beta random variable $Beta(\mu,\nu)$, on fractal curve,
namely,

$E[A^{2}]=\frac{\mu}{\mu+\nu},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
Var[A^{2}]=\frac{\mu\nu}{(\mu+\nu)^{2}(\mu+\nu+1)}$ (89)

and it is independent from $X_{0}$ and $X_{1}$. To find solution of Eq.(88),
we use Frobenius like-method and consider an infinite series solution as

$X(\tau)=\sum_{m=0}X_{m}J(\tau)^{m},$ (90)

where $X_{m}$ are random variables on fractal curves. The f.m.s derivative
from both sides of Eq.(90) and a computation gives

$(D_{F}^{\alpha})^{2}X(\tau)=\sum_{m=0}(m+2)(m+1)X_{m+2}J(\tau)^{m}.$ (91)

By replacing Eq.(90) and Eq.(91) into Eq.(88) we obtain

$\sum_{m=0}(m+2)(m+1)X_{m+2}+A^{2}X_{m}J(\tau)^{m}=0$ (92)

It follows that

$X_{m+2}=-\frac{A^{2}X_{m}}{(m+2)(m+1)}.$ (93)

By some manipulations, and using initial conditions, we get the solution as

$X(\tau)=X_{0}\cos(AJ(\tau))+\frac{X_{1}}{A}\sin(AJ(\tau))$ (94)

where

$\displaystyle\cos(AJ(\tau))$
$\displaystyle=\sum_{m=0}\frac{(-1)^{m}A^{2m}}{(2m)!}J(\tau)^{2m},$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\sin(AJ(\tau))$
$\displaystyle=\sum_{m=0}\frac{(-1)^{m}A^{2m+1}}{(2m+1)!}J(\tau)^{2m+1}$ (95)

The truncated Taylor series of solution Eq.(94) is

$X_{N}(\tau)=X_{0}\sum_{m=0}^{N}\frac{(-1)^{m}A^{2m}}{(2m)!}J(\tau)^{2m}+X_{1}\sum_{m=0}^{N}\frac{(-1)^{m}A^{2m}}{(2m+1)!}J(\tau)^{2m+1}$
(96)

The mean of $X_{N}(\tau)$ by Definition 3.4 is

$E[X_{N}(\tau)]=E[X_{0}]\sum_{m=0}^{N}\frac{(-1)^{m}E[A^{2m}]}{(2m)!}J(\tau)^{2m}+E[X_{1}]\sum_{m=0}^{N}\frac{(-1)^{m}E[A^{2m}]}{(2m+1)!}J(\tau)^{2m+1}.$
(97)

To derive variance of $X(\tau)$, let us first calculate

$\displaystyle E[X_{N}(\tau)^{2}]$
$\displaystyle=\sum_{m=0}^{N}\bigg{(}\frac{E[X_{0}]^{2}E[(A^{2})^{2m}]}{((2m)!)^{2}}J(\tau)^{4m}+\frac{E[X_{1}^{2}]E[(A^{2})^{2m}]}{((2m+1)!)^{2}}\bigg{)}J(\tau)^{4m+2}$
$\displaystyle+2E[X_{0}X_{1}]\sum_{n=0}^{N}\sum_{m=0}^{N}\frac{(-1)^{n+m}E[(A^{2})^{n+m}]}{(2n)!(2m+1)!}J(\tau)^{2(n+m)+1}$
(98)

By approximating Eqs.(97) and (4), we arrive at

$\displaystyle E[X_{N}(\tau)]$
$\displaystyle=E[X_{0}]-\frac{E[A^{2}]}{2}J(\tau)^{2}+E[X_{1}]J(\tau)-\frac{E[A^{2}]}{3!}J(\tau)^{3}+\frac{E[A^{4}]}{4!}J(\tau)^{4}+\frac{E[A^{4}]}{5!}J(\tau)^{5}+...$
$\displaystyle E[X_{N}(\tau)^{2}]$
$\displaystyle=E[X_{0}^{2}]+2E[X_{0}X_{1}]J(\tau)+E[X_{1}^{2}]J(\tau)^{2}+...$
(99)

If we choose $E[X_{0}]=1,\leavevmode\nobreak\
E[A^{2}]=2/3,\leavevmode\nobreak\ E[X_{1}]=1,\leavevmode\nobreak\
E[X_{0}^{2}],\leavevmode\nobreak\ E[X_{1}^{2}]=1,\leavevmode\nobreak\
E[X_{0}X_{1}]=1$, we have

$\displaystyle E[X_{N}(\tau)]$
$\displaystyle=1+J(\tau)-\frac{1}{3}J(\tau)^{2}-\frac{1}{9}J(\tau)^{3}+...$
$\displaystyle E[X_{N}(\tau)^{2}]$ $\displaystyle=1+2J(\tau)+J(\tau)^{2}+...$
(100)

Figure 2: Graph of Eq. (97) choosing $E[X_{1}]=0,E[X_{0}]=1,E[A^{2}]=4$

In Figure 2, we have plotted Eq. (97).
Acknowledgment  Cristina Serpa acknowledges partial funding by national funds
through FCT - Foundation for Science and Technology, project reference:
UIDB/04561/2020.

##  5 Conclusion

In this work, we have generalized random variables and processed them on
fractal curves by defining mean, variance, and correlation functions. The
second random variable, the mean square convergent, fundamental fractal in
probability, converges in probability, mean square $F^{\alpha}$-continuous,
mean square $F^{\alpha}$-derivative, and mean square $F^{\alpha}$-integral are
defined to formulate a new framework on fractal curves. This framework is
based on new mathematical models for science and physics to apply processes
with fractal structure.

