#  Noise Reinforced Lévy Processes:
Lévy-Itô Decomposition and Applications

Alejandro<EMAIL_ADDRESS>
(Institute of Mathematics, University of Zürich 222Research supported by the
Swiss National Science Foundation (SNSF).)

###### Abstract

A step reinforced random walk is a discrete time process with memory such that
at each time step, with fixed probability $p\in(0,1)$, it repeats a previously
performed step chosen uniformly at random while with complementary probability
$1-p$, it performs an independent step with fixed law. In the continuum, the
main result of Bertoin in [7] states that the random walk constructed from the
discrete-time skeleton of a Lévy process for a time partition of mesh-size
$1/n$ converges, as $n\uparrow\infty$ in the sense of finite dimensional
distributions, to a process $\hat{\xi}$ referred to as a noise reinforced Lévy
process. Our first main result states that a noise reinforced Lévy processes
has rcll paths and satisfies a noise reinforced Lévy Itô decomposition in
terms of the noise reinforced Poisson point process of its jumps. We introduce
the joint distribution of a Lévy process and its reinforced version
$(\xi,\hat{\xi})$ and show that the pair, conformed by the skeleton of the
Lévy process and its step reinforced version, converge towards
$(\xi,\hat{\xi})$ as the mesh size tend to $0$. As an application, we analyse
the rate of growth of $\hat{\xi}$ at the origin and identify its main features
as an infinitely divisible process.

##  1 Introduction

The Lévy-Itô decomposition is one of the main tools for the study of Lévy
processes. In short, any real Lévy process $\xi$ has rcll sample paths and its
jump process induces a Poisson random measure – called the jump measure
$\mathcal{N}$ of $\xi$ – whose intensity is described by its Lévy measure
$\Lambda$. Moreover, it states that $\xi$ can be written as the sum of tree
process

$\xi_{t}=\xi^{(1)}_{t}+\xi^{(2)}_{t}+\xi^{(3)}_{t},\quad\quad t\geq 0,$

of radically different nature. More precisely, the continuous part of $\xi$ is
given by $\xi^{(1)}=(at+qB_{t}:t\geq 0)$ for a Brownian motion $B$ and reals
$a,q$, while $\xi^{(2)}$ is a compound Poisson process with jump-sizes greater
than 1 and $\xi^{(3)}$ is a purely discontinuous martingale with jump-sizes
smaller than 1. Moreover, the processes $\xi^{(2)}$, $\xi^{(3)}$ can be
reconstructed from the jump measure $\mathcal{N}$. It is well known that
$\mathcal{N}$ is characterised by the two following properties: for any Borel
$A$ with $\Lambda(A)<\infty$, the counting process of jumps
$\Delta{\xi}_{s}\in A$ that we denote by ${N}_{A}$ is a Poisson process with
rate $\Lambda(A)$, and for any disjoint Borel sets $A_{1},\dots,A_{k}$ with
$\Lambda(A_{i})<\infty$, the corresponding Poisson processes
${N}_{A_{1}},\dots,{N}_{A_{k}}$ are independent. We refer to e.g. [5, 16, 23]
for a complete account on the theory of Lévy processes.

In this work, we shall give an analogous description for noise reinforced Lévy
processes (abbreviated NRLPs). This family of processes has been recently
introduced by Bertoin in [7] and correspond to weak limits of step reinforced
random walks of skeletons of Lévy process. In order to be more precise, let us
briefly recall the connection between these discrete objects and our
continuous time setting. Fix a Lévy process $\xi$ and denote, for each fixed
$n$, by $X^{(n)}_{k}:=\xi_{k/n}-\xi_{(k-1)/n}$ the $k$-th increment of ${\xi}$
for a partition of size $1/n$ of the real line. The process
$S^{(n)}_{k}:=X^{(n)}_{1}+\dots+X^{(n)}_{k}=\xi_{k/n}$ for $k\geq 1$ is a
random walk, also called the $n$-skeleton of $\xi$. Now, fix a real number
$p\in(0,1)$ that we call the reinforcement or memory parameter and let
$\hat{S}^{(n)}_{1}:=X^{(n)}_{1}$. Then, define recursively $\hat{S}^{(n)}_{k}$
for $k\geq 2$ according to the following rule: for each $k\geq 2$, set
$\hat{S}^{(n)}_{k}:=\hat{S}^{(n)}_{k-1}+\hat{X}^{(n)}_{k}$ where, with
probability $1-p$, the step $\hat{X}^{(n)}_{k}$ is the increment $X^{(n)}_{k}$
with law $\xi_{1/n}$ – and hence independent from the previously performed
steps – while with probability $p$, $\hat{X}^{(n)}_{k}$ is an increment chosen
uniformly at random from the previous ones
$\hat{X}^{(n)}_{1},\dots,\hat{X}^{(n)}_{k}$. When the former occurs, the step
is called an innovation, while in the latter case it is referred to as a
reinforcement. The process $(\hat{S}^{(n)}_{k})$ is called the step-reinforced
version of $(S^{(n)}_{k})$. It was shown in [7] that, under appropriate
assumptions on the memory parameter $p$, we have the following convergence in
the sense of finite dimensional distributions as the mesh-size tends to $0$

$(\hat{S}^{(n)}_{\lfloor
nt\rfloor})_{t\in[0,1]}\overset{f.d.d.}{\longrightarrow}\big{(}\hat{\xi}_{t}\big{)}_{t\in[0,1]},$
(1.1)

towards a process $\hat{\xi}$ identified in [7] and called a noise reinforced
Lévy process. It should be noted that the process $\hat{\xi}$ constructed in
[7] is a priori not even rcll, and this will be one of our first concerns.

We are now in position to briefly state the main results of this work. First,
we shall prove the existence of a rcll modification for $\hat{\xi}$. In
particular, this allow us to consider the jump process
$(\Delta\hat{\xi}_{s})$; a proper understanding of its nature will be crucial
for this work. In this direction, we introduce a new family of random measures
in $\mathbb{R}^{+}\times\mathbb{R}$ of independent interest under the name
noise reinforced Poisson point processes (abbreviated NRPPPs) and we study its
basic properties. This lead us towards our first main result, which is a
version of the Lévy-Itô decomposition in the reinforced setting. More
precisely, we show that the jump measure of $\hat{\xi}$ is a NRPPP and that
$\hat{\xi}$ can be written as

$\hat{\xi}_{t}=\hat{\xi}^{(1)}_{t}+\hat{\xi}^{(2)}_{t}+\hat{\xi}^{(3)}_{t},\quad
t\geq 0,$

where now, $\hat{\xi}^{(1)}=(at+q\hat{B}_{t}:t\geq 0)$ for a continuous
Gaussian process $\hat{B}$, the process $\hat{\xi}^{(2)}$ is a reinforced
compound Poisson process with jump-sizes greater than one, while
$\hat{\xi}^{(3)}$ is a purely discontinuous semimartingale. The continuous
Gaussian process $\hat{B}$ is the so-called noise reinforced Brownian motion,
a Gaussian process introduced in [8] with law singular with respect to $B$,
and arising as the universal scaling limit of noise reinforced random walks
when the law of the typical step is in $L_{2}(\mathbb{P})$ – and hence plays
the role of Brownian motion in the reinforced setting, see also [4] for
related results. Needless to say that if the starting Lévy process $\xi$ is a
Brownian motion, the limit $\hat{\xi}$ obtained in (1.1) is a noise reinforced
Brownian motion. As in the non-reinforced case, $\hat{\xi}^{(2)}$ and
$\hat{\xi}^{(3)}$ can be recovered from the jump measure $\hat{\mathcal{N}}$,
but in contrast, they are not Markovian. The terminology used for the jump
measure of $\hat{\xi}$ is justified by the following remarkable property: for
any Borel $A$ with $\Lambda(A)<\infty$, the counting process of jumps
$\Delta\hat{\xi}_{s}\in A$ that we denote by $\hat{N}_{A}$ is a reinforced
Poisson process and, more precisely, it has the law of the noise reinforced
version of $N_{A}$ (hence, the terminology $\hat{N}_{A}$ is consistent).
Moreover, for any disjoint Borel sets $A_{1},\dots,A_{k}$ with
$\Lambda(A_{i})<\infty$, the corresponding
$\hat{N}_{A_{1}},\dots,\hat{N}_{A_{k}}$ are independent noise reinforced
Poisson processes. Informally, the reinforcement induces memory on the jumps
of $\hat{\xi}$, and these are repeated at the jump times of an independent
counting process. When working on the unit interval, this counting process is
the so-called Yule-Simon process.

The second main result of this work consists in defining pathwise, the noise
reinforced version $\hat{\xi}$ of the Lévy process $\xi$. We always denote
such a pair by $(\xi,\hat{\xi})$. This is mainly achieved by transforming the
jump measure of $\xi$ into a NRPPP, by a procedure that can be interpreted as
the continuous time analogue of the reinforcement algorithm we described for
random walks. More precisely, the steps $X_{k}^{(n)}$ of the $n$-skeleton are
replaced by the jumps $\Delta\xi_{s}$ of the Lévy process; each jump of $\xi$
is shared with its reinforced version $\hat{\xi}$ with probability $1-p$,
while with probability $p$ it is discarded and remains independent of
$\hat{\xi}$. We then proceed to justify our construction by showing that the
skeleton of $\xi$ and its reinforced version $(S^{(n)}_{\lfloor
n\cdot\rfloor},\hat{S}^{(n)}_{\lfloor n\cdot\rfloor})$ converge weakly towards
$(\xi,\hat{\xi})$, strengthening (1.1) considerably.

Section 6 is devoted to applications: on the one hand, in Section 6.1 we study
the rates of growth at the origin of $\hat{\xi}$ and prove that well know
results established by Blumenthal and Getoor in [9] for Lévy processes still
hold for NRLPs. On the other hand, in Section 6.2 we analyse NRLPs under the
scope of infinitely divisible processes in the sense of [21]. We shall give a
proper description of $\hat{\xi}$ in terms of the usual terminology of
infinitely divisible processes, as well as an application, by making use of
the so-called Isomorphism theorem for infinitely divisible processes.

Let us mention that in the discrete setting, reinforcement of processes and
models has been subject of active research for a long time, see for instance
the survey by Pemantle [19] as well as e.g. [6, 3, 1, 18, 2, 11] and
references therein for related work. However, reinforcement of time-continuous
stochastic processes, which is the topic of this work, remains a rather
unexplored subject.
The rest of the work is organised as follows: in Section 2 we recall the basic
building blocs needed for the construction of NRLPs and recall the main
results that will be needed. Notably, we give a brief overview of the features
of the Yule-Simon process and present some important examples of NRLPs. In
Section 3 we show that a NRLP has a rcll modification. In Section 4 we
construct NRPPPs, study their main properties of interest, and in Section 4.3
we prove that the jump measure of a NRLP is a NRPPP – a result that we refer
to as the ”reinforced Lévy-Itô decomposition”. In Section 5 we show that the
pair conformed by the $n$-skeleton of a Lévy process and its reinforced
version converge in distribution, as the mesh size tends to 0, towards
$(\xi,\hat{\xi})$. To achieve this, first we start by proving in Section 5.1
that a NRLP can be reconstructed from its jump measure – a result that we
refer to as the ”reinforced Lévy Itô synthesis”. Making use of this result in
Section 5.2 we define the joint law $(\xi,\hat{\xi})$ and in Section 5.3 we
establish our convergence result. Finally, Section 6 is devoted to
applications. Particular attention is given through this work at comparing,
when possible and pertinent, our results for NRLPs to the classical ones for
Lévy processes.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Yule-Simon processes
    2. 2.2 Noise reinforced Lévy processes
    3. 2.3 Building blocks: noise reinforced Brownian motion and noise reinforced compound Poisson process
  3. 3 Trajectorial regularity
  4. 4 Reinforced Lévy-Itô decomposition
    1. 4.1 The jumps of noise reinforced Poisson processes
    2. 4.2 Construction of noise reinforced Poisson point processes by decoration
    3. 4.3 Proof of Theorem 4.1 and compensator of the jump measure
  5. 5 Weak convergence of the pair of skeletons
    1. 5.1  Proof of Theorem 5.1
    2. 5.2 The joint law $(\xi,\hat{\xi})$ of a Lévy process and its reinforced version
    3. 5.3 Proof of Theorem 5.4
  6. 6 Applications
    1. 6.1 Rates of growth at the origin
    2. 6.2 Noise reinforced Lévy processes as infinitely divisible processes
  7. 7 Appendix

##  2 Preliminaries

###  2.1 Yule-Simon processes

In this section, we recall several results from [7] concerning Yule-Simon
processes needed for defining NRLPs. These results will be used frequently in
this work and are re-stated for ease of reading.

A Yule-Simon process on the interval $[0,1]$ is a counting process, started
from $0$, with first jump time uniformly distributed in $[0,1]$, and behaving
afterwards as a (deterministically) time-changed standard Yule process. More
precisely, for fixed $p\in(0,1)$, if $U$ is a uniform random variable in
$[0,1]$ and $Z$ a standard Yule process,

$Y(t):=\mathds{1}_{\\{U\leq t\\}}Z_{p(\ln(t)-\ln(U))},\quad t\in[0,1],$ (2.1)

is a Yule-Simon process with parameter $1/p$. Its law in $D[0,1]$, the space
of $\mathbb{R}$-valued rcll functions in the unit interval endowed with the
Skorokhod topology, will be denoted by $\mathbb{Q}$. It readily follows from
the definition that this is a time-inhomogeneaous Markov process, with time-
dependent birth rates given at time $t$ by $\lambda_{0}(t)=1/(1-t)$ and
$\lambda_{k}(t)=pk/t$ for $k\in\\{1,2,\dots\\}$. Remark as well that we have
$\mathds{P}\left(Y(t)\geq 1\right)=t$. In our work, only $p\in(0,1)$ will be
used, and it always corresponds to the reinforcement parameter. The Yule-Simon
process with parameter $1/p$ is closely related to the Yule-Simon distribution
with parameter $1/p$, i.e. the probability measure supported on
$\\{1,2,\dots\\}$ with probability mass function given in terms of the Beta
function B$(x,y)$ by

$p^{-1}\text{B}(k,1/p+1)=p^{-1}\int_{0}^{1}u^{p}(1-u)^{k-1}{\mathrm{d}}u,\quad\quad\text{
for }k\geq 1.$ (2.2)

The relation with the Yule process is simply that $Y(1)$ is distributed Yule-
Simon with parameter $1/p$. In this work, we refer to $p\in(0,1)$ as a
reinforcement or memory parameter, for reasons that will be explained shortly.
In the following lemma we state for further use the conditional self-
similarity property of the Yule-Simon process, a key feature that will be used
frequently.

######  Lemma 2.1.

_[7, Corollary 2.3]_
Let $Y$ be a Yule-Simon process with parameter $1/p$ and fix $t\in(0,1]$.
Then, the process $(Y(rt))_{r\in[0,1]}$ conditionally on $\\{Y(t)\geq 1\\}$
has the same distribution $\mathbb{Q}$ as $Y$.

In particular, conditionally on $\\{Y(t)\geq 1\\},\,Y(t)$ is distributed Yule-
Simon with parameter $1/p$ and it follows that for every $t\in[0,1]$, $Y(t)$
has finite moments only of order $r<1/p$. Moreover, by the previous lemma and
the Markov property of the standard Yule process $Z$, we deduce that if $Y$ is
a Yule-Simon process with parameter $1/p$ with $p\in(0,1)$ and $k\geq 1$, we
have

$\mathds{E}\left[Y(t)\right]=(1-p)^{-1}t\quad\quad\text{ and
}\quad\quad\mathds{E}\left[Y(t)|Y(s)=k\right]=k(t/s)^{p}\quad\text{ for any
}0<s\leq t\leq 1,$ (2.3)

while if $1/p>2$,

$\mathds{E}\left[Y(s)Y(t)\right]=\frac{1}{(1-p)(1-2p)}s^{1-p}t^{p}.$ (2.4)

More details on these statements can be found in Section 2 of [7].

###  2.2 Noise reinforced Lévy processes

Now, we turn our attention to the main ingredients involved in the
construction of NRLPs. For the rest of the section, fix a real valued Lévy
process $\xi$ of characteristic triplet $(a,q^{2},\Lambda)$, where $\Lambda$
is the Lévy measure, and recall that its characteristic exponent
$\Psi(\lambda):=\log\mathds{E}\left[e^{i\lambda\xi_{1}}\right]$ is given by
the Lévy-Khintchine formula

$\Psi(\lambda)=ia\lambda-\frac{q^{2}}{2}\lambda^{2}+\int_{\mathbb{R}}\left(e^{i\lambda
x}-1-ix\lambda\mathds{1}_{\\{|x|\leq 1\\}}\right)\Lambda({\mathrm{d}}x).$
(2.5)

The constraints on the reinforcement parameter $p$ are given in terms of the
following two indices introduced by Blumenthal and Getoor: the Blumenthal-
Getoor (upper) index $\beta(\Lambda)$ of the Lévy measure $\Lambda$ is defined
as

$\beta(\Lambda):=\inf\big{\\{}r>0:\int_{[0,1]}|x|^{r}\Lambda({\mathrm{d}}x)<\infty\big{\\}},$
(2.6)

while the Blumenthal-Getoor index $\beta$ of the Lévy process $\xi$ is defined
by the relation

$\beta:=\begin{cases}\beta(\Lambda)\quad\text{ if }q^{2}=0\\\ 2\hskip
25.60747pt\text{ if }q^{2}\neq 0.\end{cases}$ (2.7)

When $\xi$ has no Gaussian component, we have $\beta=\beta(\Lambda)$ and both
notations will be used indifferently. We say that a memory parameter
$p\in(0,1)$ is admissible for the triplet $(a,q^{2},\Lambda)$ if $p\beta<1$.
Now, fix $p$ an admissible memory parameter for $\xi$. If $(S^{(n)}_{k})$ is
the $n$-skeleton of the Lévy process $\xi$, the sequence of reinforced
versions with parameter $p$,

$(\hat{S}^{(n)}_{\lfloor nt\rfloor})_{t\in[0,1]},\hskip 22.76219ptn\geq 1,$

converge in the sense of finite dimensional distributions, as the mesh-size
tends to 0, towards a process whose law was identified in [7] and called the
noise reinforced Lévy process $\hat{\xi}$ of characteristics
$(a,q^{2},\Lambda,p)$. In the sequel, when considering a NRLP with parameter
$p$, it will be implicitly assumed that $p$ is admissible for the
corresponding triplet. For instance, when working with a memory parameter
$p\geq 1/2$ it is implicitly assumed that $q=0$. It was shown in [7, Corollary
2.11] that the finite-dimensional distributions of $\hat{\xi}$ can be
expressed in terms of the Yule-Simon process $Y$ with parameter $1/p$ and the
characteristic exponent $\Psi$ as follows:

$\mathds{E}\left[\exp\left\\{i\sum_{i=1}^{k}\lambda_{i}\hat{\xi}_{s_{i}}\right\\}\right]=\exp\left\\{(1-p)\mathds{E}\left[\Psi\left(\sum_{i=1}^{k}\lambda_{i}Y(s_{i})\right)\right]\right\\},$
(2.8)

for $0<s_{1}<\dots<s_{k}\leq 1$. Now we turn our attention at defining NRLPs
in $\mathbb{R}^{+}$. Notice that the construction given in the unit interval
in [7] can not be directly extended to the real line since it relies on
Poissonian sums of Yule-Simon processes, and these are only defined on the
unit interval.

######  Proposition 2.2.

_(NRLPs in $\mathbb{R}^{+}$)_
Let $(a,q^{2},\Lambda)$ be the triplet of a Lévy process of exponent $\Psi$
and consider an admissible memory parameter $p\in(0,1)$. There exists a
process $\hat{\xi}=(\hat{\xi}_{s})_{s\in\mathbb{R}^{+}}$ whose finite
dimensional distributions satisfy that, for any $0<s_{1}<\dots<s_{k}\leq t$,

$\mathds{E}\left[\exp\left\\{i\sum_{i=1}^{k}\lambda_{i}\hat{\xi}_{s_{i}}\right\\}\right]=\exp\left\\{(1-p)t\mathds{E}\left[\Psi\left(\sum_{i=1}^{k}\lambda_{i}Y(s_{i}/t)\right)\right]\right\\},$
(2.9)

where the right-hand side does not depend on the choice of $t$. The process
$\hat{\xi}$ is called a noise reinforced Lévy process with characteristics
$(a,q^{2},\Lambda,p)$.

