# Affine statistical bundle modeled on a Gaussian Orlicz-Sobolev space

Giovanni Pistone<EMAIL_ADDRESS>de Castro Statistics,
Collegio Carlo Alberto, Piazza Vincenzo Arbarello 8, Torino 10122 Italy
giannidiorestino.it

###### Abstract.

The dually flat structure of statistical manifolds can be derived in a non-
parametric way from a particular case of affine space defined on a qualified
set of probability measures. The statistically natural displacement mapping of
the affine space depends on the notion of Fisher’s score. The model space must
be carefully defined if the state space is not finite. Among various options,
we discuss how to use Orlicz-Sobolev spaces with Gaussian weight. Such a fully
non-parametric set-up provides tools to discuss intrinsically infinite-
dimensional evolution problems. keywords Information geometry, Gaussian
Orlicz-Sobolev space, Statistical Bundle, Exponential Manifold, Dually Flat
Affine Manifold

###### Contents

  1. 1 Non-parametric statistical bundle
  2. 2 Gaussian Orlicz-Sobolev model space
    1. 2.1 Orlicz spaces
    2. 2.2 Sub-exponential and sub-Gaussian random variables
    3. 2.3 Calculus of the Gaussian space
    4. 2.4 Exponential statistical bundle
    5. 2.5 Orlicz spaces with derivatives
    6. 2.6 Orlicz-Sobolev space with Gaussian weight
  3. 3 Conclusion
    1. 3.1 Sub-exponential random variables
    2. 3.2 Hyvärinen divergence
    3. 3.3 Otto’s metric
    4. 3.4 Boltzmann equation
    5. 3.5 Other possible applications

##  1\. Non-parametric statistical bundle

Professor S.-I. Amari clearly stated in a 1987 conference paper [3] the notion
of a non-parametric fiber bundle in Information Geometry (IG). He says

> _A fibre bundle is constructed on a finite-dimensional parametric
> statistical model with a Hilbert space as the fibre space. The Hilbert space
> represents the tangent directions of the set of probability distributions in
> the function space. A pair of dual linear connections is introduced in the
> Hilbert bundle._

A related journal paper [6] shows the applications to the statistics of semi-
parametric statistical models. R. E. Kass and P. W. Vos, in their 1997
monograph [25, 10.3], further explain the construction by describing the
tangent fiber $T_{p}\mathcal{M}$ of a statistical model $\mathcal{M}$ as a
vector space of random variables:

> _The tangent space $T_{p}\mathcal{M}$ is an inner product space of random
> variables having zero expectation and finite variance that describe
> $\mathcal{M}$ at $p$. Finite-dimensional vector bundles enlarge
> $T_{p}\mathcal{M}$ by adding other random variables, each of which has mean
> zero and finite variance. The Hilbert bundle carries this process to its
> conclusion by enlarging $T_{p}\mathcal{M}$ to the space of all random
> variables having zero mean and finite variance._

I started to work on a non-parametric version of professor Amari’s ideas in
the early nineties [47]. In doing that, the self-imposed prescription to avoid
parameters led to the use of differential geometry as presented by N. Bourbaki
[12] and S. Lang [26]. Specifically, the approach involves the expression of
the tangent spaces as vector spaces of random variables. I give below an
updated summary of such a construction.

If $\mu$ is any probability measure, $\mu\in\mathcal{M}_{1}$, we define as
fiber at $\mu$ the Hilbert space $H_{\mu}$ of $\mu$-square integrable random
variable with zero $\mu$-mean, $H_{\mu}=L^{2}_{0}(\mu)$. The set of all
couples $(\mu,u)$ with $\mu\in\mathcal{M}_{1}$ and $u\in H_{\mu}$ is the
(maximal) (statistical) Hilbert bundle $H\mathcal{M}_{1}$. The restriction to
$\mu$-centered random variables is intended to reflect the affine constraint
$\mu(X)=1$ satisfied by $\mathcal{M}_{1}$ as a subset of the vector space
$\mathcal{S}$ of finite signed measures.

In which sense does each fiber $H_{\mu}$ contain the tangents to statistical
models? Clearly, $M_{1}$ is a convex set whose tangent space is the set
$S_{0}$ of signed finite measure with total charge 0. But there is another
sense to consider. Assume the 1-dimensional statistical model model
$\theta\mapsto\mu(\theta)$ is set-wise continuously differentiable, that is,
for each measurable set $A\in\mathcal{X}$ the real map
$\theta\mapsto\mu(A;\theta)$ is continuously differentiable with derivative
$\theta\mapsto\dot{\mu}(A;\theta)$. Assume that
$\theta\mapsto\dot{\mu}(\theta)$ is a curve in the Banach space
$\mathcal{S}_{0}$ of signed finite measures with 0 total charge and total
variation norm. Clearly, $\dot{\mu}(A;\theta)=0$ if $\dot{\mu}(A;\theta)=0$,
hence $\dot{\mu}(\theta)$ is absolutely continuous with respect to
$\mu(\theta)$. See this argument in [9] and, in full detail, in [27].

The Radon-Nikodym derivative $\ell(\theta)=d\dot{\mu}(\theta)/d\mu(\theta)$ is
such that $\int\ell(\theta)\,d\mu(\theta)=\int d\dot{\mu}(\theta)=0$. If
moreover $\int\ell(\theta)^{2}\,d\mu(\theta)=\int l(\theta)d\dot{\mu}(\theta)$
is finite, then $\ell(\theta)\in H_{\theta}=H_{\mu_{\theta}}$. Under the
classical statistical assumption of a regular likelihood,
$\mu_{\theta}=p_{\theta}\cdot\mu$, $p_{\theta}>0$, it holds

$\ell(\theta)=\frac{\dot{p}_{\theta}}{p_{\theta}}=\frac{d}{d\theta}\log
p_{\theta}$

That is, $\ell_{\theta}$ is the Fisher’s score of the statistical model, a
statistically natural expression of the rate of variation [22, 4.2]. In this
sense, it is an expression of the tangent vector.

Conversely, let be given an element $u\in H_{\mu}$ such that
$\int\mathrm{e}^{\theta u}\,d\mu<\infty$ for all $\theta$ in a neighborhood of
0. The statistical model $\mu(\theta)\propto\mathrm{e}^{\theta u}\cdot\mu$ is
such that the Fisher score at 0 is $u$. This, again, is quite a classical
argument. It depends only on the classical theory of exponential families, see
[14] or [22, 5.5].

Let us fix a reference measure $\mu$ and restrict our attention to the set
$\mathcal{P}\cdot\mu$, $\mathcal{P}$ strictly positive probability densities.
If the sample space is finite, all conditions outlined above are met. The
Hilbert bundle is an expression of the tangent bundle where Fisher’s score
expresses the velocity of one-dimensional curves.

If the sample space is not finite, one must introduce restrictions on both the
set of probabilities and the random variables in the fiber. One option is to
restrict to a convenient subset of positive densities
$\mathcal{E}\subset\mathcal{P}$ in such a way there is a vector space
$B(\mathcal{E})\subset\cap_{\mu\in\mathcal{E}}L^{2}(\mu)$ that contains the
scores of all one-dimensional statistical models. Then, each fiber is defined
to be

$S_{p}\mathcal{E}=\left\\{u\in B(\mathcal{E})\,\middle|\,\int
u\,p\,d\mu=0\right\\}\ .$

For _example_ , assume the sample space is a compact set $K$, and $\mu$ is a
diffuse measure. Let $\mathcal{E}$ be the set of all continuous, strictly
positive probability densities. $\mathcal{E}$ is an open convex set of $C(K)$.
If $\theta\mapsto p_{\theta}$ is a differentiable curve in $\mathcal{E}$, then
the score $\frac{d}{d\theta}\log p_{\theta}$ is a curve in
$B(\mathcal{E})=C(K)$ such that for each $\theta$ it holds
$\int\frac{d}{d\theta}\log p_{\theta}\,d\mu=0$. And conversely, for each $u\in
B_{\theta}$ the curve $\theta\mapsto p_{\theta}\propto\mathrm{e}^{tu}$ has
values in $\mathcal{E}$ and its score at $\theta$ equals $u$. See [10] for an
approach to non-parametric IG based on smooth densities.

A similar but different approach will be used below. Both depend on the idea
of finding a convenient class of random variables $u$ such that the
exponential family proportional to $\mathrm{e}^{\theta u}$ is conveniently
defined.

For my purpose, the most important of professor Amari’s contributions has been
the _definition_ a couple of affine connections in a fully nonparametric way,
[3, Th. 1 and 2]. In the notations used here, he considers two types of
transport between fibers,

(1) $\displaystyle\prescript{\text{e}}{}{\mathbb{U}}_{\mu}^{\nu}\colon
S_{\mu}\mathcal{E}\ni u\mapsto u-\int u\,d\nu\in S_{\nu}\mathcal{E}$ (2)
$\displaystyle\prescript{\text{m}}{}{\mathbb{U}}_{\mu}^{\nu}\colon
S_{\mu}\mathcal{E}\ni u\mapsto\frac{d\mu}{d\nu}u\,\in S_{\nu}\mathcal{E}\ ,$

and proves the duality result

$\left\langle\prescript{\text{e}}{}{\mathbb{U}}_{\mu}^{\nu}u,v\right\rangle_{\nu}=\left\langle
u,\prescript{\text{m}}{}{\mathbb{U}}_{\nu}^{\mu}v\right\rangle_{\mu}\ .$

Also, there is a transport of the inner product from one fiber to the other,

$\left\langle
u,v\right\rangle_{\mu}=\left\langle\prescript{\text{e}}{}{\mathbb{U}}_{\nu}^{\mu}\prescript{\text{e}}{}{\mathbb{U}}_{\mu}^{\nu}u,v\right\rangle_{\mu}=\left\langle\prescript{\text{e}}{}{\mathbb{U}}_{\mu}^{\nu}u,\prescript{\text{m}}{}{\mathbb{U}}_{\mu}^{\nu}v\right\rangle_{\nu}\
.$

The equations above clearly define a geometry of probability measures that is
related but different from the previously studied Riemannian geometry based on
the notion of Fisher-Rao information matrix taken as an expression of an inner
product between tangent vectors. This new geometry originated, a least in the
statistical community, with the idea of defining the geometry of curved
exponential models as embedded in a larger exponential family [55, 20, 21, 2].

Such a theory has been known for a long time in statistical mechanics. The
main difference is that R. Fisher and other statisticians of the same period
used to think about parsimoniously parameterized models. In contrast,
physicists such as Boltzmann and Gibbs used to think in terms of simple
relations between statistical observables. The exponential family appears as a
model with peculiar invariance properties in statistical mechanics.111The
difference in terminology between Mathematical Statistics and Physics is
sometimes confusing. The tutorial [50] and the textbook [8] should be helpful.

Consider, for _example_ , the Hamiltonian $H(y,x)=\frac{y^{2}}{2m}+V(x)$,
$y,x\in\mathbb{R}$. Let the associated flow be $(t,y,x)\mapsto
T_{t}(y,x)\in\mathbb{R}^{2}$, that is,

$\displaystyle\frac{d}{dt}T_{t}(y,x)=\omega H(T_{t}(y,x))\ ,\quad\omega
H=\left(-\frac{\partial}{\partial x}H,\frac{\partial}{\partial y}H\right)\ ,$
$\displaystyle T_{0}(y,x)=(y,x)\ .$

The evolution of a probability measure $f\circ m$, $m(dy,dx)=dydx$, $f>0$,
under the action of the flow is

$\mu(t)=(T_{t})_{\\#}f\cdot m=f\circ S_{t}\det JS_{t}\cdot m=f_{t}\cdot m\
,\quad S_{t}=T_{t}^{-1}\ ,$

provided the flow is a global diffeomorphism. The curve $t\mapsto f_{t}$ is
controlled by the continuity equation,

$\frac{\partial}{\partial t}f_{t}+\omega H\cdot\nabla f_{t}=0\ .$

The score is $d\dot{\mu}_{t}/d\mu_{t}=d\log f_{t}/f_{t}$ and the continuity
equation can be written

$\frac{\partial}{\partial t}\log f_{t}+\omega H\cdot\nabla\log f_{t}=0\ .$

In particular, $f_{t}=f$ if $f$ is a function of $H$. Among all invariant
probability densities, the curve $\theta\mapsto p_{t}=\mathrm{e}^{\theta
H}/Z(\theta)$ represents an evolution in the class of invariant probabilities.
The score of the model is

$\frac{d}{dt}\log\left(\mathrm{e}^{\theta
H}/Z(\theta)\right)=H-\frac{d}{dt}\log Z(t)=H-\int H\,p_{t}\,dm\ .$

In statistical mechanics, the score is interpreted as the fluctuation of the
Hamiltonian.

The differential geometry of the dual collections of H. Nagaoka [7, 8.4]
naturally follows from the various elements discussed above. In particular,
the differential notion of connection can be derived from the notion of
parallel transport in an affine setting. In the non-parametric setting, it is
convenient to base the affine structure on a variation of the original notion
of affine space of [54]. Below is a summary of the presentation in [19].

The word “affine” above refers to the geometrical construction of vectors
associated with displacement according to classical H. Weyl’s _axioms_ of an
affine space. Let be given a set $M$ and a real finite-dimensional vector
space $V$. A displacement mapping is a mapping

$M\times M\ni(P,Q)\mapsto\overrightarrow{PQ}\in V\ ,$

such that

  1. (1)

for each fixed $P$ the partial mapping $s_{P}\colon
Q\mapsto\overrightarrow{PQ}$ is _1-to-1 and onto_ , and

  2. (2)

the parallelogram law,
$\overrightarrow{PQ}+\overrightarrow{QR}=\overrightarrow{PR}$, holds true.

The structure $(M,V,\overrightarrow{\phantom{pq}})$ is, by _definition_ , the
affine space. The corresponding affine manifold is derived from the atlas of
charts $s_{P}\colon M\to V$, $P\in M$. Notice that the change of chart is the
choice of a new origin. Such a structure supports a full geometrical
development, see [36].

Weyl’s axioms suggest the following _definition_.

_Let $M$ be a set and let $B_{\mu}$, $\mu\in M$, be a family of real
topological vector spaces. Let $({\mathbb{U}}_{\nu}^{\mu})$, $\nu,\mu\in M$ be
a family of isomorphism ${\mathbb{U}}_{\nu}^{\mu}\colon B_{\nu}\to B_{\mu}$
satisfying the cocycle condition,_

AF0:

${\mathbb{U}}_{\mu}^{\mu}=I$_and_
${\mathbb{U}}_{\nu}^{\rho}{\mathbb{U}}_{\mu}^{\nu}={\mathbb{U}}_{\mu}^{\rho}$,
where ${\mathbb{U}}_{\nu}^{\mu}$ is the transport from $B_{\nu}$ onto
$B_{\mu}$.

_Consider a displacement mapping_

$\operatorname{\mathbb{S}}\colon(\nu,\mu)\mapsto s_{\nu}(\mu)\in B_{\nu}$

_defined on a subset of the product space
$\operatorname{dom}\left(\operatorname{\mathbb{S}}\right)\subset M\times M$.
Assume_

AF1:

_For each fixed_ $\nu$ the partial mapping $M_{\nu}\ni\mu\mapsto
s_{\nu}(\mu)=\operatorname{\mathbb{S}}(\nu,\mu)$ is injective.

