# Stochastic Burgers equation with fractional derivative driven by
multiplicative noise

Guang-an Zou<EMAIL_ADDRESS>Bo Wang  School of Mathematics and
Statistics, Henan University, Kaifeng 475004, China

###### Abstract

This article is devoted to the study of the existence and uniqueness of mild
solution to time- and space-fractional stochastic Burgers equation perturbed
by multiplicative white noise. The required results are obtained by stochastic
analysis techniques, fractional calculus and semigroup theory. We also proved
the regularity properties of mild solution for this generalized Burgers
equation.

###### keywords:

Stochastic Burgers equation, fractional derivative, mild solution, regularity
properties.

††journal: Computers and Mathematics with Applications

##  1 Introduction

Stochastic Burgers equation (SBE) plays an important role in the modeling of
many phenomena in different fields, such as fluid dynamics, nonlinear
acoustics, hydrodynamics, cosmology, astrophysics and statistical physics, and
so on. In the last decade, SBE has gained a great development in both theory
and application and a large volume of literature is available on this subject
(see e.g.[1-4] and references therein). It is particularly mentioned that when
the Laplacian operator $\Delta$ in SBE is replaced by fractional derivative,
which can be used to describe anomalous diffusion processes in fractal flow
and acoustic waves propagation in porous media [6,22,23]. Sugimoto [5] have
studied the generalized Burgers-type equation with a fractal power of
Laplacian in the principal part, which described the unidirectional
propagation of acoustic waves through a gas-filled tube with a boundary layer.
Besides, the space-fractional SBE also can be used to study the acoustic waves
propagation in tunnels during the passage of the trains, which may yield a
memory effect and other types of resonance phenomena [6]. On the other hand,
time-fractional differential equations are found to be quite effective in
modelling anomalous diffusion processes as its can characterize the long
memory processes [14-16,21,24]. Hence, Burgers equation with time-fractional
can be adapted to describe the memory effect of the wall friction through the
boundary layer [7]. Furthermore, the analytical solutions of the time- and
space-fractional Burgers equations have been investigated by variational
iteration method [7] and Adomian decomposition method [8].

In this study, we focus on the following generalized SBE with time-space
fractional derivative on a bounded domain $D\subset\mathds{R}^{d}(1\leq d\leq
3)$:

${}^{C}D_{t}^{\beta}u-u\cdot\nabla
u+(-\triangle)^{\frac{\alpha}{2}}u=g(u)\dot{W}(t),(t,x)\in(0,T]\times D,$
(1.1)

subject to the initial condition:

$\displaystyle u(0,x)=u_{0}(x),x\in D,$ (1.2)

and the Dirichlet boundary conditions:

$\displaystyle u(t,x)|_{\partial D}=0,t\in[0,T],$ (1.3)

in which the term $g(u)\dot{W}(t)=g(u)\frac{dW(t)}{dt}$ describes a state
dependent random noise, where $W(t)_{t\in[0,T]}$ is a
$\mathcal{F}_{t}$-adapted Wiener process defined on a completed probability
space $(\Omega,\mathcal{F},\mathds{P})$ with the expectation $\mathbb{E}$, and
associate with the normal filtration $\mathcal{F}_{t}=\sigma\\{W(s):0\leq
s\leq t\\}$; The operator $(-\triangle)^{\frac{\alpha}{2}},\alpha\in(1,2)$
stands for the fractional power of the Laplacian (see [25]); We denote by
${}^{C}D_{t}^{\beta}$ the Caputo derivative of order $\beta$, which is defined
by (see [9])

${}^{C}D_{t}^{\beta}u(t,x)=\begin{cases}\frac{1}{\Gamma(1-\beta)}\int_{0}^{t}\frac{\partial
u(s,x)}{\partial s}\frac{ds}{(t-s)^{\beta}},~{}0<\beta<1,\\\ \frac{\partial
u(t,x)}{\partial t},\qquad\qquad\qquad\qquad\beta=1,\\\ \end{cases}$ (1.4)

where $\Gamma(\cdot)$ stands for the gamma function
$\Gamma(\beta)=\int_{0}^{\infty}t^{\beta-1}e^{-t}dt$.

Eq.(1.1) might be used to model anomalous diffusion processes in disordered
media, and describe the acoustic wave propagation in porous media with memory
effect and with random effects. Notice that the study of space-fractional SBE
can be found in some literatures. For details, Brzeźniak and Debbi [10] proved
existence and uniqueness of a mild global solution to the Cauchy problem for
the stochastic fractional Burgers equation. Brzeźniak et al.[6] studied the
ergodic properties of the solution for space-fractional SBE. Yang [11]
proposed some estimates on the solution of space-fractional SBE and given the
invariant measure. Lv and Duan [12] investigated the existence of martingale
solutions and weak solutions for space-fractional SBE on a bounded domain.
However, to the best of our knowledge, there are no existing works for the
time- and space-fractional SBE, which is a fascinating and useful problem.

The main contribution of this paper is to establish the existence, uniqueness,
and regularity properties of mild solution to time-space fractional SBE driven
by multiplicative noise, which generalizes many previous works [6,11,12]. The
rest of the paper is organized as follows. In Section 2, we will introduce
some notations and preliminaries, which play a crucial role in our theorem
analysis. In Section 3, the existence and uniqueness of mild solution to the
problems of time-space fractional SBE are obtained by stochastic analysis
techniques, fractional calculus and semigroup theory. Finally, the spatial and
temporal regularity properties of mild solution to this time-space fractional
SBE are proved.

##  2 Notations and preliminaries

Denote the basic functional space $L^{p}(D),1\leq p<\infty$ and $H^{s}(D)$ by
the usual Lebesgue and Sobolev spaces, respectively. We assume that $A$ is the
negative Laplacian $-\Delta$ in a bounded domain $D$ with zero Dirichlet
boundary conditions in a Hilbert space $H=L^{2}(D)$, which are given by

$\displaystyle A=-\Delta,~{}\mathcal{D}(A)=H_{0}^{1}(D)\cap H^{2}(D).$

Since the operator $A$ is self-adjoint on $H$ with discrete spectral, i.e.,
there exists the eigenvectors $e_{n}$ with corresponding eigenvalues
$\lambda_{n}$ such that

$\displaystyle
Ae_{n}=\lambda_{n}e_{n},e_{n}=\sqrt{2}\sin(n\pi),\lambda_{n}=\pi^{2}n^{2},n\in
N^{+}.$

For any $s>0$, let $\dot{H}^{s}$ be the domain of the fractional power
$A^{\frac{s}{2}}=(-\triangle)^{\frac{s}{2}}$, which can be defined by

$\displaystyle
A^{\frac{s}{2}}e_{n}=\lambda_{n}^{\frac{s}{2}}e_{n},n=1,2,\ldots,$

and

$\displaystyle\dot{H}^{s}=\mathcal{D}(A^{\frac{s}{2}})=\\{v\in
L^{2}(D),s.t.\|v\|_{\dot{H}^{s}}^{2}=\sum\limits_{n=1}^{\infty}\lambda_{n}^{\frac{s}{2}}v_{n}^{2}<\infty\\},$

where $v_{n}:=\langle v,e_{n}\rangle$ with the inner product
$\langle\cdot,\cdot\rangle$ in $L^{2}(D)$. We denote that
$\|v\|_{\dot{H}^{s}}=\|A^{\frac{s}{2}}v\|$, and the corresponding dual space
$\dot{H}^{-s}$ with the inverse operator $A^{-\frac{s}{2}}$. We also denote
$A_{s}$ for $A^{\frac{s}{2}}$ and the bilinear operator $B(u,v)=u\cdot\nabla
v$, and $\mathcal{D}(B)=H_{0}^{1}(D)$ with a slight abuse of notation
$B(u):=B(u,u)$. Then the Eqs.(1.1)-(1.3) can be rewritten as the following
abstract formulation:

$\displaystyle\begin{cases}{}^{C}D_{t}^{\beta}u(t)=-A_{\alpha}u(t)+B(u(t))+g(u(t))\frac{dW(t)}{dt},t>0,\\\
u(0)=u_{0},\end{cases}$ (2.1)

where $\\{W(t)\\}_{t\geq 0}$ is a $Q$-Wiener process with linear bounded
covariance operator $Q$ such that $\mathrm{Tr}(Q)<\infty$. Further, there
exists the eigenvalues $\lambda_{n}$ and corresponding eigenfunctions $e_{n}$
satisfy $Qe_{n}=\lambda_{n}e_{n},n=1,2,\ldots$, then the Wiener process is
given by

$\displaystyle
W(t)=\sum\limits_{n=1}^{\infty}\lambda^{1/2}_{n}\beta_{n}(t)e_{n},$

in which $\\{\beta_{n}\\}_{n\geq 1}$ is a sequence of real-valued standard
Brownian motions.

Let $L_{0}^{2}=L^{2}(Q^{1/2}(H),H)$ denote the space of Hillbert-Schmidt
operators from $Q^{1/2}(H)$ to $H$ with the norm $\|\phi\|_{L_{0}^{2}}:=\|\phi
Q^{1/2}\|_{HS}=(\sum\limits_{n=1}^{\infty}\phi Q^{1/2}e_{n})^{1/2}$, i.e.,
$L_{0}^{2}=\\{\phi\in L(H):\sum\limits_{n=1}^{\infty}\|\phi
Q^{1/2}e_{n})\|^{2}<\infty\\}$, where $L(H)$ is the space of bounded linear
operators from $H$ to $H$.

For an arbitrary Banach space $B$, we denote $\|\cdot\|_{L^{p}(\Omega;B)}$ by
the norm in $L^{p}(\Omega,\mathcal{F},\mathds{P};B)$, which defined as

$\displaystyle\|v\|_{L^{p}(\Omega;B)}=(\mathbb{E}[\|v\|_{B}^{p}])^{\frac{1}{p}},\forall~{}v\in
L^{p}(\Omega,\mathcal{F},\mathds{P};B),$

for any $p\geq 2$.

We shall also need the following result with respect to the fractional
operator $A_{\alpha}$ (see Ref.[11]).

Lemma 2.1. For any $\alpha>0$, an analytic semigroup
$S_{\alpha}(t)=e^{-tA_{\alpha}},t\geq 0$ is generated by the operator
$-A_{\alpha}$ on $L^{p}$, and for any $\nu\geq 0$, there exists a constant
$C_{\alpha,\nu}$ dependent on $\alpha$ and $\nu$ such that

$\displaystyle\|A_{\nu}S_{\alpha}(t)\|_{\mathcal{L}(L^{p})}\leq
C_{\alpha,\nu}t^{-\frac{\nu}{\alpha}},t>0,$ (2.2)

in which $\mathcal{L}(B)$ denotes the Banach space of all linear bounded
operators from $B$ to itself.

Next, we will introduce the following lemma to estimate the stochastic
integrals, which contains the Burkholder-Davis-Gundy’s inequality.

Lemma 2.2.([13]) For any $0\leq t_{1}<t_{2}\leq T$ and $p\geq 2$, and for any
predictable stochastic process $v:[0,T]\times\Omega\rightarrow L_{0}^{2}$,
which satisfies

$\displaystyle\mathbb{E}[(\int_{0}^{T}\|v(s)\|_{L_{0}^{2}}^{2}ds)^{\frac{p}{2}}]<\infty,$

then we have

$\displaystyle\mathbb{E}[\|\int_{t_{1}}^{t_{2}}v(s)dW(s)\|^{p}]\leq
C(p)\mathbb{E}[(\int_{t_{1}}^{t_{2}}\|v(s)\|_{L_{0}^{2}}^{2}ds)^{\frac{p}{2}}],$
(2.3)

where
$C(p)=[\frac{p(p-1)}{2}]^{\frac{p}{2}}(\frac{p}{p-1})^{p(\frac{p}{2}-1)}$ is a
constant.

Inspired by the definition of the mild solution to the time-fractional
differential equations (see Refs.[14-18]), we give the following definition of
mild solution for our time-space fractional stochastic Burgers equation.

Definition 2.1. A $\mathcal{F}_{t}$-adapted process $(u(t))_{t\in[0,T]}$ is
called a mild solution to (2.1), if $(u(t))_{t\in[0,T]}\in
C([0,T];\dot{H}^{\nu})$ $\mathds{P}$-a.e., and it holds

$\displaystyle u(t)$
$\displaystyle=E_{\beta}(t)u_{0}+\int_{0}^{t}(t-s)^{\beta-1}E_{\beta,\beta}(t-s)B(u(s))ds$
$\displaystyle\hskip
5.69054pt+\int_{0}^{t}(t-s)^{\beta-1}E_{\beta,\beta}(t-s)g(u(s))dW(s),$ (2.4)

for a.s. $\omega\in\Omega$, where the generalized Mittag-Leffler operators
$E_{\beta}(t)$ and $E_{\beta,\beta}(t)$ are defined as

$\displaystyle
E_{\beta}(t)=\int_{0}^{\infty}M_{\beta}(\theta)S_{\alpha}(t^{\beta}\theta)d\theta,$

and

$\displaystyle E_{\beta,\beta}(t)=\int_{0}^{\infty}\beta\theta
M_{\beta}(\theta)S_{\alpha}(t^{\beta}\theta)d\theta,$

which contain the Mainardi’s Wright-type function with $\beta\in(0,1)$ given
by

$\displaystyle
M_{\beta}(\theta)=\sum_{n=0}^{\infty}\frac{(-1)^{n}\theta^{n}}{n!\Gamma(1-\beta(1+n))},$

in which the Mainardi function $M_{\beta}(\theta)$ act as a bridge between the
classical integral-order and fractional derivatives of differential equations,
for more details see [19,20]. Here, the derivation of mild solution (2.4) can
be found in Appendix A.