###### Proof.

First, let us show that the right-hand side of (2.9) does not depend on $t$.
To prove this, pick another arbitrary $T>t$ and write $r_{i}=s_{i}/t\in[0,1]$.
From conditioning on $\\{Y_{t/T}\geq 1\\}$, an event with probability $t/T$,
by Lemma 2.1 we get

$\displaystyle
T\mathds{E}\left[\Psi\left(\sum_{i=1}^{k}\lambda_{i}Y(s_{i}/T)\right)\right]$
$\displaystyle=t(T/t)\mathds{E}\left[\Psi\left(\sum_{i=1}^{k}\lambda_{i}Y(r_{i}\cdot(t/T))\right)\right]$
$\displaystyle=t\mathds{E}\left[\Psi\left(\sum_{i=1}^{k}\lambda_{i}Y(r_{i}\cdot(t/T))\right)\Big{|}Y(t/T)\geq
1\right]$
$\displaystyle=t\mathds{E}\left[\Psi\left(\sum_{i=1}^{k}\lambda_{i}Y(s_{i}/t)\right)\right],$
(2.10)

proving our claim, and where in the second equality we used that $\Psi(0)=0$.
Now, let us establish the existence of a process with finite-dimensional
distributions characterised by (2.9). Remark that by Kolmogorov’s consistency
theorem, it suffices to show that for arbitrary $1\leq S<T$, there exists
processes $\hat{X}^{S}=(\hat{X}^{S}_{t})_{t\in[0,S]}$,
$\hat{X}^{T}:=(\hat{X}^{T}_{t})_{t\in[0,T]}$ with finite dimensional
distributions characterised by the identity (2.9) for $(s_{i})$ in $[0,S]$,
$t=S$ and $(s_{i})$ in $[0,T]$, $t=T$ respectively – and hence satisfying that
$(\hat{X}^{T}_{t})_{t\in[0,S]}\overset{\mathscr{L}}{=}(\hat{X}^{S}_{t})_{t\in[0,S]}$.
Write $\hat{\xi}^{S}=(\hat{\xi}^{S}_{t})_{t\in[0,1]}$ the reinforced version
of the Lévy process $(\xi_{tS})_{t\in[0,1]}$, remark that the latter has
characteristic exponent $S\Psi$, and set
$(\hat{X}^{S}_{t})_{t\in[0,S]}:=(\hat{\xi}^{S}_{t/S})_{t\in[0,S]}$. From the
identity (2.8), we deduce that, for any $0<s_{1}<\dots<s_{k}$ in the interval
$[0,S]$, we have:

$\displaystyle\mathds{E}\left[\exp\left\\{i\sum_{i=1}^{k}\lambda_{i}\hat{X}^{S}({s_{i}})\right\\}\right]=\exp\left\\{(1-p)S\mathds{E}\left[\Psi\left(\sum_{i=1}^{k}\lambda_{i}Y(s_{i}/S)\right)\right]\right\\}.$
(2.11)

In particular $\hat{X}^{S}$ restricted to the interval $[0,1]$ has the same
distribution as $(\hat{\xi}_{t})_{t\in[0,1]}$ by the first part of the proof
and (2.8). If we consider the restriction of $(\hat{X}^{T})_{t\in[0,T]}$ to
the interval $[0,S]$, we obtain similarly and by applying (2.2) that, for any
$0<s_{1}<\dots<s_{k}\leq S$,

$\displaystyle\mathds{E}\left[\exp\left\\{i\sum_{i=1}^{k}\lambda_{i}\hat{X}^{T}({s_{i}})\right\\}\right]$
$\displaystyle=\exp\left\\{(1-p)T\mathds{E}\left[\Psi\left(\sum_{i=1}^{k}\lambda_{i}Y(s_{i}/T)\right)\right]\right\\}$
$\displaystyle=\exp\left\\{(1-p)S\mathds{E}\left[\Psi\left(\sum_{i=1}^{k}\lambda_{i}Y(s_{i}/S)\right)\right]\right\\},$

and it follows that $\hat{X}^{T}$ restricted to $[0,S]$ has the same
distribution as $\hat{X}^{S}$. Since this holds for any $1\leq S<T$, we deduce
by Kolmogorov’s consistency theorem the existence of a process satisfying for
any $0<s_{1}<\dots<s_{k}\leq t$, the identity (2.9). In particular, from
taking the value $t=1$, it follows that this process satisfies that its
restriction to $[0,1]$ has the same law as $\hat{\xi}$ by (2.8). ∎

For later use, notice from (2.9) that for any fixed $t\in\mathbb{R}^{+}$, we
have the following equality in law

$(\hat{\xi}_{st})_{s\in[0,1]}\overset{\mathscr{L}}{=}(\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[width("\xi_{\cdot
t}")]{\kern 0.1pt\mathchar 866\relax\kern
0.1pt}{\rule{0.0pt}{505.89pt}}}{}}{2.4ex}}\stackon[-6.9pt]{\xi_{\cdot
t}}{\tmpbox})_{s\in[0,1]}$ (2.12)

where the right-hand side stands for the noise-reinforced version of the Lévy
process $(\xi_{st})_{s\in[0,1]}$. In particular,
$(\hat{\xi}_{st})_{s\in[0,1]}$ is the NRLP associated to the exponent $t\Psi$
with same reinforcement parameter.

###  2.3 Building blocks: noise reinforced Brownian motion and noise
reinforced compound Poisson process

The characteristic exponent $\Psi$ can be naturally decomposed in tree terms,

$\Psi(\lambda)=\big{(}ia\lambda-q\frac{\lambda^{2}}{2}\big{)}+\Phi^{(2)}(\lambda)+\Phi^{(3)}(\lambda),$
(2.13)

where respectively, we write

$\Phi^{(2)}(\lambda):=\int_{\\{|x|\geq 1\\}}\left(e^{i\lambda
x}-1\right)\Lambda({\mathrm{d}}x)\quad\text{ and
}\quad\Phi^{(3)}(\lambda):=\int_{\\{|x|<1\\}}\left(e^{i\lambda x}-1-i\lambda
x\right)\Lambda({\mathrm{d}}x).$

This decomposition yields that the Lévy process $\xi$ can be written as the
sum of tree independent Lévy process of radically different nature. Namely, we
have $\xi_{t}=(at+qB_{t})+\xi^{(2)}_{t}+\xi^{(3)}_{t}$, for $t\geq 0$, where
$B$ is a Brownian motion, $\xi^{(2)}$ is a compound Poisson process with
exponent $\Phi^{(2)}$ and $\xi^{(3)}$ is the so-called compensated sum of
jumps with characteristic exponent $\Phi^{(3)}$. In the reinforced setting, it
readily follows from the identity (2.9) that an analogous decomposition holds
for NRLPs. More precisely, the NRLP $\hat{\xi}$ of characteristics
$(a,q^{2},\Lambda,p)$ can be written as a sum of three independent NRLPs,

$\hat{\xi}_{t}\overset{\mathscr{L}}{=}(at+q\hat{B}_{t})+\hat{\xi}^{(2)}_{t}+\hat{\xi}^{(3)}_{t},\quad
t\geq 0,$ (2.14)

the equality holding in law, and where we denoted respectively by $\hat{B}$,
$\hat{\xi}^{(2)}$, $\hat{\xi}^{(3)}$, independent reinforced versions of the
Lévy processes $B$, ${\xi}^{(2)}$, ${\xi}^{(3)}$. Notice that their respective
characteristics are given by $(a,q^{2},0,p)$,
$(0,0,\mathds{1}_{(-1,1)^{c}}\Lambda,p)$ and
$(0,0,\mathds{1}_{(-1,1)}\Lambda,p)$. Let us now give a brief description of
these three building blocks separately:
$\circ$ Noise reinforced Brownian motion: Assume $p<1/2$, consider a Brownian
motion $B$ and set ${\xi}:={B}$. In that case, we simply have
$\Psi(\lambda)=-\lambda^{2}/2$ and we write $\hat{B}$ for the corresponding
noise reinforced Lévy process $\hat{\xi}$. The process $\hat{B}$ is the so-
called noise reinforced Brownian motion (abbreviated NRBM) with reinforcement
parameter $p$, a centred Gaussian process with covariance given by:

$\mathds{E}\left[\hat{B}_{t}\hat{B}_{s}\right]=\frac{(t\vee s)^{p}(t\wedge
s)^{1-p}}{1-2p}.$ (2.15)

Indeed, recalling (2.4), observe first that for any $0\leq t,s<T$ the
covariance (2.15) can be written in terms the Yule-Simon process $Y$ with
parameter $1/p$ as follows:

$\mathds{E}\left[\hat{B}_{t}\hat{B}_{s}\right]=(1-p)T\cdot\mathds{E}\left[Y(t/T)Y(s/T)\right].$
(2.16)

It is now straightforward to deduce from (2.9) with
$\Psi(\lambda)=-\lambda^{2}/2$ that the noise reinforced version of $B$
corresponds to the Gaussian process with covariance (2.15). The noise
reinforced Brownian motion admits a simple representation as a Wiener
integral. More precisely, the process

$t^{p}\int_{0}^{t}s^{-p}{\mathrm{d}}B_{s},\quad\quad t\geq 0,$ (2.17)

has the law of a noise reinforced Brownian motion with parameter $p$. Remark
that when $p=0$, there is no reinforcement and we recover a Brownian motion in
(2.17). As was already mentioned, noise reinforced Brownian motion plays the
role of Brownian motion in the reinforced setting, since it is the scaling
limit of noise reinforced random walks under mild assumptions on the law of
the typical step. We refer to [8, 4] for a detailed discussion.
$\circ$ Noise reinforced compound Poisson process: If $\xi$ is a compound
Poisson process with rate $c>0$ and jumps with law $P_{X}$, then its Lévy
measure is just $\Lambda({\mathrm{d}}x)=cP_{X}({\mathrm{d}}x)$, and any
$p\in(0,1)$ is admissible. When working in $[0,1]$, the noise reinforced
compound Poisson process $\hat{\xi}$ admits a simple representation in terms
of Poissonian sums of Yule-Simon processes. In this direction, let
$\mathbb{Q}$ be the law of the Yule Simon process with parameter $1/p$ and
consider a Poisson random measure $\mathcal{M}$ in $\mathbb{R}^{+}\times
D[0,1]$ with intensity $(1-p)\Lambda\otimes\mathbb{Q}$. If we denote its atoms
by $(x_{i},Y_{i})$, the process

$\hat{\xi}_{t}=\sum_{i}x_{i}Y_{i}(t),\quad t\in[0,1],$ (2.18)

has the law of the noise reinforced version of $\xi$ with reinforcement
parameter $p$ – as can be easily verified by Campbell’s formula and was
already established in [7, Corollary 2.11]. Notice that (2.18) is a finite
variation process and its jump sizes are dictated by $P_{X}({\mathrm{d}}x)$.
Getting back to (2.14), it readily follows form our discussion that the NRLP
$\hat{\xi}^{(2)}$ associated with the exponent $\Phi^{(2)}$ is a reinforced
compound Poisson process and its jumps-sizes are greater than one. Finally,
notice that if $P_{X}=\delta_{1}$, the Lévy process $\xi$ is just a Poisson
process with rate $c$ and we deduce from the last display a simple
representation for the reinforced Poisson process $\hat{N}$ in $[0,1]$.
Observe that it is a counting process, since the atoms $x_{i}$ are then
identically equal to 1.
$\circ$ Noise reinforced compensated compound Poisson process:  Let us now
introduce properly $\hat{\xi}^{(3)}$, viz. the noise reinforced version of the
compensated martingale ${\xi}^{(3)}$. When working in $[0,1]$, this process
also admits a representation in terms of random series of Yule-Simon
processes. In this direction, consider
${\mathcal{M}}:=\sum_{i}\delta_{(x_{i},Y_{i})}$ a Poisson random measure with
intensity $(1-p)\Lambda\otimes\mathbb{Q}$ and for each $a\in[0,1]$, set

$\hat{\xi}^{(3)}_{a,1}(t):=\sum_{i}\mathds{1}_{\\{a\leq|x_{i}|<1\\}}x_{i}Y_{i}(t)-t\int_{\\{a\leq|x|<1\\}}x\Lambda({\mathrm{d}}x),\quad
t\in[0,1].$ (2.19)

In the terminology of [7, Section 2], the process $\hat{\xi}^{(3)}_{a,1}$ is a
Yule-Simon compensated series 333The notation used in [7] for
$\hat{\xi}^{(2)}_{t}$ and $\hat{\xi}^{(3)}_{t}$ is respectively
$\Sigma_{1,\infty}(t)$ and $\Sigma_{0,1}^{(c)}(t)$. These are respectively
referred to as Yule-Simon series and compensated Yule-Simon series., and note
that $\mathbb{E}[\hat{\xi}^{(3)}_{a,1}(t)]=0$ for every $t\in[0,1]$. Moreover,
the following family indexed by $a\in(0,1)$,

$\hat{\xi}^{(3)}_{a,1}(t),\quad\text{ for }t\in[0,1],$ (2.20)

is a collection of NRLPs with memory parameter $p$, Lévy measure
$\mathds{1}_{\\{a\leq|x|<1\\}}\Lambda({\mathrm{d}}x)$ and the corresponding
exponent writes:

$\Phi_{a}^{(3)}(\lambda):=\int_{\\{a\leq|x|<1\\}}\left(e^{i\lambda
x}-1-i\lambda x\right)\Lambda({\mathrm{d}}x).$

Notice that for each $a>0$, the process $\hat{\xi}^{(3)}_{a,1}$ is rcll and
with jump-sizes in $[a,1]$. Now, the process defined at each fixed $t$ as the
pointwise and $L_{1}(\mathbb{P})$-limit

$\hat{\xi}^{(3)}_{t}:=\lim_{a\downarrow 0}\hat{\xi}^{(3)}_{a,1}(t),$ (2.21)

is a NRLP with characteristics $(0,0,\mathds{1}_{\\{|x|<1\\}}\Lambda)$. In
contrast with $\xi^{(3)}$, the noise reinforced version $\hat{\xi}^{(3)}$ is
no longer a martingale, we shall discuss this point in the next section in
detail. For latter use, we point out from [7, Section 2] that the convergence
in the previous display also holds in $L_{r}(\mathbb{P})$, for $r$ chosen
according to

$r\in(\beta(\Lambda)\vee 1,1/p),\text{ if }1/p\leq 2\hskip 14.22636pt\text{
and }\hskip 14.22636ptr=2,\text{ if }1/p>2.$ (2.22)

In particular, we have $\hat{\xi}^{(3)}_{t}\in L_{r}(\mathbb{P})$ and
$\mathbb{E}[\hat{\xi}^{(3)}_{t}]=0$ for every $t$. We refer to [7] for a
complete account on this construction and for a proof of the convergence in
(2.21). The convergence in (2.21) will be strengthen in the sequel, by showing
that it holds uniformly in $[0,1]$. At this point, we have introduced the main
ingredients needed for this work.

##  3 Trajectorial regularity

The purpose of this short section is to establish the following regularity
theorem:

######  Theorem 3.1.

A noise reinforced Lévy process $\hat{\xi}$ has a rcll modification, that we
still denote by $\hat{\xi}$. Moreover, if for $\varepsilon\in(0,1)$,
$\hat{\xi}^{(3)}_{0,\varepsilon}$ denotes a NRLP with characteristics
$(0,0,\mathds{1}_{\\{|x|<\varepsilon\\}}\Lambda,p)$, then for any $t>0$ we
have:

$\lim_{\varepsilon\downarrow 0}\mathds{E}\left[\sup_{s\leq
t}|\hat{\xi}^{(3)}_{0,\varepsilon}(s)|\right]=0.$ (3.1)

Before proving this result, let us explain the role of (3.1). Working in
$[0,1]$ and with the construction (2.21) for $\hat{\xi}^{(3)}$, remark that
for any $\varepsilon\in(0,1)$ we can write
$\hat{\xi}^{(3)}=\hat{\xi}^{(3)}_{0,\varepsilon}+\hat{\xi}^{(3)}_{\varepsilon,1}$,
where $|\Delta\hat{\xi}_{\varepsilon,1}^{(3)}(t)|\geq\varepsilon$ for every
jump-time $t\in[0,1]$ by construction. Now, the convergence (3.1) shows that
in fact, the jumps of $\hat{\xi}$ of size greater than $\varepsilon$ are
precisely the jumps of $\hat{\xi}^{(2)}+\hat{\xi}^{(3)}_{\varepsilon,1}$.
Hence, when working in $[0,1]$, the jumps of $\hat{\xi}^{(3)}$ are precisely
the jumps of the weighted Yule-Simon processes $x_{i}Y_{i}(t)$ –
heuristically, this is the continuous-time analogue of the dynamics described
for the noise reinforced random walk. This fact will be used in Section 4.3.
Moreover, (3.1) allow us to improve the convergence stated in (2.21) towards
$\hat{\xi}^{(3)}$. Namely, it follows that for some subsequence $(a_{n})$ with
$a_{n}\downarrow 0$ as $n\uparrow\infty$, the convergence

$\lim_{n\rightarrow\infty}(\hat{\xi}_{a_{n},1}^{(3)}(s))_{s\in[0,1]}=(\hat{\xi}^{(3)}_{s})_{s\in[0,1]},$

holds a.s. uniformly in $[0,1]$. Remark that the convergence in the previous
display was only stated when working in $[0,1]$ since, so far, the only
explicit construction of NRLPs is the one in the unit interval we recalled
from [7]. In Section 5.1 we shall address this point.

The rest of the section is devoted to the proof of Theorem 3.1. Recalling the
building blocks introduced in Section 2.3 and the identity in distribution
(2.14), $\hat{\xi}^{(2)}$ is a reinforced compound Poisson process and hence
has finite variation rcll trajectories, while $\hat{B}$ is continuous. It is
then clear that the only difficulty consists in establishing the regularity of
the process $\hat{\xi}^{(3)}$ and we rely on a remarkable martingale
associated with centred NRLPs, that we now introduce. This martingale will
play a key role in this work.

######  Proposition 3.2.

Consider a Lévy process $\xi$ with characteristic exponent $\Psi$ satisfying
$\Psi^{\prime}(0)=0$ and Lévy measure fulfilling the integrability condition
$\int_{\\{|x|\geq 1\\}}x\Lambda({\mathrm{d}}x)<\infty$. Then, the process
$M=(M_{t})_{t\in\mathbb{R}^{+}}$ defined as $M_{0}=0$ and for $t>0$, as
$M_{t}=t^{-p}\hat{\xi}_{t}$, is a martingale. Consequently, $M$ has a rcll
modification.

###### Proof.

Recall from (2.14) that in that case, $\hat{\xi}$ can be written as a sum of
two independent processes $\hat{\xi}=q\hat{B}+\hat{\xi}^{(3)}$, where
$\hat{B}$ is a noise reinforced Brownian motion. Recalling the representation
(2.17) for $\hat{B}$, it follows that
$(t^{-p}\hat{B}_{t})_{t\in\mathbb{R}^{+}}$ is a continuous martingale and we
assume therefore that $q=0$.

Turning our attention to $\hat{\xi}^{(3)}$, notice that $M_{t}$ is in
$L_{r}(\mathbb{P})$ for $r$ chosen according to (2.22) and that $E[M_{t}]=0$
since, as we discussed after (2.22), we have $\hat{\xi}^{(3)}_{t}\in
L_{r}(\mathbb{P})$, $\mathbb{E}[\hat{\xi}^{(3)}_{t}]=0$. Now, it remains to
show that $(M_{t})_{t\in(0,1]}$ satisfies the martingale property. In this
direction it is enough to check that for any $0<t_{0}<\dots<t_{k}<t$ and
$\lambda_{1},\dots,\lambda_{k-1}\in\mathbb{R}$, we have

$\mathds{E}\left[t_{k}^{-p}\hat{\xi}^{(3)}_{t_{k}}\exp\left\\{i\sum_{i=1}^{k-1}\lambda_{i}\hat{\xi}^{(3)}_{t_{i}}\right\\}\right]=\mathds{E}\left[t_{k-1}^{-p}\hat{\xi}^{(3)}_{t_{k-1}}\exp\left\\{i\sum_{i=1}^{k-1}\lambda_{i}\hat{\xi}^{(3)}_{t_{i}}\right\\}\right].$
(3.2)

On the one hand, under our standing assumptions, the left-hand side of (3.2)
corresponds to the derivative at $\lambda_{k}=0$ of (2.9) multiplied by
$-it^{-p}_{k}$ and hence equals:

$-it(1-p)\exp\left\\{t(1-p)\mathds{E}\left[\Psi\left(\sum_{j=1}^{k-1}\lambda_{j}Y(t_{j}/t)\right)\right]\right\\}\cdot\mathds{E}\left[H\left(Y(s)\,:s\leq
t_{k-1}/t\right)Y(t_{k}/t)\right]t_{k}^{-p},$

for $H$ defined as

$\quad H\left(Y(s)\,:s\leq
t_{k-1}/t\right):=\Psi^{\prime}\left(\sum_{j=1}^{k-1}\lambda_{i}Y(t_{j}/t)\right).$

Remark that this is a $\sigma(Y(s):s\leq t_{k-1}/t)$-measurable random
variable. On the other hand, the right-hand side of (3.2) corresponds to the
derivative with respect to $\lambda_{k-1}$ of (2.9) multiplied by
$-it_{k-1}^{p}$ for $\lambda_{k}=0$ and similarly, we deduce that the right-
hand side of (3.2) writes:

$-it(1-p)\exp\left\\{t(1-p)\mathds{E}\left[\Psi\left(\sum_{j=1}^{k-1}\lambda_{j}Y(t_{j}/t)\right)\right]\right\\}\cdot\mathds{E}\left[H\left(Y(s)\,:s\leq
t_{k-1}/t\right)Y(t_{k-1}/t)\right]t_{k-1}^{-p}.$

Now, it only remains to show that:

$\mathds{E}\left[H(Y(s)\,:s\leq
t_{k-1}/t)Y(t_{k}/t)\right]t_{k}^{-p}=\mathds{E}\left[H\left(Y(s)\,:s\leq
t_{k-1}/t\right)Y(t_{k-1}/t)\right]t_{k-1}^{-p}.$ (3.3)

Notice that since $\Psi^{\prime}(0)=0$ and $Y$ is increasing,
$H\left(Y(s)\,:s\leq t_{k-1}/t\right)$ vanishes if $Y({t_{k-1}}/t)=0$. This
allows us to restrict the terms inside the expectations in (3.3) to
$\\{Y(t_{k-1}/t)\geq 1\\}$ and to apply the Markov property (2.3) at time
$t_{k-1}/t$ to get:

$\displaystyle\mathds{E}\left[H(Y(r):r\leq
t_{k-1}/t)Y(t_{k}/t)\right]t^{-p}_{k}$
$\displaystyle=\sum_{j=1}^{\infty}\mathds{E}\left[H(Y(r):r\leq
t_{k-1}/t)\mathds{E}\left[Y(t_{k}/t)|Y(t_{k-1}/t)=j\right]\mathds{1}_{\\{Y(t_{k-1}/t)=j\\}}\right]t^{-p}_{k}$
$\displaystyle=\sum_{j=1}^{\infty}\mathds{E}\left[H(Y(r):r\leq t_{k-1}/t)\cdot
j(t_{k}/t_{k-1})^{p}\mathds{1}_{\\{Y(t_{k-1}/t)=j\\}}\right]t^{-p}_{k}$
$\displaystyle=\mathds{E}\left[H(Y(r):r\leq
t_{k-1}/t)Y({t_{k-1}}/t)\right]t^{-p}_{k-1},$ (3.4)

proving the claim. ∎

Let us now conclude the proof of Theorem 3.1.