AF2:

$\operatorname{\mathbb{S}}(\mu_{1},\mu_{2})+{\mathbb{U}}_{\mu_{2}}^{\mu_{1}}\operatorname{\mathbb{S}}(\mu_{2},\mu_{3})=\operatorname{\mathbb{S}}(\mu_{1},\mu_{3})$_._

_The structure $(M,(B_{\mu})_{\mu\in M},({\mathbb{U}}_{\nu}^{\mu})_{\mu,\nu\in
M},\operatorname{\mathbb{S}})$ is an affine bundle._

The affine bundle provides a family of candidates to charts $s_{\nu}\colon
M_{\nu}\to B_{\nu}$, $\nu\in M$, to for an atlas. Let
$\left(M,(B_{\mu})_{\mu\in M},({\mathbb{U}}_{\nu}^{\mu})_{\mu,\nu\in
M},\operatorname{\mathbb{S}}\right)$ be an affine space and assume _AF3_ For
each $\nu$, the image set $s_{\nu}(M_{\nu})$ is a neighborhood of 0 in
$B_{\mu}$. That is, its interior $s_{\nu}(M_{\nu})^{\circ}$ is an open set
containing $s_{\nu}(\nu)=0$. Define the coordinates domains as
$U_{\nu}=s_{\nu}^{-1}\left(s_{\nu}(M)^{\circ}\right)$, so that
$(s_{\nu},U_{\nu},B_{\nu})$ is a chart on $M$. Such a chart is said to have
_origin_ $\nu$. Such charts are compatible, and the resulting manifold

$\mathcal{M}=\left(M,(B_{\mu})_{\mu\in
M},({\mathbb{U}}_{\nu}^{\mu})_{\mu,\nu\in M},(s_{\mu})_{\mu\in M}\right)$

is, by _definition_ , the affine manifold associated with the given affine
bundle.

Here is our main instance. Consider the exponential transport of eq. 1 and
define $s_{p}(q)=\log\frac{q}{p}-\int\log\frac{q}{p}\ p\cdot dm$. The
parallelogram identity is

$\left(\log\frac{q}{p}-\int\log\frac{q}{p}\ p\cdot dm\right)+\\\
\left(\log\frac{r}{q}-\int\log\frac{r}{q}\ p\cdot
dm-\int\left(\log\frac{r}{q}-\int\log\frac{r}{q}\ p\cdot dm\right)\
dm\right)=\\\ \left(\log\frac{r}{p}-\int\log\frac{r}{p}\ p\cdot dm\right)$

The inverse of the chart is easily seen to be

$s_{p}^{-1}(u)=\exp\left(u-K_{p}(u)\right)\cdot p\ ,\quad
K_{p}(u)=\log\int\mathrm{e}^{u}\ p\ dm\ ,\quad u\in B_{p}\ .$

The dual instance is associated with the mixture transport of eq. 2 and
$s_{p}(q)=\frac{q}{p}-1$. The parallelogram identity is

$\frac{q}{p}-1+\frac{q}{p}\left(\frac{r}{q}-1\right)=\frac{r}{p}-1\ .$

Given an affine manifold $\mathcal{M}$, the affine bundle is again an affine
manifold. In

$S\mathcal{M}=\left\\{(\mu,v)\,\middle|\,\mu\in M,v\in B_{\mu}\right\\}$

the equation

$S\mathcal{M}\times
S\mathcal{M}\ni((\nu,u),(\mu,v))\mapsto(s_{\nu}(\mu),{\mathbb{U}}_{\mu}^{\nu}v)\in
B_{\nu}\times B_{\nu}$

defines a displacement on the bundle. For each $\nu$ define the chart

$s_{\nu}\colon SM\ni(\mu,v)\mapsto(s_{\nu}(\mu),{\mathbb{U}}_{\mu}^{\nu}v)\in
B_{\nu}\times B_{\nu}$

to define the affine bundle $S\mathcal{M}$ as a manifold. Equivalently, we can
say that $S\mathcal{M}$ is the bundle with trivialization

$s_{\nu}\colon(\mu,v)\mapsto(s_{\nu}(\mu),{\mathbb{U}}_{\mu}^{\nu}v)\ .$

The affine bundle is a convenient expression of the tangent bundle of the
affine manifold if we _define_ the velocity as follows. The velocity of the
smooth curve $t\mapsto\gamma(t)$ of the affine manifold $\mathcal{M}$ is the
curve $t\mapsto(\gamma(t),\accentset{\star}{\gamma}(t))$ of the affine bundle
whose second component is

$\accentset{\star}{\gamma}(t)=\lim_{h\to
0}h^{-1}(s_{\gamma(t)}(\gamma(t+h))=\left.\frac{d}{dh}s_{\gamma(t)}(\gamma(t+h))\right|_{h=0}\
.$

By assumption _AF2_ applied to the points, the expression in the chart
centered at $\nu$ of $\accentset{\star}{\gamma}(t)$ is
${\mathbb{U}}_{\gamma(t)}^{\nu}\accentset{\star}{\gamma}(t)=s_{\nu}(\gamma(t))$.

For _example_ , in the exponential manifold, it holds

$\accentset{\star}{\gamma}(t)=\prescript{\text{e}}{}{\mathbb{U}}_{p}^{\gamma(t)}\frac{d}{dt}\left(\log\frac{\gamma(t)}{p}-\operatorname{\mathbb{E}}_{p}\left[\log\frac{\gamma(t)}{p}\right]\right)=\frac{d}{dt}\log\gamma(t)\
,$

so that the (affine) velocity in the exponential manifold equals Fisher’s
score.

Let $F$ be a section of the affine manifold, that is,
$\mu\mapsto(\mu,F(\mu))\in\mathcal{S}\mathcal{M}$. An _integral curve_ of the
section $F$ is a curve $t\mapsto\gamma(t)$ such that
$\accentset{\star}{\gamma}(t)=F(\gamma(t))$. A _flow_ of the section $F$ is a
mapping

$M\times I\ni(\nu,t)\mapsto\gamma_{t}(\nu)$

such that for each $\nu$ the curve $t\mapsto\gamma_{t}(\nu)$ is an integral
curve and $\gamma(0,\nu)=\nu$.

The following _proposition_ gives a characterization of affine geodesics. _The
following statements are equivalent._

  1. (1)

_The curve_ $I\colon t\mapsto\gamma(t)$ is auto parallel, that is,
$\accentset{\star}{\gamma}(t)={\mathbb{U}}_{\gamma(s)}^{\gamma(t)}\accentset{\star}{\gamma}(s)$,
$s,t\in I$.

  2. (2)

_The expression of the curve in each chart is affine._

  3. (3)

_For all_ $s,t$

$\gamma(t)=S^{-1}_{\gamma(s)}\left((t-s)\accentset{\star}{\gamma}(s)\right)$

The acceleration is _defined_ as a velocity in the affine bundle. Consider the
curve $t\mapsto\mu(t)$ with velocity $t\mapsto\accentset{\star}{\mu}(t)$. The
acceleration $t\mapsto\accentset{**}{\mu}(t)$ is the velocity
$t\mapsto(\mu(t),\accentset{\star}{\mu}(t))$.

$(\accentset{\star}{\mu}(t),\accentset{**}{\mu}(t))=\lim_{h\to
0}h^{-1}s_{\mu(t),\accentset{\star}{\mu}(t)}(\mu(t+h),\accentset{\star}{\mu}(t+h))\
.$

Especially, for all $\mu\in M$,

$\accentset{**}{\mu}(t)={\mathbb{U}}_{\mu}^{\mu(t)}\frac{d}{dt}{\mathbb{U}}_{\mu(t)}^{\mu}\accentset{\star}{\mu}(t)\
.$

This equation shows that a curve with 0 acceleration is auto-parallel.

In the exponential _example_ , the acceleration is computed as follows: In the
exponential case,

$\accentset{**}{p}(t)=\prescript{\text{e}}{}{\mathbb{U}}_{p}^{p(t)}\frac{d}{dt}\prescript{\text{e}}{}{\mathbb{U}}_{p(t)}^{p}\accentset{\star}{p}(t)=\prescript{\text{e}}{}{\mathbb{U}}_{p}^{p(t)}\frac{d}{dt}\left(\frac{\dot{p}(t)}{p(t)}-\int\frac{\dot{p}(t)}{p(t)}\
p\ dm\right)=\\\
\frac{\ddot{p}(t)}{p(t)}-\left(\frac{\dot{p}(t)}{p(t)}\right)^{2}-\int\left(\frac{\ddot{p}(t)}{p(t)}-\left(\frac{\dot{p}(t)}{p(t)}\right)^{2}\right)\
p(t)\ dm=\\\
\frac{\ddot{p}(t)}{p(t)}-\left(\frac{\dot{p}(t)}{p(t)}\right)^{2}+\int\left(\frac{\dot{p}(t)}{p(t)}\right)^{2}\
p(t)\ dm\ .$

##  2\. Gaussian Orlicz-Sobolev model space

Above, we have discussed in general terms how to define an affine Banach
manifold. We now proceed to instantiate the general formalism into a specific
case of Gaussian space. In doing that, the usual toolbox of IG should be
extended with other analytical notions. A general reference is [11]. We now
restrict our attention to a particular instance of model Banach space.
Precisely, we are going to use the generalization of Lebesgue spaces called
Orlicz spaces. General references are the monographs [35, Ch. II] and [1, Ch.
VII]. The basic technical tools is are the notion of conjugation between
convex functions and the analysis of the Gaussian space. I will use my
conference paper [45].

###  2.1. Orlicz spaces

Assume $\phi\in C[0,+\infty[$ is null at 0, $\phi(0)=0$, strictly increasing,
and $\lim_{u\to+\infty}\phi(u)=+\infty$. Let $\Phi$ be its primitive function
with $\Phi(0)=0$. We call such a function a Young function. The inverse
function $\psi=\phi^{-1}$ has the same properties as $\phi$, so that the
primitive $\Psi$ with $\Psi(0)=0$ is again a Young function. The couple
$(\Phi,\Psi)$, is a couple of conjugate Young functions. The relation is
symmetric and we write both $\Psi=\Phi_{*}$ and $\Phi=\Psi_{*}$. The Young
inequality holds true,

$\Phi(x)+\Psi(y)\geq xy\ ,\quad x,y\geq 0\ ,$

and the Legendre equality holds true,

$\Phi(x)+\Psi(\phi(x))=x\phi(x)\ ,\quad x\geq 0\ .$

Here are my notations for specific cases we are going to use:

(3) $\displaystyle\Phi(x)={x^{p}}/p\ ,\quad\Psi(y)={y^{q}}/q\ ,\quad p,q>1\
,\quad 1/p+1/q=1\ ;$ (4) $\displaystyle\exp_{2}(x)=\mathrm{e}^{x}-1-x\
,\quad(\exp_{2})_{*}(y)=(1+y)\log(1+y)-y\ ;$ (5)
$\displaystyle\cosh_{2}(x)=\cosh x-1\
,\quad(\cosh_{2})_{*}(y)=\int_{0}^{y}\sinh^{-1}(v)\ dv\ ;$ (6)
$\displaystyle{\operatorname{gauss}_{2}}(x)=\exp\left(\frac{1}{2}x^{2}\right)-1\
.$

Given a Young function $\Phi$, and a probability measure $\mu$, the Orlicz
space $L_{\Phi}\left(\mu\right)$ is the Banach space whose closed unit ball is

$\left\\{f\in L^{0}(\mu)\,\middle|\,\int\Phi(\left|f\right|)\ d\mu\leq
1\right\\}\ .$

The corresponding Minkowski norm is usually called Luxemburg norm,

$\left\|f\right\|_{L_{\Phi}\left(\mu\right)}=\inf\left\\{\alpha>0\,\middle|\,\int\Phi(\alpha^{-1}\left|f\right|)\
d\mu\leq 1\right\\}\ .$

The Young inequality provides a separating duality $\left\langle
u,v\right\rangle_{\mu}=\int uv\ d\mu$ of $L_{\Phi}\left(\mu\right)$ and
$L_{\Phi_{*}}\left(\mu\right)$ such that $\left\langle
u,v\right\rangle_{\mu}\leq
2\left\|u\right\|_{L_{\Phi}\left(\mu\right)}\left\|v\right\|_{L_{\Phi_{*}}\left(\mu\right)}$.
The dual norm is called the Orlicz norm and is equivalent to the Luxembourg
norm.

Domination relation between Young functions implies continuous injection
properties for the corresponding Orlicz spaces. We say that $\Phi_{2}$
_eventually dominates_ $\Phi_{1}$, written $\Phi_{1}\prec\Phi_{2}$, if there
is a constant $\kappa$ such that $\Phi_{1}(x)\leq\Phi_{2}(\kappa x)$ for all
$x$ larger than some $\bar{x}$. As, in our case, $\mu$ is a probability
measure, the continuous embedding $L_{\Phi_{2}}\left(\mu\right)\to
L_{\Phi_{1}}\left(\mu\right)$ holds if, and only if, $\Phi_{1}\prec\Phi_{2}$.
See proof in [1, Th. 8.2]. If $\Phi_{1}\prec\Phi_{2}$, then
$(\Phi_{2})_{*}\prec(\Phi_{1})_{*}$. Looking at the examples above, $\exp_{2}$
(4) and $\cosh_{2}$ (5) are equivalent, they both are eventually dominated by
$\operatorname{gauss}_{2}$ (6) and eventually dominate all powers (3).

A special case occurs when there exists a function $C$ such that $\Phi(ax)\leq
C(a)\Phi(x)$ for all $a\geq 0$. This is true, for example, for a power
function and in the case of the functions $({\exp_{2}})_{*}$ and
$({\cosh_{2}})_{*}$. In such a case, the conjugate space is the dual space and
and bounded functions form a dense set.

The spaces corresponding to case (3) are ordinary Lebesgue spaces. The cases
(4) and (5) provide isomorphic Banach spaces, which are of special interest to
us as they provide the model spaces for our non-parametric version of IG. In
fact, a random variable $u$ belongs to $L_{\exp_{2}}\left(\mu\right)$ if, and
only if, the exponential family $p_{\theta}\propto\mathrm{e}^{\theta u}$ is
defined in a neighborhood of $\theta=0$. In the conjugate space, a strictly
positive probability density $f$ has finite entropy if, and only if, the
random variable $v=f-1$ belongs to $L_{(\exp_{2})_{*}}\left(\mu\right)$.

###  2.2. Sub-exponential and sub-Gaussian random variables

There is another important feature of the class
$L_{\cosh_{2}}\left(\mu\right)$. Such a class coincides with the class of
_sub-exponential_ random variables, that is, those for which there exist
constants $C_{1},C_{2}>0$ such that the large deviations admit an exponential
bound

$\operatorname{\mathbb{P}}_{\mu}\left(\left|f\right|\geq t\right)\leq
C_{1}\exp\left(-C_{2}t\right)\ ,\quad t\geq 0\ .$

Sub-exponential random variables are of special interest in statistical
applications because they admit explicit exponential bounds in the Law of
Large Numbers. There is a large amount of literature on this subject; see, for
example, [15, 51, 53].

