Case 2 |  $\mathbb{Q}\big{(}\sqrt{-6}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/2$ and $\mathfrak{U}_{2}=(2,w)$
Case 3 |  $\mathbb{Q}\big{(}\sqrt{-10}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/2$ and $\mathfrak{U}_{2}=(2,w)$
Case 4 |  $\mathbb{Q}\big{(}\sqrt{-13}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/2$ and $\mathfrak{U}_{2}=(2,1+w)$
Case 5 |  $\mathbb{Q}\big{(}\sqrt{-15}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/2$ and $\mathfrak{U}_{2}=(2,1+w)$
Case 6 |  $\mathbb{Q}\big{(}\sqrt{-22}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/2$ and $\mathfrak{U}_{2}=(2,w)$
Case 7 |  $\mathbb{Q}\big{(}\sqrt{-35}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/5$ and $\mathfrak{U}_{2}=(5,2+w)$
Case 8 |  $\mathbb{Q}\big{(}\sqrt{-37}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/2$ and $\mathfrak{U}_{2}=(2,1+w)$
Case 9 |  $\mathbb{Q}\big{(}\sqrt{-51}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/5$ and $\mathfrak{U}_{2}=(5,1+w)$
Case 10 |  $\mathbb{Q}\big{(}\sqrt{-58}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/2$ and $\mathfrak{U}_{2}=(2,w)$
Case 11 |  $\mathbb{Q}\big{(}\sqrt{-91}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/7$ and $\mathfrak{U}_{2}=(7,3+w)$
Case 12 |  $\mathbb{Q}\big{(}\sqrt{-115}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/5$ and $\mathfrak{U}_{2}=(5,-3+w)$
Case 13 |  $\mathbb{Q}\big{(}\sqrt{-123}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/3$ and $\mathfrak{U}_{2}=(3,1+w)$
Case 14 |  $\mathbb{Q}\big{(}\sqrt{-187}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/7$ and $\mathfrak{U}_{2}=(7,-2+w)$
Case 15 |  $\mathbb{Q}\big{(}\sqrt{-235}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/5$ and $\mathfrak{U}_{2}=(5,2+w)$
Case 16 |  $\mathbb{Q}\big{(}\sqrt{-267}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/3$ and $\mathfrak{U}_{2}=(3,1+w)$
Case 17 |  $\mathbb{Q}\big{(}\sqrt{-403}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/11$ and $\mathfrak{U}_{2}=(11,6+w)$
Case 18 |  $\mathbb{Q}\big{(}\sqrt{-427}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/7$ and $\mathfrak{U}_{2}=(7,3+w)$

Table 1. Non-free unary unimodular Hermitian lattices of class number $2$.

##  6 Appendix 2

Below, we record a representative from each different isometry class of non-
free unary unimodular Hermitian lattices over
$E=\mathbb{Q}\big{(}\sqrt{-d}\big{)}$, with class number $3$. One can find
details in the work of Arno, Robinson and Wheeler [1].

Cases |  $\mathbb{Q}\big{(}\sqrt{-d}\big{)}$ |  Isometry class representative
---|---|---
Case 1 |  $\mathbb{Q}\big{(}\sqrt{-23}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/2$ and $\mathfrak{U}_{2}=(2,w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/2$ and $\mathfrak{U}_{3}=(2,-1+w)$
Case 2 |  $\mathbb{Q}\big{(}\sqrt{-31}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/2$ and $\mathfrak{U}_{2}=(2,w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/2$ and $\mathfrak{U}_{3}=(2,-1+w)$
Case 3 |  $\mathbb{Q}\big{(}\sqrt{-59}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/3$ and $\mathfrak{U}_{2}=(3,w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/3$ and $\mathfrak{U}_{3}=(3,-1+w)$
Case 4 |  $\mathbb{Q}\big{(}\sqrt{-83}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/3$ and $\mathfrak{U}_{2}=(3,w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/3$ and $\mathfrak{U}_{3}=(3,-1+w)$
Case 5 |  $\mathbb{Q}\big{(}\sqrt{-107}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/3$ and $\mathfrak{U}_{2}=(3,w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/3$ and $\mathfrak{U}_{3}=(3,-1+w)$
Case 6 |  $\mathbb{Q}\big{(}\sqrt{-139}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/5$ and $\mathfrak{U}_{2}=(5,w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/5$ and $\mathfrak{U}_{3}=(5,-1+w)$
Case 7 |  $\mathbb{Q}\big{(}\sqrt{-211}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/5$ and $\mathfrak{U}_{2}=(5,1+w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/5$ and $\mathfrak{U}_{3}=(5,-2+w)$
Case 8 |  $\mathbb{Q}\big{(}\sqrt{-283}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/7$ and $\mathfrak{U}_{2}=(7,2+w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/7$ and $\mathfrak{U}_{3}=(7,-3+w)$
Case 9 |  $\mathbb{Q}\big{(}\sqrt{-307}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/7$ and $\mathfrak{U}_{2}=(7,w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/7$ and $\mathfrak{U}_{3}=(7,-1+w)$
Case 10 |  $\mathbb{Q}\big{(}\sqrt{-331}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/5$ and $\mathfrak{U}_{2}=(5,1+w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/5$ and $\mathfrak{U}_{3}=(5,-2+w)$
Case 11 |  $\mathbb{Q}\big{(}\sqrt{-379}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/5$ and $\mathfrak{U}_{2}=(5,w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/5$ and $\mathfrak{U}_{3}=(5,-1+w)$
Case 12 |  $\mathbb{Q}\big{(}\sqrt{-499}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/5$ and $\mathfrak{U}_{2}=(5,w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/5$ and $\mathfrak{U}_{3}=(5,-1+w)$
Case 13 |  $\mathbb{Q}\big{(}\sqrt{-547}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/11$ and $\mathfrak{U}_{2}=(11,2+w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/11$ and $\mathfrak{U}_{3}=(11,-3+w)$
Case 14 |  $\mathbb{Q}\big{(}\sqrt{-643}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/7$ and $\mathfrak{U}_{2}=(7,w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/7$ and $\mathfrak{U}_{3}=(7,-1+w)$
Case 15 |  $\mathbb{Q}\big{(}\sqrt{-883}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/13$ and $\mathfrak{U}_{2}=(13,w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/13$ and $\mathfrak{U}_{3}=(13,-1+w)$
Case 16 |  $\mathbb{Q}\big{(}\sqrt{-907}\big{)}$ |  $L_{2}=\mathfrak{U}_{2}v_{2}$ with $h(v_{2})=1/13$ and $\mathfrak{U}_{2}=(13,4+w)$
|  |  $L_{3}=\mathfrak{U}_{3}v_{3}$ with $h(v_{3})=1/13$ and $\mathfrak{U}_{3}=(13,-5+w)$

Table 2. Non-free unary unimodular Hermitian lattices of class number $3$.

Acknowledgments. I truly appreciate the anonymous referees and Professor
Kirsten Eisenträger for their invaluable suggestions, and acknowledge the
support of $2020-2021$ Faculty Research Fellowship from College of Arts &
Sciences at Texas A&M University-San Antonio.

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#  $\alpha$-stable Lévy processes entering the half space or a slab

Andreas E. Kyprianou, Sonny Medina and Juan Carlos Pardo  Department of
Statistics University of Warwick Coventry CV4 7AL, UK. E-mail:
<EMAIL_ADDRESS>Department of Mathematical Sciences University
of Bath Bath BA2 7AY, UK. E-mail<EMAIL_ADDRESS>A. C., Calle Jalisco
s/n, Col. Valenciana, A. P. 402, C.P. 36000, Guanajuato, Gto., Mexico. Email
<EMAIL_ADDRESS>
###### Abstract

Recently a series of publications, including e.g. [4, 10, 9, 8, 7], considered
a number of new fluctuation identities for $\alpha$-stable Lévy processes in
one and higher dimensions by appealing to underlying Lamperti-type path
decompositions. In the setting of $d$-dimensional isotropic processes, [9] in
particular, developed so called $n$-tuple laws for first entrance and exit of
balls. Fundamental to these works is the notion that the paths can be
decomposed via generalised spherical polar coordinates revealing an underlying
Markov Additive Process (MAP) for which a more advanced form of excursion
theory (in the sense of [11]) can be exploited.

Inspired by this approach, we give a different decomposition of the
$d$-dimensional isotropic $\alpha$-stable Lévy processes in terms of
orthogonal coordinates. Accordingly we are able to develop a number of
$n$-tuple laws for first entrance into a half-space bounded by an
$\mathbb{R}^{d-1}$ hyperplane, expanding on existing results of [3, 13]. This
gives us the opportunity to numerically construct the law of first entry of
the process into a slab of the form $(-1,1)\times\mathbb{R}^{d-1}$ using a
‘walk-on-half-spaces’ Monte Carlo approach in the spirit of the ‘walk-on-
spheres’ Monte Carlo method given in [5].

Key words: Stable processes, first passage problems, walk-on-spheres Monte
Carlo

MSC 2020: Primary 60G18, 60G52; Secondary 60G51

##  1 Introduction

Let $X=(X_{t},t\geq 0)$ be a $d$-dimensional $\alpha$-stable Lévy process
(henceforth referred to as just a stable process) with probabilities
$\mathbb{P}:=(\mathbb{P}_{x},x\in\mathbb{R}^{d})$. This means that $X$ has
càdlàg paths with stationary and independent increments as well as respecting
a property of self-similarity: There is an $\alpha>0$ such that, for $c>0,$
and $x\in{\mathbb{R}^{d}},$ under $\mathbb{P}_{x}$, the law of
$(cX_{c^{-\alpha}t},t\geq 0)$ is equal to $\mathbb{P}_{cx}$. It turns out that
stable Lévy processes necessarily have the scaling index $\alpha\in(0,2]$. The
case $\alpha=2$ pertains to a standard $d$-dimensional Brownian motion, thus
has a continuous path. Similar results are possible in the diffusive setting
but we nonetheless restrict ourselves to the isotropic pure jump setting of
$\alpha\in(0,2)$ in dimension $d\geq 2$ in order to bring forward features of
the proof which are dependent on the jump structure.