###### Proof of Theorem 3.1..

The first assertion is now a consequence of the following simple observation:
denoting by $\,\overline{\\!{M}}$ the rcll modification of the martingale
$M=(t^{-p}\hat{\xi}^{(3)}_{t})_{t\in\mathbb{R}^{+}}$, it is then clear that
the process $\hat{J}^{(3)}:=t^{p}\,\overline{\\!{M}}_{t}$, for $t\geq 0$, is a
rcll modification of $\hat{\xi}^{(3)}$. Notice by intergrating by parts that
consequently, the process $\hat{\xi}^{(3)}$ is a semimartingale, this will be
needed in Section 4.3. To prove the second claim, remark that by the
observation right after (2.12), it suffices to work on the time interval
$[0,1]$. Moreover, by Proposition 3.2, for each $\varepsilon>0$, the process

$M^{(\varepsilon)}:=(s^{-p}\hat{\xi}_{0,\varepsilon}^{(3)}(s))_{s\in(0,1]}$

with $M^{(\varepsilon)}_{0}=0$, is a $L_{r}(\mathbb{P})$ rcll martingale in
$[0,1]$, for $r$ chosen according to (2.22). Since $r>1$, by Doob’s inequality
at time $t=1$ we have

$\displaystyle\mathds{E}\left[\sup_{s\leq
t}|\hat{\xi}^{(3)}_{0,\varepsilon}(s)|^{r}\right]\leq\mathds{E}\left[\sup_{s\leq
t}\big{|}s^{-p}\hat{\xi}^{(3)}_{0,\varepsilon}(s)\big{|}^{r}\right]\leq
C_{r}\mathds{E}\left[|\hat{\xi}^{(3)}_{0,\varepsilon}(1)|^{r}\right],$

for some constant $C_{r}>0$, and it remains to show that the right-hand side
converges to $0$ as $\varepsilon\downarrow 0$. However, this is a consequence
of (2.21). More precisely, recalling the construction detailed in (2.19), note
that $\hat{\xi}^{(3)}_{t}-\hat{\xi}^{(3)}_{\varepsilon,1}(t)$ has the same
distribution as $\hat{\xi}^{(3)}_{0,\varepsilon}(t)$ for every $t\in[0,1]$ and
$\varepsilon>0$. Since the convergence (2.21) still holds in
$L_{r}(\mathbb{P})$, the result follows by taking the limit as
$\varepsilon\downarrow 0$. ∎

Now that we have established that a NRLP is a rcll process, in the next
section we study the structure of its jump process $(\Delta\hat{\xi}_{t})$.
Since it will share striking similarities with the jump process of a Lévy
process, before concluding the section we recall well known results on
$(\Delta\xi_{t})$. Namely, if $\xi$ is a Lévy process with Lévy measure
$\Lambda$, its jump measure

$\mu({\mathrm{d}}t,{\mathrm{d}}x)=\sum_{s}\mathds{1}_{\\{\Delta{\xi}_{s}\neq
0\\}}\delta_{(s,\Delta\xi_{s})}({\mathrm{d}}t,{\mathrm{d}}x),$ (3.5)

is a homogeneous Poisson point process (abbreviated PPP) with characteristic
measure $\Lambda({\mathrm{d}}x)$. Such a PPP can be constructed by decorating
the point process of jumps of a Poisson process, and it is classic that (3.5)
is determined by the following two properties:

  1. (i)

For any Borelian $A$ with $\Lambda(A)<\infty$, the counting process of jumps
$\Delta\xi_{s}\in A$ occurring until time $t$, defined as

$N_{A}(t)=\\#\big{\\{}(s,\Delta\xi_{s})\in[0,t]\times A\big{\\}},\quad\quad
t\geq 0,$

is a Poisson process with rate $\Lambda(A)$.

  2. (ii)

If $A_{1},\dots A_{k}$ are disjoint Borelians with $\Lambda(A_{i})<\infty$ for
all $i\in\\{1,\dots,k\\}$, the processes ${N}_{A_{1}},\dots,{N}_{A_{k}}$ are
independent.

In particular, from (i), it follows that
$(N_{A}(t)-\Lambda(A)t)_{t\in\mathbb{R}^{+}}$ is a martingale.

##  4 Reinforced Lévy-Itô decomposition

This section is devoted to the study of the jump process
$(\Delta\hat{\xi}_{s})_{s\in\mathbb{R}^{+}}$ and the associated jump measure
in $\mathbb{R}^{+}\times\mathbb{R}$, viz.

$\hat{\mu}({\mathrm{d}}t,{\mathrm{d}}x):=\sum_{s}\mathds{1}_{\\{\Delta\hat{\xi}_{s}\neq
0\\}}\delta_{(s,\Delta\hat{\xi}_{s})}({\mathrm{d}}t,{\mathrm{d}}x).$ (4.1)

In this direction, we shall introduce in Definition 4.5 below a family of
random measures in $\mathbb{R}^{+}\times\mathbb{R}$ under the name noise
reinforced Poisson point processes – abbreviated NRPPPs – that will play the
analogous role of PPPs for the jump measure of Lévy processes. Each element of
this family of measures is parametrized by a sigma finite measure $\Lambda$ in
$\mathbb{R}$, that we refer to as its characteristic measure, and a positive
value $p\in(0,1)$, that we call its reinforcement parameter. The construction
of NRPPPs consists essentially in the reinforced version of the one of PPPs.
More precisely, we shall construct them by decorating the point process of
jumps of a reinforced Poisson process. The main result of this section is the
reinforced version of the celebrated Lévy-Itô decomposition:

######  Theorem 4.1.

_(Reinforced Lévy-Itô decomposition)_
The jump measure $\hat{\mu}$ of $\hat{\xi}$ is a noise reinforced Poisson
point process with characteristic measure $\Lambda({\mathrm{d}}x)$ and
reinforcement parameter $p$.

The rest of the section is organised as follows: In Section 4.1 we restrict
our study to the jump process of reinforced Poisson processes. In Section 4.2,
we construct NRPPPs by decorating the jump process of reinforced Poisson
processes and then study its basic properties. For instance, in Proposition
4.8 we prove a characterisation in the same vein as the one holding for PPPs,
recalled at the end of Section 3. Finally, in Section 4.3 we prove Theorem 4.1
and in Proposition 4.11 we identify the predictable compensator of
$\hat{\mu}$.

###  4.1 The jumps of noise reinforced Poisson processes

Let us start by introducing the basic building block of this section.
$\circ$ Noise reinforced Poisson process: When $\xi$ is a Poisson process $N$
with rate $c$, any reinforcement parameter $p\in(0,1)$ is admissible and
recall from the discussion following (2.18) that $\hat{N}$ is a counting
process. Moreover, the corresponding noise reinforced Poisson process
(abbreviated NRPP) with rate $p$ has finite dimensional distributions
characterised, for any $0<s_{1}<\dots<s_{k}\leq t$ and
$\lambda_{j}\in\mathbb{R}$, by the identity

$\mathds{E}\left[\exp\left\\{i\sum_{i=1}^{k}\lambda_{i}\hat{N}_{s_{i}}\right\\}\right]=\exp\left\\{(1-p)ct\mathds{E}\left[\left(\exp\bigg{\\{}{i\sum_{i=1}^{k}\lambda_{i}Y(s_{i}/t)}\bigg{\\}}-1\right)\right]\right\\}.$
(4.2)

A Poisson process with rate $c$ has associated to it the random measure
${\mathrm{d}}N_{s}$, also called its point process of jumps. This is a Poisson
random measure in $\mathbb{R}^{+}$ with intensity $c{\mathrm{d}}t$ and it has
a natural reinforced counterpart: namely, the random measure
${\mathrm{d}}\hat{N}_{s}$, that we shall now study in detail.
To do so, we start by introducing some standard notation for point processes.
We shall identify discrete random sets
$D=\\{t_{1},t_{2},\dots\\}\subset\mathbb{R}$ with counting measures
$\sum_{t\in D}\delta_{t}$ and for $f:\mathbb{R}\mapsto\mathbb{R}$, we use the
notation $\langle D,f\rangle$ for $\sum_{t\in D}f(t)$. The collection of
counting measures in $\mathbb{R}$ is denoted by $\mathcal{M}_{c}$. We will
make use of the following two basic transformations: for $x\in\mathbb{R}$, we
denote by $\mathcal{T}_{x}D$ the translated point process $\\{t+x:t\in D\\}$
and for $f:\mathbb{R}\mapsto\mathbb{R}$, we write $D\circ f^{-1}$ the push-
forwarded point process $\\{f(t):t\in D\\}$.

Now, consider an increasing sequence of random times
$0=T_{0}<T_{1}<T_{2}<...$, such that the increments $(T_{n}-T_{n-1}:n\geq 1)$
are independent and for any $n\geq 1$, $T_{n}-T_{n-1}$ is exponentially
distributed with parameter $pn$. Write $D:=\\{0,T_{1},T_{2},...\\}$ the point
process associated to this family and we denote its law in $\mathcal{M}_{c}$
by $\mathbb{D}({\mathrm{d}}\mu)$. From these ingredients, we define a
decorated measure as follows: first, consider $\mathscr{E}$ a Poisson point
process with intensity $c(1-p)e^{t}{\mathrm{d}}t$ in $\mathbb{R}$ and, for
each atom $u\in\mathscr{E}$, let $D_{u}$ be an independent copy of $D$. Then,
we set

$\mathcal{L}({\mathrm{d}}s):=\sum_{u\in\mathscr{E}}\sum_{t\in
D_{u}}\delta_{u+t}=\sum_{u\in\mathscr{E}}\mathcal{T}_{u}D_{u}.$ (4.3)

Remark that if $(Z_{t})$ is a standard Yule process started from $1$, $D$ has
the same law as the point process induced by the jump-times of $(Z_{tp})$,
with a Dirac mass at $0$. The next proposition shows that the law of the point
process of jumps of a noise reinforced Poisson process with rate $c$ is
precisely $\mathcal{L}\circ\exp^{-1}$, the pushforward of $\mathcal{L}$ by the
exponential function.

######  Proposition 4.2.

The following properties hold:

  1. _(i)_

Let $\hat{N}$ be a noise-reinforced Poisson process with rate $c$ and write
$\hat{\mathscr{P}}:={\mathrm{d}}\hat{N}_{s}$ the point process of its jump-
times in $\mathbb{R}^{+}$. Then, we have the equality in distribution
$\hat{\mathscr{P}}\overset{\mathscr{L}}{=}\mathcal{L}\circ\exp^{-1}$. We will
still refer to $\hat{\mathscr{P}}$ as a reinforced Poisson process with rate
$c$ and reinforcement parameter $p$.

  2. _(ii)_

If $Y$ is a Yule-Simon process with parameter $1/p$, for any
$f:\mathbb{R}^{+}\mapsto\mathbb{R}^{+}$ we have

$-\log\mathds{E}\left[\exp\bigg{\\{}-\langle\hat{\mathscr{P}},\mathds{1}_{(0,t]}f\rangle\bigg{\\}}\right]=tc\cdot(1-p)\mathds{E}\left[1-e^{-\int_{0}^{1}f(st){\mathrm{d}}Y(s)}\right].$
(4.4)

In particular, from (4.3) and (i) we deduce the following identity in
distribution: if $\mathscr{P}$ is a Poisson process in $\mathbb{R}_{+}$ with
intensity $c(1-p){\mathrm{d}}t$, we have

$\hat{\mathscr{P}}=\sum_{s\in\mathbb{R}^{+}}\mathds{1}_{\\{s:\Delta\hat{N}_{s}=1\\}}\delta_{s}\overset{\mathscr{L}}{=}\sum_{u\in\mathscr{P}}\sum_{t\in
D_{u}}\delta_{ue^{t}}.$ (4.5)

Roughly speaking, the jumps of $\hat{N}$ consist in Poissonian jumps
$u\in\mathscr{P}$ which – in analogy with the discrete setting – we refer to
as innovations, and each $u$ has attached to it a family $\\{ue^{t}:t\in
D_{u},\,t\neq 0\\}$ which should be interpreted as repetitions of the original
$u$ through time.

Figure 1: Sketch of the jumps of a noise reinforced Poisson process. We marked
by  x the jumps corresponding to innovations, while each linked o is a
repetition of the former.

Notice that the time at which $u$ occurs affects the rate of the subsequent
repetitions, slowing the rate down as $u$ grows. This is closely related to
what happens to the rate at which a step is repeated in a step reinforced
random walk, depending on its first time of appearance. For later use, remark
that for fixed $u\in\mathbb{R}^{+}$, the atoms of $\sum_{t\in
D}\delta_{ue^{t}}$ are distributed as the jump times of the counting process

$\mathds{1}_{\\{u\leq s\\}}Z_{p(\ln(s)-\ln(u))}\,,\quad\quad s\geq 0.$ (4.6)

###### Proof.

To establish the identity in distribution stated in (i), we compute the
respective Laplace functional of both random measures. Starting with
$\hat{\mathscr{P}}$, fix $t\geq 0$ and recall from the identity in
distribution (2.12) that $(\hat{N}_{ts})_{s\in[0,1]}$ has the same law as a
noise reinforced Poisson process with same reinforcement parameter $p$ and
rate $tc$, say $(\hat{N}^{(t)}_{s})_{s\in[0,1]}$. This NRLP is defined in
$[0,1]$ and hence admits a simple representation in terms of Poisson random
measures: by (2.18), if $\sum_{i}\delta_{Y_{i}}$ is a Poisson random measure
in $D[0,1]$ with intensity $tc(1-p)\mathbb{Q}$, the process
$(\sum_{i}Y_{i}(s):s\in[0,1])$ has the same distribution as
$(\hat{N}^{(t)}_{s})_{s\in[0,1]}$. In particular, we have

$\int_{0}^{t}f(s){\mathrm{d}}\hat{N}_{s}=\int_{0}^{1}f(st){\mathrm{d}}\hat{N}_{st}\overset{\mathscr{L}}{=}\sum_{i}\int_{0}^{1}f(st){\mathrm{d}}Y_{i}(s).$

Putting everything together, we deduce (4.4) by making use of the Laplace
formula for integrals with respect to Poisson random measures – we invite the
reader to compare (4.4) with the identity (2.9) for the finite-dimensional
distributions of NRLPs – and it remains to show that the Laplace functional of
$\mathcal{L}\circ\exp^{-1}$ coincide with this expression.

In this direction, recall the observation made in (4.6) and denote by Z the
law of the standard Yule process $Z$. It follows that the law of
$\langle\mathcal{L}\circ\exp^{-1},1_{(0,t]}f\rangle$ can be expressed in terms
of the Poisson random measure $\mathcal{M}:=\sum_{i}\delta_{(u_{i},Z^{(i)})}$
in $\mathbb{R}^{+}\times D[0,1]$, with intensity
$c(1-p){\mathrm{d}}t\otimes\textbf{{Z}}$, by considering the functional

$\sum_{i}\int_{(0,t]}f(s)\,{\mathrm{d}}\left(\mathds{1}_{\\{u_{i}\leq
s\\}}Z^{(i)}_{p(\ln(s)-\ln(u_{i}))}\right),$

where the integrals in the previous expression are respectively with respect
to the Stieltjes measure associated to the counting process
$s\mapsto\mathds{1}_{\\{u_{i}\leq s\\}}Z^{(i)}_{p(\ln(s)-\ln(u_{i}))}$. It now
follows also by the exponential formula that

$\displaystyle-\log\mathds{E}\left[e^{-\langle\mathcal{L}\circ\exp^{-1},1_{\\{\cdot\leq
t\\}}f\rangle}\right]$
$\displaystyle=(1-p)c\int_{\mathbb{R}}{\mathrm{d}}u\,\mathds{E}\left[1-\exp\bigg{\\{}-\int_{0}^{t}f(s){\mathrm{d}}\left(\mathds{1}_{\\{u\leq
s\\}}Z_{p(\ln(s)-\ln(u))}\right)\bigg{\\}}\right]$
$\displaystyle=(1-p)c\int_{0}^{t}{\mathrm{d}}u\,\mathds{E}\left[1-\exp\bigg{\\{}-\int_{0}^{t}f(s){\mathrm{d}}\left(\mathds{1}_{\\{u\leq
s\\}}Z_{p(\ln(s)-\ln(u))}\right)\bigg{\\}}\right]$
$\displaystyle=t(1-p)c\cdot\textbf{{Z}}^{\bullet}\left(1-\exp\bigg{\\{}\int_{0}^{1}f(st){\mathrm{d}}\left(\mathds{1}_{\\{u\leq
st\\}}Z_{p(\ln(st)-\ln(u))}\right)\bigg{\\}}\big{|}u\leq t\right),$ (4.7)

where we denoted in the last line by $\textbf{{Z}}^{\bullet}(\,\cdot\,|u\leq
t)$ the integral in $\mathbb{R}^{+}\times D[0,\infty)$ with respect to the
probability measure

$\textbf{{Z}}^{\bullet}(\,\cdot\,|u\leq t):=\frac{\mathds{1}_{\\{u\leq
t\\}}}{t}{\mathrm{d}}u\,\textbf{{Z}}({\mathrm{d}}Z).$

Now, we deduce by Lemma 7.1 \- (ii) in the Appendix that (4.1) is precisely
(4.4). ∎

Finally, for later use we state the following equivalent expression for the
Laplace functional associated to the random measure
$\mathcal{L}\circ\exp^{-1}$.

######  Lemma 4.3.

For any measurable $f:\mathbb{R}^{+}\mapsto\mathbb{R}^{+}$, we have

$-\log\mathds{E}\left[\exp\bigg{\\{}-\langle\mathcal{L}\circ\exp^{-1},f\rangle\bigg{\\}}\right]=(1-p)c\int_{0}^{\infty}{\mathrm{d}}u\int_{\mathcal{M}_{c}}1-e^{-\langle
T_{\log(u)}\mu,f\circ\,\exp\rangle}\mathbb{D}({\mathrm{d}}\mu).$ (4.8)

###### Proof.

The proof follows from the equality
$\langle\mathcal{L}\circ\exp^{-1},f\rangle=\langle\mathcal{L},f\circ\exp\rangle$
and the identity:

$-\log\mathds{E}\left[\exp\left\\{-\langle\mathcal{L},h\rangle\right\\}\right]=(1-p)c\int_{0}^{\infty}{\mathrm{d}}u\int_{\mathcal{M}_{c}}1-e^{-\langle\mathcal{T}_{\log(u)}\mu,h\rangle}\mathbb{D}({\mathrm{d}}\mu),$

holding for any measurable $h:\mathbb{R}^{+}\mapsto\mathbb{R}^{+}$. The proof
of the later is just a straightforward consequence of (4.3) and the
exponential formula for Poisson random measures. ∎

######  Remark 4.4.

Notice from (4.5) that the reinforced Poisson process with rate $c$ can be
interpreted as a Yule-Simon process with immigration: this is, a process
modelling the evolution of a population where new independent immigrants
arrive according to a Poisson point process with intensity
$(1-p)c\cdot{\mathrm{d}}t$ and reproduce according to a time changed Yule
process, independent of the rest.

