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# Nonextensive Itô-Langevin Dynamics

Leonardo Santos<EMAIL_ADDRESS>Naturwissenschaftlich-
Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, 57068 Siegen,
Germany  Departament of Mathematical Physics, Institute of Physics, University
of São Paulo, São Paulo 05508-090, São Paulo, Brazil

###### Abstract

We study generalizations of Itô-Langevin dynamics consistent within
nonextensive thermostatistics. The corresponding stochastic differential
equations are shown to be connected with a wide class of nonlinear Fokker-
Planck equations describing correlated anomalous diffusion in fractals. A
generalized central limit theorem is proposed in order to demonstrate how such
equations emerge as a limit of correlated random variables. In doing so, we
connect microscopic and macroscopic descriptions of correlated anomalous
diffusion in a mathematically sound way and shed some light in explaining why
$q$-Gaussian distributions appear quite often in nature.

Keywords: Nonextensive Statistical Mechanics, Tsallis Distributions, Nonlinear
Fokker-Planck Equation, Langevin Dynamics, Central Limit Theorems.

##  I Introduction

Since the Langevin’s seminal contribution to the Brownian motion theory
Langevin08 , stochastic differential equations (SDEs) have played a
fundamental role in non-equilibrium thermodynamics UO30 ; Kubo66 ; GM13 ; TO15
. The Langevin’s fundamental idea was to describe the Brownian particle motion
in terms of the combination of deterministic forces, coming from damping and
external potentials, and stochastic fluctuations, modeling collisions with
fluid molecules. In a more general perspective, this work introduced a simple
but powerful idea: the description of systems subjected to irregular and
erratic fluctuations (noises) may be much more enlightening when modeled by an
effective stochastic dynamics rather than Hamiltonian dynamics of many-body
systems. Nowadays, Langevin’s equations – and their version in the Itô
stochastic calculus formalism, the Itô-Langevin equations (ILEs) – are
workhorses of stochastic modeling in various branches of science, both in
classical and quantum domains CK96 ; Gardiner ; GZ99 ; BP12 .

The ILEs can be applied to a variety of problems, however their simplifying
power is manifested mainly in the description of memoryless (Markovian)
stochastic dynamics. The paradigmatic example is the diffusion of colloidal
particles under the action of a memoryless noise. From a macroscopic point of
view, the application of Fick’s law together with local conservation of
particles yields the well-known linear Fokker-Planck equation (FPE) Risken .
This system can be equivalently described by a local-in-time SDE, the standard
Itô-Langevin equation

$\displaystyle\mathrm{d}X_{t}=F(X_{t},t)\mathrm{d}t+\sqrt{2D}\mathrm{d}W_{t},$
(1)

where $X_{t}$ is a stochastic process representing the system state, $F$ is
the deterministic force acting on the system, $D$ is the diffusion matrix and
$W_{t}$ is the standard Wiener process. These equations typically yields a
Gaussian distributions. In particular, in the free diffusion regime, the
covariance matrix linearly depends on time. These two properties define the
Gaussian normal diffusion regime, a signature of memoryless dynamics.

The standard ILE (1) provides a satisfactory description for the stochastic
dynamics of a wide variety of systems Kampen81 . This is because we often deal
with situations where the noises come from weak and uncontrollable
interactions with the surrounding environment. In general, such environments
are much larger, complex and chaotic than the analyzed system. This prevents
system information that flowed into the environment from returning at a later
time, in such a way that the system dynamics is (at least approximately)
Markovian. Similarly, invoking the central limit theorem (CLT) Durrett10 , we
would expect a Gaussian process modelling the effective interaction.
Significant deviations from this regime often occur in complex systems
Vemuri78 , which implies that the stochastic dynamics may be neither Markovian
nor Gaussian. In this case, the ILE needs to be generalized, e.g. by
introducing memory kernels and/or multiplicative noises dosSantos19 .

In this paper, we propose a generalization of the standard Itô-Langevin
dynamics consistent within nonextensive (NE) statistical mechanics – a
possible generalization of the Boltzmann-Gibbs (BG) theory based on Tsallis
non-additive entropy Tsallis88 . The motivation for such a generalization is
manifold. Plastino-Plastino PP94  demonstrated that equilibrium distributions
for systems in contact with a finite thermal bath obey the Tsallis statistics
rather than BG (which emerges only in the limit of an infinite thermal bath).
It was demonstrated by Plastino-Plastino PP95  and Tsallis-Bukman TB96  that
normalized scale invariant solutions of a power-law-type nonlinear FPE are
distributions that maximize Tsallis entropy. Such distributions also emerge in
the weak chaos regime K12 ; TB16 ; RTBT17a ; RTBT17b ; CTB22 , in a wide class
of systems featuring long-ranged interactions AT98 ; CAT14 ; CTB14 , in
anomalous transport in optical lattices Lutz03 ; LR13 , anomalous diffusion in
granular media CRSA15 , diffusion of Hydra cell in cellular aggregates URGS01
, finances Borland02 , high-energy experiments CMS10a ; CMS10b ; ALICE10 ,
etc. TsallisTemuco .

Based on the mathematical formalism of $q$-statistics UT22 , we propose a wide
class of SDEs that generalize the ILEs (1). The formalism is based on
generalizations of the standard Wiener process and Itô stochastic calculus, in
such a way that the corresponding SDEs are local-in-time and can be solved
and/or simulated in a similar fashion as Eq. (1). As particular cases, these
equations describe the stochastic dynamics of linear and a wide class of
nonlinear FPEs describing correlated anomalous diffusion in fractals, without
multiplicative noises nor couplings with the macroscopic evolution, extending
ideas presented in Ref. Santos21  and in contrast to the approaches of Refs.
Borland98 ; Frank01 . Furthermore, we demonstrate how such processes can
emerge as a limit of correlated random variables by extending the generalized
CLT by Umarov-Tsallis-Steinberg (UTS) UTS08 . In addition to supporting NE
stochastic dynamics, we believe that this last result also sheds light on
explaining why $q$-Gaussians appear so frequently in nature.

We have organized the paper as follows. In Sec. II we present the mathematical
foundations of our formalism. We have chosen to present such results in a
concise manner where the technical details are discussed in Appendix A. In
Sec. III we discuss our generalization of the Itô-Langevin dynamics. We finish
the paper with a short discussion in Sec. IV.

##  II Mathematical Foundations

###  II.1 $\theta$-Gaussian Distributions

The Tsallis entropy $S_{q}$ of a real random variable $X$, taking values in a
$d$-dimensional space with probability density function (PDF) $p$, is defined
through

$\displaystyle S_{q}[X]=\int_{\mathrm{Supp}\hskip
1.42271ptp}p(x)\log_{q}\frac{1}{p(x)}\mathrm{d}x,$ (2)

where $q$ is a real parameter and $\log_{q}u=(u^{q-1}-1)/(q-1)$, $u\geq 0$, is
the $q$-logarithm. The Boltzmann-Gibbs (BG) entropy is recovered when $q\to
1$. The maximization of $S_{q}$, under appropriated norm and width constrains,
yields a generalized Gaussian distribution often referred to as $q$-Gaussian
distribution CT91 ; Tsallis09 ; GT04

$\displaystyle
G_{q}(x;\mu,\Sigma)=\frac{\exp_{q}\left[-\frac{1}{2_{qd}}(x-\mu)^{\mathrm{T}}\Sigma^{-1}(x-\mu)\right]}{\sqrt{(2_{qd}\pi_{qd})^{d}\det\Sigma}},$
(3)

where $\exp_{q}u=[\max\\{0,1+(1-q)u\\}]^{\frac{1}{1-q}}$ is the
$q$-exponential, and $2_{qd}$ and $\pi_{qd}$ are normalization parameters such
that $2_{qd}\to 2$ and $\pi_{qd}\to\pi$ when $q\to 1$, thus recovering the
usual Gaussian distribution in such a limit. In this paper, we consider a
generalization of $G_{q}$ defined through

$\displaystyle
G_{\theta}(x;\mu,\Sigma)=\frac{\exp_{q}\left\\{-\frac{1}{2_{\theta}}[(x-\mu)^{\mathrm{T}}\Sigma^{-1}(x-\mu)]^{\eta}\right\\}}{\sqrt{(2_{\theta}\hskip
1.42271pt\pi_{\theta})^{d}\det\Sigma}},$ (4)

where $\theta=(q,\eta,d)\in\mathbb{R}^{2}\times\mathbb{N}$. Clearly,
$G_{\theta}\to G_{q}$ when $\eta\to 1$ if we set $2_{\theta}\to 2_{qd}$ and
$\pi_{\theta}\to\pi_{qd}$ in such a limit. We denote $X\sim
N_{\theta}(\mu;\Sigma)$ to refer to a $\theta$-Gaussian random variable with
expectation value $\mu=E[X]$ and covariance matrix
$\Sigma=E[XX^{\mathrm{T}}]$. We shall only consider those values of $\theta$
such that $G_{\theta}$ is normalized and $\mathrm{Tr}\hskip
1.42271pt\Sigma<\infty$. A short calculation shows that such conditions are
fulfilled if, and only if, $\eta>0$ and $(d+2)(q-1)<2\eta$.

Both parameters $q$ and $\eta$ modify the properties of $G_{\theta}$ in
relation to the Gaussian distribution. The parameter $q$ is more significant
in this sense as it breaks the exponential decay when $q>1$ and limits the
support of the distribution when $q<1$. Indeed, for $q\geq 1$, the support of
$G_{\theta}$ is the whole $\mathbb{R}^{d}$ whereas for $q<1$ it has a compact
support given by $\mathrm{Supp}\hskip
1.42271ptG_{\theta}=\\{(x-\mu)^{\mathrm{T}}\Sigma^{-1}(x-\mu)<(1-q)^{-1/\eta}\\}$.
For the one-dimensional case, Eq. (3) reduces to

$\displaystyle G_{\theta}(x;\mu,\sigma^{2})=\frac{1}{\sqrt{2_{\theta}\hskip
1.42271pt\pi_{\theta}\hskip
1.42271pt\sigma^{2}}}\exp_{q}\left(-\frac{|x-\mu|^{2\eta}}{2_{\theta}\hskip
1.42271pt\sigma^{2}}\right),$ (5)

where $\sigma^{2}$ denotes variance. This distribution has a power-law
asymptotic behavior $G_{\theta}\sim x^{\frac{2\eta}{1-q}}$ if $q>1$ and its
support is the interval defined by $|x-\mu|^{\eta}<(1-q)\sigma^{2}$ if $q<1$
(see Fig. 1).

