#  Langevin process reflected on a partially elastic boundary II

Emmanuel Jacob111 email<EMAIL_ADDRESS>
website. http://www.proba.jussieu.fr/pageperso/jacob

_Laboratoire de Probabilités et Modèles Aléatoires_
_Université Pierre et Marie Curie_
_4 place Jussieu, 75005 Paris, France_

Abstract

A particle subject to a white noise external forcing moves like a Langevin
process. Consider now that the particle is reflected at a boundary which
restores a portion $c$ of the incoming speed at each bounce. For $c$ strictly
smaller than the critical value $c_{crit}=\exp(-\pi/\sqrt{3})$, the bounces of
the reflected process accumulate in a finite time. We show that nonetheless
the particle is not necessarily absorbed after this time. We define a
“resurrected” reflected process as a recurrent extension of the absorbed
process, and study some of its properties. We also prove that this resurrected
reflected process is the unique solution to the stochastic partial
differential equation describing the model. Our approach consists in defining
the process conditioned on never being absorbed, via an $h-$transform, and
then giving the Itō excursion measure of the recurrent extension thanks to a
formula fairly similar to Imhof’s relation.

Key words. Langevin process, second order reflection, recurrent extension,
excursion measure, stochastic partial differential equation, $h$-transform.

A.M.S classification. (MSC2010) 60J50, 60H15

##  1 Introduction

Consider a particle in a one-dimensional space, submitted to a white noise
external forcing. Its velocity is then well-defined and given by a Brownian
motion, while its position is given by a so-called Langevin process. The
Langevin process is non-Markov, therefore its study is often based on that of
the Kolmogorov process, which is Markov. This Kolmogorov process is simply the
two-dimensional process, whose first coordinate is a Langevin process, and
second coordinate its derivative. We refer to Lachal [12] for a detailed
account about it. Further, suppose that the particle is constrained to stay in
$[0,+\infty[$ by a boundary at 0 characterized by an elasticity coefficient
$c\geq 0$. That is, the boundary restores a portion $c$ of the incoming
velocity at each bounce, and the equation of motion that we consider is the
following:

$(SOR)\qquad\left\\{\begin{array}[]{ccl}X_{t}&=&X_{0}+\displaystyle\int_{0}^{t}\dot{X}_{s}\mathrm{d}s\\\
\\\ \dot{X}_{t}&=&\dot{X}_{0}+B_{t}-(1+c)\sum_{0<s\leq
t}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0},\end{array}\right.$

where $B$ is a standard Brownian motion and $(X_{0},\dot{X}_{0})$ is called
the initial or starting condition. This stochastic partial differential
equation is nice outside the point $(0,0)$. Indeed, if the starting condition
is different from $(0,0)$, there is a simple pathwise construction of the
solution to this equation system, until time $\zeta_{\infty}$, the hitting
time of $(0,0)$ for the process $(X,\dot{X})$. However there is a tough
problem at $(0,0)$. Indeed, there exists an old literature about a
deterministic analogue to theses equations, where the white noise force is
replaced by a deterministic force. See Ballard [1] for a vast review. As early
as in 1960, Bressan [6] pointed out that multiple solutions may occur, even
when the force is $\mathcal{C}^{\infty}$. It appears that the introduction of
a white noise allows to get back a weak uniqueness result. We refer to [4]
(see also [3], [11]) for the particular case $c=0$.

In [10], we have shown for $c>0$ the existence of two different regimes, the
critical elasticity being $c_{crit}:=\exp(-\pi/\sqrt{3})$. It is critical in
the sense that when the starting condition is different from $(0,0)$, then we
have $\zeta_{\infty}=+\infty$ almost surely if $c\geq c_{crit}$, and
$\zeta_{\infty}<+\infty$ almost surely if $c<c_{crit}$. Further, we studied
the super-critical and the critical regimes. In this paper, we study the sub-
critical regime $c<c_{crit}$. The finite time $\zeta_{\infty}$ corresponds to
an accumulation of bounces in a finite time. We write $\mathbb{P}_{x,u}^{c}$
for the law of the reflected Kolmogorov process, with starting condition
$(x,u)\neq(0,0)$, elasticity coefficient $c$, and _killed at time
$\zeta_{\infty}$_. It is the unique strong solution to $(SOR)$ equations, up
to time $\zeta_{\infty}$. We also write $\mathrm{P}_{t}^{c}$ for the
associated semigroup. We will devote ourselves to prove the existence of a
unique recurrent extension to this process that leaves $(0,0)$ continuously.
Moreover, we will prove that this extension gives the unique solution, in the
weak sense, to $(SOR)$ equations.

We point out that this model was encountered by Bect in his thesis ([2],
section III.4.B). He observed the existence of the critical elasticity and
asked several questions on the different regimes. We answer to all of them.

In this work we will be largely inspired by a paper of Rivero [15], in which
he studies the recurrent extensions of a self-similar Markov process with
semigroup $\mathrm{P}_{t}$. Briefly, first, he recalls that recurrent
extensions are equivalent to excursion measures compatible with
$\mathrm{P}_{t}$, thanks to Itō’s program. Then a change of probability allows
him to define the Markov process conditioned on never hitting $0$, where this
conditioning is in the sense of Doob, via an $h-$transform. An inverse
$h-$transform on the Markov process conditioned on never hitting zero _and
starting from 0_ then gives the construction of the excursion measure.

We will not recall it at each step throughout the paper, but a lot of
parallels can be made. However, it is a two-dimensional Markov process that we
consider here. Further, its study will rely on an underlying random walk
$(S_{n})_{n\in\mathbb{N}}$ constructed from the velocities at bouncing times.

In the Preliminaries, we introduce this random walk and use it to estimate the
tail of the variable $\zeta_{\infty}$ under $\mathbb{P}_{0,1}^{c}$. In the
Section 3, we introduce a change of probability, via an $h-$transform, to
define $\widetilde{\mathbb{P}}_{x,u}$, law of a process which can be viewed as
the reflected Kolmogorov process conditioned on never being killed. We then
show in Subsection 3.2 that this law has a weak limit
$\widetilde{\mathbb{P}}_{0^{+}}$ when $(x,u)$ goes to $(0,0)$, using the same
method that was used in [10] to show that for $c>c_{crit}$, the laws
$\mathbb{P}_{0,u}^{c}$ have the weak limit $\mathbb{P}_{0+}^{c}$ when $u$ goes
to zero. All this section can be seen as a long digression to prepare the
construction of the excursion measure in Section 4. This excursion measure is
defined by a formula similar to Imhof’s relation (see [9]), connecting the
excursion measure of Brownian motion and the law of a Bessel(3) process. But
our formula involves the law $\mathbb{P}_{0+}^{c}$ and determines the unique
excursion measure compatible with the semigroup $\mathrm{P}_{t}^{c}$ . We call
_resurrected Kolmogorov process_ the corresponding recurrent extension.
Finally, we prove that this is the (weakly) unique solution to $(SOR)$
equations when the starting condition is $(0,0)$.

##  2 Preliminaries

We largely use the same notations as in [10]. For the sake of simplicity, we
use the same notation (say $P$) for a probability measure and for the
expectation under this measure. We will even authorize ourselves to write
$P(f,A)$ for the quantity $P(f\mathbbm{1}_{A})$, when $f$ is a measurable
functional and $A$ an event. We introduce
$D=(\\{0\\}\times\mathbb{R}_{+}^{*})\cup(\mathbb{R}_{+}^{*}\times\mathbb{R})$
and $D^{0}:=D\cup\\{(0,0)\\}$. Our working space is $\mathcal{C}$, the space
of càdlàg trajectories $(x,\dot{x}):[0,\infty)\to D^{0}$, which satisfy

$x(t)=x(0)+\displaystyle\int_{0}^{t}\dot{x}(s)\mathrm{d}s.$

That space is endowed with the $\sigma-$algebra generated by the coordinate
maps and with the topology induced by the following injection:

$\begin{array}[]{ccc}\mathcal{C}&\to&\mathbb{R}_{+}\times\mathbb{D}\\\
(x,\dot{x})&\mapsto&\big{(}x(0),\dot{x}\big{)},\end{array}$

where $\mathbb{D}$ is the space of càdlàg trajectories on $\mathbb{R}_{+}$,
equipped with Skorohod topology.

We denote by $(X,\dot{X})$ the canonical process and by
$(\mathfrak{F}_{t},t\geq 0)$ its natural filtration, satisfying the usual
conditions of right continuity and completeness. For an initial condition
$(x,u)\in D$, the $(SOR)$ equations

$\left\\{\begin{array}[]{ccl}X_{t}&=&x+\displaystyle\int_{0}^{t}\dot{X}_{s}\mathrm{d}s\\\
\\\ \dot{X}_{t}&=&u+B_{t}-(1+c)\sum_{0<s\leq
t}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0}\end{array}\right.$

have a unique solution, at least up to the random time

$\zeta_{\infty}:=\inf\\{t>0,X_{t}=0,\dot{X}_{t}=0\\}.$

We call (killed) reflected Kolmogorov process this solution killed at time
$\zeta_{\infty}$, and write $\mathbb{P}_{x,u}^{c}$ for its law. It is Markov.
We also call reflected Langevin process the first coordinate of this process,
which is no longer Markov.

Call $\zeta_{1}$ the first hitting time of zero for the reflected Langevin
process $X$, that is $\zeta_{1}:=\inf\\{t>0,X_{t}=0\\}$. More generally, the
sequence of the successive hitting times of zero $(\zeta_{n})_{n\geq 1}$ is
defined recursively by $\zeta_{n+1}:=\inf\\{t>\zeta_{n},X_{t}=0\\}$. We write
$(V_{n})_{n\geq 1}:=(\dot{X}_{\zeta_{n}})_{n\geq 1}$ for the sequence of the
velocities of the process at these hitting times. That means outgoing
velocities, as we are dealing with right-continuous processes. Finally, when
the starting position is $x=0$, we will simply write $\mathbb{P}^{c}_{u}$ for
$\mathbb{P}^{c}_{0,u}$, and we will also define $\zeta_{0}=0$ and
$V_{0}=\dot{X}_{0}$. We insist on the fact that in each case the starting
condition $(x,u)$ is different from $(0,0)$. Then it is not difficult to see
that $\zeta_{\infty}$ coincides almost surely with $\sup\zeta_{n}$. But we can
say much more.

The sequence
$\left(\dfrac{\zeta_{n+1}-\zeta_{n}}{V_{n}^{2}},\dfrac{V_{n+1}}{V_{n}}\right)_{n\geq
0}$ is i.i.d. and of law independent of $u$, which can be deduced from the
following density:

$\frac{1}{\mathrm{d}s\mathrm{d}v}\mathbb{P}_{1}^{c}\left(({\zeta_{1}},{V_{1}}/c)\in(\mathrm{d}s,\mathrm{d}v)\right)=\frac{3v}{\pi\sqrt{2}s^{2}}\exp(-2\frac{v^{2}-v+1}{s})\int_{0}^{4v/s}e^{-\frac{3\theta}{2}}\frac{\mathrm{d}\theta}{\sqrt{\pi\theta}},$
(2.1)

given by McKean [13]. The second marginal of this density is

$\mathbb{P}_{1}^{c}({V_{1}}/c\in\mathrm{d}v)=\frac{3}{2\pi}\frac{v^{\frac{3}{2}}}{1+v^{3}}\mathrm{d}v.$
(2.2)

In particular, the sequence $S_{n}:=\ln(V_{n})$ is a random walk, with drift

$\mathbb{P}_{1}^{c}(S_{1}-S_{0})=\ln(c)+\frac{\pi}{\sqrt{3}},$

which is zero for the critical value $c_{crit}=\exp(-\pi/\sqrt{3})$. In this
paper we lie in the subcritical case $c<c_{crit}$, when the drift is negative.
A thorough study allows to not only deduce the finiteness of $\zeta_{\infty}$,
but also estimate its tail.