Firstly, let us state the property of the special Mainardi function
$M_{\beta}(\theta)$. Further, the properties of generalized Mittag-Leffler
operators $E_{\beta}(t)$ and $E_{\beta,\beta}(t)$ are proved.

Lemma 2.3. (see [15]) For any $\beta\in(0,1)$ and $-1<\varepsilon<\infty$, it
is not difficult to verity that

$\displaystyle M_{\beta}(\theta)\geq
0,~{}and~{}\int_{0}^{\infty}\theta^{\varepsilon}M_{\beta}(\theta)d\theta=\frac{\Gamma(1+\varepsilon)}{\Gamma(1+\beta\varepsilon)},$
(2.5)

for all $\theta\geq 0$.

Theorem 2.1. For any $t>0$, $E_{\beta}(t)$ and $E_{\beta,\beta}(t)$ are linear
and bounded operators. Moreover, for $0\leq\nu<\alpha<2$, there exist
constants $C_{\alpha}=C(\alpha,\beta,\nu)>0$ and
$C_{\beta}=C(\alpha,\beta,\nu)>0$ such that

$\displaystyle\|E_{\beta}(t)v\|_{\dot{H}^{\nu}}\leq
C_{\alpha}t^{-\frac{\beta\nu}{\alpha}}\|v\|,\|E_{\beta,\beta}(t)v\|_{\dot{H}^{\nu}}\leq
C_{\beta}t^{-\frac{\beta\nu}{\alpha}}\|v\|.$ (2.6)

Proof. For $t>0$ and $0\leq\nu<\alpha<2$, by means of the Lemma 2.1 and Lemma
2.3, we have

$\displaystyle\|E_{\beta}(t)v\|_{\dot{H}^{\nu}}$
$\displaystyle\leq\int_{0}^{\infty}M_{\beta}(\theta)\|A_{\nu}S_{\alpha}(t^{\beta}\theta)v\|d\theta$
$\displaystyle\leq\int_{0}^{\infty}C_{\alpha,\nu}t^{-\frac{\beta\nu}{\alpha}}\theta^{-\frac{\nu}{\alpha}}M_{\beta}(\theta)\|v\|d\theta$
$\displaystyle=\frac{C_{\alpha,\nu}\Gamma(1-\frac{\nu}{\alpha})}{\Gamma(1-\frac{\beta\nu}{\alpha})}t^{-\frac{\beta\nu}{\alpha}}\|v\|,v\in
L^{2}(D),$

and

$\displaystyle\|E_{\beta,\beta}(t)v\|_{\dot{H}^{\nu}}$
$\displaystyle\leq\int_{0}^{\infty}\beta\theta
M_{\beta}(\theta)\|A_{\nu}S_{\alpha}(t^{\beta}\theta)v\|d\theta$
$\displaystyle\leq\int_{0}^{\infty}C_{\alpha,\nu}\beta
t^{-\frac{\beta\nu}{\alpha}}\theta^{1-\frac{\nu}{\alpha}}M_{\beta}(\theta)\|v\|d\theta$
$\displaystyle=\frac{C_{\alpha,\nu}\beta\Gamma(2-\frac{\nu}{\alpha})}{\Gamma(1+\beta(1-\frac{\nu}{\alpha}))}t^{-\frac{\beta\nu}{\alpha}}\|v\|,v\in
L^{2}(D),$

which imply that the estimates (2.6) hold, so it is easy to know that
$E_{\beta}(t)$ and $E_{\beta,\beta}(t)$ are linear and bounded operators.

Theorem 2.2. For any $t>0$, the operators $E_{\beta}(t)$ and
$E_{\beta,\beta}(t)$ are strongly continuous. Moreover, for any $0\leq
t_{1}<t_{2}\leq T$ and for $0<\nu<\alpha<2$, there exist constants
$C_{\alpha\nu}=C(\alpha,\beta,\nu)>0$ and $C_{\beta\nu}=C(\alpha,\beta,\nu)>0$
such that

$\displaystyle\|(E_{\beta}(t_{2})-E_{\beta}(t_{1}))v\|_{\dot{H}^{\nu}}\leq
C_{\alpha\nu}(t_{2}-t_{1})^{\frac{\beta\nu}{\alpha}}\|v\|,$ (2.7)

and

$\displaystyle\|(E_{\beta,\beta}(t_{2})-E_{\beta,\beta}(t_{1}))v\|_{\dot{H}^{\nu}}\leq
C_{\beta\nu}(t_{2}-t_{1})^{\frac{\beta\nu}{\alpha}}\|v\|.$ (2.8)

Proof. For any $0<T_{0}\leq t_{1}<t_{2}\leq T$, it is easy to deduce that

$\displaystyle\int_{t_{1}}^{t_{2}}\frac{dS_{\alpha}(t^{\beta}\theta)}{dt}dt$
$\displaystyle=S_{\alpha}(t_{2}^{\beta}\theta)-S_{\alpha}(t_{1}^{\beta}\theta)$
$\displaystyle=-\int_{t_{1}}^{t_{2}}\beta t^{\beta-1}\theta
A_{\alpha}S_{\alpha}(t^{\beta}\theta)dt.$ (2.9)

For $0<\nu<\alpha<2$, making use of the above expression, the Lemma 2.1 and
Lemma 2.3, we can arrive at

$\displaystyle\|(E_{\beta}(t_{2})-E_{\beta}(t_{1}))v\|_{\dot{H}^{\nu}}$
$\displaystyle=\|A_{\nu}(E_{\beta}(t_{2})-E_{\beta}(t_{1}))v\|$
$\displaystyle=\|\int_{0}^{\infty}M_{\beta}(\theta)A_{\nu}(S_{\alpha}(t_{2}^{\beta}\theta)-S_{\alpha}(t_{1}^{\beta}\theta)vd\theta\|$
$\displaystyle\leq\int_{0}^{\infty}\beta\theta
M_{\beta}(\theta)\int_{t_{1}}^{t_{2}}t^{\beta-1}\|A_{\alpha+\nu}S_{\alpha}(t^{\beta}\theta)v\|_{L^{2}}dtd\theta$
$\displaystyle\leq\int_{0}^{\infty}C_{\alpha,\nu}\beta\theta^{-\frac{\nu}{\alpha}}M_{\beta}(\theta)(\int_{t_{1}}^{t_{2}}t^{-\frac{\beta\nu}{\alpha}-1}dt)\|v\|d\theta$
$\displaystyle=\frac{\alpha
C_{\alpha,\nu}\Gamma(1-\frac{\nu}{\alpha})}{\nu\Gamma(1-\frac{\beta\nu}{\alpha})}(t_{1}^{-\frac{\beta\nu}{\alpha}}-t_{2}^{-\frac{\beta\nu}{\alpha}})\|v\|$
$\displaystyle\leq\frac{\alpha C_{\alpha,\nu}\Gamma(1-\frac{\nu}{\alpha})}{\nu
T_{0}^{\frac{2\beta\nu}{\alpha}}\Gamma(1-\frac{\beta\nu}{\alpha})}(t_{2}-t_{1})^{\frac{\beta\nu}{\alpha}}\|v\|,v\in
L^{2}(D),$

and

$\displaystyle\|(E_{\beta,\beta}(t_{2})-E_{\beta,\beta}(t_{1}))v\|_{\dot{H}^{\nu}}$
$\displaystyle=\|A_{\nu}(E_{\beta,\beta}(t_{2})-E_{\beta,\beta}(t_{1}))v\|$
$\displaystyle=\|\int_{0}^{\infty}\beta\theta
M_{\beta}(\theta)A_{\nu}(S_{\alpha}(t_{2}^{\beta}\theta)-S_{\alpha}(t_{1}^{\beta}\theta)vd\theta\|$
$\displaystyle\leq\int_{0}^{\infty}\beta^{2}\theta^{2}M_{\beta}(\theta)\int_{t_{1}}^{t_{2}}t^{\beta-1}\|A_{\alpha+\nu}S_{\alpha}(t^{\beta}\theta)v\|dtd\theta$
$\displaystyle\leq\int_{0}^{\infty}C_{\alpha,\nu}\beta^{2}\theta^{1-\frac{\nu}{\alpha}}M_{\beta}(\theta)(\int_{t_{1}}^{t_{2}}t^{-\frac{\beta\nu}{\alpha}-1}dt)\|v\|d\theta$
$\displaystyle=\frac{\alpha\beta
C_{\alpha,\nu}\Gamma(2-\frac{\nu}{\alpha})}{\nu\Gamma(1+\beta(1-\frac{\nu}{\alpha}))}(t_{1}^{-\frac{\beta\nu}{\alpha}}-t_{2}^{-\frac{\beta\nu}{\alpha}})\|v\|$
$\displaystyle\leq\frac{\alpha\beta
C_{\alpha,\nu}\Gamma(2-\frac{\nu}{\alpha})}{\nu
T_{0}^{\frac{2\beta\nu}{\alpha}}\Gamma(1+\beta(1-\frac{\nu}{\alpha}))}(t_{2}-t_{1})^{\frac{\beta\nu}{\alpha}}\|v\|,v\in
L^{2}(D).$

It is obviously to see that the term
$\|(E_{\beta}(t_{2})-E_{\beta}(t_{1}))v\|_{\dot{H}^{\nu}}\rightarrow 0$ and
$\|(E_{\beta,\beta}(t_{2})-E_{\beta,\beta}(t_{1}))v\|_{\dot{H}^{\nu}}\rightarrow
0$ as $t_{1}\rightarrow t_{2}$, which mean that the operators $E_{\beta}(t)$
and $E_{\beta,\beta}(t)$ are strongly continuous.

Remark. Assume $\nu=0$ in Theorem 2.2, then there exist constants
$C_{\alpha}=C(\alpha,\beta)>0$ and $C_{\beta}=C(\alpha,\beta)>0$ such that

$\displaystyle\|(E_{\beta}(t_{2})-E_{\beta}(t_{1}))v\|\leq
C_{\alpha}(t_{2}-t_{1})\|v\|,$ (2.10)

and

$\displaystyle\|(E_{\beta,\beta}(t_{2})-E_{\beta,\beta}(t_{1}))v\|\leq
C_{\beta}(t_{2}-t_{1})\|v\|.$ (2.11)

Proof. For any $0<T_{0}\leq t_{1}<t_{2}\leq T$, the same as the proof of
Theorem 2.2, we get

$\displaystyle\|(E_{\beta}(t_{2})-E_{\beta}(t_{1}))v\|$
$\displaystyle=\|\int_{0}^{\infty}M_{\beta}(\theta)(S_{\alpha}(t_{2}^{\beta}\theta)-S_{\alpha}(t_{1}^{\beta}\theta)vd\theta\|_{L^{2}}$
$\displaystyle\leq\int_{0}^{\infty}\beta\theta
M_{\beta}(\theta)\int_{t_{1}}^{t_{2}}t^{\beta-1}\|A_{\alpha}S_{\alpha}(t^{\beta}\theta)v\|dtd\theta$
$\displaystyle\leq\int_{0}^{\infty}C_{\alpha,\alpha}\beta
M_{\beta}(\theta)(\int_{t_{1}}^{t_{2}}t^{-1}dt)\|v\|d\theta$
$\displaystyle=C_{\alpha,\alpha}\beta(\ln t_{2}-\ln t_{1})\|v\|$
$\displaystyle\leq\frac{C_{\alpha,\alpha}\beta}{T_{0}}(t_{2}-t_{1})\|v\|,v\in
L^{2}(D),$

and

$\displaystyle\|(E_{\beta,\beta}(t_{2})-E_{\beta,\beta}(t_{1}))v\|$
$\displaystyle=\|\int_{0}^{\infty}\beta\theta
M_{\beta}(\theta)(S_{\alpha}(t_{2}^{\beta}\theta)-S_{\alpha}(t_{1}^{\beta}\theta)vd\theta\|$
$\displaystyle\leq\int_{0}^{\infty}\beta^{2}\theta^{2}M_{\beta}(\theta)\int_{t_{1}}^{t_{2}}t^{\beta-1}\|A_{\alpha}S_{\alpha}(t^{\beta}\theta)v\|dtd\theta$
$\displaystyle\leq\int_{0}^{\infty}C_{\alpha,\alpha}\beta^{2}\theta
M_{\beta}(\theta)(\int_{t_{1}}^{t_{2}}t^{-1}dt)\|v\|d\theta$
$\displaystyle=\frac{C_{\alpha,\alpha}\beta^{2}\Gamma(2)}{\Gamma(1+\beta)}(\ln
t_{2}-\ln t_{1})\|v\|$
$\displaystyle\leq\frac{C_{\alpha,\alpha}\beta^{2}\Gamma(2)}{T_{0}\Gamma(1+\beta)}(t_{2}-t_{1})\|v\|,v\in
L^{2}(D).$

This completes the proof.

