# Smoothing effect and Derivative formulas for Ornstein–Uhlenbeck processes
driven by subordinated cylindrical Brownian noises

Alessandro Bondi

###### Abstract

We investigate the concept of cylindrical Wiener process subordinated to a
strictly $\alpha$–stable Lévy process, with $\alpha\in\left(0,1\right)$, in an
infinite dimensional, separable Hilbert space, and consider the related
stochastic convolution. We then introduce the corresponding Ornstein–Uhlenbeck
process, focusing on the regularizing properties of the Markov transition
semigroup defined by it. In particular, we provide an explicit, original
formula –which is not of Bismut–Elworthy–Li’s type– for the Gateaux
derivatives of the functions generated by the operators of the semigroup, as
well as an upper bound for the norm of their gradients. In the case
$\alpha\in\left(\frac{1}{2},1\right)$, this estimate represents the starting
point for studying the Kolmogorov equation in its mild formulation.

###### keywords:

Subordinated cylindrical Wiener process , Isotropic $\alpha$–stable processes,
Markov transition semigroup, Derivative formulas , Gradient estimates.

##  1 Introduction

The aim of the paper is to analyze the Ornstein–Uhlenbeck processes
$Z^{x},\,x\in H$, being $H$ an infinite dimensional, separable Hilbert space.
They are defined as the $H$–valued, mild solutions of the linear stochastic
differential equations

$dZ^{x}_{t}=AZ_{t}^{x}dt+\sqrt{Q}\,dW_{L_{t}},\quad Z^{x}_{0}=x\in H,$

where $A\colon\mathcal{D}\left(A\right)\subset H\to H$ is a linear,
selfadjoint, negative definite, unbounded operator, and $Q\colon H\to H$ is a
linear, bounded, nonnegative definite operator. By construction, $A$ and $Q$
share a common CONS of eigenvectors for $H$: it is denoted by
$\left(e_{n}\right)_{n}$. The main novelty of our work consists in the
structure of the noise $W_{L}$. Intuitively speaking, it can be thought of as

$W_{L_{t}}=\sum_{n=1}^{\infty}\beta^{n}_{L_{t}}e_{n},\quad t\geq 0,$

where $\left(\beta^{n}\right)_{n}$ is a sequence of independent Brownian
motions and $L=\left(L_{t}\right)_{t}$ is an independent, strictly
$\alpha$–stable subordinator representing the random time change, for
$\alpha\in\left(0,1\right)$. Therefore $W_{L}$ is nothing else than a
subordinated cylindrical Wiener process, even if, of course, the convergence
of the series needs to be formally investigated.

In literature the canonical case is the Gaussian one, which involves a
cylindrical Wiener process
$W_{t}=\sum_{n=1}^{\infty}\beta^{n}_{t}e_{n},\,t\geq 0.$ There is a
well–established theory concerning this setting, and we may refer to the book
[5] for an extensive collection of results on the subject. Another important
framework is the one proposed by [11], where the authors deal with a
cylindrical, $\alpha$–stable Lévy process
$Z_{t}=\sum_{n=1}^{\infty}\zeta^{n}_{t}e_{n},\,t\geq 0$: here
$\left(\zeta^{n}\right)_{n}$ are independent, real–valued, symmetric
$\alpha$–stable Lévy processes, for $\alpha\in\left(0,2\right)$. Despite the
interesting generalization offered by this approach, the structure of the
noise could be questionable in some applications, especially in physics. In
fact, fixing $t>0$ and $N\in\mathbb{N}$, the corresponding Galerkin projection
of $Z_{t}$ has characteristic function

$\mathbb{E}\left[e^{i\left\langle
h,\sum_{n=1}^{N}\zeta^{n}_{t}e_{n}\right\rangle}\right]=e^{-t\gamma^{\alpha}\sum_{n=1}^{N}\left|\left\langle
h,e_{n}\right\rangle\right|^{\alpha}},\quad h\in H,$

for some constant $\gamma>0$. Therefore with respect to the Brownian case we
lose the isotropy, that is, the rotational stability of the noise, which is a
property as desirable as realistic for a random perturbation.

Motivated by this argument, it is worth studying the results contained in the
aforementioned works also for the subordinated process $W_{L}$, since its
Galerkin projections are $2\alpha$–stable, isotropic Lévy processes, as we
shall discuss in Section 2. With this purpose in mind, the present paper just
focuses on the linear case, i.e., the Ornstein–Ulenbeck (henceforth
abbreviated as _OU_) one. A number of complications arises from the approach
that we suggest, the most evident being the lack of independence of the
processes $\left(\beta^{n}_{L}\right)_{n}$, which in general makes the
techniques used in the other cases unfeasible. Nevertheless, the structure of
the noise allows to construct the objects of our interest and to carry out our
arguments with the intuition that, conditioning on the $\sigma$–algebra
generated by the subordinator $L$, we are dealing with time–shifted Brownian
motions.

The paper is structured as follows. In Section 2 we carefully describe the
theoretical framework of our analysis and suggest a natural procedure
–essentially relying on _Markov’s inequality_ – to construct both the
subordinated cylindrical Wiener process $W_{L}$, or, more precisely,
$\sqrt{Q}W_{L}$, which in general takes values in a Hilbert space bigger than
$H$, and the stochastic convolution $\tilde{Z}_{A,Q}$, which is a $H$–valued
random process instead.
In Section 3 we are concerned with the smoothing effect of the Markov
transition semigroup $R=\left(R_{t}\right)_{t\geq 0}$ associated with
$\left(Z^{x}\right)_{x\in H}$, defined by

$R_{t}\phi\left(x\right)\coloneqq\mathbb{E}\left[\phi\left(Z^{x}_{t}\right)\right],\quad
x\in H,\,\phi\in\mathcal{B}_{b}\left(H\right),\,t\geq 0.$

We first study the finite–dimensional case, starting with a deterministic time
change (see Theorem 4) and subsequently recovering the random time shift in
Theorem 6. This way to proceed is customary while working with subordinated
Brownian motions (see, e.g., [9, 15]). Taking advantage of the linear
structure of our model, we are able to get a derivation formula for
$R_{t}\phi$ (see Equation (18)) with a density argument, shunning an
application of the Bismut–Elworthy–Li’s type formula provided by [15]. This is
a remarkable fact, also because it is consistent with the Gaussian framework,
where it is preferable to use the _Bismut–Elworthy–Li’s formula_ only in the
nonlinear case. Then in Theorem 7 we pass to the general, infinite–dimensional
setting under suitable assumptions. A subtle difference between the finite–
and infinite–dimensional cases is that in the former we get an expression for
the Gateaux derivative of $R_{t}\phi$ for every
$\phi\in\mathcal{B}_{b}\left(H\right)$, whereas in the latter such a formula
(see Equation (26)) holds true only for $\phi\in C_{b}\left(H\right)$. In
addition, in Corollary 8 we provide a gradient estimate that, for
$\alpha\in\left(\frac{1}{2},1\right)$, represents the starting point for the
analysis of the Kolmogorov equation in its mild form with fixed–point
arguments, analysis which will be the topic of a future research.
Each of the previous two sections is closed by an example which studies a
concrete framework, namely $H=L^{2}_{0}\left(\mathbb{T}^{d}\right)$, with
$\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$ being the $d$–dimensional
torus. Herein we discuss the hypotheses required by the several theorems of
the paper, with explicit computations that offer a parallel with the
corresponding, well–known results of the Gaussian setting.

##  2 Subordinated Cylindrical Wiener Process and Stochastic Convolution

Let $H$ be a separable Hilbert space and $\left(e_{n}\right)_{n}$ be a
complete orthonormal system. We consider a complete probability space
$\left(\Omega,\mathcal{F},\mathbb{P}\right)$ and introduce a sequence of
independent Brownian motions $\left(\beta^{n}\right)_{n}$ on it. Let
$L=\left(L_{t}\right)_{t}$ be a strictly $\alpha$–stable subordinator, i.e.,
an increasing Lévy process where the distribution of $L_{1}\sim\mu$ is
characterized by

$\hat{\mu}\left(u\right)=\exp\left\\{-\bar{c}\left|u\right|^{\alpha}\left(1-i\tan\frac{\pi\alpha}{2}\,\text{sign}\left(u\right)\right)\right\\},\quad
u\in\mathbb{R},$ (1)

with $\bar{c}>0,\,\alpha\in\left(0,1\right)$. The Laplace transform of $\mu$
is given by

$L_{\mu}\left(u\right)=\mathbb{E}\left[e^{-uL_{1}}\right]=e^{-c^{\prime}u^{\alpha}},\quad
u\geq 0,$ (2)

where $c^{\prime}$ is a constant depending on $\bar{c}$ (for an expression of
$c^{\prime}$ we refer to [13, Example $24.12$], but it is of no use in our
work). Let us introduce the subordinated Brownian motions
$\left(\beta^{n}_{L_{t}}\right)_{t},n\in\mathbb{N}$: assuming $L$ to be
independent from $\left(\beta^{n}\right)_{n}$, [13, Theorem $30.1$] implies
that $\left(\beta^{n}_{L}\right)_{n}$ are real–valued Lévy processes.

Denoting by $\mathcal{N}$ the family of $\mathcal{F}$–negligible sets, we
introduce the augmented $\sigma$–algebra
$\mathcal{F}^{L}\coloneqq\sigma\left(\mathcal{F}^{L}_{0}\cup\mathcal{N}\right)$,
where $\mathcal{F}^{L}_{0}$ is the natural $\sigma$–algebra generated by the
subordinator. Analogously, we consider the augmented $\sigma$–algebras
$\mathcal{F}^{\beta^{n}}$ generated by the Brownian motions. Thanks to the
hypotheses of independence that we have assumed on the processes, we have that
$\mathcal{F}^{L},\,\left(\mathcal{F}^{\beta^{n}}\right)_{n}$ are mutually
independent. In our context, it is natural to deal with different filtrations.
Specifically, for every $n\in\mathbb{N}$ let
$\mathbb{F}^{n}=\left(\mathcal{F}^{\beta^{n}}_{t}\right)_{t}$ be the minimal
augmented filtration generated by $\beta^{n}$, that is,
$\mathcal{F}^{\beta^{n}}_{t}\coloneqq\sigma\left(\left({\mathcal{F}_{0}^{\beta^{n}}}\right)_{t}\cup\mathcal{N}\right)$
for every $t\geq 0$, where
$\left(\left({\mathcal{F}_{0}^{\beta^{n}}}\right)_{t}\right)_{t}$ is the
natural filtration of the process. According to [12, Theorem I$.31$],
$\mathbb{F}^{n}$ satisfies the usual hypotheses. Then we construct a complete
filtration associated with the subordinated Brownian motions. It is denoted by
$\mathbb{F}_{L}=\left(\mathcal{F}_{t}\right)_{t}$, where we define

$\mathcal{F}_{t}\coloneqq\sigma\left(\bigcup_{n\in\mathbb{N}}\mathcal{F}^{\beta_{L}^{n}}_{t}\right),\quad
t\geq 0,$

with $\mathbb{F}^{n}_{L}=\left(\mathcal{F}^{\beta_{L}^{n}}_{t}\right)_{t}$
being the minimal augmented filtration associated with $\beta^{n}_{L}$.

######  Remark 1.

In the finite dimensional case, we denote by
$W_{L}^{N}=\left(W^{N}_{L_{t}}\right)_{t}$ the subordinated,
$\mathbb{R}^{N}$–valued Brownian motion, meaning that

$W^{N}_{L_{t}}=\left[\begin{matrix}\beta^{1}_{L_{t}}&\cdots&\beta^{N}_{L_{t}}\end{matrix}\right]^{T},\quad
t\geq 0.$

By [13, Theorem $30.1$], $W^{N}_{L}$ is an $\mathbb{R}^{N}$–valued Lévy
process, and it is easy to verify that its minimal augmented filtration
$\left(\mathcal{F}^{W_{L}^{N}}_{t}\right)_{t}$ coincides with
$\mathbb{F}_{L}$. This fact shows that the construction that we have carried
out for $\mathbb{F}_{L}$ is natural.

Using the notation we have just introduced, in the general case the
$\sigma$–algebras constituting $\mathbb{F}_{L}$ can be expressed as follows:

$\mathcal{F}_{t}=\sigma\left(\bigcup_{N\in\mathbb{N}}\mathcal{F}^{W_{L}^{N}}_{t}\right),\quad
t\geq 0.$

###  2.1 Subordinated Cylindrical Wiener Process

The aim of this section is to give a rigorous meaning to the formal notation
$W_{L_{t}}=\sum_{n=1}^{\infty}\beta^{n}_{L_{t}}e_{n},t>0$.

First, fix $h\in H,\,t>0$ and notice that the series
$\sum_{n=1}^{\infty}\beta^{n}_{L_{t}}\left\langle h,e_{n}\right\rangle$
converges in distribution. Indeed, even if the random variables
$\left(\beta^{n}_{L_{t}}\right)_{n\in\mathbb{N}}$ are not independent due to
the presence of the subordinator, we can still exploit the mutual independence
of the $\sigma$–algebras $\left(\mathcal{F}^{\beta^{n}}\right)_{n}$ by
conditioning with respect to $\mathcal{F}^{L}$, which in turn is independent
from the previous ones. In order to do so, we use the law of total expectation
together with (2) to get, for every $u\in\mathbb{R}$,

$\displaystyle\mathbb{E}\left[\exp\left\\{iu\sum_{n=1}^{N}\beta^{n}_{L_{t}}\left\langle
h,e_{n}\right\rangle\right\\}\right]$
$\displaystyle\\!=\mathbb{E}\left[\mathbb{E}\left[{\left.\kern-1.2pt\exp\left\\{iu\\!\sum_{n=1}^{N}\beta^{n}_{r}\left\langle
h,e_{n}\right\rangle\right\\}\vphantom{\big{|}}\right|_{r=L_{t}}}\Bigg{|}\mathcal{F}^{L}\right]\right]$
$\displaystyle=\mathbb{E}\left[{\left.\kern-1.2pt\mathbb{E}\left[\exp\left\\{iu\\!\sum_{n=1}^{N}\beta^{n}_{r}\left\langle
h,e_{n}\right\rangle\right\\}\right]\vphantom{\big{|}}\right|_{r=L_{t}}}\right]\\!\\!=\mathbb{E}\left[\prod_{n=1}^{N}\exp\left\\{-\frac{1}{2}L_{t}\left|u\right|^{2}\left|\left\langle
h,e_{n}\right\rangle\right|^{2}\right\\}\right]$
$\displaystyle=\exp\left\\{-tc^{\prime}\frac{1}{2^{\alpha}}\left|u\right|^{2\alpha}\left(\sum_{n=1}^{N}\left|\left\langle
h,e_{n}\right\rangle\right|^{2}\right)^{\alpha}\right\\}\underset{N\to\infty}{\longrightarrow}\exp\left\\{-t\frac{c^{\prime}}{2^{\alpha}}\left\lVert
h\right\rVert^{2\alpha}_{H}\left|u\right|^{2\alpha}\right\\}.$ (3)

Hence applying Lévy’s continuity theorem we see that the series
$\sum_{n=1}^{\infty}\beta^{n}_{L_{t}}\left\langle h,e_{n}\right\rangle$
converges in distribution to a symmetric, $2\alpha$–stable random variable.
Moreover, for every $n\in\mathbb{N}$, choosing $h=e_{n}$ and $N>n$ the
computations in (2.1) provide the distribution of the Lévy process
$\beta^{n}_{L}$, namely

$\mathbb{E}\left[e^{iu\beta^{n}_{L_{t}}}\right]=\exp\left\\{-t\frac{c^{\prime}}{2^{\alpha}}\left|u\right|^{2\alpha}\right\\},\quad
u\in\mathbb{R},\,\text{ for any }t>0.$ (4)

The process $W_{L}=\left(W_{L_{t}}\right)_{t}$ is a _subordinated cylindrical
Wiener process_ , but we might also call it _cylindrical, $2\alpha$–stable
isotropic process_. In fact, for every $N\in\mathbb{N}$ and $t>0$, if we
denote by $\pi_{N}$ the projection onto the first $N$ Fourier components and
by $H_{N}$ its range, an argument analogous to the one in (2.1) yields:

$\mathbb{E}\left[\exp\left\\{i\left\langle
z,\sum_{n=1}^{N}\beta^{n}_{L_{t}}e_{n}\right\rangle\right\\}\right]=\exp\left\\{-t\frac{c^{\prime}}{2^{\alpha}}\left(\sum_{n=1}^{N}\left|\left\langle
z,e_{n}\right\rangle\right|^{2}\right)^{\alpha}\right\\},\quad z\in{H}.$

Hence canonically identifying $H_{N}$ with $\mathbb{R}^{N}$, the Galerkin
projection $\left(\sum_{n=1}^{N}\beta^{n}_{L_{t}}e_{n}\right)_{t}$ can be read
as an $\mathbb{R}^{N}$–valued, $2\alpha$–stable, isotropic Lévy process.