######  Lemma 1.

We have

$\mathbb{P}_{1}^{c}\left(V_{1}^{x}\right)=\frac{c^{x}}{2\cos(\frac{x+1}{3}\pi)}\textrm{
for }x<1/2.$ (2.3)

There exists a unique $k=k(c)$ in $(0,1/4)$ such that
$\mathbb{P}_{1}^{c}\left(V_{1}^{2k}\right)=1$, and

$\mathbb{P}_{1}^{c}(\zeta_{\infty}>t)\underset{t\to\infty}{\sim}C_{1}t^{-k},$
(2.4)

where $C_{1}=C_{1}(c)\in(0,\infty)$ is a constant depending only on $c$, given
by

$C_{1}=\frac{\mathbb{P}_{1}^{c}\left(\zeta_{\infty}^{k}-(\zeta_{\infty}-\zeta_{1})^{k}\right)}{k\mathbb{P}_{1}^{c}(V_{1}^{2k}\ln(V_{1}^{2}))}.$
(2.5)

In other words, $k(c)$ is given implicitly as the unique solution in
$]0,\frac{1}{4}]$ of the equation

$c=\left[2\cos\left(\frac{2k+1}{3}\pi\right)\right]^{\frac{1}{2k}}.$ (2.6)

The upper bound $1/4$ stems from the fact that
$\mathbb{P}_{1}^{c}\left(V_{1}^{2k}\right)$ becomes infinite for $k=1/4$. The
value of $k(c)$ converges to $1/4$ when $c$ goes to 0, and to $0$ when $c$
goes to $c_{crit}$, as illustrated by Figure 1. We may notice that Formula
(2.4) remains true for $c=0$ and $k=1/4$ (and for $c=c_{crit}$ and $k=0$, in a
certain sense).

Figure 1: Graph of the exponent k(c)

###### Proof.

Formula (2.3) is not new. For the convenience of the reader, we still provide
the following calculation. From Formula (2.2), it follows, for $x<1/2$,

$\displaystyle\mathbb{P}_{1}^{c}\left(\left({V_{1}}/c\right)^{x}\right)$
$\displaystyle=$
$\displaystyle\frac{3}{2\pi}\int_{0}^{\infty}\frac{t^{x+3/2}}{1+t^{3}}\mathrm{d}t=\frac{1}{2\pi}\int_{0}^{\infty}\frac{t^{\frac{x}{3}-\frac{1}{6}}}{1+t}\mathrm{d}t.$

Note $\cos(\frac{x+1}{3}\pi)=\sin(\pi y)$, where $y=\frac{x}{3}+\frac{5}{6}$.
Using the variable $y$, which belongs to $(0,1)$, Equation (2.3) becomes

$\int_{0}^{\infty}\frac{t^{y-1}}{1+t}\mathrm{d}t=\frac{\pi}{\sin(\pi y)},$

and follows from:

$\displaystyle\int_{0}^{\infty}\frac{t^{y-1}}{1+t}\mathrm{d}t$
$\displaystyle=$ $\displaystyle\int_{0}^{1}t^{y}(1-t)^{1-y}\mathrm{d}t$
$\displaystyle=$ $\displaystyle\mathrm{B}(y,1-y)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(y)\Gamma(1-y)}{\Gamma(1)}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\sin(\pi y)}.$

where $\mathrm{B}$ and $\Gamma$ are the usual Beta and Gamma function,
respectively.

Now, the function $x\mapsto\mathbb{P}_{1}^{c}\left(V_{1}^{x}\right)$ is
convex, takes value 1 at $x=0$ and becomes infinite at $x=1/2$. Its derivative
at 0 is equal to $\mathbb{P}_{1}^{c}(S_{1}-S_{0})<0.$ We deduce that there is
indeed a unique $k(c)$ in $(0,\frac{1}{4})$ such that
$\mathbb{P}_{1}^{c}\left(V_{1}^{2k}\right)=1.$

Estimate (2.4) will appear as a particular case of an “implicit renewal
theory” result of Goldie [7]. Let us express $\zeta_{\infty}$ as the series:

$\displaystyle\zeta_{\infty}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\frac{\zeta_{n}-\zeta_{n-1}}{V_{n-1}^{2}}V_{n-1}^{2},$

with
$V_{n}^{2}:=V_{1}^{2}\frac{V_{2}^{2}}{V_{1}^{2}}\cdot\cdot\cdot\frac{V_{n}^{2}}{V_{n-1}^{2}}$,
and where
$\left(\dfrac{\zeta_{n}-\zeta_{n-1}}{V_{n-1}^{2}},\dfrac{V_{n}^{2}}{V_{n-1}^{2}}\right)_{n\geq
1}$ is i.i.d. We lie in the setting of Section 4 of Goldie’s paper [7], and
can apply its Theorem (4.1). Indeed, all the following conditions are
satisfied:

$\mathbb{P}_{1}^{c}(V_{1}^{2k})=1,$
$\mathbb{P}_{1}^{c}(V_{1}^{2k}\ln(V_{1}^{2}))<\infty,$
$\mathbb{P}_{1}^{c}(\zeta_{1}^{k})<\infty,$

the last one being a consequence of the inequality $k<1/4$ and of the
following estimate of the queue of the variable $\zeta_{1}$,

$\mathbb{P}_{1}^{c}(\zeta_{1}>t)\underset{t\to\infty}{\sim}c^{\prime}t^{-\frac{1}{4}},$
(2.7)

which was already pointed out in Lemma 1 in [10]. All this is enough to apply
the theorem of Goldie and deduce the requested result, namely

$\mathbb{P}_{1}^{c}(\zeta_{\infty}>t)\underset{t\to\infty}{\sim}C_{1}t^{-k},$

where $C_{1}$ is the constant defined by (2.5), and belongs to $]0,\infty[$. ∎

Next section is devoted to the definition and study of the reflected
Kolmogorov process, _conditioned on never hitting $(0,0)$_. This process will
be of great use for studying the recurrent extensions of the reflected
Kolmogorov process in Section 4.

##  3 The reflected Kolmogorov process conditioned on never hitting $(0,0)$

###  3.1 Definition via an $h-$transform

Recall that under $\mathbb{P}_{1}^{c}$, the sequence $(S_{n})_{n\geq
0}=(\ln(V_{n}))_{n\geq 0}$ is a random walk starting from 0, and write
$\mathbf{P}_{0}$ for its law. The important fact
$\mathbb{P}_{1}^{c}(V_{1}^{2k})=1$ implies $\mathbb{P}_{1}^{c}(V_{n}^{2k})=1$
for any $n>0$, and can be rewritten $\mathbf{P}_{0}(\theta^{S_{n}})=1$, with
$\theta:=\exp(2k)$.

The sequence $\theta^{S_{n}}$ being a martingale, we introduce the change of
probability

$\widetilde{\mathbf{P}}_{0}(S_{n}\in\mathrm{d}t)=\theta^{t}\mathbf{P}_{0}(S_{n}\in\mathrm{d}t).$

Under $\widetilde{\mathbf{P}}_{0}$, $(S_{n})_{n\geq 0}$ becomes a random walk
drifting to $+\infty$. Informally, it can be viewed as being the law of the
random walk $S_{n}$ under $\mathbf{P}_{0}$ conditioned on hitting arbitrary
high levels.

There is a corresponding change of probability for the reflected Kolmogorov
process and its law $\mathbb{P}_{1}^{c}$. We introduce the law
$\widetilde{\mathbb{P}}_{1}$ determined by

$\widetilde{\mathbb{P}}_{1}(A\mathbbm{1}_{\zeta_{n}>T})=\mathbb{P}^{c}_{1}(A\mathbbm{1}_{\zeta_{n}>T}\mathbb{P}_{1}^{c}(V_{n}^{2k}|\mathfrak{F}_{T})),$

for any $n>0$, stopping-time $T$ and $A\in\mathfrak{F}_{T}$. By the strong
Markov property we have

$\mathbb{P}_{1}^{c}(V_{n}^{2k}|\mathfrak{F}_{T})=\mathbb{P}_{X_{T},\dot{X}_{T}}^{c}(V_{1}^{2k})\qquad\text{on
the event }\\{\zeta_{n}>T\\},$

so that there is the identity

$\widetilde{\mathbb{P}}_{1}(A\mathbbm{1}_{\zeta_{n}>T})=\mathbb{P}^{c}_{1}(A\mathbbm{1}_{\zeta_{n}>T}H(X_{T},\dot{X}_{T})),$

where we have written

$H(x,u):=\mathbb{P}_{x,u}^{c}(V_{1}^{2k}).$

Note that $H(0,u)=u^{2k}$. Letting $n$ go to infinity, we get:

$\widetilde{\mathbb{P}}_{1}(A\mathbbm{1}_{\zeta_{\infty}>T})=\mathbb{P}^{c}_{1}(A\mathbbm{1}_{\zeta_{\infty}>T}H(X_{T},\dot{X}_{T})).$

We have $H(0,1)=1$, the function $H$ is harmonic for the semigroup of the
reflected Kolmogorov process, and the process $\widetilde{\mathbb{P}}_{1}$ is
the $h-$transform of $\mathbb{P}_{1}^{c}$, in the sense of Doob.

Under $\widetilde{\mathbb{P}}_{1}$, the law of the sequence $(S_{n})_{n\geq
0}$ is $\widetilde{\mathbf{P}}_{0}$, thus this sequence is diverging to
$+\infty$, and as a consequence the time $\zeta_{\infty}$ is infinite
$\widetilde{\mathbb{P}}_{1}-$almost surely. The term
$\mathbbm{1}_{\zeta_{\infty}>T}$ in
$\widetilde{\mathbb{P}}_{1}(A\mathbbm{1}_{\zeta_{\infty}>T})$ is thus
unnecessary. We may now give a more general definition of this change of
probability, as an $h-$transform, for any starting position $(x,u)$.

######  Definition 1.

The reflected Kolmogorov process _conditioned on never hitting $(0,0)$_ is the
Markov process given by its law $\widetilde{\mathbb{P}}_{x,u}$, for any
starting condition $(x,u)\in D$, which is the unique measure such that for
every stopping-time $T$ we have

$\widetilde{\mathbb{P}}_{x,u}(A)=\frac{1}{H(x,u)}\mathbb{P}_{x,u}^{c}(AH(X_{T},\dot{X}_{T}),T<\zeta_{\infty}),$
(3.1)

for any $A\in\mathfrak{F}_{T}$. We write $\widetilde{\mathrm{P}}_{t}$ its
associated semigroup, and we also write $\widetilde{\mathbb{P}}_{u}$ for
$\widetilde{\mathbb{P}}_{0,u}$.

This denomination is justified by the following proposition.

######  Proposition 1.

For any $(x,u)\in D$ and $t>0$, we have

$\widetilde{\mathbb{P}}_{x,u}(A)=\lim_{s\to\infty}\mathbb{P}^{c}_{x,u}(A|\zeta_{\infty}>s),$
(3.2)

for any $A\in\mathfrak{F}_{t}$.

We stress that in [15], Proposition 2, Rivero defines in a similar way the
self-similar Markov process conditioned on never hitting 0. Incidentally, you
can find in [11] a thorough study of other $h-$transforms regarding the
Kolmogorov process killed at time $\zeta_{1}$.