##  3 Existence and uniqueness of mild solution

Our main purpose of this section is to prove the existence and uniqueness of
mild solution to the problem (2.1). To do this, the following assumptions are
imposed.

Assumption 3.1. The measurable function $g:\Omega\times H\rightarrow
L_{0}^{2}$ satisfies the following global Lipschitz and growth conditions:

$\displaystyle\|g(v)\|_{L_{0}^{2}}\leq C\|v\|,\|g(u)-g(v)\|_{L_{0}^{2}}\leq
C\|u-v\|,$ (3.1)

for any $u,v\in H$.

Assumption 3.2. Let $C>0$ be a real number, then the bounded bilinear operator
$B:L^{2}(D)\rightarrow\dot{H}^{-1}(D)$ satisfies the following properties:

$\displaystyle\|B(u)\|_{\dot{H}^{-1}}\leq C\|u\|^{2},$ (3.2)

and

$\displaystyle\|B(u)-B(v)\|_{\dot{H}^{-1}}\leq C(\|u\|+\|v\|)\|u-v\|,$ (3.3)

for any $u,v\in L^{2}(D)$.

Assumption 3.3. Assume that the initial value
$u_{0}:\Omega\rightarrow\dot{H}^{\nu}$ is a $\mathcal{F}_{0}-$measurable
random variable, it holds that

$\displaystyle\|u_{0}\|_{L^{p}(\Omega;\dot{H}^{\nu})}<\infty,$ (3.4)

for any $0\leq\nu<\alpha<2$.

Theorem 3.1. Let Assumptions 3.1 to 3.3 be satisfied for some $p\geq 2$, then
there exists a unique mild solution $(u(t))_{t\in[0,T]}$ in the space
$L^{p}(\Omega;\dot{H}^{\nu})$ with $0\leq\nu<\alpha<2$.

Proof. We fix an $\omega\in\Omega$ and use the standard Picard’s iteration
argument to prove the existence of mild solution. To begin with, the sequence
of stochastic process $\\{u_{n}(t)\\}_{n\geq 0}$ is constructed as

$\displaystyle\begin{cases}u_{n+1}(t)=E_{\beta}(t)u_{0}+N_{1}(u_{n}(t))+N_{2}(u_{n}(t)),\\\
u_{0}(t)=u_{0},\end{cases}$ (3.5)

where

$\displaystyle\begin{cases}N_{1}(u_{n}(t))=\int_{0}^{t}(t-s)^{\beta-1}E_{\beta,\beta}(t-s)B(u_{n}(s))ds,\\\
N_{2}(u_{n}(t))=\int_{0}^{t}(t-s)^{\beta-1}E_{\beta,\beta}(t-s)g(u_{n}(s))dW(s).\end{cases}$
(3.6)

The proof will be split into three steps.

Step 1: For each $n\geq 0$, we show that

$\displaystyle\sup\limits_{t\in[0,T]}\mathbb{E}[\|u_{n}(t)\|_{\dot{H}^{\nu}}^{p}]<\infty.$

Note that

$\displaystyle\mathbb{E}[\|u_{n+1}(t)\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle\leq
3^{p-1}\mathbb{E}[\|E_{\beta}(t)u_{0}\|_{\dot{H}^{\nu}}^{p}]+3^{p-1}\mathbb{E}[\|N_{1}(u_{n}(t))\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle+3^{p-1}\mathbb{E}[\|N_{2}(u_{n}(t))\|_{\dot{H}^{\nu}}^{p}].$
(3.7)

The application of the Lemma 2.1 gives

$\displaystyle\mathbb{E}[\|E_{\beta}(t)u_{0}\|_{\dot{H}^{\nu}}]$
$\displaystyle\leq\mathbb{E}[\int_{0}^{\infty}M_{\beta}(\theta)(\|A_{\nu}S_{\alpha}(t^{\beta}\theta)u_{0}\|^{2})^{\frac{1}{2}}d\theta]$
$\displaystyle=\mathbb{E}[\int_{0}^{\infty}M_{\beta}(\theta)(\sum\limits_{n=1}^{\infty}\langle
A_{\nu}e^{-t^{\beta}\theta
A_{\alpha}}u_{0},e_{n}\rangle^{2})^{\frac{1}{2}}d\theta]$
$\displaystyle=\mathbb{E}[\int_{0}^{\infty}M_{\beta}(\theta)(\sum\limits_{n=1}^{\infty}\langle
A_{\nu}u_{0},e^{-t^{\beta}\theta\lambda_{n}^{\frac{\alpha}{2}}}e_{n}\rangle^{2})^{\frac{1}{2}}d\theta]$
$\displaystyle\leq\mathbb{E}[\int_{0}^{\infty}M_{\beta}(\theta)\|u_{0}\|_{\dot{H}^{\nu}}d\theta]=\mathbb{E}[\|u_{0}\|_{\dot{H}^{\nu}}].$
(3.8)

Applying the following Hölder inequality to the second term of the right-hand
side of (3.7)

$\displaystyle\int_{a}^{b}|f(t)g(t)|dt\leq(\int_{a}^{b}|f(t)|^{p}dt)^{\frac{1}{p}}(\int_{a}^{b}|g(t)|^{q}dt)^{\frac{1}{q}},\frac{1}{p}+\frac{1}{q}=1,$

where $p,q\in(1,\infty)$, we infer

$\displaystyle\mathbb{E}[\|N_{1}(u_{n}(t))\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle\leq\mathbb{E}[(\int_{0}^{t}\|(t-s)^{\beta-1}A_{1}E_{\beta,\beta}(t-s)A_{\nu-1}B(u_{n}(s))\|ds)^{p}]$
$\displaystyle\leq
C_{\beta}^{p}(\int_{0}^{t}(t-s)^{\frac{p(\beta-1-\frac{\beta}{\alpha})}{p-1}}ds)^{p-1}\int_{0}^{t}\mathbb{E}[\|A_{\nu-1}B(u_{n}(s))\|^{p}]ds$
$\displaystyle\leq
K_{1}\int_{0}^{t}\mathbb{E}[\|u_{n}(s)\|_{\dot{H}^{\nu}}^{p}]ds,$ (3.9)

where
$K_{1}=C_{\beta}^{p}C^{p}[\frac{p-1}{p(\beta-\frac{\beta}{\alpha})-1}]^{p-1}T^{p(\beta-\frac{\beta}{\alpha})-1}(\max\limits_{t\in[0,T]}\mathbb{E}[\|u_{n}(t)\|_{\dot{H}^{\nu}}^{p}])$.

Making use of the Hölder inequality and Lemma 2.2 to the third term of the
right-hand side of (3.7), we get

$\displaystyle\mathbb{E}[\|N_{2}(u_{n}(t))\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle\leq
C(p)\mathbb{E}[(\int_{0}^{t}\|(t-s)^{\beta-1}E_{\beta,\beta}(t-s)\|^{2}\|A_{\nu}g(u_{n}(s))\|_{L_{0}^{2}}^{2}ds)^{\frac{p}{2}}]$
$\displaystyle\leq
C(p)C_{\beta}^{p}(\int_{0}^{t}(t-s)^{\frac{2p(\beta-1)}{p-2}}ds)^{\frac{p-2}{2}}\int_{0}^{t}\mathbb{E}\|A_{\nu}g(u_{n}(s))\|_{L_{0}^{2}}^{p}ds$
$\displaystyle\leq
K_{2}\int_{0}^{t}\mathbb{E}[\|u_{n}(s)\|_{\dot{H}^{\nu}}^{p}]ds,$ (3.10)

where
$K_{2}=C(p)C_{\beta}^{p}C^{p}[\frac{p-2}{p(2\beta-1)-2}]^{\frac{p-2}{2}}T^{\frac{p(2\beta-1)-2}{2}}$.

Using the above estimates (3.7)-(3.10), we have

$\displaystyle\mathbb{E}[\|u_{n+1}(t)\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle\leq
3^{p-1}\mathbb{E}[\|u_{0}\|_{\dot{H}^{\nu}}^{p}]+3^{p-1}(K_{1}+K_{2})\int_{0}^{t}\mathbb{E}[\|u_{n}(s)\|_{\dot{H}^{\nu}}^{p}]ds.$

By means of the extension of Gronwall’s lemma, it holds that

$\displaystyle\sup\limits_{t\in[0,T]}\mathbb{E}[\|u_{n}(t)\|_{\dot{H}^{\nu}}^{p}]<\infty,$

for each $n\geq 0$.

Step 2: Show that the sequence $\\{u_{n}(t)\\}_{n\geq 0}$ is a Cauchy sequence
in the space $L^{p}(\Omega;\dot{H}^{\nu})$.

For any $n\geq m\geq 1$, applying the similar arguments employed to obtain
(3.9) and (3.10), we get

$\displaystyle\mathbb{E}[\|u_{n}(t)-u_{m}(t)\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle\leq
2^{p-1}\mathbb{E}[\|N_{1}(u_{n-1}(t))-N_{1}(u_{m-1}(t))\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle\hskip
5.69054pt+2^{p-1}\mathbb{E}[\|N_{2}(u_{n-1}(t))-N_{2}(u_{m-1}(t))\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle\leq
K\int_{0}^{t}\mathbb{E}[\|u_{n-1}(s)-u_{m-1}(s)\|_{\dot{H}^{\nu}}^{p}]ds,$
(3.11)

in which

$\displaystyle K$
$\displaystyle=2^{p-1}\\{C_{\beta}^{p}C^{p}[\frac{p-1}{p(\beta-\frac{\beta}{\alpha})-1}]^{p-1}T^{p(\beta-\frac{\beta}{\alpha})-1}(\max\limits_{t\in[0,T]}\mathbb{E}[\|u_{n-1}(t)\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle\hskip
5.69054pt+\max\limits_{t\in[0,T]}\mathbb{E}[\|u_{m-1}(t)\|_{\dot{H}^{\nu}}^{p}])+C(p)C_{\beta}^{p}C^{p}[\frac{p-2}{p(2\beta-1)-2}]^{\frac{p-2}{2}}T^{\frac{p(2\beta-1)-2}{2}}\\}.$

A direct application of Gronwall’s lemma yields

$\displaystyle\sup\limits_{t\in[0,T]}\mathbb{E}[\|u_{n}(t)-u_{m}(t)\|_{\dot{H}^{\nu}}^{p}]=0.$

As a result, the sequence $\\{u_{n}(t)\\}_{n\geq 0}$ is a Cauchy sequence in
the space $L^{p}(\Omega;\dot{H}^{\nu})$. Further, there exists a $u(t)\in
L^{p}(\Omega;\dot{H}^{\nu})$ such that

$\displaystyle\sup\limits_{t\in[0,T]}\mathbb{E}[\|u_{n}(t)-u(t)\|_{\dot{H}^{\nu}}^{p}]=0,$

for all $T>0$.

Taking limits to the stochastic sequence $\\{u_{n}(t)\\}_{n\geq 0}$ in (3.5)
as $n\rightarrow\infty$, we finish the proof of the existence of mild solution
to (2.1).

Step 3: We show the uniqueness of mild solution.

Assume $u$ and $v$ are two mild solutions of the problem (2.1), using the
similar calculations as in Step 2, we can obtain

$\displaystyle\sup\limits_{t\in[0,T]}\mathbb{E}[\|u(t)-v(t)\|_{\dot{H}^{\nu}}^{p}]=0,$
(3.12)

for all $T>0$, which implies that $u=v$, it follows that the uniqueness of
mild solution.

Obviously, when $\nu=0$, the above three steps still work. Thus the proof of
Theorem 3.1 is completed.

##  4 Regularity of mild solution

In this section, we will prove the spatial and temporal regularity properties
of mild solution to time-space fractional SBE based on the analytic semigroup.

