# Maximally distributed random fields under sublinear expectation

Xinpeng Li & Shige Peng 111Research Center for Mathematics and
Interdisciplinary Sciences, Shandong University, 266237, Qingdao, China and
School of Mathematics, Shandong University, 250100, Jinan, China. Email:
<EMAIL_ADDRESS>(Shige Peng)

###### Abstract

This paper focuses on the maximal distribution on sublinear expectation space
and introduces a new type of random fields with the maximally distributed
finite-dimensional distribution. The corresponding spatial maximally
distributed white noise is constructed, which includes the temporal-spatial
situation as a special case due to the symmetrical independence property of
maximal distribution. In addition, the stochastic integrals with respect to
the spatial or temporal-spatial maximally distributed white noises are
established in a quite direct way without the usual assumption of adaptability
for integrand.

Keywords: sublinear expectation, maximal distribution, maximally distributed
random field, maximally distributed white noise, stochastic integral

##  1 Introduction

In mathematics and physics, a random field is a type of parameterized family
of random variables. When the parameter is time $t\in\mathbb{R}^{+}$, we call
it a stochastic process, or a temporal random field. Quite often the parameter
is space $x\in\mathbb{R}^{d}$, or time-space
$(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{d}$. In this case, we call it a
spatial or temporal-spatial random field. A typical example is the
electromagnetic wave dynamically spread everywhere in our
$\mathbb{R}^{3}$-space or more exactly, in
$\mathbb{R}^{+}\times\mathbb{R}^{3}$-time-space. In principle, it is
impossible to know the exact state of the electromagnetic wave of our real
world , namely, it is a nontrivial random field parameterized by the time-
space $(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{3}$.

Classically, a random field is defined on a given probability space
$(\Omega,{\mathcal{F}},P)$. But for the above problem, can we really get to
know the probability $P$? This involves the so called problem of uncertainty
of probabilities.

Over the past few decades, non-additive probabilities or nonlinear
expectations have become active domains for studying uncertainties, and
received more and more attention in many research fields, such as mathematical
economics, mathematical finance, statistics, quantum mechanics. A typical
example of nonlinear expectation is sublinear one, which is used to model the
uncertainty phenomenon characterized by a family of probability measures
$\\{P_{\theta}\\}_{\theta\in\Theta}$ in which the true measure is unknown, and
such sublinear expectation is usually defined by

$\mathbb{E}[X]:=\sup_{\theta\in\Theta}E_{P_{\theta}}[X].$

This notion is also known as the upper expectation in robust statistics (see
Huber [9]), or the upper prevision in the theory of imprecise probabilities
(see Walley [20]), and has the closed relation with coherent risk measures
(see Artzner et al. [1], Delbaen [4], Föllmer and Schied [6]). A first
dynamical nonlinear expectation, called $g$-expectation was initiated by Peng
[12].

The foundation of sublinear expectation theory with a new type of $G$-Brownian
motion and the corresponding Itô’s stochastic calculus was laid in Peng [13],
which keeps the rich and elegant properties of classical probability theory
except linearity of expectation. Peng [15] initially defined the notion of
independence and identical distribution (i.i.d.) based on the notion of
nonlinear expectation instead of the capacity. Based on the notion of new
notions, the most important distribution called $G$-normal distribution
introduced, which can be characterized by the so-called $G$-heat equation. The
notions of $G$-expectation and $G$-Brownian motion can be regarded as a
nonlinear generalization of Wiener measure and classical Brownian motion. The
corresponding limit theorems as well as stochastic calculus of Itô’s type
under $G$-expectation are systematically developed in Peng [18]. Besides that,
there is also another important distribution, called maximal distribution. The
distribution of maximally distributed random variable $X$ can be calculated
simply by

$\mathbb{E}[\varphi(X)]=\max_{v\in[-\mathbb{E}[-X],\mathbb{E}[X]]}\varphi(v),\
\ \varphi\in C_{b}(\mathbb{R}).$

The law of large numbers under sublinear expectation (see Peng [18]) shows
that if $\\{X_{i}\\}_{i=1}^{\infty}$ is a sequence of independent and
identical distributed random variables with
$\lim_{c\rightarrow\infty}\mathbb{E}[(|X_{1}|-c)^{+}]=0$, then the sample
average converges to maximal distribution in law, i.e.,

$\lim_{n\rightarrow\infty}\mathbb{E}[\varphi(\frac{X_{1}+\cdots+X_{n}}{n})]=\max_{v\in[-\mathbb{E}[-X_{1}],\mathbb{E}[X_{1}]]}\varphi(v),\
\ \forall\varphi\in C_{b}(\mathbb{R}).$

We note that the finite-dimensional distribution for quadratic variation
process of $G$-Brownian motion is also maximal distributed.

Recently, Ji and Peng [10] introduced a new $G$-Gaussian random fields, which
contains a type of spatial white noise as a special case. Such white noise is
a natural generalization of the classical Gaussian white noise (for example,
see Walsh [21], Dalang [2] and Da Prato and Zabczyk [3]). As pointed in [10],
the space-indexed increments do not satisfy the property of independence. Once
the sublinear $G$-expectation degenerates to linear case, the property of
independence for the space-indexed part turns out to be true as in the
classical probability theory.

In this paper, we introduce a very special but also typical random field,
called maximally distributed random field, in which the finite-dimensional
distribution is maximally distributed. The corresponding space-indexed white
noise is also constructed. It is worth mentioning that the space-indexed
increments of maximal white noise is independent, which is essentially
different from the case of $G$-Gaussian white noise. Thanks to the symmetrical
independence of maximally distributed white noise, it is natural to view the
temporal-spatial maximally distributed white noise as a special case of the
space-indexed maximally distributed white noise. The stochastic integrals with
respect to spatial and temporal-spatial maximally distributed white noises can
be constructed in a quite simple way, which generalize the stochastic integral
with respect to quadratic variation process of $G$-Brownian motion introduced
in Peng [18]. Furthermore, due to the boundedness of maximally distributed
random field, the usual assumption of adaptability for integrand can be
dropped. We emphasize that the structure of maximally distributed white noise
is quite simple, it can be determined by only two parameters $\underline{\mu}$
and $\overline{\mu}$, and the calculation of the corresponding finite-
dimensional distribution is taking the maximum of continuous function on the
domain determined by $\underline{\mu}$ and $\overline{\mu}$. The use of
maximally distributed random fields for modelling purposes in applications can
be explained mainly by the simplicity of their construction and analytic
tractability combined with the maximal distributions of marginal which
describe many real phenomena due to the law of large numbers with uncertainty.

This paper is organized as follows. In Section 2, we review basic notions and
results of nonlinear expectation theory and the notion and properties of
maximal distribution. In Section 3, we first recall the general setting of
random fields under nonlinear expectations, and then introduce the maximally
distributed random fields. In Section 4, we construct the spatial maximally
distributed white noise and study the corresponding properties. The properties
of spatial as well as temporal-spatial maximally distributed white noise and
the related stochastic integrals are established in Section 5.

##  2 Preliminaries

In this section, we recall some basic notions and properties in the nonlinear
expectation theory. More details can be found in Denis et al. [5], Hu and Peng
[8] and Peng [13, 14, 15, 16, 18, 19].

Let $\Omega$ be a given nonempty set and $\mathcal{H}$ be a linear space of
real-valued functions on $\Omega$ such that if $X\in\mathcal{H}$, then
$|X|\in\mathcal{H}$. $\mathcal{H}$ can be regarded as the space of random
variables. In this paper, we consider a more convenient assumption: if random
variables $X_{1}$,$\cdots$,$X_{d}\in\mathcal{H}$, then
$\varphi(X_{1},X_{2},\cdots,X_{d})\in\mathcal{H}$ for each $\varphi\in
C_{b.Lip}(\mathbb{R}^{d})$. Here $C_{b.Lip}(\mathbb{R}^{d})$ is the space of
all bounded and Lipschitz functions on $\mathbb{R}^{d}$.

We call $X=(X_{1},\cdots,X_{n})$, $X_{i}\in\mathcal{H}$, $1\leq i\leq n$, an
$n$-dimensional random vector, denoted by $X\in\mathcal{H}^{n}$.

###### Definition 2.1

A nonlinear expectation $\hat{E}$ on $\mathcal{H}$ is a functional
$\hat{E}:\mathcal{H}\rightarrow\mathbb{R}$ satisfying the following
properties: for each $X,Y\in\mathcal{H}$,

(i)

Monotonicity: $\hat{E}[X]\geq\hat{E}[Y]\ \ \text{if}\ X\geq Y$;

(ii)

Constant preserving: $\hat{E}[c]=c\ \ \text{for}\ c\in\mathbb{R}$;

The triplet $(\Omega,\mathcal{H},\hat{E})$ is called a nonlinear expectation
space. If we further assume that

(iii)

Sub-additivity: $\hat{E}[X+Y]\leq\hat{E}[X]+\hat{E}[Y]$;

(iv)

Positive homogeneity: $\hat{E}[\lambda X]=\lambda\hat{E}[X]\ \ \text{for}\
\lambda\geq 0$.

Then $\hat{E}$ is called a sublinear expectation, and the corresponding
triplet $(\Omega,\mathcal{H},\hat{E})$ is called a sublinear expectation
space.

Let $(\Omega,\mathcal{H},\hat{E})$ be a nonlinear (resp., sublinear)
expectation space. For each given $n$-dimensional random vector $X$, we define
a functional on $C_{b.Lip}(\mathbb{R}^{n})$ by

$\mathbb{{F}}_{X}[\varphi]:=\hat{E}[\varphi(X)],\text{ for each }\varphi\in
C_{b.Lip}(\mathbb{R}^{n}).$

$\mathbb{{F}}_{X}$ is called the distribution of $X$. It is easily seen that
$(\mathbb{R}^{n},C_{b.Lip}(\mathbb{R}^{n}),\mathbb{{F}}_{X})$ forms a
nonlinear (resp., sublinear) expectation space. If $\mathbb{{F}}_{X}$ is not a
linear functional on $C_{b.Lip}(\mathbb{R}^{n})$, we say $X$ has
distributional uncertainty.