In order to get Formula (3.2), we first prove the following lemma, which is a
slight improvement of (2.4):

######  Lemma 2.

For any $(x,u)\in D$,

$s^{k}\mathbb{P}^{c}_{x,u}(\zeta_{\infty}>s)\underset{s\to\infty}{\longrightarrow}H(x,u)C_{1}.$
(3.3)

###### Proof.

For $(x,u)=(0,1)$, this is (2.4). For $x=0$, the rescaling invariance property
yields immediately

$s^{k}\mathbb{P}^{c}_{0,u}(\zeta_{\infty}>s)=s^{k}\mathbb{P}^{c}_{0,1}(\zeta_{\infty}>su^{-2})\underset{s\to\infty}{\longrightarrow}u^{2k}C_{1}=H(0,u)C_{1}.$

For $(x,u)\in D$, the Markov property at time $\zeta_{1}$ yields

$\displaystyle s^{k}\mathbb{P}^{c}_{x,u}(\zeta_{\infty}>s)$ $\displaystyle=$
$\displaystyle\mathbb{P}^{c}_{x,u}(s^{k}\mathbb{P}^{c}_{0,V_{1}}(\zeta_{\infty}>s-\zeta_{1}))$
$\displaystyle\underset{s\to\infty}{\longrightarrow}$
$\displaystyle\mathbb{P}^{c}_{x,u}(H(0,V_{1})C_{1})=H(x,u)C_{1},$

where the convergence holds by dominated convergence. The lemma is proved. ∎

Formula (3.2) then results from:

$\displaystyle\mathbb{P}^{c}_{x,u}(A|\zeta_{\infty}>s)$ $\displaystyle=$
$\displaystyle\frac{1}{\mathbb{P}^{c}_{x,u}(\zeta_{\infty}>s)}\mathbb{P}^{c}_{x,u}\left(A\mathbb{P}^{c}_{X_{t},\dot{X}_{t}}(\zeta_{\infty}>s-t),\zeta_{\infty}>t\right)$
$\displaystyle\underset{s\to\infty}{\longrightarrow}$
$\displaystyle\frac{1}{H(x,u)}\mathbb{P}^{c}_{x,u}\left(AH(X_{T},\dot{X}_{T}),\zeta_{\infty}>t\right)$
$\displaystyle=$ $\displaystyle\ \widetilde{\mathbb{P}}_{x,u}(A).$

###  3.2 Starting the conditioned process from $(0,0)$

The study of the reflected Kolmogorov process conditioned on never hitting
$(0,0)$ will happen to be very similar to that of the reflected Kolmogorov
process in the supercritical case $c>c_{crit}$, done in [10]. Observe the
following similarities between the laws $\widetilde{\mathbb{P}}_{u}$, and
$\mathbb{P}^{c}_{u}$ when $c>c_{crit}$: the sequence
$\left(\dfrac{\zeta_{n+1}-\zeta_{n}}{V_{n}^{2}},\dfrac{V_{n+1}}{V_{n}}\right)_{n\geq
0}$ is i.i.d., we know its law explicitly, and the sequence $S_{n}=\ln(V_{n})$
is a random walk with positive drift. It follows that a major part of [10] can
be transcribed _mutatis mutandis_. In particular we will get a convergence
result for the probabilities $\widetilde{\mathbb{P}}_{u}$ when $u$ goes to
zero, similar to Theorem 1 of [10].

Under $\widetilde{\mathbb{P}}_{1}$, the sequence $(S_{n})_{n\geq 0}$ is a
random walk of law $\widetilde{\mathbf{P}}_{0}$. Write $\mu$ for its drift,
that is the expectation of its jump distribution, which is positive and
finite. The associated strictly ascending ladder height process
$(H_{n})_{n\geq 0}$, defined by $H_{k}=S_{n_{k}}$, where $n_{0}=0$ and
$n_{k}=\inf\\{n>n_{k-1},S_{n}>S_{n_{k-1}}\\}$, is a random walk with positive
jumps. Its jump distribution also has positive and finite expectation
$\mu_{H}\geq\mu$. The measure

$m(\mathrm{d}y):=\frac{1}{\mu_{H}}\widetilde{\mathbf{P}}_{0}(H_{1}>y)\mathrm{d}y.$
(3.4)

is the “stationary law of the overshoot”, both for the random walks
$(S_{n})_{n\geq 0}$ and $(H_{n})_{n\geq 0}$. The following proposition holds.

######  Proposition 2.

The family of probability measures $(\widetilde{\mathbb{P}}_{x,u})_{(x,u)\in
D}$ on $\mathcal{C}$ has a weak limit when $(x,u)\to(0,0)$, which we denote by
$\widetilde{\mathbb{P}}_{0^{+}}$. More precisely, write $\tau_{v}$ for the
instant of the first bounce with speed greater than $v$, that is
$\tau_{v}:=\inf\\{t>0,X_{t}=0,\dot{X}_{t}>v\\}.$ Then the law
$\widetilde{\mathbb{P}}_{0^{+}}$ satisfies the following properties:

$\begin{array}[]{ll}(*)&\left\\{\begin{array}[]{l}\displaystyle\lim_{v\to
0^{+}}\tau_{v}=0\quad\text{almost surely}.\\\ \text{For any }u,v>0\text{, and
conditionally on }\dot{X}_{\tau_{v}}=u\text{, the process }\\\
(X_{\tau_{v}+t},\dot{X}_{\tau_{v}+t})_{t\geq 0}\text{ is independent of
}(X_{s},\dot{X}_{s})_{s<\tau_{v}}\text{ and has law
}\widetilde{\mathbb{P}}_{u}.\end{array}\right.\\\ \\\ (**)&\text{For any
}v>0,\text{ the law of }\ln(\dot{X}_{\tau_{v}}/v)\text{ is }m.\end{array}$

In the proof of this proposition we can take $x=0$ and just prove the
convergence result for the laws $\widetilde{\mathbb{P}}_{u}$ when $u\to 0+$.
The general result will follow as an application of the Markov property at
time $\zeta_{1}$.

The complete proof follows mainly the proof of Theorem 1 in [10] and takes
many pages. Here, the reader has three choices. Skip this proof and go
directly to next section about the resurrected process. Or read the following
for an overview of the ideas of the proof, with details given only when
significantly different from that in [10]. Or, read [10] and the following, if
(s)he wants to get the complete proof.

Call $T_{y}(S)$ the hitting time of $(y,\infty)$ for the random walk $S$
starting from $x<y$. Call $\widetilde{\mathbf{P}}_{\mu}$ the law of
$(S_{n})_{n\geq 0}$ obtained by taking $S_{0}$ and $(S_{n}-S_{0})_{n\geq 0}$
independent, with law $m$ and $\widetilde{\mathbf{P}}_{0}$, respectively. That
is, we allow the starting position to be nonconstant and distributed according
to $\mu$. A result of renewal theory states that the law of the overshoot
$(S_{n+T_{y}}-y)_{n\geq 0}$ under $\widetilde{\mathbf{P}}_{x}$, when $x$ goes
to $-\infty$, converges to $\widetilde{\mathbf{P}}_{m}$. Now, for a process
indexed by $I$ an interval of $\mathbb{Z}$, we define a spatial translation
operator by $\Theta^{sp}_{y}((S_{n})_{n\in
I})=(S_{n+T_{y}}-y)_{n\in{I-T_{y}}}$. We get that under
$\widetilde{\mathbf{P}}_{x}$ and when $x$ goes to $-\infty$, the translated
process $\Theta^{sp}_{y}(S)$ converges to a process called the “spatially
stationary random walk", a process indexed by $\mathbb{Z}$ which is spatially
stationary and whose restriction to $\mathbb{N}$ is
$\widetilde{\mathbf{P}}_{m}$ (see [10]). We write $\widetilde{\mathbf{P}}$ for
the law of this spatially stationary random walk.

There exists a link between the law $\widetilde{\mathbf{P}}_{x}$ and the law
$\widetilde{\mathbb{P}}_{e^{x}}$: the first one is the law of the underlying
random walk $(S_{n})_{n\geq 0}=(\ln V_{n})_{n\geq 0}$ for a process
$(X,\dot{X})$ following the second one. Now, in a very brief shortcut, we can
say that the law $\widetilde{\mathbf{P}}$ is linked to a law written
$\widetilde{\mathbb{P}}_{0^{+}}^{*}$. And the convergence results of
$\widetilde{\mathbf{P}}_{x}\circ\Theta^{sp}_{y}$ to $\mathbf{P}$ when
$x\to-\infty$ provide convergence results of $\widetilde{\mathbb{P}}_{u}$ to
$\widetilde{\mathbb{P}}_{0^{+}}^{*}$ when $u\to 0$.

However, this link is different, as the spatially stationary random walk, of
law $\widetilde{\mathbf{P}}$, is a process indexed by $\mathbb{Z}$. The value
$S_{0}$ is thus not equal to the logarithm of the velocity of the process at
time 0, but at time $\tau_{1}$ (recall that
$\tau_{1}=\inf\\{t>0,X_{t}=0,\dot{X}_{t}\geq 1\\}$ is the instant of the first
bounce with speed no less than one). The sequence $(S_{n})_{n\geq 0}$ is then
the sequence of the logarithms of the velocities of the process at the
bouncing times, starting from that bounce. The sequence $(S_{-n})_{n\geq 0}$
is the sequence of the logarithms of the velocities of the process at the
bouncing times happening _before_ that bounce.

The law $\widetilde{\mathbb{P}}_{0^{+}}^{*}$ is the law of a process indexed
by $\mathbb{R}_{+}^{*}$, but we actually construct it “from the random time
$\tau_{1}$”. In order for the definition to be clean, we have to prove that
the random time $\tau_{1}$ is finite a.s. In [10], we used the fact that if
$(\zeta_{1,k})_{k\geq 0}$ is a sequence of i.i.d random variables, with common
law that of $\zeta_{1}$ under $\mathbb{P}^{c}_{1}$, then for any
$\varepsilon>0$ there is almost surely only a finite number of indexes $k$
such that $\ln(\zeta_{1,k})\geq\varepsilon k.$ This was based on Formula 2.7,
which, we recall, states

$\mathbb{P}_{1}^{c}(\zeta_{1}>t)\underset{t\to\infty}{\sim}c^{\prime}t^{-\frac{1}{4}},$

where $c^{\prime}$ is some positive constant. Here the same results holds with
replacing $\mathbb{P}^{c}_{1}$ by $\widetilde{\mathbb{P}}_{1}$ and is a
consequence from the following lemma.

######  Lemma 3.

We have

$\widetilde{\mathbb{P}}_{1}(\zeta_{1}>t)\underset{t\to\infty}{\sim}c^{\prime}t^{k-\frac{1}{4}},$
(3.5)

where $c^{\prime}$ is some positive constant.

###### Proof.

From (3.1) and (2.1), we get that the density of $(\zeta_{1},V_{1}/c)$ under
$\widetilde{\mathbb{P}}_{1}$ is given by

$\displaystyle
f(s,v):=\frac{1}{\mathrm{d}s\mathrm{d}v}\widetilde{\mathbb{P}}_{1}((\zeta_{1},V_{1}/c)\in\mathrm{d}s\mathrm{d}v)$
$\displaystyle=$
$\displaystyle(cv)^{2k}\frac{3v}{\pi\sqrt{2}s^{2}}\exp(-2\frac{v^{2}-v+1}{s})\int_{0}^{\frac{4v}{s}}e^{-\frac{3\theta}{2}}\frac{\mathrm{d}\theta}{\sqrt{\pi\theta}}.$

Thanks to the inequality

$4\sqrt{\frac{v}{s\pi}}e^{-\frac{6v}{s}}\leq\int_{0}^{\frac{4v}{s}}e^{-\frac{3\theta}{2}}\frac{\mathrm{d}\theta}{\sqrt{\pi\theta}}\leq
4\sqrt{\frac{v}{s\pi}},$

we may write

$\displaystyle f(s,v)$ $\displaystyle=$
$\displaystyle(6\sqrt{2}.\pi^{-\frac{3}{2}}c^{2k})s^{-\frac{5}{2}}v^{\frac{3}{2}+2k}e^{-2\frac{v^{2}}{s}+\frac{v}{s}K(s,v)},$

where $(s,v)\mapsto K(s,v)$ is continuous and bounded. The marginal density of
$\zeta_{1}$ is thus given by

$\displaystyle\frac{1}{\mathrm{d}s}\widetilde{\mathbb{P}}_{1}(\zeta_{1}\in\mathrm{d}s)$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}_{+}}f(s,v)\mathrm{d}v$
$\displaystyle=$
$\displaystyle(3\sqrt{2}.\pi^{-\frac{3}{2}}c^{2k})s^{-\frac{5}{4}+k}\int_{\mathbb{R}_{+}}w^{\frac{1}{4}+k}e^{-2w+K(s,\sqrt{sw})\sqrt{w/s}}\mathrm{d}w$
$\displaystyle\underset{s\to\infty}{\sim}$
$\displaystyle(3\sqrt{2}.\pi^{-\frac{3}{2}}c^{2k})s^{-\frac{5}{4}+k}\int_{\mathbb{R}_{+}}w^{\frac{1}{4}+k}e^{-2w}\mathrm{d}w,$

where we used successively the change of variables $w=v^{2}/s$ and dominated
convergence theorem. Just integrate this equivalence in the neighborhood of
$+\infty$ to get

$\widetilde{\mathbb{P}}_{1}(\zeta_{1}>t)\underset{t\to\infty}{\sim}c^{\prime}t^{k-\frac{1}{4}},$

with the constant

$c^{\prime}=\frac{3\sqrt{2}.\pi^{-\frac{3}{2}}c^{2k}}{\frac{1}{4}-k}\int_{\mathbb{R}_{+}}w^{\frac{1}{4}+k}e^{-2w}\mathrm{d}w=\frac{3c^{2k}}{\pi^{\frac{3}{2}}2^{\frac{3}{4}+k}}\cdot\frac{1+4k}{1-4k}\
\Gamma\left(\frac{1}{4}+k\right).$

∎

For now, we have introduced $\widetilde{\mathbb{P}}_{0^{+}}^{*}$, law of a
process $(X,\dot{X})$ indexed by $\mathbb{R}_{+}^{*}$. We keep on following
the proof of [10]. First, we get that this law satisfies conditions $(*)$ and
$(**)$, and that for any $v>0$, the joint law of $\tau_{v}$ and
$(X_{\tau_{v}+t},\dot{X}_{\tau_{v}+t})_{t\geq 0}$ under
$\widetilde{\mathbb{P}}_{u}$ converges to that under
$\widetilde{\mathbb{P}}_{0^{+}}^{*}$. Then we establish Proposition 2 by
controlling the behavior of the process just after time 0, through the two
following lemmas:

######  Lemma 4.

Under $\widetilde{\mathbb{P}}_{0^{+}}^{*}$, we have almost surely
$(X_{t},\dot{X}_{t})\underset{t\to 0}{\longrightarrow}(0,0).$

This lemma allows in particular to extend $\widetilde{\mathbb{P}}_{0^{+}}^{*}$
to $\mathbb{R}_{+}$. We call $\widetilde{\mathbb{P}}_{0^{+}}$ this extension.
The second lemma is more technical and controls the behavior of the process on
$[0,\tau_{v}[$ under $\widetilde{\mathbb{P}}_{u}$.