To be more precise, this means, for all orthogonal transformations
$U:\mathbb{R}^{d}\mapsto\mathbb{R}^{d}$ and $x\in\mathbb{R}^{d},$

$\quad\textit{the law of}\quad(UX_{t},t\geq 0)\textit{
under}\quad\mathbb{P}_{x}\textit{ is equal to }(X_{t},t\geq 0)\textit{ under
}\mathbb{P}_{Ux}.$

The stable Lévy process has a the jump measure $\Pi$ that satisfies

(1)
$\Pi(B)=\frac{2^{\alpha}\Gamma((d+\alpha)/2)}{\pi^{d/2}|\Gamma(-\alpha/2)|}\int_{B}\frac{1}{|y|^{\alpha+d}}{\rm
d}y,\quad B\subseteq\mathcal{B}(\mathbb{R}^{d}).$

The constant in the definition of $\Pi(B)$ can be arbitrary, however, our
choice corresponds to the one that allows us to identify the characteristic
exponent Lévy process as

(2) $\Psi(\theta)=-\frac{1}{t}\log\mathbb{E}({\rm e}^{\mathrm{i}\theta\cdot
X_{t}})=|\theta|^{\alpha},\quad\theta\in\mathbb{R}^{d},$

where we write $\mathbb{P}$ in preference to $\mathbb{P}_{0}$; more precisely,
the coefficient of $|\theta|^{\alpha}$ is one.

We are interested in describing how the paths of $X=(X^{(1)},\cdots,X^{(d)})$
behave in relation to hyperplanes and the half-spaces that they bound. Because
$X$ has stationary and independent increments and is isotropic, it suffices to
focus on the hyperplane $\mathbb{H}_{0}:=\\{z\in\mathbb{R}^{d}:z^{(1)}=0\\}$
and the two open half spaces that it partitions,
$\mathbb{H}_{0\downarrow}:=\\{z\in\mathbb{R}^{d}:z^{(1)}<0\\}$ and
$\mathbb{H}_{0\uparrow}:=\\{z\in\mathbb{R}^{d}:z^{(1)}>0\\}$.

As such, we need to point out that a subset of $k\in\\{1,2,\dots,d\\}$
coordinates of $X$, under $\mathbb{P}_{x}$, is in fact a $k$-dimensional
isotropic stable process. This follows by evaluating (2) on the vector
$\theta_{k}:=(\theta^{(1)},\theta^{(2)},\dots,\theta^{(k)},0,\dots,0)$ and
noting that

$\Psi(\theta_{k})=|\theta_{k}|^{\alpha}.$

In particular $X^{(1)}$ is a symmetric real valued stable process. Moreover,
$X^{(2:d)}:=(X^{(2)},\cdots,X^{(d)})$ is an isotropic stable process in
dimension $d-1$. By using isotropy, the projective properties of $X$ onto
lower dimensions, and recalling some path properties of symmetric real-valued
stable processes, it is worth noting the following facts, which will be
important for our forthcoming computations.

  * •

The isotropic stable process touches $\mathbb{H}_{0}$ (i.e. $X^{(1)}$ touches
0) in finite time with probability one when $\alpha\in(1,2)$ and with
probability zero when $\alpha\in(0,1]$. Moreover, in the almost sure sense,
when $\alpha=1$, $\liminf_{t\to\infty}|X^{(1)}|=0$, whereas, when
$\alpha\in(0,1)$, $\liminf_{t\to\infty}|X^{(1)}|>0$.

  * •

Due to the oscillatory behaviour of $X^{(1)}$, i.e.
$\limsup_{t\to\infty}X^{(1)}_{t}=-\liminf_{t\to\infty}X^{(1)}_{t}=+\infty$,
for any $x\in\mathbb{R}^{d}$ the process $\mathbb{P}_{x}$-almost surely visits
$\mathbb{H}_{0\downarrow}:=\\{z\in\mathbb{R}^{d}:z^{(1)}<0\\}$ and
$\mathbb{H}_{0\uparrow}:=\\{z\in\mathbb{R}^{d}:z^{(1)}>0\\}$ the ‘lower’ and
‘upper’ half-spaces an infinite number of times.

  * •

The process $X$ started in $x=(x^{(1)},\dots,x^{(d)})$ is regular for both the
half spaces $(x^{(1)},\infty)\times\mathbb{R}^{d-1}$ and
$(-\infty,x^{(1)})\times\mathbb{R}^{d-1}$. That is, for
$\tau_{x^{(1)}}^{\wedge}:=\inf\\{t>0:X^{(1)}>x^{(1)}\\}$ and
$\tau_{x^{(1)}}^{\vee}:=\inf\\{t>0:X^{(1)}<x^{(1)}\\}$, we necessarily have
$\mathbb{P}_{x}(\tau_{x^{(1)}}^{\wedge}=0)=\mathbb{P}_{x}(\tau_{x^{(1)}}^{\vee}=0)=1$.

  * •

Crossing a hyperplane does not occur by creeping. This means, for $r\neq
x^{(1)}$, where $x^{(1)}$ is the first component of the point of issue,
$x\in\mathbb{R}^{d}$, of $X$, with
$\tau_{r}^{\vee}=\inf\\{t>0:X_{t}^{(1)}<r\\}$ and
$\tau_{r}^{\wedge}=\inf\\{t>0:X_{t}^{(1)}>r\\}$,

$\mathbb{P}_{x}(X_{\tau_{r}^{\vee}}^{(1)}<r)=\mathbb{P}_{x}(X_{\tau_{r}^{\wedge}}^{(1)}>r)=1.$

Our results, given in the next two sections, pertain to the first passage
problem around the aforementioned stopping times
$\tau_{r}^{\vee}=\inf\\{t>0:X_{t}^{(1)}<r\\}$ and
$\tau_{r}^{\wedge}=\inf\\{t>0:X_{t}^{(1)}>r\\}$. With the above properties in
mind we will develop a theory of excursions from a directional
minimum/maximum, which are related to the excursions from a hyperplane studied
by [2]. Our excursion process is ‘indexed’ by the set of times at which the
process reaches a new minimum/maximum in the direction of $X^{(1)}$. What is
important however is that such excursions are now indexed by the value of
$X^{(2:d)}$ at the time the minimum/maximum is reached.

The results we will achieve for first entry into the half-space
$\\{x\in\mathbb{R}^{d}:x^{(1)}<0\\}$ will give us some interesting insights
into the much more difficult problem of understanding the first passage
problem into the slab $S:=(-1,1)\times\mathbb{R}^{d-1}$, based on the stopping
time

$\tau_{S}=\inf\\{t>0:X_{t}\in S\\}=\inf\\{t>0:X^{(1)}_{t}\in(-1,1)\\}.$

That is, we are interested in the law of $X_{\tau_{S}}$. This problem was
recently solved in one dimension by [8], in other words the distribution of
$X^{(1)}_{\tau_{S}}$ is already known. Understanding the distribution of the
additional variables $X^{(2:d)}_{\tau_{S}}$ and their correlation to
$X^{(1)}_{\tau_{S}}$ (which is specifically unlike the setting of Brownian
motion where there is independence) is a much harder problem.

Although we have been unable to provide a closed form expression for the
distribution of $X_{\tau_{S}}$, we will show how the law of
$(X^{(1)}_{t},t\leq\tau_{S})$ and $X^{(2:d)}_{\tau_{S}}$ are correlated
through a conditional dependency which relies on the $(d-1)$-dimensional
Cauchy distribution. Moreover, we introduce a new Monte Carlo method, based on
the method of ‘walks-on-spheres’ (cf. [5]), albeit now a method of ‘walks-on-
half-spaces’, which allows us to numerically compute the law of
$X_{\tau_{S}}$.

##  2 First crossing of a hyperplane

Our first result considers the one-sided point of closest reach to the
hyperplane $\mathbb{H}_{0}:=\\{x\in\mathbb{R}^{d}:x^{(1)}=0\\}$, before
crossing it for the first time. Specifically, we are interested in
$X_{G(\tau_{0}^{\vee})}$, where

$G(\tau_{r}^{\vee})=\sup\\{s<\tau_{r}^{\vee}:X^{(1)}_{s}=\inf_{u\leq\tau_{r}^{\vee}}X^{(1)}_{u}\\},$

and we recall $\tau_{0}^{\vee}:=\inf\\{s>0:X_{s}^{(1)}<0\\}$.

######  Theorem 1 (One-sided point of closest reach to $\mathbb{H}_{0}$ at
first crossing).

Let $x,y\in\mathbb{H}_{0\uparrow}$, then

(3) $\mathbb{P}_{x}(X_{G(\tau_{0}^{\vee})}\in{\rm
d}y)=C_{\alpha,d}\frac{|y^{(1)}-x^{(1)}|^{\alpha/2}}{|y^{(1)}|^{\alpha/2}}|x-y|^{-d}{\rm
d}y\quad\text{ for }0<y^{(1)}<x^{(1)},$

where $C_{\alpha,d}=\pi^{-(d/2+1)}\Gamma(d/2)\sin(\alpha\pi/2)$.

The concept of ‘point of closest reach’ comes from recent work in [9], which
considered the point on the path of an isotropic stable process which is
radially closest to the origin. This concept was later used in [6] to
condition the same process to continuously hit a patch on the unit sphere from
outside.

######  Remark 1.

If we consider $x,y\in\mathbb{H}_{0\downarrow}$ such that $x^{(1)}<y^{(1)}<0$,
replace $\tau_{0}^{\vee}$ by the stopping time
$\tau_{0}^{\wedge}=\inf\\{s>0:X_{s}^{(1)}>0\\}$ and define
$\overline{G}(\tau_{0}^{\wedge}):=\sup\\{s<\tau_{0}^{\wedge}:X_{s}^{(1)}=\sup_{u<\tau_{0}^{\vee}}X_{u}^{(1)}\\}$
(the point of closest reach from below of $\mathbb{H}_{0}$) by symmetry, the
kernel in (3) is also the density of $X_{\overline{G}(\tau_{0}^{\wedge})}$.