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11institutetext:  Departament de Física Quàntica i Astrofísica, Institut de
Ciències del Cosmos, Universitat de Barcelona, Martí i Franquès 1, E-08028
Barcelona, Spain
11email<EMAIL_ADDRESS>22institutetext: ICREA, Pg. Lluís Companys 23,
E08010 Barcelona, Spain 33institutetext: TAPIR, California Institute of
Technology, Pasadena, CA 91125, USA

# Dynamical formation of Gaia BH3 in the progenitor globular cluster of the
ED-2 stream

Daniel Marín Pina 11 Sara Rastello 11 Mark Gieles 1122 Kyle Kremer 33 Laura
Fitzgerald 11 Bruno Rando 11

(Received XX; accepted XX)

###### Abstract

Context.  The star-black hole (S-BH) binary discovered by the Gaia
Collaboration – Gaia BH3 – is chemically and kinematically associated with the
metal-poor ED-2 stream in the Milky Way halo.

Aims.  We explore the possibility that Gaia BH3 was assembled dynamically in
the progenitor globular cluster (GC) of the ED-2 stream.

Methods.  We use a public suite of star-by-star dynamical Monte Carlo models
by Kremer et al. (2020) to identify S-BH binaries in GCs with different
initial masses and (half-mass) radii.

Results.  We show that a likely progenitor of the ED-2 stream was a relatively
low-mass ($\lesssim 10^{5}\,{\rm M}_{\rm\odot}$) GC with an initial half-mass
radius of $2-4\,$pc. Such a GC can dynamically retain a large fraction of its
BH population and dissolve on the orbit of ED-2. From the suite of models we
find that GCs produce $\sim 3-30$ S-BH binaries, approximately independent of
initial GC mass and inversely correlated with initial cluster radius. Scaling
the results to the Milky Way GC population, we find that $\sim 75\%$ of the
S-BH binaries formed in GCs are ejected from their host GC in the early phases
of evolution ($\lesssim 1\,$Gyr); these are expected to no longer be close to
the stream. The $\sim 25\%$ of S-BH binaries retained until dissolution are
expected to form part of streams, such that for an initial mass of the
progenitor of ED-2 of a few $10^{4}\,{\rm M}_{\rm\odot}$, we expect $\sim 2-3$
S-BH to end up in the stream. GC models with metallicities similar to Gaia BH3
($\lesssim 1\%$ solar) include S-BH binaries with similar BH masses ($\gtrsim
30\,{\rm M}_{\rm\odot}$), orbital periods, and eccentricities.

Conclusions.  We predict the Galactic halo contains of order $10^{5}$ S-BH
binaries that formed dynamically in GCs, a fraction of which may readily be
detected in Gaia DR4. The detection of these sources provides valuable tests
of BH dynamics in clusters and the possible role in formation of gravitational
wave sources.

###### Key Words.:

Stars: black holes – Stars: Population II - Galaxy: kinematics and dynamics –
Galaxy: halo-globular clusters

##  1 Introduction

The discovery of a star-black hole (S-BH) binary by the Gaia collaboration
(Gaia Collaboration, 2024) – Gaia BH3 – constitutes another glimpse into the
population of dormant BHs. Gaia BH3 is composed of a $33\,\mathrm{{\rm
M}_{\rm\odot}}$ BH with a $0.76\,\mathrm{{\rm M}_{\rm\odot}}$ star. The star
is a metal-poor giant, with metallicity ${\rm[Fe/H]}\simeq-2.6$. The binary
has an eccentricity of $e\simeq 0.73$, and a wide orbit with period $P\simeq
4200\,\mathrm{d}$. Gaia BH3 is on a halo orbit, similar to the debris of the
Sequoia accretion event (Gaia Collaboration, 2024).

The previously discovered Gaia BH1 (El-Badry et al., 2023b) is a binary with a
Sun-like star orbiting a BH with a mass $m_{\rm\mathrm{BH}}\simeq
10\,\mathrm{{\rm M}_{\rm\odot}}$, an eccentricity $e=0.45$, and a period of
$P\simeq 186\,\mathrm{d}$. Gaia BH2 (El-Badry et al., 2023a) is a binary with
a red giant orbiting a $m_{\rm\mathrm{BH}}\simeq 9\,\mathrm{{\rm
M}_{\rm\odot}}$ BH, with an eccentricity $e\simeq 0.5$, and a relatively long
period of $P\simeq 1280\,\mathrm{d}$. Their metallicities
(${\rm[Fe/H]}\simeq-0.2$) and their inferred orbits in the Galactic plane
point towards a formation in a young stellar population, possibly an open
cluster (Rastello et al., 2023; Di Carlo et al., 2024; Tanikawa et al., 2024;
Arca Sedda et al., 2024) or an isolated stellar binary (Kotko et al., 2024).

The association of Gaia BH3 with the stellar halo calls for another formation
scenario. Under the assumption that the unseen companion is a single BH111The
available data can not rule out that the unseen object is a binary BH (BBH).,
Gaia Collaboration (2024) argue that formation of Gaia BH3 in isolation is
unlikely. The star does not present strong chemical peculiarities, indicating
a lack of pollution by the BH progenitor in its evolved stage. This makes a
dynamical formation scenario, where the system is assembled after the BH
progenitor has exploded as a supernova, more plausible. Its orbit in the
Galactic halo and low metallicity suggest that Gaia BH3 formed in a globular
cluster (GC).