###  4.2 Construction of noise reinforced Poisson point processes by
decoration

This section is devoted to the construction of noise reinforced Poisson point
processes and to establishing their first properties. From here, we fix
$p\in(0,1)$.
$\,\bullet$ Step 1: Suppose first that $0<\Lambda(\mathbb{R})<\infty$. With
the same notation of Section 4.1, denote by $\mathscr{E}$ a Poisson random
measure in $\mathbb{R}$ with intensity
$\Lambda(\mathbb{R})(1-p)e^{t}{\mathrm{d}}t$ and consider the Poisson point
process $\Sigma_{u\in\mathscr{E}}\delta_{(u,x_{u})}$ in
$\mathbb{R}\times\mathbb{R}$ with intensity
$(1-p)e^{t}{\mathrm{d}}t\otimes\Lambda({\mathrm{d}}x)$. Now, for each
$u\in\mathscr{E}$, consider an independent copy $D_{u}$ of $D$ and set

$\mathcal{L}^{x}({\mathrm{d}}s,{\mathrm{d}}x):=\sum_{u\in\mathscr{E}}\sum_{t\in
D_{u}}\delta_{(u+t,x_{u})}.$ (4.9)

This is just the point process $\mathcal{L}$ from (4.3) with
$c:=\Lambda(\mathbb{R})$, marked by a collection of i.i.d. random variables
with law $\Lambda({\mathrm{d}}x)/\Lambda(\mathbb{R})$. Formula (4.9) defines a
random measure in $\mathbb{R}\times\mathbb{R}$ and if we consider its push
forward by $(t,x)\mapsto(\exp(t),x)$, that we denote as
$\hat{\mathcal{N}}:=\mathcal{L}^{x}\circ(\exp,\text{Id})^{-1}$, we obtain the
measure in $\mathbb{R}^{+}\times\mathbb{R}$ given by

$\hat{\mathcal{N}}({\mathrm{d}}s,{\mathrm{d}}x):=\sum_{u\in\mathscr{P}}\sum_{t\in
D_{u}}\delta_{(ue^{t},x_{u})},$ (4.10)

where $\mathscr{P}:=\mathscr{E}\circ\exp^{-1}$ is a Poisson point process in
$\mathbb{R}_{+}$ with intensity $\Lambda(\mathbb{R})(1-p){\mathrm{d}}t$. We
refer to the measure in the previous display as a NRPPP with (finite)
characteristic measure $\Lambda$ and reinforcement parameter $p$.
$\bullet$ Step 2:  If we no longer assume $\Lambda(\mathbb{R})<\infty$, we
proceed by superposition. More precisely, let $(A_{j})_{j\in\mathcal{I}}$ be a
disjoint partition of $\mathbb{R}\setminus\\{0\\}$ such that
$\Lambda(A_{j})<\infty$. Consider a collection of independent NRPPPs
$(\hat{\mathcal{N}}_{j}({\mathrm{d}}s,{\mathrm{d}}x):j\in\mathcal{I})$ with
respective characteristic measures $(\Lambda(\,\cdot\cap
A_{j}):j\in\mathcal{I})$ constructed as in (4.10), respectively in terms of:
\- independent Poisson random measures
$\sum_{u\in\mathscr{P}_{j}}\delta_{(u,x_{u})}$ with intensities
$(1-p){\mathrm{d}}t\otimes\Lambda(\,\cdot\cap A_{j})$.
\- independent collections $(D_{u})_{u\in\mathscr{P}_{j}}$ of i.i.d. copies of
$D$.
Finally, set $\mathscr{P}:=\sum_{j}\mathscr{P}_{j}$. Now we are in position to
introduce NRPPPs with sigma-finite characteristic measures:

######  Definition 4.5.

_(Noise Reinforced Poisson Point Process - NRPPP)_
The random measure
$\hat{\mathcal{N}}({\mathrm{d}}s,{\mathrm{d}}x):=\sum_{j\in\mathcal{I}}\hat{\mathcal{N}}_{j}({\mathrm{d}}s,{\mathrm{d}}x)$
is called a reinforced Poisson point process with reinforcement (or memory)
parameter $p$ and characteristic measure $\Lambda$. Moreover,
$\hat{\mathcal{N}}$ writes

$\hat{\mathcal{N}}({\mathrm{d}}s,{\mathrm{d}}x)=\sum_{u\in\mathscr{P}}\sum_{t\in
D_{u}}\delta_{(ue^{t},x_{u})}.$ (4.11)

From the identity in the previous display and recalling that the first element
of $D$ is just $0$, the measure $\hat{\mathcal{N}}$ naturally decomposes as
$\hat{\mathcal{N}}=\mathcal{N}^{\prime}+{\mathcal{N}}^{\prime\prime}$, where
$\mathcal{N}^{\prime}$ is a PPP with intensity
$(1-p){\mathrm{d}}t\otimes\Lambda$. Moreover, the following properties readily
follow from our construction:

######  Lemma 4.6.

Let $\hat{\mathcal{N}}$ be a NRPPP with characteristic measure $\Lambda$ and
reinforcement parameter $p$.

  1. _(i)_

If $A\in\mathscr{B}(\mathbb{R})$, the restriction
$\mathds{1}_{A}(x)\hat{\mathcal{N}}({\mathrm{d}}s,{\mathrm{d}}x)$ is a NRPPP
with characteristic measure $\mathds{1}_{A}\Lambda$ and parameter $p$.

  2. _(ii)_

If $A_{1},A_{2}\in\mathscr{B}(\mathbb{R})$ are disjoints, then
$\mathds{1}_{A_{1}}(x)\hat{\mathcal{N}}({\mathrm{d}}s,{\mathrm{d}}x)$,
$\mathds{1}_{A_{2}}(x)\hat{\mathcal{N}}({\mathrm{d}}s,{\mathrm{d}}x)$ are
independent.

  3. _(iii)_

If $\hat{\mathcal{N}}_{1}$, $\hat{\mathcal{N}}_{2}$ are independent NRPPPs
with respective characteristic measures $\Lambda_{1}$, $\Lambda_{2}$ and same
reinforcement parameter $p$, then
$\hat{\mathcal{N}}_{1}+\hat{\mathcal{N}}_{2}$ is a NRPPP with characteristic
measure $\Lambda_{1}+\Lambda_{2}$ and parameter $p$.

The following lemma shows that the intensity measure of a NRPPP with
characteristic measure $\Lambda$ and parameter $p$, coincides with the one of
a PPP with characteristic measure $\Lambda$.

######  Lemma 4.7.

Let $\hat{\mathcal{N}}$ be a NRPPP with characteristic measure $\Lambda$ and
reinforcement parameter $p$. For any measurable
$f:\mathbb{R}^{+}\times\mathbb{R}\mapsto\mathbb{R}^{+}$, we have
$\mathbb{E}[\langle
f,\hat{\mathcal{N}}\rangle]=\int_{0}^{\infty}{\mathrm{d}}s\int_{\mathbb{R}}\Lambda({\mathrm{d}}x)f(s,x).$

###### Proof.

Suppose first that $\Lambda(\mathbb{R})<\infty$ and recall from (4.6) that for
fixed $u\in\mathbb{R}_{+}$, the atoms of the measure $\sum_{t\in
D}\delta_{ue^{t}}$ are precisely the jumps of the time-changed Yule process
(4.6). Hence, if $\sum_{u\in\mathscr{P}}\delta_{(u,x_{u})}$ is a Poisson
random measure with intensity
$(1-p){\mathrm{d}}t\otimes\Lambda({\mathrm{d}}x)$ and
$(Z^{(u)})_{u\in\mathscr{P}}$ is an independent collection with law Z, it is
then clear from our construction in the finite case (4.10) that we can write

$\mathds{E}\left[\hat{\mathcal{N}}(0,T]\times
A\right]=\mathds{E}\left[\sum_{u\in\mathscr{P}}\mathds{1}_{\\{u\leq
T\\}}Z^{(u)}_{\\{p(\ln(T)-\ln(u))\\}}\mathds{1}_{\\{x_{u}\in A\\}}\right],$

where the random measure $\sum_{u\in\mathscr{P}}\delta_{(u,x_{u},Z^{(u)})}$ is
Poisson with intensity $(1-p){\mathrm{d}}t\otimes\Lambda\otimes\textbf{{Z}}$.
Consequently, recalling that $E[Z_{t}]=e^{t}$, by Campbell’s formula we obtain
that

$\mathds{E}\left[\hat{\mathcal{N}}(0,T]\times A\right]=T\cdot\Lambda(A),$

and we deduce that the intensity measure of $\hat{\mathcal{N}}$ is given by
${\mathrm{d}}t\otimes\Lambda$. When $\Lambda(\mathbb{R})=\infty$, we can
proceed by superposition. ∎

We now identify the law of $\hat{\mathcal{N}}$ by computing its exponential
functionals.

######  Proposition 4.8.

Let $\hat{\mathcal{N}}$ be a NRPPP with characteristic measure $\Lambda$ and
reinforcement parameter $p$.

  1. _(i)_

For every measurable $f:\mathbb{R}^{+}\times\mathbb{R}\mapsto\mathbb{R}^{+}$
and $t\geq 0$ we have

$\mathds{E}\left[\exp\bigg{\\{}-\int_{(0,t]\times\mathbb{R}}f(s,x)\hat{\mathcal{N}}({\mathrm{d}}s,{\mathrm{d}}x)\bigg{\\}}\right]=\exp\bigg{\\{}-t(1-p)\int_{\mathbb{R}}\Lambda({\mathrm{d}}x)\mathds{E}\left[1-\exp\left(-\int_{0}^{1}f(st,x){\mathrm{d}}Y(s)\right)\right]\bigg{\\}}.$
(4.12)

  2. _(ii)_

If we no longer assume that $f$ is non-negative, under the condition
$\int_{0}^{t}{\mathrm{d}}s\int_{\mathbb{R}}\Lambda({\mathrm{d}}x)|f(s,x)|<\infty$
we have:

$\mathds{E}\left[\exp\bigg{\\{}i\int_{(0,t]\times\mathbb{R}}f(s,x)\hat{\mathcal{N}}({\mathrm{d}}s,{\mathrm{d}}x)\bigg{\\}}\right]=\exp\bigg{\\{}t(1-p)\int_{\mathbb{R}}\Lambda({\mathrm{d}}x)\mathds{E}\left[\exp\left(i\int_{0}^{1}f(st,x){\mathrm{d}}Y(s)\right)-1\right]\bigg{\\}}.$
(4.13)

###### Proof.

(i) We start by considering the finite case $\Lambda(\mathbb{R})<\infty$ and
we make use of the notations introduced in (4.9); for instance, recall that
$\langle\hat{\mathcal{N}},f\rangle=\langle\mathcal{L}^{x},f\circ(\exp,\rm{\text{Id}})\rangle$.
We start showing the result for $f$ of the form $f(s,x)=h(s)g(x)$, for non-
negatives $h:\mathbb{R}^{+}\mapsto\mathbb{R}^{+}$ and
$g:\mathbb{R}\mapsto\mathbb{R}^{+}$, in which case we can write

$\displaystyle\langle\mathcal{L}^{x},(h\circ\exp)g\rangle=\sum_{u\in\mathscr{E}}\sum_{t\in
D_{u}}h\circ\exp(u+t)g(x_{u})=\sum_{u\in\mathscr{E}}g(x_{u})\langle\mathcal{T}_{u}D_{u},h\circ\exp\rangle.$
(4.14)

Now, we deduce from the formula for the Laplace transform of Poisson integrals
and a change of variable that

$-\log\mathds{E}\left[e^{-\langle\mathcal{L}^{x},(h\circ\exp)g\rangle}\right]=(1-p)\int_{\mathbb{R}}\Lambda({\mathrm{d}}x)\int_{\mathbb{R}^{+}}{\mathrm{d}}u\int_{\mathcal{M}_{c}}1-e^{-g(x)\langle\mathcal{T}_{\log(u)}\mu,h\circ\exp\rangle}\mathbb{D}({\mathrm{d}}\mu).$

If we now replace $h$ by $h\mathds{1}_{\\{\cdot\leq t\\}}$, making use of the
equivalent identities (4.8) and (4.4), we obtain that the previous display
writes:

$t\cdot(1-p)\int_{\mathbb{R}}\Lambda({\mathrm{d}}x)\mathds{E}\left[1-e^{-g(x)\int_{0}^{1}h(st){\mathrm{d}}Y(s)}\right],$

proving the claim. Now, still under the hypothesis
$\Lambda(\mathbb{R}^{+})<\infty$, fix arbitrary
$\alpha_{i,j}\in\mathbb{R}^{+}$, consider $0=t_{1}<\dots<t_{k+1}<t$ as well as
disjoint subsets $A_{1},\dots,A_{n}$ of $\mathbb{R}^{+}$. Further, suppose
that $f$ is of the form

$f(s,x):=\sum_{j=1}^{n}\sum_{i=1}^{k}\alpha_{i,j}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathds{1}_{A_{j}}(x)\quad\quad\text{
and write }\quad
g_{j}(s,x):=\sum_{i=1}^{k}\alpha_{i,j}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathds{1}_{A_{j}}(x).$
(4.15)

Recall from Lemma 4.6 that the restrictions
$\mathds{1}_{A_{1}}\hat{\mathcal{N}},\dots,\mathds{1}_{A_{n}}\hat{\mathcal{N}}$
are independent NRPPPs with respective characteristic measures
$\Lambda(\cdot\cap A_{i})$. By independence and applying the previous case to
each $g_{j}$, we deduce that

$\displaystyle\mathds{E}\left[\exp\bigg{\\{}-\langle\hat{\mathcal{N}},\mathds{1}_{\\{\cdot\leq
t\\}}f\rangle\bigg{\\}}\right]$
$\displaystyle=\prod_{j=1}^{n}\mathds{E}\left[\exp\bigg{\\{}-\langle\hat{\mathcal{N}},\mathds{1}_{\\{\cdot\leq
t\\}}g_{j}\rangle\bigg{\\}}\right]$
$\displaystyle=\prod_{j=1}^{n}\exp\bigg{\\{}t(1-p)\Lambda(A_{j})\mathds{E}\left[1-\exp\bigg{\\{}-\int_{0}^{1}\sum_{i=1}^{k}\alpha_{i,j}\mathds{1}_{(t_{i}<st\leq
t_{i+1}]}(s){\mathrm{d}}Y(s)\bigg{\\}}\right]\bigg{\\}}$
$\displaystyle=\exp\bigg{\\{}t(1-p)\int_{\mathbb{R}}\Lambda({\mathrm{d}}x)\mathds{E}\left[1-\exp\bigg{\\{}-\int_{0}^{1}f(st,x){\mathrm{d}}Y(s)\bigg{\\}}\right]\bigg{\\}},$

and once again we recover (4.12). Finally, if $f$ is non-negative and bounded
with support in $[0,t]\times\mathbb{R}$, it can be approximated by a bounded
sequence of functions $(f_{n})$ of the form (4.15), the convergence holding
${\mathrm{d}}t\Lambda({\mathrm{d}}x)$ a.e. For each $n$, we have

$\mathds{E}\left[\exp\bigg{\\{}-\langle\hat{\mathcal{N}},f_{n}\rangle\bigg{\\}}\right]=\exp\bigg{\\{}t(1-p)\int_{\mathbb{R}}\Lambda({\mathrm{d}}x)\mathds{E}\left[1-\exp\bigg{\\{}-\int_{0}^{1}f_{n}(st,x){\mathrm{d}}Y(s)\bigg{\\}}\right]\bigg{\\}},$
(4.16)

and by Lipschitz-continuity, it follows that

$\displaystyle\mathds{E}\left[|\exp\big{\\{}-\langle\hat{\mathcal{N}},f\rangle\big{\\}}-\exp\big{\\{}-\langle\hat{\mathcal{N}},f_{n}\rangle\big{\\}}|\right]$
$\displaystyle\leq\mathds{E}\left[\int_{[0,t]\times\mathbb{R}}|f(s,x)-f_{n}(s,x)|\hat{\mathcal{N}}({\mathrm{d}}s,{\mathrm{d}}x)\right]$
$\displaystyle=\int_{0}^{t}{\mathrm{d}}s\int_{\mathbb{R}}\Lambda({\mathrm{d}}x)|f(s,x)-f_{n}(s,x)|\rightarrow
0\,\text{ as }n\uparrow\infty.$

In the last equality we used Lemma 4.7. From the same arguments we also obtain
that

$\displaystyle\int\Lambda({\mathrm{d}}x)\mathds{E}\left[|e^{-\int_{0}^{1}f(st,x){\mathrm{d}}Y(s)}-e^{-\int_{0}^{1}f_{n}(st,x){\mathrm{d}}Y(s)}|\right]$
$\displaystyle\leq\int\Lambda({\mathrm{d}}x)\mathds{E}\left[\int_{0}^{1}|f(st,x)-f_{n}(st,x)|{\mathrm{d}}Y(s)\right]$
$\displaystyle=(1-p)^{-1}\int_{0}^{1}{\mathrm{d}}s\int_{\mathbb{R}}\Lambda({\mathrm{d}}x)|f(st,x)-f_{n}(st,x)|\rightarrow
0\,$

as $n\uparrow\infty$. Now, we deduce from taking the limit as
$n\uparrow\infty$ in (4.16) that the identity (4.12) also holds for $f$.

If we suppose that $\Lambda(\mathbb{R})=\infty$, the proof follows by
superposition. Namely, with the same notation used for constructing (4.11),
the random measures $(\hat{\mathcal{N}}_{j})_{j\in\mathcal{I}}$ are
independent NRPPPs with respective finite characteristic measures
$\Lambda(\,\cdot\,\cap A_{j})$ and by definition we have
$\hat{\mathcal{N}}=\sum_{j}\hat{\mathcal{N}}_{j}$. From the formula for the
Laplace transform we just proved in the finite case and independence it
follows that

$\displaystyle\mathds{E}\left[e^{-\langle\hat{\mathcal{N}},f\mathds{1}_{\\{\cdot\leq
t\\}}\rangle}\right]$
$\displaystyle=\prod_{j\in\mathcal{I}}\mathds{E}\left[e^{-\langle\hat{\mathcal{N}}_{j},f\mathds{1}_{\\{\cdot\leq
t\\}}\rangle}\right]$
$\displaystyle=\prod_{j\in\mathcal{I}}\exp\bigg{\\{}-t\cdot(1-p)\int_{{A}_{j}}{\Lambda({\mathrm{d}}x)}\mathds{E}\left[\left(1-e^{-\int_{0}^{1}f(st,x){\mathrm{d}}Y(s)}\right)\right]\bigg{\\}},$

proving (i). Now (ii) follows from similar arguments, by making use of the
formula for the characteristic function for Poissonian integrals and the
inequality $|e^{ib}-e^{ia}|\leq|a-b|$ for $a,b\in\mathbb{R}$, we omit the
details. ∎

The following result is the reinforced analogue of the well known
characterisation result for Poisson point processes. The arguments we use are
similar to the ones in the non-reinforced case.

######  Proposition 4.9.

Let $\hat{\mathcal{N}}$ be a point process in $\mathbb{R}^{+}\times\mathbb{R}$
and for any Borelian $A\subset\mathbb{R}$, set

$\hat{N}_{A}(t):=\hat{\mathcal{N}}([0,t]\times A),\quad t\geq 0.$

Then, $\hat{\mathcal{N}}$ is a noise reinforced Poisson point process with
characteristic measure $\Lambda$ and parameter $p$ if and only if the two
following conditions are satisfied:

  1. _(i)_

For any Borelian $A$ with $\Lambda(A)<\infty$, the process $\hat{N}_{A}$ is a
noise reinforced Poisson process with rate $\Lambda(A)$ and reinforcement
parameter $p$.

  2. _(ii)_

If $A_{1},\dots A_{k}$ are disjoint Borelians with $\Lambda(A_{i})<\infty$ for
all $i\in\\{1,\dots,k\\}$, the processes
$\hat{N}_{A_{1}},\dots,\hat{N}_{A_{k}}$ are independent.