Theorem 4.1. Let Assumptions 3.1 to 3.3 hold with $1\leq\nu<\alpha<2$ and
$p\geq 2$, let $u(t)$ be a unique mild solution of the problem (2.1) with
$\mathds{P}(u(t)\in\dot{H}^{\nu})=1$ for any $t\in[0,T]$, then there exists a
constant $C$ such that

$\displaystyle\sup\limits_{t\in[0,T]}\|u(t)\|_{L^{p}(\Omega;\dot{H}^{\nu})}\leq
C(\|u_{0}\|_{L^{p}(\Omega;H)}+\sup\limits_{t\in[0,T]}\|u(t)\|_{L^{p}(\Omega;\dot{H}^{1})})$
(4.1)

Proof. For any $0\leq t\leq T$ and $1\leq\nu<\alpha<2$, we have

$\displaystyle\|u(t)\|_{L^{p}(\Omega;\dot{H}^{\nu})}$
$\displaystyle=(\mathbb{E}[\|u(t)\|_{\dot{H}^{\nu}}^{p}])^{\frac{1}{p}}=\|A_{\nu}u(t)\|_{L^{p}(\Omega;H)}$
$\displaystyle\leq\|A_{\nu}E_{\beta}(t)u_{0}\|_{L^{p}(\Omega;H)}$
$\displaystyle\hskip
5.69054pt+\|A_{\nu}\int_{0}^{t}(t-s)^{\beta-1}E_{\beta,\beta}(t-s)B(u(s))ds\|_{L^{p}(\Omega;H)}$
$\displaystyle\hskip
5.69054pt+\|A_{\nu}\int_{0}^{t}(t-s)^{\beta-1}E_{\beta,\beta}(t-s)g(u(s))dW(s)\|_{L^{p}(\Omega;H)}$
$\displaystyle=I+II+III.$ (4.2)

Using Theorem 2.1, the first term can be estimated by

$\displaystyle I=\|A_{\nu}E_{\beta}(t)u_{0}\|_{L^{p}(\Omega;H)}\leq
C_{\alpha}t^{-\frac{\beta\nu}{\alpha}}\|u_{0}\|_{L^{p}(\Omega;H)}<\infty.$
(4.3)

It is easy to know that

$\displaystyle\int_{0}^{T}C_{\alpha}t^{-\frac{\beta\nu}{\alpha}}\|u_{0}\|_{L^{p}(\Omega;H)}dt=\frac{\alpha
C_{\alpha}}{\alpha-\beta\nu}T^{\frac{\alpha-\beta\nu}{\alpha}}\|u_{0}\|_{L^{p}(\Omega;H)}.$
(4.4)

The application of Theorem 2.1 and Assumptions 3.2, we get

$\displaystyle(II)^{p}$
$\displaystyle\leq\mathbb{E}[(\int_{0}^{t}\|(t-s)^{\beta-1}A_{\nu}E_{\beta,\beta}(t-s)B(u(s))\|ds)^{p}]$
$\displaystyle\leq
C_{\beta}^{p}(\int_{0}^{t}(t-s)^{\frac{p[\beta-1-\frac{\beta(\nu+1)}{\alpha}]}{p-1}}ds)^{p-1}\int_{0}^{t}\mathbb{E}[\|A_{-1}B(u(s))\|_{\dot{H}^{1}}^{p}]ds$
$\displaystyle\leq
C_{1}\sup\limits_{t\in[0,T]}\mathbb{E}[\|u(s)\|_{\dot{H}^{1}}^{p}],$ (4.5)

where
$C_{1}=C_{\beta}^{p}C^{p}\\{\frac{p-1}{p[\beta-\frac{\beta(\nu+1)}{\alpha}]-1}\\}^{p-1}T^{p[\beta-\frac{\beta(\nu+1)}{\alpha}]-1}(\max\limits_{t\in[0,T]}\mathbb{E}[\|u(t)\|_{\dot{H}^{1}}])$.

By means of Theorem 2.1, Assumptions 3.1 and Lemma 2.2, we can deduce

$\displaystyle(III)^{p}$ $\displaystyle\leq
C(p)\mathbb{E}[(\int_{0}^{t}\|(t-s)^{\beta-1}A_{\nu-1}E_{\beta,\beta}(t-s)\|^{2}\|A_{1}g(u(s))\|_{L_{2}^{0}}^{2}ds)^{\frac{p}{2}}]$
$\displaystyle\leq
C(p)C_{\beta}^{p}(\int_{0}^{t}(t-s)^{\frac{2p[\beta-1-\frac{\beta(\nu-1)}{\alpha}]}{p-2}}ds)^{\frac{p-2}{2}}\int_{0}^{t}\mathbb{E}\|A_{1}g(u(s))\|_{L_{2}^{0}}^{p}ds$
$\displaystyle\leq
C_{2}\sup\limits_{t\in[0,T]}\mathbb{E}[\|u(s)\|_{\dot{H}^{1}}^{p}],$ (4.6)

where
$C_{2}=C(p)C_{\beta}^{p}C^{p}[\frac{p-2}{p(2\beta-1-\frac{\beta(\nu-1)}{\alpha})-2}]^{\frac{p-2}{2}}T^{\frac{p[2\beta-1-\frac{\beta(\nu-1)}{\alpha}]-2}{2}}$.

Thus, we conclude the proof of Theorem 4.1 by combining with the estimates
(4.2)-(4.6).

Next, we will devote to the temporal regularity of the mild solution.

Theorem 4.2. Let Assumptions 3.1 to 3.3 be fulfilled with $0<\nu<\alpha<2$ and
$p\geq 2$, for any $0\leq t_{1}<t_{2}\leq T$, the unique mild solution $u(t)$
to the problem (2.1) is Hölder continuous with respect to the norm
$\|\cdot\|_{L^{p}(\Omega;\dot{H}^{\nu})}$ and satisfies

$\displaystyle\|u(t_{2})-u(t_{1})\|_{L^{p}(\Omega;\dot{H}^{\nu})}\leq
C(t_{2}-t_{1})^{\gamma}.$ (4.7)

Proof. For any $0\leq t_{1}<t_{2}\leq T$, from the mild solution (2.4), we
have

$\displaystyle u(t_{2})-u(t_{1})$
$\displaystyle=E_{\beta}(t_{2})u_{0}-E_{\beta}(t_{1})u_{0}$
$\displaystyle\hskip
5.69054pt+\int_{0}^{t_{2}}(t_{2}-s)^{\beta-1}E_{\beta,\beta}(t_{2}-s)B(u(s))ds$
$\displaystyle\hskip
5.69054pt-\int_{0}^{t_{1}}(t_{1}-s)^{\beta-1}E_{\beta,\beta}(t_{1}-s)B(u(s))ds$
$\displaystyle\hskip
5.69054pt+\int_{0}^{t_{2}}(t_{2}-s)^{\beta-1}E_{\beta,\beta}(t_{2}-s)g(u(s))dW(s)$
$\displaystyle\hskip
5.69054pt-\int_{0}^{t_{1}}(t_{1}-s)^{\beta-1}E_{\beta,\beta}(t_{1}-s)g(u(s))dW(s)$
$\displaystyle=:I_{1}+I_{2}+I_{3},$ (4.8)

where

$\displaystyle I_{1}=E_{\beta}(t_{2})u_{0}-E_{\beta}(t_{1})u_{0},$
$\displaystyle I_{2}$
$\displaystyle=\int_{0}^{t_{2}}(t_{2}-s)^{\beta-1}E_{\beta,\beta}(t_{2}-s)B(u(s))ds$
$\displaystyle\hskip
5.69054pt-\int_{0}^{t_{1}}(t_{1}-s)^{\beta-1}E_{\beta,\beta}(t_{1}-s)B(u(s))ds$
$\displaystyle=\int_{0}^{t_{1}}(t_{1}-s)^{\beta-1}[E_{\beta,\beta}(t_{2}-s)-E_{\beta,\beta}(t_{1}-s)]B(u(s))ds$
$\displaystyle\hskip
5.69054pt+\int_{0}^{t_{1}}[(t_{2}-s)^{\beta-1}-(t_{1}-s)^{\beta-1}]E_{\beta,\beta}(t_{2}-s)B(u(s))ds$
$\displaystyle\hskip
5.69054pt+\int_{t_{1}}^{t_{2}}(t_{2}-s)^{\beta-1}E_{\beta,\beta}(t_{2}-s)B(u(s))ds$
$\displaystyle=:I_{21}+I_{22}+I_{23}.$ (4.9)

and

$\displaystyle I_{3}$
$\displaystyle=\int_{0}^{t_{2}}(t_{2}-s)^{\beta-1}E_{\beta,\beta}(t_{2}-s)g(u(s))dW(s)$
$\displaystyle\hskip
5.69054pt-\int_{0}^{t_{1}}(t_{1}-s)^{\beta-1}E_{\beta,\beta}(t_{1}-s)g(u(s))dW(s)$
$\displaystyle=\int_{0}^{t_{1}}(t_{1}-s)^{\beta-1}[E_{\beta,\beta}(t_{2}-s)-E_{\beta,\beta}(t_{1}-s)]g(u(s))dW(s)$
$\displaystyle\hskip
5.69054pt+\int_{0}^{t_{1}}[(t_{2}-s)^{\beta-1}-(t_{1}-s)^{\beta-1}]E_{\beta,\beta}(t_{2}-s)g(u(s))dW(s)$
$\displaystyle\hskip
5.69054pt+\int_{t_{1}}^{t_{2}}(t_{2}-s)^{\beta-1}E_{\beta,\beta}(t_{2}-s)g(u(s))dW(s)$
$\displaystyle=:I_{31}+I_{32}+I_{33}.$ (4.10)

For any $0<\nu<\alpha<2$ and $p\geq 2$, by virtue of Theorem 2.2, it follows
that

$\displaystyle\mathbb{E}[\|I_{1}\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle=\mathbb{E}[\|A_{\nu}[E_{\beta}(t_{2})-E_{\beta}(t_{1})]u_{0}\|^{p}]$
$\displaystyle\leq
C_{\alpha\nu}^{p}(t_{2}-t_{1})^{\frac{p\beta\nu}{\alpha}}\mathbb{E}[\|u_{0}\|^{p}].$
(4.11)

For the first term $I_{21}$ in (4.9), applying the Assumptions 3.2 and Theorem
2.2 and Hölder’s inequality, we have

$\displaystyle\mathbb{E}[\|I_{21}\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle=\mathbb{E}[\|\int_{0}^{t_{1}}(t_{1}-s)^{\beta-1}A_{\nu}[E_{\beta,\beta}(t_{2}-s)-E_{\beta,\beta}(t_{1}-s)]B(u(s))ds\|^{p}]$
$\displaystyle\leq
C_{\beta\nu}^{p}(t_{2}-t_{1})^{\frac{p\beta(\nu+1)}{\alpha}}(\int_{0}^{t_{1}}(t_{1}-s)^{\frac{p(\beta-1)}{p-1}}ds)^{p-1}\int_{0}^{t}\mathbb{E}[\|A_{-1}B(u(s))\|_{\dot{H}^{1}}^{p}]ds$
$\displaystyle\leq
C^{p}C_{\beta\nu}^{p}T^{p\beta}(\frac{p-1}{p\beta-1})^{p-1}(\sup\limits_{t\in[0,T]}\mathbb{E}[\|u(s)\|_{\dot{H}^{1}}^{2p}])(t_{2}-t_{1})^{\frac{p\beta(\nu+1)}{\alpha}}.$
(4.12)

Using the Assumptions 3.2, Theorem 2.1 and Hölder’s inequality, we get

$\displaystyle\mathbb{E}[\|I_{22}\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle=\mathbb{E}[\|\int_{0}^{t_{1}}[(t_{2}-s)^{\beta-1}-(t_{1}-s)^{\beta-1}]A_{\nu}E_{\beta,\beta}(t_{2}-s)B(u(s))ds\|^{p}]$
$\displaystyle\leq
C_{\beta}^{p}(\int_{0}^{t_{1}}\\{[(t_{2}-s)^{\beta-1}-(t_{1}-s)^{\beta-1}]\times(t_{2}-s)^{-\frac{\beta(\nu+1)}{\alpha}}\\}^{\frac{p}{p-1}}ds)^{p-1}$
$\displaystyle\hskip
5.69054pt\times\int_{0}^{t_{1}}\mathbb{E}[\|A_{-1}B(u(s))\|_{\dot{H}^{1}}^{p}]ds$
$\displaystyle\leq
C^{p}C_{\beta}^{p}T\\{\frac{p-1}{p[\beta-\frac{\beta(\nu+1)}{\alpha}]-1}\\}^{p-1}(\sup\limits_{t\in[0,T]}\mathbb{E}[\|u(s)\|_{\dot{H}^{1}}^{2p}])(t_{2}-t_{1})^{\frac{p\beta(\alpha-\nu-1)-\alpha}{\alpha}},$
(4.13)

and

$\displaystyle\mathbb{E}[\|I_{23}\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle=\mathbb{E}[\|\int_{t_{1}}^{t_{2}}(t_{2}-s)^{\beta-1}A_{\nu}E_{\beta,\beta}(t_{2}-s)B(u(s))ds\|^{p}]$
$\displaystyle\leq
C_{\beta}^{p}(\int_{t_{1}}^{t_{2}}[(t_{2}-s)^{\beta-1-\frac{\beta(\nu+1)}{\alpha}}]^{\frac{p}{p-1}}ds)^{p-1}\int_{t_{1}}^{t_{2}}\mathbb{E}[\|A_{-1}B(u(s))\|_{\dot{H}^{1}}^{p}]ds$
$\displaystyle\leq
C^{p}C_{\beta}^{p}\\{\frac{p-1}{p[\beta-\frac{\beta(\nu+1)}{\alpha}]-1}\\}^{p-1}(\sup\limits_{t\in[0,T]}\mathbb{E}[\|u(s)\|_{\dot{H}^{1}}^{2p}])(t_{2}-t_{1})^{\frac{p\beta(\alpha-\nu-1)}{\alpha}}.$
(4.14)