###### Definition 2.2

Two $n$-dimensional random vectors $X_{1}$ and $X_{2}$ defined on nonlinear
expectation spaces $(\Omega_{1},\mathcal{H}_{1},\hat{E}_{1})$ and
$(\Omega_{2},\mathcal{H}_{2},\hat{E}_{2})$ respectively, are called
identically distributed, denoted by $X_{1}\overset{d}{=}X_{2}$, if
$\mathbb{{F}}_{X_{1}}=\mathbb{{F}}_{X_{2}}$, i.e.,

$\hat{E}_{1}[\varphi(X_{1})]=\hat{E}_{2}[\varphi(X_{2})],\ \ \forall\varphi\in
C_{b.Lip}(\mathbb{R}^{n}).$

###### Definition 2.3

Let $(\Omega,\mathcal{H},\hat{E})$ be a nonlinear expectation space. An
$n$-dimensional random vector $Y$ is said to be independent from another
$m$-dimensional random vector $X$ under the expectation $\hat{E}$ if, for each
test function $\varphi\in C_{b.Lip}(\mathbb{R}^{m+n})$, we have

$\hat{E}[\varphi(X,Y)]=\hat{E}[\hat{E}[\varphi(x,Y)]_{x=X}].$

###### Remark 2.4

Peng [15] (see also Peng [18]) introduced the notions of the distribution and
the independence of random variables under a nonlinear expectation, which play
a crucially important role in the nonlinear expectation theory.

For simplicity, the sequence $\\{X_{i}\\}_{i=1}^{n}$ is called independence if
$X_{i+1}$ is independent from $(X_{1},\cdots,X_{i})$ for $i=1,2,\cdots,n-1$.
Let $\bar{X}$ and $X$ be two $n$-dimensional random vectors on
$(\Omega,\mathcal{H},\hat{E})$. $\bar{X}$ is called an independent copy of
$X$, if $\bar{X}\overset{d}{=}X$ and $\bar{X}$ is independent from $X$.

###### Remark 2.5

It is important to note that “$Y$ is independent from $X$” does not imply that
“$X$ is independent from $Y$” (see Peng [18]).

In this paper, we focus on an important distribution on sublinear expectation
space $(\Omega,\mathcal{H},\hat{E})$, called maximal distribution.

###### Definition 2.6

An $n$-dimensional random vector $X=(X_{1},\cdots,X_{n})$ on a sublinear
expectation space $(\Omega,\mathcal{H},\hat{E})$ is said to be maximally
distributed, if there exists a bounded and closed convex subset
$\Lambda\subset\mathbb{R}^{n}$ such that, for every continuous function
$\varphi\in C(\mathbb{R}^{n})$,

$\hat{E}[\varphi(X)]=\max_{x\in\Lambda}\varphi(x).$

###### Remark 2.7

Here $\Lambda$ characterizes the uncertainty of $X$. It is easy to check that
this maximally distributed random vector $X$ satisfies

$X+\bar{X}\overset{d}{=}2X,$

where $\bar{X}$ is an independent copy of $X$. Conversely, suppose a random
variable $X$ satisfying $X+\bar{X}\overset{d}{=}2X$, if we further assume the
uniform convergence condition
$\lim_{c\rightarrow\infty}\hat{E}[(|X|-c)^{+}]=0$ holds, then we can deduce
that $X$ is maximally distributed by the law of large numbers (see Peng [18]).
An interesting problem is that is $X$ still maximally distributed without such
uniform convergence condition? We emphasize that the law of large numbers does
not hold in this case, a counterexample can be found in Li and Zong [11].

###### Proposition 2.8

Let $g(p)=\max_{v\in\Lambda}v\cdot p$ be given. Then an $n$-dimensional random
variable is maximally distributed if and only if for each $\varphi\in
C(\mathbb{R}^{n})$, the following function

$u(t,x):=\hat{E}[\varphi(x+tX)]=\max_{v\in\Lambda}\varphi(x+tv),\
(t,x)\in[0,\infty)\times\mathbb{R}^{n}$ (1)

is the unique viscosity solution of the the following nonlinear partial
differential equation

$\partial_{t}u-g(D_{x}u)=0,\ \ u|_{t=0}=\varphi(x).$ (2)

This property implies that, each sublinear function $g$ on $\mathbb{R}^{n}$
determines uniquely a maximal distribution. The following property is easy to
check.

###### Proposition 2.9

Let $X$ be an $n$-dimensional maximally distributed random vector
characterized by its generating function

$g(p):=\hat{E}[X\cdot p],\,\,\,p\in\mathbb{R}^{n}.$

Then, for any function $\psi\in C(\mathbb{R}^{n})$, $Y=\psi(X)$ is also an
$\mathbb{R}$-valued maximally distributed random variable:

$\mathbb{E}[\varphi(Y)]=\max_{v\in[\underline{{\rho}},\overline{\rho}]}\varphi(v),\qquad\overline{\rho}=\max_{\gamma\in\Lambda}\psi(\gamma),\quad\underline{{\rho}}=\min_{\gamma\in\Lambda}\psi(\gamma).$

###### Proposition 2.10

Let $X=(X_{1},\cdots,X_{n})$ be an $n$-dimensional maximal distribution on a
sublinear expectation space $(\Omega,\mathcal{H},\hat{E}).$ If the
corresponding generating function satisfies, for all
$p=(p_{1},\cdots,p_{n})\in\mathbb{R}^{n},$

$g(p)=\hat{E}[X_{1}p_{1}+\cdots+X_{n}p_{n}]=\hat{E}[X_{1}p_{1}]+\cdots+\hat{E}[X_{n}p_{n}],$

then $\\{X_{i}\\}_{i=1}^{n}$ is a sequence of independent maximally
distributed random variables.

Moreover, for any permutation $\pi$ of $\\{1,2,\cdots,n\\}$, the sequence
$\\{X_{\pi(i)}\\}_{i=1}^{n}$ is also independent.

Proof. For $i=1,\cdots,n$, we denote $\overline{\mu}_{i}=\hat{E}[X_{i}]$ and
$\underline{\mu}_{i}=-\hat{E}[-X_{i}]$. Since

$\displaystyle g(p)$ $\displaystyle=\hat{E}[X_{1}\cdot p_{1}+\cdots+X_{n}\cdot
p_{n}]=\hat{E}[X_{1}\cdot p_{1}]+\hat{E}[X_{2}\cdot
p_{2}]+\cdots+\hat{E}[X_{n}\cdot p_{n}]$
$\displaystyle=\sum_{i=1}^{n}\max_{v_{i}\in[\underline{\mu}_{i},\overline{\mu}_{i}]}p_{i}v_{i}=\max_{(v_{1},\cdots,v_{n})\in\otimes_{i=1}^{n}[\underline{\mu}_{i},\overline{\mu}_{i}]}(p_{1}v_{1}+\cdots+p_{n}v_{n}),$

it follows Proposition 2.8 that $(X_{1},\cdots,X_{n})$ is an $n$-dimensional
maximally distributed random vector such that, $\forall\varphi\in
C(\mathbb{R}^{n})$,

$\hat{E}[\varphi(X_{1},\cdots,X_{n})]=\max_{(v_{1},\cdots,v_{n})\in\otimes_{i=1}^{n}[\underline{\mu}_{i},\overline{\mu}_{i}]}\varphi(v_{1},\cdots,v_{n}).$

It is easy to check that $\\{X_{i}\\}_{i=1}^{n}$ is independent, and so does
the permuted sequence $\\{X_{\pi(i)}\\}_{i=1}^{n}$.

###### Remark 2.11

The independence of maximally distributed random variables is symmetrical.
But, as discussed in Remark 2.5, under a sublinear expectation, $X$ is
independent from $Y$ does not automatically imply that $Y$ is also independent
from $X$. In fact, Hu and Li [7] proved that, if $X$ is independent from $Y$,
and $Y$ is also independent from $X$, and both of $X$ and $Y$ have
distributional uncertainty, then $(X,Y)$ must be maximally distributed.

##  3 Maximally distributed random fields

In this section, we first recall the general setting of random fields defined
on a nonlinear expectation space introduced by Ji and Peng [10].

###### Definition 3.1

Under a given nonlinear expectation space $(\Omega,\mathcal{H},\hat{E})$, a
collection of $m$-dimensional random vectors
$W=(W_{\gamma})_{\gamma\in\Gamma}$ is called an $m$-dimensional random field
indexed by $\Gamma$, if for each $\gamma\in\Gamma$,
$W_{\gamma}\in\mathcal{H}^{m}$.

In order to introduce the notion of finite-dimensional distribution of a
random field $W$, we denote the family of all sets of finite indices by

$\mathcal{J}_{\Gamma}:=\\{\underline{\gamma}=(\gamma_{1},\cdots,\gamma_{n}):\
\forall n\in\mathbb{N},\ \gamma_{1},\cdots,\gamma_{n}\in\Gamma,\
{\gamma_{i}\neq\gamma_{j}\ \text{if}\ i\neq j}\\}.$

###### Definition 3.2

Let $(W_{\gamma})_{\gamma\in\Gamma}$ be an $m$-dimensional random field
defined on a nonlinear expectation space $(\Omega,\mathcal{H},\hat{E})$. For
each
$\underline{\gamma}=(\gamma_{1},\cdots,\gamma_{n})\in\mathcal{J}_{\Gamma}$ and
the corresponding random vector
$W_{\underline{\gamma}}=(W_{\gamma_{1}},\cdots,W_{\gamma_{n}})$, we define a
functional on $C_{b.Lip}(\mathbb{R}^{n\times m})$ by

$\mathbb{F}_{\underline{\gamma}}^{W}[\varphi]=\hat{E}[\varphi(W_{\underline{\gamma}})]$

The collection
$(\mathbb{F}_{\underline{\gamma}}^{W}[\varphi])_{\underline{\gamma}\in\mathcal{J}_{\Gamma}}$
is called the family of finite-dimensional distributions of
$(W_{\gamma})_{\gamma\in\Gamma}$.

It is clear that, for each $\underline{\gamma}\in\mathcal{J}_{\Gamma}$, the
triple $(\mathbb{R}^{n\times m},C_{b.Lip}(\mathbb{R}^{n\times
m}),\mathbb{F}_{\underline{\gamma}}^{W})$ constitutes a nonlinear expectation
space.