Random variables whose square is sub-exponential are called sub-Gaussian. For
each Young function $\Phi$, the function $\overline{\Phi}(x)=\Phi(x^{2})$ is
again a Young function, and
$\left\|f\right\|_{L_{\overline{\Phi}}\left(\mu\right)}\leq\lambda$ if, and
only if,
$\left\|\left|f\right|^{2}\right\|_{L_{\Phi}\left(\mu\right)}\leq\lambda^{2}$.
We denote the resulting space by $L_{\Phi}^{2}\left(\mu\right)$. For example,
${\operatorname{gauss}_{2}}$ and $\overline{\cosh_{2}}$ are $\prec$-equivalent
, hence the isomorphisn
$L_{{\operatorname{gauss}_{2}}}\left(\mu\right)\leftrightarrow
L_{{\cosh_{2}}}^{2}\left(\mu\right)$.

As an application, consider that for each increasing convex $\Phi$ it holds
$\Phi(fg)\leq\Phi((f^{2}+g^{2})/2)\leq(\Phi(f^{2})+\Phi(g^{2}))/2$. It follows
that when the $L_{2}^{\Phi}\left(\mu\right)$-norm of $f$ and of $g$ is bounded
by one, the $L_{\Phi}\left(\mu\right)$-norm of $f$, $g$, and $fg$, are all
bounded by one. The need to control the product of two random variables in
$L_{{\cosh_{2}}}\left(\mu\right)$ appears, for example, in the study of the
covariant derivatives of the statistical bundle, see [23, 30, 44, 18].

###  2.3. Calculus of the Gaussian space

From now on, the base probability space is the Gaussian probability space
$(\mathbb{R}^{n},\gamma)$,
$\gamma(z)=(2\pi)^{n/2}\exp\left(-\left|z\right|^{2}/2\right)$. Let us recall
a few simple facts about the analysis of the Gaussian space, see [32, Ch. V].

Let us denote by $C^{k}_{\text{poly}}(\mathbb{R}^{n})$, $k=0,1,\dots$, the
vector space of functions which are differentiable up to order $k$ and which
are bounded, together with all derivatives, by a polynomial. This class of
functions is dense in $L^{2}(\gamma)$. For each couple $f,g\in
C^{1}_{\text{poly}}\left(\mathbb{R}^{n}\right)$, we have

$\int{f(x)\ \partial_{i}g(x)}\,\gamma(x)\,dx\,=\int{\delta_{i}f(x)\
g(x)}\,\gamma(x)\,dx\,\ ,$

where the divergence operator $\delta_{i}$ is defined by
$\delta_{i}f(x)=x_{i}f(x)-\partial_{i}f(x)$. Vector case is

$\int{\nabla f(x)\cdot\nabla g(x)}\,\gamma(x)\,dx\,=\int{f(x)\
\delta\cdot\nabla g(x)}\,\gamma(x)\,dx\,\ ,\quad f,g\in
C^{2}_{\text{poly}}\left(\mathbb{R}^{n}\right)\ ,$

with $\delta\cdot\nabla g(x)=x\cdot\nabla g(x)-\Delta g(x)$.

Hermite polynomials $H_{\alpha}=\delta^{\alpha}1$ provide an orthogonal basis
for $L^{2}(\gamma)$ such that $\partial_{i}H_{\alpha}=\alpha_{i}H_{\alpha-
e_{i}}$, $e_{1}$ the $i$-th element of the standard basis of $\mathbb{R}^{n}$.
In turn, this provides a way to prove that there is a closure of both operator
$\partial_{i}$ and $\delta_{i}$ on a domain which is an Hilbert subspace of
$L^{2}(\gamma)$. Such a space is denoted by $D^{2}$ in [32]. Moreover, the
closure of $\partial_{i}$ is the infinitesimal generator of the translation
operator, [33, 11]. The space $D^{2}$ is a Sobolev Space with Gaussian weight
based on the $L^{2}$ norm, [1].

###  2.4. Exponential statistical bundle

We refer to section 1 and [42, 43] for the definition of maximal exponential
manifold $\operatorname{\mathcal{E}}\left(\gamma\right)$, and of statistical
bundle $S\operatorname{\mathcal{E}}\left(\gamma\right)$. Below we report the
results that are necessary in the context of the present paper.

A key result is the proof of the following _proposition_ , see [47, 16, 17]
and [48, Th. 4.7].

_For all $p,q\in\operatorname{\mathcal{E}}\left(\gamma\right)$ it holds
$q=\mathrm{e}^{u-K_{p}(u)}\cdot p$, where $u\in
L_{{\cosh_{2}}}\left(\gamma\right)$,
$\operatorname{\mathbb{E}}_{p}\left[u\right]=0$, and $u$ belongs to the
interior of the proper domain of the convex function $K_{p}$. This property is
equivalent to any of the following:_

  1. (1)

$p$_and_ $q$ are connected by an open exponential arc;

  2. (2)

$L_{{\cosh_{2}}}\left(p\right)=L_{{\cosh_{2}}}\left(q\right)$_and the norms
are equivalent;_

  3. (3)

$p/q\in\cup_{a>1}L^{a}(q)$_and_ $q/p\in\cup_{a>1}L^{a}(p)$.

Item 2 ensures that all the fibers of the statistical bundle, namely
$S_{p}\operatorname{\mathcal{E}}\left(\gamma\right)$,
$p\in\operatorname{\mathcal{E}}\left(\gamma\right)$, are isomorphic. Item 3
gives a explicit description of the exponential manifold. For _example_ , let
$p$ be a positive probability density with respect to $\gamma$, and take $q=1$
and $a=2$. Then a sufficient condition for
$p\in\operatorname{\mathcal{E}}\left(\gamma\right)$ is

$\int{p(x)^{2}}\,\gamma(x)\,dx\,<\infty\quad\text{and}\quad\int{\frac{1}{p(x)}}\,\gamma(x)\,dx\,<\infty\
.$

###  2.5. Orlicz spaces with derivatives

By replacing the $L^{2}$-norm with a ${\cosh_{2}}$-Orlicz norm, a set-up for
IG obtains [29, 43]. Precisely, we have exponential families with weakly
differentiable densities and a Gaussian reference probability measure. The
main outcome is the possibility to discuss topics related to the analytical
picture of IG.

Let us start with a class of inequalities related to the classical Gauss-
Poincaré inequality,

$\int{\left(f(x)-\int{f(y)}\,\gamma(y)\,dy\,\right)^{2}}\,\gamma(x)\,dx\,\leq\int{\left|\nabla
f(x)\right|^{2}}\,\gamma(x)\,dx\,\ ,$

where $f\in C^{1}_{\text{poly}}(\mathbb{R}^{n})$. See a proof in [37, 1.4]. In
terms of norms, the inequality above is equivalent to
$\left\|f-\overline{f}\right\|_{L^{2}(\gamma)}\leq\left\|\left|\nabla
f\right|\right\|_{L^{2}(\gamma)}$, where
$\overline{f}=\int{f(y)}\,\gamma(y)\,dy\,$.

For _example_ , if $p\in C^{2}_{\text{poly}}$ is a probability density with
respect to $\gamma$, then the $\chi^{2}$-divergence of $P=p\cdot\gamma$ from
$\gamma$ is bounded by

$D_{\chi^{2}}(P|\gamma)=\int{(p(x)-1)^{2}}\,\gamma(x)\,dx\,\leq\int{(\delta\cdot\nabla
p(x))^{2}}\,\gamma(x)\,dx\,\ .$

Generalizations of the Gauss-Poincaré inequality follow from the properties of
the Ornstein-Uhlenbeck (OU) semigroup

$P_{t}f(x)=\int{f(\mathrm{e}^{-t}x+\sqrt{1-\mathrm{e}^{-2t}}y)}\,\gamma(y)\,dy\,,\quad
t\geq 0,\quad f\in C^{k}_{\text{poly}}(\mathbb{R}^{n})\ ,$

see [32, V-1.5] and [37, 1.3]. Notice that the OU semigroup interpolates
between $P_{0}f=f$ and $P_{\infty}f=\overline{f}$. If $X$, $Y$ are independent
standard Gaussian random variables in $\mathbb{R}^{n}$, then

$X_{t}=\mathrm{e}^{-t}X+\sqrt{1-\mathrm{e}^{-2t}}Y,\quad
Y_{t}=\sqrt{1-\mathrm{e}^{-2t}}X-\mathrm{e}^{-t}Y$

are independent standard Gaussian random variables for all $t\geq 0$. By the
change of variable $(X,Y)\to(X_{t},Y_{t})$ and Jensen’s inequality, it follows
for each convex $\Phi$ that

$\int{\Phi(P_{t}f(x))}\,\gamma(x)\,dx\,\leq\int{\Phi(f(x))}\,\gamma(x)\,dx\,\
.$

That is, for all $t\geq 0$, the mapping $f\mapsto P_{t}f$ is non-expansive for
the norm of each Orlicz space $L_{\Phi}\left(\gamma\right)$.

For all $\Phi\colon\mathbb{R}$ convex and all $f\in
C^{1}_{\text{poly}}(\mathbb{R}^{n})$, it holds

(7)
$\int{\Phi\left(f(x)-\int{f(y)}\,\gamma(y)\,dy\,\right)}\,\gamma(x)\,dx\,\leq\\\
\iint\Phi\left(\frac{\pi}{2}\nabla f(x)\cdot y\right)\ \gamma(x)\gamma(y)\
dxdy=\\\ \frac{1}{\sqrt{2\pi}}\iint\Phi\left(\frac{\pi}{2}\left|\nabla
f(x)\right|z\right)\ \mathrm{e}^{-z^{2}/2}\gamma(x)\ dzdx=\\\
\int{\widetilde{\Phi}\left(\left|\nabla f(x)\right|\right)}\,\gamma(x)\,dx\,\
,$

where $\widetilde{\Phi}$ is the convex function

$\widetilde{\Phi}(a)=\int{\Phi\left(\frac{\pi}{2}az\right)}\,\gamma(z)\,dz\,\
.$

The first example of convex function is $\Phi(s)=\mathrm{e}^{s}$, with
$\widetilde{\Phi}(a)=\exp\left(\frac{\pi^{2}a^{2}}{8}\right)$, so that the
inequality applied to $\frac{2\kappa}{\pi}f$ becomes

(8)
$\int{\exp\left(\frac{2\kappa}{\pi}\left(f(x)-\overline{f}\right)\right)}\,\gamma(x)\,dx\,\leq\int{\exp\left(\frac{\kappa^{2}}{2}\left|\nabla
f(x)\right|^{2}\right)}\,\gamma(x)\,dx\,$

If the function $f$ is Lipschitz with norm $\kappa^{-1}$, then the RHS is
finite.

The first case of bound for Orlicz norms we is the $\Phi(s)=s^{2p}$, $p>1/2$.
In such a case,

$\widetilde{\Phi}(a)=\left(\frac{\pi}{2}\right)^{2p}m(2p)\ a^{2p}\ ,$

where $m(2p)$ is the $2p$-moment of the standard Gaussian distribution. It
follows that

$\left\|f-\int{f(y)}\,\gamma(y)\,dy\,\right\|_{L^{2p}(\gamma)}\leq\frac{\pi}{2}(m(2p))^{1/2p}\left\|\left|\nabla
f\right|\right\|_{L^{2p}(\gamma)}\ .$

The cases $\Phi(a)=a^{2p}$ are special in that we can use them in the proof
the multiplicative property $\Phi(ab)=\Phi(a)\Phi(b)$. The argument
generalizes to the case where the convex function $\Phi$ is a Young function
whose increase is controlled through a function $C$, $\Phi(uv)\leq
C(u)\Phi(v)$, and such that there exists a $\kappa>0$ for which

$\int{C\left(\frac{\pi}{2}\kappa u\right)}\,\gamma(u)\,du\,\leq 1\ ,$

so that section 2.5 becomes

$\widetilde{\Phi}(\kappa a)=\int{\Phi\left(\frac{\pi}{2}\kappa
az\right)}\,\gamma(z)\,dz\,\leq\int{C\left(\frac{\pi}{2}\kappa
z\right)}\,\gamma(z)\,dz\,\ \Phi(a)\leq\Phi(a)\ .$

By using this bound in eq. 7, we get

$\int{\Phi\left(\kappa\left(f(x)-\int{f(y)}\,\gamma(y)\,dy\,\right)\right)}\,\gamma(x)\,dx\,\leq\int{\Phi\left(\left|\nabla
f(x)\right|\right)}\,\gamma(x)\,dx\,\ .$

Assume now that $\left\|\left|\nabla
f\right|\right\|_{L_{\Phi}\left(\gamma\right)}\leq 1$ so that the LHS does not
exceed 1. Then
$\kappa\left\|f-\overline{f}\right\|_{L_{\Phi}\left(\gamma\right)}\leq 1$,
which, in turn, implies the inequality

$\left\|f-\overline{f}\right\|_{L_{\Phi}\left(\gamma\right)}\leq\kappa^{-1}\left\|\left|\nabla
f\right|\right\|_{L_{\Phi}\left(\gamma\right)}\ .$

It is of particular interest the case of the Young function $\Phi=\cosh-1$,
for which there is no such bound. Instead, we use eq. 8 with $\kappa$ and
$-\kappa$ to get

(9)
$\int{(\cosh-1)\left(\frac{2\kappa}{\pi}\left(f(x)-\overline{f}\right)\right)}\,\gamma(x)\,dx\,\leq\\\
\int{\operatorname{gauss}_{2}\left(\kappa\left|\nabla
f(x)\right|\right)}\,\gamma(x)\,dx\,\ .$

Now, if $\kappa=\left\|\left|\nabla
f\right|\right\|_{L_{\operatorname{gauss}_{2}}\left(\gamma\right)}^{-1}$, then
the LHS is smaller or equal then 1, and hence
$2\kappa/\pi\left\|f-\overline{f}\right\|_{L_{\cosh-1}\left(\gamma\right)}\leq
1$. It follows that

$\left\|f-\overline{f}\right\|_{L_{\cosh-1}\left(\gamma\right)}\leq\frac{\pi}{2}\left\|\left|\nabla
f\right|\right\|_{L_{\operatorname{gauss}_{2}}\left(\gamma\right)}\ .$

In the following _proposition_ , we summarize the inequalities proved so far.

_There exists constants $C_{1}$, $C_{2}(p)$, $C_{3}$ such that for all $f\in
C^{1}_{\text{\emph{poly}}}(\mathbb{R}^{n})$ the following inequalities hold:_

(10)
$\left\|f-\int{f(y)}\,\gamma(y)\,dy\,\right\|_{L_{(\exp_{2})_{*}}(\gamma)}\leq
C_{1}\left\|\left|\nabla f\right|\right\|_{L_{(\exp_{2})_{*}}(\gamma)}\ .$
(11) $\left\|f-\int{f(y)}\,\gamma(y)\,dy\,\right\|_{L^{2p}(\gamma)}\leq
C_{2}(p)\left\|\left|\nabla f\right|\right\|_{L^{2p}(\gamma)}\ ,\quad p>1/2\
.$ (12)
$\left\|f-\int{f(y)}\,\gamma(y)\,dy\,\right\|_{L_{{\cosh_{2}}}\left(\gamma\right)}\leq
C_{3}\left\|\left|\nabla
f\right|\right\|_{L_{\operatorname{gauss}_{2}}\left(\gamma\right)}\ .$

Other equivalent norms could be used in the inequalities above. For example,
$L_{(\exp_{2})_{*}}\left(\gamma\right)\leftrightarrow
L_{(\cosh-1)_{*}}\left(\gamma\right)$ and
$L_{\operatorname{gauss}_{2}}\left(\gamma\right)\leftrightarrow
L_{2}^{\cosh-1}\left(\gamma\right)$.