######  Lemma 5.

Write $M_{v}=\sup\\{|\dot{X}_{t}|,t\in[0,\tau_{v}[\\}$. Then,

$\forall\varepsilon>0,\forall\delta>0,\exists v_{0}>0,\exists u_{0}>0,\forall
0<u\leq
u_{0},\quad\widetilde{\mathbb{P}}_{u}(M_{v_{0}}\geq\delta)\leq\varepsilon,$
(3.6)

In [10], we proved these two results by using the stochastic partial
differential equation satisfied by the laws $\mathbb{P}^{c}$. They are of
course not available for the laws $\widetilde{\mathbb{P}}$, and we need a new
proof. We start by showing a rather simple but really useful inequality:

######  Lemma 6.

The following inequality holds for any $(x,u)\in D$,

$\widetilde{\mathbb{P}}_{x,u}\left(V_{1}/c\geq\frac{|u|}{2}\right)\geq
1-\frac{\sqrt{3}}{\pi}.$ (3.7)

For us, the important fact is that the probability is bounded below by a
positive constant, uniformly in $x$ and $u$. The constant $1-\sqrt{3}/\pi$ is
not intended to be the optimal one. Note that this inequality will also be
used again later on in this paper.

###### Proof of Lemma 6.

For $u=0$, there is nothing to prove. By a scaling invariance property we may
suppose $u\in\\{-1,1\\}$, what we do.

The density $f_{x,u}$ of $V_{1}/c$ under $\mathbb{P}_{x,u}^{c}$ is given in
Gor’kov [8]. If you write $p_{t}(x,u;y,v)$ for the transition densities of the
(free) Kolmogorov process, given by

$p_{t}(x,u;y,v)=\frac{\sqrt{3}}{\pi t^{2}}\exp\Big{[}-\frac{6}{t^{3}}(y-x-
tu)^{2}+\frac{6}{t^{2}}(y-x-tu)(v-u)-\frac{2}{t}(v-u)^{2}\Big{]},$

and $\Phi(x,u;y,v)$ for its total occupation time densities, defined by

$\Phi(x,u;y,v):=\int_{0}^{\infty}p_{t}(x,u;y,v)dt,$

then the density $f_{x,u}$ is given by

$\displaystyle f_{x,u}(v)$ $\displaystyle=$ $\displaystyle
v\Big{[}\Phi(x,u;0,-v)-\frac{3}{2\pi}\int_{0}^{\infty}\frac{\mu^{\frac{3}{2}}}{\mu^{3}+1}\Phi(x,u;0,\mu
v)d\mu\Big{]}.\ \ \ \ \ \ \ \ $ (3.8)

Now, knowing the density of $V_{1}$ under $\mathbb{P}_{x,u}^{c}$, we get that
of $V_{1}$ under $\widetilde{\mathbb{P}}_{x,u}$ by multiplying it by the
increasing function $v\mapsto v^{2k}$. This necessarily increases the
probability of being greater than $c/2$. Consequently, it is enough to prove

$\mathbb{P}_{x,u}^{c}(V_{1}/c\geq\frac{1}{2})\geq K^{\prime}$

as soon as $u\in\\{-1,1\\}$. But very rough bounds give

$\displaystyle f_{x,u}(v)$ $\displaystyle\leq$ $\displaystyle v\Phi(x,u;0,-v)$
$\displaystyle\leq$ $\displaystyle v\int_{0}^{\infty}\frac{\sqrt{3}}{\pi
t^{2}}\exp(-\frac{(u+v)^{2}}{2t})\mathrm{d}t.$

For $u\in\\{-1,1\\}$ and $v\in[0,1/2]$ we have $|u+v|\geq 1/2$ and thus

$f_{x,u}(v)\leq\frac{v\sqrt{3}}{\pi}\int_{0}^{\infty}\frac{1}{t^{2}}\exp(-\frac{1}{8t})\mathrm{d}t=\frac{8\sqrt{3}}{\pi}v.$

Consequently,

$\mathbb{P}_{x,u}(V_{1}/c\geq\frac{1}{2})\geq
1-\int_{0}^{1/2}\frac{8\sqrt{3}}{\pi}v\mathrm{d}v=1-\frac{\sqrt{3}}{\pi}>0.$

∎

###### Proof of Lemma 4.

First, observe that conditions $(*)$ and $(**)$ imply that the variables
$\tau_{v}=\inf\\{t>0,X_{t}=0,\dot{X}_{t}>v\\}$ and
$\tau_{v}^{-}:=\sup\\{t<\tau_{v},X_{t}=0\\}$ are almost surely strictly
positive and go to zero when $v$ goes to zero. Then, observe that is is enough
to show the almost sure convergence of $\dot{X}_{t}$ to 0 when $t\to 0$, and
suppose on the contrary that this does not hold.

Then, there would exist a positive $x$ such that
$\widetilde{\mathbb{P}}_{0^{+}}^{*}(T_{x}=0)>0$, where we have written
$T_{x}:=\inf\\{t>0,|\dot{X}_{t}|>x\\}.$ By self-similarity this would be true
for any $x>0$ and in particular we would have

$K:=\widetilde{\mathbb{P}}_{0^{+}}^{*}(T_{1}=0)>0.$ (3.9)

Informally, this, together with (3.7), should induce that $\tau_{c/2}^{-}$
takes the value zero with probability at least $(1-\sqrt{3}/\pi)K$, and give
the desired contradiction. However it is not straightforward, because we
cannot use a Markov property at time $T_{1}$, which can take value 0, while
the process is still not defined at time 0. Consider the stopping time
$T_{1}^{\varepsilon}:=\inf\\{t>\varepsilon,|\dot{X}_{t}|>x\\}$. For any
$\eta>0$, we have

$\liminf_{\varepsilon\to
0}\widetilde{\mathbb{P}}_{0^{+}}^{*}(T_{1}^{\varepsilon}<\eta)\geq\widetilde{\mathbb{P}}_{0^{+}}^{*}(\liminf_{\varepsilon\to
0}\\{T_{1}^{\varepsilon}<\eta\\})\geq\widetilde{\mathbb{P}}_{0^{+}}^{*}(T_{1}<\eta)\geq
K,$

and in particular there is some $\varepsilon_{0}(\eta)$ such that for any
$\varepsilon<\varepsilon_{0}(\eta)$,

$\widetilde{\mathbb{P}}_{0^{+}}^{*}(T_{1}^{\varepsilon}<\eta)\geq\frac{K}{2}.$
(3.10)

Now, write $\theta$ for the translation operator defined by
$\theta_{x}((X_{t})_{t\geq 0})=(X_{x+t})_{t\geq 0}$, so that
$V_{1}\circ\theta_{T_{1}^{\varepsilon}}$ denotes the velocity of the process
at its first bounce after time $T_{1}^{\varepsilon}$. From (3.10) and Lemma 6,
a Markov property gives, for $\varepsilon<\varepsilon_{0}(\eta)$,

$\widetilde{\mathbb{P}}_{0^{+}}^{*}\left(T_{1}^{\varepsilon}<\eta,V_{1}\circ\theta_{T_{1}^{\varepsilon}}\geq\frac{c}{2}\right)\geq
K^{\prime}:=\Bigg{(}1-\frac{\sqrt{3}}{\pi}\Bigg{)}\frac{K}{2}.$

We have _a fortiori_
$\widetilde{\mathbb{P}}_{0^{+}}^{*}(\tau_{c/2}^{-}\leq\eta)\geq K^{\prime}.$
This result true for any $\eta>0$ leads to
$\widetilde{\mathbb{P}}_{0^{+}}^{*}(\tau_{c/2}^{-}=0)\geq K^{\prime}>0$, and
we get a contradiction. This shows $(X_{t},\dot{X}_{t})\underset{t\to
0}{\longrightarrow}(0,0)$ under $\widetilde{\mathbb{P}}_{0^{+}}^{*}$, as
requested. ∎

###### Proof of Lemma 5.

We should prove (3.6). Fix $\varepsilon,\delta>0$. The event
$\\{M_{v}\geq\delta\\}$ coincides with the event $T_{\delta}\leq\tau_{v}$.
From a Markov property at time $T_{\delta}$ and (3.7), we get, for any
$v<c\delta/2$, and any $u$,

$(1-\sqrt{3}/\pi)\widetilde{\mathbb{P}}_{u}(M_{v}\geq\delta)\leq\widetilde{\mathbb{P}}_{u}(\dot{X}_{\tau_{v}}\geq
c\delta/2).$

Choose $v_{0}$ such that
$\widetilde{\mathbb{P}}_{0^{+}}(\dot{X}_{\tau_{v_{0}}}\geq
c\delta/2)\leq\varepsilon$. Then, from the convergence of the law of
$\dot{X}_{\tau_{v_{0}}}$ under $\widetilde{\mathbb{P}}_{u}$ to that under
$\widetilde{\mathbb{P}}_{0^{+}}$, we get, for $u$ small enough,

$\widetilde{\mathbb{P}}_{u}(\dot{X}_{\tau_{v_{0}}}\geq c\delta/2)\leq
2\varepsilon,$

and hence

$\widetilde{\mathbb{P}}_{v}(M_{v_{0}}\geq\delta)\leq\frac{2}{1-\sqrt{3}/\pi}\
\varepsilon.$

∎

In conclusion, all this suffices to show Proposition 2.

##  4 The resurrected process

###  4.1 Itō excursion measure, recurrent extensions,
and $(SOR)$ equations

We finally tackle the problem of interest, that is the recurrent extensions of
the reflected Kolmogorov process. A recurrent extension of the latter is a
Markov process that behaves like the reflected Kolmogorov process until
$\zeta_{\infty}$, the hitting time of $(0,0)$, but that is defined for any
positive times and does not stay at $(0,0)$, in the sense that the Lebesgue
measure of the set of times when the process is at $(0,0)$ is almost surely 0.
More concisely, we will call such a process a resurrected reflected process.

We recall that Itō’s program and results of Blumenthal [5] establish an
equivalence between the law of recurrent extensions of a Markov process and
excursion measures compatible with its semigroup, here $\mathrm{P}_{t}^{c}$
(where as usually in Itō’s excursion theory we identify the measures which are
equal up to a multiplicative constant). The _set of excursions_ $\mathcal{E}$
is defined by

$\mathcal{E}:=\\{(x,\dot{x})\in\mathcal{C}|\zeta_{\infty}>0\text{ and
}x_{t}\mathbbm{1}_{t\geq\zeta_{\infty}}=0\\}.$

An excursion measure $n$ compatible with the semigroup $\mathrm{P}_{t}^{c}$ is
defined by the three following properties:

  1. 1.

The measure $n$ is carried by $\mathcal{E}$.

  2. 2.