Let $(W_{\gamma}^{(1)})_{\gamma\in\Gamma}$ and
$({W}_{\gamma}^{(2)})_{\gamma\in\Gamma}$ be two $m$-dimensional random fields
defined on nonlinear expectation spaces
$(\Omega_{1},\mathcal{H}_{1},\hat{E}_{1})$ and
$(\Omega_{2},\mathcal{H}_{2},\hat{E}_{2})$ respectively. They are said to be
identically distributed, denoted by
$(W_{\gamma}^{(1)})_{\gamma\in\Gamma}\overset{d}{=}({W}_{\gamma}^{(2)})_{\gamma\in\Gamma}$,
or simply $W^{(1)}\overset{d}{=}W^{(2)}$, if for each
$\underline{\gamma}=(\gamma_{1},\cdots,\gamma_{n})\in\mathcal{J}_{\Gamma}$,

$\hat{E}_{1}[\varphi(W_{\underline{\gamma}}^{(1)})]=\hat{E}_{2}[\varphi({W}_{\underline{\gamma}}^{(2)})],\
\ \forall\varphi\in C_{b.Lip}(\mathbb{R}^{n\times m}).$

For any given $m$-dimensional random field $W=(W_{\gamma})_{\gamma\in\Gamma}$,
the family of its finite-dimensional distributions satisfies the following
properties of consistency:

(1) Compatibility:

For each $(\gamma_{1},\cdots,\gamma_{n},\gamma_{n+1})\in\mathcal{J}_{\Gamma}$
and $\varphi\in C_{b.Lip}(\mathbb{R}^{n\times m}),$

$\mathbb{F}_{\gamma_{1},\cdots,\gamma_{n}}^{W}[\varphi]=\mathbb{F}_{\gamma_{1},\cdots,\gamma_{n},\gamma_{n+1}}^{W}[\widetilde{\varphi}],$
(3)

where the function $\widetilde{\varphi}$ is a function on
$\mathbb{R}^{(n+1)\times m}$ defined for any
$y_{1},\cdots,y_{n},y_{n+1}\in\mathbb{R}^{m}$,

$\widetilde{\varphi}(y_{1},\cdots,y_{n},y_{n+1})=\varphi(y_{1},\cdots,y_{n});$

(2) Symmetry:

For each $(\gamma_{1},\cdots,\gamma_{n})\in\mathcal{J}_{\Gamma}$, $\varphi\in
C_{b.Lip}(\mathbb{R}^{n\times m})$ and each permutation $\pi$ of
$\\{1,\cdots,n\\}$,

$\mathbb{F}_{\gamma_{\pi(1)},\cdots,\gamma_{\pi(n)}}^{W}[\varphi]=\mathbb{F}_{\gamma_{1},\cdots,\gamma_{n}}^{W}[\varphi_{\pi}]$
(4)

where we denote
$\varphi_{\pi}(y_{1},\cdots,y_{n})=\varphi(y_{\pi(1)},\cdots,y_{\pi(n)})$, for
$y_{1},\cdots,y_{n}\in\mathbb{R}^{m}$.

The following theorem generalizes the classical Kolmogorov’s existence theorem
to the situation of sublinear expectation space, which is a variant of Theorem
3.8 in Peng [17]. The proof can be founded in Ji and Peng [10].

###### Theorem 3.3

Let
$\\{\mathbb{F}_{\underline{\gamma}},\underline{\gamma}\in\mathcal{J}_{\Gamma}\\}$
be a family of finite-dimensional distributions satisfying the compatibility
condition (3) and the symmetry condition (4). Then there exists an
$m$-dimensional random field $W=(W_{\gamma})_{\gamma\in\Gamma}$ defined on a
nonlinear expectation space $(\Omega,\mathcal{H},\hat{E})$ whose family of
finite-dimensional distributions coincides with
$\\{\mathbb{F}_{\underline{\gamma}},\underline{\gamma}\in\mathcal{J}_{\Gamma}\\}$.
Moreover, if we assume that each $\mathbb{F}_{\underline{\gamma}}$ in
$\\{\mathbb{F}_{\underline{\gamma}},\underline{\gamma}\in\mathcal{J}_{\Gamma}\\}$
is sublinear, then the corresponding expectation $\hat{E}$ on the space of
random variables $(\Omega,\mathcal{H})$ is also sublinear.

Now we consider a new random fields under a sublinear expectation space.

###### Definition 3.4

Let $(W_{\gamma})_{\gamma\in\Gamma}$ be an $m$-dimensional random field,
indexed by $\Gamma$, defined on a sublinear expectation space
$(\Omega,\mathcal{H},\hat{E})$. $(W_{\gamma})_{\gamma\in\Gamma}$ is called a
maximally distributed random field if for each
$\underline{\gamma}=(\gamma_{1},\cdots,\gamma_{n})\in\mathcal{J}_{\Gamma}$,
the following $(n\times m)$-dimensional random vector

$\displaystyle W_{\underline{\gamma}}=$
$\displaystyle(W_{\gamma_{1}},\cdots,W_{\gamma_{n}})$ $\displaystyle=$
$\displaystyle(W_{\gamma_{1}}^{(1)},\cdots
W_{\gamma_{1}}^{(m)},\cdots,W_{\gamma_{n}}^{(1)},\cdots,W_{\gamma_{n}}^{(m)}),\,\,W_{\gamma_{i}}^{(j)}\in\mathcal{H},$

is maximally distributed.

For each
$\underline{\gamma}=(\gamma_{1},\cdots,\gamma_{n})\in\mathcal{J}_{\Gamma}$, we
define

$g^{W}_{\underline{\gamma}}(p)=\hat{E}[W_{\underline{\gamma}}\cdot
p],\,\,\,\,p\in\mathbb{R}^{n\times m},$

Then
$(g^{W}_{\underline{\gamma}})_{\underline{\gamma}\in\mathcal{J}_{\Gamma}}$
constitutes a family of sublinear functions:

$g^{W}_{\underline{\gamma}}:\mathbb{R}^{n\times
m}\mapsto\mathbb{R},\,\,\,\,\underline{\gamma}=(\gamma_{1},\cdots,\gamma_{n}),\,\,\gamma_{i}\in\Gamma,\,\,1\leq
i\leq n,\,\,n\in\mathbb{N},$

which satisfies the properties of consistency in the following sense:

(1) Compatibility:

For any $(\gamma_{1},\cdots,\gamma_{n},\gamma_{n+1})\in\mathcal{J}_{\Gamma}$
and $p=(p_{i})_{i=1}^{n\times m}\in\mathbb{R}^{n\times m}$,

$g^{W}_{\gamma_{1},\cdots,\gamma_{n},\gamma_{n+1}}(\bar{p})=g^{W}_{{\gamma_{1}},\cdots,W_{\gamma_{n}}}(p),$
(5)

where $\bar{p}=\left(\begin{array}[]{c}p\\\
0\end{array}\right)\in\mathbb{R}^{(n+1)\times m}$;

(2) Symmetry:

For any permutation $\pi$ of $\\{1,\cdots,n\\}$,

$g^{W}_{\gamma_{\pi(1)},\cdots,\gamma_{\pi(n)}}(p)=g^{W}_{\gamma_{1},\cdots,\gamma_{n}}(\pi^{-1}(p)),$
(6)

where $\pi^{-1}(p)=(p^{(1)},\ldots,p^{(n)}),$

$p^{(i)}=(p_{(\pi^{-1}(i)-1)m+1},\ldots,p_{(\pi^{-1}(i)-1)m+m})\ ,\ 1\leq
i\leq n.$

If the above type of family of sublinear functions
$(g_{\underline{\gamma}})_{\underline{\gamma}\in\mathcal{J}_{\Gamma}}$ is
given, following the construction procedure in the proof of Theorem 3.5 in Ji
and Peng [10], we can construct a maximally distributed random field on
sublinear expectation space.

###### Theorem 3.5

Let $(g_{\underline{\gamma}})_{\underline{\gamma}\in\mathcal{J}_{\Gamma}}$ be
a family of real-valued functions such that, for each
$\underline{\gamma}=(\gamma_{1},\cdots,\gamma_{n})\in\mathcal{J}_{\Gamma}$,
the real function $g_{\underline{\gamma}}$ is defined on $\mathbb{R}^{n\times
m}\mapsto\mathbb{R}$ and satisfies the sub-linearity. Moreover, this family
$(g_{\underline{\gamma}})_{\underline{\gamma}\in\mathcal{J}_{\Gamma}}$
satisfies the compatibility condition (5) and symmetry condition (6). Then
there exists an $m$-dimensional maximally distributed random field
$(W_{\gamma})_{\gamma\in\Gamma}$ on a sublinear expectation space
$(\Omega,\mathcal{H},\hat{E})$ such that for each
$\underline{\gamma}=(\gamma_{1},\cdots,\gamma_{n})\in\mathcal{J}_{\Gamma}$,
$W_{\underline{\gamma}}=(W_{\gamma_{1}},\cdots,W_{\gamma_{n}})$ is maximally
distributed with generating function

$g^{W}_{{\underline{\gamma}}}(p)=\hat{E}[W_{\underline{\gamma}}\cdot
p]=g_{\underline{\gamma}}(p),\,\,\text{ for any }\,p\in\mathbb{R}^{n\times
m}.$

Furthermore, if there exists another maximally distributed random field
$({\bar{W}}_{\gamma})_{\gamma\in\Gamma}$, with the same index set $\Gamma$,
defined on a sublinear expectation space
$(\bar{\Omega},\bar{\mathcal{H}},\bar{E})$ such that for each
$\underline{\gamma}=(\gamma_{1},\cdots,\gamma_{n})\in\mathcal{J}_{\Gamma}$,
$\bar{W}_{\underline{\gamma}}$ is maximally distributed with the same
generating function $g_{\underline{\gamma}}$, namely,

$\bar{E}[\bar{W}_{\underline{\gamma}}\cdot
p]=g_{\underline{\gamma}}(p)\,\,\text{ for any }\ p\in\mathbb{R}^{n\times m},$

then we have $W\overset{d}{=}\bar{W}$.

##  4 Maximally distributed white noise

In this section, we formulate a new type of maximally distributed white noise
on $\mathbb{R}^{d}$.

Given sublinear expectation space $\Omega,\mathcal{H},\hat{E}$, let
$\mathbb{L}^{p}(\Omega)$ be the completion of $\mathcal{H}$ under the Banach
norm $\|X\|:=\hat{E}[|X|^{p}]^{\frac{1}{p}}$. For any
$X,Y\in\mathbb{L}^{1}(\Omega)$, we say that $X=Y$ if $\hat{E}[|X-Y|]=0$. As
shown in Chapter 1 of Peng [18], $\hat{E}$ can be continuously extended to the
mapping from $\mathbb{L}^{1}(\Omega)$ to $\mathbb{R}$ and properties (i)-(iv)
of Definition 2.1 still hold. Moreover,
$(\Omega,\mathbb{L}^{1}(\Omega),\hat{E})$ also forms a sublinear expectation
space, which is called the complete sublinear expectation space.