We now consider a further set of inequalities based on the use of
infinitesimal generator $-\delta\cdot\nabla$ of the OU semigroup [37, 1.3.7].

We have, for all $f\in C^{2}_{\text{poly}}\left(\mathbb{R}^{n}\right)$, that

(13) $f(x)-\overline{f}=-\int_{0}^{\infty}\frac{d}{dt}P_{t}f(x)\
dt=\int_{0}^{\infty}\delta\cdot\nabla P_{t}f(x)\ dt\ .$

Note that

$\nabla
P_{t}f(x)=\nabla\int{f(\mathrm{e}^{-t}x+\sqrt{1-\mathrm{e}^{-2t}}y)}\,\gamma(y)\,dy\,=\\\
\mathrm{e}^{-t}\int{\nabla
f(\mathrm{e}^{-t}x+\sqrt{1-\mathrm{e}^{-2t}}y)}\,\gamma(y)\,dy\,=\mathrm{e}^{-t}P_{t}\nabla
f(x)\ ,$

so that

$P_{t}\delta\cdot\nabla f(x)=\delta\cdot\nabla
P_{t}f(x)=\mathrm{e}^{-t}\delta\cdot P_{t}\nabla f(x)\ .$

Now, eq. 13 becomes

(14) $f(x)-\overline{f}=\int_{0}^{\infty}\mathrm{e}^{-t}\delta\cdot
P_{t}\nabla f(x)\ dt\ .$

As

$\int{\delta\cdot\nabla f(x)}\,\gamma(x)\,dx\,=0\ ,$

the covariance of $f,g\in C^{0}_{\text{poly}}\left(\mathbb{R}^{n}\right)$ is

$\operatorname{Cov}_{\gamma}\left(f,g\right)=\\\
\int{(f(x)-\overline{f})g(x)}\,\gamma(x)\,dx\,=\int{(f(x)-\overline{f})(g(x)-\overline{g})}\,\gamma(x)\,dx\,\
.$

It follows that for all $f,g\in
C^{2}_{\text{poly}}\left(\mathbb{R}^{n}\right)$ we derive from eq. 14

(15)
$\operatorname{Cov}_{\gamma}\left(f,g\right)=\int_{0}^{\infty}\mathrm{e}^{-t}\int{P_{t}\nabla
f(x)\cdot\nabla g(x)}\,\gamma(x)\,dx\,\ dt\ .$

We use here a result of [43, Prop. 5]. Let $\left|\cdot\right|_{1}$ and
$\left|\cdot\right|_{2}$ be two norms on $\mathbb{R}^{n}$, such that
$\left|x\cdot y\right|\leq\left|x\right|_{1}\left|y\right|_{2}$. For a Young
function $\Phi$, consider the norm of $L_{\Phi}\left(\gamma\right)$ and the
conjugate space endowed with the dual norm,

$\left\|f\right\|_{L_{\Psi,*}\left(\gamma\right)}=\sup\left\\{\int fg\
\gamma\,\middle|\,\int\Phi(g)\ \gamma\leq 1\right\\}\ .$

The following _proposition_ includes the standard Poincaré case provided
$\Phi(u)=u^{2}/2$.

_Given a couple of conjugate Young function $\Phi$, $\Psi$, and norms
$\left|\cdot\right|_{1}$, $\left|\cdot\right|_{2}$ on $\mathbb{R}^{n}$ such
that $x\cdot y\leq\left|x\right|_{1}\left|y\right|_{2}$,
$x,y\in\mathbb{R}^{n}$, for all $f,g\in
C^{1}_{\text{\emph{poly}}}(\mathbb{R}^{n})$, it holds_

$\left|\operatorname{Cov}_{\gamma}\left(f,g\right)\right|\leq\left\|\left|\nabla
f\right|_{1}\right\|_{L_{\Phi}\left(\gamma\right)}\left\|\left|\nabla
g\right|_{2}\right\|_{L_{\Psi,*}\left(\gamma\right)}\ .$

The case of our interest here is $\Phi=\cosh-1$, $\Psi=(\cosh-1)_{*}$. As
$(\cos-1)_{*}\prec(\cosh-1)$, it follows, in particular, that
$\operatorname{Cov}_{\gamma}\left(f,f\right)$ is bounded by constant times
$\left\|\left|\nabla f\right|\right\|_{L_{\cosh-1}\left(\gamma\right)}^{2}$.

###  2.6. Orlicz-Sobolev space with Gaussian weight

A reasonable option for our model space is to assume densities
$f=\mathrm{e}^{u-K_{\gamma}(u)}\cdot\gamma$ in the Gaussian maximal
exponential family, $f\in\operatorname{\mathcal{E}}\left(\gamma\right)$, and,
moreover, assume differentiability in the form $u\in
L_{{\cosh_{2}}}^{2}\left(\gamma\right)=L_{{\operatorname{gauss}_{2}}}\left(\gamma\right)$,
that is, $u^{2}\in L_{{\cosh_{2}}}\left(\gamma\right)$, see section 2.2.

Precisely, the exponential and the mixture Orlicz-Sobolev-Gauss (OSG) spaces
of interest are, respectively,

(16) $\displaystyle W^{1,2}_{{\cosh_{2}}}\left(\gamma\right)$
$\displaystyle=\left\\{v\in
L_{{\cosh_{2}}}\left(\gamma\right)\,\middle|\,\partial_{j}v\in
L_{{\operatorname{gauss}_{2}}}\left(\gamma\right)\right\\}\ ,$ (17)
$\displaystyle W^{1,2}_{{\cosh_{2}}_{*}}\left(\gamma\right)$
$\displaystyle=\left\\{\eta\in
L_{{\cosh_{2}}_{*}}\left(\gamma\right)\,\middle|\,\partial_{j}f\in
L_{{\operatorname{gauss}_{2}}_{*}}\left(\gamma\right)\right\\}\ ,$

where $\partial_{j}$, $j=1,\dots,n$, is the partial derivative in the sense of
distributions.

As $\phi\in C_{0}^{\infty}\left(\mathbb{R}^{n}\right)$ implies $\phi\gamma\in
C_{0}^{\infty}\left(\mathbb{R}^{n}\right)$, for each $f\in
W^{1,2}_{{\cosh_{2}}_{*}}\left(\gamma\right)$ it holds

$\left\langle\partial_{j}f,\phi\right\rangle_{\gamma}=\left\langle\partial_{j}f,\phi\gamma\right\rangle=-\left\langle
f,\partial_{j}(\phi\gamma)\right\rangle=\left\langle
f,\gamma(u_{j}-\partial_{j})\phi\right\rangle=\left\langle
f,\delta_{j}\phi\right\rangle_{\gamma}\ ,$

with $\delta_{j}\phi=(u_{j}-\partial_{j})\phi$. Here, the _Stein operator_
$\delta_{i}$ acts on $C_{0}^{\infty}\left(\mathbb{R}^{n}\right)$ [13].

The meaning of both operators $\partial_{j}$ and
$\delta_{j}=(u_{j}-\partial_{j})$ when acting on square-integrable random
variables of the Gaussian space is well known. Still, here we are specifically
interested in the action on OSG spaces. Let us denote by
$C_{\mathrm{p}}^{\infty}\left(\mathbb{R}^{n}\right)$ the space of infinitely
differentiable functions with polynomial growth of all derivatives. Polynomial
growth implies the existence of $\gamma$-moments of all derivatives, hence
$C_{\mathrm{p}}^{\infty}\left(\mathbb{R}^{n}\right)\subset
W^{1,2}_{{\cosh_{2}}_{*}}\left(\gamma\right)$. If $f\in
C_{\mathrm{p}}^{\infty}\left(\mathbb{R}^{n}\right)$, then the distributional
derivative and the ordinary derivative are equal and moreover $\delta_{j}f\in
C_{\mathrm{p}}^{\infty}\left(\mathbb{R}^{n}\right)$. For each $\phi\in
C_{0}^{\infty}\left(\mathbb{R}^{n}\right)$ we have
$\left\langle\phi,\delta_{j}f\right\rangle_{\gamma}=\left\langle\partial_{j}\phi,f\right\rangle_{\gamma}$.

The OSG spaces $W^{1,2}_{{\cosh_{2}}}\left(\gamma\right)$ and
$W^{1,2}_{{\cosh_{2}}_{*}}\left(\gamma\right)$ are both Banach spaces [35,
Sec. 10]. The norm is the graph norm,

$\displaystyle\left\|v\right\|_{W^{1,2}_{{\cosh_{2}}}\left(\gamma\right)}=\left\|v\right\|_{L_{{\cosh_{2}}}\left(\gamma\right)}+\sum_{j=1}^{n}\left\|\partial_{j}v\right\|_{L_{{\operatorname{gauss}_{2}}}\left(\gamma\right)}\
,$
$\displaystyle\left\|\eta\right\|_{W^{1,2}_{{\cosh_{2}}_{*}}\left(\gamma\right)}=\left\|\eta\right\|_{L_{{\cosh_{2}}_{*}}\left(\gamma\right)}+\sum_{j=1}^{n}\left\|\partial_{j}\eta\right\|_{L_{{\operatorname{gauss}_{2}}_{*}}\left(\gamma\right)}\
.$

In the cases of null integral, eq. 12 shows that the second term only provides
an equivalent norm for $W^{1,2}_{{\cosh_{2}}}\left(\gamma\right)$.

We review some relations between OSG spaces and Sobolev spaces without weight
[1] in the following _proposition_. For each ball radius $R>0$,

$(2\pi)^{-\frac{n}{2}}\geq\gamma(x)\geq\gamma(x)(\left|x\right|<R)\geq(2\pi)^{-\frac{n}{2}}\mathrm{e}^{-\frac{R^{2}}{2}}(\left|x\right|<R),\quad
x\in\mathbb{R}^{n}.$

_Let $\Omega_{R}$ denote the open sphere of radius $R>0$ and consider the
restriction $u\mapsto u_{R}$ of $u$ to $\Omega_{R}$._

  1. (1)

_We have the continuous mappings_

$W^{1,(\cosh-1)}\left(\mathbb{R}^{n}\right)\subset
W^{1,(\cosh-1)}\left(\gamma\right)\rightarrow W^{1,p}(\Omega_{R}),\quad p\geq
1.$

  2. (2)

_We have the continuous mappings_

$W^{1,p}(\mathbb{R}^{n})\subset
W^{1,(\cosh-1)_{*}}\left(\mathbb{R}^{n}\right)\subset
W^{1,(\cosh-1)_{*}}\left(\gamma\right)\rightarrow W^{1,1}(\Omega_{R}),\quad
p>1.$

  3. (3)

_Each_ $u\in W^{1,(\cosh-1)}\left(\gamma\right)$ is a.s. Hölder of all orders
on each $\overline{\Omega}_{R}$ and hence a.s. continuous. The restriction
$W^{1,(\cosh-1)}\left(\gamma\right)\to C(\overline{\Omega}_{R})$ is compact.

For _example_ ,if $q=\mathrm{e}^{v-K_{\gamma}(v)}\cdot\gamma$ and
$p=\mathrm{e}^{u-K_{\gamma}(u)}\cdot\gamma$ with
$q,p\in{\mathcal{E}}\left(\gamma\right)$ and $v,u\in
W^{1,2}_{{\cosh_{2}}}\left(\gamma\right)$, the the Hyvärinen divergence is

$\operatorname{DH}\left(p\middle|q\right)=\frac{1}{2}\int{\left\|\nabla(u-v)\right\|^{2}p(x)}\,\gamma(x)\,dx\,<+\infty$

because $\nabla(u-v)\in
L_{{\cosh_{2}}}\left(\gamma\right)=L_{{\cosh_{2}}}\left(p\right)\subset
L^{2}(p\cdot\gamma)$.

##  3\. Conclusion

In this final section, I suggest a few applications of my infinite-dimensional
setup of professor Amari’s ideas I have considered recently, starting from
[29] and [43]. I will conclude by mentioning a few other topics, for which it
is probably possible to extend rigorous results from the finite state space to
the Gaussian space.

###  3.1. Sub-exponential random variables

Let $f\in C^{2}_{\text{poly}}\left(\mathbb{R}^{n}\right)$ and assume $f$ is
globally Lipschitz, that is, $\left|\nabla
f(x)\right|\leq\left\|f\right\|_{\operatorname{Lip}(\mathbb{R}^{n})}$, where
$\left\|f\right\|_{\operatorname{Lip}(\mathbb{R}^{n})}$ is the Lipschitz semi-
norm. It follows from eq. 8 that $f\in L_{{\cosh_{2}}}\left(\gamma\right)$ and
the norm admits a computable bound. If $p$ is any probability density of the
maximal exponential model of $\gamma$, that is, it is connected to 1 by an
open exponential arc, then the proposition in section 2.4 implies that $f\in
L_{{\cosh_{2}}}\left(p\right)$, that is, $f$ is sub-exponential under the
distribution $P=p\cdot\gamma$. If the sequence $(X_{n})_{n=1}^{\infty}$ is an
independent sample of $p\cdot\gamma$, then the sequence of sample means will
converge,

$\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}f(X_{j})=\int{f(x)\
p(x)}\,\gamma(x)\,dx\,\ ,$

with an exponential bound on the tail probability. See [51, 2.8] and [49].

###  3.2. Hyvärinen divergence

I adapt [24], [39], and [5, 13.6.2] to my Gaussian case. Consider the
Hyvärinen divergence of section 2.6 in the Gaussian case, that is,
$P=p\cdot\gamma$ and $Q=q\cdot\gamma$. As a function of $q$, the divergence is

$q\mapsto\operatorname{DH}\left(q\cdot\gamma\middle|p\cdot\gamma\right)=\frac{1}{2}\int{\left|\nabla\log
p(x)\right|^{2}p(x)}\,\gamma(x)\,dx\,+\\\ \frac{1}{2}\int{\left|\nabla\log
q(x)\right|^{2}p(x)}\,\gamma(x)\,dx\,-\int{\nabla\log p(x)\cdot\nabla\log
q(x)\ p(x)}\,\gamma(x)\,dx\,\ ,$

where the first term does not depend on $q$ and the second term is a
$p\cdot\gamma$-expectation. As $\nabla\log p=p^{-1}\nabla p$, the third term
equals

$-\int{\delta\cdot\nabla\log q(x)\ p(x)}\,\gamma(x)\,dx\,\ ,$

which is again a $p$-expectation. To minimize the Hyvärinen divergence we must
minimize the $p$-expected value of the local score

$S(q,x)=\frac{1}{2}\left|\nabla\log q(x)\right|^{2}-\delta\cdot\nabla\log
q(x)$

If $p$ and $q$ belong to the maximal exponential model of $\gamma$, then
$q=\mathrm{e}^{u-K(u)}$ with $u\in L_{{\cosh_{2}}}\left(\gamma\right)$ and
$\int{u(x)}\,\gamma(x)\,dx\,=0$. The local score becomes
$\frac{1}{2}\left|\nabla u\right|^{2}-\delta\cdot\nabla u$. To compute the
$p$-expected value of the score with an independent sample of $p\cdot\gamma$,
we have an interest to assume that the score is in
$L_{{\cosh_{2}}}\left(\gamma\right)$, because this assumption implies the good
convergence of the empirical means for all $p$. Assume, for _example_ ,
$\nabla u\in
L_{{\cosh_{2}}}^{2}\left(\gamma\right)=L_{{\operatorname{gauss}_{2}}}\left(\gamma\right)$.
This implies directly $\left|\nabla u\right|^{2}\in
L_{(\cosh-1)}\left(\gamma\right)$. Moreover, we must assume that the
$L_{{\cosh_{2}}}\left(\gamma\right)$-norm of $\delta\cdot\nabla u$ is finite.
Under such assumptions, one hopes that the minimization of a suitable model of
the sample expectation of the Hyvärinen score is consistent.