Figure 1: One dimensional $q$-Gaussians with $\mu=0$ and $\sigma^{2}=1$ [Eqs.
(4) and (5)].

###  II.2 Central Limit Theorems

As already mentioned, $\theta$-Gaussian distributions appear in a variety of
situations in complex systems PMMS09 . In the last decades, several models
have been proposed to explain the ubiquity of such distributions in nature
Tsallis01 ; Tsallis05 ; MTG06 . In this paper, we shall demonstrate here how
these PDFs can emerge as the limit of correlated random variables from a
generalization of the central limit theorem (CLT) based on the Umarov-Tsallis-
Steinberg (UTS) approach UTS08  (see also Ref. UTGS10 ). The first definition
we introduce in this direction is that of a $\vartheta$-characteristic
function of a random variable $X$, which is the $\vartheta$-Fourier transform
(FT) of its PDF $p$:

$\displaystyle\textrm{F}^{\hskip
0.71114pt\vartheta}_{X}(k)=\int_{\mathrm{Supp}\hskip
1.42271ptp}p(x)\exp_{q}\left\\{\mathrm{i}\hskip
0.71114pt(k^{\mathrm{T}}x)\hskip
0.71114pt[p(x)]^{\frac{1}{d}(1-\alpha)}\right\\}\mathrm{d}x,$ (6)

where $\vartheta=(\theta,\lambda)$, $\lambda\in\mathbb{R}$, and

$\displaystyle\alpha(\vartheta)=\eta+d\lambda(1-q),$ (7)

$\alpha\equiv\alpha(\vartheta)$. This integral transform generalizes the usual
FT and UTS’ $q$-FT. In what follows two properties of $\textrm{F}^{\hskip
0.71114pt\vartheta}$ shall be relevant. First, the scaling property:

$\displaystyle\textrm{F}_{rX}^{\hskip
0.71114pt\vartheta}(k)=\textrm{F}_{X}^{\hskip
0.71114pt\vartheta}(r^{\alpha}k),$ (8)

$r>0$. The other property is that the $\vartheta$-characteristic function of a
$\theta$-Gaussian random variable $X\sim N_{\theta}(0,\Sigma)$ is of the form

$\displaystyle\textrm{F}^{\hskip
0.71114pt\vartheta}_{X}(k)=\mathrm{Exp}_{\vartheta}\left[-\frac{2_{\theta}}{4(2_{\theta}\pi_{\theta})^{1-\alpha}}\frac{k^{\mathrm{T}}\Sigma
k}{(\det\Sigma)^{\frac{1}{d}(1-\alpha)}}\right],$ (9)

where $\mathrm{Exp}_{\vartheta}(x)$ is a non-negative real function such that
$\mathrm{Exp}_{\vartheta}(0)=1$ (see Appendix A.1).

It is often useful to describe power-law decaying PDFs in terms of escort
distributions and moments BS93 ; TPA09 . We define the escort distribution of
order $s$ of a random variable $X$ with PDF $p$ as

$\displaystyle p_{s}(x)=\frac{[p(x)]^{s}}{v_{s}[p]},$ (10)

where

$\displaystyle v_{s}[p]=\int_{\mathrm{Supp}\hskip
1.42271ptp}[p(x)]^{s}\mathrm{d}x.$ (11)

Similarly, the $s$-expectation value is defined as the expectation value with
respect to the escort distribution of order $s$. A short calculation shows
that the $\vartheta$-FT of $X$ can be expanded in terms of its escort moments:

$\displaystyle\textrm{F}^{\hskip 0.71114pt\vartheta}_{X}(k)=1$
$\displaystyle+$
$\displaystyle\mathrm{i}v_{s_{1}}[p]k^{\mathrm{T}}E_{s_{1}}[X]$ (12)
$\displaystyle-$ $\displaystyle\frac{q\hskip 1.42271ptv_{s_{2}}[p]\hskip
1.42271ptk^{\mathrm{T}}}{2}E_{s_{2}}[XX^{\mathrm{T}}]k+\dots,$

where $s_{n}=1+n(1-\alpha)/d$, $n\geq 1$.

We now introduce a notion of independence for correlated random variables, the
$\vartheta$-independence. Given a sequence of random variables $X_{n}$, taking
values in some $\mathbb{R}^{d}$ with sufficiently well-behaved PDFs, this
sequence is $\vartheta$-independent if for all $n\geq 1$ the
$\vartheta$-characteristic function of the sum $X_{1}+\dots+X_{n}$
$\vartheta$-factorizes as follows:

$\displaystyle\textrm{F}^{\hskip
0.71114pt\vartheta}_{X_{1}+\dots+X_{n}}(k)=\textrm{F}^{\hskip
0.71114pt\vartheta}_{X_{1}}(k)\hskip 1.42271pt\otimes_{\vartheta}\hskip
1.42271pt\dots\otimes_{\vartheta}\hskip 1.42271pt\textrm{F}^{\hskip
0.71114pt\vartheta}_{X_{n}}(k),$ (13)

where $\otimes_{\vartheta}$ is defined as

$\displaystyle
a\otimes_{\vartheta}b=\mathrm{Exp}_{\vartheta}\left(\mathrm{Log}_{\vartheta}\hskip
0.71114pta+\mathrm{Log}_{\vartheta}\hskip 0.71114ptb\right),$ (14)

$a,b>0$ and $\mathrm{Log}_{\vartheta}$ is the $\vartheta$-logarithm, the
inverse of $\mathrm{Exp}_{\vartheta}$. Finally, if a sequence $X_{n}$ of
random variables is such that the sequence of $\vartheta$-characteristic
functions $\textrm{F}_{X_{n}}^{\hskip 0.71114pt\vartheta}$ pointwise converges
to $\textrm{F}_{X}^{\hskip 0.71114pt\vartheta}$, for a random variable $X$, we
then say that $X_{n}$ $\vartheta$-converges to $X$.

We now consider a sequence of random variables $X_{n}$ that are
$\vartheta$-independent and identically distributed with PDF $p$. We assume
that

$\displaystyle\int_{\mathrm{Supp}\hskip
1.42271ptp}\|{x}\|^{2}[p({x})]^{s_{2}}\mathrm{d}{x}<\infty,$ (15)

Then it is possible to demonstrate (see Appendix A.1) that the sequence
$\tilde{X}_{n}$,

$\displaystyle\tilde{X}_{n}=(nz)^{-\frac{1}{2\alpha}}\left(S_{n}-nE_{s_{1}}[X_{n}]\right),$
(16)

$S_{n}=X_{1}+\dots+X_{n}$, $\vartheta$-converges to a $\theta$-Gaussian random
variable ${X}$ with zero expectation value and covariance matrix ${\Sigma}$
satisfying

$\displaystyle\frac{\Sigma}{(\det\Sigma)^{\frac{1-\alpha}{d}}}=\frac{2_{\theta^{\prime}}v_{s_{2}}[p]}{2_{\theta}(2_{\theta}\pi_{\theta})^{\alpha-1}}\left(\frac{\pi_{\theta}s_{2}}{\pi_{\theta^{\prime}}}\right)^{\frac{d}{2}}E_{s_{2}}[X_{1}X_{1}^{\mathrm{T}}],$
(17)

where $\theta^{\prime}=(q^{\prime},\eta,d)$, $q^{\prime}=[s_{2}-(1-q)]/s_{2}$,
$\mathbb{I}_{d}$ denotes the $d\times d$ identity matrix and
$s_{n}=1+n(1-\alpha)/d$. In short, the attractor of $\vartheta$-independent
random variables is the $\theta$-Gaussian distribution. The standard CLT is
recovered when $q=\eta\to 1$. When $\eta=1$ but $q\neq 1$, we have a
multivariate version of UTS $q$-CLT UTS08  (cf. Ref. UT07 ).

###  II.3 Generalized Wiener Processes

The generalized CLT discussed above has several possible consequences worthy
to be explored. From now on, however, we shall focus only on generalizations
of the Wiener processes. In particular, we define $Q_{t}^{\vartheta}$, for
$1\leq q<\min\\{(\lambda d+\eta)/\lambda d,(d+2\eta+2)/(d+2)\\}$, as a
stochastic process in $\mathbb{R}^{d}$ satisfying the following conditions:

  1. (1)

${Q}_{0}^{\vartheta}={0}$ almost surely;

  2. (2)

The paths $t\mapsto{Q}_{t}^{\vartheta}$ are continuous with probability one;

  3. (3)

${Q}_{t}^{\vartheta}-{Q}_{s}^{\vartheta}\sim
N_{\theta}\big{[}0,(t-s)^{\frac{1}{\alpha}}\mathbb{I}_{d}\big{]}$, $0\leq
s\leq t$;

  4. (4)

${Q}_{t}^{\vartheta}-{Q}_{s}^{\vartheta}$, $0\leq s\leq t$, is
${\vartheta}$-independent of ${Q}_{s}^{\vartheta}$;

  5. (5)

Given $t,s\geq 0$, the covariance between the components of
${Q}_{t}^{\vartheta}$ and ${Q}_{s}^{\vartheta}$ satisfies

$\displaystyle 2\hskip 1.42271pt\mathrm{Cov}[Q_{t;\hskip
1.42271pt\mu}^{{\vartheta}},Q_{s;\hskip
1.42271pt\nu}^{{\vartheta}}]=\delta_{\mu\nu}(t^{\frac{1}{\alpha}}+s^{\frac{1}{\alpha}}-|t-s|^{\frac{1}{\alpha}}),$
(18)

where $\delta_{\mu\nu}$ denotes the Kronecker’s delta.

In the Appendix A.2 we demonstrate, for the cases where $\eta=1$, how such
processes emerge as a limit of correlated random variables through the
generalized CLT presented above.