Next, by following the similar arguments as in the proof of (4.12)-(4.14) and
using the Lemma 2.2, there holds

$\displaystyle\mathbb{E}[\|I_{31}\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle=\mathbb{E}[\|\int_{0}^{t_{1}}(t_{1}-s)^{\beta-1}A_{\nu}[E_{\beta,\beta}(t_{2}-s)-E_{\beta,\beta}(t_{1}-s)]g(u(s))dW(s)\|^{p}]$
$\displaystyle\leq
C(p)\mathbb{E}[(\int_{0}^{t_{1}}\|(t_{1}-s)^{\beta-1}A_{\nu}[E_{\beta,\beta}(t_{2}-s)-E_{\beta,\beta}(t_{1}-s)]\|^{2}\|g(u(s))\|_{L_{0}^{2}}^{2}ds)^{\frac{p}{2}}]$
$\displaystyle\leq
C(p)C_{\beta\nu}^{p}(t_{2}-t_{1})^{\frac{p\beta\nu}{\alpha}}(\int_{0}^{t_{1}}(t_{1}-s)^{\frac{2p(\beta-1)}{p-2}}ds)^{\frac{p-2}{2}}\int_{0}^{t_{1}}\mathbb{E}\|g(u(s))\|_{L_{0}^{2}}^{p}ds$
$\displaystyle\leq
C(p)C^{p}C_{\beta\nu}^{p}T^{\frac{2p\beta-p-1}{2}}(\frac{p-1}{2p\beta-p-2})^{p-1}(\sup\limits_{t\in[0,T]}\mathbb{E}[\|u(t)\|^{p}])(t_{2}-t_{1})^{\frac{p\beta\nu}{\alpha}},$
(4.15)

and

$\displaystyle\mathbb{E}[\|I_{32}\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle=\mathbb{E}[\|\int_{0}^{t_{1}}[(t_{2}-s)^{\beta-1}-(t_{1}-s)^{\beta-1}]A_{\nu}E_{\beta,\beta}(t_{2}-s)g(u(s))dW(s)\|^{p}]$
$\displaystyle\leq
C(p)\mathbb{E}[(\int_{0}^{t_{1}}\|[(t_{2}-s)^{\beta-1}-(t_{1}-s)^{\beta-1}]A_{\nu}E_{\beta,\beta}(t_{2}-s)\|^{2}\|g(u(s))\|_{L_{0}^{2}}^{2}ds)^{\frac{p}{2}}]$
$\displaystyle\leq
C(p)C_{\beta}^{p}(\int_{0}^{t_{1}}\\{[(t_{2}-s)^{\beta-1}-(t_{1}-s)^{\beta-1}]\times(t_{2}-s)^{-\frac{\beta\nu}{\alpha}}\\}^{\frac{2p}{p-2}}ds)^{\frac{p-2}{2}}$
$\displaystyle\hskip
5.69054pt\times\int_{0}^{t_{1}}\mathbb{E}\|g(u(s))\|_{L_{0}^{2}}^{p}ds$
$\displaystyle\leq
C(p)C^{p}C_{\beta}^{p}T[\frac{\alpha(p-2)}{2p\beta(\alpha-\nu)-(p+2)\alpha}]^{\frac{p-2}{2}}(\sup\limits_{t\in[0,T]}\mathbb{E}[\|u(t)\|^{p}])(t_{2}-t_{1})^{\frac{2p\beta(\alpha-\nu)-(p+2)\alpha}{2\alpha}},$
(4.16)

and

$\displaystyle\mathbb{E}[\|I_{33}\|_{\dot{H}^{\nu}}^{p}]$
$\displaystyle=\mathbb{E}[\|\int_{t_{1}}^{t_{2}}(t_{2}-s)^{\beta-1}A_{\nu}E_{\beta,\beta}(t_{2}-s)g(u(s))dW(s)\|^{p}]$
$\displaystyle\leq
C(p)\mathbb{E}[(\int_{t_{1}}^{t_{2}}\|(t_{2}-s)^{\beta-1}A_{\nu}E_{\beta,\beta}(t_{2}-s)\|^{2}\|g(u(s))\|_{L_{0}^{2}}^{2}ds)^{\frac{p}{2}}]$
$\displaystyle\leq
C(p)C_{\beta}^{p}(\int_{t_{1}}^{t_{2}}[(t_{2}-s)^{\beta-1-\frac{\beta\nu}{\alpha}}]^{\frac{2p}{p-2}}ds)^{\frac{p-2}{2}}\int_{t_{1}}^{t_{2}}\mathbb{E}\|g(u(s))\|_{L_{0}^{2}}^{p}ds$
$\displaystyle\leq
C(p)C^{p}C_{\beta}^{p}[\frac{\alpha(p-2)}{2p\beta(\alpha-\nu)-(p+2)\alpha}]^{\frac{p-2}{2}}(\sup\limits_{t\in[0,T]}\mathbb{E}[\|u(t)\|^{p}])(t_{2}-t_{1})^{\frac{2p\beta(\alpha-\nu)-p\alpha}{2\alpha}}.$
(4.17)

Taking expectation on the both side of (4.8), and in view of the estimates
(4.11)-(4.17), we conclude that

$\displaystyle\|u(t_{2})-u(t_{1})\|_{L^{p}(\Omega;\dot{H}^{\nu})}\leq
C(t_{2}-t_{1})^{\gamma},$ (4.18)

in which we take
$\gamma=\min\\{\frac{\beta\nu}{\alpha},\frac{p\beta(\alpha-\nu-1)-\alpha}{p\alpha},\frac{2p\beta(\alpha-\nu)-(p+2)\alpha}{2p\alpha}\\}$
when $0<t_{2}-t_{1}<1$. Otherwise, if $t_{2}-t_{1}\geq 1$, then we set
$\gamma=\max\\{\frac{\beta(\nu+1)}{\alpha},\frac{\beta(\alpha-\nu-1)}{\alpha},\frac{2p\beta(\alpha-\nu)-p\alpha}{2p\alpha}\\}$.

This completes the proof of Theorem 4.2.

## Acknowledgements

The authors are very thankful to the anonymous referees for their valuable
comments and constructive suggestions, which helped us to improve the
manuscript. Guang-an Zou is supported by National Nature Science Foundation of
China (Grant No. 11626085), Bo Wang is supported by the foundation for Young
University Key Teacher by the Educational Department of Henan Province
(No.2014GGJS-021).

## Appendix A

Considering the following abstract formulation of time-space fractional
stochastic Burgers equation:

$\displaystyle\begin{cases}{}^{C}D_{t}^{\beta}u(t)=-A_{\alpha}u(t)+B(u(t))+g(u(t))\frac{dW(t)}{dt},t>0,\\\
u(0)=u_{0}.\end{cases}$ (A1)

We derive the mild solution to (A1) by means of Laplace transform, which
denoted by $\widehat{}$ . Let $\lambda>0$, and we define that

$\displaystyle\widehat{u}(\lambda)=\int_{0}^{\infty}e^{-\lambda
s}u(s)ds,~{}\widehat{B}(\lambda)=\int_{0}^{\infty}e^{-\lambda s}B(u(s))ds,$

and

$\displaystyle\widehat{G}(\lambda)=\int_{0}^{\infty}e^{-\lambda
s}[g(u(s))\frac{dW(s)}{ds}]ds=\int_{0}^{\infty}e^{-\lambda s}g(u(s))dW(s).$

Upon Laplace transform, using the formula
$\widehat{{}^{C}D_{t}^{\beta}u}=\lambda^{\beta}\widehat{u}-\lambda^{\beta-1}u_{0}$.
Then applying the Laplace transform to (A1), we obtain

$\displaystyle\widehat{u}(\lambda)$
$\displaystyle=\frac{1}{\lambda}u_{0}+\frac{1}{\lambda^{\beta}}(-A_{\alpha})\widehat{u}(\lambda)+\frac{1}{\lambda^{\beta}}[\widehat{B}(\lambda)+\widehat{G}(\lambda)]$
$\displaystyle=\lambda^{\beta-1}(\lambda^{\beta}I+A_{\alpha})^{-1}u_{0}+(\lambda^{\beta}I+A_{\alpha})^{-1}[\widehat{B}(\lambda)+\widehat{G}(\lambda)]$
$\displaystyle=\lambda^{\beta-1}\int_{0}^{\infty}e^{-\lambda^{\beta}s}S_{\alpha}(s)u_{0}ds+\int_{0}^{\infty}e^{-\lambda^{\beta}s}S_{\alpha}(s)[\widehat{B}(\lambda)+\widehat{G}(\lambda)]ds,$
(A2)

in which $I$ is the identity operator, and $S_{\alpha}(t)=e^{-tA_{\alpha}}$ is
an analytic semigroup generated by the operator $-A_{\alpha}$.

We introduce the following one-sided stable probability density function:

$\displaystyle\omega_{\beta}(\theta)=\frac{1}{\pi}\sum_{n=1}^{\infty}(-1)^{n-1}\theta^{-\beta
n-1}\frac{\Gamma(\beta n+1)}{n!}\sin(n\pi\beta),\theta\in(0,+\infty),$ (A3)

whose Laplace transform is given by

$\displaystyle\int_{0}^{\infty}e^{-\lambda\theta}\omega_{\beta}(\theta)d\theta=e^{-\lambda^{\beta}},0<\beta<1.$
(A4)

Making use of above expression (A4), then the terms on the right-hand side of
(A2) can be written as

$\displaystyle\lambda^{\beta-1}\int_{0}^{\infty}e^{-\lambda^{\beta}s}S_{\alpha}(s)u_{0}ds$
$\displaystyle=\int_{0}^{\infty}\lambda^{\beta-1}e^{-\lambda^{\beta}t^{\beta}}S_{\alpha}(t^{\beta})u_{0}d(t^{\beta})$
$\displaystyle=\int_{0}^{\infty}\beta(\lambda t)^{\beta-1}e^{-(\lambda
t)^{\beta}}S_{\alpha}(t^{\beta})u_{0}dt$
$\displaystyle=\int_{0}^{\infty}-\frac{1}{\lambda}\frac{d}{dt}[e^{-(\lambda
t)^{\beta}}]S_{\alpha}(t^{\beta})u_{0}dt$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\theta\omega_{\beta}(\theta)e^{-\lambda
t\theta}S_{\alpha}(t^{\beta})u_{0}d\theta dt$
$\displaystyle=\int_{0}^{\infty}e^{-\lambda
t}[\int_{0}^{\infty}\omega_{\beta}(\theta)S_{\alpha}(\frac{t^{\beta}}{\theta^{\beta}})u_{0}d\theta]dt,$
(A5)

and

$\displaystyle\int_{0}^{\infty}e^{-\lambda^{\beta}s}S_{\alpha}(s)\widehat{B}(\lambda)ds$
$\displaystyle=\int_{0}^{\infty}\beta t^{\beta-1}e^{-(\lambda
t)^{\beta}}S_{\alpha}(t^{\beta})\widehat{B}(\lambda)dt$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\beta t^{\beta-1}e^{-(\lambda
t)^{\beta}}S_{\alpha}(t^{\beta})e^{-\lambda s}B(u(s))dsdt$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\beta\omega_{\beta}(\theta)e^{-\lambda
t\theta}S_{\alpha}(t^{\beta})e^{-\lambda s}t^{\beta-1}B(u(s))d\theta dsdt$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\beta\omega_{\beta}(\theta)e^{-\lambda(t+s)}S_{\alpha}(\frac{t^{\beta}}{\theta^{\beta}})\frac{t^{\beta-1}}{\theta^{\beta}}B(u(s))d\theta
dsdt$ $\displaystyle=\int_{0}^{\infty}e^{-\lambda
t}[\beta\int_{0}^{t}\int_{0}^{\infty}\omega_{\beta}(\theta)S_{\alpha}(\frac{(t-s)^{\beta}}{\theta^{\beta}})\frac{(t-s)^{\beta-1}}{\theta^{\beta}}B(u(s))d\theta
ds]dt.$ (A6)

and

$\displaystyle\int_{0}^{\infty}e^{-\lambda^{\beta}s}S_{\alpha}(s)\widehat{G}(\lambda)ds$
$\displaystyle=\int_{0}^{\infty}\beta t^{\beta-1}e^{-(\lambda
t)^{\beta}}S_{\alpha}(t^{\beta})\widehat{G}(\lambda)dt$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\beta t^{\beta-1}e^{-(\lambda
t)^{\beta}}S_{\alpha}(t^{\beta})e^{-\lambda s}g(u(s))dW(s)dt$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\beta\omega_{\beta}(\theta)e^{-\lambda
t\theta}S_{\alpha}(t^{\beta})e^{-\lambda s}t^{\beta-1}g(u(s))d\theta dW(s)dt$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\beta\omega_{\beta}(\theta)e^{-\lambda(t+s)}S_{\alpha}(\frac{t^{\beta}}{\theta^{\beta}})\frac{t^{\beta-1}}{\theta^{\beta}}g(u(s))d\theta
dW(s)dt$ $\displaystyle=\int_{0}^{\infty}e^{-\lambda
t}[\beta\int_{0}^{t}\int_{0}^{\infty}\omega_{\beta}(\theta)S_{\alpha}(\frac{(t-s)^{\beta}}{\theta^{\beta}})\frac{(t-s)^{\beta-1}}{\theta^{\beta}}g(u(s))d\theta
dW(s)]dt.$ (A7)