The use of an excursion framework to characterise directional minima, together
with explicit expressions for functionals under the excursion measure, allow
us to obtain joint laws of related quantities at first passage times in
regards half-spaces. These so-called $n$-tuple laws at first entry/exit of a
half-space provide a non-trivial extension to the simple undershoot-overshoot
law.

######  Theorem 2 (Triple law at the first crossing of a hyperplane).

Fix $r\in\mathbb{R}$ and define for $w,x,y\in\mathbb{H}_{r\uparrow}$ and
$z\in\mathbb{H}_{r\downarrow}$, the following density

$\Xi_{x}(w,y,z):=A_{\alpha,d}\frac{|x^{(1)}-w^{(1)}|^{\alpha/2}}{|x-w|^{d}}\frac{|w^{(1)}-y^{(1)}|^{\alpha/2}}{|w-y|^{d}}|y-z|^{-\alpha-d},$

where

$A_{\alpha,d}=\frac{2^{\alpha}\Gamma(d/2)^{2}\Gamma((d+\alpha)/2)}{\pi^{3d/2}\Gamma(\alpha/2)^{2}|\Gamma(-\alpha/2)|}.$

Then, for $x^{(1)}>w^{(1)}>r,$ $y^{(1)}>w^{(1)}$ and $z^{(1)}<r$,

(4) $\mathbb{P}_{x}(X_{G(\tau_{r}^{\vee})}\in{\rm
d}w,X_{\tau_{r}^{\vee}-}\in{\rm d}y,X_{\tau_{r}^{\vee}}\in{\rm
d}z)=\Xi_{x}(w,y,z)\,{\rm d}w\,{\rm d}y\,{\rm d}z.$

Once again, by symmetry, the previous result can be easily extended to the
case where the point of issue is in $\mathbb{H}_{r\downarrow}$ and upward
crossing occurs at $\tau_{r}^{\wedge}$.

Technically speaking, the next two Lemmas could be derived by marginalising
(4) and, as such may be regarded as corollaries of Theorem 2. However, it
turns out they can be derived directly in a more straightforward way. We
therefore state them as lemmas rather than corollaries.

######  Lemma 1 (Overshoot and undershoot at first crossing of a hyperplane).

Fix $r\in\mathbb{R}$ and define for $x,y\in\mathbb{H}_{r\uparrow}$ and
$z\in\mathbb{H}_{r\downarrow}$

$\Xi_{x}(\bullet,y,z):={B}_{\alpha,d}\bigg{(}|x-y|^{\alpha-d}\int_{0}^{\zeta_{r}^{\vee}(x,y)}(u+1)^{-d/2}u^{\alpha/2-1}{\rm
d}u\bigg{)}|z-y|^{-(\alpha+d)}$

where

$\zeta_{r}^{\vee}(x,y):=\frac{4(x^{(1)}-r)(y^{(1)}-r)}{|x-y|^{2}}\quad\text{
and
}\quad{B}_{\alpha,d}=\frac{\Gamma((d+\alpha)/2)\Gamma(d/2)}{\pi^{d}|\Gamma(\alpha/2)^{2}\Gamma(-\alpha/2)|}.$

Then, for $y^{(1)}>r>z^{(1)}$

(5) $\mathbb{P}_{x}(X_{\tau_{r}^{\vee}-}\in{\rm d}y,X_{\tau_{r}^{\vee}}\in{\rm
d}z)=\Xi_{x}(\bullet,y,z){\rm d}y{\rm d}z.$

Finally, in a second lemma below we give the law of just the overshoot
distribution. However, we note that its proof again avoids performing a
marginalisation of, this time, the density (5), and we use a third approach
for its proof.

######  Lemma 2 (Overshoot at first crossing of a hyperplane).

Fix $r\in\mathbb{R}$ and define for $x\in\mathbb{H}_{r\uparrow}$ and
$z\in\mathbb{H}_{r\downarrow}$

$\Xi_{x}(\bullet,\bullet,z):=C_{\alpha,d}|x-z|^{-d}\frac{|r-x^{(1)}|^{\alpha/2}}{|r-z^{(1)}|^{\alpha/2}},$

where $C_{\alpha,d}=\pi^{-(d/2+1)}\Gamma(d/2)\sin(\alpha\pi/2)$. Then

(6) $\quad\mathbb{P}_{x}(X_{\tau^{\vee}_{r}}\in{\rm
d}z)=\Xi_{x}(\bullet,\bullet,z)\mathrm{d}z\quad z^{(1)}<r<x^{(1)}.$

######  Remark 2.

Lemma 2 is known already from [3] and Lemma 1 is also implicitly known from
the same reference, albeit stated up to multiplication by the stable jump
measure as the Green function for $X$ killed on exiting
$\mathbb{H}_{r\uparrow}$. The approach taken by [3] uses potential analysis
rather than the fluctuation theory approach that we use here.

##  3 Conditional law and numerics for first entry into a slab

The objective of this section is to give a partial distributional
decomposition of the density of the first hit into the infinite slab
$S=(-1,1)\times\mathbb{R}^{d-1}$ of $(X,\mathbb{P})$, together with a Monte
Carlo method for numerically determining it. Pre-emptively assuming it exists,
the density we are interested in is the bivariate function $H(x,y)$ such that
for $x\in\mathbb{R}^{d}\setminus S$,

$\mathbb{P}_{x}(X_{\tau_{S}}\in\mathrm{d}y,\tau_{S}<\infty)=H(x,y)\mathrm{d}y,\quad
y\in(-1,1)\times\mathbb{R}^{d-1}.$

It is known that in the case $\alpha\in(1,2)$ the process $X$ will hit the
slab with probability one because $X^{(1)}$, its first coordinate, will hit
any interval with probability one; cf. [8]. However, in the case
$\alpha\in(0,1)$, we must necessarily work on the event
$\\{\tau_{S}<\infty\\}$.

Because stable processes always make first passage into a half space by a jump
(i.e. no creeping) we know that the coordinate $X^{(1)}$ must make a random
number of crossings of $(-1,1)$ before entering it. To this end, without loss
of generality, we can start with $x^{(1)}>1$ and define $\sigma_{0}=0$. Then
for $k=1,2,\cdots$, iteratively define,

$\displaystyle\sigma_{k}$
$\displaystyle=\begin{cases}\inf\\{t>\sigma_{k-1}:X^{(1)}_{t}>-1\\}&\text{if
$k$ is even }\\\ \inf\\{t>\sigma_{k-1}:X^{(1)}_{t}<1\\}&\text{if $k$ is odd
},\\\ \end{cases}$

and the random index $N$ is such that $\\{N=k\\}$ agrees with
$\tau_{S}=\sigma_{k}$.

Our first result concerns the conditional distribution of
$X^{(2:d)}_{\tau_{S}}$ given $(X^{(1)}_{t},t\leq\tau_{S})$. More precisely, we
are interested in the conditional distribution of $X^{(2:d)}_{\tau_{S}}$ given
$(X^{(1)}_{\sigma_{k}},k=0,1,\cdots,N)$. To this end, let us define
$\mathtt{Cauchy}_{d-1}(0,\gamma)$, a $(d-1)$-dimensional Cauchy distribution
centred at 0 with scaling $\gamma>0$. That is to say, a random variable on
$\mathbb{R}^{d-1}$ with density

$\frac{\Gamma(d/2)}{\pi^{d/2}}\frac{\gamma}{(\gamma^{2}+x^{2})^{d/2}},\qquad
x\in\mathbb{R}^{d-1}.$

######  Theorem 3 (Conditional law of first entry into a slab).

We have

$\emph{\text{Law}}\left(X_{\tau_{S}}^{(2:d)}\bigg{|}(X^{(1)}_{t},t\leq\tau_{S})\right)\sim
x^{(2:d)}+\mathtt{Cauchy}_{d-1}\left(0,\,\sum_{k=1}^{N}|X^{(1)}_{\sigma_{k}}-X^{(1)}_{\sigma_{k-1}}|\right).$

###### Proof.

By Lemma 2 in the case $d\geq 2$,

$\mathbb{P}_{x}(X_{\tau^{\vee}_{r}}\in{\rm
d}y)=C_{\alpha,d}|x-y|^{-d}\frac{|r-x^{(1)}|^{\alpha/2}}{|r-y^{(1)}|^{\alpha/2}}{\rm
d}y\quad\text{ for }y^{(1)}<r<x^{(1)}$

where $C_{\alpha,d}=\pi^{-(d/2+1)}\Gamma(d/2)\sin(\alpha\pi/2)$. Recalling
that the same formula is true for the case $d=1$; cf. Corollary 3.5 in Chapter
3 of [7] we have,

$\mathbb{P}_{x}(X^{(1)}_{\tau^{\vee}_{r}}\in{\rm
d}y^{(1)})=C_{\alpha,1}|x^{(1)}-y^{(1)}|^{-1}\frac{|r-x^{(1)}|^{\alpha/2}}{|r-y^{(1)}|^{\alpha/2}}{\rm
d}y^{(1)}\quad\text{ for }y^{(1)}<r<x^{(1)}.$

Thus for $y^{(2:d)}\in\mathbb{R}^{d-1}$,

(7)
$\mathbb{P}_{x}(X_{\tau_{r}^{\vee}}^{(2:d)}\in\mathrm{d}y^{(2:d)}|X_{\tau_{r}^{\vee}}^{(1)}=y^{(1)})=\frac{\Gamma(d/2)}{\pi^{d/2}}\frac{|x^{(1)}-y^{(1)}|}{\left((x^{(1)}-y^{(1)})^{2}+|x^{(2:d)}-y^{(2:d)}|^{2}\right)^{d/2}}\,\mathrm{d}y^{(2:d)}.$

That is to say, under $\mathbb{P}_{x}$,

$\displaystyle\text{Law}\left(X^{(2:d)}_{\tau_{r}^{\vee}}-x^{(2:d)}\bigg{|}X^{(1)}_{\tau_{r}^{\vee}}=y^{(1)}\right)$
$\displaystyle\sim\mathtt{Cauchy}_{d-1}\left(0,\,|x^{(1)}-y^{(1)}|\right).$

By symmetry we can show that for $x^{(1)}<r<y^{(1)}$,
Law$(X^{(2)}_{\tau_{r}^{\wedge}}|X^{(1)}_{\tau_{r}^{\wedge}}=y^{(1)})$ under
$\mathbb{P}_{x}$, follows a centered Cauchy distribution with scale parameter
$|y^{(1)}-x^{(1)}|$.