###  3.3. Otto’s metric

This metric was originally defined in [38]. Let be given in the maximal
exponential model of $\gamma$,
$p\in\operatorname{\mathcal{E}}\left(\gamma\right)$, and let $f$ and $g$ be
given in the $p$-fiber of the statistical statistical bundle, that is, $f,g\in
W^{1,2}_{{\cosh_{2}}}\left(p\right)=W^{1,2}_{{\cosh_{2}}}\left(\gamma\right)$
and $\int{f(x)\,p}\,\gamma(x)\,dx\,=\int{g(x)\,p}\,\gamma(x)\,dx\,=0$. Otto’s
inner product is

$(f,g)\mapsto\langle\langle f,g\rangle\rangle_{p}=\int{\nabla f(x)\cdot\nabla
g(x)\ p(x)}\,\gamma(x)\,dx\,=\\\ \int{f(x)\ \delta\cdot(p(x)\nabla
g(x))}\,\gamma(x)\,dx\,=\left\langle f,\delta\cdot(p\nabla
g)\right\rangle_{p}\ .$

The LHS is well defined and continuous if $\nabla f,\nabla g\in
L_{{\cosh_{2}}}^{2}\left(\gamma\right)$, because, in such a case,
$\left|\nabla f\right|^{2},\left|\nabla g\right|^{2}\in
L_{{\cosh_{2}}}\left(\gamma\right)=L_{{\cosh_{2}}}\left(p\right)$. The RHS, if
defined, is an inner product in $L_{{\cosh_{2}}}\left(\gamma\right)$. Note
that the mapping $g\mapsto\delta\cdot(p\nabla g)$ is 1-to-1 if $g$ because by
$\int{g(x)p(x)}\,\gamma(x)\,dx\,=0$. The inverse of this mapping provides the
natural gradient of Otto’s inner product in the sense of [4] and [28].

###  3.4. Boltzmann equation

The space-homogeneous Boltzmann operator with angular collision kernel
$B(z,x)=\left|x^{\prime}z\right|$ is discussed, for example, in [52]. I
briefly show below how to use Gaussian Orlicz spaces in this context, see [41]
and [29].

The mixture bundle
$\prescript{*}{}{S}\operatorname{\mathcal{E}}\left(\gamma\right)$ is the set
of all couples $(f,\eta)$ with
$f\in\operatorname{\mathcal{E}}\left(\gamma\right)$, $\eta\in
L_{{\cosh_{2}}_{*}}\left(\gamma\right)$, and
$\int{\eta(x)\,f(x)}\,\gamma(x)\,dx\,=0$. The ∗-notation recalls that the dual
of $L_{{\cosh_{2}}_{*}}\left(\gamma\right)$ is
$L_{{\cosh_{2}}}\left(\gamma\right)$. The Boltzmann operator is the mapping
$\operatorname{\mathcal{E}}\left(\gamma\right)\mapsto Q(f)$ with

$Q(f)(v)=\\\
\int_{\mathbb{R}^{3}}\int_{S^{2}}(f(v-x{x^{\prime}}(v-w))f(w+x{x^{\prime}}(v-w))-f(v)f(w))\left|{x^{\prime}}(v-w)\right|\
\sigma(dx)\ dw\ ,$

where $x^{\prime}$ is the transpose of the column vector $x$, $S_{2}$ is the
unit sphere of $\mathbb{R}^{3}$, and $\sigma$ is the uniform probability on
$S_{2}$. One can prove that $f\mapsto Q(f)/f$ is a section of the mixture
bundle. The Boltzmann equation can be seen as the equation
$\accentset{\star}{f}=Q(f)/f$.

The smoothness of the Boltzmann section follows from a superposition of
operators:

  1. (1)

Product: ${\mathcal{E}}\left(f_{0}\right)\ni f\mapsto f\otimes
f\in{\mathcal{E}}\left(f_{0}\otimes f_{0}\right)$;

  2. (2)

Interaction: ${\mathcal{E}}\left(f_{0}\otimes f_{0}\right)\ni f\otimes
f\mapsto g=Bf\otimes f\in{\mathcal{E}}\left(f_{0}\otimes f_{0}\right)$;

  3. (3)

Conditioning: ${\mathcal{E}}\left(f_{0}\otimes f_{0}\right)\ni
g\mapsto\int_{S_{2}}{g\circ A_{x}}\
{\sigma(dx)}\in{\mathcal{E}}\left(f_{0}\otimes f_{0}\right)$;

  4. (4)

Marginalization.

There is a weak form of the Boltzmann section. Let $v,w$ be a couple of
velocities before collision and let us denote by $(v_{x},w_{x})$ the
velocities after collision, see [52]. For
$f\in\operatorname{\mathcal{E}}\left(\gamma\right)$ and $g\in
L_{{\cosh_{2}}}\left(\gamma\right)$, define the operator $A$ with

$Ag(v,w)=\int_{S^{2}}{\frac{1}{2}(g(v_{x})+g(w_{x}))}\
{\sigma(dx)}-\frac{1}{2}(g(v)+g(w))\ .$

$Ag$ is in $L_{{\cosh_{2}}}\left(\gamma^{\otimes 2}\right)$ and $\left\langle
g,Q(f)/f\right\rangle_{f}=\operatorname{\mathbb{E}}_{f\otimes
f}\left[Ag\right]$.

###  3.5. Other possible applications

The two cases below are open suggestions.

  1. (1)

The transport problem [40] is discussed from the point of view compatible with
IG in [31, 34]. But it does not fit the strictly positive probability
densities assumption. However, suppose the given margins belong to the
exponential manifold. In that case, it is possible to consider the exponential
sub-bundle with the given margins and discuss the gradient flow of the given
optimization problem. This is done in a particular finite case in [46].

  2. (2)

The dual couple $W^{1,2}_{{\cosh_{2}}}\left(\gamma\right)$ and
$W^{1,2}_{{\cosh_{2}}_{*}}\left(\gamma\right)$ define a couple of dual bundles
on the set $\operatorname{\mathcal{E}}\left(\gamma\right)$ where velocity,
acceleration, moment, gradient, Hessian, are all well defined from the affine
structure. A real function on the exponential bundle is a Lagrangian function,
and a real function on the mixture bundle is a Hamiltonian function. The
conjugation relation holds, and the mechanic’s equations provide a dynamic
picture of the statistical bundle. For example, one can consider the
Lagrangian function, where entropy takes the role of the potential energy, and
Fisher’s metric takes the role of the kinetic energy. This was done in the
finite case in [44] and [18].

### Acknowledgments

The author is partially supported by de Castro Statistics and Collegio Carlo
Alberto. He is a member of GNAMPA-INDAM and former faculti of DISMA
Politecnico di Torino.

## References

  * [1] Robert A. Adams and John J. F. Fournier, _Sobolev spaces_ , second ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078 (2009e:46025)
  * [2] Shun-Ichi Amari, _Differential Geometry of Curved Exponential Families-Curvatures and Information Loss_ , The Annals of Statistics 10 (1982), no. 2, 357 – 385.
  * [3] Shun-ichi Amari, _Dual connections on the Hilbert bundles of statistical models_ , Geometrization of statistical theory (Lancaster, 1987) (Lancaster) (C. T. J. Dodson, ed.), ULDM Publ., 1987, pp. 123–151.
  * [4] Shun-Ichi Amari, _Natural gradient works efficiently in learning_ , Neural Computation 10 (1998), no. 2, 251–276.
  * [5] Shun-ichi Amari, _Information geometry and its applications_ , Applied Mathematical Sciences, vol. 194, Springer, Tokyo], 2016. MR 3495836
  * [6] Shun-ichi Amari and Masayuki Kumon, _Estimation in the presence of infinitely many nuisance parameters—geometry of estimating functions_ , Ann. Statist. 16 (1988), no. 3, 1044–1068.
  * [7] Shun-ichi Amari and Hiroshi Nagaoka, _Methods of information geometry_ , Translations of Mathematical Monographs, vol. 191, American Mathematical Society, Providence, RI; Oxford University Press, Oxford, 2000, Translated from the 1993 Japanese original by Daishi Harada. MR 1800071
  * [8] V. I. Arnold, _Mathematical methods of classical mechanics_ , Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989, Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition. MR 1345386
  * [9] Nihat Ay, Jürgen Jost, Hông Vân Lê, and Lorenz Schwachhöfer, _Information geometry_ , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 64, Springer, Cham, 2017. MR 3701408
  * [10] Martin Bauer, Martins Bruveris, and Peter W. Michor, _Uniqueness of the Fisher-Rao metric on the space of smooth densities_ , Bull. Lond. Math. Soc. 48 (2016), no. 3, 499–506. MR 3509909
  * [11] Vladimir I. Bogachev, _Differentiable measures and the Malliavin calculus_ , Mathematical Surveys and Monographs, vol. 164, American Mathematical Society, Providence, RI, 2010. MR 2663405
  * [12] Nicolas Bourbaki, _Variétés differentielles et analytiques. fascicule de résultats / paragraphes 1 à 7_ , Éléments de mathématiques, no. XXXIII, Hermann, Paris, 1971.
  * [13] Haim Brezis, _Functional analysis, Sobolev spaces and partial differential equations_ , Universitext, Springer, New York, 2011. MR 2759829 (2012a:35002)
  * [14] Lawrence D. Brown, _Fundamentals of statistical exponential families with applications in statistical decision theory_ , IMS Lecture Notes. Monograph Series, no. 9, Institute of Mathematical Statistics, Hayward, 1986. MR MR882001 (88h:62018)
  * [15] V. V. Buldygin and Yu. V. Kozachenko, _Metric characterization of random variables and random processes_ , Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000, Translated from the 1998 Russian original by V. Zaiats. MR 1743716
  * [16] Alberto Cena, _Geometric structures on the non-parametric statistical manifold_ , Ph.D. thesis, Università degli Studi di Milano, 2002.
  * [17] Alberto Cena and Giovanni Pistone, _Exponential statistical manifold_ , Ann. Inst. Statist. Math. 59 (2007), no. 1, 27–56. MR MR2396032 (2009b:62011)
  * [18] Goffredo Chirco, Luigi Malagò, and Giovanni Pistone, _Lagrangian and Hamiltonian dynamics for probabilities on the statistical bundle_ , International Journal of Geometric Methods in Modern Physics (2022).
  * [19] Goffredo Chirco and Giovanni Pistone, _Dually affine information geometry modeled on a banach space_ , 2022.
  * [20] Bradley Efron, _Defining the curvature of a statistical problem (with applications to second order efficiency)_ , Ann. Statist. 3 (1975), no. 6, 1189–1242, With a discussion by C. R. Rao, Don A. Pierce, D. R. Cox, D. V. Lindley, Lucien LeCam, J. K. Ghosh, J. Pfanzagl, Niels Keiding, A. P. Dawid, Jim Reeds and with a reply by the author. MR MR0428531 (55 #1552)
  * [21] by same author, _The geometry of exponential families_ , Ann. Statist. 6 (1978), no. 2, 362–376. MR 57 #10890
  * [22] Bradley Efron and Trevor Hastie, _Computer age statistical inference_ , Institute of Mathematical Statistics (IMS) Monographs, vol. 5, Cambridge University Press, New York, 2016, Algorithms, evidence, and data science. MR 3523956
  * [23] Paolo Gibilisco and Giovanni Pistone, _Connections on non-parametric statistical manifolds by Orlicz space geometry_ , IDAQP 1 (1998), no. 2, 325–347. MR 1 628 177
  * [24] Aapo Hyvärinen, _Estimation of non-normalized statistical models by score matching_ , J. Mach. Learn. Res. 6 (2005), 695–709. MR 2249836
  * [25] Robert E. Kass and Paul W. Vos, _Geometrical foundations of asymptotic inference_ , Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, New York, 1997.
  * [26] Serge Lang, _Differential and Riemannian manifolds_ , third ed., Graduate Texts in Mathematics, vol. 160, Springer-Verlag, New York, 1995. MR 96d:53001
  * [27] Hông Vân Lê, _Natural differentiable structures on statistical models and the fisher metric_ , 2022.
  * [28] Wuchen Li and Guido Montúfar, _Natural gradient via optimal transport_ , Information Geometry 1 (2018), no. 2, 181–214.
  * [29] Betrand Lods and Giovanni Pistone, _Information geometry formalism for the spatially homogeneous Boltzmann equation_ , Entropy 17 (2015), no. 6, 4323–4363.
  * [30] John Lott, _Some geometric calculations on Wasserstein space_ , Comm. Math. Phys. 277 (2008), no. 2, 423–437. MR 2358290
  * [31] Luigi Malagò, Luigi Montrucchio, and Giovanni Pistone, _Wasserstein riemannian geometry of gaussian densities_ , Information Geometry 1 (2018), no. 2, 137–179.
  * [32] Paul Malliavin, _Integration and probability_ , Graduate Texts in Mathematics, vol. 157, Springer-Verlag, New York, 1995, With the collaboration of Héléne Airault, Leslie Kay and Gérard Letac, Edited and translated from the French by Kay, With a foreword by Mark Pinsky. MR MR1335234 (97f:28001a)
  * [33] by same author, _Stochastic analysis_ , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 313, Springer-Verlag, Berlin, 1997. MR MR1450093 (99b:60073)
  * [34] Luigi Montrucchio and Giovanni Pistone, _Kantorovich distance on finite metric spaces: Arens–eells norm and cut norms_ , Information Geometry (2021).
  * [35] Julian Musielak, _Orlicz spaces and modular spaces_ , Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983.
  * [36] Katsumi Nomizu and Takeshi Sasaki, _Affine differential geometry: geometry of affine immersions_ , Cambridge Tracts in Mathematics, no. 111, Cambridge University Press, Cambridge, 1994.
  * [37] Ivan Nourdin and Giovanni Peccati, _Normal approximations with Malliavin calculus. from stein’s method to universality_ , Cambridge Tracts in Mathematics, vol. 192, Cambridge University Press, Cambridge, 2012.
  * [38] Felix Otto, _The geometry of dissipative evolution equations: the porous medium equation_ , Comm. Partial Differential Equations 26 (2001), no. 1-2, 101–174. MR 2002j:35180
  * [39] Matthew Parry, A. Philip Dawid, and Steffen Lauritzen, _Proper local scoring rules_ , Ann. Statist. 40 (2012), no. 1, 561–592. MR 3014317
  * [40] Gabriel Peyré and Marco Cuturi, _Computational optimal transport_ , Foundations and Trends in Machine Learning 11 (2019), no. 5–6, 355–607, arXiv:1803.00567.
  * [41] Giovanni Pistone, _Examples of the application of nonparametric information geometry to statistical physics_ , Entropy 15 (2013), no. 10, 4042–4065. MR 3130268
  * [42] by same author, _Nonparametric information geometry_ , Geometric science of information (Frank Nielsen and Frédéric Barbaresco, eds.), Lecture Notes in Comput. Sci., vol. 8085, Springer, Heidelberg, 2013, First International Conference, GSI 2013 Paris, France, August 28-30, 2013 Proceedings, pp. 5–36. MR 3126029
  * [43] by same author, _Information geometry of the Gaussian space_ , Information geometry and its applications, Springer Proc. Math. Stat., vol. 252, Springer, Cham, 2018, pp. 119–155. MR 3876116
  * [44] by same author, _Lagrangian function on the finite state space statistical bundle_ , Entropy 20 (2018), no. 2, 139.
  * [45] by same author, _Information geometry of smooth densities on the gaussian space: Poincaré inequalities_ , pp. 1–17, Springer International Publishing, Cham, 2021.
  * [46] by same author, _Statistical bundle of the transport model_ , Geometric science of information, Lecture Notes in Comput. Sci., vol. 12829, Springer, Cham, [2021] ©2021, pp. 752–759. MR 4424383
  * [47] Giovanni Pistone and Carlo Sempi, _An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one_ , Ann. Statist. 23 (1995), no. 5, 1543–1561. MR 97j:62006
  * [48] Marina Santacroce, Paola Siri, and Barbara Trivellato, _New results on mixture and exponential models by Orlicz spaces_ , Bernoulli 22 (2016), no. 3, 1431–1447. MR 3474821
  * [49] Paola Siri and Barbara Trivellato, _Robust concentration inequalities in maximal exponential models_ , Statistics & Probability Letters 170 (2021), 109001.
  * [50] Leonard Susskind and George Hrabovsky, _The theoretical minimum: What you need to know to start doing physics_ , Basic Books, New York, 2013.
  * [51] Roman Vershynin, _High-dimensional probability: an introduction with applications in data science_ , Cambridge Series in Statistical and Probabilistic Mathematics, vol. 47, Cambridge University Press, Cambridge, 2018, With a foreword by Sara van de Geer. MR 3837109
  * [52] Cd́ric Villani, _A review of mathematical topics in collisional kinetic theory_ , Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 2002, pp. 71–305. MR 1942465 (2003k:82087)
  * [53] Martin J. Wainwright, _High-dimensional statistics: a non-asymptotic viewpoint_ , Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2019.
  * [54] Hermann Weyl, _Space Time Matter_ , Dover, New York, 1952 (eng), translation of the 1921 RAUM ZEIT MATERIE.
  * [55] N. N. Čencov, _Statistical decision rules and optimal inference_ , Translations of Mathematical Monographs, vol. 53, American Mathematical Society, Providence, R.I., 1982, Translation from the Russian edited by Lev J. Leifman. MR MR645898 (83g:62004)