Balbinot et al. (2024) show that Gaia BH3 is both chemically and kinematically
associated with the ED-2 stream (Dodd et al., 2023). The progenitor of the
ED-2 stream was most likely a GC, given its low velocity dispersion (Balbinot
et al., 2023) and the absence of an ${\rm[Fe/H]}$ spread
($<0.04\,\mathrm{dex}$, Balbinot et al., 2024). These authors also find a
negligible spread in Na and Al, suggesting an initial cluster mass $\lesssim
4\times 10^{4}\,{\rm M}_{\rm\odot}$. The maximum initial mass of a cluster
with BHs that can dissolve in 13 Gyr on the orbit of ED-2 is $\sim 9\times
10^{4}\,{\rm M}_{\rm\odot}$ (Gieles & Gnedin, 2023).

So far, only two (possibly three) detached stellar-mass BH binary candidates
have been found in GCs (in NGC 3201, Giesers et al., 2018, 2019)222Additional
BH candidates are observed in GCs as accreting sources via their X-ray/radio
emission (e.g., Maccarone et al., 2007; Strader et al., 2012; Miller-Jones et
al., 2015).. Only line-of-sight velocities are available for these binaries,
providing a minimum BH mass of $\gtrsim 5\,{\rm M}_{\rm\odot}$. Kremer et al.
(2018b) showed that these binaries were likely formed and shaped via dynamical
exchange interactions. Evidence for massive BHs ($\gtrsim 30\,{\rm
M}_{\rm\odot}$) in a (dissolved) GC is critical to understand the contribution
of dynamically-produced BBHs to the growing sample of gravitational wave
sources (e.g., Portegies Zwart & McMillan, 2000; Rodriguez et al., 2016; Askar
et al., 2017; Antonini et al., 2023).

In this Letter we show that Gaia BH3 could have formed dynamically in the
progenitor GC of the ED-2 stream. In Sec. 2 we describe the dynamical models
of GCs that we use to search for S-BH binaries. In Sect. 3 we present the
properties of S-BH in the models and the implications are discussed in Sect.
4. A discussion and conclusions are presented in Sect. 5 and 6, respectively.

##  2 Methods

We use the suite of GC simulations run with the Cluster Monte Carlo (CMC)
code. CMC is a Hénon-type Monte Carlo code for the long term evolution of
collisional stellar systems (Rodriguez et al., 2022, and references therein).
The integration is done on a star-by-star basis, following gravitational
interactions, the internal evolution of stars and binary stars, strong few-
body interactions and dynamical ejections (Weatherford et al., 2023). Further
details on the code can be found in Rodriguez et al. (2022).

For this work, we use the set of 148 CMC models presented in Kremer et al.
(2020). This catalog is meant to sample the relevant parameter space of local-
universe GCs varying the following initial conditions: the number of stars
$N$, virial radius $r_{\rm v}$333For the initial density profile of these
models $r_{\rm v}\simeq 1.25r_{\rm h}$, where $r_{\rm h}$ is the half-mass
radius., metallicity $Z$, and Galactocentric distance $R_{\rm G}$. The choice
of values for the above parameters is $N\in[2,4,8,16]\times 10^{5}$, $r_{\rm
v}\in[0.5,1,2,4]\,\mathrm{pc}$, $Z\in[0.0002,0.002,0.02]$, and $R_{\rm
G}\in[2,8,20]\,\mathrm{kpc}$, which account for 144 models, plus four extra
models with $N=3.2\times 10^{6}$, $R_{\rm G}=20\,\mathrm{kpc}$,
$Z\in[0.0002,0.02]$, and $r_{\rm v}\in[1,2]\,\mathrm{pc}$.

We first look at the raw results of all S-BH occurring in the database in
Sect. 3. We also make the CMC database representative for the Milky Way GC
system, by applying weights to the result of GCs with different $N$ by a
factor $N^{-1}$, such that the logarithmically spaced $N$ values represent an
initial GC mass function of the form $M_{\rm 0}^{-2}$. We then only consider
the low metallicity models ($Z=2\times 10^{-4}$ and $Z=2\times 10^{-3}$), with
relative weights 4:1, representative for the Milky Way GC population (e.g.,
Harris, 1996).

We only consider hard binaries (Heggie, 1975), defined as those whose absolute
value of the binding energy is greater than the average kinetic energy of
particles in the GC core. This is done to exclude loosely bound pairs that are
constantly being assembled and destroyed.

As clusters evolve they lose mass and stars due to internal two-body
relaxation and interactions with the Galactic tidal field. Therefore, they
dissolve over a period which depends on the cluster relaxation time scale and
their position within the Galactic potential. In our models, clusters are
assumed to be completely dissolved once
$t_{\rm\rm{rlx}}>M_{\rm\rm{SC}}/\dot{M}_{\rm\rm{SC}}$ where $t_{\rm\rm{rlx}}$
and $M_{\rm\rm{SC}}$ indicate the relaxation time and the total mass of the
clusters, respectively (Kremer et al., 2020).

##  3 Origin of Gaia BH3-like binaries

Figure 1: Probability distribution functions for the masses of the BH and star
($m_{\rm\mathrm{BH}}$ and $m_{\rm\star}$, respectively) and orbital parameters
(period $P$ and eccentricity $e$) for all the detached S-BH binaries in the
cluster simulations of Kremer et al. (2020). For ejected binaries, we show
their parameters at the time of ejection. The contribution of each binary to
the histogram is weighted as explained in Sec. 2. In red (continuous line for
1D histograms, star symbol for 2D histograms), the values for Gaia BH3. In
dashed red line, the range for the analysis of Gaia BH3-like binaries in Sec.
3. The green lines in the 1D histogram include only the S-BH binaries that are
not ejected from their parent cluster.