Since we know that

$\mathtt{Cauchy}_{d-1}(0,\gamma)=\gamma\times\mathtt{Cauchy}_{d-1}(0,1),$

we can thus identify, for $k=1,\cdots,N$, with $\sigma_{0}=0$, and hence
$x^{(1)}=X_{\sigma_{0}}$,

$\text{Law}\left(X^{(2:d)}_{\sigma_{k}}-x^{(2:d)}|X^{(1)}_{\sigma_{j}},\,j=0,1,\cdots,k\right)\sim\sum_{i=1}^{k}|X^{(1)}_{\sigma_{i}}-X^{(1)}_{\sigma_{i-1}}|\times\mathtt{Cauchy}^{(i)}_{d-1}(0,\,1),$

where $\mathtt{Cauchy}^{(i)}_{d-1}(0,\,1)$ are iid copies of
$\mathtt{Cauchy}_{d-1}(0,\,1)$ and the result follows by infinite divisibility
and scaling of the Cauchy distribution. ∎

The proof of Theorem 3 inspires the following numerical approach to simulating
the law of $X_{\tau_{S}}$ using an approach which is reminiscent of the so-
called ‘walk-on-spheres’ Monte Carlo approach described for stable processes
in [5]. As such we refer to it as the ‘walk-on-half-spaces’ Monte Carlo
approach.

Algorithm 1 (simulation of $X_{\tau_{S}}$ under $\mathbb{P}_{x}$ for
$x\in\mathbb{R}^{d}$ with $x^{(1)}>1$, $\alpha\in[1,2)$).

sample $x_{1}$ from the distribution of $X_{\tau_{1}^{\vee}}$ under
$\mathbb{P}_{x}$;

set $k=1$;

while $x_{k}\notin S$

if $x_{k}^{(1)}<-1$ then

sample $x_{k+1}$ from the distribution of $X_{\tau_{-1}^{\wedge}}$ under
$\mathbb{P}_{x_{k}}$;

set: $k=k+1$;

end if

if $x_{k}^{(1)}>1$ then

sample $x_{k+1}$ from the distribution of $X_{\tau_{1}^{\vee}}$ under
$\mathbb{P}_{x_{k}}$;

set $k=k+1$;

end if

end while

return $N=k$;

return $X_{\tau_{S}}=x_{k}$;

More generally, the Algorithm above stores the information
$(x_{1},\cdots,x_{N})$, which corresponds precisely to a simulation of
$(X_{\sigma_{1}},\cdots,X_{\sigma_{N}})$.

One of the problems with Algorithm 1 is that, in the setting $\alpha\in(0,1)$,
the while loop may never end on account of the fact that
$\mathbb{P}_{x}(\tau_{S}=\infty)>0$, by Corollary 1.2 of [8]. Nonetheless, it
is possible to ‘force’ a stable process to hit $S$ in the case
$\alpha\in(0,1)$ improving the efficiency of the walk-on-half-spaces algorithm
without compromising the exactness of the algorithm.

We recall from Chapter 12.2 of [7] that, when $\alpha\in(0,1)$, the change of
measure

$\frac{\mathrm{d}\mathbb{P}^{\circ}_{x}}{\mathrm{d}\mathbb{P}_{x}}\Bigg{|}_{(X_{s},s\leq
t)}=\left|\frac{X^{(1)}_{t}}{x^{(1)}}\right|^{\alpha-1}$

corresponds to conditioning the process $X^{(1)}$ to continuously absorb at
the origin in an almost surely finite time. It follows that

$\mathbb{P}_{x}(X_{\tau_{S}}\in\mathrm{d}y,\tau_{S}<\infty)=\frac{|x^{(1)}|^{\alpha-1}}{|y^{(1)}|^{\alpha-1}}\mathbb{P}^{\circ}_{x}(X_{\tau_{S}}\in\mathrm{d}y),\qquad
x^{(1)}\not\in(-1,1),y^{(1)}\in(-1,1).$

This suggests we can numerically reconstruct the law of $X_{\tau_{S}}$ by
undertaking Algorithm 1 albeit with sampling taken under $\mathbb{P}^{\circ}$
rather than $\mathbb{P}$. Fundamentally, this means sampling from the two
distributions

(8)
$\mathbb{P}_{x}^{\circ}(X_{\tau_{1}^{\vee}}\in\mathrm{d}y)=C_{\alpha,d}\frac{|y^{(1)}|^{\alpha-1}}{(x^{(1)})^{\alpha-1}}|x-y|^{-d}\frac{(x^{(1)}-1)^{\alpha/2}}{(1-y^{(1)})^{\alpha/2}}{\rm
d}y,\qquad y^{(1)}<1<x^{(1)},$

and

$\mathbb{P}_{x}^{\circ}(X_{\tau_{-1}^{\wedge}}\in\mathrm{d}y)=C_{\alpha,d}\frac{|y^{(1)}|^{\alpha-1}}{(-x^{(1)})^{\alpha-1}}|x-y|^{-d}\frac{(-1-x^{(1)})^{\alpha/2}}{(y^{(1)}+1)^{\alpha/2}}{\rm
d}y,\qquad x^{(1)}<-1<y^{(1)}.$

##  4 A numerical example

We consider the implementation of the walk-on-half-spaces algorithm for a
stable process on $\mathbb{R}^{2}$. To sample from the distribution of
$X_{\tau_{1}^{\vee}}=(X_{\tau_{1}^{\vee}}^{(1)},X_{\tau_{1}^{\vee}}^{(2)})$
under $\mathbb{P}_{x}$, $x^{(1)}>1$, we split the procedure into two steps.

First, we simulate an instance of $X_{\tau_{1}^{\vee}}^{(1)}$. This can be
done by treating this as a first passage problem of the one-dimensional
symmetric stable process $X^{(1)}$ and noting from Lemma 2 that

$\mathbb{E}_{x^{(1)}}\left[f\left(\frac{1-X_{\tau_{1}^{\vee}}^{(1)}}{x^{(1)}-X_{\tau_{1}^{\vee}}^{(1)}}\right)\right]=\frac{\sin(\alpha\pi/2)}{\pi}\int_{0}^{1}f(u)u^{-\alpha/2}(1-u)^{\alpha/2-1}\,\mathrm{d}u,$

which said in another way means
$(1-X_{\tau_{1}^{\vee}}^{(1)})(X_{\tau_{1}^{\vee}}^{(1)}-x^{(1)})^{-1}$, when
$X^{(1)}_{0}=x^{(1)}$, follows a Beta$(1-\alpha/2,\alpha/2)$ distribution.
Observe that this transform is a bijection, hence we can simulate from the
Beta kernel and map it back to obtain $X_{\tau_{1}^{\vee}}^{(1)}$.

Second, we use equation (7), which under the event
$\\{X_{\tau_{1}^{\vee}}^{(1)}=y_{1}^{(1)}\\}$, allows us to sample
$X_{\tau_{1}^{\vee}}^{(2)}-x^{(2:d)}$ by sampling from a
$\mathtt{Cauchy}_{1}(0,|X_{\tau_{1}^{\vee}}^{(1)}-x^{(1)}|)$ distribution. By
symmetry, the same can be done for sampling from $X_{\tau_{-1}^{\wedge}}$
under $\mathbb{P}_{x^{(1)}}$, $x^{(1)}<-1$. Except for these details on how to
simulate $X_{\tau_{1}^{\vee}}$ and $X_{\tau_{-1}^{\wedge}}$, the rest of the
algorithm implementation is straightforward, see Figure 1.

In the numerical examples, cf. Figure 2, we consider, without loss of
generality, that $x^{(2)}=0$. To illustrate the effect of choosing different
parameters on the empirical distribution of $X_{\tau_{S}}$ we vary the point
of issue $x^{(1)}$ and the scale parameter $\alpha$. In Figure 2 (left) we
observe the empirical distribution of 5 million first-hitting points in the
strip, for 4 combinations of parameters $x^{(1)}$ and $\alpha$. We observe
that an initial point closer to the boundary of the strip leads to a sample
more concentrated in the horizontal axis neighbouring the starting point.
Similarly, a scale index closer to $\alpha=2$ leads to a first-hitting point
more concentrated around the starting position, but also increases the
probability of hitting the strip near its boundaries. These effects can also
be seen in the marginal histograms of these samples, Figure 2 (centre). It is
worth noting that the marginal distribution of the first coordinate is known
explicitly (cf. Theorem 1.1 in [8]). The code for these simulations can be
found in the following github repository https://github.com/sdaphnis/WoHS/.

$X_{0}$$X_{\sigma_{1}}=X_{\tau_{1}^{\lor}}$$X_{\sigma_{2}}$$X_{\sigma_{3}}=X_{\tau_{S}}$$X^{(1)}$$X^{(2:d)}$$-1$$1$
Figure 1: Representation of the Walk on half-spaces algorithm. In this
illustration the process starts above the strip, the first jump out of
$\mathbb{H}_{1\uparrow}$ misses the strip, we start the algorithm again taking
as initial position the first hit into $\mathbb{H}_{-1\downarrow}$. The
process repeats and after $N=3$ crossings the process finally enters the
strip. The red lines correspond to the $\mathtt{Cauchy}_{1}$ horizontal
displacements which add up to $X^{(2:d)}_{\tau_{S}}$.