It is easy to verify that the standard Wiener process ${W}_{t}$ corresponds to
$q=\eta\to 1$. These $\vartheta$-generalizations differ from ${W}_{t}$ in
fundamental aspects such as the fact that they have increments that are not
Gaussian nor independent. However, some properties of ${W}_{t}$ are also
present in ${Q}_{t}^{\vartheta}$, e.g. conditions (1) and (2). Also, the paths
of ${Q}_{t}^{\vartheta}$ are in general continuous but not smooth and hence
non-differentiable MP10 . In fact, notice that, given $\Delta t$ and $A$, both
positive, the probability that $\|\Delta{Q}_{t}^{\vartheta}\|/\Delta t$ takes
a value bigger than $A$ is

$\displaystyle\mathrm{Pr}\left[\left\|\frac{\Delta{Q}_{t}^{\vartheta}}{\Delta
t}\right\|>A\right]$
$\displaystyle=\int_{\|{x}\|>A}G_{\theta}\big{(}{x};{0},(\Delta
t)^{\frac{1}{\alpha}-2}\mathbb{I}_{d}\big{)}\mathrm{d}{x}$
$\displaystyle=\int_{\|{u}\|>(\Delta
t)^{1-\frac{1}{2\alpha}}A}G_{\theta}({u};{0},\mathbb{I}_{d})\mathrm{d}{u}.$
(19)

In the limit $\Delta t\to 0$, this probability equals one for any $A>0$ if
$\vartheta=(q,\eta,d,\lambda)$ satisfies

$\displaystyle q<\frac{2\eta-1+2d\lambda}{2d\lambda}.$ (20)

In other words, $\|\Delta{Q}^{\vartheta}\|/\Delta t$ is bigger than any $A>0$
with probability one and thus the paths of ${Q}_{t}^{\vartheta}$ cannot be
differentiable. In addition, the paths of ${Q}_{t}^{\vartheta}$ obey the
following self-similarity law:

$\displaystyle{Q}_{t}^{\vartheta}=c^{-\frac{1}{2\alpha}}{Q}_{t/c}^{\vartheta},$
(21)

for any $c>0$. The last two mentioned properties are also present in the
standard Wiener process.

We finish this section by defining the stochastic integral of a function $f$
with respect to ${Q}_{t}^{\vartheta}$ as a stochastic process ${I}_{t}$,

$\displaystyle{I}_{t}=\int_{0}^{t}f\big{(}{Q}_{\tau}^{\vartheta},\tau\big{)}\mathrm{d}{Q}_{\tau}^{\vartheta},$
(22)

whose PDF is obtained by

$\displaystyle\mathrm{F}^{\hskip
1.42271pt\vartheta}_{{I}_{t}}({k})=\lim_{n\to\infty}\bigotimes_{\vartheta;\hskip
1.42271pt[t_{j-1},t_{j}]\in\mathsf{p}_{n}}\mathrm{F}^{\hskip
1.42271pt\vartheta}_{f({Q}^{\vartheta}_{t_{j-1}},\hskip
1.42271ptt_{j-1})\Delta{Q}_{t_{j}}^{\vartheta}}({k}),$ (23)

where $\mathsf{p}_{n}$ is a sequence of partitions of the interval $[0,t]$
with mesh going to zero,
$\Delta{Q}^{\vartheta}_{t_{j}}\equiv{Q}^{\vartheta}_{t_{j}}-{Q}_{t_{j-1}}^{\vartheta}$,
and the product in the right-hand-side is the $\vartheta$-product indexed by
the elements of $\mathsf{p}_{n}$. This definition generalizes (in
distribution) the usual stochastic integral (cf. Ref. Privault13 ) with
respect to the Wiener process. Moreover, it is possible to demonstrate (see
appendix A.3) that this stochastic integral obeys a generalized Itô formula:

$\displaystyle I_{t}\sim N_{\theta}\left[{0},\hskip
2.84544pt\left(\int_{0}^{t}|f(\tau)|^{2\alpha}\mathrm{d}\tau\right)^{\frac{1}{\alpha}}\mathbb{I}_{d}\right],$
(24)

where $f(t)\in L^{2\alpha}([0,t])$, $t\geq 0$.

##  III Nonextensive Itô-Langevin Equations

In this section we shall discuss a possible generalization of the standard
Itô-Langevin equation (1) describing a stochastic system subjected to a non-
Markovian $\theta$-Gaussian noise. The state of the system is represented by a
stochastic process $X_{t}$ satisfying the following SDE:

$\displaystyle\mathrm{d}X_{t}=F(X_{t},t)\mathrm{d}t+B(t)\mathrm{d}W^{\vartheta}_{t},$
(25)

where $B$ is a (possible time dependent) $d\times d$ matrix,
$W_{t}^{\vartheta}=\sqrt{\varepsilon_{\vartheta}}Q^{\vartheta}_{t}$,
$\varepsilon_{\vartheta}$ is a non-negative number such that
$\varepsilon_{\vartheta}\to 1$ when $q=\eta\to 1$, ensuring that the standard
Wiener process $W_{t}$ and Itô-Langevin equation (1) are recovered in such a
limit. The solutions of this equations can be extracted through the
generalized Itô formula (24):

$\displaystyle\int_{0}^{t}f(t^{\prime})\mathrm{d}W_{t}^{\vartheta}\sim
N_{\theta}\left[0,\hskip
1.42271pt\varepsilon_{\vartheta}\left(\int_{0}^{t}|f(\tau)|^{2\alpha}\mathrm{d}\tau\right)^{\frac{1}{\alpha}}\right].$
(26)

In general it is very difficult to solve Eq. (25) given a general
deterministic force $F$, even in the standard form Gardiner . In this paper,
we shall focus mainly on linear systems, for which we write

$\displaystyle F(x,t)=Ax+b(t),$ (27)

where $A$ is $d\times d$ time independent matrix and $b(t)$ is a
$d$-dimensional time dependent vector. We also make the reasonable assumption
that the eigenvalues of $A$ have negative real part, thence guaranteeing the
stability of the solutions. Plugging Eq. (27) into Eq. (25), we can obtain
(after some manipulations) the following expression

$\displaystyle
X_{t}=\mathrm{e}^{At}X_{0}+\int_{0}^{t}\mathrm{e}^{A(t-t^{\prime})}b(t^{\prime})\mathrm{d}t+\int_{0}^{t}\mathrm{e}^{A(t-\tau)}B(\tau)\mathrm{d}W_{\tau}^{\hskip
0.71114pt\vartheta}.$ (28)

From the application of Eq. (26) in the above equation together with a point
source initial condition centered at $\bar{x}_{0}$, i.e. $X_{0}=\bar{x}_{0}$
(almost surely), we conclude that $X_{t}$ is a $\theta$-Gaussian stochastic
process with expectation value $\bar{x}(t)=E[X_{t}]$ and covariance matrix
$\Sigma(t)=E[X_{t}X_{t}^{\mathrm{T}}]$ satisfying

$\displaystyle\bar{x}(t)=\bar{x}_{0}+\int_{0}^{t}\mathrm{e}^{A(t-\tau)}b(\tau)\mathrm{d}\tau,$
(29)

and

$\displaystyle\Sigma(t)=\varepsilon_{\vartheta}\left(\int_{0}^{t}\left\\{\mathrm{e}^{A(t-\tau)}[2D(\tau)]^{\frac{1}{\alpha}}\mathrm{e}^{A^{\mathrm{T}}(t-\tau)}\right\\}^{\alpha}\mathrm{d}\tau\right)^{\frac{1}{\alpha}},$
(30)

where $D$, $2D=(BB^{\mathrm{T}})^{\alpha}$, is a positive semi-definite
matrix, the diffusion matrix.

In the simplest possible situation we have the free diffusion scenario, where
the system evolves in the absence of deterministic forces (i.e., $F=0$). Under
this assumption, Eq. (30) reduces to

$\displaystyle\Sigma(t)=\varepsilon_{\theta}\left(\int_{0}^{t}2D(\tau)\mathrm{d}\tau\right)^{\frac{1}{\alpha}}.$
(31)

In the particular case where the diffusion matrix is time independent, the
covariance matrix equals

$\displaystyle\Sigma(t)=\varepsilon_{\vartheta}(2Dt)^{\frac{1}{\alpha}}.$ (32)

In summary, in the free diffusion scenario the stochastic system described by
Eq. (25) corresponds to a $\theta$-Gaussian PDF with a power-law time-
dependence in the covariance matrix. These two properties are signatures of a
correlated anomalous diffusion regime, which is characterized by the exponent

$\displaystyle\alpha^{-1}=[\eta+d\lambda(1-q)]^{-1}.$ (33)

Normal Gaussian diffusion corresponds to $q=\eta=1$, which yields
$\alpha^{-1}=1$ in the above equation. Otherwise, the system is super-
diffusive if $\alpha^{-1}>1$, sub-diffusive if $\alpha^{-1}<1$ and normal if
$\alpha^{-1}=1$.