In this section, we explore the formation of S-BH binaries in the CMC
catalogue that are the most similar to Gaia BH3. From the 1837 S-BH binaries
in the catalogue, we filter out 565 binaries that are accreting. The remaining
1435 detached binaries are shown in Fig. 1, where the contribution of each of
the models is weighted as explained in Sec. 2. From these binaries, 78% are
ejected and the rest remain in the cluster, either until it dissolves or the
simulation ends. We assume that S-BH binaries that survive until 13 Gyr in
non-dissolved clusters will remain in the cluster until dissolution, which is
justified by the decreasing velocity dispersion of the cluster, making it less
likely to break up the binary in its final stages.

From the sample of detached binaries, we select those most similar to Gaia
BH3, which we define as those whose parameters all fall within the red dashed
lines in the subplots of Fig. 1. The red lines span the following ranges:
$m_{\rm\mathrm{BH}}\in[25,45]\,\mathrm{{\rm M}_{\rm\odot}}$,
$m_{\rm\star}\in[0.2,1]\,\mathrm{{\rm M}_{\rm\odot}}$,
$P\in[400,40000]\,\mathrm{d}$, and $e\in[0.63,0.83]\,\mathrm{{\rm
M}_{\rm\odot}}$. These ranges are much larger than the typical uncertainties
on the observationally inferred parameters of Gaia BH3 – which are of order
1-2% – and are the chosen to (i) cover a roughly similar fraction of the
ranges of the parameters found in the models and (ii) end up with a reasonable
number of S-BH.

This criteria selects nine binaries. Next we refine this sample to match the
model GC properties with what we know of the ED-2 stream, but before we do
that, it is interesting to note that Gaia BH3-like binaries exist and that all
of them have a dynamical origin.

From the nine resembling S-BH binaries, we narrow down our sample by selecting
only those binaries that are ejected at very late times or remain in the
cluster until dissolution. We note that there are no escapers (of these nine)
after the first $\sim 500\,\mathrm{Myr}$. This filtering is done because early
escapers drift away from the cluster. For the orbit of ED-2, escapers have an
average drift velocity with respect to the progenitor cluster of $\sim
1\,\mathrm{km/s}$ (e.g., Küpper et al., 2010; Roberts et al., 2024), such that
escapers in the first few Gyr of the cluster’s life would no longer be
associated with the stream, which contains stars that were lost in the final
$\sim 3\,$ Gyr of the cluster’s life (Erkal et al., 2017; Gieles et al.,
2021).

Two out of the nine binaries in our sample pass this further criteria. We
refer to these two candidate Gaia BH3-like binaries as cSBH1 and cSBH2. For
cSBH1 (cSBH2), the masses of the components are $m_{\rm\mathrm{BH}}=29.9\
(40.5)\,\mathrm{{\rm M}_{\rm\odot}}$ and $m_{\rm\star}=0.21\
(0.44)\,\mathrm{{\rm M}_{\rm\odot}}$. The semimajor axis is $a=69\
(57)\,\mathrm{AU}$ – which implies a period of $P=37760\ (24791)\,\mathrm{d}$
– and the eccentricity is $e=0.72\ (0.80)$. Both binaries originate from the
cluster model with initial parameters $N=2\times 10^{5}$, $r_{\rm
v}=4\,\mathrm{pc}$. The initial mass for this model is $M_{\rm 0}=1.2\times
10^{5}\,\mathrm{{\rm M}_{\rm\odot}}$, which is the lowest mass GC in the CMC
catalogue. This is within a factor of a few of the mass estimate provided by
Balbinot et al. (2024) derived by the light elements spreads. Combined with
the upper limits from the abundances and dissolution time (Sect. 1), we
therefore conclude that the models prefer an initial GC mass $\lesssim
10^{5}\,{\rm M}_{\rm\odot}$.

Both of these binaries form dynamically roughly $100\,\mathrm{Myr}$ after GC
formation, corresponding to the moment of core collapse of the BH subsystem.
The classic picture is that a binary forms in a three-body interaction
(Heggie, 1975; Atallah et al., 2024). We note that the outcome of core
collapse is the production of a dynamically active BBH, likely involving $\sim
5-10$ particles (Tanikawa et al., 2012). The S-BH appears to form in an
intermediate step, involving two stars and a BH, and it seems to be a by-
product of BBH formation. Developing a full understanding of the relation
between the formation of the S-BH and the BBH is beyond the scope of this
work. We here simply note it as a curiosity worth looking into in future work,
and in Sect. 4.3 we discuss the implications.

In the case of cSBH1, it is directly formed in a three-body interaction,
whereas cSBH2 is the exchange of two three-body binaries. Both S-BH systems
remain in the cluster until it dissolves. After formation, these binaries
undergo 18 and 7 strong interactions respectively, implying that even if
formed from binary evolution, dynamical interactions in the ED-2 progenitor
need to be included to understand their final properties.

If we do not exclude early escapers from our sample (i.e., if we consider the
nine non-accreting binaries whose parameters all fall within the red boxes in
Fig. 1), we find a similar interaction history. Except for cSBH1, all nine
S-BH binaries are formed via three-body encounters. The ejected binaries
undergo multiple strong interactions (ranging from one to twelve) until they
escape the cluster. From these, all but one are ejected following interactions
with BBHs. A high rate of binary-binary interactions is expected in clusters
with relatively low number of BHs (Marín Pina & Gieles, 2024).

##  4 Implications

###  4.1 Formation efficiency

To make the result more quantatitive, we now extend our analysis to compute
how many S-BH binaries are expected to be ejected in the ED-2 stream. We begin
by defining the formation efficiency, $\eta$, as the number of ejected S-BH
binaries in our model per unit of initial cluster mass ($M_{\rm 0}$). In the
CMC models, we find a clear dependence of $\eta$ on the initial cluster mass
and virial radius, shown in Fig. 2. The formation efficiency is higher in
denser and less massive clusters; however, this trend does not extend to lower
masses ($\lesssim 10^{4}\,\mathrm{{\rm M}_{\rm\odot}}$). There, it levels off,
as there is a hard limit where the number of ejected S-BH binaries, $N_{\rm
S-BH}$, equals the initial number of BHs in the cluster, $N_{\rm\mathrm{BH}}$.
For more massive clusters, formation of S-BH binaries is relatively inhibited
by pronounced dynamical heating from their large central BH populations
(Kremer et al., 2018a).