Figure 2: (left): Histograms of the simulations of first-hits into the strip
for different values of $X_{0}=(x^{(1)},0)$ and $\alpha$. The green point
represents the starting point $X_{0}$. (centre): Histograms of the second
coordinate at the instant of first-hit into the strip. Different values for
the point of issue and $\alpha$ are considered. Instead of drawing solid bars,
the histograms are drawn as step-like curves that outline the shape of the
histogram, i.e. we see the edges of each bin without any filled area
underneath. (right): A 3d histogram of the first hits into the strip.

##  5 Proofs of Lemmas 1 and 2

In order to prove Lemmas 1 and 2 we use a method of converting results for
first entry into a ball into first entry into a half space by expanding the
radius of the sphere asymptotically so that, with re-centring, the interior of
the ball becomes a half-space. We call this the flat earth theory, in the
sense that, from an infinitesimally localised perspective, the surface of a
sphere appears to be that of a flat surface.

The lemma we present below combines a geometric argument with the Lévy and
scaling properties of the stable process $(X,\mathbb{P})$.

######  Lemma 3 (Flat earth approximation).

Let $\tau_{r}^{\oplus}=\inf\\{t>0:|X_{t}|<r\\}$ and suppose
$f:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}$ is bounded and measurable.
Then

(9)
$\mathbb{E}_{x}\left[f(X_{\tau_{r}^{\vee}},X_{\tau_{r}^{\vee}-})\right]=\lim_{R\to\infty}\mathbb{E}_{x+R\mathbf{e}_{1}}\left[f(X_{\tau_{R+r}^{\oplus}}-R\mathbf{e}_{1},X_{\tau_{R+r}^{\oplus}-}-R\mathbf{e}_{1}),\tau_{R}^{\oplus}<\infty\right],$

where $\mathbf{e}_{1}=(1,0,\cdots,0)\in\mathbb{R}^{d}$.

###### Proof.

We start by approximating
$\mathbb{H}_{r\downarrow}=\\{z\in\mathbb{R}^{d}:z^{(1)}<r\\}$, $r>0$, by an
increasing sequence of balls
$\mathbb{B}_{(-R\mathbf{e}_{1},R+r)}=\\{z\in\mathbb{R}^{d}:|z+R\mathbf{e}_{1}|<R+r\\}\subseteq\mathbb{H}_{r\downarrow}$
where $R>0$. We have

$\lim_{R\to\infty}\mathbb{B}_{(-R\mathbf{e}_{1},R+r)}=\mathbb{H}_{r\downarrow}.$

To see why, we only have to show that
$\mathbb{H}_{r\downarrow}\subseteq\lim_{R\to\infty}\mathbb{B}_{(-R\mathbf{e}_{1},R+r)}$.
Indeed, take an arbitrary point in $\mathbb{H}_{r\downarrow}$ written
$y=(r-r_{0},y^{(2:d)})$. An elementary calculation shows that if
$R>(2r_{0})^{-1}(r_{0}^{2}+|y^{(2:d)}|^{2})-r$, then
$y\in\mathbb{B}_{(-R\mathbf{e}_{1},R+r)}$.

$r$$0$$\mathbb{H}_{r\downarrow}$$\mathbb{B}_{(-Re1,R+r)}$ Figure 3:
Approximation of $\mathbb{H}_{r\downarrow}$ by
$\mathbb{B}_{(-R\mathbf{e}_{1},R+r)}$.

The next assertion is that we can define a decreasing sequence of stopping
times which will limit to $\tau_{r}^{\vee}$. If
$\tau_{\mathbb{B}_{R+r}}=\inf\\{t>0:X_{t}\in\mathbb{B}_{(-R\mathbf{e}_{1},R+r)}\\}$,
we claim that, for all $x\in\mathbb{R}^{d}$ with $x^{(1)}>r$,
$\mathbb{P}_{x}$-almost surely it holds that

$\tau_{\mathbb{B}_{R+r}}\downarrow\tau_{r}^{\vee},$

as $R\to\infty$.

The first thing we need to point out is that $(\tau_{\mathbb{B}_{R+r}},R>0)$
is a decreasing sequence of stopping times, hence its limit is a stopping
time. For $\mathbb{P}_{x}$-almost every $\omega$ let
$y(\omega)=X_{\tau_{r}^{\vee}(\omega)}(\omega)$. Then there exists a
$R_{0}(\omega)>0$ such that
$y(\omega)\in\mathbb{B}_{(-R_{0}(\omega)\mathbf{e}_{1},R_{0}(\omega)+r)}$,
implying by convexity that
$\tau_{\mathbb{B}_{R+r}}(\omega)=\tau_{r}^{\vee}(\omega)$ for all $R>R_{0}.$
In which case
$X_{\tau_{r}^{\vee}(\omega)}(\omega)=X_{\tau_{\mathbb{B}_{R+r}}(\omega)}(\omega)$
and
$X_{\tau_{r}^{\vee}(\omega)-}(\omega)=X_{\tau_{\mathbb{B}_{R+r}-}(\omega)}(\omega)$.

Hence, for any $x\in\mathbb{H}_{r\uparrow}$ appealing to dominated convergence
(as $f$ is bounded),

$\displaystyle\mathbb{E}_{x}\left[f(X_{\tau_{r}^{\vee}},X_{\tau_{r}^{\vee}-})\right]$
$\displaystyle=\mathbb{E}_{x}\left[\lim_{R\to\infty}f(X_{\tau_{\mathbb{B}_{R+r}}},X_{\tau_{\mathbb{B}_{R+r}}-})\right]$
$\displaystyle=\lim_{R\to\infty}\mathbb{E}_{x}\left[f(X_{\tau_{\mathbb{B}_{R+r}}},X_{\tau_{\mathbb{B}_{R+r}}-}),\tau_{\mathbb{B}_{R+r}}<\infty\right]$
$\displaystyle=\lim_{R\to\infty}\mathbb{E}_{x+R\mathbf{e}_{1}}\left[f(X_{\tau_{R+r}^{\oplus}}-R\mathbf{e}_{1},X_{\tau_{R+r}^{\oplus}-}-R\mathbf{e}_{1}),\tau_{R+r}^{\oplus}<\infty\right],$

where the last equality is given by stationary and independent increments. ∎

Next we give the proof of Lemma 2 which is almost trivial in light of Lemma 3.
We leave the proof of Lemma 1 to the reader, pointing to Corollary 1.4 of [9]
as the starting expression from which limits can be taken.

###### Proof.

Applying Lemma 3 using the explicit formula for the distribution of the first
hit inside a ball found in [1], we have that

$\displaystyle\mathbb{E}_{x}[f(X_{\tau_{r}^{\vee}})]$
$\displaystyle=\lim_{R\to\infty}\mathbb{E}_{x+R\mathbf{e}_{1}}\left[f(X_{\tau_{R+r}^{\oplus}}-R\mathbf{e}_{1});\tau_{R+r}^{\oplus}<\infty\right]$
$\displaystyle=\lim_{R\to\infty}C_{\alpha,d}\int_{|y|<R+r}f(y-R\mathbf{e}_{1})\frac{|(R+r)^{2}-|x+R\mathbf{e}_{1}|^{2}|^{\alpha/2}}{|(R+r)^{2}-|y|^{2}|^{\alpha/2}}|(x+R\mathbf{e}_{1})-y|^{-d}{\rm
d}y$
$\displaystyle=\lim_{R\to\infty}C_{\alpha,d}\int_{|z+R\mathbf{e}_{1}|<R+r}f(z)\frac{|(R+r)^{2}-|x+R\mathbf{e}_{1}|^{2}|^{\alpha/2}}{|(R+r)^{2}-|y+R\mathbf{e}_{1}|^{2}|^{\alpha/2}}|x-z|^{-d}{\rm
d}y$
$\displaystyle=C_{\alpha,d}\int_{\mathbb{H}_{r\downarrow}}f(z)|x-y|^{-d}\lim_{R\to\infty}\frac{|2R(r-x^{(1)})+r^{2}-|x|^{2}|^{\alpha/2}}{|2R(r-y^{(1)})+r^{2}-|y|^{2}|^{\alpha/2}}{\rm
d}z$
$\displaystyle=C_{\alpha,d}\int_{\mathbb{H}_{r\downarrow}}f(z)|x-y|^{-d}\frac{|r-x^{(1)}|^{\alpha/2}}{|r-y^{(1)}|^{\alpha/2}}{\rm
d}y,$

which completes the proof. ∎

######  Remark 3.

To complete this section, we address the rather obvious question as to why
flat Earth theory cannot be used to address Theorem 1. In theory this may well
be possible but we could not see a way to ensure that the point of closest
reach to the origin preceding first entry to the ball
$\mathbb{B}_{(-R\mathbf{e}_{1},R+r)}$ asymptotically converges to
$X_{G(\tau^{\vee}_{r})}$. For this reason, we take a different approach to
proving Theorem 1, which uses the so-called directional excursion theory in
the spirit of [2].

##  6 Directional excursion theory and proof of Theorem 1

Let $\underline{X}^{(1)}_{t}:=\inf_{s\leq t}X^{(1)}_{s}$ be the running
infimum of the first coordinate of $X$, and
$R^{(1)}_{t}:=X^{(1)}_{t}-\underline{X}^{(1)}_{t}$ the reflection at the
infimum of $X^{(1)}$. Under $\mathbb{P}_{x}$, the process
$R^{(1)}=(R^{(1)}_{t}:t\geq 0)$ is a strong Markov process for which $0$ is
regular with respect to $(0,\infty)$. Moreover, the joint process
$(R^{(1)}_{t},X_{t})_{t\geq 0}$ is a
$\mathbb{R}^{+}\times\mathbb{R}^{d}$-valued strong Markov process with
transition semigroup

$P_{t}f(u,x):=\mathbb{E}_{x}\left[f(u\vee R^{(1)}_{t},X_{t})\right],$

for every $t>0$ and every nonnegative measurable function
$f:\mathbb{R}^{+}\times\mathbb{R}^{d}\to\mathbb{R}^{+}$. We shall work with
the canonical realization of $(R^{(1)}_{t},X_{t})_{t\geq 0}$ on the sample
space $\mathbb{D}(\mathbb{R}^{+}\times\mathbb{R}^{d})$ of càdlàg paths mapping
$\mathbb{R}^{+}$ to $\mathbb{R}^{d}$ with appended cemetery state, which is
where the path is sent at its lifetime (which may be infinite).