For any $\mathfrak{F}_{\infty}-$measurable function $F$ and any $t>0$, any
$A\in\mathfrak{F}_{t}$,

$n(F\circ\theta_{t},A\cap\\{t<\zeta_{\infty}\\})=n(\mathbb{P}_{X_{t},\dot{X}_{t}}^{c}(F),A\cap\\{t<\zeta_{\infty}\\}).$

  3. 3.

$n(1-e^{-\zeta_{\infty}})<\infty.$

We also say that $n$ is a pseudo-excursion measure compatible with the
semigroup $\mathrm{P}_{t}^{c}$ if only the two first properties are satisfied
and not necessarily the third one. We recall that the third property is the
necessary condition in Itō’s program in order for the lengths of the
excursions to be summable, hence in order for Itō’s program to succeed.
Besides, we are here interested in recurrent extensions which leave $(0,0)$
continuously. These extensions correspond to excursion measures $n$ which
satisfy the additional condition $n((X_{0},\dot{X}_{0})\neq(0,0))=0$. Our main
results are the following:

######  Theorem 1.

There exists, up to a multiplicative constant, a unique excursion measure
$\mathbf{n}$ compatible with the semigroup $\mathrm{P}_{t}^{c}$ and such that
$\mathbf{n}((X_{0},\dot{X}_{0})\neq(0,0))=0$. We may choose $\mathbf{n}$ such
that

$\mathbf{n}(\zeta_{\infty}>s)=C_{1}s^{-k},$ (4.1)

where $C_{1}$ is the constant defined by (2.5), and $k=k(c)$ has been
introduced in Lemma 1. The measure $\mathbf{n}$ is then characterized by any
of the two following formulas:

$\displaystyle\mathbf{n}(f(X,\dot{X}),\zeta_{\infty}>T)$ $\displaystyle=$
$\displaystyle\widetilde{\mathbb{P}}_{0^{+}}(f(X,\dot{X})H(X_{T},\dot{X}_{T})^{-1}),$
(4.2)

for any $\mathfrak{F}_{t}-$stopping time $T$ and any $f$ positive measurable
functional depending only on $(X_{t},\dot{X}_{t})_{0\leq t\leq T}$.

$\displaystyle\mathbf{n}(f(X,\dot{X}),\zeta_{\infty}>T)$ $\displaystyle=$
$\displaystyle\lim_{(x,u)\to(0,0)}H(x,u)^{-1}\mathbb{P}_{x,u}^{c}(f(X,\dot{X}),\zeta_{\infty}>T),$
(4.3)

for any $\mathfrak{F}_{t}-$stopping time $T$ and any $f$ positive _continuous_
functional depending only on $(X_{t},\dot{X}_{t})_{0\leq t\leq T}$.

So Itō’s program constructs a Markov process with associated Itō excursion
measure $\mathbf{n}$ and that spends no time at $(0,0)$, that is a recurrent
extension, that is a resurrected reflected process. We call its law
$\mathbb{P}_{0}^{r}$. The second theorem will be the weak existence and
solution to equations $(SOR)$, the law of any solution being given by
$\mathbb{P}_{0}^{r}$. It is implicit in this theorem and until the end of the
paper that the initial condition is $(0,0)$, though this generalizes easily to
any other initial condition $(x,u)\in D$.

######  Theorem 2.

The law $\mathbb{P}_{0}^{r}$ gives the unique solution, in the weak sense, of
equations $(SOR)$:

$\bullet$ Consider $(X,\dot{X})$ a process of law $\mathbb{P}_{0}^{r}$. Then
the jumps of $\dot{X}$ on any finite interval are summable and the process $W$
defined by

$W_{t}=\dot{X}_{t}+(1+c)\sum_{0<s\leq t}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0}$

is a Brownian motion. As a consequence the triplet $(X,\dot{X},W)$ is a
solution to $(SOR)$.

$\bullet$ For any solution $(X,\dot{X},W)$ to $(SOR)$, the law of
$(X,\dot{X})$ is $\mathbb{P}_{0}^{r}$.

Before we tackle the proof these theorems, let us write some comments and
consequences. First, the Itō excursion measure $\mathbf{n}$ is entirely
determined by its entrance law, which is defined by

$\mathbf{n}_{s}(\mathrm{d}x,\mathrm{d}u):=n((X_{s},\dot{X}_{s})\in\mathrm{d}x\otimes\mathrm{d}u,s<\zeta_{\infty})$

for $s>0.$ But Theorem 1 implies that it is characterized by any of the two
following formulas:

$\displaystyle\mathbf{n}_{s}(f)$ $\displaystyle=$
$\displaystyle\widetilde{\mathbb{P}}_{0^{+}}(f(X_{s},\dot{X}_{s})H(X_{s},\dot{X}_{s})^{-1}),\quad
s>0,$ (4.4)

for $f:D^{0}\to\mathbb{R}_{+}$ measurable.

$\displaystyle\mathbf{n}_{s}(f)$ $\displaystyle=$
$\displaystyle\lim_{(x,u)\to(0,0)}H(x,u)^{-1}\mathbb{P}_{x,u}^{c}(f(X_{s},\dot{X}_{s}),\zeta_{\infty}>s),\quad
s>0,$ (4.5)

for $f:D^{0}\to\mathbb{R}_{+}$ continuous.

Formulas similar to these are found in the case of self-similar Markov
processes studied by Rivero [15]. This ends the parallel between our works.
Rivero underlined that the self-similar Markov process conditioned on never
hitting 0 that he introduced plays the same role as the Bessel process for the
Brownian motion. In our model, this role is played by the reflected Kolmogorov
process conditioned on never hitting $(0,0)$. Here is a short presentation of
this parallel. Write $P_{x}$ for the law of a Brownian motion starting from
position $x$, $\widetilde{P}_{x}$ for the law of the “three-dimensional”
Bessel process starting from $x$. Write $n$ for the Itō excursion measure of
the absolute value of the Brownian motion (that is, the Brownian motion
reflected at 0), and $\zeta$ for the hitting time of 0. Then the inverse
function is excessive (i.e nonnegative and superharmonic) for the Bessel
process and we have the two well-known formulas

$\displaystyle\mathbf{n}(f(X),\zeta>T)$ $\displaystyle=$
$\displaystyle\widetilde{P}_{0}(f(X)/X_{T})$
$\displaystyle\mathbf{n}(f(X),\zeta>T)$ $\displaystyle=$
$\displaystyle\lim_{x\to 0}\frac{1}{x}P_{x}(f(X),\zeta>T),$

for any $\mathfrak{F}_{t}-$stopping time $T$ and any $f$ positive measurable
functional (resp. continuous functional for the second formula) depending only
on $(X_{t})_{0\leq t\leq T}$.

Now, let us give an application of Formula (4.1). Write $l$ for the local time
spent by $X$ at zero, under $\mathbb{P}_{0}^{r}$. Formula (4.1) implies that
the inverse local time $l^{-1}$ is a subordinator with jumping measure $\Pi$
satisfying $\Pi(\zeta_{\infty}>s)\propto s^{-k}.$ That is, it is a stable
subordinator of index $k$. A well-known result of Taylor and Wendel [16] then
gives that the exact Hausdorff function of the closure of its range (the range
is the image of $\mathbb{R}_{+}$ by $l^{-1}$) is given by
$\phi(\varepsilon)=\varepsilon^{k}(\ln\ln 1/\varepsilon)^{1-k}$ almost surely.
The closure of the range of $l^{-1}$ being equal to the zero set
$\mathcal{Z}:=\\{t\geq 0:X_{t}=\dot{X}_{t}=0\\}$, we get the following
corollary:

######  Corollary 1.

The exact Hausdorff function of the set of the passage times to $(0,0)$ of the
resurrected reflected Kolmogorov process is
$\phi(\varepsilon)=\varepsilon^{k}(\ln\ln 1/\varepsilon)^{1-k}$ almost surely.

It is also clear that the set of the bouncing times of the resurrected
reflected Langevin process – the moments when the process is at zero with a
nonzero speed – is countable. Therefore the zero set of the resurrected
reflected Langevin process has the same exact Hausdorff function.

Finally, we should mention that the self-similarity property enjoyed by the
Kolmogorov process easily spreads to all the processes we introduced. If $a$
is a positive constant, denote by $(X^{a},\dot{X}^{a})$ the process
$(a^{3}X_{a^{-2}t},aX_{a^{-2}t})_{t\geq 0}$. Then the law of
$(X^{a},\dot{X}^{a})$ under $\mathbb{P}^{c}_{x,u}$ is simply
$\mathbb{P}^{c}_{a^{3}x,au}$. We have $H(a^{3}x,au)=a^{2k}H(x,u).$ The law of
$(X^{a},\dot{X}^{a})$ under $\widetilde{\mathbb{P}}_{x,u}$, resp.
$\widetilde{\mathbb{P}}_{0^{+}}$, is simply
$\widetilde{\mathbb{P}}_{a^{3}x,au}$, resp. $\widetilde{\mathbb{P}}_{0^{+}}$.
Finally, the measure of $(X^{a},\dot{X}^{a})$ under $\mathbf{n}$ is simply
$a^{2k}\mathbf{n}$.

Last two subsections are devoted to the proof of the two theorems.

###  4.2 The unique recurrent extension compatible with $\mathrm{P}_{t}^{c}$

#### Construction of the excursion measure

The function $1/H$ is excessive for the semigroup $\widetilde{\mathrm{P}}_{t}$
and the corresponding $h-$transform is $\mathrm{P}_{t}^{c}$ (see Definition
1). Write $\mathbf{n}$ for the $h-$tranform of
$\widetilde{\mathbb{P}}_{0^{+}}$ via this excessive function $1/H$. That is,
$\mathbf{n}$ is the unique measure on $\mathcal{C}$ carried by
$\\{\zeta_{\infty}>0\\}$ such that under $\mathbf{n}$ the coordinate process
is Markovian with semigroup $\mathrm{P}_{t}^{c}$, and for any
$\mathfrak{F}_{t}-$stopping time $T$ and any $A_{T}$ in $\mathfrak{F}_{T}$, we
have

$\mathbf{n}(A_{T},T<\zeta_{\infty})=\widetilde{\mathbb{P}}_{0^{+}}(A_{T},H(X_{T},\dot{X}_{T})^{-1}).$

Then, $\mathbf{n}$ is a pseudo-excursion measure compatible with semigroup
$\mathrm{P}_{t}^{c}$, which verifies
$\mathbf{n}((X_{0},\dot{X}_{0})\neq(0,0))=0$ and satisfies Formula (4.2). For
$f$ continuous functional depending only on $(X_{t},\dot{X}_{t})_{t\leq T}$,
we have

$\displaystyle\widetilde{\mathbb{P}}_{0^{+}}(f(X_{s},\dot{X}_{s})H(X_{s},\dot{X}_{s})^{-1})$
$\displaystyle=$
$\displaystyle\lim_{(x,u)\to(0,0)}\widetilde{\mathbb{P}}_{x,u}(f(X_{s},\dot{X}_{s})H(X_{s},\dot{X}_{s})^{-1})$
$\displaystyle=$
$\displaystyle\lim_{(x,u)\to(0,0)}\frac{1}{H(x,u)}\mathbb{P}_{x,u}^{c}(f(X_{s},\dot{X}_{s}),\zeta_{\infty}>s),$

so that the pseudo-excursion measure $\mathbf{n}$ also satisfies Formula
(4.3). In particular, taking $T=s$ and $f=1$, and considering the limit along
the half-line $x=0$, this gives

$\mathbf{n}(\zeta_{\infty}>s)=\lim_{u\to
0}u^{-2k}\mathbb{P}_{0,u}(\zeta_{\infty}>s).$

Using Lemma 2 and the scaling invariance property, we get

$\mathbf{n}(\zeta_{\infty}>s)=C_{1}s^{-k},$

where $C_{1}$ is the constant defined by (2.5). This is exactly Formula (4.1).
This formula gives, in particular,

$\mathbf{n}(1-e^{-\zeta_{\infty}})=C_{1}\Gamma(1-k),$

where $\Gamma$ denotes the usual Gamma function. Hence, $\mathbf{n}$ is an
excursion measure.