In the absence of a deterministic force, the system will never reach the
equilibrium since the covariance matrix has a power-law dependence on time. If
we consider a linear force $F=Ax+b(t)$, however, the system will eventually
reach the equilibrium whenever $A$ is stable (i.e. its eigenvalues have
negative real part) and $b$ is time independent. In these regime, the
equilibrium expectation is read immediately from Eq. (29):

$\displaystyle\bar{x}_{\textrm{eq}}=\bar{x}_{0}+A^{-1}b.$ (34)

To obtain the equilibrium covariance matrix, notice the time derivative of Eq.
(30) yields

$\displaystyle\frac{\mathrm{d}\Sigma}{\mathrm{d}t}$ $\displaystyle=$
$\displaystyle A\Sigma+\Sigma A^{\mathrm{T}}$ (35) $\displaystyle+$
$\displaystyle\frac{\varepsilon_{\vartheta}^{\alpha}}{\alpha}\mathrm{e}^{At}\left(\mathrm{e}^{-At}\Sigma\mathrm{e}^{-A^{\mathrm{T}}t}\right)^{1-\alpha}\left[\mathrm{e}^{-At}(2D)^{\frac{1}{\alpha}}\mathrm{e}^{-A^{\mathrm{T}}t}\right]^{\alpha}\mathrm{e}^{A^{\mathrm{T}}t},$

where we have assumed a time independent diffusion matrix. Since we are
assuming $A$ stable, the equilibrium covariance matrix is then obtained
through the following matrix equation:

$\displaystyle A\Sigma_{\rm eq}+\Sigma_{\rm
eq}A^{\mathrm{T}}+\frac{2\varepsilon_{\vartheta}^{\alpha}(\Sigma_{\rm
eq})^{1-\alpha}D}{\alpha}=0.$ (36)

For $\alpha=1$, we recover the continuous in time Lyapunov equation
Simoncini16 . $\theta$-Gaussian processes whose expectation value and
covariance matrix are obtained from Eqs. (29) and (30), with both $D$ and $b$
being time independent and $A$ stable, are then NE versions of multivariate
Ohrnstein-Uhlenbeck processes. A particularly relevant case occurs when $A$ is
proportional to the identity. Denoting $A=-\gamma\mathbb{I}_{d}$, Eq. (30)
equals

$\displaystyle\Sigma(t)=\varepsilon_{\vartheta}\bigg{[}\frac{D}{\alpha\gamma}\big{(}1-\mathrm{e}^{-2\alpha\gamma
t}\big{)}\bigg{]}^{\frac{1}{\alpha}},$ (37)

which asymptotically converges to $\Sigma_{\rm
eq}=\varepsilon_{\vartheta}(D/\alpha\gamma)^{\frac{1}{\alpha}}$

We now proceed by showing how these family of generalized ILEs are related
with nonlinear Fokker-Planck equations (FPEs). First, consider $\eta=1$. Under
this restriction, the PDF $p$ is a time dependent $q$-Gaussian distribution
(4) with zero expectation value and covariance matrix satisfying Eq. (32) with
$\alpha=1+d\lambda(1-q)$, if we assume a point-source initial condition
$p(x,0)=\delta(x)$. The PDFs satisfy the following power-law-type nonlinear
FPE:

$\displaystyle\frac{\partial p}{\partial t}+\mathrm{div}\hskip
1.42271pt\left[pF-(v_{2-q}[p])^{1-2\lambda}D\nabla p^{2-q}\right]=0,$ (38)

if we define $\varepsilon_{\vartheta}$ to be

$\displaystyle\varepsilon_{\vartheta}=\frac{1}{2_{qd}\pi_{qd}}\left[2(2-q)\alpha\hskip
1.42271pt\pi_{qd}\right]^{\alpha}\hskip 14.22636pt(\eta=1).$ (39)

In the particular case where $\lambda=1/2$, Eq. (38) reduces to the well-known
porous media equation Vazquez ; Aronson

$\displaystyle\frac{\partial p}{\partial t}=\mathrm{div}\hskip
1.42271pt\big{(}pF-D\nabla p^{2-q}\big{)}.$ (40)

The exponent characterizing the anomalous diffusion is

$\displaystyle\alpha^{-1}=\frac{2}{2+d(1-q)}\hskip
14.22636pt(\eta=2\lambda=1),$ (41)

which was firstly derived by Tsallis-Bukman TB96  through the maximization of
Tsallis non-additive entropy Tsallis88  and experimentally verified within
great precision (2% of error) in a granular media diffusion experiment CRSA15
. The corresponding stochastic process, in turn, was firstly studied in Ref.
Santos21 . If $\lambda\neq 1$, Eq. (38) is related with the Sharma-Mittal
entropy FD00 ; KL12a ; KL12b ; KD21 , a two parameter non-additive
generalization of BG entropy that has Tsallis entropy as a particular case
Frank .

Figure 2: Time correlations for different values of $\alpha$ in the super-
diffusive regime for $d=1$ [Eq. (46)].

Now consider $q=2\lambda=1$ but $\eta\neq 1$. For simplicity, let us assume
that $D$ is proportional to the identity and
$\varepsilon_{\vartheta}=\eta^{2}$. The corresponding generalized ILE yields a
stochastic process $X_{t}$ whose PDF is

$\displaystyle
p(x,t)=\frac{2\eta}{\Gamma(d/2\eta)(4\eta^{2}Dt)^{d/2\eta}}\exp\bigg{(}-\frac{\|x\|^{2\eta}}{4\eta^{2}Dt}\bigg{)},$
(42)

where $D$ is now a positive number. This PDF is the solution of the
O’Shaughnessy-Procaccia equation SP85a ; SP85b

$\displaystyle\frac{\partial p}{\partial
t}=\frac{D}{r^{d-1}}\frac{\partial}{\partial
r}\left(r^{d+1-2\eta}\frac{\partial p}{\partial r}\right).$ (43)

If $d$ is interpreted as a positive real number, it plays the role of a
fractal dimension embedded in some $N$-dimensional Euclidean space. The
anomalous diffusion exponent in this case is $\alpha^{-1}=\eta^{-1}$. If we
consider $q>1$, we can mix the correlated anomalous diffusion regime described
by nonlinear FPEs (38) and O’Shaughnessy-Procaccia diffusion in fractals by
introducing a power-law nonlinearity in Eq. (43). In this case, the solution
is a $\theta$-Gaussian distribution and the anomalous diffusion exponent
$\alpha^{-1}$ equals

$\displaystyle\alpha^{-1}=\frac{2}{2\eta+d(1-q)}.$ (44)

This problem has been studied in detail in Refs. MMPL01 ; PMML02  from the
point of view of FPEs, where $\theta$-Gaussian distributions arise from a
Barenblatt-Pattle-type ansatz (scale invariant solutions) Barenblatt ; Pattle
. Our formalism then provides a microscopic model for this scenario and
extends it with the introduction of the parameter $\lambda$, which may control
the memory of the process.

In the traditional Langevin prescription, Eq. (1) is often written as

$\displaystyle\frac{\mathrm{d}x}{\mathrm{d}t}=F(x,t)+\sqrt{2D}\xi(t),$ (45)

where $\xi(t)=(\xi_{1}(t),\dots,\xi_{d}(t))^{\mathrm{T}}$ is the
$d$-dimensional Gaussian white noise, heuristically defined through
$\xi(t)\mathrm{d}t=\mathrm{d}W_{t}$. The memoryless action of the stochastic
force is read by the fact that $\xi$ exhibits delta correlation, i.e.
$\langle\xi_{a}(t)\xi_{b}(s)\rangle=d\delta_{ab}\delta(t-s)$. It is possible
to define the NE analog of $\xi$,
$\xi^{\vartheta}(t)=(\xi_{1}^{\vartheta}(t),\dots,\xi_{d}^{\vartheta}(t))^{\mathrm{T}}$,
as $\xi^{\vartheta}(t)\mathrm{d}t=\mathrm{d}W_{t}^{\vartheta}$. Now, we have
memory effects characterized by the fact that the noise $\xi^{\vartheta}$ is
not $\delta$-correlated but instead

$\displaystyle\langle\xi_{a}^{\vartheta}(t)\xi_{b}^{\vartheta}(s)\rangle=d\delta_{ab}\frac{\varepsilon_{\vartheta}(1-\alpha)}{4\alpha^{2}}|t-s|^{-\frac{2\alpha-1}{\alpha}},$
(46)

as a consequence of Eq. (18). The “tail” presented in these correlators
quantifies the temporal correlations of the noise (see Fig. 2). Furthermore,
the signal of $\langle\xi_{a}^{\vartheta}(t)\xi_{b}^{\vartheta}(s)\rangle$
explains super and sub diffusive regimes. Indeed, notice that if
$\langle\xi_{a}^{\vartheta}(t)\xi_{b}^{\vartheta}(s)\rangle$ is
positive(negative) then the action of the stochastic force at an instant
$t+\Delta t$, $\Delta t>0$, is positively(negatively) correlated with the
action at a previous instant $t$, tending to amplify(reduce) it. In this case,
we would expect a super(sub) diffusive regime, a fact that is consistent with
our discussion: the correlator
$\langle\xi_{a}^{\vartheta}(t)\xi_{b}^{\vartheta}(s)\rangle$ is
positive(negative) if, and only if, $\alpha^{-1}>1$($\alpha^{-1}<1$).
Furthermore, Eqs. (46) and (33) show us that our formalism verifies the
Hurst’s law with Hurst exponent

$\displaystyle H=\frac{1}{2\alpha},$ (47)

as the fractional Brownian motion MvN68 . In terms of the $\theta$-Gaussian
colored noise $\xi^{\vartheta}$, Eq. (45) reads

$\displaystyle\frac{\mathrm{d}x}{\mathrm{d}t}=F(x,t)+(2D)^{\frac{1}{2\alpha}}\xi^{\vartheta}(t).$
(48)

The above Langevin equation for $x(t)$ considers an additive colored noise
whose solutions can be extracted through the formalism presented in this
section. If $\eta=2\lambda=1$, the same dynamical behavior is obtained from
the following Langevin-type equation (cf. Ref. Borland98 )

$\displaystyle\frac{\mathrm{d}x}{\mathrm{d}t}=F(x,t)+\sqrt{2D}[p(x,t)]^{\frac{1-q}{2}}\xi(t),$
(49)

where $p$ is obtained through Eq. (40). In the above equation, we have a
multiplicative noise for the stochastic dynamics of $x(t)$, which is described
in terms of the Gaussian white noise $\eta$. In a sense, the coupling between
microscopic and macroscopic dynamics, i.e. the fact that Eq. (49) explicitly
depends on $p$, arises because we are trying to describe a non-Markovian and
non-Gaussian system in terms of Gaussian white noise (or equivalently, the
Wiener process).

Figure 3: Stochastic trajectories of $W_{t}^{\vartheta}$ (left) and numerical
estimation of variance (right) with $q=1.5$, $\eta=1$, $d=1$ and
$\lambda=1/2$.