Secondly, we consider a linear, bounded, nonnegative definite operator $Q:H\to
H$ such that $e_{n}$ is one of its eigenvectors corresponding to the
eigenvalue $\sigma_{n}^{2}\geq 0$, $n\in\mathbb{N}$. We study the convergence
in probability –on an appropriate space– of the series:

$\sqrt{Q}W_{L_{t}}=\sum_{n=1}^{\infty}\sigma_{n}\beta^{n}_{L_{t}}e_{n},\quad
t>0.$

Let us introduce a bounded sequence $\left(\rho_{n}\right)_{n}$ of strictly
positive numbers such that
$\sum_{n=1}^{\infty}\rho_{n}^{2r}\sigma_{n}^{2r}<\infty$ for some
$r\in\left(0,\alpha\right)$, and consider the corresponding Hilbert space
$\left(V,\left\langle\cdot,\cdot\right\rangle_{V}\right)$, where

$V\coloneqq\left\\{h\in H:\sum_{n=1}^{\infty}\rho_{n}^{-2}\left|\left\langle
h,e_{n}\right\rangle\right|^{2}<\infty\right\\}\quad\text{and}\quad\left\langle
v,w\right\rangle_{V}\coloneqq\sum_{n=1}^{\infty}\rho_{n}^{-2}\left\langle
v,e_{n}\right\rangle\left\langle w,e_{n}\right\rangle,\quad v,w\in V.$ (5)

Evidently $V\subset H$ with dense and continuous embedding, therefore using
the concept of _Gelfand triple_ we can think a generic $h\in H$ as an object
in $V^{\prime}$, namely

$\left\langle h,v\right\rangle_{V^{\prime},V}=\sum_{n=1}^{\infty}\left\langle
h,e_{n}\right\rangle\left\langle v,e_{n}\right\rangle,\quad v\in V.$

Noticing that $\left\langle
h,\cdot\right\rangle_{V^{\prime},V}=\left\langle\tilde{v},\cdot\right\rangle_{V}$,
where $\tilde{v}\coloneqq\sum_{n=1}^{\infty}\rho_{n}^{2}\left\langle
h,e_{n}\right\rangle e_{n}\in V$, we can apply _Riesz representation theorem_
to get $\left\lVert
h\right\rVert^{2}_{V^{\prime}}=\sum_{n=1}^{\infty}\rho_{n}^{2}\left|\left\langle
h,e_{n}\right\rangle\right|^{2}.$ Now we fix $t>0$ and show that
$\left(\sum_{n=1}^{N}\sigma_{n}\beta^{n}_{L_{t}}e_{n}\right)_{N}\subset
V^{\prime}$ is a Cauchy sequence in probability. Indeed, applying _Markov’s
inequality_ and using the fact that the function
$\phi\left(x\right)=x^{r},\,x\geq 0$, is subadditive and strictly increasing
as $0<r<\alpha<1$, for every $\epsilon>0$ we get:

$\displaystyle\mathbb{P}$
$\displaystyle\left(\left\lVert\sum_{n=p}^{q}\sigma_{n}\beta^{n}_{L_{t}}e_{n}\right\rVert_{V^{\prime}}>\epsilon\right)\leq\mathbb{P}\left(\phi\left(\left\lVert\sum_{n=p}^{q}\sigma_{n}\beta^{n}_{L_{t}}e_{n}\right\rVert_{V^{\prime}}^{2}\right)>\phi\left(\epsilon^{2}\right)\right)\leq\frac{1}{\epsilon^{2r}}\mathbb{E}\left[\phi\left(\left\lVert\sum_{n=p}^{q}\sigma_{n}\beta^{n}_{L_{t}}e_{n}\right\rVert_{V^{\prime}}^{2}\right)\right]$
$\displaystyle=\\!\epsilon^{-2r}\mathbb{E}\left[\phi\left(\sum_{n=p}^{q}\sigma_{n}^{2}\rho_{n}^{2}\left|\beta^{n}_{L_{t}}\right|^{2}\right)\right]\\!\leq\epsilon^{-2r}\sum_{n=p}^{q}\mathbb{E}\left[\left(\sigma_{n}^{2r}\rho_{n}^{2r}\left|\beta^{n}_{L_{t}}\right|^{2r}\right)\right]\\!=\\!\epsilon^{-2r}\mathbb{E}\left[\left|\beta^{1}_{L_{t}}\right|^{2r}\right]\left(\sum_{n=p}^{q}\sigma_{n}^{2r}\rho_{n}^{2r}\right)\\!\underset{p,q\to\infty}{\longrightarrow}0,$

where we use that by construction
$\beta^{n}_{L_{t}}\sim\beta^{1}_{L_{t}},n\in\mathbb{N},$ and that by (4) they
all generate a $2\alpha$–stable distribution, which has finite moment of order
$2r$ (see also Remark 2). By completeness, we can conclude the existence of an
a.s. unique, $V^{\prime}$–valued random variable $\sqrt{Q}W_{L_{t}}$ such that

$\sqrt{Q}W_{L_{t}}=\mathbb{P}-\lim_{N\to\infty}\sum_{n=1}^{N}\sigma_{n}\beta^{n}_{L_{t}}e_{n}\quad\text{in
}V^{\prime}.$

Actually such a convergence in probability is true also in the
$\mathbb{P}-$a.s. sense, as the following, easy and general lemma proves.

######  Lemma 1.

Let $\left(X^{n}\right)_{n}$ be a sequence of real–valued random variables
defined on a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$
and $H$ be a separable Hilbert space admitting $\left(e_{n}\right)_{n}$ as
CONS. If $\sum_{n=1}^{\infty}X^{n}e_{n}$ converges in probability, then it
converges $\mathbb{P}-$a.s.

###### Proof.

Let
$S\coloneqq\mathbb{P}-\lim_{N\to\infty}\sum_{n=1}^{N}X^{n}e_{n}\colon\Omega\to
H$. Obviously

$S\left(\omega\right)=\sum_{n=1}^{\infty}\left\langle
S\left(\omega\right),e_{n}\right\rangle
e_{n}=H-\\!\\!\lim_{N\to\infty}\sum_{n=1}^{N}\left\langle
S\left(\omega\right),e_{n}\right\rangle e_{n},\quad\omega\in\Omega.$ (6)

Convergence in measure implies a.s. convergence along a subsequence, hence we
have

$S\left(\omega\right)=H-\\!\\!\lim_{k\to\infty}\sum_{n=1}^{N_{k}}X^{n}\left(\omega\right)e_{n}\quad\text{for
}\mathbb{P}-\text{a.e. }\omega\in\Omega.$

Therefore, for $\mathbb{P}-$a.e. $\omega\in\Omega$, we see that the Fourier
components of $S$ are

$\left\langle
S\left(\omega\right),e_{\bar{n}}\right\rangle=\lim_{k\to\infty}\left\langle\sum_{n=1}^{N_{k}}X^{n}\left(\omega\right)e_{n},e_{\bar{n}}\right\rangle=X^{\bar{n}}\left(\omega\right)\quad\text{for
every }\bar{n}\in\mathbb{N}.$

Substituting in (6) we conclude

$S\left(\omega\right)=H-\\!\\!\lim_{N\to\infty}\sum_{n=1}^{N}X^{n}\left(\omega\right)e_{n}\quad\text{for
}\mathbb{P}-\text{a.e. }\omega\in\Omega,$

as we stated. ∎

Going back to $\sqrt{Q}W_{L_{t}}$, since $\left(\rho_{n}e_{n}\right)_{n}$ is a
CONS for the Hilbert space $V$, Lemma 1 allows to write

$\sqrt{Q}W_{L_{t}}=\lim_{N\to\infty}\left\langle\sum_{n=1}^{N}\sigma_{n}\beta^{n}_{L_{t}}e_{n},\cdot\right\rangle_{V^{\prime},V}=\lim_{N\to\infty}\sum_{n=1}^{N}\rho_{n}\sigma_{n}\beta^{n}_{L_{t}}\left\langle\left(\rho_{n}e_{n}\right),\cdot\right\rangle_{V}\quad\mathbb{P}-\text{a.s.}$

It then follows that
$\left\langle\sqrt{Q}W_{L_{t}},v\right\rangle_{V^{\prime},V}=\lim_{N\to\infty}\sum_{n=1}^{N}\sigma_{n}\beta^{n}_{L_{t}}\left\langle
v,e_{n}\right\rangle$ for every $v\in V,\,\mathbb{P}-$a.s. Combining this with
(2.1), we can see that
$\left\langle\sqrt{Q}W_{L_{t}},v\right\rangle_{V^{\prime},V}$ has a symmetric,
$2\alpha$–stable distribution. We collect the previous results in the next
theorem.

######  Theorem 2.

  1. 1.

Given $h\in H$ and $t>0$, the series
$\sum_{n=1}^{\infty}\beta^{n}_{L_{t}}\left\langle h,e_{n}\right\rangle$
converges in distribution to a real–valued, symmetric, $2\alpha$–stable random
variable $X_{t}$ whose characteristic function is

$\mathbb{E}\left[e^{iuX_{t}}\right]=\exp\left\\{-t\frac{c^{\prime}}{2^{\alpha}}\left\lVert
h\right\rVert^{2\alpha}_{H}\left|u\right|^{2\alpha}\right\\},\quad
u\in\mathbb{R}.$

  2. 2.

Consider a linear, bounded, nonnegative definite operator $Q:H\to H$ such that
$\left(e_{n}\right)_{n}$ is a basis of its eigenvectors corresponding to the
eigenvalues
$\left(\sigma_{n}^{2}\right)_{n}\left(\subset\mathbb{R}_{+}\right)$. Let
$\left(\rho_{n}\right)_{n}$ be a bounded sequence of strictly positive weights
such that $\sum_{n=1}^{\infty}\rho_{n}^{2r}\sigma_{n}^{2r}<\infty$ for some
$0<r<{\alpha}$. Then the corresponding Hilbert space
$\left(V,\left\langle\cdot,\cdot\right\rangle_{V}\right)$ defined in (5) is
continuously embedded with density in $H$ and, for every $t>0$, the random
variable $\sqrt{Q}W_{L_{t}}\colon\Omega\to V^{\prime}$ is defined as

$\sqrt{Q}W_{L_{t}}\coloneqq\lim_{N\to\infty}\sum_{n=1}^{N}\sigma_{n}\beta^{n}_{L_{t}}e_{n}\quad\mathbb{P}-\text{a.s.}$

In particular, for every $v\in V$,

$\sum_{n=1}^{N}\beta^{n}_{L_{t}}\left\langle\sqrt{Q}v,e_{n}\right\rangle\underset{N\to\infty}{\longrightarrow}\left\langle\sqrt{Q}W_{L_{t}},v\right\rangle_{V^{\prime},V}\quad\mathbb{P}-\text{a.s.}$

######  Remark 2.

We can state the finiteness of the moment of order $2r$ of the random variable
$\beta^{1}_{L_{t}}$ without explicitly knowing its distribution, i.e., without
using (4). In fact, we can proceed as follows:

$\mathbb{E}\left[\left|\beta^{1}_{L_{t}}\right|^{2r}\right]=\mathbb{E}\left[\mathbb{E}\left[\left|\beta^{1}_{L_{t}}\right|^{2r}\Bigr{|}\mathcal{F}_{L}\right]\right]=\frac{2^{r}}{\sqrt{\pi}}\Gamma\left(\frac{2r+1}{2}\right)\mathbb{E}\left[L_{t}^{r}\right]<\infty,$

since we are dealing with $0<r<\alpha.$ For the second equality we refer to
[14, Equation $\left(17\right)$].

###  2.2 Stochastic Convolution

Let $A:\mathcal{D}\left(A\right)\subset H\to H$ be a linear, selfadjoint,
negative definite, unbounded operator that shares with $Q$ a common basis of
eigenvectors $\left(e_{n}\right)_{n}$. We denote by
$\left(-\lambda_{n}\right)_{n}$, with
$0<\lambda_{1}\leq\lambda_{2}\leq\dots\leq\lambda_{n}\leq\cdots$ the
corresponding eigenvalues, i.e., $Ae_{n}=-\lambda_{n}e_{n},\,n\in\mathbb{N}$.
Recalling that $\alpha\in\left(0,1\right)$ has been fixed at the beginning of
Section 2, it is convenient to introduce the shorthand notation
$X\sim\text{stable}\left(\alpha,\beta,\gamma,\delta\right)$ to denote a random
variable $X$ with characteristic function given by

$\mathbb{E}\left[e^{iuX}\right]=\exp\left\\{-\gamma^{\alpha}\left|u\right|^{\alpha}\left(1-i\beta\tan\frac{\pi\alpha}{2}\text{sign}\left(u\right)\right)+i\delta
u\right\\},\quad u\in\mathbb{R}.$

where $\left|\beta\right|\leq 1,\gamma>0$ and $\delta\in\mathbb{R}$. Hence by
(4), for every $n\in\mathbb{N}$ the Lévy process $\beta^{n}_{L}$ has random
variables distributed as

$\beta^{n}_{L_{t}}\sim\text{stable}\left(2\alpha,0,\left(t\frac{c^{\prime}}{2^{\alpha}}\right)^{{1}/\left({2\alpha}\right)},0\right),\quad
t>0.$

We denote by $U^{n}=\left(U_{t}^{n}\right)_{t\geq 0}$ the OU–process
$U_{t}^{n}\coloneqq\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}\sigma_{n}\,d\beta^{n}_{L_{s}},\,t\geq
0:$ this is the unique (up to evanescence) solution of the one dimensional
stochastic differential equation

$dU^{n}_{t}=-\lambda_{n}U^{n}_{t}dt+\sigma_{n}\,d\beta^{n}_{L_{t}},\quad
U^{n}_{0}=0.$ (7)

The processes $\left(U^{n}\right)_{n}$ are càdlàg and adapted to the
filtration $\mathbb{F}_{L}$, and direct computations (see, e.g., [1,
Proposition $3.2$]) show that
$U_{t}^{n}\sim\text{stable}(2\alpha,0,\gamma_{n}\left(t\right),0)$, where

$\gamma_{n}\left(t\right)\coloneqq\left(\frac{c^{\prime}}{2^{\alpha}}\right)^{{1}/\left({2\alpha}\right)}\\!\\!\left(\int_{0}^{t}\\!\\!e^{-2\alpha\lambda_{n}\left(t-s\right)}\sigma_{n}^{2\alpha}\,ds\right)^{{1}/\left({2\alpha}\right)}\\!\\!\\!=\sigma_{n}\left(\frac{c^{\prime}}{2^{\alpha+1}\alpha}\right)^{{1}/\left({2\alpha}\right)}\\!\\!\left(\frac{1-e^{-2\alpha\lambda_{n}t}}{\lambda_{n}}\right)^{{1}/\left({2\alpha}\right)}\\!\\!,\quad
t>0,\,n\in\mathbb{N}.$

We are now in position to construct the _stochastic convolution_ and the
corresponding OU–process.