Together with (A2) and (A5)-(A7) helps us to get

$\displaystyle\widehat{u}(\lambda)$
$\displaystyle=\int_{0}^{\infty}e^{-\lambda
t}[\int_{0}^{\infty}\omega_{\beta}(\theta)S_{\alpha}(\frac{t^{\beta}}{\theta^{\beta}})u_{0}d\theta]dt$
$\displaystyle\hskip 5.69054pt+\int_{0}^{\infty}e^{-\lambda
t}[\beta\int_{0}^{t}\int_{0}^{\infty}\omega_{\beta}(\theta)S_{\alpha}(\frac{(t-s)^{\beta}}{\theta^{\beta}})\frac{(t-s)^{\beta-1}}{\theta^{\beta}}B(u(s))d\theta
ds]dt$ $\displaystyle\hskip 5.69054pt+\int_{0}^{\infty}e^{-\lambda
t}[\beta\int_{0}^{t}\int_{0}^{\infty}\omega_{\beta}(\theta)S_{\alpha}(\frac{(t-s)^{\beta}}{\theta^{\beta}})\frac{(t-s)^{\beta-1}}{\theta^{\beta}}g(u(s))d\theta
dW(s)]dt.$ (A8)

Now, by means of inverse Laplace transform to (A8), we have achieved that

$\displaystyle u(t)$
$\displaystyle=\int_{0}^{\infty}\omega_{\beta}(\theta)S_{\alpha}(\frac{t^{\beta}}{\theta^{\beta}})u_{0}d\theta$
$\displaystyle\hskip
5.69054pt+\beta\int_{0}^{t}\int_{0}^{\infty}\omega_{\beta}(\theta)S_{\alpha}(\frac{(t-s)^{\beta}}{\theta^{\beta}})\frac{(t-s)^{\beta-1}}{\theta^{\beta}}B(u(s))d\theta
ds$ $\displaystyle\hskip
5.69054pt+\beta\int_{0}^{t}\int_{0}^{\infty}\omega_{\beta}(\theta)S_{\alpha}(\frac{(t-s)^{\beta}}{\theta^{\beta}})\frac{(t-s)^{\beta-1}}{\theta^{\beta}}g(u(s))d\theta
dW(s)$
$\displaystyle=\int_{0}^{\infty}\frac{1}{\beta}\theta^{-\frac{1}{\beta}-1}\omega_{\beta}(\theta^{-\frac{1}{\beta}})S_{\alpha}(t^{\beta}\theta)u_{0}d\theta$
$\displaystyle\hskip
5.69054pt+\int_{0}^{t}\int_{0}^{\infty}\theta^{-\frac{1}{\beta}}\omega_{\beta}(\theta^{-\frac{1}{\beta}})S_{\alpha}((t-s)^{\beta}\theta)(t-s)^{\beta-1}B(u(s))d\theta
ds$ $\displaystyle\hskip
5.69054pt+\int_{0}^{t}\int_{0}^{\infty}\theta^{-\frac{1}{\beta}}\omega_{\beta}(\theta^{-\frac{1}{\beta}})S_{\alpha}((t-s)^{\beta}\theta)(t-s)^{\beta-1}g(u(s))d\theta
dW(s).$ (A9)

Here, we also introduce the Mainardi’s Wright-type function

$\displaystyle M_{\beta}(\theta)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n}\theta^{n}}{n!\Gamma(1-\beta(1+n))}$
$\displaystyle=\frac{1}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}\theta^{n-1}}{(n-1)!}\Gamma(n\beta)\sin(n\pi\beta),$

where $0<\beta<1$ and $\theta\in(0,+\infty)$. Further, the relationships
between the probability density function $\omega_{\beta}(\theta)$ and
Mainardi’s Wright-type function $M_{\beta}(\theta)$ are shown that

$\displaystyle
M_{\beta}(\theta)=\frac{1}{\beta}\theta^{-\frac{1}{\beta}-1}\omega_{\beta}(\theta^{-\frac{1}{\beta}}).$

We denote the generalized Mittag-Leffler operators $E_{\beta}(t)$ and
$E_{\beta,\beta}(t)$ as

$\displaystyle
E_{\beta}(t)=\int_{0}^{\infty}M_{\beta}(\theta)S_{\alpha}(t^{\beta}\theta)d\theta,$

and

$\displaystyle E_{\beta,\beta}(t)=\int_{0}^{\infty}\beta\theta
M_{\beta}(\theta)S_{\alpha}(t^{\beta}\theta)d\theta.$

Therefore, the equation (A9) can be written as

$\displaystyle u(t)$
$\displaystyle=E_{\beta}(t)u_{0}+\int_{0}^{t}(t-s)^{\beta-1}E_{\beta,\beta}(t-s)B(u(s))ds$
$\displaystyle\hskip
5.69054pt+\int_{0}^{t}(t-s)^{\beta-1}E_{\beta,\beta}(t-s)g(u(s))dW(s).$ (A10)

Up to now, we have deduced the mild solution (A10) to the time-space
fractional stochastic Burgers equation (A1).

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# Understanding partonic energy loss from measured light charged particles and
jets in PbPb collisions at LHC energies

Prashant Shukla1,2† and Kapil Saraswat3,∗ 1Nuclear Physics Division, Bhabha
Atomic Research Center, Mumbai 400085, India.  2Homi Bhabha National
Institute, Anushakti Nagar, Mumbai 400094, India.  3Department of Physics, DSB
Campus, Kumaun University, Nainital - 263001, INDIA.  $^†$
<EMAIL_ADDRESS>$^∗$<EMAIL_ADDRESS>
###### Abstract

We perform a comprehensive study of partonic energy loss reflected in the
nuclear modification factors of charged particles and jets measured in PbPb
collisions at $\sqrt{s_{\rm NN}}$ = 2.76 and 5.02 TeV in wide transverse
momentum ($p_{\rm T}$) and centrality range. The $p_{\rm T}$ distributions in
pp collisions are fitted with a modified power law and the nuclear
modification factor in PbPb collisions can be obtained using effective shift
($\Delta p_{\rm T}$) in the spectrum measured at different centralities.
Driven by physics consideration, the functional form of energy loss given by
$\Delta p_{\rm T}$ can be assumed as power law with different power indices in
three different $p_{\rm T}$ regions. The power indices and the boundaries of
three $p_{\rm T}$ regions are obtained by fitting the measured nuclear
modification factor as a function of $p_{\rm T}$ in all collision centralities
simultaneously. The energy loss in different collisions centralities are
described in terms of fractional power of number of participants. It is
demanded that the power law functions in three $p_{T}$ regions and their
derivatives are continuous at the $p_{\rm T}$ boundaries. The $\Delta p_{\rm
T}$ for light charged particles is found to increase linearly with $p_{\rm T}$
in low $p_{\rm T}$ region below $\sim 5-6$ GeV/$c$ and approaches a constant
value in high $p_{\rm T}$ region above $\sim 22-29$ GeV/$c$ with an
intermediate power law connecting the two regions. The method is also used for
jets and it is found that for jets, the $\Delta p_{\rm T}$ increases
approximately linearly even at very high $p_{\rm T}$.

##  1 Introduction

The collisions of heavy ions at ultra relativistic energies are performed to
create and study bulk strongly interacting matter at high temperatures. The
data from RHIC and LHC provide strong evidences of formation of a new state of
matter known as quark gluon plasma (QGP) in these collisions [1].

The light charged hadrons and jets transverse momentum ($p_{\rm{T}}$) spectra
give insight into the particle production mechanism in pp collisions. The
partonic energy energy loss is reflected in these particles when measured in
heavy ion collisions due to jet-quenching [2] which measures the opacity of
the medium. A modified power law distribution [3, 4, 5] describes the
$p_{\rm{T}}$ spectra of the hadrons in pp collisions in terms of a power index
$n$ which determines the initial production in partonic collisions. In
Ref.[6], the power law function is applied to heavy ion collisions as well
which includes the transverse flow in low $p_{\rm{T}}$ region and the in-
medium energy loss (also in terms of power law) in high $p_{\rm{T}}$ region.

The spectra of hadrons are measured in both pp and AA collisions and nuclear
modification factor ($R_{\rm{AA}}$) is obtained. The energy loss of partons
can be connected to horizontal shift in the scaled hadron spectra in AA with
respect to pp spectra as done by PHENIX measurement [7]. Their measurements of
neutral pions upto $p_{\rm T}\sim 10$ GeV/$c$ are consistent with the scenario
where the momentum shift $\Delta p_{\rm T}$ is proportional to $p_{\rm T}$. In
similar approach, the authors in Ref. [8], extracted the fractional energy
loss from the measured nuclear modification factor of hadrons as a function of
$p_{\rm{T}}$ below 10 GeV/$c$ in AuAu collisions at $\sqrt{s_{\rm{NN}}}$ = 200
GeV. They also considered that the energy loss increases linearly with $p_{\rm
T}$. In recent PHENIX work [9], fractional energy loss was obtained in the
hadron spectrum measured upto $p_{\rm T}=20$ GeV/$c$ in heavy ions collisions
at RHIC and LHC energy and is not found to be constant. This means that a
constant fractional energy loss (energy loss varying linearly with $p_{\rm
T}$) can be applicable only to low $p_{\rm T}$ RHIC measurements.

There are many recent studies which use so-called shift formalism to study the
energy loss. The work in Ref.[10] is based on shift formalism and describes
the transverse momentum ($p_{\rm{T}}$), rapidity ($y$) and centrality
dependences of the measured jet nuclear modification factor ($R_{\rm{AA}}$) in
PbPb collisions at LHC. They assume that the energy loss is given by a power
law in terms of $p_{\rm T}$, the value of power index is obtained between 0.4
to 0.8 by fitting the $R_{\rm{AA}}$ as a function of $p_{\rm{T}}$ and
centralities. They also found that the energy loss linearly increases with
number of participants. Using the same method they study the magnitude and the
colour charge dependence of the energy loss in PbPb collisions at LHC energies
using the measured data of the inclusive jet suppression [11]. The authors of
the Ref.[12] work on inclusive charged particle spectra measured in the range
($5<p_{\rm T}<20$ GeV/$c$) in heavy ion collisions at RHIC and LHC. They
assume that the energy loss linearly increases with $p_{\rm T}$ and
pathlength.

There are detailed calculations of energy loss of partons in the hot medium
[see e.g. Refs. [13, 14]. Phenomenological models tend to define simple
dependence of the radiative energy loss of the parton on the energy of the
parton inside the medium [for a discussions see Ref. [15]]. The energy loss
can be characterized in terms of coherence length $l_{\rm{coh}}$, which is
associated with the formation time of gluon radiation by a group of scattering
centres. If $l_{\rm{coh}}$ is less then the mean free path $(\lambda)$ of the
parton, the energy loss is proportional to the energy of the parton. If
$l_{\rm{coh}}$ is greater than $\lambda$ but less than the path length ($L$)
of the parton ($\lambda<l_{\rm{coh}}<L)$, the energy loss is proportional to
the square root of the energy of the parton. In the complete coherent regime,
$l_{\rm{coh}}>L$, the energy loss per unit length is independent on energy but
proportional to the parton path length implying that $\Delta p_{T}$ is
proportional to square of pathlength. There is a nice description of charged
particle spectra at RHIC and LHC using such a prescription by dividing the
$p_{\rm T}$ spectra in three regions [16, 17]. For low and intermediate energy
partons, $\Delta p_{T}$ is assumed to be linearly dependent on $L$ [18]. The
work in Ref. [19] studies the energy loss of jets in terms of exponent of the
number of participants. It should be remembered that the fragmentation changes
the momentum between the partonic stage (at which energy is lost) and hadron
formation. There are models which say that softening occurs at fragmentation
stage due to color dynamics [See e.g. Ref. [20]].

In general, one can assume that the energy loss of partons in the hot medium
as a function of parton energy is in the form of power law where the power
index ranges from 0 (constant) to 1 (linear). Guided by these considerations,
in the present work, the $p_{\rm T}$ loss has been assumed as power law with
different power indices in three different $p_{\rm T}$ regions. The energy
loss in different collisions centralities are described in terms of fractional
power of number of participants. The $p_{\rm T}$ distributions in pp
collisions are fitted with a modified power law and $R_{\rm AA}$ in PbPb
collisions can be obtained using effective shift ($\Delta p_{\rm T}$) in the
$p_{\rm T}$ spectrum measured at different centralities. The power index and
the boundaries of three $p_{\rm T}$ regions are obtained by fitting the
measured $R_{\rm{AA}}$ of charged particles and jets in PbPb collisions at
$\sqrt{s_{\rm NN}}$ = 2.76 and 5.02 TeV in large transverse momentum ($p_{\rm
T}$) and centrality range. The shift $\Delta p_{\rm{T}}$ includes the medium
effect, mainly energy loss of parent quark inside the plasma. The shift
$\Delta p_{T}$ can be approximatively understood as the partonic energy loss
in the case of jets while in case of hadrons it is not simple due to
complicated correlations. Often we refer to the shift $\Delta p_{\rm{T}}$ as
the energy loss.