We now proceed by showing how the stochastic trajectories of
$W_{t}^{\vartheta}$ can be generated. For this notice that, as a consequence
of Itô’s lemma Privault13  together with the definition of
$W_{t}^{\vartheta}$, we have the following equality in distribution:

$\displaystyle\mathrm{d}W_{t}^{\vartheta}=\sqrt{\mathrm{d}t}L(W_{t}^{\vartheta},t)\mathfrak{g}_{t},$
(50)

where, for each $t$, $\mathfrak{g}_{t}\sim N(0,1)$ and

$\displaystyle L(x,t)=$
$\displaystyle\|x\|^{2(\eta-1)}\Big{\\{}v_{2-q}\Big{[}G_{\theta}\big{(}-;0,\varepsilon_{\vartheta}t^{\frac{1}{\alpha}}\big{)}\Big{]}\Big{\\}}^{2\lambda-1}$
$\displaystyle\times
G_{\theta}\big{(}x;0,\varepsilon_{\vartheta}t^{\frac{1}{\alpha}}\big{)}.$ (51)

We can thus generate the stochastic trajectories of $W_{t}^{\vartheta}$, e.g.
from $t=0$ to $t=1$, by generating $n$ independent Gaussian random variables
$\mathfrak{g}_{j}$ (e.g., through the Box-Muller method) in such a way that

$\displaystyle
W_{j+1}^{\vartheta}=\frac{L(W_{j}^{\vartheta},j/n)}{\sqrt{n}}\mathfrak{g}_{j}$
(52)

together with $W_{0}^{\vartheta}=0$ define a sequence of random variables that
approximates a realization of $W_{t}^{\vartheta}$ in the interval $[0,1]$.

##  IV Discussion

The goal of this paper was to study nonextensive generalizations of Itô-
Langevin dynamics. We believe this is of value for the following reasons.
First, nonextensive statistical mechanics is a fundamental theory underlying
many complex system phenomena in physics and beyond Tsallis09 . This includes
those phenomena associated with $q$-Gaussian distributions and nonlinear
Fokker-Planck equations, as discussed in the throughout this paper. In our
contribution we have developed a consistent generalization of the standard
Itô-Langevin formalism. This allows us to extract analytical solutions in a
simple manner and other properties such as time-correlations [Eqs. (18) and
(46)], Hurst exponent [Eq. (47)], and self-similarity laws [Eq. (21)] of
stochastic processes describing correlated anomalous diffusion in fractals.

From a more theoretical or mathematical point of view, nonextensive
statistical mechanics also provides significant insights UTBook . In
particular, we highlight the role of generalized central limit theorems to
explain why there are so many $q$-Gaussian-like distributions in nature
Tsallis09a . Our contribution in this direction was to extend the results of
Ref. UTS08 . This generalization was needed to properly justify the emergency
of generalized Wiener processes. We believe that this is indeed a significant
step towards a definite theory explaining the appearance of $q$-Gaussian
distributions. Nevertheless, we have shown that strict
$\vartheta$-independence is a sufficient condition for the emergency of
$q$-Gaussian random variables. It is then natural to search for necessary
conditions [at this point, possible extensions of the formalism for
$(q,\alpha)$-stable distributions of Ref. UTGS10  would be warmly welcome].
Other equally interesting directions to investigate are the scale-invariant
probabilistic models and large deviation theory (see, e.g., Refs. RST08 ;
HTT09 ; STRT15 ; TTA21 ), that have been shown promise in explaining the
emergence of $q$-Gaussians, and their possible connection with the results
developed here.

## Acknowledgments

The author thanks Prof. Constantino Tsallis, Prof. Otfried Gühne, Prof. Stefan
Nimmrichter, Prof. Gabriel Landi. Prof. Jorge Anderson Ramos, Prof. Márcio
Bortoloti and Carlos de Gois for their insightful comments and suggestions.
This work was supported by the Conselho Nacional de Pesquisa Científica (CNPq)
(Grants No. GM:155496/2019-0 and GD:141212/2021-7), the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation, project numbers
447948357 and 440958198), the Sino-German Center for Research Promotion
(Project M-0294) and the ERC (Consolidator Grant 683107/TempoQ).

##  Appendix A Technicalities of Section 2

###  A.1 Generalized Gaussian Distributions and Central Limit Theorems

In section 2, we presented the $\theta$-Gaussian distributions and discussed
how it can emerge as a limit of correlated random variables. In this
supplemental material we shall discuss the technical part of this section.
First, $\theta$-Gaussian distributions were defined as

$\displaystyle
G_{\theta}(x;\mu,\Sigma)=\frac{\exp_{q}\left\\{-\frac{1}{2_{\theta}}[(x-\mu)^{\mathrm{T}}\Sigma^{-1}(x-\mu)]^{\eta}\right\\}}{\sqrt{(2_{\theta}\hskip
1.42271pt\pi_{\theta})^{d}\det\Sigma}},$ (53)

where $\theta=(q,\eta,d)\in\mathbb{R}^{2}\times\mathbb{N}$ and
$\exp_{q}(x)=[\max\\{1+(1-q)x,0\\}]^{\frac{1}{1-q}}$. The normalization
constants $2_{\theta}$ and $\pi_{\theta}$ can be defined through

$\displaystyle\pi_{\theta}=\left(\int_{\mathbb{R}^{d}}\mathrm{e}_{q}^{-\|x\|^{2\eta}}\mathrm{d}x\right)^{\frac{2}{d}}\hskip
28.45274pt\textrm{and}\hskip
28.45274pt2_{\theta}=d(\pi_{\theta})^{\frac{d}{2}}\left(\int_{\mathbb{R}^{d}}\|x\|^{2}\hskip
1.42271pt\mathrm{e}_{q}^{-\|x\|^{2\eta}}\mathrm{d}x\right)^{-1}.$ (54)

It is easy to verify that $2_{\theta}\to 2$ and $\pi_{\theta}\to\pi$ when
$q=\eta\to 1$, as stated in the main text. Given $\vartheta=(\theta,\lambda)$,
$\lambda\in\mathbb{R}$, we defined the $\vartheta$-characteristic function of
a random variable $X$ with probability density function (PDF) $p$ as

$\displaystyle\mathrm{F}_{X}^{\hskip
1.42271pt\vartheta}(k)=\int_{\mathrm{Supp}\hskip
1.42271ptp}p(x)\exp_{q}\left\\{\mathrm{i}{k}^{\mathrm{T}}{x}[p(x)]^{\frac{1-\alpha}{d}}\right\\}\mathrm{d}x,$
(55)

where $\alpha=\eta+d(1-q)$. In the main text we mention two properties of this
integral transform. First, the scaling property:

$\displaystyle\mathrm{F}_{rX}^{\hskip
1.42271pt\vartheta}(k)=\mathrm{F}_{X}^{\hskip
1.42271pt\vartheta}(r^{\alpha}k),$ (56)

$r>0$. This is a direct consequence of the fact that the PDF of $rX$ is
$\tilde{p}(x)=r^{-d}p(x/r)$, where $p$ is the PDF of $X$. The second property
is related with $\theta$-Gaussian random variables. Let $X\sim
N_{\theta}(0,\Sigma)$. The $\vartheta$-characteristic function of $X$ is

$\displaystyle\mathrm{F}_{X}^{\hskip 1.42271pt\vartheta}(k)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}G_{\theta}(x;0,\Sigma)\exp_{q}\left\\{\mathrm{i}{k}^{\mathrm{T}}{x}[G_{q}({x};{0},{\Sigma})]^{\frac{1-\alpha}{d}}\right\\}\mathrm{d}{x}$
(57) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\frac{\mathrm{e}_{q}^{-\|{u}\|^{2\eta}}}{(\pi_{\theta})^{\frac{d}{2}}}\left\\{1-\mathrm{i}(q-1)\left(a\mathrm{e}_{q}^{-\|u\|^{2\eta}}\right)^{\frac{1-\alpha}{d}}\tilde{{k}}^{\mathrm{T}}{u}\right\\}^{\frac{1}{1-q}}\mathrm{d}{u}$
$\displaystyle=$
$\displaystyle\mathrm{Exp}_{\theta}\left(-\frac{2_{\theta}}{4(2_{\theta}\pi_{\theta})^{1-\alpha}}\frac{{k}^{\mathrm{T}}{\Sigma}{k}}{(\det{\Sigma})^{\frac{1-\alpha}{d}}}\right),$

where

$\displaystyle\mathrm{Exp}_{\vartheta}(x)=\frac{1}{(\pi_{\theta})^{\frac{d}{2}}}\int_{\mathbb{R}^{d}}\mathrm{e}_{q}^{-\|{u}\|^{2\eta}}\left[1-2\sqrt{x}(q-1)(\mathrm{e}_{q}^{-\|u\|^{2\eta}})^{\frac{1-\alpha}{d}}v^{\mathrm{T}}{u}\right]^{\frac{1}{1-q}}\mathrm{d}{u},$
(58)

$x\in\mathbb{R}$, and ${v}$ an arbitrary norm-1 vector in $\mathbb{R}^{d}$. In
Eq. (57) we have used the fact $\mathrm{F}^{\hskip 1.42271pt\vartheta}_{{X}}$
depends only on the norm square of $\mathrm{i}a^{\lambda(q-1)}\tilde{{k}}$.
This is a direct consequence of the fact that the odd escort moments of the
$\theta$-Gaussian distribution $\propto\mathrm{e}_{q}^{-\|x\|^{2\eta}}$ are
zero. From Eq. (58) we directly conclude that $\mathrm{Exp}_{\vartheta}(x)$ is
non-negative and $\mathrm{Exp}_{\vartheta}(0)=1$, where the former fact is a
consequence of the power-law-type decay of $\mathrm{e}_{q}^{-\|x\|^{2\eta}}$
for $q>1$ and the latter follows from the normalization of the
$\theta$-Gaussian.