######  Theorem 3.

Assume that

$\sum_{n=1}^{\infty}\dfrac{\sigma_{n}^{2r}}{\lambda_{n}^{{r}/{\alpha}}}<\infty\quad\text{for
some }r\in\left(0,\alpha\right).$ (i)

Then, for all $t\\!>\\!0$, the series $\sum_{n=1}^{\infty}\\!U^{n}_{t}e_{n}$
converges $\mathbb{P}-$a.s. to a random variable
$\tilde{Z}_{A,Q}\\!\left(t\right)=\\!\int_{0}^{t}e^{\left(t-s\right)A}\sqrt{Q}\,dW_{L_{s}}$.
The resulting process
$\tilde{Z}_{A,Q}=\left(\tilde{Z}_{A,Q}\left(t\right)\right)_{t}$ is
$\mathbb{F}_{L}$–adapted and is called _stochastic convolution_.

The corresponding OU–process starting at $x\in H$, denoted by
$Z^{x}=\left(Z_{t}^{x}\right)_{t}$ and defined by

$Z_{t}^{x}\coloneqq
e^{tA}x+\int_{0}^{t}e^{\left(t-s\right)A}\sqrt{Q}\,dW_{L_{s}}=e^{tA}x+\tilde{Z}_{A,Q}\left(t\right),\quad
t\geq 0,$

is $\mathbb{F}_{L}$–adapted and Markovian with homogeneity in time.

###### Proof.

Fix $t>0$. Thanks to the preceding discussion, we know that
$U^{n}_{t}\sim\gamma_{n}\left(t\right)X,\,n\in\mathbb{N}$, where $X$ is a
random variable such that $X\sim\text{stable}\left(2\alpha,0,1,0\right)$. Then
an application of Markov’s inequality entails:

$\mathbb{P}\left(\left\lVert\sum_{n=p}^{q}U^{n}_{t}e_{n}\right\rVert_{H}\\!\\!\\!>\epsilon\right)\\!\\!\leq\epsilon^{-2r}\mathbb{E}\left[\phi\left(\left\lVert\sum_{n=p}^{q}U^{n}_{t}e_{n}\right\rVert_{H}^{2}\right)\right]\\!\\!=\epsilon^{-2r}\mathbb{E}\left[\phi\left(\sum_{n=p}^{q}\left|U^{n}_{t}\right|^{2}\right)\right]\\!\\!\leq\epsilon^{-2r}\sum_{n=p}^{q}\mathbb{E}\left[\left(\left|U^{n}_{t}\right|^{2r}\right)\right]\\\
=\epsilon^{-2r}\mathbb{E}\left[\left|X\right|^{2r}\right]\left(\frac{c^{\prime}}{2^{\alpha+1}\alpha}\right)^{r/\alpha}\left(\sum_{n=p}^{q}\frac{\sigma_{n}^{2r}}{\lambda_{n}^{r/\alpha}}\left(1-e^{-2\alpha\lambda_{n}t}\right)^{r/\alpha}\right)\leq
c\left(\epsilon\right)\left(\sum_{n=p}^{q}\frac{\sigma_{n}^{2r}}{\lambda_{n}^{r/\alpha}}\right)\underset{p,q\to\infty}{\longrightarrow}0,\quad\epsilon>0,$

with
$c\left(\epsilon\right)\coloneqq\epsilon^{-2r}\mathbb{E}\left[\left|X\right|^{2r}\right]\left(\dfrac{c^{\prime}}{2^{\alpha+1}\alpha}\right)^{{r}/{\alpha}}$
and $\phi\left(x\right)=x^{r}$, as above. Therefore the series converges in
probability:

$\tilde{Z}_{A,Q}\left(t\right)=\int_{0}^{t}e^{\left(t-s\right)A}\sqrt{Q}\,dW_{L_{s}}\coloneqq\mathbb{P}-\lim_{N\to\infty}\sum_{n=1}^{N}U^{n}_{t}e_{n}.$

An application of Lemma 1 shows that such convergence is true in the
$\mathbb{P}-$a.s. sense, as well. Obviously $\tilde{Z}_{A,Q}$ is an
$\mathbb{F}_{L}$–adapted process, since the space
$\left(\Omega,\mathcal{F},\mathbb{P}\right)$ is complete by hypothesis,
$\mathbb{F}_{L}$ is complete by construction and the one dimensional
OU–processes $U^{n}$ are $\mathbb{F}_{L}$–adapted.

Concerning the OU–processes, for every $x\in H$ we can express the random
variables of $Z^{x}=\left(Z_{t}^{x}\right)_{t}$ as follows:

$Z^{x}_{t+h}\overset{a.s.}{=}e^{hA}Z^{x}_{t}+\int_{t}^{t+h}e^{\left(t+h-s\right)A}\sqrt{Q}\,dW_{L_{s}}=e^{hA}Z^{x}_{t}+\sum_{n=1}^{\infty}\left(\int_{t}^{t+h}e^{-\lambda_{n}\left(t+h-s\right)}\sigma_{n}\,d\beta_{L_{s}}^{n}\right)e_{n},\quad
t,h\geq 0.$

This immediately implies the Markovianity of the process, recalling the
independence of the increments of the Lévy processes
$\left(\beta^{n}_{L}\right)_{n}$. The time homogeneity is obtained by a
standard argument relying on the stationarity of the increments of the same
processes and the fact that the coefficients of the one–dimensional SDEs in
(7) are time–autonomous. The proof is then complete. ∎

We close this section with an example which analyzes a common framework in
applications (see, e.g., [6]).

######  Example 1.

Let $\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$ be the $d$–dimensional
torus and denote by $e_{k}$ the functions

$e_{k}\left(x\right)\coloneqq\begin{cases}\cos\left(2\pi k\cdot
x\right),&k\in\mathbb{Z}^{d}_{+}\\\ \sin\left(2\pi k\cdot
x\right),&k\in\mathbb{Z}^{d}_{-}\end{cases},\quad x\in\mathbb{T}^{d},$

where $\mathbb{Z}^{d}_{+}\coloneqq\left\\{\left(k_{1}>0\right)\text{ or
}\left(k_{1}=0\text{ and }k_{j}>0\text{ for }j=2,\dots,d\right)\right\\}$ and
$\mathbb{Z}^{d}_{-}\coloneqq-\mathbb{Z}^{d}_{+}$. Then
$\left\\{e_{k}:k\in\mathbb{Z}_{0}^{d}\right\\}$ constitute a complete
orthonormal system for the Hilbert space

$H=L^{2}_{0}\left(\mathbb{T}^{d};\mathbb{R}\right)\coloneqq\left\\{f\in
L^{2}\left(\mathbb{T}^{d};\mathbb{R}\right):\int_{\mathbb{T}^{d}}f\left(x\right)dx=0\right\\},$

where of course
$\mathbb{Z}_{0}^{d}\coloneqq\mathbb{Z}^{d}\setminus\left\\{0\right\\}.$ In
particular, for every $f\in H$, we have

$f=\sum_{k\in\mathbb{Z}^{d}_{0}}\hat{f}_{k}e_{k},\quad\hat{f}_{k}\coloneqq\int_{\mathbb{T}^{d}}f\left(x\right)e_{k}\left(x\right)\,dx,\quad
k\in\mathbb{Z}^{d}_{0}.$

We first introduce the Sobolev spaces

$W_{0}^{\beta,2}\left(\mathbb{T}^{d}\right)\coloneqq\left\\{f\in
H:\sum_{k\in\mathbb{Z}^{d}_{0}}\left|k\right|^{2\beta}\hat{f}_{k}^{2}<\infty\right\\},\quad\left\lVert
f\right\rVert_{W_{0}^{\beta,2}}^{2}\coloneqq\sum_{k\in\mathbb{Z}^{d}_{0}}\left|k\right|^{2\beta}\hat{f}_{k}^{2},$

and then define the linear operator $A$ as follows:

$A\colon W_{0}^{2,2}\left(\mathbb{T}^{d}\right)\to H\quad\text{ such that
}\quad Af=\Delta
f=-\left(2\pi\right)^{2}\sum_{k\in\mathbb{Z}^{d}_{0}}\left|k\right|^{2}\hat{f}_{k}e_{k}.$

In particular, the eigenvalues of $A$ corresponding to $e_{k}$ are
$-\lambda_{k}=-\left(2\pi\right)^{2}\left|k\right|^{2}$, hence $A$ is
unbounded and negative definite. Moreover it is selfadjoint, as well. Now we
analyze Hypothesis (i) for two specifications of the linear, bounded, positive
semidefinite operator $Q\colon H\to H$.

  * •

Let $Q=\text{Id}$. Then $\sigma_{k}=1,\,k\in\mathbb{Z}^{d}_{0}$, and (i) reads

$\frac{1}{\left(2\pi\right)^{2r/\alpha}}\sum_{k\in\mathbb{Z}^{d}_{0}}\frac{1}{\left|k\right|^{2r/\alpha}}<\infty\quad\text{for
some }r\in\left(0,\alpha\right),$

which is satisfied if and only if $d=1$. Hence the stochastic convolution is
defined only in dimension $d=1$.

  * •

Set $Q=Q_{\eta}=\left(-\Delta\right)^{-\eta}$ for $\eta>0$, the negative
fractional power of the Laplacian, defined as an operator $Q_{\eta}\colon H\to
H$ such that

$Q_{\eta}f=\frac{1}{\left(2\pi\right)^{2\eta}}\sum_{k\in\mathbb{Z}^{d}_{0}}\frac{1}{\left|k\right|^{2\eta}}\hat{f}_{k}e_{k},\quad
f\in H.$

In this case the convergence of the infinite sum in (i) amounts to requiring
$\eta>\left(\frac{d}{2r}-\frac{1}{\alpha}\right)\vee 0$. Since $r$ is chosen
freely in the interval $\left(0,\alpha\right)$, Hypothesis (i) is satisfied if
and only if

$\eta>\left(\frac{d-2}{2\alpha}\right)\vee 0.$ (8)

This fact can be interpreted as follows: the higher the dimension $d$, the
weaker the effect of the noise on the high Fourier modes needs to be in order
to have the well–posedness of the stochastic convolution.

##  3 Smoothing effect of the Markov Transition Semigroup

Let us introduce the Markov transition semigroup $R=\left(R_{t}\right)_{t}$
associated with the OU–processes $\left(Z^{x}\right)_{x\in H}$, which is given
by

$R_{t}\phi\left(x\right)\coloneqq\mathbb{E}\left[\phi\left(Z^{x}_{t}\right)\right],\quad
x\in H,\,\phi\in\mathcal{B}_{b}\left(H\right),\,t\geq 0,$

where $B_{b}\left(H\right)$ is the space of bounded, real–valued,
Borel–measurable functions in $H$. Evidently, each $R_{t}$ is linear and
bounded from $C_{b}\left(H\right)$ into itself and $R_{0}$ is the identity.
Our aim is to prove that, under suitable conditions, the operator $R_{t}$ has
a smoothing effect for every $t>0$. Specifically, given a function $\phi\in
B_{b}\left(H\right)$, in the case $\alpha\in\left(\frac{1}{2},1\right)$ we are
going to show that $R_{t}\phi\in C^{1}_{b}\left(H\right)$ and that the
following gradient estimate holds:

$\sup_{x\in H}\left\lVert\nabla
R_{t}\phi\left(x\right)\right\rVert_{H}\leq\frac{C}{t^{{\gamma}}}\left\lVert\phi\right\rVert_{\infty}\quad\text{for
every $t>0$, for some }0<{\gamma}<1,\,C>0.$ (9)

###  3.1 Finite dimensional case $H=\mathbb{R}^{N}$

Let $H=\mathbb{R}^{N}$ and
$W^{N}=\left[\begin{matrix}\beta^{1}&\cdots&\beta^{N}\end{matrix}\right]^{T}$.
We start by presenting a theorem which allows to obtain an original derivation
formula for the semigroup corresponding to the finite–dimensional OU processes
$Z_{t}^{\ell}\left(x\right)$. They are defined as the unique, càdlàg solutions
of the linear SDEs
$dZ^{\ell}_{t}\left(x\right)=AZ^{\ell}_{t}\left(x\right)dt+\sqrt{Q}\,dW^{N}_{\ell_{t}},\,Z^{\ell}_{0}\left(x\right)=x,$
and can be expressed by the variation of constant formula as follows:

$Z^{\ell}_{t}\left(x\right)=e^{tA}x+\int_{0}^{t}e^{\left(t-s\right)A}\sqrt{Q}\,dW^{N}_{\ell_{s}},\quad
t\geq 0,\,\mathbb{P}-\text{a.s.},$ (10)

where $x\in\mathbb{R}^{N}$ and $\ell\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ is
an increasing, càdlàg function such that $\ell_{0}=0$ and $\ell_{t}>0$ for
every positive $t$: the set of functions with these properties will be denoted
by $\mathbb{S}$. Note that, for every $\ell\in\mathbb{S}$,
$W^{N}_{\ell}=\left(W^{N}_{\ell_{t}}\right)_{t}$ is a càdlàg martingale with
respect to the filtration $\left(\mathcal{F}^{N}_{\ell_{t}}\right)_{t}$, where
$\left(\mathcal{F}^{N}_{t}\right)_{t}$ is the minimal augmented filtration
generated by $W^{N}$. Analogously, for every $\ell\in\mathbb{S}$, we introduce
the filtrations
$\mathbb{F}^{n}_{\ell}=\left(\mathcal{F}^{\beta^{n}}_{\ell_{t}}\right)_{t},\,n\in\mathbb{N}$,
and observe that $\beta^{n}_{\ell}=\left(\beta^{n}_{\ell_{t}}\right)_{t}$ is a
càdlàg, $\mathbb{F}^{n}_{\ell}$–martingale. The proof of such theorem is
essentially based on the deterministic time–change procedure described by
Zhang in [15, Section $2$], but exploits the linear nature of our setting to
avoid the application of the _Bismut–Elworthy–Li’s formula_ (see, e.g., [3,
Proposition $8.21$]). For the sake of completeness we report its main
passages.