We define $M:=\\{t\geq 0:R^{(1)}_{t}=0\\}$ and $\bar{M}$ its closure in
$\mathbb{R}^{+}$. The set $\mathbb{R}\setminus\bar{M}$ is an open set and can
be written as a union of intervals. We use $\mathtt{G}$ and $\mathtt{D}$ to
denote the sets of left and right endpoints of such intervals. If we define
$\mathcal{R}:=\inf\\{t>0:t\in\bar{M}\\}$, the downward regularity of $X^{(1)}$
implies that for every $x\in\mathbb{R}^{d}$, with probability one
$\mathbb{P}_{x}(\mathcal{R}=0)=1$.

By Theorem 4.1 in [11] and stationary independent increments, there exist a
continuous additive functional $t\mapsto L_{t}$ of $(R^{(1)},X)$ carried by
$\\{0\\}\times\mathbb{R}^{d}$ and a kernel $\mathtt{N}$ from
$\mathbb{R}^{+}\times\mathbb{R}^{d}$ into
$\mathbb{D}(\mathbb{R}^{+}\times\mathbb{R}^{d})$ satisfying
$\mathtt{N}_{x}(\mathcal{R}=0)=0$ and $\mathtt{N}_{x}\left[1-{\rm
e}^{-\mathcal{R}}\right]\leq 1$ such that for any nonnegative predictable
process $Z$ and any nonnegative function $f$ which is measurable with respect
to $\sigma((R^{(1)}_{t},X_{t})_{t\geq 0})$ then

(10) $\mathbb{E}_{x}\left[\sum_{s\in
G}Z_{s}f\circ\theta_{s}\right]=\mathbb{E}_{x}\left[\int_{0}^{\infty}Z_{s}\,\mathtt{N}_{X_{s}}(f)\mathrm{d}L_{s}\right],$

where $\theta_{s}$ is the Markov shift operator and $\mathtt{N}_{x}(f)=\int
f\mathrm{d}\mathtt{N}_{x}$. In the setting that $f$ depends only on the
process $(R^{(1)},X^{(2:d)})$ we can reduce the form of
$\mathtt{N}_{x}=\mathtt{N}_{(x^{(1)},x^{(2:d)})}$ to being dependent only on
$x^{(2:d)}$, in which case we will write it as $\mathbb{N}_{x^{(2:d)}}$.

To define the excursion process we start by expressing the endpoints of the
intervals of excursions. Let $\mathtt{G}=\cup_{t>0}\mathtt{g}_{t}$ and
$\mathtt{D}=\cup_{t>0}\mathtt{d}_{t}$ where
$\mathtt{g}_{t}:=\sup\\{s<t:X^{(1)}_{s}=\underline{X}^{(1)}_{s}\\}$ and
$\mathtt{d}_{t}:=\inf\\{s>t:X^{(1)}_{s}=\underline{X}^{(1)}_{s}\\}$. Hence,
for all $t>0$ such that $\mathtt{d}_{t}>\mathtt{g}_{t}$, the excursions of $X$
from its directional minimum in the first axis, or equivalently the excursions
of $(X^{(1)}-\underline{X}^{(1)},X^{(2:d)})$ from the set
$\\{0\\}\times\mathbb{R}^{d-1}$ are encoded in the following process

(11)
$(\epsilon^{(1)}_{\mathtt{g}_{t}}(s),\epsilon^{(2:d)}_{\mathtt{g}_{t}}(s)):=(X^{(1)}_{\mathtt{g}_{t}+s}-X^{(1)}_{\mathtt{g}_{t}},X^{(2:d)}_{\mathtt{g}_{t}+s})\quad
0\leq s\leq\mathtt{d}_{t}-\mathtt{g}_{t}=:\zeta_{\mathtt{g}_{t}}.$

Such excursions live in the space
$\mathbb{D}(\mathbb{R}^{+}\times{\mathbb{R}^{d-1}})$ of càdlàg paths in
$\mathbb{R}^{+}\times{\mathbb{R}^{d-1}}$, such that

$(\epsilon^{(1)},\epsilon^{(2:d)})=\left((\epsilon^{(1)}(t),\epsilon^{(2:d)}(t)):t\leq\zeta\right)\text{
with lifetime }\zeta=\inf\\{s>0:\epsilon^{(1)}(s)<0\\}$

such that
$\left(\epsilon^{(1)}(0),\epsilon^{(2:d)}(0)\right)\in\\{0\\}\times\mathbb{R}^{d-1},$
$(\epsilon^{(1)}(s),\epsilon^{(2:d)}(s))\in(0,\infty)\times\mathbb{R}^{d-1}$
and
$\left(\epsilon^{(1)}(\zeta),\epsilon^{(2:d)}(\zeta)\right)\in(-\infty,0)\times\mathbb{R}^{d-1}.$

Reinterpreting (10) get the so-called last exit formula

$\displaystyle\mathbb{E}_{(x^{(1)},x^{(2:d)})}$
$\displaystyle\left[\sum_{g\in\mathtt{G}}F\left(X_{s},s<g\right)H\left((\epsilon^{(1)}_{g},\epsilon^{(2:d)}_{g})\right)\right]$
(12)
$\displaystyle=\mathbb{E}_{(x^{(1)},x^{(2:d)})}\left[\int_{0}^{\infty}F\left(X_{s},s<t\right)\mathbb{N}_{X^{(2:d)}_{t}}\left(H(\epsilon^{(1)},\epsilon^{(2:d)})\right){\rm
d}L_{t}\right],$

where $F$ is non-negative and bounded on the space of càdlàg paths on
$\mathbb{R}^{d}$ and $H$ is measurable on
$\mathbb{D}(\mathbb{R}^{+}\times{\mathbb{R}^{d-1}})$. Under the measures
$(\mathbb{N}_{x^{(2:d)}},x^{(2:d)}\in\mathbb{R}^{d-1})$, the process
$(\epsilon^{(1)}(s),\epsilon^{(2:d)}(s);s<\zeta)$ is Markovian with the same
semigroup as $X=(X^{(1)},X^{(2:d)})$ killed at its first hitting time of
$(-\infty,0]\times\mathbb{R}^{d-1}$.

The couple $(L,\mathbb{N}_{\cdot})$ is called an exit system. This pair is not
unique, but once $L$ is chosen, the measures,
$(\mathbb{N}_{x^{(2:d)}},x^{(2:d)}\in\mathbb{R}^{d-1})$ are determined but for
a $L$-neglectable set, i.e. a set $A$ such that

$\mathbb{E}_{x,\theta}\left[\int_{0}^{\infty}\mathbf{1}\left((X^{(1)}_{s},X^{(2:d)}_{s})\in
A\right){\rm d}Ls\right]=0.$

###### Proof of Theorem 1.

Let $x\in\mathbb{H}_{0\uparrow}$, the exit formula (12) now gives us, for
bounded measurable $g$ on $\mathbb{H}_{0\uparrow}$,

$\displaystyle\mathbb{E}_{x}\left[g(X_{G(\tau_{0}^{\vee})})\right]$
$\displaystyle=\mathbb{E}_{x}\left[\sum_{t\in\mathtt{G}}g(X_{t})\mathbf{1}(X^{(1)}_{t}+\epsilon^{(1)}_{t}(\zeta)<0)\right]$
$\displaystyle=\mathbb{E}_{x}\left[\int_{0}^{\infty}g(X_{t})\mathbb{N}_{X_{t}^{(2:d)}}(\epsilon^{(1)}_{t}(\zeta)<-y)\big{|}_{y=X^{(1)}_{t}}{\rm
d}L_{t}\right].$

In the above expression, the event in the excursion measure depends only on
the first coordinate of the excursion process. Hence it is independent of the
excursion measure index $X^{(2:d)}_{t}$ and can be written in terms of the
jump measure, say $\Pi_{-}^{(1)}$, of the descending ladder process associated
to $X^{(1)}$, which is known to be an $\alpha/2$-stable subordinator. More
precisely,

$\mathbb{N}_{x^{(2:d)}}(\epsilon^{(1)}_{t}(\zeta)<-y)=\Pi^{(1)}_{-}(y,\infty)=\frac{1}{\Gamma(1-(\alpha/2))}y^{-\alpha/2},\qquad
y>0.$

As such, we have

$\displaystyle\mathbb{E}_{x}\left[g(X_{G(\tau_{0}^{\vee})})\right]$
$\displaystyle=\mathbb{E}_{x}\left[\int_{0}^{\infty}g(X_{t})\frac{1}{\Gamma(1-(\alpha/2))}(X^{(1)}_{t})^{-\alpha/2}{\rm
d}L_{t}\right]$
$\displaystyle=\frac{1}{\Gamma(1-(\alpha/2))}\mathbb{E}_{x}\left[\int_{0}^{\infty}g(H_{t}^{(1)-},H^{(2:d)-}_{t})(H^{(1)-}_{t})^{-\alpha/2}{\rm
d}t\right]$ (13)
$\displaystyle=\frac{1}{\Gamma(1-(\alpha/2))}\int_{\\{y^{(1)}<x^{(1)}\\}}g(y^{(1)},y^{(2:d)})(y^{(1)})^{-\alpha/2}U_{x}^{-}({\rm
d}y),$

where $H^{(1)-}_{t}:=X^{(1)}_{L^{-1}_{t}}$ and
$H^{(2:d)-}_{t}:=X^{(2:d)}_{L_{t}^{-1}}$, $t\geq 0$, and we define the
descending ladder potential measure

(14) $U_{x}^{-}({\rm
d}y):=\int_{0}^{\infty}\mathbb{P}_{x}\left((H_{t}^{(1)-},H^{(2:d)-}_{t})\in{\rm
d}y\right){\rm d}t\quad y^{(1)}<x^{(1)},y^{(2:d)}\in\mathbb{R}^{d-1}.$

We note that the integral on the right-hand side of (13) is over the whole of
$\mathbb{R}^{d}$ with the restriction that $\\{y^{(1)}<x^{(1)}\\}$. In the
sequel, we will see many such integrals where it is implicit that we are
integrating over $\mathbb{R}^{d}$ and only the restriction over this space is
indicated in the delimiter.