In the particular case where $X$ is a $q$-Gaussian random variable and
$\lambda=1$, Eq. (57) equals

$\displaystyle\mathrm{F}_{{X}}^{\hskip 1.42271ptq}({k})$ $\displaystyle=$
$\displaystyle\frac{1}{(\pi_{qd})^{\frac{d}{2}}}\int_{\mathbb{R}^{d}}\mathrm{e}_{q}^{-\|{u}\|^{2}}\left\\{1-\frac{\mathrm{i}(q-1)a^{q-1}\tilde{{k}}^{\mathrm{T}}{u}}{1+(q-1)\|{u}\|^{2}}\right\\}^{\frac{1}{1-q}}\mathrm{d}{u}$
(59) $\displaystyle=$
$\displaystyle\left[1-(1-q)\frac{2_{qd}}{4(2_{qd}\pi_{qd})^{d(q-1)}}\frac{{k}^{\mathrm{T}}{\Sigma}{k}}{(\det{\Sigma})^{q-1}}\right]^{\frac{2+d(1-q)}{2(1-q)}}$
$\displaystyle=$
$\displaystyle\exp_{\tilde{q}_{d}}\left\\{-\frac{2_{qd}[2+d(1-q)]}{8(2_{qd}\pi_{qd})^{d(q-1)}}\frac{{k}^{\mathrm{T}}{\Sigma}{k}}{(\det{\Sigma})^{q-1}}\right\\},$

where $\tilde{q}_{d}=[2+(d+2)(1-q)]/[2+d(1-q)]$ and $\mathrm{F}^{\hskip
1.42271ptq}$ denotes the UTS’ $q$-characteristic function. This is the
multivariate version of the Lemma 2.5 of Ref. UTS08  (cf. Ref. UT07 ). In this
particular case, the $q$-FT of a $q$-Gaussian is a $\tilde{q}_{d}$-Gaussian
(up to normalization) if $q\geq 1$. In the cases where $q<1$, it was
demonstrated by Umarov-Queirós UQ10  that the $q$-FT of a $q$-Gaussian in
general does not correspond to another $q$-Gaussian [even with a different
value of $q$ as in Eq. (59)]. In our discussion, the undesirable behavior
presented by $\theta$-FTs when $q<1$ is that the corresponding
$\vartheta$-Exponential can assume negative values.

Figure 4: Generalized $\vartheta$-exponential with $d=\eta=1$ [Eq. (58)].

We have introduced in Eq. (58) a new deformed exponential function, the
$\vartheta$-exponential. This is a three parameter generalization of the
exponential function $\mathrm{e}^{x}$ (which is recovered when $q=\eta\to 1$).
In general it is not easy to find closed expressions for
$\vartheta$-exponential. As illustration, we compute the
$\vartheta$-exponentials corresponding to some values of $\vartheta$ (see Fig.
4). First, if $\eta=\lambda=1$ we have

$\displaystyle\mathrm{Exp}_{(q,\hskip 1.42271ptd,\hskip 1.42271pt1,\hskip
1.42271pt1)}(x)=\exp_{\tilde{q}_{d}}\left[\frac{2+d(1-q)}{2}x\right],$ (60)

$\tilde{q}_{d}=[2+(d+2)(1-q)]/[2+d(1-q)]$ [cf. Eq. (59)]. Similarly, if
$\eta=d=2\lambda=1$

$\displaystyle\mathrm{Exp}_{(q,\hskip 1.42271pt1,\hskip 1.42271pt1,\hskip
1.42271pt1/2)}(x)$
$\displaystyle={}_{2}F_{1}\left(\frac{1}{2(q-1)},\frac{q}{2(q-1)},\frac{1}{q-1},4(q-1)x\right),$
(61)

where ${}_{2}F_{1}$ denotes the Gaussian hypergeometric function. Finally, if
$q=d=1$ we have

$\displaystyle\mathrm{Exp}_{(1,\hskip 1.42271pt1,\hskip 1.42271pt\eta,\hskip
1.42271pt\lambda)}^{\hskip
1.42271ptq}(x)=\frac{1}{\Gamma\left(\frac{2\eta+1}{2\eta}\right)}\int_{0}^{\infty}\mathrm{e}^{-|u|^{2\eta}}\exp\left(2u\sqrt{x}\hskip
1.42271pt\mathrm{e}^{-(1-\eta)|u|^{2\eta}}\right)\mathrm{d}u.$ (62)

As the exponential, for all $\lambda>0$ and $\eta=1$,
$\mathrm{Exp}_{\vartheta}(x)>0$, $\mathrm{Exp}_{\vartheta}(0)=1$ and $x\geq y$
implies $\mathrm{Exp}_{\vartheta}(x)\geq\mathrm{Exp}_{\vartheta}(y)$. For
$\lambda\leq 0$ or $\eta\neq 1$, in turn, $\mathrm{Exp}_{\vartheta}(x)>0$ only
if $x\leq 0$ and $\mathrm{Exp}_{\vartheta}(x)=0$ if $x>0$. We define the
$\vartheta$-Logarithm as the inverse of $\mathrm{Exp}_{\vartheta}$ in the
appropriated domain. The usual (natural) logarithm is recovered when $q\to 1$,
while for $\lambda=1$ we have, in terms of the $\tilde{q}_{d}$-logarithm,
$\mathrm{Log}_{\vartheta}(x)=2[2+d(1-q)]^{-1}\log_{\tilde{q}_{d}}(x)$, where
$\log_{q}$ is the $q$-logarithm of NE statistical mechanics Tsallis09

$\displaystyle\log_{q}u=\frac{u^{q-1}-1}{q-1}.$ (63)

From the above results, we would conclude that $\vartheta$-Fourier transform
(FT) are natural NE extensions of the usual FT. The proposed generalization,
however, does not have all the good properties of its extensive counterpart as
already noted for the UTS’ $q$-FT. One the most mentioned problems is its
(lack of) invertibility, since it is not invertible in the full space of PDFs
for $q>1$. In fact, as demonstrated by Hilhorst Hilhorst10 , the following
PDFs

$\displaystyle H_{r;\hskip
1.42271ptq}(x)=\begin{cases}\frac{\big{(}|x|^{\frac{q-2}{q-1}}-r\big{)}^{\frac{1}{q-2}}}{\sqrt{2_{q}\pi_{q}}|x|^{\frac{1}{q-1}}}\exp_{q}\left[-\frac{1}{2_{q}}\left(|x|^{\frac{q-2}{q-1}}-r\right)^{\frac{2(q-1)}{q-2}}\right]\hskip
2.84544pt&\textrm{if }|x|^{\frac{q-2}{q-1}}>r\\\ 0\hskip
7.11317pt&\textrm{otherwise}\end{cases},$ (64)

$1<q<2$ and $r\in\mathbb{R}$, share the same UTS’ $q$-FT of $G_{q}(x;0,1)$,
$q>1$. This occurs because the $\vartheta$-FT associates a family of functions
to each real function in $L_{1}$, instead only one as the usual FT. By using
the same arguments of Refs. JT11 ; ST17 , it is possible to demonstrate that

$\displaystyle
f({x})=\left(\frac{1}{c_{\vartheta}}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}f({y})\mathrm{e}_{q}^{\mathrm{i}{k}^{\mathrm{T}}({y}-{x})[f({y})]^{\frac{1-\alpha}{d}}}\mathrm{d}{y}\mathrm{d}{k}\right)^{\frac{1}{\alpha}},$
(65)

for all ${x}$ in the interior of the support of $f$, $1\leq q<(1+d)/d$,
$c_{\vartheta}>0$. Eq. (65) shows that in order to recover the original
function $f$ we need to know its $\vartheta$-FT and all possible translations
of it. The only exception occurs when $q=\eta\to 1$, since in this case we can
factorize the exponential in the argument of the integral obtaining the
inverse of the usual FT.

With these definitions on hand, we are now ready to generalize CLT.

######  Theorem 1.

Let $X_{n}$ be a sequence of $\vartheta$-independent and identically
distributed random variables. Suppose also

$\displaystyle\int_{\mathrm{Supp}\hskip
1.42271ptp}\|{x}\|^{2}[p({x})]^{s_{2}}\mathrm{d}{x}<\infty,$ (66)

where $p$ denotes the PDF of ${X}_{n}$. Then the sequence $\tilde{X}_{n}$,

$\displaystyle\tilde{X}_{n}=(nz)^{-\frac{1}{2\alpha}}\left(S_{n}-nE_{s_{1}}[X_{n}]\right),$
(67)

$z>0$, $S_{n}=X_{1}+\dots+X_{n}$, $\vartheta$-converges to a $\theta$-Gaussian
random variable ${X}$ with zero expectation value and covariance matrix
${\Sigma}$ satisfying

$\displaystyle\frac{\Sigma}{(\det\Sigma)^{\frac{1-\alpha}{d}}}=\frac{2_{\theta^{\prime}}v_{s_{2}}[p]}{2_{\theta}(2_{\theta}\pi_{\theta})^{\alpha-1}}\left(\frac{\pi_{\theta}s_{2}}{\pi_{\theta^{\prime}}}\right)^{\frac{d}{2}}E_{s_{2}}[X_{1}X_{1}^{\mathrm{T}}],$
(68)

where $\mathbb{I}_{d}$ denotes the $d\times d$ identity matrix and
$s_{n}=1+n(1-\alpha)/d$.

###### Proof.

From condition (67), we can (without loss of generality) assume
$E_{s_{1}}[X_{i}]=0$ for all $i\geq 1$, since $E_{s_{1}}[X-E_{s_{1}}[X]]=0$
for any random variable $X$. The $\vartheta$-characteristic function of
$\tilde{X}_{n}=(nz)^{-\frac{1}{2\alpha}}(X_{1}+\dots+X_{n})$ then reads

$\displaystyle\mathrm{F}^{\hskip 1.42271pt\vartheta}_{\tilde{{X}_{n}}}({k})$
$\displaystyle=$ $\displaystyle\mathrm{F}^{\hskip
1.42271pt\vartheta}_{{X}_{1}}({k}/\sqrt{nz})\otimes_{\vartheta}\dots\otimes_{\vartheta}\mathrm{F}^{\hskip
1.42271pt\vartheta}_{{X}_{n}}({k}/\sqrt{nz}),$ (69)

where we have made use of the scaling property of Eq. (56) together with the
fact that the sequence is supposed to be $\vartheta$-independent. Since the
sequence is also supposed to identically distributed, taking the
$\vartheta$-Log in both sides of Eq. (69) we obtain

$\displaystyle\mathrm{Log}_{\vartheta}\big{[}\mathrm{F}^{\hskip
1.42271pt\vartheta}_{\tilde{{X}}_{n}}({k})\big{]}=n\mathrm{Log}_{\vartheta}\big{[}\mathrm{F}^{\hskip
1.42271pt\vartheta}_{{{X}}_{j}}\big{(}{k}/\sqrt{nz}\big{)}\big{]}.$ (70)

With some manipulation, the $\vartheta$-characteristic function can be
expanded as