######  Theorem 4.

Let $t>0,\,\phi\in C_{b}\left(\mathbb{R}^{N}\right),\,\ell\in\mathbb{S}$ and
assume that $\sigma_{n}^{2}>0,\,n=1,\dots,N$. Then the function
$\mathbb{E}\left[\phi\left(Z^{\ell}_{t}\left(\cdot\right)\right)\right]$ is
differentiable at any point $x\in\mathbb{R}^{N}$ in every direction
$h\in\mathbb{R}^{N}$, and

$\left\langle\nabla\mathbb{E}\left[\phi\left(Z^{\ell}_{t}\left(x\right)\right)\right],h\right\rangle=\mathbb{E}\left[\phi\left(Z_{t}^{\ell}\left(x\right)\right)\left(\sum_{n=1}^{N}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}d\ell_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}d\beta_{\ell_{s}}^{n}\right)\right].$
(11)

###### Proof.

For every $\epsilon>0$ denote by
$\ell^{\epsilon}\left(t\right)\coloneqq\frac{1}{\epsilon}\int_{t}^{t+\epsilon}\ell_{s}\,ds,\,t\geq
0$, the _Steklov’s averages_ of $\ell$. They are strictly increasing,
absolutely continuous functions such that, for every $t\geq 0$,
$\ell^{\epsilon}_{t}\downarrow\ell_{t}$ as $\epsilon\downarrow 0$. Let
$\gamma^{\epsilon}\coloneqq\left(\ell^{\epsilon}\right)^{-1}\colon\left[\ell_{0}^{\epsilon},\infty\right)\to\mathbb{R}_{+}$
and define $Z^{\ell^{\epsilon}}\left(x\right)$ as in (10), i.e., for every
$x\in\mathbb{R}^{N}$ the process $Z^{\ell^{\epsilon}}\left(x\right)$ is the
unique solution of the linear SDE
$dZ^{\ell^{\epsilon}}_{t}\left(x\right)=AZ^{\ell^{\epsilon}}_{t}\left(x\right)dt+\sqrt{Q}\,dW^{N}_{\ell^{\epsilon}_{t}},\,Z^{\ell^{\epsilon}}_{0}=x$.
Now introduce the time–shifted processes
$Y_{t}^{\ell^{\epsilon}}\left(x\right)\coloneqq
Z^{\ell^{\epsilon}}_{\gamma^{\epsilon}_{t}}\left(x\right),\,t\geq\ell^{\epsilon}_{0}$,
and observe that

$Y^{\ell^{\epsilon}}_{t}\left(x\right)=x+\int_{\ell_{0}^{\epsilon}}^{t}AY_{s}^{\ell^{\epsilon}}\left(x\right)\dot{\gamma^{\epsilon}_{s}}\,ds+\sqrt{Q}\left(W^{N}_{t}-W^{N}_{\ell_{0}^{\epsilon}}\right),\quad
t\geq\ell_{0}^{\epsilon},\,\mathbb{P}-\text{a.s.},$

which shows that
$dY_{t}^{\ell^{\epsilon}}\left(x\right)=AY_{t}^{\ell^{\epsilon}}\left(x\right)\dot{\gamma^{\epsilon}_{t}}dt+\sqrt{Q}\,dW^{N}_{t},\,Y_{\ell_{0}^{\epsilon}}^{\ell^{\epsilon}}\left(x\right)=x$.
Therefore,

$Y_{t}^{\ell^{\epsilon}}\left(x\right)=e^{A\gamma_{t}^{\epsilon}}x+\int_{\ell_{0}^{\epsilon}}^{t}e^{A\left(\gamma_{t}^{\epsilon}-\gamma_{s}^{\epsilon}\right)}\sqrt{Q}\,dW^{N}_{s},\quad
t\geq\ell_{0}^{\epsilon},\,\mathbb{P}-\text{a.s.}$

In particular, since
$\int_{\ell_{0}^{\epsilon}}^{\ell_{t}^{\epsilon}}e^{2A\left(t-\gamma^{\epsilon}_{s}\right)}Q\,ds=\int_{0}^{t}e^{2A\left(t-s\right)}Q\,d\ell^{\epsilon}_{s}$,
where the integral is to be interpreted entrywise, we have

$Z_{t}^{\ell^{\epsilon}}\left(x\right)=Y_{\ell_{t}^{\epsilon}}^{\ell^{\epsilon}}\left(x\right)\sim\mathcal{N}\left(e^{At}x,\int_{0}^{t}e^{2A\left(t-s\right)}Q\,d\ell^{\epsilon}_{s}\right).$

At this point, we fix a generic $t>0,\,x\in\mathbb{R}^{N}$ and use [15,
Equation ($2.6$)] (it is just an application of _Gronwall lemma_) to get the
convergence, in the $L^{2}$–sense, of
$Z_{t}^{\ell^{\epsilon}}\left(x\right)\to Z_{t}^{\ell}\left(x\right)$ as
$\epsilon\downarrow 0$. Moreover, recalling that
$\ell^{\epsilon}_{t}\downarrow\ell_{t}$ as $\epsilon\downarrow 0$, we invoke
_Helly’s second theorem_ (see [10, Theorem $7.3$]) to get
$\int_{0}^{t}e^{2A\left(t-s\right)}Q\,d\ell^{\epsilon}_{s}\to\int_{0}^{t}e^{2A\left(t-s\right)}Q\,d\ell_{s}$
as $\epsilon\downarrow 0$. Whence,

$Z_{t}^{\ell}\left(x\right)\sim\mathcal{N}\left(e^{At}x,\int_{0}^{t}e^{2A\left(t-s\right)}Q\,d\ell_{s}\right).$
(12)

If we take $\phi\in C_{b}\left(\mathbb{R}^{N}\right)$, an explicit computation
simply based on the derivation of the normal density function implies, for
every direction $h\in\mathbb{R}^{N}$,

$\left\langle\nabla\mathbb{E}\left[\phi\left(Z^{\ell}_{t}\left(x\right)\right)\right],h\right\rangle=\mathbb{E}\left[\phi\left(Z_{t}^{\ell}\left(x\right)\right)\left\langle\left(\int_{0}^{t}e^{2A\left(t-s\right)}Q\,d\ell_{s}\right)^{-1}\left(\int_{0}^{t}e^{A\left(t-s\right)}\sqrt{Q}\,dW^{N}_{\ell_{s}}\right),e^{tA}h\right\rangle\right],$

which coincides with (11) upon expanding the notation. ∎

######  Remark 3.

The previous proof does not need the continuity of the function $\phi$.
Therefore, _Theorem_ 4 holds true for every
$\phi\in\mathcal{B}_{b}\left(\mathbb{R}^{N}\right)$.

Now we investigate the subordinated Brownian motion case. The intuition behind
the argument is to condition with respect to the $\sigma$–algebra
$\mathcal{F}^{L}$, so that it is possible to apply the deterministic
time–shift result we have just obtained in Theorem 4 upon changing the
reference probability space. Let us denote by $\mathbb{W}$ the space of
continuous functions from $\mathbb{R}_{+}$ to $\mathbb{R}^{N}$ vanishing at
$0$ and endow it with the Borel $\sigma$–algebra
$\mathcal{B}\left(\mathbb{W}\right)$ associated with the topology of locally
uniform convergence. The pushforward probability measure generated by
$W^{N}\left(\cdot\right)\colon\left(\Omega,\mathcal{F},\mathbb{P}\right)\to\left(\mathbb{W},\mathcal{B}\left(\mathbb{W}\right)\right)$
is denoted by $\mathbb{P}^{\mathbb{W}}$ and makes the canonical process
$\mathfrak{x}=\left(x_{t}\right)_{t}$ a Brownian motion, where by definition

$x_{t}\left(w\right)\coloneqq w_{t},\quad w\in\mathbb{W},\,t\geq 0.$

We work with the usual completion
$\left(\mathbb{W},\overline{\mathcal{B}\left(\mathbb{W}\right)},\overline{\mathbb{P}^{\mathbb{W}}}\right)$
of this probability space: by [8, Theorem $7.9$], $\mathfrak{x}$ is still a
Brownian motion with respect to its minimal augmented filtration, which in
turn satisfies the usual hypotheses and is denoted by
$\mathbb{F}^{\mathbb{W}}$. In particular, note that the completeness of the
space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ implies the measurability
of
$W^{N}\left(\cdot\right)\colon\left(\Omega,\mathcal{F},\mathbb{P}\right)\to\left(\mathbb{W},\overline{\mathcal{B}\left(\mathbb{W}\right)}\right)$
and the fact that $\overline{\mathbb{P}^{\mathbb{W}}}$ is still the
pushforward probability measure generated by $W^{N}\left(\cdot\right)$.
Obviously, $W^{N}\left(\cdot\right)$ is independent from $\mathcal{F}^{L}$: as
a consequence, a regular conditional distribution of $W^{N}\left(\cdot\right)$
given $\mathcal{F}^{L}$ is the probability kernel

$\mathbb{P}\left(W^{N}\\!\left(\cdot\right)\in\cdot\big{|}\mathcal{F}^{L}\right)\colon\Omega\times\overline{\mathcal{B}\left(\mathbb{W}\right)}\to\left[0,1\right]\,\,\text{
such that }\,\,\mathbb{P}\left(W^{N}\\!\left(\cdot\right)\in
A\right)\left(w\right)\coloneqq\overline{\mathbb{P}^{\mathbb{W}}}\left(A\right),\,\omega\in{\Omega},\,A\in\overline{\mathcal{B}\left(\mathbb{W}\right)}.$
(13)

As regards the space $\mathbb{S}$, for every $t\geq 0$ we introduce the map
$y_{t}\colon\mathbb{S}\to\mathbb{R}$ defined by
$y_{t}\left(\ell\right)\coloneqq\ell_{t},\,\ell\in\mathbb{S}$, and consider
the $\sigma$–algebra
$\mathcal{F}^{\mathbb{S}}\coloneqq\sigma\left(y^{-1}_{t}\left(B\right),\,B\in\mathcal{B}\left(\mathbb{R}\right),\,t\geq
0\right)$. Since
$L\left(\cdot\right)\colon\left(\Omega,\mathcal{F}^{L},\mathbb{P}\right)\to\left(\mathbb{S},\mathcal{F}^{\mathbb{S}}\right)$
is measurable, we can construct the pushforward probability measure
$\mathbb{P}^{\mathbb{S}}$ on
$\left(\mathbb{S},\mathcal{F}^{\mathbb{S}}\right)$. At this point we take into
account the product space
$\left(\mathbb{W}\times\mathbb{S},\overline{\mathcal{B}\left(\mathbb{W}\right)}\otimes\mathcal{F}^{\mathbb{S}},\overline{\mathbb{P}^{\mathbb{W}}}\otimes\mathbb{P}^{\mathbb{S}}\right)$
and note that, thanks to the mutual independence of $W^{N}\left(\cdot\right)$
and $L\left(\cdot\right)$, the product measure
$\overline{\mathbb{P}^{\mathbb{W}}}\otimes\mathbb{P}^{\mathbb{S}}$ is indeed
the pushforward probability measure generated by
$\psi\colon\Omega\to\mathbb{W}\times\mathbb{S},\,\psi\left(\omega\right)\coloneqq\left(W^{N}_{\cdot}\left(\omega\right),L_{\cdot}\left(\omega\right)\right)$.
Finally, we take the process $z=\left(z_{t}\right)_{t}$ defined by

$z_{t}\left(w,\ell\right)\coloneqq
w_{\ell_{t}},\quad\left(w,\ell\right)\in\mathbb{W}\times\mathbb{S},\,t\geq 0,$

and denote by $\mathbb{F}^{z}=\left(\mathcal{F}^{z}_{t}\right)_{t}$ its
natural filtration. By construction, $W^{N}_{L_{t}}=z_{t}\circ\psi$ for every
$t\geq 0$. Putting together all these properties, we can conclude that $z$ is
a Lévy process with respect to the right–continuous filtration
$\mathbb{F}^{z}_{+}=\left(\mathcal{F}^{z}_{t+}\right)_{t}$, where

$\mathcal{F}^{z}_{t+}\coloneqq\bigcap_{\epsilon>0}\mathcal{F}^{z}_{t+\epsilon},\quad
t\geq 0.$

Endowing the product space with this filtration, the stochastic integral of
suitable processes with respect to $z$ is well defined. Let us consider then a
deterministic, continuous, bounded, $\mathbb{R}^{N}$–valued process
$\xi=\left(\xi_{t}\right)_{t}$: weaker assumptions can be done on it, but in
our framework these are sufficient. Clearly the subordinated Brownian motion
$W^{N}_{L}$ is adapted with respect to the right–continuous filtration
$\psi^{-1}\left(\mathbb{F}^{z}_{+}\right)$, therefore the usual rules of
change of probability space (see, e.g., [7, §X-$2$]) entail

$\int_{0}^{t}\xi_{s}\cdot dW^{N}_{L_{s}}=\left(\int_{0}^{t}\xi_{s}\cdot
dz_{s}\right)\circ\psi,\quad t\geq 0,\,\mathbb{P}-\text{a.s.}$ (14)

We conclude this preliminary discussion with an important substitution
formula.