Finally, in order to establish Theorem 1 we just should prove that
$\mathbf{n}$ is the only excursion measure compatible with the semigroup
$\mathrm{P}_{t}^{c}$ such that $\mathbf{n}((X_{0},\dot{X}_{0})\neq(0,0))=0$.
That is, we should show the uniqueness of the law of the resurrected reflected
process.

#### Uniqueness of the excursion measure

Let $\mathbf{n}^{\prime}$ be such an excursion measure, compatible with the
semigroup $\mathrm{P}_{t}^{c}$, and satisfying
$\mathbf{n}^{\prime}((X_{0},\dot{X}_{0})\neq(0,0))=0$. We will prove that
$\mathbf{n}$ and $\mathbf{n}^{\prime}$ coincide, up to a multiplicative
constant. Recall that $\zeta_{1}$ is defined as the infimum of
$\\{t>0,X_{t}=0\\}$.

######  Lemma 7.

The measure $\mathbf{n}^{\prime}$ satisfies:

$\mathbf{n}^{\prime}(\zeta_{1}\neq 0)=0$

###### Proof.

This condition will appear to be necessary to have the third property of
excursion measures, that is
$\mathbf{n}^{\prime}(1-e^{-\zeta_{\infty}})<\infty.$ Suppose on the contrary
that $\mathbf{n}^{\prime}(\zeta_{1}\neq 0)>0$ and write
$\tilde{\mathbf{n}}(\cdot)=\mathbf{n}^{\prime}(\cdot\mathbbm{1}_{\zeta_{1}\neq
0})$. The measure $\tilde{\mathbf{n}}$ is an excursion measure compatible with
the semigroup $\mathrm{P}_{t}^{c}$ such that
$\tilde{\mathbf{n}}((X_{0},\dot{X}_{0})\neq(0,0))=0$, satisfying
$\tilde{\mathbf{n}}(\zeta_{1}=0)=0.$ Consider
$\overline{\mathbf{n}}((X_{t},\dot{X}_{t})_{t\geq
0}):=\tilde{\mathbf{n}}((X_{t}\mathbbm{1}_{t<\zeta_{1}},\dot{X}_{t}\mathbbm{1}_{t<\zeta_{1}})_{t\geq
0})$ the excursion measure of the process killed at time $\zeta_{1}$.

The measure $\overline{\mathbf{n}}$ is an excursion measure compatible with
the semigroup $\mathrm{P}_{t}^{0}$, semigroup of the Kolmogorov process killed
at time $\zeta_{1}$ (the first hitting time of $\\{0\\}\times\mathbb{R}$).
Therefore its first marginal must be the excursion measure of the Langevin
process reflected on an inelastic boundary, introduced and studied in [3]. In
particular, under $\overline{\mathbf{n}}$, the absolute value of the incoming
speed at time $\zeta_{1}$, or $|\dot{X}_{\zeta_{1}-}|$, is distributed
proportionally to $v^{-\frac{3}{2}}\mathrm{d}v$ (see [3], Corollary 2, (ii)).
This stays true under $\tilde{\mathbf{n}}$ and implies that
$V_{1}=c|\dot{X}_{\zeta_{1}-}|$ is also distributed proportionally to
$v^{-\frac{3}{2}}\mathrm{d}v$. Now, a Markov property at the stopping time
$\zeta_{1}$ under $\tilde{\mathbf{n}}$ gives

$\tilde{\mathbf{n}}(\zeta_{\infty}-\zeta_{1}>t|V_{1}=v)=\mathbb{P}_{v}^{c}(\zeta_{\infty}>t)=\mathbb{P}_{1}^{c}(\zeta_{\infty}>v^{-2}t)\underset{v^{-2}t\to\infty}{\sim}Cv^{2k}t^{-k}$

As a consequence the function $v\mapsto
v^{-\frac{3}{2}}\tilde{\mathbf{n}}(\zeta_{\infty}-\zeta_{1}>t|V_{1}=v)$ is not
integrable in the neighborhood of 0\. That is
$\tilde{\mathbf{n}}(\zeta_{\infty}-\zeta_{1}>t)=+\infty$, we get a
contradiction. ∎

Recall that we owe to prove that $\mathbf{n}^{\prime}$ and $\mathbf{n}$ are
equal, up to a multiplicative constant. Let us work on the corresponding
entrance laws. Take $s>0$ and $f$ a bounded continuous function. It is
sufficient to prove $\mathbf{n}^{\prime}_{s}(f)=C\mathbf{n}_{s}(f)$, where $C$
is a constant independent of $s$ and $f$.

By reformulating Lemma 7, time $\zeta_{1}$ is zero
$\mathbf{n}^{\prime}$-almost surely, in the sense that the
$\mathbf{n}^{\prime}$-measure of the complementary event is 0. That is,
$\mathbf{n}^{\prime}$-a.s., the first coordinate of the process comes back to
zero just after the initial time, while the second coordinate cannot be zero,
for the simple reason that we are working on an excursion outside from
$(0,0)$. This, together with the fact that the velocity starts from
$\dot{X}_{0}=0$ and is right-continuous, implies that
$\mathbf{n}^{\prime}$-almost surely, the time $\tau_{v}$ (which, we recall, is
the instant of the first bounce with speed greater than $v$) is going to $0$
when $v$ is going to 0.

We deduce, by dominated convergence, from the continuity of $f$, and, again,
from the right-continuity of the paths, that

$\mathbf{n}^{\prime}_{s}(f)=\lim_{u\to
0}\mathbf{n}^{\prime}(f(X_{s+\tau_{v}},\dot{X}_{s+\tau_{v}})\mathbbm{1}_{\tau_{v}<\infty,\zeta_{\infty}>s+\tau_{v}}).$
(4.6)

An application of the Markov property gives

$\displaystyle\mathbf{n}^{\prime}(f(X_{s+\tau_{v}},\dot{X}_{s+\tau_{v}})\mathbbm{1}_{\tau_{v}<\infty,\zeta_{\infty}>s+\tau_{v}})$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}_{+}}\mathbf{n}^{\prime}(\dot{X}_{\tau_{v}}\in\mathrm{d}u)\mathbb{P}^{c}_{u}(f(X_{s},\dot{X}_{s})\mathbbm{1}_{\zeta_{\infty}>s})$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}_{+}}\mathbf{n}^{\prime}(\dot{X}_{\tau_{v}}\in\mathrm{d}u)u^{2k}g(u),$

where
$g(u)=u^{-2k}\mathbb{P}^{c}_{u}(f(X_{s},\dot{X}_{s})\mathbbm{1}_{\zeta_{\infty}>s})=H(0,u)^{-1}\mathbb{P}^{c}_{u}(f(X_{s},\dot{X}_{s})\mathbbm{1}_{\zeta_{\infty}>s})$
converges to $\mathbf{n}_{s}(f)$ when $u\to 0$, by Formula (4.3). Moreover the
function $u^{2k}g(u)$ is bounded by $\|f\|_{\infty}$, and for any
$\varepsilon>0$ we have
$\mathbf{n}^{\prime}(\dot{X}_{\tau_{v}}>\varepsilon)\to 0$ when $v\to 0$.
Informally, all this explains that when $v$ is small, all the mass in the
integral is concentrated in the neighborhood of $0$, where we can replace
$g(u)$ by $\mathbf{n}_{s}(f)$. More precisely, write

$\int_{\mathbb{R}_{+}}\mathbf{n}^{\prime}(\dot{X}_{\tau_{v}}\in\mathrm{d}u)u^{2k}g(u)=I(v)+J(v),$

where

$\displaystyle I(v)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\mathbf{n}^{\prime}(\dot{X}_{\tau_{v}}\in\mathrm{d}u)u^{2k}\mathbf{n}_{s}(f),$
$\displaystyle J(v)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathbf{n}^{\prime}(\dot{X}_{\tau_{v}}\in\mathrm{d}u)u^{2k}(g(u)-\mathbf{n}_{s}(f)\mathbbm{1}_{u\leq
1}).$

By splitting the integral defining $J(v)$, we deduce that $J(v)$ is negligible
compared to $1\vee I(v).$ Recalling that the sum $I(v)+J(v)$ converges to
$\mathbf{n}^{\prime}_{s}(f)$ (Formula (4.6)), we get that $I(v)$ converges to
$\mathbf{n}^{\prime}_{s}(f)$ when $v\to 0$, while $J(v)$ converges to 0.

We thus have

$\mathbf{n}^{\prime}_{s}(f)=C\mathbf{n}_{s}(f),$

where $C$ is independent of $s$ and $f$ and given by

$C=\lim_{v\to
0}\int_{0}^{1}\mathbf{n}^{\prime}(\dot{X}_{\tau_{v}}\in\mathrm{d}u)u^{2k}.$

Uniqueness follows. Theorem 1 is proved.

###  4.3 The weak unique solution to the $(SOR)$ equations

We now prove Theorem 2.

#### Weak solution

We consider, under $\mathbb{P}_{0}^{r}$, the coordinate process $(X,\dot{X})$,
and its natural filtration $(\mathfrak{F}_{t})_{t\geq 0}$. We first prove that
the jumps of $\dot{X}$ are almost-surely summable on any finite interval. As
there are (a.s.) only finitely many jumps of amplitude greater than a given
constant on any finite interval, it is enough to prove that the jumps of
amplitude less than a given constant are (a.s.) summable. Write $L$ for a
local time of the process $(X,\dot{X})$ in $(0,0)$, $L^{-1}$ its inverse, and
$\mathbf{n}$ the associated excursion measure. It is sufficient to prove that
the expectation of the sum of the jumps of amplitude less than $1+1/c$ (jumps
at the bouncing times for which the outgoing velocity is less than one), and
occurring before time $L^{-1}(1)$, is finite. This expectation is equal to

$(1+\frac{1}{c})\int_{0}^{1}\mathbf{n}(N_{[v,1]}(X,\dot{X}))\mathrm{d}v,$

where we write $N_{I}(X,\dot{X})$ for the number of bounces of the process
$(X,\dot{X})$ with outgoing speed included in the interval $I$. For a fixed
$v$, introduce the sequence of stopping times defined by $\tau^{v}_{0}=0$ and
$\tau^{v}_{n+1}=\inf\\{t>\tau^{v}_{n},X_{t}=0,\dot{X}_{t}\in[v,1]\\}$ for
$n\geq 0$. Then $N_{[v,1]}(X,\dot{X})$ is also equal to
$\sup\\{n,\tau^{v}_{n}<\zeta_{\infty}\\}$. Thanks to formula (4.2), for any
$n>0$, we have:

$\displaystyle\mathbf{n}(\zeta_{\infty}>\tau^{v}_{n})$ $\displaystyle=$
$\displaystyle\widetilde{\mathbb{P}}_{0^{+}}(H(X_{\tau^{v}_{n}},\dot{X}_{\tau^{v}_{n}})^{-1}\mathbbm{1}_{\tau^{v}_{n}<\infty})$
$\displaystyle=$
$\displaystyle\widetilde{\mathbb{P}}_{0^{+}}(\dot{X}_{\tau^{v}_{n}}^{-2k}\mathbbm{1}_{\tau^{v}_{n}<\infty})$
$\displaystyle\leq$ $\displaystyle
v^{-2k}\widetilde{\mathbb{P}}_{0^{+}}(\tau^{v}_{n}<\infty).$

As a consequence, we have

$\displaystyle\mathbf{n}(N_{[v,1]}(X,\dot{X}))$ $\displaystyle\leq$
$\displaystyle
v^{-2k}\widetilde{\mathbb{P}}_{0^{+}}(\sup\\{n,\tau^{v}_{n}<\zeta_{\infty}\\})$
$\displaystyle\leq$ $\displaystyle v^{-2k}\widetilde{\mathbf{P}}(N_{[\ln
v,0]}^{d}(S)),$

where we have written $N_{[\ln v,0]}^{d}(S)$ for the number of instants
$n\in\mathbb{Z}$ such that $S_{n}\in[\ln v,0]$. Recall also that
$\widetilde{\mathbf{P}}$ is the law of the spatially stationary random walk.
It is now a simple verification that $\widetilde{\mathbf{P}}(N_{[\ln
v,0]}^{d}(S))$ is finite and proportional to the length of the interval
$[\ln(v),0]$, that is $-\ln v$. It follows

$\mathbf{n}(N_{[v,1]}(X,\dot{X}))\underset{v\to 0}{=}O(v^{-2k}\ln(1/v))$

and (recall $k<1/4$)

$(1+\frac{1}{c})\int_{0}^{1}\mathbf{n}(N_{[v,1]}(X,\dot{X}))\mathrm{d}v<\infty.$

The jumps are summable.