###### Proof.

First, let us prove that NRPPP do satisfy (i) and (ii). Remark that (ii) is
just a consequence of Lemma 4.6 \- (ii) and we focus on (i). Fix $A$ as in (i)
as well as times $0<t_{1}<\dots<t_{k}\leq t$, and we proceed by computing the
characteristic function of the finite dimensional distributions of
$\hat{N}_{A}$. This can now be done by considering the function
$f(s,x):=\sum_{i=1}^{k}\lambda_{i}\mathds{1}_{\\{s\leq
t_{i}\\}}\mathds{1}_{A}(x)$ and applying the exponential formula (4.13),
yielding

$\displaystyle\mathds{E}\left[\exp\left\\{i\sum_{i=1}^{k}\lambda_{i}\hat{N}_{A}(t_{i})\right\\}\right]$
$\displaystyle=\exp\bigg{\\{}t(1-p)\int_{\mathbb{R}}\Lambda({\mathrm{d}}x)\mathds{E}\left[\exp\left(i\sum_{i=1}^{k}\int_{0}^{1}\lambda_{i}\mathds{1}_{\\{st\leq
t_{i}\\}}\mathds{1}_{A}(x){\mathrm{d}}Y(s)\right)-1\right]\bigg{\\}}.$
$\displaystyle=\exp\left\\{t(1-p)\Lambda(A)\mathds{E}\left[\left(\exp\bigg{\\{}{i\sum_{i=1}^{k}\lambda_{i}Y(t_{i}/t)}\bigg{\\}}-1\right)\right]\right\\}.$

Recalling the identity (4.2), we deduce that $\hat{N}_{A}$ is a noise
reinforced Poisson process with rate $\Lambda(A)$ and reinforcement $p$.

Now, we argue that if $\hat{\mathcal{N}}$ is a random measure satisfying (i)
and (ii), then it is a NRPPP. We will establish this claim by showing that
$\hat{\mathcal{N}}$ satisfies the exponential formula (4.13). First, observe
that (i) implies that $\mathbb{E}[\hat{N}_{A}(t)]=t\Lambda(A)$, for example by
making use of Lemma 4.7 and the fact that if $\hat{\mathcal{M}}$ is a NRPPP
with characteristic measure $\Lambda$ and parameter $p$, then
$(\hat{\mathcal{M}}([0,t]\times A):t\geq 0)$ is a reinforced Poisson process
with rate $\Lambda(A)$ and parameter $p$. We deduce by a monotone class
argument that $\hat{\mathcal{N}}$ satisfies, for any measurable
$f:\mathbb{R}^{+}\times\mathbb{R}\mapsto\mathbb{R}^{+}$, the identity:

$\mathds{E}\left[\int_{[0,t]\times\mathbb{R}}f(s,x)\hat{\mathcal{N}}({\mathrm{d}}s,{\mathrm{d}}x)\right]=\int_{0}^{t}{\mathrm{d}}s\int\Lambda({\mathrm{d}}x)f(s,x).$
(4.17)

Still for $A$ as in (i) and for an arbitrary collection of times
$0=t_{1}<t_{2}<\dots<t_{k+1}<t$, we set

$g(s,x):=\sum_{i=1}^{k}\alpha_{i}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathds{1}_{A}(x).$
(4.18)

Since by hypothesis $(\hat{N}_{A}(t))_{t\in\mathbb{R}^{+}}$ is a NRPP with
rate $\Lambda(A)$, by the formula (4.2) for the characteristic function of the
finite dimensional distributions of reinforced Poisson processes, we obtain
that

$\displaystyle\mathds{E}\left[\exp\bigg{\\{}i\langle\hat{\mathcal{N}},\mathds{1}_{\\{\cdot\leq
t\\}}g\rangle\bigg{\\}}\right]$
$\displaystyle=\mathds{E}\left[\exp\bigg{\\{}i\sum_{i=1}^{k}\alpha_{i}(N_{A}(t_{i+1})-N_{A}(t_{i}))\bigg{\\}}\right]$
$\displaystyle=\exp\bigg{\\{}t(1-p)\Lambda(A)\mathds{E}\left[\exp\left(i\sum_{i=1}^{k}\alpha_{i}(Y(t_{i+1}/t)-Y(t_{i}/t))\right)-1\right]\bigg{\\}}$
$\displaystyle=\exp\bigg{\\{}t(1-p)\int_{\mathbb{R}}\Lambda({\mathrm{d}}x)\mathds{E}\left[\exp\left(i\int_{0}^{1}\sum_{i=1}^{k}\alpha_{i}\mathds{1}_{\\{t_{i}<st\leq
t_{i+1}\\}}\mathds{1}_{A}(x){\mathrm{d}}Y(s)\right)-1\right]\bigg{\\}}.$

Remark that this is precisely the identity (ii) of Proposition 4.8 for our
choice of $g$. Making use of the independence hypothesis of
$\hat{N}_{A_{1}},\dots\hat{N}_{A_{k}}$ for disjoints $A_{1},\dots,A_{k}$ with
$\Lambda(A_{i})<\infty$, we can also show that the identity holds for $f$ as
in (4.15) for such collection of sets. Now, if $f$ is non-negative, bounded
and supported on $[0,t]\times A$ with $\Lambda(A)<\infty$, making use of
(4.17), we can proceed as in (4.16) for the proof of Proposition 4.8,
approximating $f$ by a bounded sequence of the form (4.15), and show that the
exponential formula (4.13) still holds. The general case follows by sigma
finiteness of $\Lambda$ and we deduce that $\hat{\mathcal{N}}$ is a NRPPP with
the desired parameters. ∎

###  4.3 Proof of Theorem 4.1 and compensator of the jump measure

Let us now establish Theorem 4.1. Remark that paired with Proposition 4.9, it
entails that the role of the counting process of jumps $\Delta\hat{\xi}_{s}\in
A$ for fixed $A\in\mathscr{B}(\mathbb{R})$ is played precisely by noise-
reinforced Poisson processes, in analogy with the non-reinforced setting.

###### Proof of Theorem 4.1.

The result will follow as soon as we establish (i) and (ii) of Proposition 4.9
for

$\hat{N}_{A}(t):=\\#\big{\\{}(s,\Delta\hat{\xi}_{s})\in[0,t]\times
A\big{\\}},\quad\quad t\geq 0,$ (4.19)

where $A$ is an arbitrary Borelian satisfying $\Lambda(A)<\infty$. By the
identity in distribution (2.12), we can restrict our arguments to the unit
interval and hence we can make use of the explicit construction of NRLPs in
$[0,1]$ that we recalled in Section 2.3, in terms of Yule-Simon series.

Denote by ${\mathcal{M}}:=\sum_{i}\delta_{(x_{i},Y_{i})}$ the Poisson random
measure with intensity $(1-p)\Lambda\otimes\mathbb{Q}$ and recall the
discussion following Theorem 3.1. If $(x_{i},Y_{i})$ is an atom of
${\mathcal{M}}$, then at time $U_{i}=\inf\\{t\geq 0:Y_{i}(t)=1\\}$, the
process $\hat{\xi}$ performs the jump $x_{i}$ for the first time, i.e.
$\Delta\hat{\xi}_{U_{i}}=x_{i}$ and this precise jump $x_{i}$ is repeated in
the interval $[0,1]$ at each jump time of $Y_{i}$. It follows that for any
$f:\mathbb{R}\mapsto\mathbb{R}^{+}$ we have:

$\sum_{s\leq t}f(\Delta\hat{\xi}_{s})=\sum_{i}f(x_{i})Y_{i}(t),$ (4.20)

and in particular, we get:

$\hat{N}_{A}(t)=\sum_{i}\mathds{1}_{\\{x_{i}\in A\\}}Y_{i}(t).$

Hence, by the independence property of Poisson random measures, the processes
$\hat{N}_{A_{1}},\dots,\hat{N}_{A_{n}}$ are independent as soon as $A_{i}\cap
A_{j}=\emptyset$ for all $i\neq j$. Now, if we fix
$\lambda_{1},\dots,\lambda_{k}\in\mathbb{R}$, $0\leq t_{1}<\dots<t_{k}\leq 1$,
we deduce from the formula for the characteristic function for Poisson
integrals the equality:

$\mathds{E}\left[\exp\bigg{\\{}i\sum_{j=1}^{k}\lambda_{j}\hat{N}_{A}({t_{j}})\bigg{\\}}\right]=\exp\bigg{\\{}(1-p)\Lambda(A)\mathds{E}\left[\exp\bigg{\\{}i\sum_{j=1}^{k}\lambda_{j}Y(t_{j})\bigg{\\}}-1\right]\bigg{\\}}.$

Comparing with $(\ref{definition:noiseReinfPoisson})$, we get that the right-
hand side in the previous display is precisely the characteristic function of
the finite dimensional distributions at times $t_{1},\dots,t_{k}$ of a
reinforced Poisson process with rate $\Lambda(A)$ and parameter $p$. ∎

Recalling the explicit construction of NRPPPs from Definition 4.5, we stress
that Theorem 4.1 formalises the idea that the jumps of NRLPs are jumps that
are repeated through time, similarly to the dynamics of noise reinforced
random walks – we refer to the beginning of Section 5.2 for a brief
introduction to the later. Our terminology and notation for the reinforced
measure $\hat{\mu}$ can now be justified by the following: if $\mu$ is the
jump measure of $\xi$, the counting process $(\mu([0,t]\times A):t\geq 0)$ is
a Poisson process with rate $\Lambda(A)$ while $(\hat{\mu}([0,t]\times
A):t\geq 0)$ is a reinforced Poisson process with rate $\Lambda(A)$. Said
otherwise, the following identity holds in distribution:

$\hat{\mu}([0,\cdot\,]\times
A)\overset{\mathscr{L}}{=}\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[width("{\mu}([0,\cdot\,]\times
A)")]{\kern 0.1pt\mathchar 866\relax\kern
0.1pt}{\rule{0.0pt}{505.89pt}}}{}}{2.4ex}}\stackon[-6.9pt]{{\mu}([0,\cdot\,]\times
A)}{\tmpbox}.$ (4.21)

Now that the key result of the section has been established, we continue our
study of the jump process of NRLPs. In this direction, we start by briefly
recalling notions of semi-martingale theory that will be needed. Let $X$ be a
semimartingale defined on a probability space
$(\Omega,\mathscr{F},(\mathscr{F}_{t}),\mathbb{P}).$ Its jump measure
$\mu_{X}$ is an integer valued random measure on
$(\mathbb{R}^{+}\times\mathbb{R},\mathscr{B}(\mathbb{R}^{+})\otimes\mathscr{B}(\mathbb{R}))$,
in the sense of [12, Chapter II-1.13]. Denote the predictable sigma-field on
$\Omega\times\mathbb{R}^{+}$ by $\mathscr{P}\text{r}$. If $H$ is a
$\mathscr{P}\text{r}\otimes\mathscr{B}(\mathbb{R}^{+})$-measurable function,
we simply write $H*\mu_{X}$ for the process defined at each
$t\in\mathbb{R}^{+}$ as

$(H*\mu_{X})_{t}(\omega):=\int_{(0,t]\times\mathbb{R}}\mu_{X}(\omega;{\mathrm{d}}s,{\mathrm{d}}x)H_{s}(\omega;x),\quad\quad\text{
if
}\quad\int_{(0,t]\times\mathbb{R}}\mu_{X}(\omega;{\mathrm{d}}s,{\mathrm{d}}x)|H_{s}(\omega;x)|<\infty,$
(4.22)

and $\infty$ otherwise. Both notations for the integral will be used
indifferently. Further, we denote by $\mathscr{A}^{+}$ the class of
increasing, adapted rcll finite-variation processes $(A_{t})$, with $A_{0}=0$
such that $\mathds{E}\left[A_{\infty}\right]<\infty$, and by
$\mathscr{A}_{loc}^{+}$ its localisation class. The jump measure $\mu_{X}$
posses a predictable compensator, this is, a random measure $\mu_{X}^{\tt{p}}$
on
$(\mathbb{R}^{+}\times\mathbb{R},\mathscr{B}(\mathbb{R}^{+})\otimes\mathscr{B}(\mathbb{R}))$
unique up to a $\mathbb{P}$-null set, characterised by being the unique
predictable random measure (in the sense of [12, Chapter II-1.6]) satisfying
that for any non-negative
$H\in\mathscr{P}\text{r}\otimes\mathscr{B}(\mathbb{R})$, the equality

$\mathds{E}\left[(H*\mu_{X})_{\infty}\right]=\mathds{E}\left[(H*\mu_{X}^{\tt{p}})_{\infty}\right]$

holds. Equivalently, for any
$H\in\mathscr{P}\text{r}\otimes\mathscr{B}(\mathbb{R})$ such that
$|H|*\mu_{X}\in\mathscr{A}_{loc}^{+}$, the process $|H|*\mu_{X}^{\tt{p}}$
belongs to $\mathscr{A}_{loc}^{+}$ and $H*\mu_{X}^{\tt{p}}$ is the predictable
compensator of $H*\mu_{X}$. Said otherwise, $H*\mu_{X}-H*\mu_{X}^{\tt{p}}$ is
a local martingale.

Recall that by Proposition 3.2, the process $\hat{\xi}$ is a semimartingale.
Hence, we can consider $\hat{\mu}^{\tt{p}}$, the predictable compensator of
its jump measure $\hat{\mu}$, and our purpose is to identify explicitly
$\hat{\mu}^{\tt{p}}$. In contrast, it might be worth mentioning that if $\xi$
is a Lévy process with Lévy measure $\Lambda$, the compensator of its jump
measure $\mu$ is just the deterministic measure
$\mu^{\tt{p}}={\mathrm{d}}t\otimes\Lambda({\mathrm{d}}x)$. The first step
consists in observing the following:

######  Lemma 4.10.

Let $A\in\mathscr{B}(\mathbb{R})$ be a Borel set that doesn’t intersect some
open neighbourhood of the origin. If we denote by $(\mathscr{F}_{t}^{A})$ the
natural filtration of $\hat{N}_{A}$, then the process
$M_{A}=(M_{A}(t))_{t\in\mathbb{R}^{+}}$ defined as $M_{A}(0)=0$ and

$M_{A}(t)=t^{-p}\left(\hat{N}_{A}(t)-t\Lambda(A)\right),\quad\quad t\geq 0,$

is a finite variation $(\mathscr{F}^{A}_{t})$-martingale.

Remark that this is just a special case of Proposition 3.2 for a Lévy measure
of the form $\Lambda(A)\delta_{1}$ with $q=0$. Now we can state:

######  Proposition 4.11.

_(Compensation formula)_
Denote by $(\mathscr{F}_{t})$ the natural filtration of $\hat{\xi}$ and by
${\hat{\mu}}$ its jump measure. The predictable compensator
${\hat{\mu}}^{\tt{p}}$ of ${\hat{\mu}}$ is given by

$\hat{\mu}^{\tt{p}}(\omega;{\mathrm{d}}t,{\mathrm{d}}x)=(1-p){\mathrm{d}}t\otimes\Lambda({\mathrm{d}}x)+p\frac{{\mathrm{d}}t}{t}\mathscr{E}_{t}(\omega;{\mathrm{d}}x),$
(4.23)

where
$\mathscr{E}_{t}({\mathrm{d}}x)=\sum_{s<t}\delta_{\Delta\hat{\xi}_{s}}({\mathrm{d}}x)$
is the empirical measure of jumps that occurred strictly before time $t$.

Consequently, for any predictable process
$H\in\mathscr{P}\text{r}\otimes\mathscr{B}(\mathbb{R})$ such that
$|H|*\hat{\mu}\in\mathscr{A}_{loc}^{+}$, we have
$|H|*\hat{\mu}^{\tt{p}}\in\mathscr{A}_{loc}^{+}$ and the following process is
a local martingale:

$M_{t}=\sum_{s\leq
t}H_{s}(\cdot\,,\Delta\hat{\xi}_{s})-(1-p)\int_{0}^{t}{\mathrm{d}}s\int_{\mathbb{R}}\Lambda({\mathrm{d}}x)H_{s}(\cdot\,,x)-\int_{0}^{t}\sum_{r<s}H_{s}(\cdot\,,\Delta\hat{\xi}_{r})\frac{p}{s}{\mathrm{d}}s,\quad
t\geq 0.$ (4.24)

The first compensating term appearing in (4.24) is compensating innovations,
i.e. atoms appearing for the first time, while the second one should be
interpreted as the compensator of the memory part of $\hat{\mu}$. Notice that
Proposition 4.11 holds if $p=0$. Indeed, in that case $\hat{\xi}$ is a Lévy
process and its jump process $\mu$ is the Poisson point process (3.5). The
compensator (4.23) is just the deterministic compensator
${\mathrm{d}}t\otimes\Lambda({\mathrm{d}}x)$ for the Poisson point processes
with characteristic measure $\Lambda$ and in (4.24) we recover the celebrated
compensation formula, see e.g. [5, Chapter 1]. Remark that since the intensity
of both $\mu$ and $\hat{\mu}$ is ${\mathrm{d}}t\otimes\Lambda({\mathrm{d}}x)$,
we have, for both $X$ a Lévy process and its associated NRLP, the equality
$\mathds{E}\left[\sum_{s\leq t}f(s,\Delta
X_{s})\right]=\int_{0}^{t}{\mathrm{d}}s\int_{\mathbb{R}}\Lambda({\mathrm{d}}x)f(s,x)$
for any $f:\mathbb{R}\times\mathbb{R}\mapsto\mathbb{R}^{+}$. When $X:=\xi$, by
the compensation formula, this identity holds also if we replace $f$ by a non-
negative predictable process
$H\in\mathscr{P}\text{r}\otimes\mathscr{B}(\mathbb{R})$, viz.

$\mathds{E}\left[\sum_{s\leq
t}H_{s}(\cdot,\Delta\xi_{s})\right]=\mathds{E}\left[\int_{0}^{t}{\mathrm{d}}s\int_{\mathbb{R}}\Lambda({\mathrm{d}}x)H_{s}(\cdot,x)\right].$
(4.25)

However, we point out that if we replace in (4.25) the Lévy process by its
reinforced version $\hat{\xi}$, the identity no longer holds. Indeed, if such
formula was satisfied, the exact same proof for the exponential formula of
PPPs of XII-1.12 in [20] would hold in our reinforced setting, and since
random measures are characterised by their Laplace functional, this would lead
us to the conclusion that the law of $\hat{\mu}$ coincides with the law of
$\mu$.

###### Proof.

(i) In order to establish (4.23), by (i) of Theorem II-1.8 of [12], it
suffices to show that for any nonnegative predictable process
$H\in\mathscr{P}\text{r}\otimes\mathscr{B}(\mathbb{R})$,

$\mathds{E}\left[(H*\hat{\mu})_{\infty}\right]=\mathds{E}\left[(H*\hat{\mu}^{\tt{p}})_{\infty}\right],$
(4.26)

and the first step consists in showing the result for deterministic
$H_{s}(\omega,x)=\mathds{1}_{B}(x)$ for $B\in\mathscr{B}(\mathbb{R})$.
Maintaining the notation introduced in Lemma 4.1 for the process
$\hat{N}_{B}$, consider $B$ an arbitrary interval not containing a
neighbourhood of the origin as well as the associated martingale,

$M_{B}(t)=t^{-p}\hat{N}_{B}^{(c)}(t)=t^{-p}\left(\hat{N}_{B}(t)-t\Lambda(B)\right).$

Integrating by parts, we get

$t^{p}M_{B}(t)=\int_{0}^{t}s^{p}{\mathrm{d}}M_{B}(s)+\int_{0}^{t}pM_{B}(s)s^{p-1}{\mathrm{d}}s,$

and consequently,

$\displaystyle\hat{N}_{B}(t)-t\Lambda(B)$
$\displaystyle=\int_{0}^{t}s^{p}{\mathrm{d}}M_{B}(s)+\int_{0}^{t}pM_{B}(s)s^{p-1}{\mathrm{d}}s$
$\displaystyle=\int_{0}^{t}s^{p}{\mathrm{d}}M_{B}(s)+\int_{0}^{t}\left(\hat{N}_{B}(s)-s\Lambda(B)\right)ps^{-1}{\mathrm{d}}s\cdot$

Said otherwise,

$\hat{N}_{B}(t)-t(1-p)\Lambda(B)-\int_{0}^{t}\hat{N}_{B}(s)ps^{-1}{\mathrm{d}}s=\int_{0}^{t}s^{p}{\mathrm{d}}M_{B}(s),$

is a martingale. Since $(N_{B}(\omega;s))_{s\in\mathbb{R}^{+}}$ and
$(N_{B}(\omega;{s-}))_{s\in\mathbb{R}^{+}}$ differ in a set of null Lebesgue
measure, the equality still holds replacing
$\int_{0}^{t}\hat{N}_{B}(s)ps^{-1}{\mathrm{d}}s$ by
$\int_{0}^{t}\hat{N}_{B}(s-)ps^{-1}{\mathrm{d}}s$ and we obtain precisely
(4.24) for $H_{s}(\omega,x)=\mathds{1}_{B}(x)$. Now we can proceed as in the
proof of II-2.21 from [12]. Concretely, pick any positive Borel-measurable
deterministic function $h=h(x)$, $x\in\mathbb{R}$ such that
$h*\hat{\mu}-h*\hat{\mu}^{\tt{p}}$ is a local martingale and let $T$ be an
arbitrary stopping time. With the same terminology as in I.1.22 of [12] denote
by $\llbracket 0,T\rrbracket$ the subset of $\Omega\times\mathbb{R}^{+}$
defined by

$\llbracket 0,T\rrbracket=\\{(\omega,s):0\leq s\leq T(\omega)\\}.$

In particular, $(h*\hat{\mu})^{T}=\mathds{1}_{\llbracket
0,T\rrbracket}h*\hat{\mu}$ where the process $\mathds{1}_{\llbracket
0,T\rrbracket}$ is predictable (since left continuous) and moreover, by
Theorem I 2.2 of [12], the sigma field generated by the collection