In conclusion, we showed that the law of $X_{G(\tau_{0}^{\vee})}$ under
$\mathbb{P}_{x}$ (up to a constant) is nothing but $U_{x}^{-}({\rm
d}y)(y^{(1)})^{-\alpha/2}$. We therefore focus on identifying $U_{x}^{-}({\rm
d}y)$. The proof we follow is guided by the calculations leading to Theorem
1.1 in [9].

Let us define, for $x\in\mathbb{R}^{d}$ such that $x^{(1)}>r>0$, $\delta>0$,
and continuous, positive and bounded $f$ on
$\mathbb{R}^{+}\times\mathbb{R}^{d-1}$,

(15)
$\Delta_{r}^{\delta}f(x)=\frac{1}{\delta}\mathbb{E}_{x}\big{[}f(r,X^{(2:d)}_{G(\tau_{0}^{\vee})}),X^{(1)}_{G(\tau_{0}^{\vee})}\in[r-\delta,r]\big{]}.$

We want to establish the limit as $\delta\to 0$. In order to do so, we start
by conditioning on the first entry into the half-space
$\mathbb{H}_{r\downarrow}$, this can be done by appealing to Lemma 2, which we
have already proved. We have

$\displaystyle\Delta_{r}^{\delta}f(x)$
$\displaystyle=\frac{1}{\delta}\int_{\\{r-\delta\leq y^{(1)}\leq
r\\}}\mathbb{P}_{x}(X_{\tau_{r}^{\vee}}\in{\rm
d}y)\mathbb{E}_{y}\big{[}f(r,X^{(2:d)}_{G(\tau_{0}^{\vee})}),X^{(1)}_{G(\tau_{0}^{\vee})}\in[r-\delta,y^{(1)}]\big{]}$
$\displaystyle=\frac{1}{\delta}C_{\alpha,d}\int_{\\{r-\delta\leq y^{(1)}\leq
r\\}}{\rm
d}y|x-y|^{-d}\frac{|r-x^{(1)}|^{\alpha/2}}{|r-y^{(1)}|^{\alpha/2}}\mathbb{E}_{y}\big{[}f(r,X^{(2:d)}_{G(\tau_{0}^{\vee})}),X^{(1)}_{G(\tau_{0}^{\vee})}\in[r-\delta,y^{(1)}]\big{]}$
$\displaystyle=\frac{1}{\delta}C_{\alpha,d}\int_{\\{r-\delta\leq y^{(1)}\leq
r\\}}{\rm d}y|x-y|^{-d}\frac{|r-x^{(1)}|^{\alpha/2}}{|r-y^{(1)}|^{\alpha/2}}$
(16) $\displaystyle\hskip
113.81102pt\frac{1}{\Gamma(1-(\alpha/2))}\int_{\\{r-\delta<z^{(1)}\leq
y^{(1)}\\}}f(r,z^{(2:d)})(z^{(1)})^{-\alpha/2}U_{y}^{-}({\rm d}z),$

where the third equality is given by (13).

Next we provide a limiting approximation for the inner integral of (16) as
$\delta\to 0$. From now on we will write $\tilde{U}_{x}^{-}({\rm
d}z)=(z^{(1)})^{-\alpha/2}U_{x}^{-}({\rm d}z)$, $z\in\mathbb{R}^{d}$ and
recall that this is proportional to the law of $X_{G(\tau_{0}^{\vee})}$ under
$\mathbb{P}_{x}$. We claim that

(17) $\lim_{\delta\to
0}\sup_{\\{y\in{\mathbb{R}^{d}}:y^{(1)}\in(r-\delta,r]\\}}\bigg{\lvert}\frac{\int_{\\{r-\delta<z^{(1)}\leq
y^{(1)}\\}}f(r,z^{(2:d)})\tilde{U}_{y}^{-}({\rm
d}z)}{\int_{\\{r-\delta<z^{(1)}\leq y^{(1)}\\}}\tilde{U}_{y}^{-}({\rm
d}z)}-f(r,y^{(2:d)})\bigg{\rvert}=0.$

Suppose that $H_{r,\delta,\varepsilon}(y)\subset\mathbb{R}^{d}$ is the solid
cylinder defined by
$\\{x\in\mathbb{R}^{d}:|x^{(2:d)}-y^{(2:d)}|\leq\varepsilon\text{ and
}r-\delta\leq x^{(1)}\leq r\\}$, denote by
$H_{\varepsilon}(y)\subseteq\mathbb{R}^{d}$ the section of the cylinder given
by $\\{x^{(2:d)}\in\mathbb{R}^{d-1}:|x^{(2:d)}-y^{(2:d)}|\leq\varepsilon\\}$.

Thanks to the continuity of $f$, we can choose $\varepsilon$ such that

$\sup_{z\in
H_{\varepsilon}(y)}|f(r,y^{(2:d)})-f(r,z^{(2:d)})|<\varepsilon^{\prime},$

for some $\varepsilon^{\prime}\ll 1$. Then we have

$\displaystyle\sup_{\\{y\in{\mathbb{R}^{d}}:y^{(1)}\in(r-\delta,r]\\}}\bigg{\lvert}\frac{\int_{\\{r-\delta<z^{(1)}\leq
y^{(1)}\\}}f(r,z^{(2:d)})\tilde{U}_{y}^{-}({\rm
d}z)}{\int_{\\{r-\delta<z^{(1)}\leq y^{(1)}\\}}\tilde{U}_{y}^{-}({\rm
d}z)}-f(r,y^{(2:d)})\bigg{\rvert}$ (18)
$\displaystyle\leq\varepsilon^{\prime}+2\|f\|_{\infty}\sup_{\\{y\in{\mathbb{R}^{d}}:y^{(1)}\in(r-\delta,r]\\}}\frac{\int_{\\{r-\delta<z^{(1)}\leq
y^{(1)}\\}}\tilde{U}_{y}^{-}({\rm d}z)\mathbf{1}_{\\{z^{(2:d)}\notin
H_{\varepsilon}(y)\\}}}{\int_{\\{r-\delta<z^{(1)}\leq
y^{(1)}\\}}\tilde{U}^{-}_{y}({\rm d}z)}.$

Analysing the fraction on the right-hand side of (18), we notice that

$\displaystyle\frac{\int_{\\{r-\delta<z^{(1)}\leq
y^{(1)}\\}}\tilde{U}_{y}^{-}({\rm d}z)\mathbf{1}_{\\{z^{(2:d)}\notin
H_{\varepsilon}(y)\\}}}{\int_{\\{r-\delta<z^{(1)}\leq
y^{(1)}\\}}\tilde{U}^{-}_{y}({\rm d}z)}$
$\displaystyle=\frac{\int_{\\{r-\delta<z^{(1)}\leq y^{(1)}\\}}U_{y}^{-}({\rm
d}z)(z^{(1)})^{-\alpha/2}\mathbf{1}_{\\{z^{(2:d)}\notin
H_{\varepsilon}(y)\\}}}{\int_{\\{r-\delta<z^{(1)}\leq
y^{(1)}\\}}U^{-}_{y}({\rm d}z)(z^{(1)})^{-\alpha/2}}$
$\displaystyle=c_{\alpha}\frac{\int_{\\{r-\delta<z^{(1)}\leq
y^{(1)}\\}}U_{y}^{-}({\rm
d}z)\Pi_{-}^{(1)}(z^{(1)}-(r-\delta),\infty)\left(1-\frac{(r-\delta)}{z^{(1)}})\right)^{\alpha/2}\mathbf{1}_{\\{z^{(2:d)}\notin
H_{\varepsilon}(y)\\}}}{\int_{\\{r-\delta<z^{(1)}\leq
y^{(1)}\\}}U^{-}_{y}({\rm d}z)(z^{(1)})^{-\alpha/2}}$ $\displaystyle\leq
c_{\alpha}\frac{\int_{\\{r-\delta<z^{(1)}\leq y^{(1)}\\}}U_{y}^{-}({\rm
d}z)\Pi_{-}^{(1)}(z^{(1)}-(r-\delta),\infty)\left(1-\frac{(r-\delta)}{y^{(1)}}\right)^{\alpha/2}\mathbf{1}_{\\{z^{(2:d)}\notin
H_{\varepsilon}(y)\\}}}{\int_{\\{r-\delta<z^{(1)}\leq
y^{(1)}\\}}U^{-}_{y}({\rm d}z)(y^{(1)})^{-\alpha/2}}$ (19)
$\displaystyle=c_{\alpha}\frac{\mathbb{P}_{y}(H^{(2:d)-}_{\sigma_{r-\delta}^{\vee}-}\notin
H_{\varepsilon}(y))}{\left(1-\frac{(r-\delta)}{y^{(1)}}\right)^{-\alpha/2}(y^{(1)})^{-\alpha/2}U^{(1)-}_{y^{(1)}}(r-\delta,y^{(1)}]},$

where $c_{\alpha}=\Gamma(1-(\alpha/2))$,
$\sigma_{r-\delta}^{\vee}=\inf\\{t>0:H^{(1)-}_{t}<r-\delta\\}$ and, recalling
$y=(y^{(1)},y^{(2:d)})$,

$U_{y^{(1)}}^{(1)-}({\rm
d}z^{(1)}):=\int_{0}^{\infty}\mathbb{P}_{y}(H_{t}^{(1)-}\in{\rm d}z^{(1)}){\rm
d}t\quad z^{(1)}<y^{(1)}.$

As $y\in[r-\delta,r]\times H_{\varepsilon}(y)$, if we denote by
$\nu_{\varepsilon}=\inf\\{t>0:H_{t}^{(2:d)-}\notin H_{\varepsilon}(y)\\}$,
which only depends on $y$ through $y^{(2:d)}$, by the right-continuity of the
paths of $X$ we conclude that, for the numerator in (19),