######  Lemma 5.

Let $\xi=\left(\xi_{t}\right)_{t}$ be a deterministic, continuous, bounded,
$\mathbb{R}^{N}$–valued process. Then, for any $t>0$,

$\left(\int_{0}^{t}\xi_{s}\cdot
dz_{s}\right)\left(\cdot,\ell\right)=\int_{0}^{t}\xi_{s}\cdot
dx_{\ell_{s}}\quad\overline{\mathbb{P}^{\mathbb{W}}}-\text{a.s., for
}\mathbb{P}^{\mathbb{S}}-\text{a.e. }\ell\in\mathbb{S},$

where the integral on the right–hand side of the equality is intended in the
sense of stochastic integrals by càdlàg martingales on the filtered
probability space
$\left(\mathbb{W},\overline{\mathcal{B}\left(\mathbb{W}\right)},\overline{\mathbb{P}^{\mathbb{W}}};\mathbb{F}^{\mathbb{W}}_{\ell}\right).$

###### Proof.

Fix $t>0$ and introduce the elementary, predictable (with respect to both
$\mathbb{F}^{z}_{+}$ and $\mathbb{F}^{\mathbb{W}}_{\ell},\,\ell\in\mathbb{S}$)
processes

$\xi^{m}_{s}\coloneqq\xi_{0}1_{\left\\{0\right\\}}\left(s\right)+\sum_{i=0}^{m-1}\xi_{t_{i}}1_{]t_{i},t_{i+1}]}\left(s\right),$

where $t_{i}=\frac{t}{m}i,\,i=0,\dots,m$. The continuity of $\xi$ implies that
$\xi^{m}\to\xi$ pointwise; furthermore, since $\xi$ is bounded, the sequence
$\left(\xi^{m}\right)_{m}$ is uniformly bounded. This implies that

$\int_{0}^{t}\xi_{s}\cdot
dz_{s}=\left(\overline{\mathbb{P}^{\mathbb{W}}}\otimes\mathbb{P}^{\mathbb{S}}\right)\\!-\lim_{m\to\infty}\int_{0}^{t}\xi^{m}_{s}\cdot
dz_{s}.$

Now convergence in probability implies almost–sure convergence along a
subsequence, hence we can say that for $\mathbb{P}^{\mathbb{S}}-$a.e.
$\ell\in\mathbb{S}$,

$\left(\int_{0}^{t}\xi^{m_{k}}_{s}\cdot
dz_{s}\right)\left(\cdot,\ell\right)\underset{k\to\infty}{\longrightarrow}\left(\int_{0}^{t}\xi_{s}\cdot
dz_{s}\right)\left(\cdot,\ell\right)\quad\overline{\mathbb{P}^{\mathbb{W}}}-\text{a.s.}$
(15)

With the same argument as above, we have

$\int_{0}^{t}\xi_{s}\cdot
dx_{\ell_{s}}=\overline{\mathbb{P}^{\mathbb{W}}}\\!-\lim_{k\to\infty}\int_{0}^{t}\xi^{m_{k}}_{s}\cdot
dx_{\ell_{s}}\quad\text{for every $\ell\in\mathbb{S}$}.$ (16)

On the other hand, by the very definition of stochastic integral it is
immediate to notice that, for every
$\left(w,\ell\right)\in\mathbb{W}\times\mathbb{S},$

$\left(\int_{0}^{t}\xi^{m_{k}}_{s}\cdot
dz_{s}\right)\left(w,\ell\right)=\sum_{i=0}^{{m_{k}}-1}\xi_{t_{i}}\cdot\left(z_{t_{i+1}}-z_{t_{i}}\right)\left(w,\ell\right)=\sum_{i=0}^{{m_{k}}-1}\xi_{t_{i}}\cdot\left(x_{\ell_{t_{i+1}}}-x_{\ell_{t_{i}}}\right)\left(w\right)=\left(\int_{0}^{t}\xi^{m_{k}}_{s}\cdot
dx_{\ell_{s}}\right)\left(w\right).$

Combining the last equation with (15) and (16) we get

$\left(\int_{0}^{t}\xi_{s}\cdot
dz_{s}\right)\left(\cdot,\ell\right)=\int_{0}^{t}\xi_{s}\cdot
dx_{\ell_{s}}\quad\overline{\mathbb{P}^{\mathbb{W}}}-\text{a.s., for
}\mathbb{P}^{\mathbb{S}}-\text{a.e. }\ell\in\mathbb{S},$

proving the thesis of the lemma. ∎

A useful result due to [2, Equation $\left(14\right)$] shows that there exists
a constant $c>0$ such that, for every $t>0$, the density $\eta_{t}$ of $L_{t}$
satisfies

$\eta_{t}\left(s\right)\leq c\,t\,s^{-1-\alpha}e^{-ts^{-\alpha}},\quad s>0.$

As a consequence, for every $p\geq 1$ we have that ${L_{t}}^{-1}\in L^{p}$,
with

$\mathbb{E}\left[\frac{1}{L_{t}^{p}}\right]^{1/p}\leq
c_{\alpha,p}\frac{1}{t^{1/\alpha}}\quad\text{for some }c_{\alpha,p}>0.$ (17)

We are now in position to obtain the derivation formula for the Markov
transition semigroup, together with an estimate on its gradient, in the
subordinated Brownian motion case.

######  Theorem 6.

Let $t>0,\,\phi\in C_{b}\left(\mathbb{R}^{N}\right)$ and assume that
$\sigma_{n}^{2}>0,\,n=1,\dots,N$. Then the function
$\mathbb{E}\left[\phi\left(Z^{\cdot}_{t}\right)\right]$ is differentiable at
any point $x\in\mathbb{R}^{N}$ in every direction $h\in\mathbb{R}^{N}$, and

$\left\langle\nabla\mathbb{E}\left[\phi\left(Z_{t}^{x}\right)\right],h\right\rangle=\mathbb{E}\left[\phi\left(Z_{t}^{x}\right)\left(\sum_{n=1}^{N}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}d\beta_{L_{s}}^{n}\right)\right].$
(18)

In addition, there exists $c_{\alpha}>0$ such that the following gradient
estimate holds:

$\sup_{x\in\mathbb{R}^{N}}\left|\nabla\mathbb{E}\left[\phi\left(Z_{t}^{x}\right)\right]\right|\leq
c_{\alpha}\left\lVert\phi\right\rVert_{\infty}\sup_{n=1,\dots,N}\left(\frac{1}{\sigma_{n}}\sqrt[2\alpha]{\frac{2\alpha\lambda_{n}}{1-e^{-2\alpha\lambda_{n}t}}}e^{-\lambda_{n}t}\right)\quad\text{for
every }t>0.$ (19)

###### Proof.

Fix $t>0$ and $\phi\in C_{b}\left(\mathbb{R}^{N}\right)$. In what follows, we
denote by $\mathbb{E}^{\mathbb{W}}\left[\cdot\right]$ the expected value of a
random variable defined on
$\left(\mathbb{W},\overline{\mathcal{B}\left(\mathbb{W}\right)},\overline{\mathbb{P}^{\mathbb{W}}}\right)$.
Bearing in mind that
$Z_{t}^{x}=e^{tA}x+\int_{0}^{t}e^{\left(t-s\right)A}\sqrt{Q}\,dW^{N}_{L_{s}}$,
by (14) we have, for every $x\in\mathbb{R}^{N}$,

$Z_{t}^{x}=\left(e^{tA}x+\int_{0}^{t}e^{\left(t-s\right)A}\sqrt{Q}\,dz_{s}\right)\circ\psi=\left(e^{tA}x+\int_{0}^{t}e^{\left(t-s\right)A}\sqrt{Q}\,dz_{s}\right)\left(W^{N}\left(\cdot\right),L\left(\cdot\right)\right)\quad\mathbb{P}-\text{a.s.}$

Therefore recalling the expression (13) for the regular conditional
distribution
$\mathbb{P}\left(W^{N}\\!\left(\cdot\right)\in\cdot\big{|}\mathcal{F}^{L}\right)$,
we apply the _disintegration formula_ for the conditional expectation to write

$\displaystyle\mathbb{E}\left[\phi\left(Z^{x}_{t}\right)\right]$
$\displaystyle=\mathbb{E}\left[\mathbb{E}\left[\phi\left(Z^{x}_{t}\right)\bigg{|}\mathcal{F}^{L}\right]\right]=\mathbb{E}\left[\int_{\mathbb{W}}\phi\left(\left(e^{tA}x+\int_{0}^{t}e^{\left(t-s\right)A}\sqrt{Q}\,dz_{s}\right)\left(w,L\left(\cdot\right)\right)\right)\overline{\mathbb{P}^{\mathbb{W}}}\left(dw\right)\right]$
$\displaystyle=\mathbb{E}\left[{\left.\kern-1.2pt\mathbb{E}^{\mathbb{W}}\left[\phi\left(e^{tA}x+\int_{0}^{t}e^{\left(t-s\right)A}\sqrt{Q}\,dx_{\ell_{s}}\right)\right]\vphantom{\big{|}}\right|_{\ell=L\left(\cdot\right)}}\right]=\mathbb{E}\left[{\left.\kern-1.2pt\mathbb{E}^{\mathbb{W}}\left[\phi\left(Z^{\ell}_{t}\left(x\right)\right)\right]\vphantom{\big{|}}\right|_{\ell=L\left(\cdot\right)}}\right],\quad
x\in\mathbb{R}^{N},$

where in the second–to–last equality we use Lemma 5 and the fact that
$\mathbb{P}^{\mathbb{S}}$ is the pushforward probability measure generated by
$L\left(\cdot\right)$ on $\mathbb{S}$. Take $x\in\mathbb{R}^{N}$ and a
direction $h\in\mathbb{R}^{N}$; if we can justify the derivation under the
expected value, an application of (11) immediately leads to (18), as the
following computations based on the previous argument show:

$\displaystyle\left\langle\nabla\mathbb{E}\left[\phi\left(Z_{t}^{x}\right)\right],h\right\rangle=\mathbb{E}\left[{\left.\kern-1.2pt\mathbb{E}^{\mathbb{W}}\left[\phi\left(Z_{t}^{\ell}\left(x\right)\right)\left(\sum_{n=1}^{N}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}d\ell_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}dx_{\ell_{s}}^{n}\right)\right]\vphantom{\big{|}}\right|_{\ell=L\left(\cdot\right)}}\right]$
(20)
$\displaystyle\quad\\!=\\!\mathbb{E}\\!\left[\sum_{n=1}^{N}\\!\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}\\!e^{-2\lambda_{n}\left(t-s\right)}dL_{s}}\\!\left\\{\\!\int_{\mathbb{W}}\\!\left(\\!\phi\\!\left(\\!e^{tA}x\\!+\\!\int_{0}^{t}\\!e^{\left(t-s\right)A}\sqrt{Q}\,dz_{s}\right)\\!\\!\times\\!\\!\left(\int_{0}^{t}\\!e^{-\lambda_{n}\left(t-s\right)}dz_{s}^{n}\right)\\!\\!\right)\\!\left(w,L\left(\cdot\right)\right)\overline{\mathbb{P}^{\mathbb{W}}}\\!\left(dw\right)\right\\}\\!\right]$
$\displaystyle\quad\\!=\mathbb{E}\left[\mathbb{E}\left[\phi\left(Z_{t}^{x}\right)\left(\sum_{n=1}^{N}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}d\beta_{L_{s}}^{n}\right)\Bigg{|}\mathcal{F}^{L}\right]\right].$

Indeed, such a derivation is licit, since _Jensen’s inequality_ and (12)
entail

$\displaystyle\left|{\left.\kern-1.2pt\mathbb{E}^{\mathbb{W}}\left[\phi\left(Z_{t}^{\ell}\left(x\right)\right)\left(\sum_{n=1}^{N}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}d\ell_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}dx_{\ell_{s}}^{n}\right)\right]\vphantom{\big{|}}\right|_{\ell=L\left(\cdot\right)}}\right|^{2}\leq\left\lVert\phi\right\rVert_{\infty}^{2}\sum_{n=1}^{N}\frac{1}{\sigma_{n}^{2}}\frac{e^{-2\lambda_{n}t}\left|\left\langle
h,e_{n}\right\rangle\right|^{2}}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}},$
(21)

with the right–hand side which does not depend on $x$ and is integrable. In
fact, for every $n=1,\dots,N$, recalling that
$L_{1}\sim\text{stable}\left(\alpha,1,\bar{c}^{1/\alpha},0\right)$ by (1), we
have

$\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}\sim\text{stable}\left(\alpha,1,\bar{c}^{\frac{1}{\alpha}}\left(\frac{1-e^{-2\alpha\lambda_{n}t}}{2\alpha\lambda_{n}}\right)^{1/\alpha},0\right)\Longrightarrow\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}\sim\left(\frac{1-e^{-2\alpha\lambda_{n}t}}{2\alpha\lambda_{n}}\right)^{\frac{1}{\alpha}}L_{1},$

hence by (17) there exists $c_{\alpha}>0$ such that

$\mathbb{E}\left[\sum_{n=1}^{N}\frac{1}{\sigma_{n}^{2}}\frac{e^{-2\lambda_{n}t}\left|\left\langle
h,e_{n}\right\rangle\right|^{2}}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}}\right]\leq\mathbb{E}\left[\frac{1}{L_{1}}\right]\left(\sum_{n=1}^{N}\frac{e^{-2\lambda_{n}t}}{\sigma_{n}^{2}}\left(\frac{2\alpha\lambda_{n}}{1-e^{-2\alpha\lambda_{n}t}}\right)^{\frac{1}{\alpha}}\left|\left\langle
h,e_{n}\right\rangle\right|^{2}\right)\\\ \leq
c_{\alpha}\sum_{n=1}^{N}\frac{e^{-2\lambda_{n}t}}{\sigma_{n}^{2}}\left(\frac{2\alpha\lambda_{n}}{1-e^{-2\alpha\lambda_{n}t}}\right)^{\frac{1}{\alpha}}\left|\left\langle
h,e_{n}\right\rangle\right|^{2}.$ (22)

Concerning the gradient estimate, it is sufficient to combine (20), (21) &
(22) and to recall that the $L^{1}$–norm of a random variable is smaller than
its $L^{2}$–norm to get

$\left|\left\langle\nabla\mathbb{E}\left[\phi\left(Z_{t}^{x}\right)\right],h\right\rangle\right|\leq
c_{\alpha}\left\lVert\phi\right\rVert_{\infty}\sup_{n=1,\dots,N}\left(\frac{1}{\sigma_{n}}\sqrt[2\alpha]{\frac{2\alpha\lambda_{n}}{1-e^{-2\alpha\lambda_{n}t}}}e^{-\lambda_{n}t}\right)\left|h\right|,\quad
x,h\in\mathbb{R}^{N},$

where the constant $c_{\alpha}$ is allowed to be different from the one in
(22). The desired inequality (19) is then recovered taking the $\sup$ for
$\left|h\right|\leq 1$, and the proof is complete. ∎

As in Remark 3, note that Theorem 6 holds true for every
$\phi\in\mathcal{B}_{b}\left(\mathbb{R}^{N}\right)$.

###  3.2 Infinite dimensional case

In this subsection we analyze the general case where $H$ is infinite
dimensional. Assuming $\sigma_{n}^{2}>0,\,n\in\mathbb{N}$, let us introduce
the following Hypothesis:

$\sup_{n}\left(\frac{1}{\sigma_{n}}\sqrt[2\alpha]{\frac{2\alpha\lambda_{n}}{1-e^{-2\alpha\lambda_{n}t}}}e^{-\lambda_{n}t}\right)\leq
C_{t}\quad\text{ for every }t>0,\,\text{for some function }C_{t}>0.$ (ii)

In this setting, for every $h\in H$ and $t>0$, we can define the real–valued
random variable

$\sum_{n=1}^{\infty}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}d\beta_{L_{s}}^{n}\coloneqq
L^{2}-\lim_{N\to\infty}\left(\sum_{n=1}^{N}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}d\beta_{L_{s}}^{n}\right).$

Indeed, with the same argument as the one in (22), Hypothesis (ii) yields

$\displaystyle\mathbb{E}\left[\left|\sum_{n=m}^{M}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}d\beta_{L_{s}}^{n}\right|^{2}\right]$
$\displaystyle\leq
c_{\alpha}\sum_{n=m}^{M}\frac{e^{-2\lambda_{n}t}}{\sigma_{n}^{2}}\left(\frac{2\alpha\lambda_{n}}{1-e^{-2\alpha\lambda_{n}t}}\right)^{\frac{1}{\alpha}}\left|\left\langle
h,e_{n}\right\rangle\right|^{2}$ $\displaystyle\leq
c_{\alpha}\,C_{t}^{2}\,\left(\sum_{n=m}^{M}\left|\left\langle
h,e_{n}\right\rangle\right|^{2}\right)\underset{m,M\to\infty}{\longrightarrow}0,$

where $c_{\alpha}>0$. In particular,

$\mathbb{E}\left[\left|\sum_{n=1}^{\infty}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}d\beta_{L_{s}}^{n}\right|^{2}\right]^{\frac{1}{2}}\leq\sqrt{c_{\alpha}}\,C_{t}\,\left\lVert
h\right\rVert_{H}.$ (23)

Hence the following, useful property holds:

$\sum_{n=1}^{\infty}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h_{m},e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}d\beta_{L_{s}}^{n}\overset{L^{2}}{\longrightarrow}\sum_{n=1}^{\infty}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}d\beta_{L_{s}}^{n}\quad\text{as
}h_{m}\to h.$ (24)

At this point we can present the main theorem of the paper.