In the CMC regime, the formation efficiency can be approximated by the
following power law:

$\displaystyle\eta=1.6\times 10^{-4}\,\,\mathrm{{\rm
M}_{\rm\odot}^{-1}}\left(\frac{M_{\rm 0}}{10^{5}\,\mathrm{{\rm
M}_{\rm\odot}}}\right)^{-1.2}\left(\frac{r_{\rm
v}}{1\,\mathrm{pc}}\right)^{-0.42}.$ (1)

Extrapolating to the mass estimate for the ED-2 progenitor ($5\times
10^{4}\,{\rm M}_{\rm\odot}$, Sect. 1) and $r_{\rm v}=2-4\,$pc, implies that
the ED-2 stream contains about $\sim 9-13$ S-BH binaries, of which $\sim 2-3$
are retained in the cluster. We conclude that dynamical formation of Gaia BH3
is a viable explanation for its origin.

Figure 2: Formation efficiency of S-BH in all CMC models with $Z=2\times
10^{-4}$ (bullets) and all S-BH formed in the $N$-body results of Rastello et
al. (2023) for Solar metallicity (square). Dashed lines show the maximum
$\eta$, corresponding to all BHs in the cluster being in an S-BH and the
minimum $\eta$ corresponding to clusters with one BH. The dotted lines
correspond to constant numbers of S-BH.

###  4.2 Number of S-BH in the Galactic halo

With the relation $\eta(M_{\rm 0},r_{\rm v})$ from equation (1) we can
estimate the total number of S-BH binaries produced dynamically in the
complete population of Galactic GCs. We write the initial GC mass function
(GCMF) as $\psi_{\rm 0}(M_{\rm 0})=AM_{\rm 0}^{-2}$. We then assume that all
clusters lose 40% of their mass by stellar evolution and an amount
$\Delta\simeq 2\times 10^{5}\,{\rm M}_{\rm\odot}$ by evaporation. This amount
is required to get the turn-over mass in the GCMF at $\sim 2\times
10^{5}\,{\rm M}_{\rm\odot}$ if mass is lost at a constant rate (Jordán et al.,
2007). The evolved GCMF can then be written as

$\psi(M)=\mu_{\rm\rm sev}A(M+\Delta)^{-2},$ (2)

where $\mu_{\rm\rm sev}\simeq 0.6$. The normalisation constant $A$ is found
from the total number of Milky Way GCs (157) and integrating $\psi(M)$ from
$10^{2}\,{\rm M}_{\rm\odot}$ (roughly the lowest mass Milky Way GC) to
$10^{6}\,{\rm M}_{\rm\odot}$.

The total number of S-BH binaries produced by all Milky Way GCs, including the
dissolved ones, is then

$\displaystyle N_{\rm S-BH}$ $\displaystyle=\int_{\rm 10^{4}\,{\rm
M}_{\rm\odot}}^{10^{6}\,{\rm M}_{\rm\odot}}\eta(M_{\rm 0},r_{\rm v})M_{\rm
0}\psi_{\rm 0}(M_{\rm 0}){\rm d}M_{\rm 0},$ (3) $\displaystyle\simeq 10^{5}.$
(4)

Here we weighed the results for different $r_{\rm v}$ equally. We note that
there are no CMC models in the range $10^{4}\leq M_{\rm 0}/{\rm
M}_{\rm\odot}\lesssim 10^{5}$, so the results depends on an extrapolation
outside of the range of the CMC grid. From the models of lower mass clusters
of Rastello et al. (2023) (Fig. 2), this seems reasonable for $r_{\rm
v}\gtrsim 1\,$pc.

Because most S-BH are produced by low-mass dissolved GCs, the number density
of S-BH in the Galactic halo follows the initial density distribution of GCs
in the Milky Way: $n_{\rm GC}\propto R_{\rm G}^{-4.5}$ between 1 kpc and 100
kpc (e.g. Fall & Zhang, 2001; Gieles & Gnedin, 2023). Scaling this to a total
number of $10^{5}$, we then find that at the solar radius the number density
of S-BH in the halo is $n_{\rm S-BH}\simeq 1\,{\rm kpc}^{-3}$, corresponding
to $\sim 1$ within the distance to Gaia BH3. Within a distance of $1-3$ kpc
from the Sun there are $5-130$ S-BH expected from (dissolved) GCs.

###  4.3 Connecting to gravitational wave sources

As the catalogue of BBH mergers detected via gravitational waves by
LIGO/Virgo/KAGRA (LVK) grows (Abbott et al., 2023), the details of the
astrophysical origin of these sources remain elusive. Dynamical formation
within GCs has emerged as one possible scenario and has been studied at length
using $N$-body cluster simulations including CMC (e.g., Rodriguez et al.,
2016). In GCs, the dynamical formation of S-BH binaries like Gaia BH3 and the
three BH candidates observed in NGC 3201 is an inevitable byproduct of the
same processes that lead to formation of merging BH pairs (e.g., Kremer et
al., 2018b). Thus, detection of sources like Gaia BH3 provide important
constraints on the properties of BHs in clusters, their dynamics, and the role
of GCs in forming GW sources.