Now, write

$W_{t}=\dot{X}_{t}+(1+c)\sum_{0<s\leq t}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0}.$

We aim to show that the continuous process $W$ is a Brownian motion. For
$\varepsilon>0$, we introduce the sequence of stopping times
$(T_{n}^{\varepsilon})_{n\geq 0}$ defined by $T_{0}^{\varepsilon}=0$ and, for
$n\geq 0$,

$\left\\{\begin{array}[]{rcl}T_{2n+1}^{\varepsilon}&=&\inf\\{t>T_{2n}^{\varepsilon},X_{t}=0,\dot{X}_{t}>\varepsilon\\}\\\
T_{2n+2}^{\varepsilon}&=&\inf\\{t>T_{2n+1}^{\varepsilon},X_{t}=\dot{X}_{t}=0\\}\end{array}\right.$

We also introduce $F^{\varepsilon}=\bigcup_{n\geq
0}[T_{2n}^{\varepsilon},T_{2n+1}^{\varepsilon}]$ and
$H_{t}^{\varepsilon}=\mathbbm{1}_{F^{\varepsilon}}(t)$. For
$0<\varepsilon^{\prime}<\varepsilon$, we have $H^{\varepsilon^{\prime}}\leq
H^{\varepsilon}$, or equivalently, $F^{\varepsilon^{\prime}}\subset
F^{\varepsilon}$. When $\varepsilon$ goes to $0+$, $F^{\varepsilon}$ converges
to the zero set $\mathcal{Z}=\\{t,X_{t}=\dot{X}_{t}=0\\}$, and
$H^{\varepsilon}$ converges pointwisely to $H^{0}=\mathbbm{1}_{\mathcal{Z}}$.
Note that the processes $H^{\varepsilon}$ and $H^{0}$ are
$\mathfrak{F}_{t}-$adapted. Note, also, that Corollary 1 implies in particular
that $\mathcal{Z}$ has zero Lebesgue measure. For ease of notations, we will
sometimes omit the superscript $\varepsilon$.

Conditionally on $\dot{X}_{T_{2n+1}}=u$, the process $(X_{(T_{2n+1}+t)\wedge
T_{2n+2}})_{t\geq 0}$ is independent of $\mathfrak{F}_{T_{2n+1}}$ and has law
$\mathbb{P}_{u}^{c}$. As a consequence the process $(W_{(T_{2n+1}+t)\wedge
T_{2n+2}}-W_{T_{2n+1}})_{t\geq 0}$ is a Brownian motion stopped at time
$T_{2n+2}-T_{2n+1}$. Write

$W_{t}=\int_{0}^{t}H_{s}^{\varepsilon}\mathrm{d}W_{s}+\int_{0}^{t}(1-H_{s}^{\varepsilon})\mathrm{d}W_{s}.$

The process $\int_{0}^{t}(1-H_{s}^{\varepsilon})\mathrm{d}W_{s}$ converges
almost surely to $\int_{0}^{t}(1-H_{s}^{0})\mathrm{d}W_{s}$. But the process
$\int_{0}^{t}(1-H_{s}^{0})\mathrm{d}W_{s}$ is a continuous martingale of
quadratic variation $\int_{0}^{t}(1-H_{s}^{0})\mathrm{d}s=t$ and thus a
Brownian motion. In order to prove that it actually coincides with $W$, we
just need to prove that the term
$D_{t}^{\varepsilon}:=\int_{0}^{t}H_{s}^{\varepsilon}\mathrm{d}W_{s}$ is
almost-surely converging to $0$ when $\varepsilon\to 0$. Without loss of
generality, we just prove it on the event $t\leq L^{-1}(1)$.

This term can be rewritten as

$D_{t}^{\varepsilon}=\left\\{\begin{array}[]{ll}\displaystyle\sum_{k\leq
n}\big{(}W_{T_{2k+1}}-W_{T_{2k}}\big{)}&\text{if }T_{2n+1}\leq t<T_{2n+2},\\\
\\\ \displaystyle
W_{t}-W_{T_{2n}}+\sum_{k<n}\big{(}W_{T_{2k+1}}-W_{T_{2k}}\big{)}&\text{if
}T_{2n}\leq t<T_{2n+1}.\end{array}\right.$

Now, for any $k$, we have

$W_{T_{2k+1}}-W_{T_{2k}}=\dot{X}_{T_{2k+1}}+(1+c)\sum_{T_{2k}<s\leq
T_{2k+1}}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0},$

and for any $T_{2n}\leq t<T_{2n+1}$,

$W_{t}-W_{T_{2n}}=\dot{X}_{t}+(1+c)\sum_{T_{2n}<s\leq
t}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0},$

Hence the term $D_{t}^{\varepsilon}$ involves jumps of amplitude less than
$(1+c)\varepsilon$, whose sum is going to 0 when $\varepsilon$ goes to zero,
plus the fraction $c/(1+c)$ of the jumps occurring at times $T_{2k+1}$, plus
the possible extra term $\dot{X}_{t}$, not corresponding to any jump. We will
prove nonetheless that the jumps occurring at times $T_{2k+1}$, and
$|\dot{X}_{t}|$, are all small when $\varepsilon$ is small enough. It will
follow that $D_{t}^{\varepsilon}$ tends to 0 when $\varepsilon$ goes to 0.

Fix $\eta>0$. Write $A^{\varepsilon}$ for the event

$\sup_{s\leq L^{-1}(1),s\in F^{\varepsilon}}\dot{X}_{s}\geq\eta.$

We will prove that the probability of $A^{\varepsilon}$ is going to 0 when
$\varepsilon$ goes to 0, so that we almost surely don’t lie in
$A^{\varepsilon}$ for $\varepsilon$ small enough, and as a consequence the
jumps occurring at times $T_{2k+1}$ and the possible term $|\dot{X}_{t}|$ will
then all be less than $\eta$, as requested. Write
$\widetilde{T}^{\varepsilon}$ for the infimum of $\\{t:t\in
F^{\varepsilon},|\dot{X}_{t}|\geq\eta\\}$ and $n_{\varepsilon}$ for the
supremum of $\\{n,T_{2n}\leq\widetilde{T}^{\varepsilon}\\}$. The event
$A^{\varepsilon}$ coincides with $\\{\widetilde{T}^{\varepsilon}<L^{-1}(1)\\}$
or $\\{T_{2n_{\varepsilon}+1}<L^{-1}(1)\\}$.

The Markov property at the stopping time $\widetilde{T}^{\varepsilon}$,
together with the inequality (3.7), gives

$\mathbb{P}(\\{\dot{X}_{T_{2n_{\varepsilon}+1}}\geq\eta c/2\\}\cap
A^{\varepsilon})\geq\big{(}1-{\sqrt{3}}/\pi\big{)}\mathbb{P}(A^{\varepsilon}).$

The event $\\{\dot{X}_{T_{2n_{\varepsilon}+1}}\geq\eta c/2\\}\cap
A^{\varepsilon}$ is contained in the event that there is an excursion
occurring before time $L^{-1}(1)$ for which the first bounce with speed
greater than $\varepsilon$ is actually greater than $\eta c/2$. This event has
probability

$\mathbf{n}(T_{1}^{\varepsilon}<\infty,\dot{X}_{T_{1}^{\varepsilon}}\geq\eta
c/2),$

where $T_{1}^{\varepsilon}$ is still defined as the time of the first bounce
with speed greater than $\varepsilon$, here for the excursion. We have:

$\displaystyle\mathbf{n}(\dot{X}_{T_{1}^{\varepsilon}}\geq\eta
c/2,\zeta_{\infty}>T_{1}^{\varepsilon})$ $\displaystyle=$
$\displaystyle\widetilde{\mathbb{P}}_{0^{+}}(H(0,\dot{X}_{T_{1}^{\varepsilon}})^{-1}\mathbbm{1}_{\dot{X}_{T_{1}^{\varepsilon}}\geq\eta
c/2})$ $\displaystyle\leq$ $\displaystyle(\eta
c/2)^{-2k}\widetilde{\mathbb{P}}_{0^{+}}(\dot{X}_{T_{1}^{\varepsilon}}\geq\eta
c/2)$ $\displaystyle\leq$ $\displaystyle(\eta c/2)^{-2k}m\big{(}]\ln({\eta
c}/({2\varepsilon})),\infty[\big{)},$

where we recall that $m$ is the stationary law of the overshoot appearing in
Proposition 2. This probability is thus going to 0 when $\varepsilon$ goes to
0, as well as $\mathbb{P}(A^{\varepsilon})$.

The process $W$ is a Brownian motion, and $(X,\dot{X},W)$ is a solution to
Equations $(SOR)$.

#### Weak uniqueness

Consider $(X,\dot{X},W)$, with law $\mathbb{P}$, be any solution to $(SOR)$,
and its associated filtration $(\mathfrak{F}_{t})_{t\geq 0}$. Then we have

$\dot{X}_{t}=W_{t}-(1+c)\sum_{0<s\leq t}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0},$

with $W$ a Brownian motion.

We start with the observation that the process $\dot{X}$ does not explode and
that the sum just involves positive jumps. Therefore these jumps are summable.
But the process $\sum_{0<s\leq t}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0}$ is
adapted, hence $\dot{X}$ is a semimartingale. As a consequence, it possesses
local times $(L^{a})_{a\in\mathbb{R}}$, and we have an occupation formula (see
for example [14], Theorem 70 Corollary 1, p216):

$\int_{-\infty}^{+\infty}L^{a}_{t}g(a)\mathrm{d}a=\int_{0}^{t}g(\dot{X}_{s-})\mathrm{d}s,$

for any $g$ bounded measurable function. Taking $g=\mathbbm{1}_{\\{0\\}}$
shows that $\dot{X}$ spends no time at zero. It follows that the process
$(X,\dot{X})$ spends no time at $(0,0)$.

Now, exactly as before, introduce, for $\varepsilon>0$, the sequence of
stopping times $T_{n}^{\varepsilon}$, defined by $T_{0}^{\varepsilon}=0$ and

$\left\\{\begin{array}[]{rcl}T_{2n+1}^{\varepsilon}&=&\inf\\{t>T_{2n}^{\varepsilon},X_{t}=0,\dot{X}_{t}>\varepsilon\\}\\\
T_{2n+2}^{\varepsilon}&=&\inf\\{t>T_{2n+1}^{\varepsilon},X_{t}=\dot{X}_{t}=0\\},\end{array}\right.$

as well as $F^{\varepsilon}=\bigcup_{n\geq
0}[T_{2n}^{\varepsilon},T_{2n+1}^{\varepsilon}]$ and
$H^{\varepsilon}=\mathbbm{1}_{F^{\varepsilon}}$. Finally, define the closed
set $F=\lim_{\varepsilon\to 0}F^{\varepsilon}$ and the adapted process
$H^{0}=\mathbbm{1}_{F}.$

######  Lemma 8.

The set $F$ has almost surely zero Lebesgue measure.

This result is not immediate. First, observe that the excursions of the
process may be of two types. Either an excursion bounces on the boundary just
after the initial time, or it doesn’t. We call $\mathcal{E}_{1}$ the set of
excursions of the first type, defined by

$\mathcal{E}_{1}:=\\{(x,\dot{x})\in\mathcal{E}|\zeta_{1}(x,\dot{x}):=\inf\\{t>0,x_{t}=0\\}=0\\},$

and $\mathcal{E}_{2}=\mathcal{E}\backslash\mathcal{E}_{1}$ the set of
excursions of the second type. Unlike before, we do not know _a priori_ that
all the excursions of the process lie in $\mathcal{E}_{1}$. If the process
starts an excursion at time $t$, we write $e^{t}$ for the corresponding
excursion.

A close look at $F$ shows that it contains not only the zero set
$\mathcal{Z}$, but also all the intervals $[t,t+\zeta_{1}(e^{t})]$, where $t$
is the starting time of an excursion $e^{t}\in\mathcal{E}_{2}$. Prove Lemma 8
is equivalent to prove that there is actually no excursion in
$\mathcal{E}_{2}$.