$\\{A\times\\{0\\}\text{ where }A\in\mathscr{F}_{0},\text{ and }\llbracket
0,T\rrbracket\text{ where }T\text{ is any }(\mathscr{F}_{t})\text{-stopping
time }\\}$

is precisely the predictable sigma field $\mathscr{P}\text{r}$. Then, if
$(T_{n})$ is a localising sequence for the local martingale
$h*\hat{\mu}-h*\hat{\mu}^{\tt{p}}$, it follows from Doob’s stopping theorem
that for each $n$,

$\mathds{E}\left[(h*\hat{\mu})_{\infty}^{T\wedge
T_{n}}\right]=\mathds{E}\left[(h*\hat{\mu}^{\tt{p}})_{\infty}^{T\wedge
T_{n}}\right].$

Consequently, taking the limit as $n\uparrow\infty$, we deduce by monotone
convergence that

$\mathds{E}\left[(\mathds{1}_{\llbracket
0,T\rrbracket}h*\hat{\mu})_{\infty}\right]=\mathds{E}\left[(\mathds{1}_{\llbracket
0,T\rrbracket}h*\hat{\mu}^{\tt{p}})_{\infty}\right]$

which in turn implies that (4.26) holds for any predictable process
$H=1_{B}1_{\llbracket 0,T\rrbracket}$ where $B$ is any closed interval not
containing the origin and $T$ an arbitrary stopping time. Now the claim
follows by a monotone class argument. ∎

We close our discussion on the jump process of NRLPs with the property at the
heart of the infinite divisibility of $\hat{\xi}$ as a stochastic process, a
topic that will be studied in Section 6.2. We claim that, for
$A\in\mathscr{B}(\mathbb{R})$ with $\Lambda(A)<\infty$ the point process of
jumps

$\nu_{A}({\mathrm{d}}s)=\sum_{s}\mathds{1}_{\\{\Delta\hat{\xi}_{s}\in
A\\}}\delta_{s},$ (4.27)

is an infinitely divisible point process. More precisely, the measure
$\nu_{A}$ is a reinforced Poisson point process $\hat{\mathscr{P}}$ with rate
$\Lambda(A)$ in $\mathbb{R}^{+}$ and if we consider $n$ independent copies
$\nu_{A}^{1},\dots,\nu_{A}^{n}$ of the reinforced Poisson process (4.27) but
with rate $n^{-1}\Lambda(A)$, we have the equality in distribution

$\nu_{A}\overset{\mathscr{L}}{=}\nu_{A}^{1}+\dots+\nu_{A}^{n}.$ (4.28)

To see this, consider $f:\mathbb{R}^{+}\mapsto\mathbb{R}^{+}$ a positive
function with support in $[0,t]$, and observe that

$\langle f,\nu_{A}\rangle=\sum_{s\leq t}\mathds{1}_{A}(\Delta\xi_{s})f(s).$

Now the claim follows by computing the Laplace functional of $\nu_{A}$,
$\nu_{A}^{i}$ respectively, by applying the exponential formula (4.12) and
from comparing with (4.4). For a more detailed discussion on infinitely
divisible point processes we refer to page 5 of [17].

##  5 Weak convergence of the pair of skeletons

Before stating the first result of the section, let us briefly recall the
statement of the Lévy-Itô synthesis for Lévy processes: a Lévy process $\xi$
with triplet $(a,q^{2},\Lambda)$ can be written as
$\xi=\xi^{(1)}+\xi^{(2)}+{\xi}^{(3)}$, where $\xi^{(1)}=(at+qB_{t}:t\geq 0)$
is a Brownian motion with drift while $\xi^{(2)}+{\xi}^{(3)}$ is a purely
discontinuous process that can be explicitly built from the jump measure $\mu$
defined in (3.5). More precisely, if we denote by $\mu^{(sc)}$ the compensated
measure of jumps $\mu^{(sc)}=\mu-{\mathrm{d}}t\Lambda({\mathrm{d}}x)$, we can
write

$\xi_{t}=at+qB_{t}+\int_{[0,t]\times(-1,1)^{c}}x\mu({{\mathrm{d}}s,{\mathrm{d}}x})+\int_{[0,t]\times(-1,1)}x\mu^{(sc)}({{\mathrm{d}}s,{\mathrm{d}}x}),\quad\quad
t\geq 0.$ (5.1)

The reinforced Lévy-Itô synthesis, which is the first main result of the
section, states that the analogous result holds for NRLPs where now, the PPP
$\mu$ in (5.1) has been replaced by the reinforced version $\hat{\mu}$, and
the Brownian motion $B$ by its reinforced version $\hat{B}$ (if $p<1/2)$. More
precisely, after properly defining the ”space-compensated” measure
$\hat{\mu}^{(sc)}$, we prove:

######  Theorem 5.1.

_(Reinforced Itô’s synthesis)_
Let $\hat{\mu}$ be the jump measure of a NRLP $\hat{\xi}$ of characteristics
$(a,q^{2},\Lambda,p)$. Then, a.s. we have

$\hat{\xi}_{t}=at+q\hat{B}_{t}+\int_{[0,t]\times(-1,1)^{c}}x\hat{\mu}({\mathrm{d}}s,{\mathrm{d}}x)+\int_{[0,t]\times(-1,1)}x\hat{\mu}^{(sc)}({\mathrm{d}}s,{\mathrm{d}}x),\quad\quad
t\geq 0,$

for some noise reinforced Brownian motion $\hat{B}$, with the convention that
if $p\geq 1/2$ the process $\hat{B}$ is null. Moreover, the integrals in the
previous display are NRLPs with respective characteristics
$(0,0,\mathds{1}_{(-1,1)^{c}}\Lambda,p)$,
$(0,0,\mathds{1}_{(-1,1)}\Lambda,p)$.

######  Remark 5.2.

Beware of the notation, $\hat{\mu}^{(sc)}$ stands for the space-compensated
jump measure $\hat{\mu}$ and should not be confused with the time-compensated
measure $(\mu-\mu^{p})$ in the sense of [12, Chapter II-1.27]. For instance,
we stress that $\hat{\xi}^{(3)}$ is not a local martingale. Remark that for
Lévy processes, the time and space compensation of its jump measure coincide,
since the compensating measure is the same.

After proving this result, we start settling the ground for the main result of
the section. First, making use of Theorem 5.1, we define the joint law, of a
Lévy process and its reinforced version, by introducing an appropriate
coupling $(\xi,\hat{\xi})$. We then characterise its law by computing the
characteristic function of its finite dimensional distributions:

######  Proposition 5.3.

There exists a pair $(\xi,\hat{\xi})$, where $\hat{\xi}$ has the law of a NRLP
with characteristics $(a,q^{2},\Lambda,p)$, with law determined by the
following: for all $k\geq 1$, $\lambda_{1},\dots,\lambda_{k}$,
$\beta_{1},\dots\beta_{k}$ real numbers, and $0<t_{1}<\dots<t_{k}\leq t$, we
have

$\displaystyle\mathds{E}\left[\exp\bigg{\\{}i\sum_{j=1}^{k}\big{(}\lambda_{j}\xi_{t_{j}}+\beta_{j}\hat{\xi}_{t_{j}}\big{)}\bigg{\\}}\right]=$
$\displaystyle\exp\bigg{\\{}t\cdot
p\mathds{E}\left[\Psi\Bigg{(}\sum_{j=1}^{k}\lambda_{j}\mathds{1}_{\\{U\leq
t_{j}/t\\}}\Bigg{)}\right]+t\cdot(1-p)\mathds{E}\left[\Psi\Bigg{(}\sum_{j=1}^{k}(\lambda_{j}\mathds{1}_{\\{Y(t_{j}/t)\geq
1\\}}+\beta_{i}Y(t_{j}/t))\Bigg{)}\right]\bigg{\\}},$ (5.2)

where $U$ is a uniform random variable in $[0,1]$. A pair of processes with
such distribution will always be denoted by $(\xi,\hat{\xi})$.

Now, we connect the distribution of the pair $(\xi,\hat{\xi})$ with the
discrete setting. In this direction, consider the Lévy process $\xi$ and for
each fixed $n\in\mathbb{N}$ we set

$X^{(n)}_{k}:=\Delta^{(n)}\xi_{k}=\xi_{k/n}-\xi_{(k-1)/n},\quad\text{ for
}k\geq 1.$ (5.3)

For each $n$, the sequence $(X^{(n)}_{k})$ is identically distributed with law
$\xi_{1/n}$ and the random walk
$S^{(n)}_{k}=X^{(n)}_{1}+\dots+X^{(n)}_{k}\text{ for }k\geq 1$,
$\,S^{(n)}_{0}=0$ built from these increments for a mesh of length $1/n$ is
referred to as the $n-$skeleton of the Lévy process $\xi$. This process
consists in the positions of $\xi$ observed at discrete time intervals and, if
we write $D(\mathbb{R}_{+})$ for the space of $\mathbb{R}_{+}$ indexed rcll
functions into $\mathbb{R}$ with the Skorokhod topology, we have
$S^{(n)}_{\lfloor n\cdot\rfloor}\overset{D(\mathbb{R}_{+})}{\rightarrow}\xi$
as $n\uparrow\infty$. Now, fix a memory parameter $p\in(0,1)$ and for each
$n$, consider the associated noise reinforced random walk
$(\hat{S}^{(n)}_{k})$ with parameter $p$ built from the same collection of
increments:

$\hat{S}_{k}^{(n)}:=\hat{X}_{1}^{(n)}+\dots+\hat{X}_{k}^{(n)},\quad\text{ for
}k\geq 1,$ (5.4)

where we set $\hat{S}^{(n)}_{0}:=0$. For a detailed account on the noise
reinforced random walk, we refer to the beginning of Section 5.2. The main
result in [7] states that $\hat{S}_{\lfloor
n\cdot\rfloor}\overset{f.d.d.}{\rightarrow}\hat{\xi}$, the convergence holding
in the sense of finite-dimensional distributions, and we shall now strength
this result. To simplify notation, write $D^{2}(\mathbb{R}_{+})$ the product
space $D(\mathbb{R}_{+})\times D(\mathbb{R}_{+})$ endowed with the product
topology. Now we can state the main result of the section:

######  Theorem 5.4.

Let $\xi$ be a Lévy process with characteristic triplet $(a,q^{2},\Lambda)$,
fix $p\in(0,1/2)$ an admissible memory parameter and for each $n$, let
$(S^{(n)}_{k},\hat{S}^{(n)}_{k})$ be the pair of the $n$-skeleton of $\xi$ and
its reinforced version. Then, there is weak convergence in
$D^{2}(\mathbb{R}_{+})$ as $n\uparrow\infty$

$\Big{(}S^{(n)}_{\lfloor n\cdot\rfloor},\hat{S}^{(n)}_{\lfloor
n\cdot\rfloor}\Big{)}\overset{\mathscr{L}}{\longrightarrow}(\xi,\hat{\xi}),$
(5.5)

where $(\xi,\hat{\xi})$ is a pair of processes with law (5.2).

The section is organised as follows: In Section 5.1, after introducing the
(space) compensated integral with respect to NRPPPs, we shall establish
Theorem 5.1. Making use of this result, in Section 5.2 we define the joint law
of a Lévy process and its reinforced version $(\xi,\hat{\xi})$. More
precisely, by Lévy-Itô Synthesis and its reinforced version of Theorem 5.1, it
will suffice to define the joint law of $(\mu,\hat{\mu})$ and $(B,\hat{B})$.
This is respectively the content of the construction detailed in 5.2.1 and
Definition 5.8. The construction of $\hat{\mu}$ is done explicitly in terms of
the jump measure of $\xi$ by a procedure that should be interpreted as the
continuous-time reinforcement analogue of the reinforcement algorithm for
random walks. We then introduce the joint law $(\xi,\hat{\xi})$ in Definition
5.10 and prove Proposition 5.3. Finally, Section 5.3 is devoted to the proof
of Theorem 5.4.

###  5.1  Proof of Theorem 5.1

Let us start by introducing the (space)-compensated integral with respect to
NRPPPs. Recall the identity of Lemma 4.7 for the intensity measure of NRPPPs
and for fixed $t\in\mathbb{R}$, let
$f:\mathbb{R}^{+}\times\mathbb{R}\mapsto\mathbb{R}$ be a measurable function
satisfying, for all $0<a<b$, the integrability condition

$\int_{(0,t]\times\\{a\leq|x|<b\\}}|f(s,x)|{\mathrm{d}}s\Lambda({\mathrm{d}}x)<\infty.$

Next, we set

$\int_{[0,t]\times\\{a\leq|x|<b\\}}f(s,x)\hat{\mathcal{N}}^{(\text{sc})}({\mathrm{d}}s,{\mathrm{d}}x):=\int_{[0,t]\times\\{a\leq|x|<b\\}}f(s,x)\hat{\mathcal{N}}({\mathrm{d}}s,{\mathrm{d}}x)-\int_{(0,t]\times\\{a\leq|x|<b\\}}f(s,x){\mathrm{d}}s\Lambda({\mathrm{d}}x).$
(5.6)

This is a centred random variable and if we denote it by
$\Sigma_{a,b}^{(c)}(f,t)$, from Proposition 4.5 \- (ii) we deduce that
$(\Sigma_{e^{-r},b}^{(c)}(f,t))_{r\in[-\log(b),\infty)}$ has independent
increments, and hence is a martingale. When the limit of this martingale
exists, we will write

$\int_{[0,t]\times(-b,b)}f(s,x)\hat{\mathcal{N}}^{(\text{sc})}({\mathrm{d}}s,{\mathrm{d}}x):=\lim_{r\uparrow\infty}\int_{[0,t]\times\\{e^{-r}\leq|x|<b\\}}f(s,x)\hat{\mathcal{N}}^{(\text{sc})}({\mathrm{d}}s,{\mathrm{d}}x).$
(5.7)

Recall that the characteristics of a NRLP are being considered with respect to
the cutoff function $x\mathds{1}_{\\{|x|<1\\}}$ as well as the notation
$f*\hat{\mathcal{N}}$ from (4.22). The following lemma shows that the sums of
atoms of NRPPPs are precisely purely discontinuous NRLPs:

######  Lemma 5.5.

Fix a Lévy measure $\Lambda$, a parameter $p\in(0,1)$ such that
$\beta(\Lambda)p<1$ and let $\hat{\mathcal{N}}$ be a NRPPP with characteristic
measure $\Lambda$ and reinforcement parameter $p$.

  1. _(i)_

For any $0<a<b$, the process
$\mathds{1}_{\\{a\leq|x|<b\\}}x*\hat{\mathcal{N}}$ is a noise reinforced
compound Poisson process with characteristics
$(\Lambda(\mathds{1}_{\\{a\leq|x|<1\\}}x),0,\mathds{1}_{\\{a\leq|x|<b\\}}\Lambda,p)$.

  2. _(ii)_

For each $t\in\mathbb{R}^{+}$ the compensated integral

$\int_{[0,t]\times(-1,1)}x\hat{\mathcal{N}}^{(\text{sc})}({\mathrm{d}}s,{\mathrm{d}}x):=\lim_{r\uparrow\infty}\int_{[0,t]\times\\{e^{-r}\leq|x|<1\\}}x\hat{\mathcal{N}}^{(\text{sc})}({\mathrm{d}}s,{\mathrm{d}}x)$
(5.8)

exists. The process $\mathds{1}_{(-1,1)}x*\hat{\mathcal{N}}^{(sc)}$ is a NRLP
with characteristics $(0,0,\mathds{1}_{(-1,1)}\Lambda,p)$ and hence has a rcll
modification. Moreover, the convergence (5.8) holds towards its rcll
modification uniformly in compact intervals for some subsequence $(r_{n})$,
and we shall consider it and denote it in the same way without further
comments.

###### Proof.

(i) If we consider $\hat{\xi}$ a reinforced compound Poisson process with such
characteristics and $\hat{\mu}$ is its jump measure, it is a pure jump process
and we can write it as the sum of its jumps. Our claim can now be proved
directly from the identity
$\hat{\xi}=(x*\hat{\mu})\overset{\mathscr{L}}{=}(\mathds{1}_{(a\leq|x|<b)}x*\hat{\mathcal{N}})$,
since by Proposition 4.6 \- (i), the restriction
$\mathds{1}_{(a\leq|x|<b)}\hat{\mathcal{N}}$ has the same distribution as
$\hat{\mu}$. Alternatively, this can be established by means of the
exponential formulas we obtained in Proposition 4.8, by fixing
$0<t_{1}<\dots<t_{k}<t$ and computing the characteristic function of the
finite-dimensional distributions at times $t_{1},\dots,t_{k}$ of
$\mathds{1}_{\\{a\leq|x|<b\\}}x*\hat{\mathcal{N}}$, noticing that for
$f(s,x):=\left(\sum_{j=1}^{k}\lambda_{j}\mathds{1}_{\\{s\leq
t_{j}\\}}\right)x\mathds{1}_{\\{a\leq|x|<b\\}}$ we have

$\sum_{j=1}^{k}\lambda_{j}(\mathds{1}_{\\{a\leq|x|<b\\}}x*\hat{\mathcal{N}})_{t_{j}}=\int_{[0,t]\times\mathbb{R}}f(s,x)\hat{\mathcal{N}}({\mathrm{d}}s,{\mathrm{d}}x).$

The claim follows by comparing with the identity for the characteristic
function of the finite-dimensional distributions (2.12) of $\hat{\xi}$ .
(ii) Recall the notation introduced before (5.7) for the martingale
$(\Sigma_{e^{-r},1}^{(c)}(f,t))_{r\geq 0}$. In our case, we have $f(s,x)=x$
and we just write $(\Sigma_{e^{-r},1}^{(c)}(t))_{r\geq 0}$. The fact that the
martingale $(\Sigma_{e^{-r},1}^{(c)}(t))_{r\geq 0}$ converges as
$r\uparrow\infty$ and that the limit is a NRLP with characteristics
$(0,0,\mathds{1}_{(-1,1)}\Lambda)$ can be achieved by similar arguments as in
[7] after a couple of observations. Starting with the former, recall the
definition of $\hat{\mathcal{N}}$ from (4.11), and remark that for each $r>0$
we have

$\int_{[0,t]\times\mathbb{R}}\mathds{1}_{\\{e^{-r}\leq|x|<1\\}}x\hat{\mathcal{N}}({\mathrm{d}}s,{\mathrm{d}}x)=\sum_{u\in\mathscr{P}}\mathds{1}_{\\{u\leq
t\\}}\mathds{1}_{\\{e^{-r}\leq|x_{u}|<1\\}}x_{u}\cdot\\#\big{\\{}\\{ue^{s}:s\in
D_{u}\\}\cap[0,t]\big{\\}}.$

From the discussion right after Proposition 4.5, we infer that if we we
consider $(Z^{u})_{u\in\mathscr{P}}$ an independent collection of independent,
standard Yule processes, the family $\\{ue^{s}:s\in D_{u}\\}$ has the same
distribution as the collection of jump times of the counting process
$\mathds{1}_{\\{u\leq t\\}}Z^{u}_{p(\ln(t)-\ln(u))}$, $t\geq 0$. Hence the
previous display can also be written as

$\sum_{u\in\mathscr{P}}\mathds{1}_{\\{e^{-r}\leq|x_{u}|<1\\}}x_{u}\mathds{1}_{\\{u\leq
t\\}}Z^{u}_{p(\ln(t)-\ln(u))},$

and now the proof of the convergence as $r\uparrow\infty$ of
$(\Sigma_{e^{-r},1}^{(c)}(t))_{r\geq 0}$ follows by the same arguments as in
[7, Lemma 2.6]. Alternatively, one can make use of (2.12) to restrict our
arguments to the interval $[0,1]$ and apply [7, Lemma 2.6]. Next, to see that
the process $\mathds{1}_{(-1,1)}x*\hat{\mathcal{N}}^{(sc)}$ defines a NRLP
with characteristics $(0,0,\mathds{1}_{(-1,1)}\Lambda)$, fix
$0<t_{1}<\dots<t_{k}<t$ and for $\varepsilon>0$, $\lambda\in\mathbb{R}$ set

$\Phi_{\varepsilon,1}^{(3)}(\lambda)=\int_{\\{\varepsilon\leq|x|<1\\}}\left(e^{i\lambda
x}-1-i\lambda x\right)\Lambda({\mathrm{d}}x).$

Recalling the formula (4.13) for the characteristic function of integrals with
respect to NRPPPs, we deduce from considering the function
$f(s,x):=\left(\sum_{j=1}^{k}\lambda_{j}\mathds{1}_{\\{s\leq
t_{j}\\}}\right)x\mathds{1}_{\\{\varepsilon\leq|x|<1\\}}$ that we have

$\mathds{E}\left[\exp\bigg{\\{}i\sum_{j=1}^{k}\lambda_{j}(\mathds{1}_{\\{\varepsilon\leq|x|<1\\}}x*\hat{\mathcal{N}}^{(sc)})_{t_{j}})\bigg{\\}}\right]=\exp\bigg{\\{}t(1-p)\mathds{E}\left[\Phi_{\varepsilon,1}^{(c)}\left(\sum_{j=1}^{k}\lambda_{j}Y(t_{j}/t)\right)\right]\bigg{\\}}.$

Now we can apply the exact same reasoning as in the proof of Corollary 2.8 in
[7] by writing $s_{j}=t_{j}/t\in[0,1]$ and taking the limit as
$\varepsilon\downarrow 0$. The uniform convergence in compact intervals
towards the rcll modification of
$\mathds{1}_{(-1,1)}x*\hat{\mathcal{N}}^{(sc)}$ follows from the second
statement of Theorem 3.1, since for every $\varepsilon\in(0,1)$, the process

$\int_{[0,t]\times\\{0\leq|x|<\varepsilon\\}}x\hat{\mathcal{N}}^{(\text{sc})}({\mathrm{d}}s,{\mathrm{d}}x),\quad
t\geq 0,$

is a NRLP with characteristics
$(0,0,\mathds{1}_{\\{|x|<\varepsilon\\}}\Lambda)$. ∎

It immediately follows from the previous lemma that if $\hat{\mathcal{N}}$ is
a NRPPP with characteristic measure $\Lambda$, parameter $p$ and, if $p<1/2$,
we consider $\hat{W}$ an independent NRBM with same parameter, then

$\hat{X}_{t}=at+q\hat{W}_{t}+\int_{[0,t]\times(-1,1)^{c}}x\mathcal{\hat{N}}({\mathrm{d}}s,{\mathrm{d}}x)+\int_{[0,t]\times(-1,1)}x\mathcal{\hat{N}}^{(sc)}({\mathrm{d}}s,{\mathrm{d}}x),\quad\quad
t\geq 0,$ (5.9)

defines a NRLP with characteristics $(a,q^{2},\Lambda,p)$. To obtain the a.s.
statement of Theorem 5.1 we still need a short argument.