##  2 Nuclear Modification Factor and Energy Loss

The nuclear modification factor $R_{\rm{AA}}$ of hadrons is defined as the
ratio of yield of the hadrons in AA collision and the yield in pp collision
with a suitable normalization

$R_{\rm{AA}}(p_{\rm{T}},b)=\frac{1}{T_{\rm{AA}}}{\frac{d^{2}N^{AA}(p_{\rm{T}},b)}{dp_{\rm{T}}dy}}/{\frac{d^{2}\sigma^{pp}(p_{\rm{T}},b)}{dp_{\rm{T}}dy}}~{}.$
(1)

Here, $T_{\rm{AA}}$ is the nuclear overlap function which can be calculated
from the nuclear density distribution. High $p_{\rm T}$ partons traversing the
medium loose energy and cause the suppression of hadrons at high $p_{\rm T}$
indicated by value of $R_{\rm{AA}}$ which is less than one. The transverse
momentum distribution of hadrons in pp collisions can be described by the
Hagedorn function which is a QCD-inspired summed power law [21] given as

$\,\,\,\,\,\,\,\,\,\,\,\frac{d^{2}\sigma^{\rm{pp}}}{dp_{\rm{T}}dy}=A_{n}~{}2\pi
p_{\rm{T}}~{}\Bigg{(}1+\frac{p_{\rm{T}}}{p_{0}}\Bigg{)}^{-n}~{}.$ (2)

where $n$ is the power and $A_{n}$ and $p_{0}$ are other parameters which are
obtained by fitting the experimental pp spectrum. The yield in the AA
collision can be obtained by shifting the spectrum by $\Delta p_{\rm T}$ as

$\displaystyle\,\,\,\,\,\frac{1}{T_{\rm{AA}}}\frac{d^{2}N^{\rm{AA}}}{dp_{\rm{T}}dy}$
$\displaystyle=\frac{d^{2}\sigma^{\rm{pp}}(p^{\prime}_{\rm{T}}=p_{\rm{T}}+\Delta
p_{\rm{T}})}{dp^{\prime}_{\rm{T}}dy}\frac{dp^{\prime}_{\rm{T}}}{dp_{\rm{T}}}$
(3)
$\displaystyle=\frac{d^{2}\sigma^{\rm{pp}}(p^{\prime}_{\rm{T}})}{dp^{\prime}_{\rm{T}}dy}\Bigg{(}1+\frac{d(\Delta
p_{\rm{T}})}{dp_{\rm{T}}}\Bigg{)}~{}.$

The reasoning behind writing Eq. 3 lies in the assumption that particle yield
at a given $p_{\rm{T}}$ in AA collisions would have been equal to the yield in
pp collisions at $p_{\rm{T}}+\Delta p_{\rm{T}}$. The shift $\Delta p_{\rm{T}}$
includes the medium effect, mainly energy loss of parent quark inside the
plasma.

The nuclear modification factor $R_{\rm{AA}}$ can be obtained as

$\displaystyle R_{\rm{AA}}=\left(1+{\Delta p_{\rm{T}}\over
p_{0}+p_{\rm{T}}}\right)^{-n}\,\,\left({p_{\rm{T}}+\Delta p_{\rm{T}}\over
p_{\rm{T}}}\right)\,\left(1+{d(\Delta p_{\rm{T}})\over dp_{\rm{T}}}\right)$
(4)

The energy loss given by $p_{T}$ loss, $\Delta p_{T}$ can be extracted by
fitting the experimental data on $R_{\rm AA}$ with Eq. 4. The $\Delta p_{\rm
T}$ and its derivative will go as input in the above equation and can be
assumed to be in the form of the power law with different values of power
indices in three different $p_{\rm T}$ regions as follows

$\displaystyle\Delta p_{\rm T}=\left\\{\begin{array}[]{l}a_{1}~{}(p_{\rm
T}-C_{1})^{\alpha_{1}}~{}~{}~{}{\rm for}~{}~{}~{}p_{\rm T}<p_{\rm
T_{1}}~{}~{}~{},\\\ a_{2}~{}(p_{\rm T}-C_{2})^{\alpha_{2}}~{}~{}~{}{\rm
for}~{}~{}~{}p_{\rm T_{1}}\leq p_{\rm T}<p_{\rm T_{2}}~{}~{},\\\
a_{3}~{}(p_{\rm T}-C_{3})^{\alpha_{3}}~{}~{}~{}{\rm for}~{}~{}~{}p_{\rm T}\geq
p_{\rm T_{2}}~{}.\end{array}\right\\}$ (8)

The parameter $a_{1}$ in our work contains the pathlength dependence. The
pathlength $L$ scales as the square root of number of participants as
$\sqrt{N_{\rm part}}$. For low and intermediate energy partons, $\Delta p_{T}$
can be assumed to be linearly dependent on $L$ [18]. If the scattering happens
in complete coherent regime where the whole medium acts as one coherent source
of radiation, the $\Delta p_{T}$ approaches quadratic dependence on $L$. The
work in Ref. [19] studies the energy loss of jets in terms of exponent of the
number of participants. Without complicating the calculations we can assume
that $a_{1}=M\,(N_{\rm{part}}/(2A))^{\beta}$. The exponent $\beta$ is obtained
separately for each dataset. The parameter $M$ relies on the energy density of
the medium depending on the collision energy but has a same value for all
centralities. The boundaries of the $p_{\rm T}$ regions $p_{{\rm{T}}_{1}}$,
$p_{{\rm{T}}_{2}}$ and the power indices $\alpha_{1}$, $\alpha_{2}$ and
$\alpha_{3}$ in the three different regions are used as free parameters while
fitting the $R_{\rm{AA}}$ measured at different centralities simultaneously.
The parameter $C_{1}$ is fixed to a suitable value to choose a lower $p_{\rm
T}$ cutoff and the parameters $C_{2}$, $C_{3}$, $a_{2}$ and $a_{3}$ are
obtained by assuming the function and its derivative to be continuous at
boundaries.

Demanding that the function in Eq. 8 to be continuous at
$p_{\rm{T}}=p_{{\rm{T}}_{1}}$ and at $p_{\rm{T}}=p_{{\rm{T}}_{2}}$ we obtain

$a_{2}=a_{1}~{}\frac{(p_{{\rm{T}}_{1}}-C_{1})^{\alpha_{1}}}{(p_{{\rm{T}}_{1}}-C_{2})^{\alpha_{2}}}~{}~{}.$
(9)

$a_{3}=a_{2}~{}\frac{(p_{{\rm{T}}_{2}}-C_{2})^{\alpha_{2}}}{(p_{{\rm{T}}_{2}}-C_{3})^{\alpha_{3}}}~{}~{}.$
(10)

Demanding that at $p_{\rm{T}}=p_{{\rm{T}}_{1}}$, the derivative of Eq. 8 is
continuous.

$a_{1}~{}\alpha_{1}~{}(p_{{\rm{T}}_{1}}-C_{1})^{(\alpha_{1}-1)}=a_{2}~{}\alpha_{2}~{}(p_{{\rm{T}}_{1}}-C_{2})^{(\alpha_{2}-1)}~{}~{},$
(11)

Using the value of $a_{2}$ from Eq. 9

$\frac{\alpha_{1}}{(p_{{\rm{T}}_{1}}-C_{1})}=\frac{\alpha_{2}}{(p_{{\rm{T}}_{1}}-C_{2})}~{}~{},$
(12)

$C_{2}=p_{{\rm{T}}_{1}}-\frac{\alpha_{2}}{\alpha_{1}}(p_{{\rm{T}}_{1}}-C_{1})~{}.$
(13)

Similarly, demanding the derivative to be continuous at
$p_{\rm{T}}=p_{{\rm{T}}_{2}}$, we get $C_{3}$

$C_{3}=p_{{\rm{T}}_{2}}-\frac{\alpha_{3}}{\alpha_{2}}(p_{{\rm{T}}_{2}}-C_{2})~{}.$
(14)

In case of jets we consider only one region as the data starts from very high
$p_{T}$ above 40 GeV/$c$.

##  3 Results and Discussions

Figure 1 shows the invariant yields of the charged particles as a function of
the transverse momentum $p_{\rm{T}}$ for pp collisions at $\sqrt{s}$ = 2.76
TeV measured by the ALICE experiment [22]. The solid curve is the Hagedorn
distribution fitted to the $p_{\rm{T}}$ spectra with the parameters given in
Table 1.

Figure 1: The invariant yields of the charged particles as a function of
transverse momentum $p_{\rm{T}}$ for pp collision at $\sqrt{s}$ = 2.76 TeV
measured by the ALICE experiment [22]. The solid curve is the fitted Hagedorn
function. Table 1: Parameters for the Hagedorn function obtained by fitting
the transverse momentum spectra of charged particles and jets measured in pp
collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 and 5.02 TeV.

Parameters | Charged particles | Jets
---|---|---
|  $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV |  $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV |  $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV |  $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV
$n$ | 7.26 $\pm$ 0.08 | 6.70 $\pm$ 0.14 | 8.21 $\pm$ 1.55 | 7.90 $\pm$ 0.50
$p_{0}$ (GeV/$c$) | 1.02 $\pm$ 0.04 | 0.86 $\pm$ 0.16 | 18.23 $\pm$ 1.69 | 19.21 $\pm$ 3.20
$\chi^{2}/\rm{NDF}$ | 0.15 | 0.06 | 0.23 | 0.95

Figure 2 shows the nuclear modification factor $R_{\rm{AA}}$ of the charged
particles as a function of the transverse momentum $p_{\rm{T}}$ for different
centrality classes in PbPb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV
measured by the ALICE experiment [23]. The solid lines are the function given
by Eq. 4. The modeling of centrality dependence using $N_{\rm part}^{\beta}$
with $\beta=0.58$ gives a very good description of the data. The extracted
parameters of the shift $\Delta p_{\rm{T}}$ obtained by fitting the
$R_{\rm{AA}}$ measured in different centrality classes of PbPb collisions at
$\sqrt{s_{\rm{NN}}}$ = 2.76 TeV are given in Table 2 along with value of
$\chi^{2}/\rm{NDF}$. It shows that the $\Delta p_{\rm{T}}$ increases almost
linearly ($p_{\rm{T}}^{0.97}$) upto $p_{\rm{T}}\simeq$ 5 GeV/$c$ in
confirmation with earlier studies. After that it increases slowly with power
$\alpha=0.224$ upto a $p_{\rm{T}}$ value 29 GeV/$c$ and then becomes constant
for higher values of $p_{\rm{T}}$.

Figure 2: The nuclear modification factor $R_{\rm{AA}}$ of the charged particles as a function of transverse momentum $p_{\rm{T}}$ for various centrality classes in PbPb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV measured by the ALICE experiment [23]. The solid curves are the $R_{\rm{AA}}$ fitting function (Eq. 4). Table 2: The extracted parameters of the shift $\Delta p_{\rm{T}}$ obtained by fitting the charged particles $R_{\rm{AA}}$ measured in different centrality classes of PbPb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 and 5.02 TeV. Parameters |  $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV |  $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV
---|---|---
$M$ | 0.75 $\pm$ 0.02 | 0.80 $\pm$ 0.038
$p_{{\rm{T}}_{1}}$ (GeV/$c$) | 5.03 $\pm$ 0.15 | 5.10 $\pm$ 0.22
$p_{{\rm{T}}_{2}}$ (GeV/$c$) | 29.0 $\pm$ 0.1 | 22.2 $\pm$ 4.1
$C_{1}$ (GeV/$c$) | 1.0 (fixed) | 1.0
$\alpha_{1}$ | 0.97 $\pm$ 0.02 | 0.95 $\pm$ 0.04
$\alpha_{2}$ | 0.22 $\pm$ 0.02 | 0.22 $\pm$ 0.03
$\alpha_{3}$ | 0.05 $\pm$ 0.13 | 0.05 $\pm$ 0.10
$\frac{\chi^{2}}{\rm{NDF}}$ | 0.35 | 0.38

Figure 3 shows the energy loss $\Delta p_{\rm{T}}$ of the charged particles as
a function of the transverse momentum $p_{\rm{T}}$ for different centrality
classes in PbPb collision at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV. The $\Delta
p_{\rm{T}}$ is obtained from Eq. 8 with parameters given in Table 2. The
$\Delta p_{\rm{T}}$ increases from peripheral to the most central collision
regions as per $N_{\rm part}^{0.58}$. The figure shows that the $\Delta
p_{\rm{T}}$ increases almost linearly upto $p_{\rm{T}}\sim 5$ GeV/$c$. After
that it increases slowly upto a $p_{\rm{T}}$ value 29 GeV/$c$ and then becomes
constant for higher values of $p_{\rm{T}}$.