Suppose that this fails. Then the process

$L(t)=\int_{0}^{t}H^{0}_{s}\mathrm{d}s$

is not almost surely constantly equal to zero. We introduce its right-
continuous inverse

$L^{-1}(t):=\inf\\{s>t,L(s)>t\\}.$

There exists a Brownian motion $M$ such that for $t<L(\infty),$

$M_{t}=\int_{0}^{L^{-1}(t)}H^{0}_{s}\mathrm{d}W_{s}.$

Introduce the time-changed process

$(Y_{t},\dot{Y}_{t})=(X_{L^{-1}(t)},\dot{X}_{L^{-1}(t)}),$

stopped at time $L(\infty)$. In order to simplify the redaction, we will often
omit to specify “stopped at time $L(\infty)$”. This time change induces that
the process $(Y,\dot{Y})$ also does not spend any time at zero, and that its
excursions are that of $(X,\dot{X})$ belonging to $\mathcal{E}_{2}$, and
stopped at $\zeta_{1}$ the first return time to $\\{0\\}\times\mathbb{R}$.

######  Lemma 9.

The triplet $(Y_{t},\dot{Y}_{t},M_{t})_{t\leq L(\infty)}$ under $\mathbb{P}$
is a solution of the equations $(SOR)$ with null elasticity coefficient,
stopped at time $L(\infty)$.

###### Proof.

Let $[t,t^{\prime}[$ be the interval corresponding to an excursion of
$(Y,\dot{Y})$. Then the interval $[L^{-1}(t),L^{-1}(t^{\prime}-)]$ is a
maximal interval included in $F$. It follows that the points $L^{-1}(t)$ and
$L^{-1}(t^{\prime})$ belong to $\mathcal{Z}$, and
$Y_{t}=\dot{Y}_{t}=0=Y_{t^{\prime}}=\dot{Y}_{t^{\prime}}$.

Let $s\in[t,t^{\prime}[$. As the process $X$ has no bounce in
$[L^{-1}(t),L^{-1}(s)]$ and $(X,\dot{X},W)$ is a solution to $(SOR)$, we can
write

$\dot{X}_{L^{-1}(s)}=\dot{X}_{L^{-1}(t)}+W_{L^{-1}(s)}-W_{L^{-1}(t)},$

or equivalently

$\dot{Y}_{s}=\dot{Y}_{t}+M_{s}-M_{t}.$

As a consequence, we may write

$\left\\{\begin{array}[]{ccl}Y_{s}&=&Y_{t}+\displaystyle\int_{t}^{s}\dot{Y}_{u}\mathrm{d}u\\\
\dot{Y}_{s}&=&\dot{Y}_{t}+M_{s}-M_{t}-\sum_{t<u\leq
s}\dot{Y}_{u-}\mathbbm{1}_{Y_{u}=0},\end{array}\right.$

where the sum is actually empty. Similarly,

$\left\\{\begin{array}[]{ccl}Y_{t^{\prime}}&=&0=X_{L^{-1}(t^{\prime}-)}=Y_{t^{\prime}-}=Y_{t}+\displaystyle\int_{t}^{t^{\prime}}\dot{Y}_{u}\mathrm{d}u\\\
\dot{Y}_{t^{\prime}}&=&0=\dot{Y}_{t^{\prime}-}-\dot{Y}_{t^{\prime}-}\mathbbm{1}_{Y_{t^{\prime}}=0}=\dot{Y}_{t}+M_{t^{\prime}}-M_{t}-\sum_{t<u\leq
t^{\prime}}\dot{Y}_{u-}\mathbbm{1}_{Y_{u}=0},\end{array}\right.$

where the sum now contains one term.

Adding these equalities on the excursion intervals of $(Y,\dot{Y})$, and
recalling that this process spends no time at $(0,0)$, gives

$\left\\{\begin{array}[]{ccl}Y_{s}&=&\displaystyle\int_{0}^{s}\dot{Y}_{u}\mathrm{d}u\\\
\dot{Y}_{s}&=&M_{s}-\sum_{0<u\leq
s}\dot{Y}_{u-}\mathbbm{1}_{Y_{u}=0},\end{array}\right.$

and $(Y,\dot{Y},M)$ is a solution to $(SOR)$ with null elasticity coefficient
(stopped at time $L(\infty)$). ∎

The article [4], which studied equations $(SOR)$ with null elasticity
coefficient, shows that a solution $(Y,\dot{Y})$ must be a Markov process,
with Itō excursion law $\overline{\mathbf{n}}$. We immediately introduce
another change of time, in a very similar way, but without stopping the
excursions of $\mathcal{E}_{2}$ at time $\zeta_{1}$. Define the random set

$A:=\mathcal{Z}\cup\bigcup_{\\{t|e^{t}\in\mathcal{E}_{2}\\}}[t,t+\zeta_{\infty}(e^{t})],$

and the adapted process $\widetilde{H}=\mathbbm{1}_{A}.$ Define also

$\widetilde{L}(t)=\int_{0}^{t}\widetilde{H}_{s}\mathrm{d}s,$

and $\widetilde{L}^{-1}$ for its right-continuous inverse. Then, there exists
a Brownian motion $\widetilde{M}$ such that

$\widetilde{M}_{t}=\int_{0}^{\widetilde{L}^{-1}(t)}\widetilde{H}_{s}\mathrm{d}W_{s}$

for $t<\widetilde{L}(\infty)$. Finally, the time-changed process

$(\widetilde{Y}_{t},\dot{\widetilde{Y}}_{t})=(X_{\widetilde{L}^{-1}(t)},\dot{X}_{\widetilde{L}^{-1}(t)}),$

stopped at time $\widetilde{L}(\infty)$, spends no time at zero and its
excursions are the excursions of $(X,\dot{X})$ included in $\mathcal{E}_{2}$.
Remark that we have $\widetilde{L}(\infty)\geq L(\infty)$ because $A\supset
F$. We also get the following lemma, similar to Lemma 9, and whose proof we
leave to the reader.

######  Lemma 10.

The triplet
$\Big{(}\widetilde{Y}_{t},\dot{\widetilde{Y}}_{t},\widetilde{M}_{t}\Big{)}_{t\leq\widetilde{L}(\infty)}$
under $\mathbb{P}$ is a solution of the equations $(SOR)$ (with elasticity
coefficient $c$), stopped at time $\widetilde{L}(\infty)$.

The process $(\widetilde{Y},\dot{\widetilde{Y}})$ spends no time at $0$, is a
solution to $(SOR)$, and its excursions, stopped at $\zeta_{1}$, the first
return time to $\\{0\\}\times\mathbb{R}$, are precisely that of $(Y,\dot{Y}).$
This induces that $(\widetilde{Y},\dot{\widetilde{Y}})$ is a Markov process
with Itō excursion measure $\tilde{\mathbf{n}}$ determined by

$\left\\{\begin{array}[]{ccl}\tilde{\mathbf{n}}\left((x_{t\wedge\zeta_{1}})_{t\geq
0}\in\cdot\right)&=&\overline{\mathbf{n}}(x\in\cdot)\\\
\tilde{\mathbf{n}}\left((x_{t+\zeta_{1}})_{t\geq
0}\in\cdot\right|\dot{X}_{\zeta_{1}}=v)&=&\mathbb{P}_{v}^{c}(x\in\cdot)\end{array}\right.$

Now, the result of uniqueness of the excursion measure implies that
$\tilde{\mathbf{n}}$ should be a multiple of $\mathbf{n}$, which is obviously
not the case (for example because $\tilde{\mathbf{n}}(\zeta_{\infty}=0)=0$).
Therefore $\widetilde{L}(\infty)=0=L(\infty)$ a.s. Lemma 8 is proved.

Now, introduce a third time-change,
$(L^{\varepsilon})^{-1}(t):=\inf\\{s>0,L^{\varepsilon}(s)>t\\}$. When
$\varepsilon$ goes to 0, $(L^{\varepsilon})^{-1}$ is going to $L^{-1}=Id$. It
follows that the process
$X^{\varepsilon}:=(X_{(L^{\varepsilon})^{-1}(t)})_{t\geq 0}$ is going
uniformly on compacts to $X$ when $\varepsilon$ goes to 0, almost surely. In
particular the law of $X$ is entirely determined by that of $X^{\varepsilon}$.
The law of $X^{\varepsilon}$ is in turn entirely determined by that of
$(\dot{X}_{T_{2n+1}^{\varepsilon}})_{n\geq 0}.$ We will now determine this
law, which will prove the uniqueness of the law of $X$.

In order to avoid complex notations, we just give the calculation of the law
of $\dot{X}_{T_{1}^{1}}$, which is not fundamentally different from others.
For $\varepsilon>0$ and $n\geq 0$, a Markov property for the process $W$
applied at time $T_{2n+1}^{\varepsilon}$ shows that conditionally on
$\dot{X}_{T_{2n+1}^{\varepsilon}}=u$, the process
$(X_{(T_{2n+1}^{\varepsilon}+t)\wedge T_{2n+2}^{\varepsilon}})_{t\geq 0}$ is
independent from $\mathfrak{F}_{T_{2n+1}^{\varepsilon}}$ and has law
$\mathbb{P}_{u}^{c}$. Write $n_{1}$ for the integer satisfying
$T_{2n_{1}+1}^{\varepsilon}\leq T_{1}^{1}<T_{2n_{1}+2}^{\varepsilon}$.
Conditionally on $\dot{X}_{T_{2n_{1}+1}^{\varepsilon}}=u$, the process
$(X_{(T_{2n_{1}+1}^{\varepsilon}+t)\wedge T_{2n_{1}+2}^{\varepsilon}})_{t\geq
0}$ has the law $\mathbb{P}_{u}^{c}$ conditioned on reaching a speed greater
than one after a bounce.

In other words, the law of $\dot{X}_{T_{1}^{1}}$ under
$\mathbb{P}(\cdot|\dot{X}_{T_{2n_{1}+1}^{\varepsilon}}=u)$ is equal to that of
$\dot{X}_{T_{1}^{1}}$ under $\mathbb{P}_{u}^{c}(\cdot|T_{1}^{1}<\infty)$.
Besides, it should be clear now that $\dot{X}_{T_{2n_{1}+1}^{\varepsilon}}$ is
going to 0 when $\varepsilon$ goes to 0. Recall that $\zeta_{\infty}$, the
hitting time of $(0,0)$, is the lifetime of the excursion (under
$\mathbb{P}_{u}^{c}$ as well as under $\mathbf{n}$). For any $f$ positive
continuous functional, we have:

$\displaystyle\mathbb{P}_{u}^{c}(f(\dot{X}_{T_{1}^{1}})|\
T_{1}^{1}<\zeta_{\infty})$ $\displaystyle=$
$\displaystyle\mathbb{P}_{u}^{c}\big{(}f(\dot{X}_{T_{1}^{1}})\mathbbm{1}_{T_{1}^{1}<\zeta_{\infty}}\big{)}\
/\>\mathbb{P}_{u}^{c}(\mathbbm{1}_{T_{1}^{1}<\zeta_{\infty}})$
$\displaystyle=$
$\displaystyle\widetilde{\mathbb{P}}_{u}\Big{(}f(\dot{X}_{T_{1}^{1}})(H(0,\dot{X}_{T_{1}^{1}}))^{-1}\Big{)}\
/\>\widetilde{\mathbb{P}}_{u}((H(0,\dot{X}_{T_{1}^{1}}))^{-1})$
$\displaystyle\underset{u\to 0}{\longrightarrow}$
$\displaystyle\widetilde{\mathbb{P}}_{0+}\big{(}f(\dot{X}_{T_{1}^{1}})(H(0,\dot{X}_{T_{1}^{1}}))^{-1}\big{)}\
/\>\widetilde{\mathbb{P}}_{0+}((H(0,\dot{X}_{T_{1}^{1}}))^{-1})$
$\displaystyle=$ $\displaystyle\mathbf{n}(f(\dot{X}_{T_{1}^{1}})|\
T_{1}^{1}<\zeta_{\infty}),$

where we used successively (3.1), Proposition 2 and (a generalization of)
(4.2). As a consequence, the law of $\dot{X}_{T_{1}^{1}}$ under $\mathbb{P}$
is entirely determined, and is equal to that of $\dot{X}_{T_{1}^{1}}$ under
$\mathbf{n}(\cdot|\ T_{1}^{1}<\zeta_{\infty})$. Uniqueness of the stochastic
partial differential equation follows.

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