###### Proof of Theorem 5.1..

The result will be deduced from the equality in distribution
$\hat{\xi}\overset{\mathscr{L}}{=}\hat{X}$ for $\hat{X}$ defined as in (5.9)
with same characteristics as $\hat{\xi}$. In this direction, wlog we assume
$p<1/2$, $q=1$ and we set

$\displaystyle\hat{\xi}^{\leq 1}:=\hat{\xi}_{t}-\sum_{s\leq
t}\mathds{1}_{\\{|\Delta\hat{\xi}_{s}|\geq 1\\}}\Delta\hat{\xi}_{s}\hskip
14.22636pt\text{ and }\hskip
14.22636pt\hat{\xi}^{<\varepsilon}_{t}:=\hat{\xi}_{t}^{\leq
1}-\Big{(}\sum_{s\leq
t}\mathds{1}_{\\{\varepsilon\leq|\Delta\xi_{s}|<1\\}}\Delta\hat{\xi}_{s}-t\int_{\\{\varepsilon\leq|x|<1\\}}x\Lambda({\mathrm{d}}x)\Big{)}.$

Notice that for every $\varepsilon>0$, we can write

$\hat{\xi}_{t}=\hat{\xi}_{t}^{<\varepsilon}+\Big{(}\sum_{s\leq
t}\mathds{1}_{\\{\varepsilon\leq|\Delta\xi_{s}|<1\\}}\Delta\hat{\xi}_{s}-t\int_{\\{\varepsilon\leq|x|<1\\}}x\Lambda({\mathrm{d}}x)\Big{)}+\hat{\xi}^{(2)}_{t}.$
(5.10)

Since $\hat{\mu}$ is a reinforced PPP, by Lemma 5.5 the process (5.10)
converges uniformly in compact intervals for some subsequence
$(\varepsilon_{n})$ as $\varepsilon_{n}\downarrow 0$ towards
$\hat{C}+\hat{\xi}^{(2)}+\hat{\xi}^{(3)}$, for some process
$\hat{C}:=\hat{\xi}-\hat{\xi}^{(2)}-\hat{\xi}^{(3)}$ continuous by
construction. Since $\hat{\mu}$ is a reinforced PPP, by the independence
properties of its restriction we know that $\hat{\xi}^{(2)},\hat{\xi}^{(3)}$
are independent. Hence, it remains to show that
$(\hat{\xi}^{(2)},\hat{\xi}^{(3)})$ is independent of $\hat{C}$ and that
$\hat{C}-at=:\hat{B}$ is a NRBM. Fix arbitrary $0<u<v\leq\infty$ and maintain
the notation for $\hat{W}$, $\hat{\mathcal{N}}$ used in the representation
(5.9). Since $\hat{\mathcal{N}}$ is the clearly the jump measure of $\hat{X}$,
we have the equality in distribution:

$\Big{(}\hat{\xi},\sum_{s\leq\cdot}\mathds{1}_{\\{u\leq|\Delta\hat{\xi}_{s}|<v\\}}\Delta\hat{\xi}_{s}\Big{)}\overset{\mathscr{L}}{=}\left(\hat{X},\mathds{1}_{\\{u\leq|x|<v\\}}x*\mathcal{\hat{N}}\right).$
(5.11)

Moreover, since $\hat{W}$ is independent of $\hat{\mathcal{N}}$, from the
independence of restrictions of NRPPP and (5.11) we deduce that
$\mathds{1}_{\\{\varepsilon\leq|x|<1\\}}x*\hat{\mu}^{(sc)}+\mathds{1}_{\\{1\leq|x|\\}}x*\hat{\mu}$
and
$\hat{\xi}-\mathds{1}_{\\{\varepsilon\leq|x|<1\\}}x*\hat{\mu}^{(sc)}-\mathds{1}_{\\{1\leq|x|\\}}x*\hat{\mu}$
are independent, the later having the same distribution as
$at+\hat{W}_{t}+\mathds{1}_{(-\varepsilon,\varepsilon)}x*\hat{\mathcal{N}}$.
Now the claim follows by taking the limit as $\varepsilon\downarrow 0$. ∎

###  5.2 The joint law $(\xi,\hat{\xi})$ of a Lévy process and its reinforced
version

In this section we construct explicitly, for an arbitrary fixed Lévy process
$\xi$, the process $\hat{\xi}$ in terms of $\xi$ that will be referred to as
the noise reinforced version of $\xi$. This will yield a definition for the
joint law $(\xi,\hat{\xi})$. Our construction will be justified by the weak
convergence of Theorem 5.4. Let us start by recalling the discrete setting,
since our construction is essentially the continuous-time analogue of the
dynamics that we now describe.
$\circ$ The noise reinforced random walk. Given a collection of identically
distributed random variables $(X_{n})$ with law $X$, denote by
$S_{n}:=X_{1}+\dots+X_{n},\text{ for }n\geq 1$ the corresponding random walk.
We construct, simultaneously to $(S_{n})$, a noise reinforced version using
the same sample of random variables and performing the reinforcement algorithm
at each discrete time step. In this direction, consider $(\varepsilon_{n})$
and $(U[n])$ independent sequences of Bernoulli random variables with
parameter $p\in(0,1)$ and uniform random variables on $\\{1,\dots,n\\}$
respectively. Set $\hat{X}_{1}:=X_{1}$ and, for $n\geq 1$, define

$\hat{X}_{n+1}:=X_{n+1}\mathds{1}_{\\{\varepsilon_{n+1}=0\\}}+\hat{X}_{U[n]}\mathds{1}_{\\{\varepsilon_{n+1}=1\\}}.$

Finally, we denote the corresponding partial sums by
$\hat{S}_{n}:=\hat{X}_{1}+\dots+\hat{X}_{n},\,n\geq 1$. The process
$(\hat{S}_{n})$ is the so-called noise reinforced random walk with memory
parameter $p$, and we refer to this particular construction of $(\hat{S}_{n})$
as the noise reinforced version of $(S_{n})$. The process $(\hat{S}_{n})$ can
be written in terms of the individual contributions made by each one of the
steps. In this direction, let us introduce a counting process keeping track of
the number of times each step $X_{k}$ is repeated up to time $n$. Since if the
law of $X$ has atoms, we have $\mathbb{P}({X}_{1}=X_{2})>0$, and we need to
perform a slight modification to our algorithm. Namely, for each $n\geq 1$ we
write $X_{n}^{\prime}:=(X_{n},n)$ and we perform the reinforcement algorithm
to the pairs $(X^{\prime}_{n})$. This yields a sequence that, with a slight
abuse of notation, we denote by $(\hat{X}^{\prime}_{n})$. If for every
$k,n\geq 1$ we set:

$N_{k}(n):=\\#\\{1\leq i\leq n:X_{k}^{\prime}=\hat{X}_{i}^{\prime}\\},$ (5.12)

we can write:

$\hat{S}_{n}=\sum_{k=1}^{\infty}N_{k}(n)X_{k},\quad\text{ for }n\geq 1.$
(5.13)

For convenience, we always set $S_{0}=0=\hat{S}_{0}$, and when working with
pairs of the form $(S,\hat{S})$ it will always be implicitly assumed that the
noise reinforced version has been constructed by the algorithm we described.
For instance, it is clear that at each discrete time step $n$, with
probability $1-p$, ${S}_{n}$ and $\hat{S}_{n}$ share the same increment, while
with complementary probability $p$, they perform different steps.
Roughly speaking, in the continuum, the steps $(X_{n})$ are replaced by jumps
$\Delta\xi_{s}$ of the Lévy process $\xi$. With probability $1-p$, the jump is
shared with its reinforced version $\hat{\xi}$ while with complementary
probability $p$ it is discarded and remains independent of $\hat{\xi}$. The
jumps that are not discarded by this procedure are then repeated at each jump
time of an independent counting process that will be attached to it. The
process of discarding jumps with probability $p$ is traduced in a thinning of
the jump measure of $\xi$. Let us now give a formal description of this
heuristic discussion.

####  5.2.1 Construction of the pair $(\mathcal{N},\hat{\mathcal{N}})$

For the rest of the section, we fix a Lévy process $\xi$ with non-trivial Lévy
measure $\Lambda$, denote the set of its jump times by
$\mathscr{I}:=\\{u\in\mathbb{R}^{+}:{\Delta}\xi_{u}\neq 0\\}$ and let

$\mathcal{N}({\mathrm{d}}s,{\mathrm{d}}x):=\sum_{u\in\mathscr{I}}\delta_{(u,\Delta\xi_{u})},$

be its jump measure. By the Lévy-Itô decomposition, this is a PPP with
characteristic measure $\Lambda$ and we can write $\xi=\xi^{(1)}+J$, where
$\xi^{(1)}$ is a continuous process while $J$ is a process that can be
explicitly recovered from $\mathcal{N}$, as we recalled in (5.1).

If $\hat{\xi}$ has the law of its reinforced version, by Theorem 5.1 it can
also be written as $\hat{\xi}=\hat{\xi}^{(1)}+\hat{J}$, where $\hat{J}$ is a
functional of a NRPPP $\hat{\mathcal{N}}$ with characteristic measure
$\Lambda$. Hence, the main step for defining the law of the pair $(J,\hat{J})$
consists in appropriately defining $(\mathcal{N},\hat{\mathcal{N}})$. However,
recalling the construction of NRPPPs by superposition detailed before
Definition 4.5, this can be achieved as follows: first, set
$A_{0}:=\\{1\leq|x|\\}$ and for each $j\geq 1$, let
$A_{j}:=\\{1/(j+1)\leq|x|<1/j\\}$. Next, for $j\geq 0$ consider the point
process

$\mathscr{I}_{j}:=\\{u\in\mathbb{R}^{+}:\Delta{\xi}_{u}\in A_{j}\\},$

remark that $\mathscr{I}_{j}$ is a PPP with intensity
$\Lambda(A_{j}){\mathrm{d}}t$ and write
$\mathscr{I}:=\cup_{j}\mathscr{I}_{j}$. Maintaining the notation of Section 4,
consider $(D_{u})_{u\in\mathscr{I}}$ a collection of i.i.d. copies of $D$ and
for each $j\geq 0$ we set

$\mathcal{N}_{j}({\mathrm{d}}s,{\mathrm{d}}x):=\sum_{u\in\mathscr{I}_{j}}\sum_{t\in
D_{u}}\delta_{(ue^{t},\Delta\xi_{u})}.$

The measure $\mathcal{N}_{j}$ is a NRPPP with characteristic measure
$(1-p)^{-1}\Lambda(\,\cdot\cap A_{j})$, and we can now proceed as in Section
4.2 to construct the following NRPPP with parameter $p$ by superposition of
$(\mathcal{N}_{j})_{j\geq 1}$,

$\sum_{u\in\mathscr{I}}\sum_{t\in D_{u}}\delta_{(ue^{t},\Delta{\xi}_{u})}.$
(5.14)

Notice however that its characteristic measure is $(1-p)^{-1}\Lambda$. In this
direction, we consider a sequence of independent Bernoulli random variables
$(\varepsilon_{u})_{u\in\mathscr{I}}$ with parameter $1-p$ and apply a
thinning:

$\hat{\mathcal{N}}({\mathrm{d}}s,{\mathrm{d}}x):=\sum_{u\in\mathscr{I}}\mathds{1}_{\\{\varepsilon_{u}=1\\}}\sum_{t\in
D_{u}}\delta_{(ue^{t},\Delta{\xi}_{u})}.$ (5.15)

Now, $\hat{\mathcal{N}}$ is a NRPPP with characteristic measure $\Lambda$ and
reinforcement parameter $p$ built explicitly from the jump process of ${\xi}$.
From the construction, if a jump $\Delta\xi_{u}$ occurs at time $u$, with
probability $1-p$ it is kept and repeated at each $ue^{t}$ for $t\in D_{u}$,
while with complementary probability $p$, it is discarded and remains
independent of $\hat{\mathcal{N}}$. From now on, we always consider the pair
$(\mathcal{N},\hat{\mathcal{N}})$ constructed by this procedure. Then, by
definition of $\mathcal{N}$ we can write

$J_{t}={\xi}^{(2)}_{t}+{\xi}^{(3)}_{t}=\int_{[0,t]\times\\{|x|\geq
1\\}}x\mathcal{N}({\mathrm{d}}s,{\mathrm{d}}x)+\int_{[0,t]\times(-1,1)}x\mathcal{N}^{(sc)}({\mathrm{d}}s,{\mathrm{d}}x),\quad\quad
t\geq 0,$

while on the other hand, by Theorem 5.1 the process defined as

$\hat{J}_{t}=\hat{\xi}^{(2)}_{t}+\hat{\xi}^{(3)}_{t}:=\int_{[0,t]\times\\{|x|\geq
1\\}}x\hat{\mathcal{N}}({\mathrm{d}}s,{\mathrm{d}}x)+\int_{[0,t]\times(-1,1)}x\hat{\mathcal{N}}^{(sc)}({\mathrm{d}}s,{\mathrm{d}}x),\quad\quad
t\geq 0,$ (5.16)

is a NRLP with characteristics $(0,0,\Lambda,p)$. From our construction, the
random measures $\mathcal{N}$, $\hat{\mathcal{N}}$ can be encoded in terms of
a single Poisson random measure
$\sum_{u\in\mathscr{I}}\delta_{(u,\Delta\xi_{u},D_{u},\varepsilon_{u})}$,
allowing us to compute explicitly the characteristic function of the finite
dimensional distributions of $(\xi^{(2)},\hat{\xi}^{(2)})$ and
$(\xi^{(3)},\hat{\xi}^{(3)})$. In this direction, for $\lambda\in\mathbb{R}$
recall the notation

$\Phi^{(2)}(\lambda):=\int_{\\{|x|\geq 1\\}}(e^{i\lambda
x}-1)\Lambda({\mathrm{d}}x).$ (5.17)

######  Lemma 5.6.

For all $k\geq 1$, let $\lambda_{1},\dots,\lambda_{k}$ and
$\beta_{1},\dots\beta_{k}$ be real numbers and fix times
$0<t_{1}<\dots<t_{k}<t$. Then, we have

$\displaystyle\mathds{E}\left[\exp\bigg{\\{}i\sum_{j=1}^{k}\lambda_{j}\big{(}\xi_{t_{j}}^{(2)}+\beta_{j}\hat{\xi}^{(2)}_{t_{j}}\big{)}\bigg{\\}}\right]=$
$\displaystyle\exp\bigg{\\{}tp\mathds{E}\left[\Phi^{(2)}\left(\sum_{j=1}^{k}\lambda_{j}\mathds{1}_{\\{Y(t_{j}/t)\geq
1\\}}\right)\right]+t(1-p)\mathds{E}\left[\Phi^{(2)}\left(\sum_{j=1}^{k}(\lambda_{j}\mathds{1}_{\\{Y(t_{j}/t)\geq
1\\}}+\beta_{i}Y(t_{j}/t))\right)\right]\bigg{\\}},$ (5.18)

where we denote by $Y$ a Yule-Simon process with parameter $1/p$.

Let us briefly comment on this expression. The first exponential term in
(5.18) corresponds to the characteristic function of the finite dimensional
distributions of a Lévy process with law
$(\xi_{pt}^{(2)})_{t\in\mathbb{R}^{+}}$, viz.

$\mathds{E}\left[\exp\bigg{\\{}i\sum_{j=1}^{k}\lambda_{j}\xi_{pt_{j}}^{(2)}\bigg{\\}}\right]=\exp\bigg{\\{}tp\mathds{E}\left[\Phi^{(1)}\Big{(}\sum_{j=1}^{k}\lambda_{j}\mathds{1}_{\\{U\leq
t_{j}/t\\}}\Big{)}\right]\bigg{\\}},$

where $U$ is a uniform random variable in $[0,1]$ (recall that the first jump
time of a Yule-Simon process is uniformly distributed in $[0,1]$). More
precisely, this Lévy process is built from the discarded jumps
$\sum_{u}\mathds{1}_{\\{\varepsilon_{u}=0\\}}\delta_{(u,\Delta\xi_{u})}$ and
consequently is independent of $\hat{\xi}^{(2)}$ and
$\sum_{u}\mathds{1}_{\\{\varepsilon_{u}=1\\}}\delta_{(u,\Delta\xi_{u})}$,
which explains the form of the identity (5.18).

###### Proof.

We can assume that $t_{k}<1$ by working with $t_{1}/t<\dots<t_{k}/t$ and with
the pair $(\xi_{st},\hat{\xi}_{st})_{s\in[0,1]}$, which now has Lévy measure
$t\Lambda$. Now, the proof follows by a rather long but straightforward
application of the formula for the characteristic function of integrals with
respect to Poisson random measures. ∎

We now turn our attention to the characteristic function of the finite
dimensional distributions of $(\xi^{(3)},\hat{\xi}^{(3)})$. In this direction,
for $\lambda\in\mathbb{R}$, recall the notation

$\Phi^{(3)}(\lambda):=\int_{\\{|x|<1\\}}(e^{i\lambda x}-1-i\lambda
x)\Lambda({\mathrm{d}}x).$ (5.19)

######  Lemma 5.7.

For all $k\geq 1$, let $\lambda_{1},\dots,\lambda_{k}$ and
$\beta_{1},\dots\beta_{k}$ be real numbers and fix times
$0<t_{1}<\dots<t_{k}<t$. Then, we have

$\displaystyle\mathds{E}\left[\exp\bigg{\\{}i\sum_{j=1}^{k}\big{(}\lambda_{j}\xi_{t_{j}}^{(3)}+\beta_{j}\hat{\xi}^{(3)}_{t_{j}}\big{)}\bigg{\\}}\right]=$
$\displaystyle\exp\bigg{\\{}tp\mathds{E}\left[\Phi^{(3)}\left(\sum_{j=1}^{k}\lambda_{j}\mathds{1}_{\\{Y(t_{j}/t)\geq
1\\}}\right)\right]+t(1-p)\mathds{E}\left[\Phi^{(3)}\left(\sum_{j=1}^{k}(\lambda_{j}\mathds{1}_{\\{Y(t_{j}/t)\geq
1\\}}+\beta_{i}Y(t_{j}/t))\right)\right]\bigg{\\}},$ (5.20)

where we denote by $Y$ a Yule-Simon process with parameter $1/p$.