$\displaystyle\mathrm{F}^{\hskip
1.42271pt\vartheta}_{{{X}}_{j}}\big{(}{k}/\sqrt{nz}\big{)}=1-\frac{qv_{s_{2}}[p]}{2nz}{k}^{\mathrm{T}}E_{s_{2}}[XX^{\mathrm{T}}]{k}+\mathcal{O}\big{(}n^{-3/2}\big{)},$
(71)

where we have made use of the fact $E_{s_{1}}[{X}_{j}]={0}$. The
$\theta$-Exponential, in turn, can be expanded as

$\displaystyle\mathrm{Exp}_{\vartheta}(y)$
$\displaystyle=\int_{\mathbb{R}^{d}}\frac{\mathrm{e}_{q}^{-\|{x}\|^{2\eta}}}{(\pi_{\theta})^{\frac{d}{2}}}\exp_{q}\left[2\sqrt{y}({x}^{\mathrm{T}}{v})\left(\mathrm{e}_{q}^{-\|{x}\|^{2\eta}}\right)^{\frac{1-\alpha}{d}}\right]\mathrm{d}{x}$
$\displaystyle=1+(2qy){v}^{\mathrm{T}}\left(\int_{\mathbb{R}^{d}}\frac{\left(\mathrm{e}_{q}^{-\|{x}\|^{2\eta}}\right)^{s_{2}}}{(\pi_{\theta})^{\frac{d}{2}}}{x}{x}^{\mathrm{T}}\mathrm{d}{x}\right){v}+\mathcal{O}(y^{2})$
$\displaystyle=1+(2qy)\int_{\mathbb{R}^{d}}\frac{\left(\mathrm{e}_{q}^{-\|{x}\|^{2\eta}}\right)^{s_{2}}}{(\pi_{\theta})^{\frac{d}{2}}}x_{j}^{2}\mathrm{d}{x}+\mathcal{O}\big{(}y^{2}\big{)}$
$\displaystyle=1+\frac{2q}{2_{\theta^{\prime}}}\left(\frac{\pi_{\theta^{\prime}}}{\pi_{\theta}s_{2}}\right)^{\frac{d}{2}}y+\mathcal{O}\big{(}y^{2}\big{)},$
(72)

where ${v}$ is an arbitrary norm-1 vector in $\mathbb{R}^{d}$ and
$\theta^{\prime}=(q^{\prime},\eta,d)$, $q^{\prime}=[s_{2}-(1-q)]/s_{2}$.
Comparing the above equation with Eq. (71), in the limit $n\to\infty$ we
obtain the following expression for the $\theta$-characteristic function of
${X}$:

$\displaystyle\mathrm{F}^{\hskip
1.42271pt\vartheta}_{{X}}({k})=\mathrm{Exp}_{\vartheta}\left\\{-\frac{2_{\theta^{\prime}}v_{s_{2}}[p]}{4z}\left(\frac{\pi_{\theta}s_{2}}{\pi_{\theta^{\prime}}}\right)^{\frac{d}{2}}k^{\mathrm{T}}E_{s_{2}}\big{[}X_{1}X_{1}^{\mathrm{T}}\big{]}k\right\\}.$
(73)

Comparing the above equation with Eq. (57), we therefore identify ${X}$ as a
$\theta$-Gaussian random variable with zero expectation value and covariance
matrix given by

$\displaystyle\frac{\Sigma}{(\det\Sigma)^{\frac{1-\alpha}{d}}}=\frac{2_{\theta^{\prime}}v_{s_{2}}[p]}{2_{\theta}(2_{\theta}\pi_{\theta})^{\alpha-1}}\left(\frac{\pi_{\theta}s_{2}}{\pi_{\theta^{\prime}}}\right)^{\frac{d}{2}}E_{s_{2}}[X_{1}X_{1}^{\mathrm{T}}]$
(74)

∎

######  Remark.

It is worthy to mention that the lack of invertibility of $q$-generalized
Fourier transforms and its impact on the corresponding CLTs have been widely
discussed in literature for the UTS formalism. We highlight, in particular,
the discussion of Ref. UT16 , where the authors presented various arguments
stating the uniqueness of limit distributions for the UTS’ CLTs. Fortunately,
the results and conclusions presented in such an article can be easily adapted
to the formalism presented here, since the $\vartheta$-FT has almost the same
form of the $q$-FT.

The usual CLT is recovered when $q\to 1$ whereas for $\lambda=1$ we have a
multivariate version of UTS’ CLT UTS08 , with the limit covariance matrix
being (cf. Ref. UT07 )

$\displaystyle\frac{{\Sigma}}{(\det{\Sigma})^{q-1}}=\frac{4q(2_{qd}\pi_{qd})^{d(q-1)}}{2_{qd}[2+d(1-q)]z}\int_{\mathrm{Supp}\hskip
1.42271ptp}[p({x})]^{2q-1}{x}{x}^{\mathrm{T}}\mathrm{d}{x},$ (75)

and a $q$-Gaussian distribution.

###  A.2 Generalized Wiener Process

Let us now demonstrate the existence of a well-defined stochastic process
${Q}_{t}^{\vartheta}$, constructed as a limit of $\vartheta$-independent
random variables, for $\eta=1$, $1\leq q<\min\\{(\lambda d+1)/\lambda
d,(d+4)/(d+2)\\}$ and $\lambda>0$, satisfying:

  1. (1)

${Q}_{0}^{\vartheta}={0}$ almost surely;

  2. (2)

The paths $t\mapsto{Q}_{t}^{\vartheta}$ are continuous with probability one;

  3. (3)

${Q}_{t}^{\vartheta}-{Q}_{s}^{\vartheta}\sim
N_{q}\big{[}0,(t-s)^{\frac{1}{\alpha}}\mathbb{I}_{d}\big{]}$, $0\leq s\leq t$;

  4. (4)

${Q}_{t}^{\vartheta}-{Q}_{s}^{\vartheta}$, $0\leq s\leq t$, is
$\vartheta$-independent of ${Q}_{s}^{\vartheta}$;

  5. (5)

Given $t,s\geq 0$, the covariance between the components of
${Q}_{t}^{\vartheta}$ and ${Q}_{s}^{\vartheta}$ is

$\displaystyle\mathrm{Cov}[Q_{t;\hskip 1.42271pt\mu}^{\vartheta},Q_{s;\hskip
1.42271pt\nu}^{\vartheta}]$
$\displaystyle:=\frac{\delta_{\mu\nu}}{2}\left(t^{\frac{1}{\alpha}}+s^{\frac{1}{\alpha}}-|t-s|^{\frac{1}{\alpha}}\right).$
(76)

First we will discuss the existence of such a stochastic process. For this,
let

$\displaystyle p({x}_{1},t_{1};\dots;{x}_{n},t_{n})\equiv
G_{\hat{q}_{nd}}\big{(}{x}_{1},\dots,{x}_{n};{\Lambda}(t_{1},\dots,t_{n})\big{)},$
(77)

define a joint PDF, $x_{i}\in\mathbb{R}^{d}$, where $0\leq
t_{1}<t_{2}<\dots<t_{n}$; ${\Lambda}(t_{1},\dots,t_{n})$ is a $nd\times nd$
matrix defined as

$\displaystyle{\Lambda}(t_{1},\dots,t_{n})\equiv\left(\begin{matrix}{\lambda}(t_{1};\hskip
0.28436ptt_{1})&{\lambda}(t_{1};\hskip
0.28436ptt_{2})&\dots&{\lambda}(t_{1};\hskip 0.28436ptt_{n})\\\
{\lambda}(t_{2};\hskip 0.28436ptt_{1})&{\lambda}(t_{2};\hskip
0.28436ptt_{2})&\dots&{\lambda}(t_{2};\hskip 0.28436ptt_{n})\\\
\vdots&\vdots&\ddots&\vdots\\\ {\lambda}(t_{n};\hskip
0.28436ptt_{1})&{\lambda}(t_{n};\hskip
0.28436ptt_{2})&\dots&{\lambda}(t_{n};\hskip
0.28436ptt_{n})\end{matrix}\right),$ (78)

$({\lambda}(t_{\mu},t_{\nu}))_{ab}=\mathrm{Cov}[Q_{t_{\mu};\hskip
1.42271pta}^{\vartheta},Q_{t_{\nu};\hskip 1.42271ptb}^{\vartheta}]$ given by
Eq. (76), and

$\displaystyle\hat{q}_{nd}=\frac{2q-d(n-1)(1-q)}{2-d(n-1)(1-q)}.$ (79)

A simple calculation show that $p(x,t)=G_{q}(x;\hskip
1.42271pt0,t^{\frac{1}{\alpha}}\mathbb{I}_{d})$ and

$\int_{\mathbb{R}^{d}}p({x}_{1},t_{1})\mathrm{d}{x}_{1}&=1,\\\
p({x}_{1},t_{1};\dots;{x}_{n},t_{n})\geq 0,\\\
\int_{\mathbb{R}^{d}}p({x}_{1},t_{1};\dots;{x}_{n},t_{n})\mathrm{d}x_{n}=p({x}_{1},t_{1};\dots;{x}_{n-1},t_{n-1}),\\\
p({x}_{\pi(1)},t_{\pi(1)};\dots;{x}_{\pi(n)},t_{\pi(n)})=p({x}_{1},t_{1};\dots;{x}_{n},t_{n}),$
(80)

where $\pi$ is an element of the permutation group of $\\{1,\dots,n\\}$. These
conditions allow us to apply the Kolmogorov consistency theorem Durrett10  for
the induced probability measure, guarantying thus the existence of a (probably
not unique) stochastic process $Q^{\vartheta}_{t}\sim
N_{q}(0,t^{\frac{1}{\alpha}}\mathbb{I}_{d})$ satisfying Eq. (76).