######  Theorem 7.

Assume $\sigma_{n}^{2}>0,\,n\in\mathbb{N}$, together with Hypotheses (i)&
(ii).

Then for every $\phi\in B_{b}\left(H\right)$ and $t>0$ the function
$R_{t}\phi\in C^{1}_{b}\left(H\right)$ and there exists $c_{\alpha}>0$ such
that

$\sup_{x\in H}\left\lVert\nabla R_{t}\phi\left(x\right)\right\rVert_{H}\leq
c_{\alpha}\,C_{t}\left\lVert\phi\right\rVert_{\infty}\quad\text{for every
}t>0.$ (25)

Moreover, given $\phi\in C_{b}\left(H\right)$ and $t>0$, for every $x,h\in H$
the Gateaux derivative of $R_{t}\phi$ at $x$ along the direction $h$ is given
by

$\left\langle\nabla
R_{t}\phi\left(x\right),h\right\rangle=\mathbb{E}\left[\phi\left(Z_{t}^{x}\right)\left(\sum_{n=1}^{\infty}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}d\beta_{L_{s}}^{n}\right)\right].$
(26)

###### Proof.

Fix $t>0$ and a function $\phi\in C_{b}\left(H\right)$.

We first consider the case $\dim H=N,$ identifying $H$ with $\mathbb{R}^{N}$,
as usual. Evidently (26) coincides with (18) and the map $x\mapsto\nabla
R_{t}\left(x\right)$ is a continuous function from $\mathbb{R}^{N}$ into
itself: this follows from dominated convergence, together with $\phi\in
C_{b}\left(\mathbb{R}^{N}\right)$ and $Z_{t}^{x_{n}}\to Z_{t}^{x}$ a.s. as
$x_{n}\to x$. Moreover, Hypothesis (ii) applied to (19) directly entails (25),
therefore $R_{t}\phi\in C_{b}^{1}\left(\mathbb{R}^{N}\right)$. In order to
pass to infinite dimension it is convenient to write

$\displaystyle R_{t}\phi\left(x+h\right)-R_{t}\phi\left(x\right)$
$\displaystyle=\int_{0}^{1}\left\langle\nabla
R_{t}\phi\Big{(}\left(1-\rho\right)x+\rho\left(x+h\right)\Big{)}\,,\,h\right\rangle
d\rho$ $\displaystyle=\int_{0}^{1}\mathbb{E}\left[\phi\left(Z_{t}^{x+\rho
h}\right)\left(\sum_{n=1}^{N}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}d\beta_{L_{s}}^{n}\right)\right]d\rho.$
(27)

We now consider the general case $\dim H=\infty$. Let $\pi_{N}$ be the
projection onto the first $N$ Fourier components and $H_{N}$ be its range. Due
to the diagonal structure of our model, the projections $\pi_{N}Z_{t}^{x}$ of
the OU–process are, $\mathbb{P}-$a.s.,

$\pi_{N}Z^{x}_{t}=\sum_{n=1}^{N}e^{-\lambda_{n}t}\left\langle
x,e_{n}\right\rangle
e_{n}+\sum_{n=1}^{N}\left(\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}\sigma_{n}d\beta_{L_{s}}^{n}\right)e_{n},\quad
N\in\mathbb{N}.$

Therefore introducing the operators
$A_{N}\coloneqq{\left.\kern-1.2ptA\vphantom{\big{|}}\right|_{H_{N}}}$ and
$Q_{N}\coloneqq{\left.\kern-1.2ptQ\vphantom{\big{|}}\right|_{H_{N}}}$, which
map $H_{N}$ into itself, we can write
$\pi_{N}Z^{x}_{t}=e^{tA_{N}}\left(\pi_{N}x\right)+\tilde{Z}_{A_{N},Q_{N}}\left(t\right)$:
this shows that such projections are OU–processes in $H_{N}$. Thus, the
dominated convergence theorem together with the expression in (3.2) and the
continuity of $\phi$ give

$\displaystyle R_{t}\phi\left(x+h\right)-R_{t}\phi\left(x\right)$
$\displaystyle=\lim_{N\to\infty}\mathbb{E}\left[\phi\left(\pi_{N}Z_{t}^{x+h}\right)-\phi\left(\pi_{N}Z_{t}^{x}\right)\right]$
$\displaystyle=\lim_{N\to\infty}\int_{0}^{1}\mathbb{E}\left[\phi\left(\pi_{N}Z_{t}^{x+\rho
h}\right)\left(\sum_{n=1}^{N}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}d\beta_{L_{s}}^{n}\right)\right]d\rho$
$\displaystyle=\int_{0}^{1}\mathbb{E}\left[\phi\left(Z_{t}^{x+\rho
h}\right)\left(\sum_{n=1}^{\infty}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}d\beta_{L_{s}}^{n}\right)\right]d\rho.$

Now we can define
$D_{t,x}\left(h\right)\coloneqq\mathbb{E}\left[\phi\left(Z_{t}^{x}\right)\left(\sum_{n=1}^{\infty}\frac{1}{\sigma_{n}}\frac{e^{-\lambda_{n}t}\left\langle
h,e_{n}\right\rangle}{\int_{0}^{t}e^{-2\lambda_{n}\left(t-s\right)}dL_{s}}\int_{0}^{t}e^{-\lambda_{n}\left(t-s\right)}d\beta_{L_{s}}^{n}\right)\right]$:
it is the Fréchet differential of $R_{t}\phi$ at $x$ (hence, in particular,
(26) is verified). To see this, it is sufficient to note that the linear
operator $D_{t,x}\left(\cdot\right)$ is continuous by the property in (24) and
to apply Hölder’s inequality, the dominated convergence theorem and (23) to
get, for a positive constant $c_{\alpha}$,

$\displaystyle\left|R_{t}\phi\left(x+h\right)-R_{t}\phi\left(x\right)-D_{t,x}\left(h\right)\right|\leq
c_{\alpha}\,C_{t}\left\lVert
h\right\rVert_{H}\int_{0}^{1}\mathbb{E}\left[\left|\phi\left(Z_{t}^{x+\rho
h}\right)-\phi\left(Z_{t}^{x}\right)\right|^{2}\right]^{1/2}d\rho=o\left(\left\lVert
h\right\rVert_{H}\right).$

The upper bound (25) for the norm of the gradient is then obtained by (23)
from the next, straightforward computation:

$\left\lVert\nabla R_{t}\phi\left(x\right)\right\rVert_{H}=\sup_{\left\lVert
h\right\rVert_{H}\leq 1}\left|\left\langle\nabla
R_{t}\phi\left(x\right),h\right\rangle\right|=\sup_{\left\lVert
h\right\rVert_{H}\leq 1}\left|D_{t,x}\left(h\right)\right|\leq
c_{\alpha}\,C_{t}\,\left\lVert\phi\right\rVert_{\infty},\quad x\in H.$

We also note that

$\sup_{\left\lVert h\right\rVert_{H}\leq
1}\left|\left(D_{t,x_{n}}-D_{t,x}\right)\left(h\right)\right|\leq
c_{\alpha}\,C_{t}\,\mathbb{E}\left[\left|\phi\left(Z_{t}^{x_{n}}\right)-\phi\left(Z_{t}^{x}\right)\right|^{2}\right]^{1/2}\to
0\quad\text{as }x_{n}\to x:$

this proves the continuity of the map $x\mapsto D_{t,\cdot}$, hence
$R_{t}\phi\in C^{1}_{b}\left(H\right)$.

Finally, we need to study the case where $\phi$ is just Borel measurable and
bounded, without the hypothesis of continuity. In order to do this, it is
sufficient to observe that by the _mean value theorem_ and (25) we have, for
every $\phi\in C_{b}^{2}\left(H\right),$

$\left|R_{t}\phi\left(x\right)-R_{t}\phi\left(y\right)\right|\leq
c_{\alpha}\,C_{t}\,\left\lVert\phi\right\rVert_{\infty}\left\lVert
x-y\right\rVert_{H},\quad x,y\in H.$ (28)

Being $R_{t}$ Markovian, [4, Lemma $7.1.5$] implies that the same holds true
for every $\phi\in B_{b}\left(H\right)$. In particular, $R_{t}$ maps bounded,
Borel measurable functions in bounded, Lip–continuous functions. The semigroup
law let us write $R_{t}\phi=R_{s}\left(R_{t-s}\phi\right)$ for some $0<s<t$,
which proves $R_{t}\phi\in C_{b}^{1}\left(H\right)$. The bound (25) follows
from (28), hence the proof is complete. ∎

We now focus on the gradient estimate (9). We need to substitute Hypothesis
(ii) with the following, stronger one:

$\sup_{n}\left(\frac{1}{\sigma_{n}}\sqrt[2\alpha]{\frac{2\alpha\lambda_{n}}{1-e^{-2\alpha\lambda_{n}t}}}e^{-\lambda_{n}t}\right)\leq
C_{0}\frac{1}{t^{\gamma}},\quad\text{for every }t>0,\,\text{for some
}C_{0}>0,\,0<\gamma<1.$ (iii)

In other terms, in Hypothesis (ii) we take $C_{t}\coloneqq
C_{0}\,t^{-\gamma},\,t>0$, for some $C_{0}>0,\,\gamma\in\left(0,1\right)$.

######  Remark 4.

Observe that, for every $n\in\mathbb{N}$, the term

$\frac{1}{\sigma_{n}}\sqrt[2\alpha]{\frac{2\alpha\lambda_{n}}{1-e^{-2\alpha\lambda_{n}t}}}e^{-\lambda_{n}t}\sim\frac{1}{\sigma_{n}}\frac{1}{t^{1/\left(2\alpha\right)}}\quad\text{as
}t\downarrow 0.$

Therefore, Hypothesis (iii) should be verified only in the case
$\alpha\in\left(\frac{1}{2},1\right)$ and for some
$\gamma\in\left[\frac{1}{2\alpha},1\right)$.

It is also worth noticing that Hypothesis (iii) is equivalent to the next
condition:

$\sigma_{n}\geq C_{1}\,\lambda_{n}^{\frac{1}{2\alpha}-\gamma},\quad
n\in\mathbb{N},$ (iii′)

for some $C_{1}>0$ and $\gamma\in\left[\frac{1}{2\alpha},1\right)$. A short
argument proving the latter fact is shown in [11, _Hypothesis (N)_].

At this point the next result is immediate.

######  Corollary 8.

Consider $\alpha\in\left(\frac{1}{2},1\right)$ and assume
$\sigma_{n}^{2}>0,\,n\in\mathbb{N}$, together with Hypotheses (i)& (iii).

Then for every $\phi\in\mathcal{B}_{b}\left(H\right)$ the function
$R_{t}\phi\in C_{b}^{1}\left(H\right),\,t>0,$ and the gradient estimate (9)
holds, namely there exists a constant $C>0$ such that

$\sup_{x\in H}\left\lVert\nabla
R_{t}\phi\left(x\right)\right\rVert_{H}\leq\frac{C}{t^{\gamma}}\left\lVert\phi\right\rVert_{\infty}\quad\text{for
every }t>0,$

where $\gamma\in\left[\frac{1}{2\alpha},1\right)$ is the one appearing in
Hypothesis (iii).

######  Example 2.

We investigate Hypothesis (iii), in its equivalent formulation (iii′) provided
by Remark 4, in the same framework as in Example 1. So we take $A=\Delta$
(hence
$-\lambda_{k}=-\left(2\pi\right)^{2}\left|k\right|^{2},\,k\in\mathbb{Z}_{0}^{d}$)
and study two possible choices for $Q$.

  * •

If $Q=\text{Id}$, then

$1\geq\frac{1}{\left(2\pi\left|k\right|\right)^{2\left(\gamma-\frac{1}{2\alpha}\right)}},\quad
k\in\mathbb{Z}_{0}^{d}$

for every $\gamma\in\left[\frac{1}{2\alpha},1\right)$. Therefore, in dimension
$d=1$ both conditions (i) and (iii) are satisfied. In particular, motivated by
the fact that $R_{t}$ is a regularization operator with $R_{0}=\text{Id}$, we
are interested in the behavior of $\nabla R_{t}\phi$ around $0$, where
$\phi\in\mathcal{B}_{b}\left(H\right)$. Therefore we choose
$\gamma=\frac{1}{2\alpha}$ and Corollary 8 provides the next estimate:

$\sup_{x\in H}\left\lVert\nabla R_{t}\phi\left(x\right)\right\rVert_{H}\leq
C\,\frac{1}{t^{2\alpha}}\left\lVert\phi\right\rVert_{\infty}\quad\text{for
every }t>0,$

for a positive constant $C$.

  * •

If $Q=Q_{\eta}=\left(-\Delta\right)^{-\eta}$ for $\eta>0$, then
$\sigma^{\left(\eta\right)}_{k}=\lambda_{k}^{-\eta/2},\,k\in\mathbb{Z}_{0}^{d}$,
and (iii′) holds true if and only if $\eta\leq 2\gamma-\frac{1}{\alpha}$.
Since we can take any $\gamma\in\left[\frac{1}{2\alpha},1\right)$, the
aforementioned condition holds as soon as $\eta<2-\frac{1}{\alpha}.$ Combining
this result with (8) obtained in Example 1, we conclude that Hypotheses (i)
and (iii) simultaneously hold if and only if

$\eta\in\left(\max\left\\{\frac{d-2}{2\alpha},0\right\\}\\!,\,2-\frac{1}{\alpha}\right).$

It then follows that there exist negative fractional powers of the Laplacian
$Q_{\eta}=\left(-\Delta\right)^{-\eta}$ meeting the requirements of Corollary
8 up to dimension $d=3$. Specifically, for $d=1,2$ there is a $Q_{\eta}$ with
the searched properties for every $\alpha\in\left(\frac{1}{2},1\right)$,
whereas in dimension $d=3$ we can find such a $Q_{\eta}$ only for
$\alpha\in\left(\frac{3}{4},1\right)$.

## Acknowledgment

I thank Professor Franco Flandoli for useful discussions and valuable insight
into the subject.

## References

  * [1] Benth, F. E., Benth, J. S., & Koekebakker, S. (2008). _Stochastic modelling of electricity and related markets_ (Vol. 11). World Scientific.
  * [2] Bogdan, K., Stós, A., & Sztonyk, P. (2003). Harnack inequality for stable processes on d-sets. _Studia Math, 158_(2), 163–198.
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  * [6] Flandoli, F., & Luo, D. (2020). Convergence of transport noise to Ornstein–Uhlenbeck for 2D Euler equations under the enstrophy measure. _The Annals of Probability, 48_(1), 264–295.
  * [7] Jacod, J. (2006). _Calcul stochastique et problemes de martingales_ (Vol. 714). Springer.
  * [8] Karatzas, I., & Shreve, S. E. (1998). Brownian motion. In _Brownian Motion and Stochastic Calculus_ (pp. 47–127). Springer, New York, NY.
  * [9] Kusuoka, S. (2010). Malliavin calculus for stochastic differential equations driven by subordinated Brownian motions. _Kyoto Journal of Mathematics, 50_(3), 491–520.
  * [10] Natanson, I. P. (2016)._Theory of functions of a real variable_. Courier Dover Publications.
  * [11] Priola, E., & Zabczyk, J. (2011). Structural properties of semilinear SPDEs driven by cylindrical stable processes. _Probability theory and related fields, 149_(1-2), 97–137.
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  * [13] Ken-Iti, S. (1999). _Lévy processes and infinitely divisible distributions_. Cambridge university press.
  * [14] Winkelbauer, A. (2012). Moments and absolute moments of the normal distribution. _arXiv preprint arXiv:1209.4340_
  * [15] Zhang, X. (2013). Derivative formulas and gradient estimates for SDEs driven by $\alpha$-stable processes. _Stochastic Processes and their Applications, 123_(4), 1213–1228.


monthdayyeardate[] ,

On General Sombor index

Phanjoubam Chinglensana1, Sainkupar Mn Mawiong2

1Department of Mathematics, North-Eastern Hill University,

NEHU Campus, Shillong-793022, INDIA

e-mail<EMAIL_ADDRESS>
2 Department of Basic Sciences and Social Sciences, North-Eastern Hill
University,

NEHU Campus, Shillong-793022, INDIA

e-mail<EMAIL_ADDRESS>
Abstract: We present the bounds in terms of other important graph parameters
for general Sombor index which generalises both the forgotten index and the
Sombor index. We also explore the Nordhaus-Gaddum-type result for the general
Sombor index. We present further the relations between general Sombor index
and other generalised indices: general Randić index and general sum-
connectivity index.