# Ion-acoustic shock waves in magnetized pair-ion plasma

T. Yeashna∗,1, R.K. Shikha∗∗,1, N.A. Chowdhury∗∗∗,2, A. Mannan†,1,3, S.
Sultana‡,1, and A.A. Mamun§,1 1Department of Physics, Jahangirnagar
University, Savar, Dhaka-1342, Bangladesh
2 Plasma Physics Division, Atomic Energy Centre, Dhaka-1000, Bangladesh
3 Institut für Mathematik, Martin Luther Universität Halle-Wittenberg, Halle,
Germany
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A theoretical investigation associated with obliquely propagating ion-acoustic
shock waves (IASHWs) in a three-component magnetized plasma having inertialess
non-extensive electrons, inertial warm positive and negative ions has been
performed. A Burgers equation is derived by employing the reductive
perturbation method. Our plasma model supports both positive and negative
shock structures under the consideration of non-extensive electrons. It is
found that the positive and negative shock wave potentials increase with the
oblique angle ($\delta$) which arises due to the external magnetic field. It
is also observed that the magnitude of the amplitude of positive and negative
shock waves is not effected by the variation of the ion kinematic viscosity
but the steepness of the positive and negative shock waves decreases with ion
kinematic viscosity. The implications of our findings in space and laboratory
plasmas are briefly discussed.

###### keywords:

Pair-ion , Magnetized plasma , Ion-acoustic waves , Perturbation methods ,
Shock waves.

††journal: “The European Physical Journal D”

##  1 Introduction

The pair-ion (PI) plasma can be observed in astrophysical environments such as
upper regions of Titan’s atmosphere [1, 2, 3, 4, 5, 6, 7, 8], cometary comae
[9], ($H^{+}$, $O_{2}^{-}$) and ($H^{+}$, $H^{-}$) plasmas in the D and
F-regions of Earth’s ionosphere [2, 3, 4, 5, 6, 7], and also in the laboratory
experiments namely, ($Ar^{+}$, $F^{-}$) plasma [10], ($K^{+}$, $SF_{6}^{-}$)
plasma [11, 12], neutral beam sources [13], plasma processing reactors [14],
($Ar^{+}$, $SF_{6}^{-}$) plasma [15, 16, 17, 18], combustion products [19],
plasma etching [19], ($Xe^{+}$, $F^{-}$) plasma [20], ($Ar^{+}$, $O_{2}^{-}$)
plasma, and Fullerene ($C_{60}^{+}$, $C_{60}^{-}$) plasma [21, 22, 23], etc.
Positive ions are produced by electron impact ionization, and negative ions
are produced by attachment of the low energy electrons. A number of authors
studied the nonlinear electrostatic structures in PI plasma [3, 4, 5, 6, 7,
8].

Highly energetic particles have been observed in the galaxy clusters [24], the
Earth’s bow-shock [25], in the upper ionosphere of Mars [26], in the vicinity
of the Moon [27], and in the magnetospheres of Jupiter and Saturn [28].
Maxwellian velocity distribution demonstrating the thermally equilibrium state
of particles is not appropriate for explaining the dynamics of these highly
energetic particles. Renyi [29] first introduced the non-extensive
$q$-distribution for explaining the dynamics of these highly energetic
particles, and further development of $q$-distribution has been demonstrated
by Tsallis [30]. The parameter $q$ in the non-extensive $q$-distribution
describes the deviation of the plasma particles from the thermally equilibrium
state. It should be noted that $q=1$ refers to Maxwellian, and $q<1$ ($q>1$)
refers to super-extensivity (sub-extensivity). Jannat et al. [7] investigated
the ion-acoustic (IA) shock waves (IASHWs) in PI plasma in the presence of
non-extensive electrons, and observed that the height of the positive
potential decreases (increases) with positive (negative) ion mass. Hussain et
al. [31] considered inertial PI and inertialess non-extensive electrons and
investigated IASHWs by considering kinematic viscosities of both positive and
negative ion species, and observed that the amplitude of the positive IASHWs
decreases with $q$. Tribeche et al. [32] studied IA solitary waves in a two-
component plasma, and found that the magnitude of the amplitude of positive
and negative solitary structures increases with super-extensive and sub-
extensive electrons.

A plasma medium having considerable dissipative properties dictates the
formation of shock structures [33, 34, 35]. The Landau damping, kinematic
viscosity among the plasma species, and the collision between plasma species
are the major causes of the dissipation which is mainly responsible for the
formation of shock structures in the plasma medium [33, 34, 35]. The presence
of kinematic viscosity plays a pivotal role in generating nonlinear waves [33,
34, 35]. Hafez et al. [33] observed that the steepness of the IASHWs decreases
with the increase of ion kinematic viscosity but the amplitude of IASHWs
remains unchanged. Abdelwahed et al. [34] investigated IASHWs in PI plasma and
reported that the kinematic viscosity coefficient of the ion reduces the
steepness of the IASHWs.

The external magnetic field is to be considered to change the dynamics of the
plasma medium, and associated electrostatic nonlinear structures. Hossen et
al. [35] studied the electrostatic shock structures in magnetized dusty
plasma, and found that the magnitude of the positive and negative shock
profiles increases with the oblique angle ($\delta$) which arises due to the
external magnetic field. El-Labany et al. [8] considered a three-component
plasma model having inertial PI and inertialess non-extensive electrons, and
investigated IASHWs, and found that the amplitude of the positive shock
profile decreases with $q$. To the best knowledge of the authors, no attempt
has been made to study the IASHWs in a three-component magnetized plasma by
considering kinematic viscosities of both inertial warm positive and negative
ion species, and inertialess non-extensive electrons. The aim of the present
investigation is, therefore, to derive Burgers’ equation and investigate
IASHWs in a three-component magnetized PI plasma, and to observe the effects
of various plasma parameters on the configuration of IASHWs.

The outline of the paper is as follows: The basic equations are displayed in
section 2. The Burgers equation has been derived in section 3. Results and
discussion are reported in section 4. A brief conclusion is provided in
section 5.

##  2 Governing equations

We consider a magnetized plasma system comprising inertial negatively and
positively charged warm ions, and inertialess electrons featuring
$q$-distribution. An external magnetic field $\mathbf{B}_{0}$ has been
considered in the system directed along the $z$-axis defining
$\mathbf{B}_{0}=B_{0}\hat{z}$, where $B_{0}$ and $\hat{z}$ are the strength of
the external magnetic field and unit vector directed along the $z$-axis,
respectively. The dynamics of the magnetized PI plasma system is governed by
the following set of equations [36, 37, 38, 39, 40, 41, 42]

$\displaystyle\frac{\partial\tilde{n}_{+}}{\partial\tilde{t}}+\acute{\nabla}\cdot(\tilde{n}_{+}\tilde{u}_{+})=0,$
(1)
$\displaystyle\frac{\partial\tilde{u}_{+}}{\partial\tilde{t}}+(\tilde{u}_{+}\cdot\acute{\nabla})\tilde{u}_{+}=-\frac{Z_{+}e}{m_{+}}\acute{\nabla}\tilde{\psi}+\frac{Z_{+}eB_{0}}{m_{+}}(\tilde{u}_{+}\times\hat{z})$
$\displaystyle\hskip
42.67912pt-\frac{1}{m_{+}n_{+}}\acute{\nabla}P_{+}+\tilde{\eta}_{+}\acute{\nabla}^{2}\tilde{u}_{+},$
(2)
$\displaystyle\frac{\partial\tilde{n}_{-}}{\partial\tilde{t}}+\acute{\nabla}\cdot(\tilde{n}_{-}\tilde{u}_{-})=0,$
(3)
$\displaystyle\frac{\partial\tilde{u}_{-}}{\partial\tilde{t}}+(\tilde{u}_{-}\cdot\acute{\nabla})\tilde{u}_{-}=\frac{Z_{-}e}{m_{-}}\acute{\nabla}\tilde{\psi}-\frac{Z_{-}eB_{0}}{m_{-}}(\tilde{u}_{-}\times\hat{z})$
$\displaystyle\hskip
42.67912pt-\frac{1}{m_{-}\tilde{n}_{-}}\acute{\nabla}P_{-}+\tilde{\eta}_{-}\acute{\nabla}^{2}\tilde{u}_{-},$
(4) $\displaystyle\acute{\nabla}^{2}\tilde{\psi}=4\pi
e[\tilde{n}_{e}+Z_{-}\tilde{n}_{-}-Z_{+}\tilde{n}_{+}],\ $ (5)

where $\tilde{n}_{+}$ ($\tilde{n}_{-}$) is the positive (negative) ion number
density, $m_{+}$ ($m_{-}$) is the positive (negative) ion mass, $Z_{+}$
($Z_{-}$) is the charge state of the positive (negative) ion, $e$ being the
magnitude of electron charge, $\tilde{u}_{+}$ ($\tilde{u}_{-}$) is the
positive (negative) ion fluid velocity, $\tilde{\eta}_{+}$
($\tilde{\eta}_{-}$) is the kinematic viscosity of the positive (negative)
ion, $P_{+}$ ($P_{-}$) is the pressure of positive (negative) ion, and
$\tilde{\psi}$ represents the electrostatic wave potential. Now, we are
introducing normalized variables, namely,
$n_{+}\rightarrow\tilde{n}_{+}/n_{+0}$,
$n_{-}\rightarrow\tilde{n}_{-}/n_{-0}$, and
$n_{e}\rightarrow\tilde{n}_{e}/n_{e0}$, where $n_{-0}$, $n_{+0}$, and $n_{e0}$
are the equilibrium number densities of the negative ions, positive ions, and
electrons, respectively; $u_{+}\rightarrow\tilde{u}_{+}/C_{-}$,
$u_{-}\rightarrow\tilde{u}_{-}/C_{-}$ [where
$C_{-}=(Z_{-}k_{B}T_{e}/m_{-})^{1/2}$, $k_{B}$ being the Boltzmann constant,
and $T_{e}$ being temperature of the electron];
$\psi\rightarrow\tilde{\psi}e/k_{B}T_{e}$; $t=\tilde{t}/\omega_{P_{-}}^{-1}$
[where $\omega_{P_{-}}^{-1}=(m_{-}/4\pi e^{2}Z_{-}^{2}n_{-0})^{1/2}$];
$\nabla=\acute{\nabla}/\lambda_{D}$ [where $\lambda_{D}=(k_{B}T_{e}/4\pi
e^{2}Z_{-}n_{-0})^{1/2}$]. The pressure term of the positive and negative ions
can be recognized as $P_{\pm}=P_{\pm 0}(N_{\pm}/n_{\pm 0})^{\gamma}$ with
$P_{\pm 0}=n_{\pm 0}k_{B}T_{\pm}$ being the equilibrium pressure of the
positive (for $+0$ sign) and negative (for $-0$ sign) ions, and $T_{+}$
($T_{-}$) being the temperature of warm positive (negative) ion, and
$\gamma=(N+2)/N$ (where $N$ is the degree of freedom and for three-dimensional
case $N=3$, then $\gamma=5/3$). For simplicity, we have considered
($\tilde{\eta}_{+}\approx\tilde{\eta}_{-}=\eta$), and $\eta$ is normalized by
$\omega_{p_{-}}\lambda_{D}^{2}$. The quasi-neutrality condition at equilibrium
for our plasma model can be written as $n_{e0}+Z_{-}n_{-0}\approx
Z_{+}n_{+0}$. Equations (1)$-$(5) can be expressed in the normalized form as
[7, 8]:

$\displaystyle\frac{\partial n_{+}}{\partial t}+\nabla\cdot(n_{+}u_{+})=0,$
(6) $\displaystyle\frac{\partial u_{+}}{\partial
t}+(u_{+}\cdot\nabla)u_{+}=-\alpha_{1}\nabla\psi+\alpha_{1}\Omega_{c}(u_{+}\times\hat{z})$
$\displaystyle\hskip 42.67912pt-\alpha_{2}\nabla
n_{+}^{\gamma-1}+\eta\nabla^{2}u_{+},$ (7) $\displaystyle\frac{\partial
n_{-}}{\partial t}+\nabla\cdot(n_{-}u_{-})=0,$ (8)
$\displaystyle\frac{\partial u_{-}}{\partial
t}+(u_{-}\cdot\nabla)u_{-}=\nabla\psi-\Omega_{c}(u_{-}\times\hat{z})$
$\displaystyle\hskip 42.67912pt-\alpha_{3}\nabla
n_{-}^{\gamma-1}+\eta\nabla^{2}u_{-},$ (9)
$\displaystyle\nabla^{2}\psi=\mu_{e}n_{e}-(1+\mu_{e})n_{+}+n_{-}.\ $ (10)

Other plasma parameters are defined as $\alpha_{1}=Z_{+}m_{-}/Z_{-}m_{+}$,
$\alpha_{2}=\gamma T_{+}m_{-}/(\gamma-1)Z_{-}T_{e}m_{+}$, $\alpha_{3}=\gamma
T_{-}/(\gamma-1)Z_{-}T_{e}$, $\mu_{e}=n_{e0}/Z_{-}n_{-0}$, and
$\Omega_{c}=\omega_{c}/\omega_{p_{-}}$ [where $\omega_{c}=Z_{-}eB_{0}/m_{-}$].
Now, the expression for the number density of electrons following non-
extensive $q$-distribution can be written as [8]

$\displaystyle n_{e}=\Big{[}1+(q-1)\psi\Big{]}^{\frac{q+1}{2(q-1)}},\ $ (11)

where the parameter $q$ represents the non-extensive properties of electrons.
We have neglected the effect of the external magnetic field on the non-
extensive electron distribution. This is valid due to the fact that the Larmor
radii of electrons is so small that as if the electrons are flowing along the
magnetic field lines of force. Now, by substituting Eq. (11) into the Eq.
(10), and expanding up to third order in $\psi$, we get

$\displaystyle\nabla^{2}\psi=\mu_{e}+n_{-}-(1+\mu_{e})n_{+}+\sigma_{1}\psi$
$\displaystyle+\sigma_{2}\psi^{2}+\sigma_{3}\psi^{3}+\cdot\cdot\cdot,\ $ (12)

where

$\displaystyle\sigma_{1}=[\mu_{e}(q+1)]/2,~{}~{}~{}~{}\sigma_{2}=[\mu_{e}(q+1)(3-q)]/8,$
$\displaystyle\sigma_{3}=[\mu_{e}(q+1)(3-q)(5-3q)]/48.\ $

We note that the terms containing $\sigma_{1}$, $\sigma_{2}$, and $\sigma_{3}$
are the contribution of $q$-distributed electrons.