Consider now a sequence of $\vartheta$-independent random variables
${\xi}_{j}$ in $\mathbb{R}^{d}$ such that ${\xi}_{j}\sim
N_{q}({0},\mathbb{I}_{d})$, and let

$\displaystyle{X}_{t;\hskip
1.42271ptn}=n^{-\frac{1}{2\alpha}}\sum_{j=1}^{\lfloor nt\rfloor}{\xi}_{j}.$
(81)

From generalized CLT, the sequence $({X}_{t;\hskip
1.42271ptn}:n\in\mathbb{N})$ $\vartheta$-converges to a random variable
${Q}^{\vartheta}_{t}$ with zero expectation value and covariance matrix
satisfying

$\displaystyle\frac{{\Sigma}}{(\det{\Sigma})^{\lambda(q-1)}}=t\hskip
1.42271pt\mathbb{I}_{d}.$ (82)

In the above equation we have made use of the fact that

$\displaystyle\frac{{2}_{q^{\prime}d}({2}_{qd}\pi_{qd})^{\lambda
d(q-1)}}{{2}_{qd}}\left\\{\frac{\pi_{qd}[1+2\lambda(q-1)]^{\frac{2+d}{2}}}{\pi_{q^{\prime}d}}\right\\}^{\frac{d}{2}}\int_{\mathbb{R}^{d}}[G_{q}({x};{0},\mathbb{I}_{d})]^{1+2\lambda(q-1)}{x}{x}^{\mathrm{T}}\mathrm{d}{x}=\mathbb{I}_{d},$
(83)

Therefore ${Q}_{t}^{\vartheta}\sim
N_{q}\big{(}{0},t^{\frac{1}{\alpha}}\mathbb{I}_{d}\big{)}$. In the same way,
the sequence defined by ${X}_{t;\hskip 1.42271ptn}-{X}_{s;\hskip 1.42271ptn}$,
$0\leq s\leq t$, $\vartheta$-converges to
${Q}_{t}^{\vartheta}-{Q}_{s}^{\vartheta}$. Moreover, since

$\displaystyle{X}_{n;\hskip 1.42271ptt}-{X}_{n;\hskip 1.42271pts}$
$\displaystyle=$ $\displaystyle n^{-\frac{1}{2\alpha}}\sum_{i=\lfloor
ns\rfloor}^{\lfloor nt\rfloor}{\xi}_{k},$ (84)

then ${Q}_{t}^{\vartheta}-{Q}_{s}^{\vartheta}$ is equal in distribution to
${Q}_{t-s}^{\vartheta}$. Therefore,
${Q}_{t}^{\vartheta}-{Q}_{s}^{\vartheta}\sim
N_{q}\big{[}{0},(t-s)^{\frac{1}{\alpha}}\mathbb{I}_{d}\big{]}$. This, together
with the following relation

$\displaystyle\mathrm{Cov}[A,B]=\frac{1}{2}\left(\textrm{Var}{[}A]+\textrm{Var}{[}B]-\textrm{Var}{[}B-A]\right),$
(85)

imply that the covariance between the components of
${Q}_{t}^{\vartheta}=\big{(}Q_{t;\hskip
1.42271pt1}^{\vartheta},\dots,Q_{t;\hskip
1.42271ptd}^{\vartheta}\big{)}^{\mathrm{T}}$ at different times equals Eq.
(76).

We now have on hand a well-defined stochastic process ${Q}_{t}^{\vartheta}$
obeying conditions (3) and (5). Condition (1) is also fulfilled since
${Q}_{t}^{\vartheta}\sim
N_{q}\big{(}{0},t^{\frac{1}{\alpha}}\mathbb{I}_{d}\big{)}$ and
$G_{q}({x};{0},\varphi\mathbb{I}_{d})\to\delta({x})$ when $\varphi\to 0$.
Continuity of the paths of ${Q}^{\vartheta}_{t}$ [condition (2)], in turn,
follows from Kolmogorov-Chentsov continuity theorem Chentsov . In fact, the
process $Q_{t}^{\vartheta}$ satisfies

$\displaystyle
E[\|{Q}_{t}^{\vartheta}-{Q}_{s}^{\vartheta}\|^{2}]=d|t-s|^{1+\frac{d(q-1)}{\alpha}},$
(86)

in such a way that for all $q$ such that
$1<q<\min\\{(d\lambda+1)/d\lambda,(d+4)/(d+2)\\}$, there exists a modification
of ${Q}^{\vartheta}_{t}$, $\tilde{{Q}}_{t}^{\vartheta}$, that is continuous,
i.e. a sample-continuous stochastic process $\tilde{{Q}}_{t}^{\vartheta}$ such
that

$\displaystyle\mathrm{Pr}[{Q}_{t}^{\vartheta}=\tilde{{Q}}_{t}^{\vartheta}]=1,$
(87)

for all $t\geq 0$. Furthermore, the paths of $\tilde{{Q}}^{\vartheta}_{t}$ are
$\beta$-Hölder continuous for all $\beta$ real such that

$\displaystyle 0<\beta<\frac{d\lambda(q-1)}{2\alpha}.$ (88)

Therefore the continuous modification of the process ${Q}_{t}^{\vartheta}$
(which we shall also denote by the same symbol) is a well-defined stochastic
process fulfilling conditions (1), (2), (3), and (5). Finally, the
$\vartheta$-independence between ${Q}_{t}^{\vartheta}-{Q}_{s}^{\vartheta}$ and
${Q}_{s}^{\vartheta}$, $s<t$, follows straightforwardly:

$\displaystyle\mathrm{F}^{\hskip
1.42271pt\vartheta}_{({Q}_{t}^{\vartheta}-{Q}_{s}^{\vartheta})+{Q}_{s}^{\vartheta}}({k})$
$\displaystyle=$ $\displaystyle\mathrm{F}^{\hskip
1.42271pt\vartheta}_{{Q}_{t}^{\vartheta}}({k})$ (89) $\displaystyle=$
$\displaystyle\mathrm{Exp}_{\vartheta}\left\\{-\frac{{2}_{qd}}{4({2}_{qd}\pi_{qd})^{d\lambda(q-1)}}[(t-s)+s]\|{k}\|^{2}\right\\}$
$\displaystyle=$ $\displaystyle\mathrm{F}^{\hskip
1.42271pt\vartheta}_{{Q}_{t}^{\vartheta}-{Q}_{s}^{\vartheta}}({k})\otimes_{\vartheta}\mathrm{F}^{\hskip
1.42271pt\vartheta}_{{Q}^{\vartheta}_{s}}({k}).$

###  A.3 Stochastic Integral

To finish, let us discuss the stochastic integral induced by
$Q_{t}^{\vartheta}$. The stochastic integral of a function $f$ with respect to
${Q}_{t}^{\vartheta}$ is a stochastic process ${I}_{t}$,

$\displaystyle{I}_{t}=\int_{0}^{t}f\big{(}{Q}_{\tau}^{\vartheta},\tau\big{)}\mathrm{d}{Q}_{\tau}^{\vartheta},$
(90)

whose PDF is obtained by

$\displaystyle\mathrm{F}^{\hskip
1.42271pt\vartheta}_{{I}_{t}}({k})=\lim_{n\to\infty}\bigotimes_{\vartheta;\hskip
1.42271pt[t_{j-1},t_{j}]\in\mathsf{p}_{n}}\mathrm{F}^{\hskip
1.42271pt\vartheta}_{f({Q}^{\vartheta}_{t_{j-1}},\hskip
1.42271ptt_{j-1})\Delta{Q}_{t_{j}}^{\vartheta}}({k}),$ (91)

where $\mathsf{p}_{n}$ is a sequence of partitions of the interval $[0,t]$
with mesh going to zero,
$\Delta{Q}^{\vartheta}_{t_{j}}\equiv{Q}^{\vartheta}_{t_{j}}-{Q}_{t_{j-1}}^{\vartheta}$,
and the product in the right-hand-side is the $\vartheta$-product indexed by
the elements of $\mathsf{p}_{n}$. As claimed in the main text, this stochastic
integral satisfies a generalized Itô formula:

$\displaystyle I_{t}\sim N_{\theta}\left[{0},\hskip
2.84544pt\left(\int_{0}^{t}|f(\tau)|^{2\alpha}\mathrm{d}\tau\right)^{\frac{1}{\alpha}}\mathbb{I}_{d}\right],$
(92)

where $f(t)\in L^{2\alpha}([0,t])$, $t\geq 0$. The proof of this result is
straightforward:

$\displaystyle\mathrm{F}^{\hskip 1.42271pt\vartheta}_{{I}_{t}}({k})$
$\displaystyle=\lim_{n\to\infty}\bigotimes_{\vartheta;\hskip
1.42271pt[t_{j-1},t_{j}]\in\mathsf{p}_{n}}\mathrm{F}^{\hskip
1.42271pt\vartheta}_{f(t_{j-1})\Delta{Q}_{t_{j}}^{\vartheta}}({k})$
$\displaystyle=\lim_{n\to\infty}\bigotimes_{\vartheta;\hskip
1.42271pt[t_{j-1},t_{j}]\in\mathsf{p}_{n}}\mathrm{F}^{\hskip
1.42271pt\vartheta}_{\Delta{Q}_{t_{j}}^{\vartheta}}\big{(}|f(t_{j-1})|^{\alpha}{k}\big{)}$
$\displaystyle=\lim_{n\to\infty}\bigotimes_{\vartheta;\hskip
1.42271pt[t_{j-1},t_{j}]\in\mathsf{p}_{n}}\mathrm{Exp}_{\vartheta}\left(-\frac{{2}_{\theta}}{4({2}_{\theta}\pi_{\theta})^{1-\alpha}}|f(t_{j-1})|^{2\alpha}\Delta
t_{j}\|{k}\|^{2}\right)$
$\displaystyle=\mathrm{Exp}_{\vartheta}\left(-\frac{{2}_{\theta}}{4({2}_{\theta}\pi_{\theta})^{1-\alpha}}\lim_{n\to\infty}\sum_{[t_{j-1},t_{j}]\in\mathsf{p}_{n}}|f(t_{j-1})|^{2\alpha}\Delta
t_{j}\|{k}\|^{2}\right)$
$\displaystyle=\mathrm{Exp}_{\vartheta}\left(-\frac{{2}_{\theta}}{4({2}_{\theta}\pi_{\theta})^{1-\alpha}}\int_{0}^{t}|f(\tau)|^{2\alpha}\mathrm{d}\tau\hskip
1.42271pt\|{k}\|^{2}\right),$ (93)

where Eq. (92) follows from direct comparison with Eq. (57).

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