Keywords: General Sombor index, General Randić index, General sum-connectivity
index
2010 Mathematics Subject Classification: 05C07, 05C90.

##  1 Introduction

We consider only finite simple graph in this paper. Let $G$ be a finite simple
graph on $n$ vertices and $m$ edges. We denote the vertex set and the edge set
of $G$ by $V(G)$ and $E(G)$, respectively. The degree of a vertex $u\in V(G)$
is denoted by $d_{G}(u)$ and it is defined as the number of edges incident
with $u$. The complement of $G$, denoted by $\overline{G}$, is a simple graph
on $V(G)$ in which two vertices $u$ and $v$ are adjacent, i.e., joined by an
edge $uv$, if and only if they are not adjacent in $G$. Hence, $uv\in
E(\overline{G})$ if and only if $uv\notin E(G)$. Clearly, $E(G)\cup
E(\overline{G})=E(K_{n})$, where $K_{n}$ is the complete graph on $n$ vertices
and $\overline{m}=|E(\overline{G})|={n\choose 2}-m.$ Let $\Delta$ and $\delta$
denote the maximum vertex degree and the minimum vertex degree of the graph
$G$, respectively.

In chemical graph theory, one generally considers various graph-theoretical
invariants of molecular graphs (also known as topological indices or molecular
descriptors), and study how strongly are they correlated with various
properties of the corresponding molecules. The first such topological index
was introduced in 1947 by H. Wiener [18] and is used for correlation with
boiling points of alkanes. Wiener’s index is related to the distances in
molecular graphs. Historically, the first vertex-degree-based topological
indices were the graph invariants that nowadays are called Zagreb indices.
Numerous graph invariants have been (and still continues to be) employed with
varying degrees of success in QSAR (quantitative structure-activity
relationship) and QSPR (quantitative structure-property relationship) studies.
The Zagreb indices are amongst the most studied invariants [14] and they are
defined as sums of contributions dependent on the degrees of adjacent vertices
over all edges of a graph. The Zagreb indices of a graph $G$, i.e., the first
Zagreb index $M_{1}(G)$ and the second Zagreb index $M_{2}(G)$, were
originally defined [11] as follows.

$M_{1}(G)=\sum_{u\in V(G)}{d_{G}(u)}^{2};\ \ M_{2}(G)=\sum_{uv\in
E(G)}d_{G}(u)d_{G}(v).$

The first Zagreb index of $G$ can also be expressed as

$M_{1}(G)=\sum_{uv\in E(G)}[d_{G}(u)+d_{G}(v)].$

Generalised version of the first Zagreb index have also been introduced [13],
known as the general first Zagreb index and is defined as

$M_{1}^{p}(G)=\sum_{u\in V(G)}{d_{G}(u)}^{p}.$

When $p=3$, $M_{1}^{3}(G)=\displaystyle\sum_{u\in V(G)}{d_{G}(u)}^{3}$ is
known as the forgotten index, denoted by $F(G)$, and is also equal to

$F(G)=\sum_{uv\in E(G)}[{d_{G}(u)}^{2}+{d_{G}(v)}^{2}].$

One of the highly successful and widely used indices in QSPR and QSAR is the
Randić index $R(G)$ [16]. It is defined as

$R(G)=\sum_{uv\in E(G)}\dfrac{1}{\sqrt{d_{G}(u)d_{G}(v)}}.$

The Randić index have been extended to general Randić index [11], defined as

$R_{\alpha}(G)=\sum_{uv\in E(G)}[d_{G}(u)d_{G}(v)]^{\alpha}$

for any real number $\alpha$. Motivated by Randić and Zagreb indices, Zhou and
Trinajstić defined sum-connectivity index $\chi(G)$ [19] and general sum-
connectivity index $\chi_{\alpha}(G)$ [20], which are defined as

$\chi(G)=\sum_{uv\in E(G)}\dfrac{1}{\sqrt{d_{G}(u)+d_{G}(v)}}$

and

$\chi_{\alpha}(G)=\sum_{uv\in E(G)}[d_{G}(u)+d_{G}(v)]^{\alpha}$

for any real number $\alpha$. Gutman recently introduced [9] a new vertex-
degree-based topological index called the Sombor index, which is defined as
$SO(G)=\displaystyle\sum_{uv\in E(G)}\sqrt{{d_{G}(u)}^{2}+{d_{G}(v)}^{2}}$.
Numerous work have so far been carried out on the Sombor index. The chemical
applicability of the Sombor index is found to have shown good predictive
potential [15]. Computations of Sombor index of various graphs have been
carried out, for example chemical graphs [5]. Basic properties of the Sombor
index have been presented and its relations with other topological indices:
the Zagreb indices, are investigated in [6]. In [3] and [17], in addition to
Zagreb indices, relations between Sombor index and other topological indices
are carried out.

Motivated by the extensions of Randić and sum-connectivity indices and several
works on Sombor index, we define general Sombor index $SO_{\alpha}(G)$. It is
defined as

$SO_{\alpha}(G)=\displaystyle\sum_{uv\in
E(G)}[{d_{G}(u)}^{2}+{d_{G}(v)}^{2}]^{\alpha/2}$

for any real number $\alpha$. For $\alpha=1$, we have the usual Sombor index
whereas for $\alpha=2$, we get the forgotten index. Thus $SO_{\alpha}(G)$
generalises the Sombor index and the forgotten index. In this paper, we
present the bounds of general Sombor index in terms of other important graph
parameters. We also explore the Nordhaus-Gaddum-type result for the general
Sombor index. We present further the relations between general Sombor index
and other generalised indices: general Randić index and general sum-
connectivity index.

##  2 Preliminaries

A graph $G$ is called regular if all vertices of $G$ have the same vertex
degree and it is called bi-degreed if it has two distinct vertex degrees. A
connected graph $G$ is called a bi-regular graph or semi-regular bipartite
graph if $G$ is a bipartite graph with two partite sets $A$ and $B$ such that
each vertex in $A$ has degree $\Delta$ and each vertex in $B$ has degree
$\delta.$

Now, we state a lemma which provides necessary and sufficient conditions for
non-regular graphs to be bi-regular [2].

######  Lemma 2.1.

Let $G$ be a connected non-regular graph. Then the following statements are
equivalent:

  1. 1.

$G$ is bi-regular.

  2. 2.

$G$ is bi-degreed and $|d_{G}(u)-d_{G}(v)|>0$ is constant for all edges $uv$
of $G$.

  3. 3.

$d_{G}(u)+d_{G}(v)>0$ is constant for all edges $uv$ of $G$.

######  Remark 2.2.

We note that the third condition in Lemma 2.1 can be replaced by
${d_{G}(u)}^{2}+{d_{G}(v)}^{2}>0$ is constant for all edges $uv$ of $G$. Thus
the following statements are equivalent for a connected non-regular graph $G$:

  1. 1.

$G$ is bi-regular.

  2. 2.

$G$ is bi-degreed and $|d_{G}(u)-d_{G}(v)|>0$ is constant for all edges $uv$
of $G$.

  3. 3.

${d_{G}(u)}^{2}+{d_{G}(v)}^{2}>0$ is constant for all edges $uv$ of $G$.

###### Proof of the Remark 2.2.

Notice that (2) implies (3) is clear from the definition. So, we only prove
that (3) implies (1). Now by (3) we have that
${d_{G}(u)}^{2}+{d_{G}(v)}^{2}>0$ is constant for all edges $uv$ of $G$. This
implies that any two vertices joined by a path of even length in $G$ must have
the same degree. We now show that $G$ is bipartite by contradiction. Suppose
that $G$ is not bipartite. Then $G$ has an odd cycle $C$. Let $u,v$ be two
adjacent vertices on $C$. Then $C$ contains a path of even length connecting
$u$ and $v$. Hence, $d_{G}(u)=d_{G}(v)=k$ for some positive integer $k$. Since
$G$ is connected, for any vertex $x$ of $G$, there is a path $P$ between $u$
and $x$. Let $e=ab$ be any edge on the path $P$. Then by (3),
$d_{G}(a)^{2}+d_{G}(b)^{2}=2k^{2}$ since
${d_{G}(u)}^{2}+{d_{G}(v)}^{2}=2k^{2}$ for the edge $uv$ of $G$. Also
$d_{G}(u)=k.$ Then each vertex on the path $P$ has vertex degree $k$ and so
does $x$. It follows that $G$ is regular. This contradicts the assumption that
$G$ is not regular. Hence, $G$ is bipartite. Furthermore, any two vertices
$u,v$ in the same partite set of $G$ are joined by a path of even length, and
so $d_{G}(u)=d_{G}(v).$ Thus $G$ is bi-regular. This shows that (3) implies
(1). ∎

Next, we recall the famous Jensen’s inequality (see [4]).

######  Lemma 2.3 (Jensen’s inequality).

Let $f:(a,b)\to\mathbb{R}$ be a convex function. Let $n\in\mathbb{N}$ and
$\alpha_{1},\alpha_{2},\dots,\alpha_{n}\in(0,1)$ be real numbers such that
$\alpha_{1}+\alpha_{2}+\dots+\alpha_{n}=1$. Then for any
$x_{1},x_{2},\dots,x_{n}\in(a,b)$ we have

$f\left(\sum_{i=1}^{n}\alpha_{i}x_{i}\right)\leq\sum_{i=1}^{n}\alpha_{i}f(x_{i}).$

######  Corollary 2.4.

For a positive integer $k$, if $f$ is strictly convex (i.e.,
$f^{\prime\prime}>0$), then

$f\left(\sum_{i=1}^{k}\dfrac{x_{i}}{k}\right)\leq\dfrac{1}{k}\sum_{i=1}^{k}f(x_{i})$

where the equality holds if and only if $x_{1}=x_{2}=\dots=x_{k}.$ Moreover,
the inequality is reversed if $-f$ is strictly convex.

The following inequality is due to Radon and can be found in [6].

######  Lemma 2.5 (Radon’s inequality).

If $a_{k},b_{k}>0$ for $k=1,2,\dots,m$ and $p>0$, then

$\sum_{k=1}^{m}\dfrac{{a_{k}}^{p+1}}{{b_{k}}^{p}}\geq\dfrac{\left(\displaystyle\sum_{k=1}^{m}a_{k}\right)^{p+1}}{\left(\displaystyle\sum_{k=1}^{m}b_{k}\right)^{p}}.$

Equality holds if
$\dfrac{a_{1}}{b_{1}}=\dfrac{a_{2}}{b_{2}}=\dots=\dfrac{a_{m}}{b_{m}}$.

##  3 General Sombor index and its properties

In this section, we present the bounds of general Sombor index in terms of
other important graph parameters. We also explore the Nordhaus-Gaddum-type
result for the general Sombor index. First, we give some examples of general
Sombor index.

######  Example 3.1.

  1. 1.

$SO_{\alpha}(K_{n})=2^{-1+\alpha/2}n(n-1)^{\alpha+1}=2^{\alpha/2}m(n-1)^{\alpha}$
and $SO_{\alpha}(\overline{K_{n}})=0.$

  2. 2.

$SO_{\alpha}(C_{n})=2^{(3\alpha)/2}n.$

  3. 3.

$SO_{\alpha}(P_{2})=SO_{\alpha}(K_{2})=2^{\alpha/2}.$

  4. 4.

$SO_{\alpha}(P_{n})=2\times 5^{\alpha/2}+2(n-3)2^{\alpha/2}$ for $n\geq 3$.

###  3.1 Bounds for the general Sombor index

We first present bounds for the general Sombor index in terms of the forgotten
index and numbers of edges of the graph.

######  Theorem 3.1.

Let $G$ be a graph with $m\geq 1$ edges. Then we have the following.

  1. 1.

$SO_{\alpha}(G)\geq m^{1-\alpha/2}F(G)^{\alpha/2}$ if $\alpha<0$ or $\alpha>1$
and

  2. 2.

$SO_{\alpha}(G)\leq m^{1-\alpha/2}F(G)^{\alpha/2}$ if $0<\alpha<1$.

Moreover, equality holds in either cases if and only if $G$ is bi-regular.

###### Proof.

Notice that if $\alpha<0$ or $\alpha>1$, then $x^{\alpha/2}$ for $x>0$ is
strictly convex. Thus by Corollary 2.4, we have

$\displaystyle\left[\dfrac{1}{m}F(G)\right]^{\alpha/2}=$
$\displaystyle\left[\dfrac{1}{m}\displaystyle\sum_{uv\in
E(G)}({d_{G}(u)}^{2}+{d_{G}(v)}^{2})\right]^{\alpha/2}$ $\displaystyle=$
$\displaystyle\left[\displaystyle\sum_{uv\in
E(G)}\dfrac{{d_{G}(u)}^{2}+{d_{G}(v)}^{2}}{m}\right]^{\alpha/2}$
$\displaystyle\leq$ $\displaystyle\dfrac{1}{m}\displaystyle\sum_{uv\in
E(G)}[{d_{G}(u)}^{2}+{d_{G}(v)}^{2}]^{\alpha/2}=\dfrac{1}{m}SO_{\alpha}(G)$

Thus $SO_{\alpha}(G)\geq m^{1-\alpha/2}F(G)^{\alpha/2}$ and the equality holds
if and only if ${d_{G}(u)}^{2}+{d_{G}(v)}^{2}$ is a constant for all edges
$uv$ of $G$. That is, by Remark 2.2 the equality holds if and only if $G$ is
bi-regular. Similarly, if $0<\alpha<1$, then $-x^{\alpha/2}$ for $x>0$ is
strictly convex. Hence $SO_{\alpha}(G)\leq m^{1-\alpha/2}F(G)^{\alpha/2}$,
where the equality holds if and only if $G$ is bi-regular. ∎

It is reported in [8] that $F(G)\geq\dfrac{M_{1}(G)^{2}}{2m}$ for a graph with
$m$ edges, where the equality holds if and only if $G$ is a regular graph.
Also, $M_{1}(G)\geq\dfrac{4m^{2}}{n}$, where the equality holds if and only if
$G$ is a regular graph (due to the Cauchy-Schwarz inequality) [1]. Thus we
have the following bounds for the general Sombor index in terms of $n$ and
$m$.