In Fig. 3, we show the cumulative distribution of BH masses from our CMC
models for BHs undergoing BH mergers (gray) and BHs found in S-BH binaries
(blue; including both retained and ejected binaries), again adopting the
cluster weighting scheme from Section 2. For reference, we also show
distributions of known stellar-mass BHs with mass measurements: low- and high-
mass X-ray binaries (red curve) taken from (Remillard & McClintock, 2006;
Corral-Santana et al., 2016), BH mergers (black curve) from GWTC-3 (Abbott et
al., 2023), and the three Gaia BH3 binaries444We note that the X-ray binary
and GW histograms shown in Fig. 3 are raw data and do not take into account
detection biases or astrophysical weighing (for discussion, see Fishbach &
Kalogera, 2022).. As shown, GCs naturally produce dynamically-active
populations of BHs with masses consistent with both the LVK sources and Gaia
BH3. Future detections of more S-BH binaries in the Milky Way will provide
further constraints on the potential connection between these populations.

Figure 3: Cumulative distribution of measured BH masses observed as X-ray
binaries (Remillard & McClintock, 2006; Corral-Santana et al., 2016) and as GW
sources (Abbott et al., 2023) compared to distributions of BH masses
identified in our CMC models. Vertical dashed lines show measured masses for
the three Gaia BHs.

##  5 Discussion

###  5.1 Binary fraction

The prediction of the number of S-BH might be a lower limit due to the
relatively low primordial binary fraction in the CMC models (5%). For solar-
type stars the initial binary/multiplicity fraction could be 30-40% (Moe & Di
Stefano, 2017) and for massive stars it is close to 100% (Sana et al., 2012).
A higher binary fraction leads to higher rate of binary-binary interactions
providing an additional pathway to formation of S-BH in exchange interactions
(e.g., González et al., 2021; Tanikawa et al., 2024).

###  5.2 Stellar masses

The stellar masses in the overall population of S-BH in the CMC models peaks
at relatively low masses ($\sim 0.2\,\mathrm{{\rm M}_{\rm\odot}}$), which
makes the majority of these S-BH difficult to detect. Recent analysis of the
stellar mass function Milky Way GCs (Baumgardt et al., 2023; Dickson et al.,
2023) suggests that the low-mass end of the mass function could be flatter
than what is assumed in CMC. This would lead to higher stellar masses in S-BH
binaries. Estimating this effect requires additional dynamical modelling and
is beyond the scope of this work, however see Weatherford et al. (2021) for
preliminary efforts.

##  6 Conclusions

Gaia BH3 belongs to the ED-2 stream, so its kinematics suggests that it was
formed in the ED-2 progenitor GC. In this Letter, we studied the dynamical
formation and ejection of Gaia BH3-like binaries in GCs. We analysed the
public catalogue of star-by-star GC simulations in Kremer et al. (2020), and
compared it to Gaia BH3. We found that the properties of Gaia BH3– masses,
period, eccentricity and lack of chemical pollution – are compatible with it
being assembled dynamically from a previously unbound star and a BH that had
not undergone joint stellar evolution.

Furthermore, we extend our analysis to the whole ED-2 stream in particular and
the Galactic halo in general. We predict that the stream contains of the order
of $\sim 9-13$ dynamically-assembled S-BH binaries, whereas the halo contains
$\sim 10^{5}$.

###### Acknowledgements.

The authors thank Eduardo Balbinot for discussions on ED-2 and the Gaia group
at ICCUB for useful interactions. DMP and MG acknowledge financial support
from the grants PID2021-125485NB-C22, EUR2020-112157, CEX2019-000918-M funded
by MCIN/AEI/10.13039/501100011033 (State Agency for Research of the Spanish
Ministry of Science and Innovation) and SGR-2021-01069 (AGAUR). SR
acknowledges support from the Beatriu de Pinós postdoctoral program under the
Ministry of Research and Universities of the Government of Catalonia (Grant
Reference No. 2021 BP 00213). Support for KK was provided by NASA through the
NASA Hubble Fellowship grant HST-HF2-51510 awarded by the Space Telescope
Science Institute, which is operated by the Association of Universities for
Research in Astronomy, Inc., for NASA, under contract NAS5-26555.

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11institutetext: University of Washington, Seattle
Electronics and Telecommunications Research Institute, Korea

# Observation Centric and Central Distance Recovery on Sports Player Tracking

Hsiang-Wei Huang  Cheng-Yen Yang  Jenq-Neng Hwang
Pyong-Kun Kim  Kwangju Kim  Kyoungoh Lee

###### Abstract

Multi-Object Tracking over humans has improved rapidly with the development of
object detection and re-identification. However, multi-actor tracking over
humans with similar appearance and non-linear movement can still be very
challenging even for the state-of-the-art tracking algorithm. Current motion-
based tracking algorithms often use Kalman Filter to predict the motion of an
object, however, its linear movement assumption can cause failure in tracking
when the target is not moving linearly. And for multi-players tracking over
the sports field, because the players in the same team are usually wearing the
same color of jersey, making re-identification even harder both in the short
term and long term in the tracking process. In this work, we proposed a
motion-based tracking algorithm and three post-processing pipelines for three
sports including basketball, football, and volleyball, we successfully handle
the tracking of the non-linear movement of players on the sports fields.
Experiments result on the testing set of ECCV DeeperAction Challenge SportsMOT
Dataset demonstrate the effectiveness of our method, which achieves a HOTA of
73.968, ranking 3rd place on the 2022 Sportsmot workshop final leaderboard.