###### Definition 4.1

Let $(\Omega,\mathbb{L}^{1}(\Omega),\hat{E})$ be a complete sublinear
expectation space and
$\Gamma=\mathcal{B}_{0}(\mathbb{R}^{d}):=\\{A\in\mathcal{B}(\mathbb{R}^{d}),\lambda_{A}<\infty\\}$,
where $\lambda_{A}$ denotes the Lebesgue measure of
$A\in\mathcal{B}(\mathbb{R}^{d})$. Let $g:\mathbb{R}\mapsto\mathbb{R}$ be a
given sublinear function, i.e.,

$g(p)=\overline{\mu}p^{+}-\underline{\mu}p^{-},\ \
-\infty<\underline{\mu}\leq\overline{\mu}<+\infty.$

A random field $W=\\{W_{A}\\}_{A\in\Gamma}$ is called a one-dimensional
maximally distributed white noise if

(i)

For each $A_{1},\cdots,A_{n}\in\Gamma$, $(W_{A_{1}},\cdots,W_{A_{n}})$ is a
$\mathbb{R}^{n}$-maximally distributed random vector under $\hat{E}$, and for
each $A\in\Gamma$,

$\displaystyle\hat{E}[W_{A}\cdot p]=g(p)\lambda_{A},\quad p\in\mathbb{R}.$ (7)

(ii)

Let $A_{1},A_{2},\cdots,A_{n}$ be in $\Gamma$ and mutually disjoint, then
$\\{W_{A_{i}}\\}_{i=1}^{n}$ are independent sequence, and

$\displaystyle W_{A_{1}\cup A_{2}\cup\cdots\cup
A_{n}}=W_{A_{1}}+W_{A_{2}}+{\cdots+W_{A_{n}}}.$ (8)

###### Remark 4.2

For each $A\in\Gamma$, we can restrict that $W_{A}$ takes values in
$[\lambda_{A}\underline{\mu},\lambda_{A}\overline{\mu}]$. Indeed, let

$d_{A}(x):=\min_{y\in[\lambda_{A}\underline{\mu},\lambda_{A}\overline{\mu}]}\\{|x-y|\\},$

by the definition of maximal distribution,

$\hat{E}[d_{A}(W_{A})]=\max_{v\in[\lambda_{A}\underline{\mu},\lambda_{A}\overline{\mu}]}\min_{y\in[\lambda_{A}\underline{\mu},\lambda_{A}\overline{\mu}]}\\{|v-y|\\}=0,$

which implies that $d_{A}(W_{A})=0$.

We can construct a spatial maximal white noise satisfying Definition 4.1 in
the following way.

For each $\underline{\gamma}=(A_{1},\cdots,A_{n})\in\mathcal{J}_{\Gamma}$,
$\Gamma=\mathcal{B}_{0}(\mathbb{R}^{d}),$ consider the mapping
$g_{\underline{\gamma}}(\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}$ defined as
follows:

$\displaystyle g_{\underline{\gamma}}(p){:=}\sum_{k\in\\{0,1\\}^{n}}g(k\cdot
p)\lambda_{B(k)},\ \ \ p\in\mathbb{R}^{n},$ (9)

where $k=(k_{1},\cdots,k_{n})\in\\{0,1\\}^{n}$, and
$B(k)=\cap_{j=1}^{n}B_{j}$, with

$B_{j}=\left\\{\begin{array}[]{cc}A_{j}&\text{if }k_{j}=1,\\\
A_{j}^{c}&\text{if }\;k_{j}=0.\end{array}\right.$

For example, given $A_{1},A_{2},A_{3}\in\Gamma$ and
$p=(p_{1},p_{2},p_{3})\in\mathbb{R}^{3}$,

$\displaystyle g_{A_{1},A_{2},A_{3}}(p)=g(p_{1}+p_{2}+p_{3})\lambda_{A_{1}\cap
A_{2}\cap A_{3}}$ $\displaystyle+g(p_{1}+p_{2})\lambda_{A_{1}\cap A_{2}\cap
A_{3}^{c}}+g(p_{2}+p_{3})\lambda_{A_{1}^{c}\cap A_{2}\cap
A_{3}}+g(p_{1}+p_{3})\lambda_{A_{1}\cap A_{2}^{c}\cap A_{3}}$
$\displaystyle+g(p_{1})\lambda_{A_{1}\cap A_{2}^{c}\cap
A_{3}^{c}}+g(p_{2})\lambda_{A_{1}^{c}\cap A_{2}\cap
A_{3}^{c}}+g(p_{3})\lambda_{A_{1}^{c}\cap A_{2}^{c}\cap A_{3}}.$

Obviously, for each $\underline{\gamma}=(A_{1},\cdots,A_{n})\subset\Gamma$,
$g_{\underline{\gamma}}(\cdot)$ defined by (9) is a sublinear function defined
on $\mathbb{R}^{n}$ due to the sub-linearity of function $g(\cdot)$. The
following property shows that the consistency conditions (5) and (6) also hold
for $\\{g_{\underline{\gamma}}\\}_{\gamma\in\mathcal{J}_{\Gamma}}$.

###### Proposition 4.3

The family $\\{g_{\underline{\gamma}}\\}_{\gamma\in\mathcal{J}_{\Gamma}}$
defined by (9) satisfies the consistency conditions (5) and (6).

Proof. For compatibility (5), given $A_{1},\cdots,A_{n},A_{n+1}\in\Gamma$ and
$\bar{p}^{T}=(p^{T},0)\in\mathbb{R}^{n+1}$, we have

$\displaystyle g_{A_{1},\cdots,A_{n+1}}(\bar{p})$
$\displaystyle=\sum_{k\in\\{0,1\\}^{n+1}}g(k\cdot\bar{p})\lambda_{B(k)}$
$\displaystyle=\sum_{k^{\prime}\in\\{0,1\\}^{n}}g(k^{\prime}\cdot
p)(\lambda_{B(k^{\prime})\cap A_{n+1}}+\lambda_{B(k^{\prime})\cap
A_{n+1}^{c}})$
$\displaystyle=\sum_{k^{\prime}\in\\{0,1\\}^{n}}g(k^{\prime}\cdot
p)\lambda_{B(k^{\prime})}=g_{A_{1},\cdots,A_{n}}(p).$

The symmetry (6) can be easily verified since the operators $k\cdot p$ and
$B(k)=\cap_{j=1}^{n}B_{j}$ are also symmetry.

Now we present the existence of the maximally distributed white noises under
the sublinear expectation.

###### Theorem 4.4

For each given sublinear function

$g(p)=\max_{\mu\in[\underline{\mu},\overline{\mu}]}(\mu\cdot
p)=\overline{\mu}p^{+}-\underline{\mu}p^{-},\ p\in\mathbb{R},$

there exists a one-dimensional maximally distributed random field
$(W_{\gamma})_{\gamma\in\Gamma}$ on a sublinear expectation space
$(\Omega,\mathbb{L}^{1}(\Omega),\hat{E})$ such that, for each
$\underline{\gamma}=(A_{1},\cdots,A_{n})\in\mathcal{J}_{\Gamma}$,
$W_{\underline{\gamma}}=(W_{A_{1}},\cdots,W_{A_{n}})$ is maximally
distributed.

Furthermore, $(W_{\gamma})_{\gamma\in\Gamma}$ is a spatial maximally
distributed white noise under $(\Omega,\mathbb{L}^{1}(\Omega),\hat{E})$,
namely, conditions (i) and (ii) of Definition 4.1 are satisfied.

If $(\bar{W}_{\gamma})_{\gamma\in\Gamma}$ is another maximally distributed
white noise with the same sublinear function $g$ in (9), then
${\bar{W}}\overset{d}{=}{W}$.

Proof. Thanks to Proposition 4.3 and Theorem 3.5, the existence and uniqueness
of the maximally distributed random field $W$ in a sublinear expectation space
$(\Omega,\mathbb{L}^{1}(\Omega),\hat{E})$ with the family of generating
functions defined by (9) hold. We only need to verify that the maximally
distributed random field $W$ satisfies conditions (i) and (ii) of Definition
4.1.

For each $A\in\Gamma$, $\hat{E}[W_{A}\cdot p]=g(p)\lambda_{A}$ by Theorem 3.5
and (9), thus (i) of Definition 4.1 holds.

We note that if $\\{A_{i}\\}_{i=1}^{n}$ are mutually disjoint, then for
$p=(p_{1},\cdots,p_{n})\in\mathbb{R}^{n}$, by (9), we have

$\hat{E}[p_{1}W_{A_{1}}+\cdots+p_{n}W_{A_{n}}]=g(p_{1})\lambda_{A_{1}}+\cdots+g(p_{n})\lambda_{A_{n}},$

thus the independence of $\\{W_{A_{i}}\\}_{i=1}^{n}$ can be implied by
Proposition 2.10.

In order to prove (8), we only consider the case of two disjoint sets. Suppose
that

$A_{1}\cap A_{2}=\emptyset,\quad A_{3}=A_{1}\cup A_{2},$

an easy computation of (9) shows that

$\displaystyle g_{A_{1},A_{2},A_{3}}(p)=$ $\displaystyle
g(p_{1}+p_{3})\lambda_{A_{1}}+g(p_{2}+p_{3})\lambda_{A_{2}}$ $\displaystyle=$
$\displaystyle\max_{v_{1}\in[\underline{{\mu}}\lambda_{A_{1}},\overline{\mu}\lambda_{A_{1}}]}\max_{v_{2}\in[\underline{{\mu}}\lambda_{A_{2}},\overline{\mu}\lambda_{A_{2}}]}\max_{v_{3}=v_{1}+v_{2}}(p_{1}\cdot
v_{1}+p_{2}\cdot v_{2}+p_{3}\cdot v_{3}).$

Thus, for each $\varphi\in C(\mathbb{R}^{3})$,

$\hat{E}[\varphi(W_{A_{1}},W_{A_{2}},W_{A_{3}})]=\max_{v_{1}\in[\underline{{\mu}}\lambda_{A_{1}},\overline{\mu}\lambda_{A_{1}}]}\max_{v_{2}\in[\underline{{\mu}}\lambda_{A_{2}},\overline{\mu}\lambda_{A_{2}}]}\max_{v_{3}=v_{1}+v_{2}}\varphi(v_{1},v_{2},v_{3}).$

In particular, we set $\varphi(v_{1},v_{2},v_{3})=|v_{1}+v_{2}-v_{3}|$, it
follows that

$\hat{E}[|W_{A_{1}}+W_{A_{2}}-W_{A_{1}\cup A_{2}}|]=0.$

which implies that

$W_{A_{1}\cup A_{2}}=W_{A_{1}}+W_{A_{2}}.$

Finally, (ii) of Definition 4.1 holds.

###### Remark 4.5

The finite-dimensional distribution of maximally distributed whiten noise can
be uniquely determined by two parameters $\overline{\mu}$ and
$\underline{\mu}$, which can be simply calculated by taking the maximum of the
continuous function over the domain determined by $\overline{\mu}$ and
$\underline{\mu}$.

Similar to the invariant property of $G$-Gaussian white noise introduced in Ji
and Peng [10], it also holds for maximally distributed white noise due to the
well-known invariance of the Lebesgue measure under rotation and translation.