##  3 Derivation of the Burgers’ equation

To derive the Burgers’ equation for the IASHWs propagating in a magnetized PI
plasma, first we introduce the stretched co-ordinates [35, 43]

$\displaystyle\xi=\epsilon(l_{x}x+l_{y}y+l_{z}z-v_{p}t),$ (13)
$\displaystyle\tau={\epsilon}^{2}t,\ $ (14)

where $v_{p}$ is the phase speed and $\epsilon$ is a smallness parameter
measuring the weakness of the dissipation ($0<\epsilon<1$). The $l_{x}$,
$l_{y}$, and $l_{z}$ (i.e., $l_{x}^{2}+l_{y}^{2}+l_{z}^{2}=1$) are the
directional cosines of the wave vector $k$ along $x$, $y$, and $z$-axes,
respectively. Then, the dependent variables can be expressed in power series
of $\epsilon$ as [35]

$\displaystyle n_{+}=1+\epsilon
n_{+}^{(1)}+\epsilon^{2}n_{+}^{(2)}+\epsilon^{3}n_{+}^{(3)}+\cdot\cdot\cdot,$
(15) $\displaystyle n_{-}=1+\epsilon
n_{-}^{(1)}+\epsilon^{2}n_{-}^{(2)}+\epsilon^{3}n_{-}^{(3)}+\cdot\cdot\cdot,$
(16) $\displaystyle
u_{+x,y}=\epsilon^{2}u_{+x,y}^{(1)}+\epsilon^{3}u_{+x,y}^{(2)}+\cdot\cdot\cdot,$
(17) $\displaystyle
u_{-x,y}=\epsilon^{2}u_{-x,y}^{(1)}+\epsilon^{3}u_{-x,y}^{(2)}+\cdot\cdot\cdot,$
(18) $\displaystyle u_{+z}=\epsilon
u_{+z}^{(1)}+\epsilon^{2}u_{+z}^{(2)}+\cdot\cdot\cdot,$ (19) $\displaystyle
u_{-z}=\epsilon u_{-z}^{(1)}+\epsilon^{2}u_{-z}^{(2)}+\cdot\cdot\cdot,$ (20)
$\displaystyle\psi=\epsilon\psi^{(1)}+\epsilon^{2}\psi^{(2)}+\cdot\cdot\cdot.\
$ (21)

Now, by substituting Eqs. (13)$-$(21) into Eqs. (6)$-$(9), and (12), and
collecting the terms containing $\epsilon$, the first-order equations reduce
to

$\displaystyle
n_{+}^{(1)}=\frac{3\alpha_{1}l_{z}^{2}}{3v_{p}^{2}-2\alpha_{2}l_{z}^{2}}\psi^{(1)},$
(22) $\displaystyle
u_{+z}^{(1)}=\frac{3v_{p}\alpha_{1}l_{z}}{3v_{p}^{2}-2\alpha_{2}l_{z}^{2}}\psi^{(1)},$
(23) $\displaystyle
n_{-}^{(1)}=-\frac{3l_{z}^{2}}{3v_{p}^{2}-2\alpha_{3}l_{z}^{2}}\psi^{(1)},$
(24) $\displaystyle
u_{-z}^{(1)}=-\frac{3v_{p}l_{z}}{3v_{p}^{2}-2\alpha_{3}l_{z}^{2}}\psi^{(1)}.\
$ (25)

Now, the phase speed of IASHWs can be written as

$\displaystyle v_{p}\equiv
v_{p+}=l_{z}\sqrt{{\frac{-a_{1}+\sqrt{a_{1}^{2}-36\sigma_{1}a_{2}}}{18\sigma_{1}}}},$
(26) $\displaystyle v_{p}\equiv
v_{p-}=l_{z}\sqrt{{\frac{-a_{1}-\sqrt{a_{1}^{2}-36\sigma_{1}a_{2}}}{18\sigma_{1}}}},\
$ (27)

where
$a_{1}=-9-6\alpha_{2}\sigma_{1}-6\alpha_{3}\sigma_{1}-9\alpha_{1}\mu_{e}-9\alpha_{1}$
and
$a_{2}=6\alpha_{2}+4\alpha_{2}\alpha_{3}\sigma_{1}+6\alpha_{1}\alpha_{3}\mu_{e}+6\alpha_{1}\alpha_{3}$.
The $x$ and $y$-components of the first-order momentum equations can be
manifested as

$\displaystyle
u_{+x}^{(1)}=-\frac{3l_{y}v_{p}^{2}}{\Omega_{c}(3v_{p}^{2}-2\alpha_{2}l_{z}^{2})}~{}\frac{\partial\psi^{(1)}}{\partial\xi},$
(28) $\displaystyle
u_{+y}^{(1)}=\frac{3l_{x}v_{p}^{2}}{\Omega_{c}(3v_{p}^{2}-2\alpha_{2}l_{z}^{2})}~{}\frac{\partial\psi^{(1)}}{\partial\xi},$
(29) $\displaystyle
u_{-x}^{(1)}=-\frac{3l_{y}v_{p}^{2}}{\Omega_{c}(3v_{p}^{2}-2\alpha_{3}l_{z}^{2})}~{}\frac{\partial\psi^{(1)}}{\partial\xi},$
(30) $\displaystyle
u_{-y}^{(1)}=\frac{3l_{x}v_{p}^{2}}{\Omega_{c}(3v_{p}^{2}-2\sigma_{3}l_{z}^{2})}~{}\frac{\partial\psi^{(1)}}{\partial\xi}.\
$ (31)

Now, by taking the next higher-order terms, the equation of continuity,
momentum equation, and Poisson’s equation can be written as

$\displaystyle\frac{\partial n_{+}^{(1)}}{\partial\tau}-v_{p}\frac{\partial
n_{+}^{(2)}}{\partial\xi}+l_{x}\frac{\partial
u_{+x}^{(1)}}{\partial\xi}+l_{y}\frac{\partial u_{+y}^{(1)}}{\partial\xi}$
$\displaystyle\hskip 42.67912pt+l_{z}\frac{\partial
u_{+z}^{(2)}}{\partial\xi}+l_{z}\frac{\partial}{\partial\xi}\big{(}n_{+}^{(1)}u_{+z}^{(1)}\big{)}=0,$
(32) $\displaystyle\frac{\partial
u_{+z}^{(1)}}{\partial\tau}-v_{p}\frac{\partial
u_{+z}^{(2)}}{\partial\xi}+l_{z}u_{+z}^{(1)}\frac{\partial
u_{+z}^{(1)}}{\partial\xi}+\alpha_{1}l_{z}\frac{\partial\psi^{(2)}}{\partial\xi}$
$\displaystyle\hskip
14.22636pt+\alpha_{2}l_{z}\frac{\partial}{\partial\xi}\bigg{[}\frac{2}{3}n_{+}^{(2)}-\frac{1}{9}(n_{+}^{(1)})^{2}\bigg{]}-\eta\frac{\partial^{2}u_{+z}^{(1)}}{\partial\xi^{2}}=0,$
(33) $\displaystyle\frac{\partial
n_{-}^{(1)}}{\partial\tau}-v_{p}\frac{\partial
n_{-}^{(2)}}{\partial\xi}+l_{x}\frac{\partial
u_{-x}^{(1)}}{\partial\xi}+l_{y}\frac{\partial u_{-y}^{(1)}}{\partial\xi}$
$\displaystyle\hskip 42.67912pt+l_{z}\frac{\partial
u_{-z}^{(2)}}{\partial\xi}+l_{z}\frac{\partial}{\partial\xi}\big{(}n_{-}^{(1)}u_{-z}^{(1)}\big{)}=0,$
(34) $\displaystyle\frac{\partial
u_{-z}^{(1)}}{\partial\tau}-v_{p}\frac{\partial
u_{-z}^{(2)}}{\partial\xi}+l_{z}u_{-z}^{(1)}\frac{\partial
u_{-z}^{(1)}}{\partial\xi}-l_{z}\frac{\partial\psi^{(2)}}{\partial\xi}$
$\displaystyle\hskip
14.22636pt+\alpha_{3}l_{z}\frac{\partial}{\partial\xi}\bigg{[}\frac{2}{3}n_{-}^{(2)}-\frac{1}{9}(n_{-}^{(1)})^{2}\bigg{]}-\eta\frac{\partial^{2}u_{-z}^{(1)}}{\partial\xi^{2}}=0,$
(35)
$\displaystyle\sigma_{1}\psi^{(2)}+\sigma_{2}{[\psi^{(1)}]}^{2}+n_{-}^{(2)}-(\mu_{e}+1)n_{+}^{(2)}=0.\
$ (36)

Figure 1: The variation of nonlinear coefficient $A$ with $\mu_{e}$ when
$q=1.2$ (left panel), and the variation of nonlinear coefficient $A$ with $q$
when $\mu_{e}=0.3$ (right panel). Other plasma parameters are
$\alpha_{1}=1.5$, $\alpha_{2}=0.2$, $\alpha_{3}=0.02$, $\delta=30^{\circ}$,
and $v_{p}\equiv v_{p+}$.

Finally, the next higher-order terms of Eqs. (6)$-$(9), and (12), with the
help of Eqs. (22)$-$(36), can provide the Burgers equation as

$\displaystyle\frac{\partial\Psi}{\partial\tau}+A\Psi\frac{\partial\Psi}{\partial\xi}=C\frac{\partial^{2}\Psi}{\partial\xi^{2}},\
$ (37)

where $\Psi=\psi^{(1)}$ is used for simplicity. In Eq. (37), the nonlinear
coefficient $A$ and dissipative coefficient $C$ are given by

$\displaystyle
A=\frac{81\alpha_{1}^{2}v_{p}^{2}s_{1}^{3}l_{z}^{4}+F_{1}}{18v_{p}s_{1}l_{z}^{2}s_{2}^{3}+F_{2}},~{}~{}~{}\mbox{and}~{}~{}~{}C=\frac{\eta}{2},\
$ (38)

where

$\displaystyle
F_{1}=81\mu_{e}\alpha_{1}^{2}v_{p}^{2}s_{1}^{3}l_{z}^{4}-81v_{p}^{2}s_{1}^{3}l_{z}^{4}+2\mu_{e}\alpha_{2}\alpha_{1}^{2}s_{1}^{3}l_{z}^{6}$
$\displaystyle+2\alpha_{2}\alpha_{1}^{2}s_{1}^{3}l_{z}^{6}+2\alpha_{3}s_{1}^{3}l_{z}^{6}-2\sigma_{2}s_{1}^{3}s_{2}^{3},$
$\displaystyle
F_{2}=18\alpha_{1}v_{p}s_{2}l_{z}^{2}s_{1}^{3}+18\alpha_{1}\mu_{e}v_{p}s_{2}l_{z}^{2}s_{1}^{3},$
$\displaystyle
s_{1}=3v_{p}^{2}-2\alpha_{3}l_{z}^{2},~{}~{}~{}s_{2}=3v_{p}^{2}-2\alpha_{2}l_{z}^{2}.\
$

Now, we look for stationary shock wave solution of this Burgers’ equation by
considering $\zeta=\xi-U_{0}\tau^{\prime}$ and $\tau=\tau^{\prime}$ (where
$U_{0}$ is the speed of the shock waves in the reference frame). These allow
us to write the stationary shock wave solution as [35, 44, 45]

$\displaystyle\Psi=\Psi_{m}\Big{[}1-\tanh\bigg{(}\frac{\zeta}{\Delta}\bigg{)}\Big{]},\
$ (39)

where the amplitude $\Psi_{m}$ and width $\Delta$ are given by

$\displaystyle\Psi_{m}=\frac{U_{0}}{A},~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}\Delta=\frac{2C}{U_{0}}.\
$ (40)

It is clear from Eqs. (39) and (40) that the IASHWs exist, which are formed
due to the balance between nonlinearity and dissipation, because $C>0$ and the
IASHWs with $\Psi>0$ ($\Psi<0$) exist if $A>0$ ($A<0$) because $U_{0}>0$.

Figure 2: The variation of $\Psi$ with $\zeta$ for different values of
$\delta$ under the consideration $\mu_{e}>\mu_{ec}$. Other plasma parameters
are $\alpha_{1}=1.5$, $\alpha_{2}=0.2$, $\alpha_{3}=0.02$, $\eta=0.3$,
$\mu_{e}=0.3$, $q=1.2$, $U_{0}=0.01$, and $v_{p}\equiv v_{p+}$.

Figure 3: The variation of $\Psi$ with $\zeta$ for different values of
$\delta$ under the consideration $\mu_{e}<\mu_{ec}$. Other plasma parameters
are $\alpha_{1}=1.5$, $\alpha_{2}=0.2$, $\alpha_{3}=0.02$, $\eta=0.3$,
$\mu_{e}=0.15$, $q=1.2$, $U_{0}=0.01$, and $v_{p}\equiv v_{p+}$.

Figure 4: The variation of $\Psi$ with $\zeta$ for different values of $\eta$
under the consideration $\mu_{e}>\mu_{ec}$. Other plasma parameters are
$\alpha_{1}=1.5$, $\alpha_{2}=0.2$, $\alpha_{3}=0.02$, $\delta=30^{\circ}$,
$\eta=0.3$, $\mu_{e}=0.3$, $q=1.2$, $U_{0}=0.01$, and $v_{p}\equiv v_{p+}$.