###### Keywords:

Multi-object Tracking, Occlusion, ReID, Association

##  1 Introduction

Multi-object tracking is a fundamental task in computer vision, aiming to
associate objects bounding boxes and keep track of all the identities in video
sequences. Most of the multi-object tracking datasets mainly focus on
pedestrians in crowded street scenes (e.g., MOT17/20) [1]. In these scenes,
most of the pedestrians’ movement is slow and linear and thus easy to predict,
besides that, the appearance of each identity is easy to distinguish, making
the re-identification in the tracking process easier. However, there is a lack
of multi-object tracking algorithms that can successfully handle the non-
linear movement of sports players and the challenge of similar appearances
between players on the sports field scenes. For this purpose, we introduced an
observation-centric and central distance recovery tracking algorithm that can
handle the non-linear movement of players on the sports field, and also use
appearance ReID post-processing to deal with the fragment tracklets during
tracking.

##  2 Related Works

###  2.1 Location and Motion Based Object Tracking

Modern object tracking algorithms usually follow the paradigm of tracking by
detection. Several motion-based tracking algorithms adopt the Kalman filter
[2] to formulate the moving trajectories of the target with the detections
from the object detector. However, the linear assumptions of the Kalman filter
can fall short when the target is not moving linearly. Observation-Centric
SORT [3] includes moving direction similarity between detection and tracklets
into the Hungarian Algorithm [4] cost and reforming the trajectories of the
re-identify target to prevent error accumulations in the Kalman filter update
process. These methods show effectiveness when dealing with the non-linear
movements of objects, achieving state-of-the-art tracking performance on
several multi-object tracking datasets.

###  2.2 Appearance Based Object Tracking

With the fast development in the appearance feature extractor, some tracking
algorithms incorporate target appearance during association [5,6] and utilizes
the appearance as a clue for identity recognition. However, most of these
appearance cues-based algorithms often fall short in many cases especially
when targets are occluded, when the scene is very crowded or the targets are
sharing similar appearances.

##  3 Proposed Method

Several challenges need to be tackled in the tracking of sports players.
First, is the non-linear movement of the players on the court. Given the high
intensity of sports like volleyball, basketball, and football, players need to
sprint, jump and change directions in a short time during the game, causing
the movement to be unpredictable. Second, is the heavy occlusion problem
during the sports game. The players will cluster together during the sports
game when some particular situations happened like grabbing the rebound in
basketball or blocking in volleyball. When this happened, it causes detection
to be unreliable due to occlusion, and thus causes the tracking performance to
drop if we do not take care of the recovery of the tracklets with lost
detection carefully. Third, in the sports video clips, a player can go out and
re-enter the camera view again after several seconds, when this happened, we
need to re-identify the player with the correct identity. However, given the
similar appearance between those team players within the same team, it is
difficult to re-identify them correctly by using their appearance. In our
work, we proposed several methods to deal with these three challenges.

###  3.1 Observation-Centric Tracking

The non-linear movement is causing big trouble during the tracking process. To
tackle this problem, we use Observation-Centric SORT (OCSORT) as our main
tracking algorithm. Its Observation-centric Online Smoothing strategy helps
deal with the non-linear movement of the target by rebuilding a virtual
trajectory when a lost target is associated again after a period of being
untracked, thus preventing error accumulation in the Kalman filter. Because
the players on the sports field can change moving direction in a short time,
the Observation-Centric Momentum in OCSORT also helps reduce the tracking
error caused by sudden direction changes. Lastly, after the association stage
is finished, OCSORT performs Observation-Centric Recovery, trying to associate
the last observation of an unassociated track to the detections on the new-
coming time step. This strategy helps to reduce the generation of a new
tracklet, which is usually abnormal given the fixed number of players on the
sports field.

###  3.2 Central Distance Recovery

After the association stage is done in OCSORT, there is still a chance that
observation-centric recovery failed to successfully recover the identity of
the targets and initiate a new tracklet. This is mainly due to observation-
centric recovery being based on using bounding box IoU as the Hungarian
assignment cost, while the unassociated tracklets and the unmatched detections
sometimes do not necessarily share an overlapping region given the fast-moving
speed of sports players. To deal with this, we use the Euclidean distance
between detections and the last observation of unassociated tracklets to
conduct observation-centric recovery again, as this stage is based on bounding
boxes’ central distance, we call this process “central distance recovery”.

###  3.3 Tracklets Association Post-Processing

After the tracking stage is finished, the result usually ended up in a bigger
number of identities than the number of identities in the ground truth. This
is because when a player re-enters the camera view, the tracking algorithm
usually treats the player as a new identity and does not re-identify the
player. To deal with the player re-entry problem, we incorporate different
strategies in three sports scenes according to the total number of sports
players, the size of the sports field, and several other sports
characteristics.

####  3.3.1 Post-Processing in Basketball

The total number of players in a basketball game is 10, we use this as the
main constraint to conduct the post-processing. Due to a large amount of
player’s camera re-entry in basketball compared to other sports, it is
necessary to incorporate person re-identification in the post-processing
stage. After we get the preliminary tracking result, we initiate the first 10
tracklets we get as the ten main players on the court. We keep updating the
tracklets appearance feature using exponential moving averages. Whenever a
player leaves the camera view, we put the player’s tracklet into a candidate
queue. And whenever a new tracklet (player) enters the camera view, we
associate the player to one of the tracklets in the candidate queue that
shares the highest cosine similarity between both.