###### Proof.

By the usual scaling argument we can suppose that $t_{k}<1=t$. Now, the proof
is similar to the one of Corollary 2.8 in [7]. In this direction, notice that
the processes ${\xi}^{(3)}=\mathds{1}_{(-1,1)}x*{\mathcal{N}}^{(sc)}$ and
$\hat{\xi}^{(3)}=\mathds{1}_{(-1,1)}x*\hat{\mathcal{N}}^{(sc)}$ are
respectively the limit as $\varepsilon\downarrow 0$ of

$\displaystyle\xi^{(3)}_{\varepsilon,1}:=\mathds{1}_{\\{\varepsilon\leq|x|<1\\}}x*{\mathcal{N}}-\mathds{1}_{\\{\varepsilon\leq|x|<1\\}}x*{\mathrm{d}}t\otimes\Lambda,$
(5.21)
$\hat{\xi}^{(3)}_{\varepsilon,1}:=\mathds{1}_{\\{\varepsilon\leq|x|<1\\}}x*\hat{\mathcal{N}}-\mathds{1}_{\\{\varepsilon\leq|x|<1\\}}x*{\mathrm{d}}t\otimes\Lambda,$
(5.22)

the convergence holding uniformly in compact intervals. The characteristic
function of the finite-dimensional distributions of the pair
$(\mathds{1}_{\\{\varepsilon\leq|x|<1\\}}x*{\mathcal{N}},\mathds{1}_{\\{\varepsilon\leq|x|<1\\}}x*\hat{\mathcal{N}})$
can be computed by the same arguments as in Lemma 5.6 and we obtain for each
$0<\varepsilon<1$ that

$\displaystyle\mathds{E}\left[\exp\bigg{\\{}i\sum_{j=1}^{k}(\lambda_{j}\xi_{\varepsilon,1}^{(3)}(t_{j})+\beta_{j}\hat{\xi}^{(3)}_{\varepsilon,1}{(t_{j})})\bigg{\\}}\right]$
$\displaystyle=\exp\bigg{\\{}p\mathds{E}\left[\Phi^{(3)}_{\varepsilon}\left(\sum_{j=1}^{k}\lambda_{j}\mathds{1}_{\\{Y(t_{j})\geq
1\\}}\right)\right]+(1-p)\mathds{E}\left[\Phi^{(3)}_{\varepsilon}\left(\sum_{j=1}^{k}\big{(}\lambda_{j}\mathds{1}_{\\{Y(t_{j})\geq
1\\}}+\beta_{i}Y(t_{j})\big{)}\right)\right]\bigg{\\}}.$ (5.23)

In order to establish that this expression converges as $\varepsilon\downarrow
0$ towards (5.20), we recall that since $|e^{ix}-1-ix|$ is $O(|x^{2}|)$ as
$|x|\downarrow 0$ and $O(|x|)$ as $|x|\uparrow\infty$, for any
$r\in(\beta(\Lambda)\vee 1,1/p\wedge 2)$ if $\beta(\Lambda)<2$ and $r=2$ if
$\beta(\Lambda)=2$, we have

$C:=\sup_{x\in\mathbb{R}}|x|^{-r}|e^{ix}-1-ix|<\infty.$

It follows that for all $0<\varepsilon<1$, $\lambda\in\mathbb{R}$, we can
bound

$|\Phi_{\varepsilon}^{(3)}(\lambda)|\leq\int_{\\{|x|<1\\}}|e^{i\lambda
x}-1-i\lambda x|\Lambda({\mathrm{d}}x)\leq
C|\lambda|^{r}\int_{\\{|x|<1\\}}|x|^{r}\Lambda({\mathrm{d}}x).$

Moreover, by the remark following Lemma 2.1, the random variable $Y(t)\in
L_{r}(\mathbb{P})$ for any $r<1/p$ and it follows that the term

$\sum_{j=1}^{k}\big{(}\lambda_{j}\mathds{1}_{\\{Y(t_{j})\geq
1\\}}+\beta_{i}Y(t_{j})\big{)},$

is in $L_{r}(\mathbb{P})$. Hence, by dominated convergence, (5.23) converges
towards (5.20) as $\varepsilon\downarrow 0$. On the other hand, since
$(\xi_{\varepsilon,1}^{(3)}(t_{j}),\hat{\xi}_{\varepsilon,1}^{(3)}({t_{j}}))\rightarrow(\xi_{t_{j}}^{(3)},\hat{\xi}^{(3)}_{t_{j}})$
as $\varepsilon\downarrow 0$, we obtain the desired result. ∎

####  5.2.2 The distribution of $(B,\hat{B})$ and proof of Proposition 5.3

The last ingredient needed to define the joint distribution of
$(\xi,\hat{\xi})$ is the joint distribution of a Brownian motion $B$ and its
reinforced version $\hat{B}$, that we denote as $(B,\hat{B})$. Recall from [8]
that $\hat{B}$ has the same law as the solution to the SDE

${\mathrm{d}}X_{t}={\mathrm{d}}B_{t}+\frac{p}{t}X_{t}{\mathrm{d}}t,$ (5.24)

and that $X$ can be written explicitly in terms of the stochastic integral
(2.17) with respect to the driving Brownian motion $B$. We also recall from
(2.16) that for $0<s,t<T$ the covariance of $\hat{B}$ can be expressed in
terms of the Yule Simon process as follows:

$\mathds{E}\left[\hat{B}_{t}\hat{B}_{s}\right]=\frac{(t\vee s)^{p}(t\wedge
s)^{1-p}}{1-2p}=T(1-p)\mathds{E}\left[Y(t/T)Y(s/T)\right],$ (5.25)

and for later use, we observe that

${(t\wedge s)^{1-p}s^{p}}=T(1-p)\mathds{E}\left[\mathds{1}_{\\{Y(t/T)\geq
1\\}}Y(s/T)\right].$ (5.26)

We stress that the right-hand side in the previous display do not depend on
the choice of $T$. The proof of this identity is a consequence of the
representation (2.1) of $Y$ in terms of a standard Yule process and an
independent uniform random variable.

######  Definition 5.8.

Let $(B,\hat{B})$ be a pair of Gaussian processes and fix a parameter
$0<p<1/2$. We say that the pair $(B,\hat{B})$ has the law of a Brownian motion
with its reinforced version if the respective covariances are given by

$\mathds{E}\left[B_{t}B_{s}\right]=(t\wedge
s),\quad\quad\quad\mathds{E}\left[{B}_{t}\hat{B}_{s}\right]=(t\wedge
s)^{1-p}s^{p},\quad\quad\quad\mathds{E}\left[\hat{B}_{t}\hat{B}_{s}\right]=\frac{(t\vee
s)^{p}(t\wedge s)^{1-p}}{1-2p},$ (5.27)

for any $s,t\in\mathbb{R}^{+}$.

Let us briefly explain where this definition comes from: for fixed $p$, by [4,
Theorem 1.1] the law of the pair $(B,\hat{B})$ is universal, in the sense that
it is the weak joint scaling limit of random walks paired with its reinforced
version with parameter $p$ for $p<1/2$, when the typical step is in
$L_{2}(\mathbb{P})$. For more details, we refer to [8, 4].

Given a fixed Brownian motion $B$, it is clear that we can not expect to have
an explicit construction of the reinforced version $\hat{B}$ in terms of $B$
similar to the one performed for $(J,\hat{J})$. However, we can make use of
the SDE (5.24) to get an explicit construct of $(B,\hat{B})$ with the right
covariance structure. This can be easily achieved as follows: first, let $W$
be an independent copy of $B$; if we set

$\beta_{t}:=(1-p)B_{t}+\sqrt{1-(1-p)^{2}}W_{t},$ (5.28)

then, $B$ and $\beta$ are two Brownian motions with
$\mathds{E}\left[B_{t}\beta_{s}\right]=(1-p)(t\wedge s)$. If we let $\hat{B}$
be the solution to the SDE,

${\mathrm{d}}\hat{B}_{t}={\mathrm{d}}\beta_{t}+\frac{p}{t}\hat{B}_{t}{\mathrm{d}}t,$
(5.29)

$\hat{B}$ has the law of a noise reinforced Brownian motion with reinforcement
parameter $p$, and can be written explicitly as
$\hat{B}_{t}=t^{p}\int_{0}^{t}s^{-p}{\mathrm{d}}\beta_{s}$. Moreover, it
readily follows that the covariance of the pair of Gaussian processes
$(B,\hat{B})$ satisfies (5.27). The decorrelation applied for constructing
$\beta$ is playing the role of the thinning in the construction of
$(J,\hat{J})$.

Finally, we will need for the proof of Proposition 5.3 the following
representation of the characteristic function of the finite-dimensional
distributions of the pair $(B,\hat{B})$ in terms of the Yule-Simon process:

######  Lemma 5.9.

Let $(B,\hat{B})$ be a Brownian motion with its reinforced version for a
memory parameter $p<1/2$. For all $k\geq 1$,
$\lambda_{1},\dots,\lambda_{k},\beta_{1},\dots\beta_{k}$ real numbers and
$0<t_{1}<\dots<t_{k}<t$, we have

$\displaystyle\mathds{E}\left[\exp\bigg{\\{}i\sum_{j=1}^{k}\big{(}\lambda_{j}B_{t_{j}}+\beta_{j}\hat{B}_{t_{j}}\big{)}\bigg{\\}}\right]=$
$\displaystyle\exp\bigg{\\{}-{t\cdot
p}\mathds{E}\left[\frac{q^{2}}{2}\Bigg{(}\sum_{j=1}^{k}\lambda_{j}\mathds{1}_{\\{Y(t_{j}/t)\geq
1\\}}\Bigg{)}^{2}\right]-t\cdot{(1-p)}\mathds{E}\left[\frac{q^{2}}{2}\Bigg{(}\sum_{j=1}^{k}\big{(}\lambda_{j}\mathds{1}_{\\{Y(t_{j}/t)\geq
1\\}}+\beta_{j}Y(t_{j}/t)\big{)}\Bigg{)}^{2}\right]\bigg{\\}}.$ (5.30)

###### Proof.

Since NRBM satisfies the same scaling property of Brownian Motion (see page 3
of [8]), from (5.27) we deduce
$(B_{tc},\hat{B}_{tc})_{t\in\mathbb{R}^{+}}\overset{\mathscr{L}}{=}(c^{1/2}B_{t},c^{1/2}\hat{B}_{t})_{t\in\mathbb{R}^{+}}$.
Hence, as usual we can suppose that $t_{k}<1$ and we take $t:=1$. To simplify
notation we also suppose that $q=1$. Now, the left hand side of (5.9) writes

$\displaystyle\mathds{E}\left[\exp\bigg{\\{}i\sum_{j=1}^{k}\big{(}\lambda_{j}B_{t_{j}}+\beta_{j}\hat{B}_{t_{j}}\big{)}\bigg{\\}}\right]$
$\displaystyle=\exp\bigg{\\{}-\frac{1}{2}\sum_{i,j}\lambda_{i}\lambda_{j}Cov(B_{t_{i}},B_{t_{j}})-\frac{1}{2}\sum_{i,j}\beta_{i}\beta_{j}Cov(\hat{B}_{t_{i}},\hat{B}_{t_{j}})-\sum_{i,j}\lambda_{i}\beta_{j}Cov(B_{t_{i}},\hat{B}_{t_{j}})\bigg{\\}}$
$\displaystyle=\exp\bigg{\\{}-\frac{1}{2}\mathds{E}\left[\left(\sum_{j=1}^{k}\lambda_{j}\mathds{1}_{\\{Y(t_{j})\geq
1\\}}\right)^{2}+(1-p)\left(\sum_{j=1}^{k}\beta_{j}Y(t_{j})\right)^{2}+2{(1-p)}\sum_{i,j}\lambda_{i}\beta_{j}\mathds{1}_{\\{Y(t_{i})\geq
1\\}}Y(t_{j})\right]\bigg{\\}},$

where we used respectively for each one of the covariances in order of
appearance that: the first jump time of a Yule-Simon process is uniformly
distributed, (5.25) and (5.26). However, this is precisely the right hand side
of (5.9). ∎

Now that all the ingredients have been introduced, we define the law of
$(\xi,\hat{\xi})$.
$\Game$ Recipe for reinforcing Lévy processes: consider a starting Lévy
process $\xi$ with triplet $(a,q^{2},\Lambda)$ and denote by
$\xi_{t}=at+qB_{t}+J_{t}$ for $t\geq 0$ its Lévy Itô decomposition, where $B$
and $J$ are respectively a Brownian motion and a Lévy process with triplet
$(0,0,\Lambda)$. Further, fix $p\in(0,1)$ an admissible parameter for the
triplet, denote the jump measure of $\xi$ by
$\mathcal{N}=\sum\delta_{(u,\Delta\xi_{u})}$ and consider the NRPPP
$\hat{\mathcal{N}}$ with characteristic measure $\Lambda$ and reinforcement
parameter $p$ as constructed in (5.15) in terms of $\mathcal{N}$. Denote by
$\hat{J}:=\mathds{1}_{(-1,1)}x*\hat{\mathcal{N}}^{(sc)}+\mathds{1}_{(-1,1)^{c}}x*\hat{\mathcal{N}}$
the corresponding NRLP of characteristics $(0,0,\Lambda,p)$ and finally,
consider a NRBM $\hat{B}$ independent of $(J,\hat{J)}$, such that
$(B,\hat{B})$ has the law of a Brownian motion with its reinforced version –
for example by proceeding as in (5.29).

######  Definition 5.10.

We call the noise reinforced Lévy process
$\hat{\xi}_{t}:=at+q\hat{B}_{t}+\hat{J}_{t}$ for $t\geq 0$ of characteristics
$(a,q^{2},\Lambda,p)$ the noise reinforced version of ${\xi}$, the unicity
only holding in distribution. From now on, every time we consider a pair
$(\xi,\hat{\xi})$, it will be implicitly assumed that $\hat{\xi}$ has been
constructed by the procedure we just described in terms of $\xi$.

Let us now conclude the proof of Proposition 5.3.

###### Proof of Proposition 5.3.

If $\Psi$ is the characteristic exponent of $\xi$, we can write

$\Psi(\lambda)=ia\lambda-\frac{1}{2}q^{2}\lambda^{2}+\Phi^{(2)}(\lambda)+\Phi^{(3)}(\lambda),$

for $\Phi^{(2)}$, $\Phi^{(3)}$ defined respectively by (5.17) and (5.19).
Recalling the independence between the pairs $(B,\hat{B})$,
$(\xi^{(2)},\hat{\xi}^{(2)})$, $(\xi^{(3)},\hat{\xi}^{(3)})$, the proof of
Proposition 5.3 now follows from Lemmas 5.6, 5.7, 5.9 and the previous
decomposition for the characteristic exponent $\Psi$. ∎

From the construction of $({\mathcal{N}},\hat{\mathcal{N}})$, we can sketch a
sample path of $(\xi,\hat{\xi})$, where the jumps that are not appearing on
the path of $\hat{\xi}$ are precisely the ones deleted by the thinning:

Figure 2: Sample path of a Lévy process and its reinforced version.

###  5.3 Proof of Theorem 5.4

Let us outline the proof of Theorem 5.4. First, by (2.12), it suffices to
prove the convergence in $[0,1]$ and we therefore work with
$\xi=(\xi_{t})_{t\in[0,1]}$. Next, since we are working in $D^{2}[0,1]$, it
suffices to establish tightness coordinate-wise to obtain tightness for the
sequence of pairs. The first coordinate in (5.5) converges a.s. towards $\xi$
in $D[0,1]$ (and in particular is tight) and hence it remains to establish
tightness for the sequence of reinforced $n$-skeletons. This is the content of
Section 5.3.2 and more precisely, of Proposition 5.13. This is achieved by
means of the celebrated Aldous tightness criterion and our arguments rely on
the discrete counterpart of the remarkable martingale from Proposition 3.2.
This discrete martingale is introduced in Lemma 5.11 and we recall from [4, 3]
its main features. This is the content of Section 5.3.1. Finally, the joint
convergence in the sense of finite-dimensional distributions towards
$(\xi,\hat{\xi})$ is proved in Proposition 5.16, by establishing the
convergence of the corresponding characteristic functions.

####  5.3.1 The martingale associated with a noise reinforced random walk

$\circ$ The elephant random walk and its associated martingale. Let us start
with some historical context. In [2], Bercu was interested in establishing
asymptotic convergence results for a particular random walk with memory,
called the elephant random walk. This process is defined as follows: for a
fixed $q\in(0,1)$ that we still call the reinforcement parameter, we set
$\mathcal{E}_{0}:=0$ and let $Y_{1}$ be a random variable with
$Y_{1}\in\\{-1,1\\}$. Then, the position of our elephant at time $n=1$ is
given by $\mathcal{E}_{1}=Y_{1}$ and for $n\geq 2$, it is defined recursively
by the relation $\mathcal{E}_{n+1}:=\mathcal{E}_{n}+Y_{n+1}$, for $Y_{n+1}$
constructed by selecting uniformly at random one of the previous increments
$\\{Y_{1}\,\dots\,Y_{n}\\}$, and changing its sign with probability $1-q$. The
analysis of Bercu relies on a martingale associated to the elephant random
walk, defined as $M_{1}=\mathcal{E}_{1}$ and for $n\geq 2$, as

$M_{n}:=\hat{a}_{n}\mathcal{E}_{n},\quad\quad\text{ for
}\hat{a}_{n}:=\frac{\Gamma(n)\Gamma(2q)}{\Gamma(n+2q-1)},$ (5.31)

and where $\Gamma$ stands for the Euler-Gamma function. This martingale had
already made its appearance in the literature in Coletti, Gava, Schütz [10].
As was pointed out by Kürsten [15], the key is that when $q\in[1/2,1)$, the
elephant random walk is a version of the noise reinforced random walk when the
typical step $X$ has distribution
$\mathds{P}\left(X=1\right)=\mathds{P}\left(X=-1\right)=1/2$ with memory
parameter $p=2q-1$.
Getting back to our setting, we maintain the notation introduced at the
beginning of Section 5.2 for the noise reinforced random walk for a memory
parameter $p\in(0,1)$. Our first observation is that the martingale (5.31)
associated to the elephant random walk is still a martingale in our setting –
we stress that the reinforcement parameter $q$ in [2] corresponds to the
parameter $p=2q-1$ in our context. This martingale plays a fundamental role in
our reasoning, and also played a central role in [4, 3]. More precisely, let
$a_{1}:=1$ and for $n\geq 2$ we set

$a_{n}:=\frac{\Gamma(n)}{\Gamma(n+p)}=\prod_{k=1}^{n-1}\gamma_{k}^{-1},$
(5.32)

for $\gamma_{n}:=\frac{n+p}{n}$. We write
$\mathscr{F}_{n}:=\sigma(\hat{X}_{1},\dots,\hat{X}_{n})$ the filtration
generated by the reinforced steps. The following lemma is taken from [4].

######  Lemma 5.11.

_[4, Proposition 2.1]_ Suppose that the typical step $X$ is centred and in
$L_{2}(\mathbb{P})$. Then, the process $M$ defined by $M_{0}=0$ and
$M_{n}=a_{n}\hat{S}_{n}$ for $n\geq 1$ is a square-integrable martingale with
respect to the filtration $(\mathscr{F}_{n})$.

In order to establish tightness for our sequence of reinforced skeletons, we
will make use of the explicit form of the predictable quadratic variation
$\langle M,M\rangle$ of this martingale, which is the process defined as
$\langle M,M\rangle_{0}=0$ and

$\langle M,M\rangle_{n}=\sum_{k=1}^{n}\mathds{E}\left[\left(\Delta
M_{k}\right)^{2}|\mathscr{F}_{k-1}\right],\quad\quad n\geq 1.$

In this direction, we introduce:

$\hat{V}_{n}:=\hat{X}_{1}^{2}+\dots+\hat{X}_{n}^{2},\quad\quad n\geq 1,$

with $\hat{V}_{0}=0$. The following lemma is also taken from [4] and was the
main tool for establishing the invariance principles proven in that work.

######  Lemma 5.12.

_[4, Proposition 2.1]_ The predictable quadratic variation process $\langle
M,M\rangle$ is given by $\langle M,M\rangle_{0}=0$ and for $n\geq 1$,

$\langle
M,M\rangle_{n}=\sigma^{2}+\sum_{k=2}^{n}a^{2}_{k}\left((1-p)\sigma^{2}-p^{2}\frac{\hat{S}_{k-1}^{2}}{(k-1)^{2}}+p\frac{\hat{V}_{k-1}}{k-1}\right),$
(5.33)

where the sum should be considered null for $n=1$.

####  5.3.2 Proof of tightness

We stress that the f.d.d. convergence of the sequence of reinforced skeletons
towards a NRLP $\hat{\xi}$ of characteristics $(a,q^{2},\Lambda,p)$ was
already established in Theorem 3.1 of [7].

######  Proposition 5.13.

Let $p<1/2$ be an admissible memory parameter for the triplet
$(a,q^{2},\Lambda)$. Then, the sequence of laws associated to the reinforced
skeletons

$\big{\\{}\big{(}\hat{S}^{(n)}_{\lfloor
nt\rfloor}\big{)}_{t\in[0,1]}\,:\,n\in\mathbb{N}\big{\\}}\quad\text{is tight
in }D[0,1].$ (5.34)

Therefore, the convergence $\big{(}\hat{S}^{(n)}_{\lfloor
nt\rfloor}\big{)}_{t\in[0,1]}\overset{\mathscr{L}}{\longrightarrow}(\hat{\xi}_{t})_{t\in[0,1]}$
holds in $D[0,1]$.

The reason behind the restriction $p<1/2$ and why we don’t expect our proof to
work for $p\geq 1/2$ is explained in Remark 5.15, at the end of the proof.

#### Proof of Proposition 5.13 for centred $\xi$ with compactly supported Lévy