###### Proposition 4.6

For each $p\in\mathbb{R}^{d}$ and
$O\in\mathbb{O}(d):=\\{O\in\mathbb{R}^{d\times d}:O^{T}=O^{-1}\\}$, we set

$T_{p,O}(A)=O\cdot A+p,\ \ \ A\in\Gamma.$

Then, for each $A_{1},\cdots,A_{n}\in\Gamma$,

$(W_{A_{1}},\cdots,W_{A_{n}})\overset{d}{=}(W_{T_{p,O}(A_{1})},\cdots,W_{T_{p,O}(A_{n})}).$

##  5 Spatial and temporal maximally distributed white noise and related
stochastic integral

In Ji and Peng [10], we see that a spatial $G$-white noise is essentially
different from the temporal case or the temporal-spatial case, since there is
no independence property for the spatial $G$-white noise. But for the
maximally distributed white noise, spatial or temporal-spatial maximally
distributed white noise has the independence property due to the symmetrical
independence for maximal distribution.

Combining symmetrical independence and boundedness properties of maximal
distribution, the integrand random fields can be largely extended when we
consider the stochastic integral with respect to spatial maximally distributed
white noise. For stochastic integral with respect to temporal-spatial case,
the integrand random fields can even contain the “non-adapted” situation.

###  5.1 Stochastic integral with respect to the spatial maximally distributed
white noise

We firstly define the stochastic integral with respect to the spatial
maximally distributed white noise in a quite direct way.

Let $\\{{W}_{\gamma}\\}_{\gamma\in\Gamma}$,
$\Gamma=\mathcal{B}_{0}(\mathbb{R}^{d})$, be a one-dimensional maximally
distributed white noise defined on a complete sublinear expectation space
$({\Omega},\mathbb{L}^{1}(\Omega),\hat{E})$, with
$g(p)=\overline{\mu}p^{+}-\underline{\mu}p^{-}$,
$-\infty<\underline{\mu}\leq\overline{\mu}<\infty.$ We introduce the following
type of random fields, called simple random fields.

Given $p\geq 1$, set

$\displaystyle M_{g}^{p,0}(\Omega)=\\{\eta(x,\omega)$
$\displaystyle=\sum_{i=1}^{n}\xi_{i}(\omega)1_{A_{i}}(x),\;A_{1},\cdots,A_{n}\in\Gamma\text{
are mutually disjoint}$ $\displaystyle i=1,2,\ldots,n,\ \
\xi_{1},\cdots,\xi_{n}\in\mathbb{L}^{p}(\Omega),\quad n=1,2,\cdots,\\}.$

For each simple random fields $\eta\in M_{g}^{p,0}(\Omega)$ of the form

$\eta(x,\omega)=\sum_{i=1}^{n}\xi_{i}(\omega){\textbf{1}}_{A_{i}}(x),$ (10)

the related Bohner’s integral for $\eta$ with respect to the Lebesgue measure
$\lambda$ is

$I_{B}(\eta)=\int_{\mathbb{R}^{d}}\eta(x,\omega)\lambda(dx):=\sum_{i=1}^{n}\xi_{i}(\omega)\lambda_{A_{i}}.$

It is immediate that
$I_{B}(\eta):M^{p,0}_{g}(\Omega)\mapsto\mathbb{L}^{p}(\Omega)$ is a linear and
continuous mapping under the norm for $\eta$, defined by,

$\|\eta\|_{M^{p}}=\hat{E}[\int_{\mathbb{R}^{d}}|\eta({x},\omega)|^{p}\lambda(dx)]^{\frac{1}{p}}.$

The completion of $M_{g}^{p,0}(\Omega)$ under this norm is denoted by
$M_{g}^{p}(\Omega)$ which is a Banach space. The unique extension of the
mapping $I_{B}$ is denoted by

$\int_{\mathbb{R}^{d}}\eta(x,\omega)\lambda(dx):=I_{B}(\eta),\,\,\eta\in
M_{g}^{p}(\Omega).$

Now for a simple random field $\eta\in M^{p,0}_{g}(\Omega)$ of form (10), we
define its stochastic integral with respect to $W$ as

$I_{W}(\eta):=\int_{\mathbb{R}^{d}}\eta(x,\omega)W(dx)=\sum_{i=1}^{n}\xi_{i}(\omega)W_{A_{i}}.$

With this formulation, we have the following estimation.

###### Lemma 5.1

For each $\eta\in M^{1,0}_{g}(\Omega)$ of form (10), we have

$\hat{E}\left[\left|\int_{\mathbb{R}^{d}}\eta(x,\omega)W(dx)\right|\right]\leq\kappa\hat{E}\left[\int_{\mathbb{R}^{d}}|\eta(x,\omega)|\lambda(dx)\right]$
(11)

where $\kappa=\max\\{|\underline{\mu}|,|\overline{\mu}|\\}$.

Proof. We have

$\displaystyle\hat{E}[|\int_{\mathbb{R}^{d}}\eta(x,\omega)W(dx)|]$
$\displaystyle=\hat{E}[|\sum_{i=1}^{N}\xi_{i}(\omega)W_{A_{i}}|]\leq\hat{E}[\sum_{i=1}^{N}|\xi_{i}(\omega)|\cdot|W_{A_{i}}|]$
$\displaystyle\leq\kappa\hat{E}[\sum_{i=1}^{N}|\xi_{i}(\omega)|\cdot\lambda_{A_{i}}]=\kappa\hat{E}[\|\eta\|_{M_{g}^{1}(\Omega)}].$

The last inequality is due to the boundedness of maximal distribution (see
Remark 4.2).

This lemma shows that $I_{W}:M^{1,0}_{g}(\Omega)\mapsto\mathbb{L}^{1}(\Omega)$
is a linear continuous mapping. Consequently, $I_{W}$ can be uniquely extended
to the whole domain $M^{1}_{g}(\Omega)$. We still denote this extended mapping
by

$\int_{\mathbb{R}^{d}}\eta W(dx):=I_{W}(\eta).$

###### Remark 5.2

Different from the stochastic integrals with respect to $G$-white noise in Ji
and Peng [10] which is only defined for the deterministic integrand, here the
integrand can be a random field.

###  5.2 Maximally distributed random fields of temporal-spatial types and
related stochastic integral

It is well-known that the framework of the classical white noise defined in a
probability space $(\Omega,\mathcal{F},P)$ with $1$-dimensional temporal and
$d$-dimensional spatial parameters is in fact a $\mathbb{R}^{1+d}$-indexed
space type white noise. But Peng [17] and then Ji and Peng [10] observed a new
phenomenon: Unlike the classical Gaussian white noise, the $d$-dimensional
space-indexed $G$-white noise cannot have the property of incremental
independence, thus spatial $G$-white noise is essentially different from
temporal-spatial or temporal one. Things will become much direct for the case
of maximally distributed white noise due to the incremental independence
property of maximal distributions. This means that a time-space maximally
distributed $(1+d)$-white noise is essentially a $(1+d)$-spatial white noise.
The corresponding stochastic integral is also the same. But in order to make
clear the dynamic properties, we still provide the description of the
temporal-spatial white-noise on the time-space framework:

$\displaystyle\mathbb{R}^{+}\times\mathbb{R}^{d}$
$\displaystyle=\\{(t,x_{1},\ldots,x_{d})\in\mathbb{R}^{+}\times\mathbb{R}^{d}\\},$

where the index $t\in[0,\infty)$ is specially preserved to be the index for
time.

Let
$\Gamma=\\{A\in\mathcal{B}(\mathbb{R}^{+}\times\mathbb{R}^{d}),\lambda_{A}<\infty\\}$,
the maximally distributed white noise $\\{W_{A}\\}_{A\in\Gamma}$ is just like
in the spatial case with dimension $1+d$.

More precisely, let

$\displaystyle\Omega=\\{\omega\in\mathbb{R}^{\Gamma}:\ \ $
$\displaystyle\omega(A\cup B)=\omega(A)+\omega(B),$ $\displaystyle\ \forall
A,B\in\Gamma,\ \ A\cup B=\emptyset\\},$

and $W=({W}_{\gamma}(\omega)=\omega_{\gamma})_{\gamma\in\Gamma}$ the canonical
random field.

For $T>0$, denote the temporal-spatial sets before time $T$ by

$\displaystyle\Gamma_{T}:=\\{A\in\Gamma:(s,x)\in A\Rightarrow 0\leq s<T\\}.$

Set $\mathcal{F}_{T}=\sigma\\{{W}_{A},A\in\Gamma_{T}\\}$,
$\mathcal{F}=\bigvee\limits_{T\geq 0}\mathcal{F}_{T}$, and

$\displaystyle L_{ip}(\Omega_{T})=$
$\displaystyle\\{\varphi(W_{A_{1}},\ldots,W_{A_{n}}),\,\,\forall
n\in\mathbb{N},$ $\displaystyle A_{i}\in\Gamma_{T},i=1,\ldots,n,\varphi\in
C_{b.Lip}(\mathbb{R}^{n})\\}.$

We denote

$L_{ip}(\Omega)=\bigcup_{n=1}^{\infty}L_{ip}(\Omega_{n}).$

For each $X\in L_{ip}(\Omega)$, without loss of generality, we assume $X$ has
the form

$\displaystyle X=$
$\displaystyle\varphi({W}_{A_{11}},\cdots,{W}_{{A}_{1m}},\cdots,{W}_{A_{n1}},\cdots,W_{A_{nm}}),$

where $A_{ij}=[t_{i-1},t_{i})\times A_{j}$, $1\leq i\leq n,1\leq j\leq m$,
$0=t_{0}<t_{1}<\cdots<t_{n}<\infty$,
$\\{A_{1},\cdots,A_{m}\\}\subset\mathcal{B}_{0}(\mathbb{R}^{d})$ are mutually
disjoint and $\varphi\in C_{b.Lip}(\mathbb{R}^{n\times m})$. Then the
corresponding sublinear expectation for $X$ can be defined by

$\displaystyle\hat{E}[X]$
$\displaystyle=\hat{E}[\varphi({W}_{A_{11}},\cdots,{W}_{A_{1m}},\cdots,{W}_{A_{n1}},\cdots,{W}_{A_{nm}})$
$\displaystyle=\max_{v_{ij}\in[{\underline{\mu}},\overline{\mu}]}\varphi(\lambda_{A_{11}}v_{11},\cdots,\lambda_{A_{1m}}v_{1m},\cdots,\lambda_{A_{n1}}v_{v_{n1},}\cdots,\lambda_{A_{nm}}v_{nm}),$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \ \ \ \ \ \ \
\ \ \ \ \ \ \ 1\leq i\leq m,1\leq j\leq n$

and the related conditional expectation of $X$ under $\mathcal{F}_{t}$, where
$t_{j}\leq t<t_{j+1}$, denoted by $\hat{E}[X|\mathcal{F}_{t}]$, is defined by