####  3.3.2 Post-Processing in Football

The search space of candidates in football is much bigger compared to other
sports because of the bigger number of players in a football game and the
lower ratio of players inside the camera view to the total number of players.
Thus it is necessary to incorporate the position as a constraint during the
ReID process. In football tracking, due to the uncertainty in the number of
identities in a video clip, we decided not to use the number of total players
as a constraint, instead, we only try to associate fragment tracklets together
based on their appearance similarity and entry/exit position. There is a total
of three rounds of association based on appearance similarity and location
between tracklets. The prerequisites for two tracklets to be associated
together are first based on their disappear and reappear location. The
threshold for the location distance for two tracklets to be associated is
determined by the number of frames between a tracklet disappearing and
reappearing in the camera view, which means that if a tracklet disappears and
reappears in the camera view in a short amount of time, the distance threshold
for their location will be small given the moving distance should not be far
in such a short time, and vice versa for a longer disappearing time. After
passing the threshold of location distance, we then try to compare the cosine
similarity based on the two tracklets’ appearance. We calculate two tracklets’
average frame-based embedding features distance as their final distance. If
the final distance is smaller than the embedding threshold, we consider these
two tracklets the same identity. We use three rounds of association based on a
different matching threshold of appearance similarity.

####  3.3.3 Post-Processing in Volleyball

Compared to basketball and football, volleyball player usually stays in the
camera view throughout the entire video clip, and even they disappear, they
reappear in a very close location. Due to the above reason, we do not
incorporate appearance in the post-processing of volleyball. We use a similar
strategy to basketball post-processing, we limited the number of tracklets to
12 players, and try to re-associate disappearing players to candidates only
based on the distance of their disappear and reappear location.

####  3.3.4 Interpolation

After the ReID post-processing part is done, we use linear interpolation as
our last step to produce the final tracking results.

##  4 Experiments and Results

###  4.1 Datasets

We use the training sets from SportsMOT for detector and ReID model training.
The training set contains 45 video clips from 3 categories (i.e., basketball,
football, and volleyball), which are collected from Olympic Games, NCAA
Championship, and NBA games on YouTube. Only the search results with 720P
resolution, 25 FPS, and official recordings are downloaded. All of the
selected videos are cut into clips of average 485 frames manually, in which
there is no shot change.

###  4.2 Implementation Details

####  4.2.1 Detector

We use YOLOX [7] as our detector due to its high accuracy and fast inference
speed. For the pretrained weight, we use the COCO pretrained YOLOX-X model
provided by the official GitHub repositories of YOLOX. We train the model with
Sportsmot training set for 80 epochs, following the YOLOX-X default training
process of ByteTrack’s [8] official GitHub repositories. The training duration
takes around 8 hours on 4 tesla V100 GPUs.

####  4.2.2 Observation-Centric SORT

We keep the original configuration of OCSORT, using 0.1 detection confidence
threshold, 0.3 IoU threshold, 0.7 track threshold, and a maximum tracklet age
of 30 frames for all of the sports.

####  4.2.3 Central Distance Recovery

The central distance recovery threshold is different based on the sports type.
We set basketball’s distance threshold as 200, football’s distance threshold
as 80, and volleyball’s distance threshold as 80. The choice of threshold is
based on the evaluation performance on the Sportsmot testing set.

####  4.2.4 Person ReID

We are using OSNet[9] as our backbone network for feature extraction. The
model is trained with Sportsmot training set for 10 epochs, using Adam
optimizer with 0.0003 learning rate.

####  4.2.5 Basketball Post-Processing Setting

In the post-processing of basketball, we limited the number of tracklets to
10, just like the number of players on the court, unless more than 10
detections appear at the same time, we do not initiate new tracklets. The re-
identification of players is based on the cosine similarity of their embedding
features.

####  4.2.6 Football Post-Processing Setting

In football, considering the ratio of players inside and outside the camera,
we use the cosine similarity and also players’ position to conduct re-
identification for camera re-entry players. There are three rounds of the
association stage. The association between two tracklets needs to pass through
a threshold of tracklet position distance before they have a chance to be
associated. The distance threshold is based on their disappear and reappear
time gap, for those tracklets that have a time gap of fewer than 100 frames,
the distance threshold is 100. For the tracklets that share a time gap between
100 to 500 frames, the distance threshold is 250, and for those tracklets that
share a time gap bigger than 500 frames, the distance threshold is set to 400.
In three rounds of association, we try to associate as many as tracklets we
can in a greedy style. For the first round of association, if the cosine
distance of two tracklets is smaller than 0.1, we treat them as the same
identity and conduct association. For the second round, the cosine distance
threshold is 0.2, and for the third round is 0.4.

####  4.2.7 Volleyball Post-Processing Setting

Due to the relatively low number of player camera re-entry cases in volleyball
compared to basketball and football, the search space for re-entry players is
small. So the post-processing of volleyball is simply based on their disappear
and reappear position. For the re-entry player, we select the player in the
re-associate candidate who has the closest distance between the disappearing
position of candidates and reappears position of the new player for re-
identification if their distance is lower than a threshold of 400.

###  4.3 Evaluation Results

The 2022 ECCV DeeperAction Challenge - SportsMOT Track on Multi-actor Tracking
competition is ranked according to the HOTA [10] performance. In contrast to
MOTA [11], HOTA maintains a balance between the accuracy of object detection
and association. The original OCSORT has a performance of 67.107 in HOTA,
after the incorporation of central-distance recovery, the HOTA improves to
71.764. After the ReID post-processing stage, we achieve 73.968 in HOTA,
63.460 in AssA, 86.316 in DetA, 94.832 in MOTA, 78.271 in IDF1, 2754 in IDS,
and 3592 in Frag, showing the effectiveness of our method.

##  5 Conclusions

In this paper, we modify the motion-based observation-centric SORT with an
extra central distance recovery stage, improving the performance without
adding too much computational cost and also keeping the algorithm online,
successfully tackle down the challenges of non-linear movement during
tracking. We also propose a ReID post-processing stage for each sport
according to the sports characteristics. Our final performance achieves 73.968
in HOTA, ranking 3rd place among all of the teams in the 2022 ECCV
DeeperAction Challenge - SportsMOT Track on Multi-actor Tracking.

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