Figure 5: The variation of $\Psi$ with $\zeta$ for different values of $\eta$
under the consideration $\mu_{e}<\mu_{ec}$. Other plasma parameters are
$\alpha_{1}=1.5$, $\alpha_{2}=0.2$, $\alpha_{3}=0.02$, $\delta=30^{\circ}$,
$\eta=0.3$, $\mu_{e}=0.15$, $q=1.2$, $U_{0}=0.01$, and $v_{p}\equiv v_{p+}$.

Figure 6: The variation of $\Psi$ with $\zeta$ for different values of $q$
under the consideration $\mu_{e}>\mu_{ec}$. Other plasma parameters are
$\alpha_{1}=1.5$, $\alpha_{2}=0.2$, $\alpha_{3}=0.02$, $\delta=30^{\circ}$,
$\eta=0.3$, $\mu_{e}=0.3$, $U_{0}=0.01$, and $v_{p}\equiv v_{p+}$.

Figure 7: The variation of $\Psi$ with $\zeta$ for different values of $q$
under the consideration $\mu_{e}>\mu_{ec}$. Other plasma parameters are
$\alpha_{1}=1.5$, $\alpha_{2}=0.2$, $\alpha_{3}=0.02$, $\delta=30^{\circ}$,
$\eta=0.3$, $\mu_{e}=0.3$, $U_{0}=0.01$, and $v_{p}\equiv v_{p+}$.

Figure 8: The variation of $\Psi$ with $\zeta$ for different values of $q$
under the consideration $\mu_{e}>\mu_{ec}$. Other plasma parameters are
$\alpha_{1}=1.5$, $\alpha_{2}=0.2$, $\alpha_{3}=0.02$, $\delta=30^{\circ}$,
$\eta=0.3$, $\mu_{e}=0.3$, $U_{0}=0.01$, and $v_{p}\equiv v_{p+}$.

Figure 9: The variation of $\Psi$ with $\zeta$ for different values of
$\alpha_{1}$ under the consideration $\mu_{e}>\mu_{ec}$. Other plasma
parameters are $\alpha_{2}=0.2$, $\alpha_{3}=0.02$, $\delta=30^{\circ}$,
$\eta=0.3$, $\mu_{e}=0.3$, $q=1.2$, $U_{0}=0.01$, and $v_{p}\equiv v_{p+}$.

Figure 10: The variation of $\Psi$ with $\zeta$ for different values of
$\alpha_{1}$ under the consideration $\mu_{e}>\mu_{ec}$. Other plasma
parameters are $\alpha_{2}=0.2$, $\alpha_{3}=0.02$, $\delta=30^{\circ}$,
$\eta=0.3$, $\mu_{e}=0.3$, $q=1.2$, $U_{0}=0.01$, and $v_{p}\equiv v_{p+}$.

##  4 Results and discussion

The balance between nonlinearity and dissipation leads to generate IASHWs in a
three-component magnetized PI plasma. We have numerically analyzed the
variation of $A$ with $\mu_{e}$ in the left panel of Fig. 1, and it is obvious
from this figure that (a) $A$ can be negative, zero, and positive depending on
the values of $\mu_{e}$; (b) the value of $\mu_{e}$ for which $A$ becomes zero
is known as critical value of $\mu_{e}$ (i.e., $\mu_{ec}$), and the $\mu_{ec}$
for our present analysis is almost $0.2$; and (c) the parametric regimes for
the formation of positive (i.e., $\psi>0$) and negative (i.e., $\psi<0$)
potential shock structures can be found corresponding to $A>0$ and $A<0$. The
right panel of Fig. 1 describes the variation of $A$ with $q$ when other
plasma parameters are constant and in this case, $A$ becomes zero for the
critical value of $q$ (i.e., $q=q_{c}\simeq 0.7$). The positive (negative)
potential can exist for $q>0.7$ ($q<0.7$) [Figures are not included].

Figures 3 and 3 display the variation of the positive potential shock
structure under the consideration $\mu_{e}>\mu_{ec}$ and negative potential
shock structure under the consideration $\mu_{e}<\mu_{ec}$ with the oblique
angle ($\delta$), respectively. It is clear from these figures that (a) the
magnitude of the amplitude of positive and negative potential structures
increases with an increase in the value of the $\delta$, and this result
agrees with the result of Hossen et al. [35]; (b) the magnitude of the
negative potential is always greater than the positive potential for same
plasma parameters. So, the oblique angle enhances the amplitude of the
potential profiles.

Figures 5 and 5 illustrate the effects of the ion kinematic viscosity on the
positive (under the consideration $\mu_{e}>\mu_{ec}$) and negative (under the
consideration $\mu_{e}<\mu_{ec}$) shock profiles. It is really interesting
that the magnitude of the amplitude of positive and negative shock profiles is
not effected by the variation of the ion kinematic viscosity but the steepness
of the shock profile decreases with ion kinematic viscosity, and this result
agrees with the previous work of Refs. [33, 34].

The effects of the sub-extensive electrons (i.e., $q>1$) on the positive
potential profile can be seen in Fig. 8 under the consideration
$\mu_{e}>\mu_{ec}$. The height of the positive potential decreases with $q$,
and this result is a good agreement with the result of El-Labany et al. [8]
and Hussain et al. [31]. Figures 8 and 8 illustrate the role of super-
extensive electrons (i.e., $q<1$) on the formation of the negative potential
under the consideration $\mu_{e}>\mu_{ec}$, and this is really interesting
that the existence of the super-extensive electron produces negative
potential, and the magnitude of the amplitude of negative potential increases
with $q$. So, the orientation of the potential profiles (positive and
negative) has been organized by the sign of $q$ under the consideration
$\mu_{e}>\mu_{ec}$.

It can be seen from the literature that the PI plasma system can support these
conditions: $m_{-}>m_{+}$ (i.e., $H^{+}-O_{2}^{-}$ [2, 3, 4, 5, 6, 7],
$Ar^{+}-SF_{6}^{-}$ [15, 16, 17, 18], and $Xe^{+}-SF_{6}^{-}$ [15, 16, 17,
18]), $m_{-}=m_{+}$ (i.e., $H^{+}-H^{-}$ [2, 3, 4, 5, 6, 7] and
$C_{60}^{+}-C_{60}^{-}$ [21, 22, 23]), and $m_{-}<m_{+}$ (i.e., $Ar^{+}-F^{-}$
[3, 4]). So, in our present investigation, we have graphically observed the
variation of the electrostatic positive potential with $\alpha_{1}$ under the
consideration of $m_{-}>m_{+}$ (i.e., $\alpha_{1}>1$) and $\mu_{e}>\mu_{ec}$
in Fig. 10, and it is obvious from this figure that (a) the amplitude of the
positive potential decreases with an increase in the value of the negative ion
mass but increases with an increase in the value of the positive ion mass for
a fixed value of their charge state; (b) the height of the IASHWs with
positive potential increases (decreases) with negative (positive) ion charge
state for a constant mass of positive and negative ion species. So, the mass
and charge state of the PI play an opposite role for the formation of positive
shock structure. Figure 10 describes the nature of the electrostatic negative
potential with $\alpha_{1}$ under the consideration of $m_{-}<m_{+}$ (i.e.,
$\alpha_{1}<1$) and $\mu_{e}>\mu_{ec}$. It is clear from this figure that (a)
due to the $m_{-}<m_{+}$ (i.e., $\alpha_{1}<1$), we have observed negative
potential profile even though we have considered $\mu_{e}>\mu_{ec}$ (i.e.,
$A>0$); (b) the existence of the heavy positive ion change the dynamics of the
plasma system; and (c) in this case, the magnitude of the amplitude of
negative potential increases (decreases) with negative (positive) ion mass
when other plasma parameters are constant. So, the dynamics of the PI plasma
rigourously changes with these conditions $m_{-}>m_{+}$ (i.e., $\alpha_{1}>1$)
and $m_{-}<m_{+}$ (i.e., $\alpha_{1}<1$).

##  5 Conclusion

We have studied IASHWs in a three-component magnetized PI plasma by
considering kinematic viscosities of both inertial warm positive and negative
ion species, and inertialess non-extensive electrons. The reductive
perturbation method [46] is used to derive the Burgers’ equation. The results
that have been found from our investigation can be summarized as follows:

  * 1.

The parametric regimes for the formation of positive (i.e., $\psi>0$) and
negative (i.e., $\psi<0$) potential shock structures can be found
corresponding to $A>0$ and $A<0$.

  * 2.

The magnitude of the amplitude of positive and negative shock structures
increases with the oblique angle ($\delta$) which arises due to the external
magnetic field.

  * 3.

The magnitude of the amplitude of positive and negative shock profiles is not
effected by the variation of the ion kinematic viscosity but the steepness of
the shock profile decreases with ion kinematic viscosity.

It may be noted here that the gravitational effect is very important but
beyond the scope of our present work. In future and for better understanding,
someone can investigate the nonlinear propagation in a three-component PI
plasma by considering the gravitational effect. The results of our present
investigation will be useful in understanding the nonlinear phenomena both in
astrophysical environments such as upper regions of Titan’s atmosphere [1, 2,
3, 4, 5, 6, 7, 8], cometary comae [9], ($H^{+}$, $O_{2}^{-}$) and ($H^{+}$,
$H^{-}$) plasmas in the D and F-regions of Earth’s ionosphere [2, 3, 4, 5, 6,
7], and also in the laboratory experiments, namely, ($Ar^{+}$, $F^{-}$) plasma
[10], ($K^{+}$, $SF_{6}^{-}$) plasma [11, 12], neutral beam sources [13],
plasma processing reactors [14], ($Ar^{+}$, $SF_{6}^{-}$) plasma [15, 16, 17,
18], combustion products [19], plasma etching [19], ($Xe^{+}$, $F^{-}$) plasma
[20], ($Ar^{+}$, $O_{2}^{-}$) plasma, and Fullerene ($C_{60}^{+}$,
$C_{60}^{-}$) plasma [21, 22, 23], etc.

## References

  * [1] A.J. Coates, et al., Geophys. Res. Lett. 34, L22103 (2007).
  * [2] H. Massey, Negative Ions, 3rd ed., (Cambridge University Press, Cambridge, 1976).
  * [3] R. Sabry, et al., Phys. Plasmas 16, 032302 (2009).
  * [4] H.G. Abdelwahed, et al., Phys. Plasmas 23, 022102 (2016).
  * [5] A. P. Misra, Phys. Plasmas, 16, 033702 (2009).
  * [6] A. Mushtaq, et al., Phys. Plasmas 19, 042304 (2012).
  * [7] N. Jannat, et al., Commun. Theor. Phys. 64, 479 (2015).
  * [8] S.K. El-Labany, et al., Eur. Phys. J. D 74, 104 (2020); N.A. Chowdhury, et al., Chaos 27, 093105 (2017); N. Ahmed, et al., Chaos 28, 123107 (2018); M. Hassan, et al., Commun. Theor. Phys. 71, 1017 (2019); S. Jahan, et al., Plasma Phys. Rep. 46, 90 (2020).
  * [9] P.H. Chaizy, et al., Nature (London), 349, 393 (1991).
  * [10] Y. Nakamura, I. Tsukabayashi, Phys. Rev. Lett. 52, 2356 (1984).
  * [11] B. Song, et al., Phys. Fluids B 3, 284 (1991).
  * [12] N. Sato, Plasma Sources Sci. Technol. 3, 395 (1994).
  * [13] M. Bacal, G.W. Hamilton, Phys. Rev. Lett. 42, 1538 (1979).
  * [14] R.A. Gottscho, C.E. Gaebe, IEEE Trans. Plasma Sci. 14, 92 (1986).
  * [15] A.Y. Wong, et al., Phys. Fluids 18, 1489 (1975).
  * [16] Y. Nakamura, et al., Plasma Phys. Control. Fusion 39, 105 (1997).
  * [17] J.L. Cooney, et al., Phys. Fluids B 3, 2758 (1991).
  * [18] Y. Nakamura, et al., Phys. Plasmas 6, 3466 (1999).
  * [19] D.P. Sheehan, N. Rynn, Rev. Sci. lnstrum. 59, 8 (1988).
  * [20] R. Ichiki, et al., Phys. Plasmas 9, 4481 (2002).
  * [21] W. Oohara, R. Hatakeyama, Phys. Rev. Lett. 91, 205005 (2003).
  * [22] R. Hatakeyama, W. Oohara, Phys. Scripta 116, 101 (2005).
  * [23] W. Oohara, et al., Phys. Rev. Lett. 95, 175003 (2005).
  * [24] S.H. Hansen, New Astron. 10, 371 (2005).
  * [25] J.R. Asbridge, et al., J. Geophys. Res. 73, 5777 (1968).
  * [26] R. Lundlin, et al., Nature (London) 341, 609 (1989).
  * [27] Y. Futaana, et al., J. Geophys. Res. 108, 1025 (2003).
  * [28] S.M. Krimigis, et al., J. Geophys. Res. 88, 8871 (1983).
  * [29] A. Rényi, Acta Math. Acad. Sci. Hung. 6, 285 (1955).
  * [30] C. Tsallis, J. Stat. Phys. 52, 479 (1988).
  * [31] S. Hussain, et al., Phys. Plasmas 20, 092303 (2013).
  * [32] M. Tribeche, L. Djebarni, R. Amour, Phys. Plasmas 17, 042114 (2010).
  * [33] M.G. Hafez, et al., Plasma Phys. Rep. 43, 499 (2017).
  * [34] H.G. Abdelwahed, et al., J. Exp. Theor. Phys. 122, 1111 (2016).
  * [35] M.M. Hossen, et al., High Energy Density Phys. 24, 9 (2017).
  * [36] A. Atteya, S. Sultana, R. Schlickeiser, Chin. J. Phys. 56, 1931 (2018).
  * [37] N.C. Adhikary, Phys. Lett. A 376, 1460 (2012).
  * [38] A.N. Dev, M.K. Deka, Phys. Plasmas 25, 072117 (2018).
  * [39] A.N. Dev, et al., Chin. Phys. B 25, 105202 (2016).
  * [40] A.N. Dev, et al., Commun. Theor. Phys. 62, 875 (2014).
  * [41] M.K. Deka, A.N. Dev, Plasma Phys. Rep. 44, 965 (2018).
  * [42] B. Sahu, A. Sinha, R. Roychoudhury, Phys. Plasmas 21, 103701 (2014).
  * [43] H. Washimi, T. Taniuti, Phys. Rev. Lett. 17, 996 (1966).
  * [44] V.I. Karpman, Nonlinear Waves in Dispersive Media, (Pergamon Press, Oxford, 1975).
  * [45] A. Hasegawa, Plasma Instabilities and Nonlinear Effects, (Springer-Verlag, Berlin, 1975).
  * [46] M.H. Rahman,et al., Phys. Plasmas 25, 102118 (2018); N.A. Chowdhury, et al., Phys. plasmas 24, 113701 (2017); M.H. Rahman, et al., Chin. J. Phys. 56, 2061 (2018); N.A. Chowdhury, et al., Vacuum 147, 31 (2018); R.K. Shikha, et al., Eur. Phys. J. D 73, 177 (2019); N.A. Chowdhury, et al., Contrib. Plasma Phys. 58, 870 (2018); N.A. Chowdhury, et al., Plasma Phys. Rep. 45, 459 (2019); S.K. Paul, et al., Pramana J. Phys 94, 58 (2020); T.I. Rajib, et al., Phys. plasmas 26, 123701 (2019); S. Jahan, et al., Commun. Theor. Phys. 71, 327 (2019).