$\displaystyle\hat{E}[\varphi({W}_{A_{11}},\cdots,{W}_{A_{1m}},\cdots,{W}_{A_{n1}},\cdots,{W}_{A_{nm}})|\mathcal{F}_{t}]$
$\displaystyle=$
$\displaystyle\psi({W}_{A_{11}},\cdots,{W}_{A_{1m}},\cdots,{W}_{A_{j1}},\cdots,{W}_{A_{jm}}),$

where

$\displaystyle\psi(x_{11},\cdots,x_{1m},\cdots,x_{j1},\cdots,x_{jm})=\hat{E}[\varphi(x_{11},\cdots,x_{1m},\cdots,x_{j1},\cdots,x_{jm},\tilde{{W})}].$

Here

$\tilde{W}=({W}_{A_{(j+1)1}},\cdots,{W}_{A_{(j+1)m}},\cdots,{W}_{A_{n1}},\cdots,{W}_{A_{nm}}).$

It is easy to verify that $\hat{E}[\cdot]$ defines a sublinear expectation on
$L_{ip}(\Omega)$ and the canonical process $({W}_{\gamma})_{\gamma\in\Gamma}$
is a one-dimensional temporal-spatial maximally distributed white noise on
$(\Omega,L_{ip}(\Omega),\hat{E}).$

For each $p\geq 1$, $T\geq 0$, we denote by ${L}_{g}^{p}(\Omega_{T})$(resp.,
${L}_{g}^{p}(\Omega)$) the completion of $L_{ip}(\Omega_{T})$(resp.,
$L_{ip}(\Omega)$) under the norm $\|X\|_{p}:=(\hat{E}[|X|^{p}])^{1/p}$. The
conditional expectation
$\hat{E}[\cdot\left|\mathcal{F}_{t}\right]:L_{ip}(\Omega)\rightarrow
L_{ip}(\Omega_{t})$ is a continuous mapping under $\|\cdot\|_{p}$ and can be
extended continuously to the mapping
${L}_{g}^{p}(\Omega)\rightarrow{L}_{g}^{p}(\Omega_{t})$ by

$|\hat{E}[X\left|\mathcal{F}_{t}\right]-\hat{E}[Y\left|\mathcal{F}_{t}\right]|\leq\hat{E}[|X-Y|\left|\mathcal{F}_{t}\right]\
\text{ for }X,Y\in L_{ip}(\Omega).$

It is easy to verify that the conditional expectation
$\hat{E}[\cdot|\mathcal{F}_{t}]$ satisfies the following properties, and the
proof is very similar to the corresponding one of Proposition 5.3 in Ji and
Peng [10].

###### Proposition 5.3

For each $t\geq 0$, the conditional expectation
$\hat{E}[\cdot\left|\mathcal{F}_{t}\right]:{L}_{g}^{p}(\Omega)\rightarrow{L}_{g}^{p}(\Omega_{t})$
satisfies the following properties: for any $X,Y\in{L}_{g}^{p}(\Omega)$,
$\eta\in{L}_{g}^{p}(\Omega_{t})$,

(i)

$\hat{E}[X\left|\mathcal{F}_{t}\right]\geq\hat{E}[Y\left|\mathcal{F}_{t}\right]$
for $X\geq Y$.

(ii)

$\hat{E}[\eta\left|\mathcal{F}_{t}\right]=\eta$.

(iii)

$\hat{E}[X+Y\left|\mathcal{F}_{t}\right]\leq\hat{E}[X\left|\mathcal{F}_{t}\right]+\hat{E}[Y\left|\mathcal{F}_{t}\right]$.

(iv)

$\hat{E}[\eta
X\left|\mathcal{F}_{t}\right]=\eta^{+}\hat{E}[X\left|\mathcal{F}_{t}\right]+\eta^{-}\hat{E}[-X\left|\mathcal{F}_{t}\right]$
if $\eta$ is bounded.

(v)

$\hat{E}[\hat{E}[X\left|\mathcal{F}_{t}\right]|\mathcal{F}_{s}]=\hat{E}[X|\mathcal{F}_{t\wedge
s}]$ for $s\geq 0$.

Now we define the stochastic integral with respect to the spatial-temporal
maximally distributed white noise ${W}$, which is similar to the spatial
situation.

For each given $p\geq 1$, let ${M}^{p,0}(\Omega_{T})$ be the collection of
simple processes with the form:

$f(s,x;\omega)=\sum\limits_{i=0}^{n-1}\sum_{j=1}^{m}X_{ij}(\omega)\textbf{1}_{A_{j}}(x)\textbf{1}_{[t_{i},t_{i+1})}(s),$
(12)

where $X_{ij}\in{L}_{g}^{p}(\Omega_{T})$, $i=0,\cdots,n-1$, $j=1,\cdots,m$,
$0=t_{0}<t_{1}<\cdots<t_{n}=T$, and $\\{A_{j}\\}_{j=1}^{m}\subset\Gamma$ is
mutually disjoint.

###### Remark 5.4

Since we only require $X_{ij}\in{L}_{g}^{p}(\Omega_{T})$, the integrand may
“non-adapted”. This issue is essentially different from the requirement of
adaptability in the definition of stochastic integral with respect to
temporal-spatial $G$-white noise in Ji and Peng [10].

The completion of ${M}^{p,0}(\Omega_{T})$ under the norm
$\|\cdot\|_{{M}^{p}}$, denoted by ${M}_{g}^{p}(\Omega_{T})$, is a Banach
space, where the Banach norm $\|\cdot\|_{{M}^{p}}$ is defined by

$\displaystyle\|f\|_{{M}^{p}}:=$
$\displaystyle\left(\hat{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}|f(s,x)|^{p}ds\lambda(dx)\right]\right)^{\frac{1}{p}}$
$\displaystyle=$
$\displaystyle\left\\{\hat{E}\left[\sum_{i=0}^{n-1}\sum_{j=1}^{m}|X_{ij}|^{p}(t_{i+1}-t_{i})\lambda_{A_{j}}\right]\right\\}^{\frac{1}{p}}.$

For $f\in{M}^{p,0}(\Omega_{T})$ with the form as (12), the related stochastic
integral with respect to the temporal-spatial maximally distributed white
noise ${W}$ can be defined as follows:

$I_{W}(f)=\int_{0}^{T}\int_{\mathbb{R}^{d}}f(s,x){W}(ds,dx):=\sum\limits_{i=0}^{n-1}\sum_{j=1}^{m}X_{ij}{W}([t_{j},t_{j+1})\times{A_{j}}).$
(13)

Similar to Lemma 5.1, we have

###### Lemma 5.5

For each $f\in{M}^{1,0}([0,T]\times\mathbb{R}^{d})$,

$\displaystyle\hat{E}\left[\left|\int_{0}^{T}\int_{\mathbb{R}^{d}}f(s,x){W}(ds,dx)\right|\right]\leq\kappa\hat{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}|f(s,x)|dsdx\right],$
(14)

where $\kappa=\max\\{|\overline{\mu}|,|\underline{\mu}|\\}$.

Thus $I_{W}:{M}^{1,0}(\Omega_{T})\mapsto{L}_{g}^{1}(\Omega_{T})$ is a
continuous linear mapping. Consequently, $I_{W}$ can be uniquely extend to the
domain ${M}_{g}^{1}(\Omega_{T})$. We still denote this mapping by

$\int_{0}^{T}\int_{\mathbb{R}^{d}}f(s,x){W}(ds,dx):=I_{W}(f)\ \ \text{for}\
f\in{M}_{g}^{1}(\Omega_{T}).$

###### Remark 5.6

Thanks to the boundedness of maximally distributed white noise, the domain of
integrand $M_{g}^{1}(\Omega_{T})$ is much larger since the usual requirement
of adaptability for integrand can be dropped.

It is easy to check that the stochastic integral has the following properties.

###### Proposition 5.7

For each $f,g\in{M}_{g}^{1}(\Omega_{T})$, $0\leq s\leq r\leq t\leq T$,

(i)
$\int_{s}^{t}\int_{\mathbb{R}^{d}}f(u,x){W}(du,dx)=\int_{s}^{r}\int_{\mathbb{R}^{d}}f(u,x){W}(du,dx)+\int_{r}^{t}\int_{\mathbb{R}^{d}}f(u,x){W}(du,dx).$

(ii) $\int_{s}^{t}\int_{\mathbb{R}^{d}}(\alpha f(u,x)+g(u,x)){W}(du,dx)$
$=\alpha\int_{s}^{t}\int_{\mathbb{R}^{d}}f(u,x){W}(du,dx)+\int_{s}^{t}\int_{\mathbb{R}^{d}}g(u,x){W}(du,dx)$,
where $\alpha\in{L}_{g}^{1}(\Omega_{T})$ is bounded.

###### Remark 5.8

In particular, if we only consider temporal maximally distributed white noise
and further assume that $\underline{\mu}\geq 0$. In this case, the index set
$\Gamma=\\{[s,t):0\leq s<t<\infty\\}$. The canonical process $W([0,t))$ is the
quadratic variation process of $G$-Brownian motion, more details about the
quadratic variation process can be found in Peng [18].

###### Acknowledgement 5.9

This research is partially supported by NSF of China (No.L1624032,
No.11526205, No.11601281), 111 Project of Chinese SAFEA (No.B12023), National
Key R&D Program of China (No.2018YFA0703900).

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Radio Imaging Spectropolarimetry of

CMEs and CME Progenitors

Principal Author:
Bin Chen1 New Jersey Institute of Technology
Email: mailto:binchen<EMAIL_ADDRESS>Phone: (973) 596-3565; Web:
https://binchensun.orghttps://binchensun.org

Co-authors
Timothy S. Bastian2, Sarah Gibson3, Yuhong Fan3, Stephen M. White4, Dale E.
Gary1, Angelos Vourlidas5, Sijie Yu1, Surajit Mondal1, Gregory D. Fleishman1,
Pascal Saint-Hilaire6

[1] New Jersey Institute of Technology; [2] National Radio Astronomy
Observatory; [3] High Altitude Observatory; [4] Air Force Research Laboratory;
[5] JHU Applied Physics Laboratory; [6] University of California, Berkeley

Co-Signers & Affiliations: see spreadsheet

Synopsis
Coronal mass ejections (CMEs) are the most important drivers of space weather.
Central to most CMEs is thought to be the eruption of a bundle of highly
twisted magnetic field lines known as magnetic flux ropes. A comprehensive
understanding of CMEs and their impacts hence requires detailed observations
of physical parameters that lead to the formation, destabilization, and
eventual eruption of the magnetic flux ropes. Recent advances in remote-
sensing observations of coronal cavities, filament channels, sigmoids, EUV
“hot channels,” white light CMEs, and in situ observations of magnetic clouds
points to the possibility of significant progress in understanding CMEs. In
this white paper, we provide a brief overview of the potential of radio
diagnostics for CMEs and CME progenitors, with a particular focus on the
unique means for constraining their magnetic field and energetic electron
population. Using synthetic observations based on realistic 3D MHD models, we
also demonstrate the transformative potential of advancing such diagnostics by
using broadband radio imaging spectropolarimetry with a high image dynamic
range and high image fidelity. To achieve this goal, a solar-dedicated radio
facility with such capabilities is recommended for implementation in the
coming decade.