######  Corollary 3.2.

Let $G$ be a graph on $n$ vertices and $m\geq 1$ edges. Then

  1. 1.

$SO_{\alpha}(G)\geq 8^{\alpha/2}m^{1+\alpha}n^{-\alpha}$ if $\alpha<0$ or
$\alpha>1$ and

  2. 2.

$SO_{\alpha}(G)\leq 8^{\alpha/2}m^{1+\alpha}n^{-\alpha}$ if $0<\alpha<1$.

The equality holds in either cases if and only if $G$ is regular.

Next, we present another bound for the general Sombor index.

######  Theorem 3.3.

Let $G$ be a graph on $n\geq 2$ vertices and $m$ edges. Then we have the
following.

  1. 1.

If $0<\alpha<1$, then

$SO_{\alpha}(G)\geq F(G)^{\alpha/2}$

where the equality holds if and only if $G=K_{2}\cup\overline{K_{n-2}}$ or
$G=\overline{K_{n}}.$

  2. 2.

If $\alpha<0$, then

$SO_{\alpha}(G)\leq 2^{-1+\alpha/2}n(n-1)$

where the equality holds if and only if $G=K_{2}$.

  3. 3.

If $\alpha>1$, then

$SO_{\alpha}(G)\leq 2^{\alpha/2}m(n-1)^{\alpha}$

where the equality holds if and only if $G=K_{n}$ or $G=\overline{K_{n}}.$

###### Proof.

Let $0<\alpha<1$. Then

$SO_{\alpha}(G)\geq\left[\displaystyle\sum_{uv\in
E(G)}[{d_{G}(u)}^{2}+{d_{G}(v)}^{2}]\right]^{\alpha/2}=F(G)^{\alpha/2}$

where the equality holds if and only if $|E(G)|\leq 1$, i.e.,
$G=K_{2}\cup\overline{K_{n-2}}$ or $G=\overline{K_{n}}.$ This proves (1).

Let $\alpha<0$ and $G\neq\overline{K_{n}}$. Let $G_{1}$ be the graph obtained
from $G$ by deleting the possible isolated vertices of $G$. Let
$n_{1}=|V(G_{1})|$ and let $\Delta_{1}$ and $\delta_{1}$ denote the maximum
and minimum degrees of $G_{1}$, respectively. Notice that by Handshake lemma,
we have $m_{1}\leq\dfrac{n_{1}\Delta_{1}}{2}$. Then

$SO_{\alpha}(G)\leq\displaystyle\sum_{uv\in
E(G_{1})}(2\delta_{1}^{2})^{\alpha/2}\leq\dfrac{n_{1}\Delta_{1}}{2}(2\delta_{1}^{2})^{\alpha/2}=2^{-1+\alpha/2}n_{1}\Delta_{1}{\delta_{1}}^{\alpha}\leq
2^{-1+\alpha/2}n(n-1),$

where the equality holds if and only if $G$ is regular and
$\Delta_{1}\delta_{1}^{\alpha}=n-1$, i.e., $G=K_{2}$. This proves (2).

Let $\alpha>1$. Notice that $d_{G}(u)\leq n-1$ for any vertex $u$ of $G$. Thus

$SO_{\alpha}(G)=\displaystyle\sum_{uv\in
E(G)}[{d_{G}(u)}^{2}+{d_{G}(v)}^{2}]^{\alpha/2}\leq
m[2(n-1)^{2}]^{\alpha/2}=2^{\alpha/2}m(n-1)^{\alpha}$

where the equality holds if and only if either $d_{G}(u)=d_{G}(v)=n-1$ for
every edge $uv$ of $G$ or $E(G)=\emptyset$, i.e., $G=K_{n}$ or
$G=\overline{K_{n}}$. This completes the proof. ∎

###  3.2 Nordhaus-Gaddum-type result for the general Sombor index

######  Theorem 3.4.

Let $G$ be a graph on $n\geq 2$ vertices and $m$ edges.

  1. 1.

If $\alpha>0$, then

$SO_{\alpha}(G)+SO_{\alpha}(\overline{G})\leq
2^{-1+\alpha/2}n(n-1)^{\alpha+1}$

where the equality holds if and only if $G=K_{n}$ or $G=\overline{K_{n}}$.
Further, if $\alpha\geq 1$, then

$SO_{\alpha}(G)+SO_{\alpha}(\overline{G})\geq\dfrac{n(n-1)^{1+\alpha}}{2^{1+\alpha}}$

and if $0<\alpha<1$, then

$SO_{\alpha}(G)+SO_{\alpha}(\overline{G})\geq\dfrac{n^{\alpha/2}(n-1)^{(3\alpha)/2}}{2^{(3\alpha)/2}}.$

  2. 2.

If $\alpha<0$, then

$2^{-1+\alpha/2}n(n-1)^{\alpha+1}\leq
SO_{\alpha}(G)+SO_{\alpha}(\overline{G})<2^{\alpha/2}n(n-1)$

where the left equality holds if and only if $G=K_{n}$ or
$G=\overline{K_{n}}$.

###### Proof.

Let $\alpha>0$. Notice that $d_{G}(u)\leq n-1$ for any vertex $u$ of $G$. Thus
by definition, we have

$\displaystyle SO_{\alpha}(G)+SO_{\alpha}(\overline{G})=$
$\displaystyle\displaystyle\sum_{uv\in
E(G)}[{d_{G}(u)}^{2}+{d_{G}(v)}^{2}]^{\alpha/2}+\displaystyle\sum_{uv\in
E(\overline{G})}[{d_{\overline{G}}(u)}^{2}+{d_{\overline{G}}(v)}^{2}]^{\alpha/2}$
$\displaystyle\leq$ $\displaystyle
m[2(n-1)^{2}]^{\alpha/2}+\overline{m}[2(n-1)^{2}]^{\alpha/2}$ $\displaystyle=$
$\displaystyle
2^{\alpha/2}(m+\overline{m})(n-1)^{\alpha}=2^{-1+\alpha/2}n(n-1)^{\alpha+1}$

where the equality holds if and only if either $d_{G}(u)=d_{G}(v)=n-1$ for
every edge $uv$ of $G$ or $E(G)=\emptyset$, i.e., $G=K_{n}$ or
$G=\overline{K_{n}}$. This proves the first part of (1). Similarly, if
$\alpha<0$, then $SO_{\alpha}(G)+SO_{\alpha}(\overline{G})\geq
2^{-1+\alpha/2}n(n-1)^{\alpha+1}$, where the equality holds if and only if
$G=K_{n}$ or $G=\overline{K_{n}}$. Further, by Theorem 3.3 (2), we have

$SO_{\alpha}(G)+SO_{\alpha}(\overline{G})<2^{-1+\alpha/2}n(n-1)+2^{-1+\alpha/2}n(n-1)=2^{\alpha/2}n(n-1).$

This proves (2).
Notice that $m+\overline{m}=\dfrac{n(n-1)}{2}$, thus
$F(G)\geq\dfrac{M_{1}(G)^{2}}{2m}\geq\dfrac{M_{1}(G)^{2}}{n(n-1)}$ and
$F(\overline{G})\geq\dfrac{M_{1}(\overline{G})^{2}}{2\overline{m}}\geq\dfrac{M_{1}(G)^{2}}{n(n-1)}$.
So, for $\alpha=2$,

$\displaystyle SO_{\alpha}(G)+SO_{\alpha}(\overline{G})=$ $\displaystyle
F(G)+F(\overline{G})$ $\displaystyle\geq$
$\displaystyle\dfrac{1}{n(n-1)}\left[M_{1}(G)^{2}+M_{1}(\overline{G})^{2}\right]$
$\displaystyle\geq$
$\displaystyle\dfrac{1}{n(n-1)}\dfrac{\left[M_{1}(G)+M_{1}(\overline{G})\right]^{2}}{2}$
$\displaystyle\geq$
$\displaystyle\dfrac{1}{n(n-1)}\dfrac{1}{2}\dfrac{n^{2}(n-1)^{4}}{4}=\dfrac{n(n-1)^{3}}{2^{3}}$
(1)

Now if $\alpha>1$, then $x^{\alpha/2}$ is strictly convex. Thus

$\displaystyle SO_{\alpha}(G)+SO_{\alpha}(\overline{G})$
$\displaystyle\geq(m+\overline{m})\left[\dfrac{\displaystyle\sum_{uv\in
E(G)}[{d_{G}(u)}^{2}+{d_{G}(v)}^{2}]+\displaystyle\sum_{uv\in
E(\overline{G})}[{d_{\overline{G}}(u)}^{2}+{d_{\overline{G}}(v)}^{2}]}{m+\overline{m}}\right]^{\alpha/2}$
$\displaystyle=$
$\displaystyle(m+\overline{m})^{1-\alpha/2}[F(G)+F(\overline{G})]^{\alpha/2}$
(2)

From (1) and (2), for $\alpha>1$ we have

$SO_{\alpha}(G)+SO_{\alpha}(\overline{G})\geq\dfrac{n(n-1)^{1+\alpha}}{2^{1+\alpha}}.$

Lastly, if $0<\alpha<1$, then

$\displaystyle SO_{\alpha}(G)+SO_{\alpha}(\overline{G})\geq$
$\displaystyle\left[\displaystyle\sum_{uv\in
E(G)}[{d_{G}(u)}^{2}+{d_{G}(v)}^{2}]+\displaystyle\sum_{uv\in
E(\overline{G})}[{d_{\overline{G}}(u)}^{2}+{d_{\overline{G}}(v)}^{2}]\right]^{\alpha/2}$
$\displaystyle=$ $\displaystyle[F(G)+F(\overline{G})]^{\alpha/2}$ (3)

From (1) and (3), for $0<\alpha<1$ we have

$SO_{\alpha}(G)+SO_{\alpha}(\overline{G})\geq\dfrac{n^{\alpha/2}(n-1)^{(3\alpha)/2}}{2^{(3\alpha)/2}}.$

This completes the proof. ∎

##  4 Comparison of general Sombor index with other generalised indices

In this section, we present the relations between general Sombor index and
other generalised indices: general Randić index and general sum-connectivity
index.

###  4.1 General Sombor index and general Randić index

We now present a relation between general Sombor index and Randić index.

######  Theorem 4.1.

Let $G$ be a graph on $n$ vertices and $m$ edges. Then

$\dfrac{2^{\alpha/2}}{\Delta^{\alpha}}R_{\alpha}(G)\leq
SO_{\alpha}(G)\leq\dfrac{2^{\alpha/2}}{\delta^{\alpha}}R_{\alpha}(G)$

where the equality holds if and only if $G$ is regular.

###### Proof.

Notice that

$\displaystyle SO_{\alpha}(G)=$ $\displaystyle\displaystyle\sum_{uv\in
E(G)}[{d_{G}(u)}^{2}+{d_{G}(v)}^{2}]^{\alpha/2}$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{uv\in
E(G)}[d_{G}(u)d_{G}(v)]^{\alpha}\left[\dfrac{1}{d_{G}(u)^{2}}+\dfrac{1}{d_{G}(v)^{2}}\right]^{\alpha/2}$
$\displaystyle\geq$ $\displaystyle\displaystyle\sum_{uv\in
E(G)}[d_{G}(u)d_{G}(v)]^{\alpha}\left[\dfrac{1}{\Delta^{2}}+\dfrac{1}{\Delta^{2}}\right]^{\alpha/2}$
$\displaystyle=$
$\displaystyle\dfrac{2^{\alpha/2}}{\Delta^{\alpha}}R_{\alpha}(G)$

Similarly, we obtain the upper bound. Moreover, it is easy to see that the
equality holds if and only if $G$ is regular. This completes the proof. ∎

As a consequence, letting $\alpha=1$ we get a relation for Sombor index in
terms of second Zagreb index as reported in [17].

######  Corollary 4.2.

Let $G$ be a graph on $n$ vertices and $m$ edges. Then

$\dfrac{\sqrt{2}}{\Delta}M_{2}(G)\leq
SO(G)\leq\dfrac{\sqrt{2}}{\delta}M_{2}(G)$

where the equality holds if and only if $G$ is regular.

###  4.2 General Sombor index and general sum-connectivity index

Lastly, we present a relation between general Sombor index and general sum-
connectivity index.

######  Theorem 4.3.

Let $G$ be a graph on $n$ vertices and $m$ edges. Then

$\dfrac{\chi_{\alpha}(G)}{2^{\alpha/2}}\leq
SO_{\alpha}(G)\leq\sqrt{m\Delta^{\alpha}\chi_{\alpha}(G)}$

where the left equality holds if and only if $G$ is regular.

###### Proof.

Let $V(G)=\\{v_{1},v_{2},\dots,v_{n}\\}$. Let, for simplicity,
$d_{i}=d_{G}(v_{i})$ for a vertex $v_{i}$ of $G$. Letting
$a_{k}\to[{d_{i}}^{2}+{d_{j}}^{2}]^{\alpha/2}$ and
$b_{k}\to[d_{i}+d_{j}]^{\alpha}$ in Lemma 2.5 for $p=1$ and the sums running
over the edges in $G$, we have

$\displaystyle\displaystyle\sum_{v_{i}v_{j}\in
E(G)}\left[\dfrac{{d_{i}}^{2}+{d_{j}}^{2}}{d_{i}+d_{j}}\right]^{\alpha}\geq\dfrac{\left[\displaystyle\sum_{v_{i}v_{j}\in
E(G)}[{d_{i}}^{2}+{d_{j}}^{2}]^{\alpha/2}\right]^{2}}{\displaystyle\sum_{v_{i}v_{j}\in
E(G)}[d_{i}+d_{j}]^{\alpha}}$ (4)

Notice that $\dfrac{{d_{i}}^{2}+{d_{j}}^{2}}{d_{i}+d_{j}}\leq\Delta$ for every
edge $v_{i}v_{j}$ of $G$. Thus (4) becomes

$\dfrac{{SO_{\alpha}(G)}^{2}}{\chi_{\alpha}(G)}\leq m\Delta^{\alpha}\implies
SO_{\alpha}(G)\leq\sqrt{m\Delta^{\alpha}\chi_{\alpha}(G)}.$

For the lower bound, notice that for any edge $v_{i}v_{j}$ of $G$, we have
${d_{i}}^{2}+{d_{j}}^{2}\geq\dfrac{(d_{i}+d_{j})^{2}}{2}$. Thus

$SO_{\alpha}(G)=\displaystyle\sum_{v_{i}v_{j}\in
E(G)}[{d_{i}}^{2}+{d_{j}}^{2}]^{\alpha/2}\geq\displaystyle\sum_{v_{i}v_{j}\in
E(G)}\dfrac{[d_{i}+d_{j}]^{\alpha}}{2^{\alpha/2}}=\dfrac{\chi_{\alpha}(G)}{2^{\alpha/2}}.$

∎

As a consequence, letting $\alpha=1$ we get the lower bound for Sombor index
as reported in [17] and the upper bound for Sombor index as found by the
present authors in [3].

######  Corollary 4.4.

Let $G$ be a graph on $n$ vertices and $m$ edges. Then

$\dfrac{M_{1}(G)}{\sqrt{2}}\leq SO(G)\leq\sqrt{m\Delta M_{1}(G)}$

where the left equality holds if and only if $G$ is regular.

## Acknowledgments

The second author is supported by the MATRICS project funded by DST-SERB,
Government of India under Grant no. MTR/2017/000403 dated 06/06/2018.

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