Figure 3: The energy loss $\Delta p_{\rm{T}}$ of the charged particles as a
function of transverse momentum $p_{\rm{T}}$ in PbPb collision at
$\sqrt{s_{\rm{NN}}}$ = 2.76 TeV for different centrality classes.

Figure 4 shows the invariant yields of the charged particles as a function of
the transverse momentum $p_{\rm{T}}$ for pp collisions at $\sqrt{s}$ = 5.02
TeV measured by the CMS experiment [24]. The solid lines are the Hagedorn
function fitted to the measured $p_{\rm{T}}$ spectra the parameters of which
are given in Table 1.

Figure 4: The invariant yields of the charged particles as a function of
transverse momentum $p_{\rm{T}}$ for pp collision at $\sqrt{s}$ = 5.02 TeV
measured by the CMS experiment [24]. The solid curve is the fitted Hagedorn
function.

Figure 5 shows the nuclear modification factor $R_{\rm{AA}}$ of the charged
particles as a function of the transverse momentum $p_{\rm{T}}$ for different
centrality classes in PbPb collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV
measured by the CMS experiment [24]. The solid curves are the $R_{\rm{AA}}$
fitting function (Eq. 4). Here also the modeling of centrality dependence
using $N_{\rm part}^{\beta}$ with $\beta=0.58$ gives a good description of the
data. The extracted parameters of the shift $\Delta p_{\rm{T}}$ obtained by
fitting the $R_{\rm{AA}}$ measured in different centrality classes of PbPb
collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV are given in Table 2 along with
the value of $\chi^{2}/\rm{NDF}$. It shows that the $\Delta p_{\rm{T}}$
increases almost linearly ($p_{\rm{T}}^{0.96}$) similar to the case at 2.76
TeV for $p_{\rm T}$ upto 5.1 GeV/$c$. After that it increases slowly with
power $\alpha=0.22$ upto a $p_{\rm{T}}$ value 22.2 GeV/$c$ and then becomes
constant for higher values of $p_{\rm{T}}$ right upto 160 GeV/$c$.

Figure 5: The nuclear modification factor $R_{\rm{AA}}$ of the charged
particles as a function of transverse momentum $p_{\rm{T}}$ for various
centrality classes in PbPb collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV
measured by the CMS experiment [24]. The solid lines are the $R_{\rm{AA}}$
fitting function (Eq. 4).

Figure 6 shows the energy loss $\Delta p_{\rm{T}}$ of the charged particles as
a function of the transverse momentum $p_{\rm{T}}$ for different centrality
classes in PbPb collision at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV. The $\Delta
p_{\rm{T}}$ is obtained from Eq. 8 with the parameters given in Table 2. The
$\Delta p_{\rm{T}}$ becomes constant for $p_{\rm{T}}$ in the range 22 GeV/$c$
to 160 GeV/c and increases as the collisions become more central.

Figure 6: The energy loss $\Delta p_{\rm{T}}$ of the charged particles as a
function of transverse momentum $p_{\rm{T}}$ in PbPb collision at
$\sqrt{s_{\rm{NN}}}$ = 5.02 TeV for different centrality classes.

Figure 7 shows the comparison of energy loss $\Delta p_{\rm{T}}$ of the
charged particles as a function of the transverse momentum $p_{\rm{T}}$ for 0
- 5 $\%$ centrality class in PbPb collision at $\sqrt{s_{\rm{NN}}}$ = 2.76 and
at 5.02 TeV. The $\Delta p_{\rm{T}}$ at 5.02 TeV is similar but slightly more
than that at 2.76 TeV.

Figure 7: The energy loss $\Delta p_{\rm{T}}$ of the charged particles as a
function of transverse momentum $p_{\rm{T}}$ in PbPb collision at
$\sqrt{s_{\rm{NN}}}$ = 2.76 and 5.02 TeV for 0 - 5 $\%$ centrality.

Figure 8 shows the yields of the jets as a function of the transverse momentum
$p_{\rm{T}}$ for pp collisions at $\sqrt{s}$ = 2.76 TeV measured by the ATLAS
experiment [25]. The solid curve is the Hagedorn distribution fitted to the
$p_{\rm{T}}$ spectra, the parameters of which are given in Table 1.

Figure 8: The yields of the jets as a function of transverse momentum
$p_{\rm{T}}$ for pp collision at $\sqrt{s}$ = 2.76 TeV measured by the ATLAS
experiment [25]. The solid curve is the fitted Hagedorn distribution.

Figure 9 shows the nuclear modification factor $R_{\rm{AA}}$ of the jets as a
function of the transverse momentum $p_{\rm{T}}$ for different centrality
classes in PbPb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV measured by the
ATLAS experiment [25]. The solid curves are the $R_{\rm{AA}}$ fitting function
(Eq. 4). Here the modeling of centrality dependence using $N_{\rm
part}^{\beta}$ with $\beta=0.60$ gives a good description of the data. The
extracted parameters of the energy loss obtained by fitting the $R_{\rm{AA}}$
measured in different centrality classes of PbPb collisions at
$\sqrt{s_{\rm{NN}}}$ = 2.76 TeV are given in Table 3 along with the value of
$\chi^{2}/\rm{NDF}$. It shows that the $\Delta p_{\rm{T}}$ increases as
$p_{\rm{T}}^{0.76}$ at all the values of $p_{\rm T}$ measured for jets.

Figure 9: The nuclear modification factor $R_{\rm{AA}}$ of jets as a function of transverse momentum $p_{\rm{T}}$ for various centrality classes in PbPb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV measured by the ATLAS experiment [25]. The solid curves are the $R_{\rm{AA}}$ fitting function given by Eq. 4. Table 3: The extracted parameters of the shift $\Delta p_{\rm{T}}$ obtained by fitting the jet $R_{\rm{AA}}$ measured in different centrality classes of PbPb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 and 5.02 TeV. Parameters |  $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV |  $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV
---|---|---
$M$ | 0.33 $\pm$ 0.1 | 0.40 $\pm$ 0.12
$C$ (GeV/$c$) | -55.1 $\pm$ 22.7 | -119 $\pm$ 15
$\alpha$ | 0.76 $\pm$ 0.08 | 0.72 $\pm$ 0.01
$\frac{\chi^{2}}{\rm{NDF}}$ | 0.30 | 0.25

Figure 10 shows the shift $\Delta p_{\rm{T}}$ of the jets as a function of the
transverse momentum $p_{\rm{T}}$ for different centrality classes in PbPb
collision at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV. The $\Delta p_{\rm{T}}$ is
obtained from Eq. 8 using the parameters given in Table 3. The $\Delta
p_{\rm{T}}$ increases from peripheral to the most central collision regions.
The figure shows that the $\Delta p_{\rm{T}}$ increases almost linearly at all
the values of $p_{\rm T}$ measured for jets.

Figure 10: The shift $\Delta p_{\rm{T}}$ of the jets as a function of
transverse momentum $p_{\rm{T}}$ in PbPb collision at $\sqrt{s_{\rm{NN}}}$ =
2.76 TeV for different centrality classes.

Figure 11 shows the yields of the jets as a function of the transverse
momentum $p_{\rm{T}}$ for pp collisions at $\sqrt{s}$ = 5.02 TeV measured by
the ATLAS experiment [26]. The solid curve is the Hagedorn distribution with
the parameters given in Table 1.

Figure 11: The yields of the jets as a function of transverse momentum
$p_{\rm{T}}$ for pp collision at $\sqrt{s}$ = 5.02 TeV measured by the ATLAS
experiment [26]. The solid curve is the fitted Hagedorn distribution.

Figure 12 shows the nuclear modification factor $R_{\rm{AA}}$ of the jets as a
function of the transverse momentum $p_{\rm{T}}$ for different centrality
classes in PbPb collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV measured by the
ATLAS experiment [26]. The solid curves are the $R_{\rm{AA}}$ fitting function
(Eq. 4). The modeling of centrality dependence is done with $N_{\rm
part}^{\beta}$ and the value of exponent is obtained as $\beta=0.75$. The
extracted parameters of the energy loss obtained by fitting the $R_{\rm{AA}}$
measured in different centrality classes of PbPb collisions at
$\sqrt{s_{\rm{NN}}}$ = 5.02 TeV are given in Table 3, along with the value of
$\chi^{2}/\rm{NDF}$. It shows that the $\Delta p_{\rm{T}}$ increases as
$p_{\rm{T}}^{0.72}$ at all the values of $p_{\rm T}$ measured for jets similar
to the case of jets at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV

Figure 12: The nuclear modification factor $R_{\rm{AA}}$ of the jets as a
function of transverse momentum $p_{\rm{T}}$ for various centrality classes in
PbPb collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV measured by the ATLAS
experiment [26]. The solid curves are the $R_{\rm{AA}}$ fitting function given
by Eq. 4.

Figure 13 shows the energy loss $\Delta p_{\rm{T}}$ of the jets as a function
of the transverse momentum $p_{\rm{T}}$ for different centrality classes in
PbPb collision at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV. The $\Delta p_{\rm{T}}$ is
obtained from Eq. 8 with the parameters given in Table 3. The $\Delta
p_{\rm{T}}$ increases from peripheral to the most central collision regions.
The figure shows that the $\Delta p_{\rm{T}}$ increases almost linearly at all
the values of $p_{\rm T}$ measured for jets.

Figure 13: The energy loss $\Delta p_{\rm{T}}$ of the jets as a function of
transverse momentum $p_{\rm{T}}$ in PbPb collision at $\sqrt{s_{\rm{NN}}}$ =
5.02 TeV for different centrality classes.

Figure 14 shows the energy loss $\Delta p_{\rm{T}}$ of the jets as a function
of the transverse momentum $p_{\rm{T}}$ in the most central (0-10%) PbPb
collision at $\sqrt{s_{\rm{NN}}}$ = 2.76 and 5.02 TeV. These are compared with
the $\Delta p_{\rm{T}}$ obtained for charged particles in the 0-5% centrality
class of PbPb collision at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV. The energy loss
$\Delta p_{\rm{T}}$ in case of jets for both the energies increases with
$p_{\rm{T}}$. The values of $\Delta p_{\rm{T}}$ for jets at 5.02 TeV is more
than that at 2.76 TeV. This behaviour at high $p_{\rm{T}}$ is very different
from the energy loss of charged particles which becomes constant in these
$p_{\rm{T}}$ regions. The modeling of centrality dependence of energy loss has
been done using $N_{\rm part}^{\beta}$. For charged particles, the centrality
dependence of $p_{T}$ shift is found to be $N_{\rm part}^{0.58}$ which
corresponds to $L^{1.18}$. The centrality dependence for jets at $\sqrt{s_{\rm
NN}}$ = 2.76 TeV is found to be $N_{\rm part}^{0.60}$. In case of jets at 5
TeV, the centrality dependence of energy loss is found to be $N_{\rm
part}^{0.75}$ corresponding to $L^{1.5}$ which means that the jets even at
very high energy are still away from complete coherent regime.

Figure 14: The energy loss $\Delta p_{\rm{T}}$ of the jets as a function of
transverse momentum $p_{\rm{T}}$ in the most central PbPb collision at
$\sqrt{s_{\rm{NN}}}$ = 2.76 and 5.02 TeV. The $\Delta p_{\rm{T}}$ obtained for
charged particles in the most central PbPb collision at $\sqrt{s_{\rm{NN}}}$ =
5.02 TeV is also shown.

##  4 Conclusions

We presented a study of partonic energy loss with $p_{T}$ shift extracted from
the measured $R_{\rm{AA}}$ of charged particles and jets in PbPb collisions at
$\sqrt{s_{\rm NN}}$ = 2.76 and 5.02 TeV in wide transverse momentum and
centrality range. The functional form of energy loss given by $\Delta p_{\rm
T}$ has been assumed as power law with different power indices in three
different $p_{\rm T}$ regions driven by physics considerations. The power
indices and the boundaries of three $p_{\rm T}$ regions are obtained by
fitting the experimental data of $R_{\rm{AA}}$ as a function of $p_{\rm T}$
and centrality. The energy loss for light charged particles is found to
increase linearly with $p_{\rm T}$ in low $p_{\rm T}$ region below 5-6 GeV/$c$
and approaches a constant value in high $p_{\rm T}$ region above 25 GeV/$c$
with an intermediate power law connecting the two regions. The $\Delta
p_{\rm{T}}$ at 5.02 TeV is similar but slightly more than that at 2.76 TeV. In
case of jets we consider only one $p_{T}$ region and it is found that for
jets, the energy loss increases almost linearly even at very high $p_{\rm T}$.
The modeling of centrality dependence of energy loss has been done using
$N_{\rm part}^{\beta}$. For charged particles, the centrality dependence of
$p_{T}$ shift is found to be $N_{\rm part}^{0.58}$ which corresponds to
$L^{1.18}$. The centrality dependence for jets at $\sqrt{s_{\rm NN}}$ = 2.76
TeV goes as $N_{\rm part}^{0.60}$. In case of jets at 5 TeV, the centrality
dependence of energy loss is found to be $N_{\rm part}^{0.75}$ corresponding
to $L^{1.5}$ which means that the jets even at very high energy are still away
from complete coherent regime.

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