##  1 Introduction

Coronal mass ejections (CMEs) are violent eruptions of coronal plasma and
magnetic field from the Sun, involving a few times $10^{14}$ to several
$\times 10^{16}$ grams of the material moving into the interplanetary medium
with speeds of a few hundred to more than 2,000 km s-1. CMEs can drive
significant geomagnetic storms and other space weather activity. In fact, the
strongest space weather effects are usually related to the impact of CMEs with
$B_{z}$ anti-aligned with the Earth’s magnetic field. Therefore, a detailed
understanding of the formation, initiation, eruption, and evolution of CMEs
and their subsequent geospace impacts is of vital importance for space weather
science and applications. In addition, fast CMEs are usually accompanied by
long-duration eruptive solar flares and large solar energetic particle (SEP)
events. Quantitative measurements of the physical properties of CMEs
constitute an essential piece to completing the “system science” of solar
eruptions, and hence are of fundamental importance for solar- and helio-
physics. Last but not least, the understanding of solar CMEs serves as a
“stepping stone” for addressing exo-space weather in other stellar systems and
the habitability of exoplanets.

To make progress in achieving a comprehensive picture, the following
measurements must be obtained:

  * •

The magnetic field configuration of the pre-eruption magnetic flux ropes and
quantifying their plasma environment. This measurement is crucial in
determining the formation and initiation conditions of CMEs.

  * •

Measuring the size, mass, speed, and magnetic field of CMEs from their early
eruption in the low corona to their propagation throughout the interplanetary
space. These observational inputs are essential for predicting the arrival of
CMEs and evaluating their geospace impacts.

  * •

Mapping the morphology, magnetic field configuration, and plasma parameters of
CME-driven shocks. This requirement is compulsory to study the origin of large
SEP events, which are themselves important space weather manifestations.

Traditionally white-light coronagraph observations have been the primary means
for studying CMEs. However, radio observations provide a rich variety of
complementary diagnostic methods for obtaining such observational constraints.
We refer interested readers to recent reviews by Vourlidas et al. (2020) and
Carley et al. (2020) for more comprehensive discussions on the details of
these methods and the progress made in the past two decades, as well as their
complementarity to observations at other wavelengths. For a discussion of
coronal cavity magnetic diagnostics at UV and optical/IR wavelengths, see the
white paper by Gibson et al. (2022). In this white paper, we will focus on
radio diagnostics for coronal cavities and erupting CMEs, with an emphasis on
the uniqueness in measuring the magnetic field and energetic electrons in CMEs
and CME progenitors, as well as their highly complementary nature to
optical/infrared spectropolarimetry techniques. We argue that the next major
advance in radio diagnostics calls for a solar-dedicated instrument that is
capable of performing broadband radio dynamic imaging spectroscopy and
polarimetry with a high dynamic range, high image fidelity, and high
sensitivity.

##  2 Coronal Cavities as CME Precursors

Coronal cavities have been observed in the Sun’s corona for several decades.
They are of particular interest because of their close relationship with CMEs
(Gibson et al. 2006). CMEs observed in white light often show a three‐part
structure: a bright expanding loop followed by a dark cavity containing a
bright core that corresponds to an erupting prominence. This structure is
mirrored in coronal cavities observed on the solar limb where a bright loop or
coronal streamer and dark cavity are seen in quiescence; a bright prominence
core is also commonly seen. Seen against the solar disk, coronal cavities
overlie solar filaments or, more generally magnetic polarity inversion lines.
A coronal cavity is often destabilized leading to a CME and hence they can be
considered as CME precursors. These ejections are magnetically driven, the
energy stored in twisted or sheared magnetic fields prior to eruption.

Clearly, a quantitative understanding of the physical attributes of coronal
cavities is needed, particularly of the magnetic field topology and the
magnetic free energy available. Significant progress has been made on the
temperature and density structure of coronal cavities over the past decade.
For example, Reeves et al. (2012) used the X-ray telescope (XRT) on board the
Hinode spacecraft to study the thermal soft X-ray emission in cavities with
hot cores, showing cavity temperatures of 1.5–1.65 MK and core temperatures of
1.7–2 MK. Forward-models of data from the Hinode EUV imaging Spectrometer
(EIS) and the white-light polarization brightness measured by the Mauna Loa
Solar Observatory (MLSO) Mk4 coronagraph were used (Schmit & Gibson 2011) to
infer the density of a cavity and the surrounding streamer from 1.05–1.25
$R_{\odot}$. In cavities, densities ranging from $0.5-3\times 10^{8}$ cm-3
were found over this radial range whereas streamer densities of $0.8-5\times
10^{8}$ cm-3 were found. For the magnetic field, recent results from the
Coronal Multi-Channel Polarimeter (CoMP) show that the linearly polarized
infrared (IR) emission in coronal cavities is consistent with a magnetic flux
rope topology. However, a critical missing ingredient is quantitative
measurements of the vector magnetic field.

At optical/IR wavelengths, spectropolarimetry observations have been used to
constrain the plane-of-the-sky direction of the magnetic field of coronal
cavities (based on the saturated Hanle effect; see, e.g., Bak-Stȩślicka et al.
2013). To measure the line-of-sight (LOS) component of the field, much more
sensitive spectropolarimetry measurements are necessary to detect the
extremely weak Stokes-V signal due to Zeeman splitting. The expected circular
polarization degree (V/I) of the off-limb measurements is only $\lesssim
0.1$%, or $10^{-9}$ of the brightness at the disk center. As a result, such
measurements are only possible for off-limb sources and require large-aperture
coronagraphs operating at a telescope site with excellent atmospheric
conditions. The Daniel K. Inouye Solar Telescope (DKIST) and the proposed
COronal Solar Magnetism Observatory (COSMO) are poised to make major advances
in this area (see the white paper by Tomczyk et al. 2022 for details).

At radio wavelengths, the Stokes-V signal of the coronal magnetic field via
thermal free-free emission is, in fact, prominent. In the optically thin
regime, the degree of circular polarization $p=V/I$ is given by
$p\approx[0.56B_{\rm LOS}/\nu_{\rm GHz}]$%, where $B_{\rm LOS}$ is the LOS
component of the magnetic field (weighted by the square of the density in the
source) and $\nu_{\rm GHz}$ is the observing frequency in GHz (Casini et al.
2017). For example, a $B_{\rm LOS}$ of 20 G would give rise to a polarization
degree of order 10% at 1 GHz—that is, 100 times stronger than that of the
optical/IR signal via Zeeman splitting! Thanks to their comparable radio
brightness relative to the disk, such $B_{\rm LOS}$ measurements can be made
for both the coronal cavities above the limb and, uniquely, also their on-disk
counterparts—sigmoids or filament channels. In addition, such measurements can
be obtained at a time cadence of few seconds. Therefore, the radio
measurements can be used to reveal rapid dynamic evolution of the plasma
structure and magnetic field configuration of CME precursors, which is another
unique diagnostic potential available to this technique.

As a demonstration, we use a realistic 3D MHD model of a magnetic-flux-rope-
hosting coronal cavity developed by Fan & Liu (2019) (Fig. 1) to calculate the
corresponding synthetic radio Stokes I and V emission maps at multiples
frequencies using the FORWARD package (Gibson et al. 2016). An example for 1
GHz is shown in Figs. 1(e) and (f). For this model, the Stokes V/I signal at 1
GHz reaches a peak of $\sim$7% in a region that coincides with the core of the
flux rope (located just above the dense and cool filament) where $B_{\rm LOS}$
reaches a maximum. Multi-frequency mapping enables spectral analysis (panels
g, h), and the spatial distribution and temporal variation of the Stokes V and
I maps serve as an excellent tool to derive the spatial and temporal evolution
of key physical parameters including $B_{\rm LOS}$, plasma density, and
temperature.

Figure 1: Radio observations can be used to derive maps of key physical
parameters of coronal cavities, including their plasma density, temperature,
and LOS magnetic field. (a–d): Parameters of the 3D MHD coronal cavity model
(showing those at the mid-plane), which features a filament-hosting twisted
magnetic flux rope viewed end on (from Fan & Liu 2019). (e) and (f): Synthetic
radio thermal free-free Stokes I and V/I maps at 1 GHz calculated by applying
radiative transfer to the 3D MHD model. They bear striking similarities to the
density and $B_{\rm LOS}$ distributions in the MHD model. (g) and (h): Stokes
I and V/I spectra sampled at three locations that correspond to the filament,
flux rope core, and cavity. Note the core of the flux rope features a strong
Stokes V signal of $\sim$1–7% over a wide range of frequencies from 1–20 GHz.

Despite the potential of radio observations for deriving the detailed
morphology of coronal cavities (both on the limb and against the disk) and
accurately mapping $B_{\rm LOS}$, current radio facilities do not have the
adequate imaging fidelity, dynamic range, and polarization purity necessary to
make such measurements. The figure on the cover page shows one of the best
examples of multi-frequency radio images of a coronal cavity obtained by the
Expanded Owens Valley Solar Array (EOVSA), a key pathfinder instrument for
demonstrating these techniques. At low frequencies (e.g., 1.4 GHz), the
filament-cavity system features a dark cavity with a bright rim. This is
because the optical depth in the low-density cavity is small compared to its
rim. At higher frequencies (e.g., 13.9 GHz), however, the optically thick
emission from the cooler but much denser filament starts to dominate the
emission. These multi-frequency images demonstrate the feasibility of imaging
the filament-cavity systems at radio wavelengths and, more importantly, the
potential of using spectral analysis to derive the plasma parameters. However,
owing to the small number of baselines from EOVSA’s 13 antennas, these images
can only be made with several hours of integration by taking advantage of
Earth’s rotation to improve the u-v coverage. Thus, crucial details of the
dynamic evolution of the coronal cavity structures are not recovered.
Additionally, owing to its limited spatial resolution, image fidelity, and
dynamic range, measuring the Stokes-V signal down to the required detection
level of a few percent is likely out of reach for EOVSA.

Figure 2: Simulated snapshot (1-s cadence) Stokes I and V images by an example
FASR array configuration that consists of 200 2-m-diameter antennas. (a)
Simulated full-disk image at 1 GHz. (b) and (c) Enlarged view of the Stokes I
and V/I images of the coronal cavity region (white box in (a)). The latter is
simulated from an input emission model calculated from the 3D MHD model shown
in Fig. 1. The angular resolution of the array is $\sim$20′′/GHz. See text for
details.

To fully exploit the potential of these radio diagnostics to study CME
precursors, a radio array with a large number of antenna elements (of order
100) and accurate spectropolarimetry capabilities over a frequency range of
$\sim$0.5–20 GHz is required. With the $>$2 orders of magnitude increase in
the number of baselines sampling the u-v domain compared to, e.g., EOVSA, the
instrument will provide dramatically improved image fidelity and dynamic
range. Such a superior imaging capability would allow all features with
different angular scales and brightness across the full-Sun disk to be
faithfully recovered. Fig. 2 shows simulated snapshot (1-s cadence) imaging
results based on an example 200-element log-spiral array configuration (see
Fig. 3 in Chen et al. 2022a for plots of the array configuration and
performance metrics). In order to best emulate the actual observational
conditions, the input “sky model” is made from the synthetic radio emission
maps calculated from the 3D MHD coronal cavity model, as shown in Fig. 1,
combined with a synthetic full-disk radio map. The latter is constructed by
calculating the thermal free-free radio emission using the differential
emission measure map derived from multi-band SDO/AIA EUV images (Fleishman et
al. 2021). This sky model is then “observed” with the model array
configuration to produce complex visibilities using CASA’s simobserve task,
and is then subsequently Fourier transformed and deconvolved using the multi-
scale CLEAN algorithm (Cornwell 2008). Random noise (due to the instrument’s
system temperature and residual gain errors after calibration) is introduced
in the simulation to best replicate the actual observations.

Contrasting the “ground truth” shown in Fig. 1 with the simulated
observations, it is evident that crucial details of the cavity brightness,
morphology, and Stokes-V signatures are faithfully reproduced. Therefore, with
the added broadband multi-frequency capability for performing spatially
resolved spectral analysis (based on spectra similar to those in Fig. 1(g) and
(h)), such a next-generation instrument is expected to make a giant leap
forward in constructing the key parameters of coronal cavities, including the
elusive measurements of the strength, spatial distribution, and short-time-
scale evolution of the LOS coronal magnetic field. We emphasize that, although
the discussion above focuses on coronal cavities that are long-lived
structures typically observed in quiet Sun regions, with the second-scale
snapshot imaging spectropolarimetry capability, similar techniques can be
applied to fast-evolving, flux-rope-hosting structures in active regions.
These active region flux rope structures, which can lead to major eruptive
flare events and fast CMEs, are expected to yield more significant radio
signatures because they are hotter, denser, and bears a stronger magnetic
field.

##  3 Radio Observations of CMEs and CME-driven Shocks

When CMEs erupt, in addition to the radio emission from the ejected filaments
and CME bubbles (e.g., Huang et al. 2019), energetic electrons accelerated or
trapped by the CMEs or CME-driven shocks also emit a variety of nonthermal
radio emissions.

###  3.1 Type II Radio Bursts

The most common radio signature of a fast CME is a type II radio burst,
coherent plasma radiation produced by the CME-driven super-Alfvénic shock
propagating in the corona and sometimes well out into the interplanetary
medium. Type IIs are recognized by their characteristic slow-drift signature
in dynamic spectra and have been used to constrain the density of the medium
and the shock speed. They have been imaged at discrete frequencies in past
years (e.g., Bain et al. 2012) but their relation to the underlying CME driver
has been murky. In more recent years, instruments like the general-purpose
LOFAR and the MWA have produced multi-frequency imaging of type IIs (Morosan
et al. 2019). Determining the relationship between type II radio bursts and
the shock driver is a key science goal of solar-dedicated imaging arrays being
commissioned at long wavelengths, which include Owens Valley Long Wavelength
Array (OVRO-LWA, 20-88 MHz; Chhabra et al. 2021) and the SunRISE mission
(Kasper et al. 2022), which will use a space-based interferometric array
covering 0.1–25 MHz. However, tracking the initiation and development of type
II radio bursts requires continuous coverage from the low corona to the middle
corona. Under typical coronal conditions, this translates to a frequency range
from $\sim\\!500$ MHz down to $<$20 MHz. As reviewed by Chen et al. (2022b),
currently there is no solar-dedicated radio instrument available that provides
broadband dynamic imaging spectroscopy over this frequency range, although
LOFAR, the MWA, and OVRO-LWA partially address the need.

###  3.2 Radio CMEs

Faint radio emissions that closely resemble their white light CME counterparts
are dubbed “radio CMEs” because of their similar appearance (see a recent
review by Vourlidas et al. 2020). The emission mechanism is believed to be
synchrotron emission which, when imaged spectroscopically, can be used to map
the evolving CME magnetic field strength (and possibly direction, if
polarimetry is available), non-thermal electron distribution, and the thermal
electron number density (see, e.g. Bastian et al. 2001). It should be noted
that the same plasma producing white light CMEs in the middle to high corona
also emits thermal free-free radiation. This is usually difficult to detect
due to low surface brightness (see discussion by Bastian & Gary 1997),
although detections have been reported in the literature (Gopalswamy & Kundu
1993; Ramesh et al. 2021).

[capbesideposition=right,center,capbesidewidth=4.2cm]figure[]

Figure 3: Examples of “radio CME” observations by the Nançay Radioheliograph
(Bastian et al. 2001) (a) and the MWA (Mondal et al. 2020) (b) in the middle
corona at metric wavelengths. (c) A high-frequency counterpart of radio CME
observed by EOVSA in the low corona at $\sim$4.2 GHz (Chen et al. 2020). (d)
Spectra of some reported radio CME events (Mondal et al. 2020).

Since the original detections made by the Nançay Radioheliograph operating
between 150–450 MHz (see Fig. 3(a) for an example), few radio CMEs have been
reported. Recently, emission surrounding a flux rope cavity, which could be
considered a radio CME at its initiation phase, was detected with EOVSA at
$\sim$4.2 GHz in the low corona ($\sim\\!1.2R_{\odot}$; Fig. 3(c)). More
recently, a faint radio CME event was observed from 80–240 MHz by the MWA
(Fig. 3(b); Mondal et al. 2020). The event shared a similar morphology to
those reported earlier but was associated with a much slower ($<$500 km s-1)
and fainter (of order 0.001–0.01 sfu) CME. Such an unprecedented detection was
made possible thanks to MWA’s ability to perform highly sensitive, high
dynamic range imaging with its 128 antenna elements. With spatially resolved
spectral analysis, a magnetic field strength of $\sim$10 G was derived at the
outer “shell” of the CME bubble. In addition, the number density and power-law
index of the nonthermal electron distribution were also constrained.

This outstanding example, obtained by a state-of-the-art general-purpose
facility (i.e. proposal driven), provides a glimpse of the unique power of
radio observations for constraining the magnetic field configuration and
nonthermal electron distribution of erupting CMEs with a variety of speeds and
strengths. These diagnostics, however, call for routine solar-dedicated
observations with high sensitivity and high dynamic range imaging
spectroscopy. In addition, to trace and measure the radio CMEs from their
initiation in the low corona to their evolution in the middle corona, a
continuous frequency coverage over 10s of MHz to a few GHz is required.

##  4 Observing the System: Findings and Recommendations

To conclude, radio observations can be used in a unique and wholly new manner
to constrain the formation, evolution, destabilization, and eruption of CMEs
and CME progenitors. Especially powerful will be joint optical/IR/(E)UV and
radio observations of coronal cavities as CME precursors and their
destabilization into nascent CMEs, with multi-wavelength coronal
spectropolarimetry yielding sensitivity to different parts of the emitting
coronal plasma through a variety of physical mechanisms (Gibson et al. 2021;
see e.g., Table 1), leading to the robust reconstruction of their 3D coronal
magnetic field. Equally important are the unique diagnostics offered by radio
observations on the magnetic field and nonthermal electron distribution of
erupting CMEs and the associated shocks as they lift off and propagate into
the upper corona. To briefly summarize the discussion above:

  * •

CME precursors (Sec. 2): Radio (this paper) and O/IR (white paper by Gibson et
al. 2022) observations from the ground and UV observations from space (white
paper by Casini et al. 2022) offer powerful and complementary diagnostics of
the plasma environment in and around coronal cavities. Radio observations of
thermal free-free emission and polarimetry will yield the line-of-sight
component of the magnetic field of coronal cavities and flux-rope-hosting
structures in both quiescent and active regions. Unlike optical/IR
observations, which for this goal can only be taken above the limb, radio
observations can be made of coronal cavities, filament channels, and sigmoids
on the disk.

  * •

Erupting CMEs and CME-driven shocks (Sec. 3): The erupting magnetic flux
ropes/CMEs can be imagined at radio wavelengths if it has significant numbers
of energetic electrons entrained in their magnetic field. As the eruption
evolves into an eruptive flare, the magnetic field and the electron
distribution function can be measured as a function of time and space—in the
reconnecting magnetic loops/current sheet (white paper by Chen et al. 2022a)
and the erupting CME themselves (this paper). Ground-based radio observations
can make these measurements dynamically across the evolving CME out to several
solar radii. Faraday Rotation (FR) measurements provide additional constraints
on the CME density and magnetic field (Kooi et al. 2022). Any type II radio
burst, tracing a shock that accompanies the CME, may also be imaged
simultaneously.

The above elements require radio coverage from a few 10s of MHz to 20 GHz in
order to observe them as a coupled system. To make transformative advances in
making new and unique measurements discussed above, a solar-dedicated radio
facility with a large number of elements (of order 100) is required to obtain
broadband radio imaging spectropolarimetry with high image fidelity, dynamic
range, angular resolution, and sensitivity. In the past decade, significant
investments have been made in ground-based low-frequency arrays at frequencies
$\lesssim$0.2 GHz with the operation of LOFAR, MWA, and the solar-dedicated
OVRO-LWA (see Fig. 5 of Chen et al. 2022b for a synopsis). Therefore, in the
coming decade, we suggest the focus should be placed in the $\sim$0.2–20 GHz
range.

The Frequency Agile Solar Radiotelescope (FASR) is a concept for realizing
such high sensitivity, image fidelity, and dynamic range imaging
spectropolarimetry observations in an ultra-wide frequency range of
$\sim$0.2–20 GHz. We refer readers to Gary et al. (2022) and references
therein for a more detailed discussion of the instrument and a broad range of
breakthrough science it is expected to achieve.

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