$\lim_{\delta\to
0}\sup_{\\{y\in{\mathbb{R}^{d}}:y^{(1)}\in(r-\delta,r]\\}}\mathbb{P}_{y}(H^{(2:d)-}_{\sigma_{r-\delta}^{\vee}-}\notin
H_{\varepsilon}(y))\leq\lim_{\delta\to
0}\sup_{\\{y\in{\mathbb{R}^{d}}:y^{(1)}\in(r-\delta,r]\\}}\mathbb{P}_{y}(\nu_{\varepsilon}<\sigma_{r-\delta}^{\vee})=0.$

On the other hand, recalling (cf. Chapter 3 of [7]) that the known descending
ladder height occupation measure of a one-dimensional symmetric stable process
(cf. Chapter 3 of [7]) gives us

(20) $U_{y^{(1)}}^{(1)-}({\rm
d}v)=\frac{1}{\Gamma(\alpha/2)}(y^{(1)}-v)^{(\alpha/2)-1},\qquad v\leq
y^{(1)},$

for the denominator of (19) we have

$\displaystyle\left(1-\frac{(r-\delta)}{y^{(1)}}\right)^{-\alpha/2}(y^{(1)})^{-\alpha/2}U^{(1)-}_{y^{(1)}}(r-\delta,y^{(1)}]$
$\displaystyle=\left(1-\frac{(r-\delta)}{y^{(1)}}\right)^{-\alpha/2}(y^{(1)})^{-\alpha/2}\int_{0}^{y^{(1)}-(r-\delta)}\frac{u^{(\alpha/2)-1}}{\Gamma(\alpha/2)}{\rm
d}u$ $\displaystyle=\frac{1}{\Gamma(1+(\alpha/2))}.$

Hence the denominator of (19) is nothing but a constant.

The rigth-hand side of (19) therefore goes to zero uniformly for
$y^{(1)}\in(r-\delta,r]$ as $\delta\to 0$, which, from (18), means that we
have proved the claim (17).

Now returning to (16), using (17), for small $\delta$ we have

$\displaystyle\Delta_{r}^{\delta}f(x)$
$\displaystyle=\frac{1}{\delta}C_{\alpha,d}\int_{\\{r-\delta\leq y^{(1)}\leq
r\\}}{\rm d}y|x-y|^{-d}\frac{|r-x^{(1)}|^{\alpha/2}}{|r-y^{(1)}|^{\alpha/2}}$
$\displaystyle\hskip
56.9055pt\frac{1}{\Gamma(1-(\alpha/2))}\int_{\\{r-\delta<z^{(1)}\leq
y^{(1)}\\}}\bigg{(}f(r,z^{(2:d)})-f(r,y^{(2:d)})+f(r,y^{(2:d)})\bigg{)}\tilde{U}_{y}^{-}({\rm
d}z)$ $\displaystyle=D(\varepsilon)\Delta_{r}^{\delta}1(x)$ (21)
$\displaystyle\hskip 10.0pt+\frac{1}{\delta}C_{\alpha,d}\int_{\\{r-\delta\leq
y^{(1)}\leq r\\}}{\rm
d}y|x-y|^{-d}\frac{|r-x^{(1)}|^{\alpha/2}}{|r-y^{(1)}|^{\alpha/2}}f(r,y^{(2:d)})\mathbb{P}_{y}(X_{G(\tau_{0}^{\vee})}^{(1)}>r-\delta),$

where $|D(\varepsilon)|<\varepsilon$.

We deal with the two terms on the right-hand side of (21) sequentially. First,
we note from the original definition in (15) that

$\displaystyle\lim_{\delta\to 0}\Delta_{r}^{\delta}1(x)$
$\displaystyle=\frac{1}{\delta}\mathbb{P}_{x}(X^{(1)}_{G(\tau_{0}^{\vee})}\in[r-\delta,r])$
$\displaystyle=\lim_{\delta\to
0}\frac{1}{\delta}\int_{r-\delta}^{r}\frac{(x^{(1)}-u)^{(\alpha/2)-1}u^{-\alpha/2}}{\Gamma(\alpha/2)\Gamma(1-(\alpha/2))}\mathrm{d}u$
(22)
$\displaystyle=\frac{(x^{(1)}-r)^{(\alpha/2)-1}r^{-\alpha/2}}{\Gamma(\alpha/2)\Gamma(1-(\alpha/2))}.$

where, we have appealed to the one-dimensional analogue of (13); using
$U^{(1)-}_{y_{0}}$ in place of $U^{+}_{y}$. The limit in (22) tells us that
the first term on the right-hand side of (21) can be ignored as $\delta\to 0$.

For the second term on the right-hand side of (21), we note that its limit as
$\delta\to 0$ is equal to

$\displaystyle\lim_{\delta\to
0}\Delta_{r}^{\delta}f(x)=C_{\alpha,d}\int_{\mathbb{R}^{d-1}}\mathrm{d}y^{(2:d)}\frac{|r-x^{(1)}|^{\alpha/2}f(r,y^{(2:d)})}{|(x^{(1)}-r)^{2}+(x^{(2:d)}-y^{(2:d)})|^{d/2}}$
(23) $\displaystyle\hskip 199.16928pt\times\lim_{\delta\to
0}\frac{1}{\delta}\int_{r-\delta}^{r}{\rm
d}y^{(1)}\frac{\mathbb{P}_{y}(X_{G(\tau_{0}^{\vee})}^{(1)}>r-\delta)}{(r-y^{(1)})^{\alpha/2}}.$

For the limit on the right-hand of (23), we can compute

$\displaystyle\lim_{\delta\to 0}\frac{1}{\delta}\int_{r-\delta}^{r}{\rm
d}y^{(1)}\frac{\mathbb{P}_{y}(X_{G(\tau_{0}^{\vee})}^{(1)}>r-\delta)}{(r-y^{(1)})^{\alpha/2}}$
$\displaystyle=\lim_{\delta\to 0}\frac{1}{\delta}\int_{r-\delta}^{r}{\rm
d}y^{(1)}(r-y^{(1)})^{-\alpha/2}\int_{r-\delta}^{y^{(1)}}\frac{(y^{(1)}-u)^{(\alpha/2)-1}u^{-\alpha/2}}{\Gamma(\alpha/2)\Gamma(1-(\alpha/2))}\mathrm{d}u$
$\displaystyle=\frac{r^{-\alpha/2}}{(\alpha/2)\Gamma(\alpha/2)\Gamma(1-(\alpha/2))}\lim_{\delta\to
0}\frac{1}{\delta}\int_{r-\delta}^{r}{\rm
d}y^{(1)}(r-y^{(1)})^{-\alpha/2}(y^{(1)}-r+\delta)^{\alpha/2}$
$\displaystyle=\frac{r^{-\alpha/2}}{(\alpha/2)\Gamma(\alpha/2)\Gamma(1-(\alpha/2))}\lim_{\delta\to
0}\frac{1}{\delta}\int_{0}^{\delta}v^{-\alpha/2}(\delta-v)^{\alpha/2}\mathrm{d}v$
$\displaystyle=\frac{r^{-\alpha/2}}{\Gamma(1+(\alpha/2))\Gamma(1-(\alpha/2))}\int_{0}^{1}w^{-\alpha/2}(1-w)^{\alpha/2}\mathrm{d}w$
(24) $\displaystyle=r^{-\alpha/2}.$

where the first equality uses a calculation similar to (22).

Putting (24) back into (23), we have from (15) that

$\displaystyle\mathbb{E}_{x}\Big{[}f(X^{(1)}_{G(\tau_{0}^{\vee})},X^{(2:d)}_{G(\tau_{0}^{\vee})})\Big{]}$
$\displaystyle=\int_{0}^{x^{(1)}}\lim_{\delta\to
0}\Delta_{r}^{\delta}f(x)\mathrm{d}r$ (25)
$\displaystyle=C_{\alpha,d}\int_{0}^{x^{(1)}}\mathrm{d}r\int_{\mathbb{R}^{d-1}}\mathrm{d}y^{(2:d)}\frac{|r-x^{(1)}|^{\alpha/2}f(r,y^{(2:d)})}{r^{\alpha/2}|(x^{(1)}-r)^{2}+(x^{(2:d)}-y^{(2:d)})|^{d/2}}$

which completes the proof of Theorem 1. ∎

##  7 Proof of Theorem 2

There are two fromulae we will need to first derive in order to prove Theorem
2. For the first, just as we have constructed the directional excursions from
the minimum in the first axis, we can construct directional excursions from
the maximum in the first axis resulting in an ascending ladder process, say
$(H^{(1)+},H^{(2:d)+})$. If we define the renewal function of the ascending
ladder process of the latter, say $U^{+}_{x}(\mathrm{d}y)$, in the obvious way
(cf. (14)), then it is immediate from Theorem 1, (13) and symmetry that

(26) $U_{x}^{+}({\rm
d}z)=\frac{\Gamma(d/2)}{\pi^{d/2}\Gamma(\alpha/2)}\frac{(z^{(1)}-x^{(1)})^{\alpha/2}}{|x-z|^{d}}{\rm
d}z\quad\text{ for }x^{(1)}<z^{(1)},z^{(2:d)}\in\mathbb{R}^{d-1}.$

The second formula we need is provided by the following lemma, which gives us
an occupation functional under the excursion measure.

######  Lemma 4.

For $x\in\mathbb{R}^{d}$ and continuous $g:\mathbb{R}^{d}\to\mathbb{R}$ whose
support is compactly embedded in $\mathbb{H}_{r\uparrow}$

$\displaystyle\mathbb{N}_{x^{(2:d)}}\left(\int_{0}^{\zeta}g(x^{(1)}+\epsilon^{(1)}(u),\epsilon^{(2:d)}(u)){\rm
d}u\right)$
$\displaystyle=\int_{\\{x^{(1)}<z^{(1)}\\}}g(z)\frac{\Gamma(d/2)}{\pi^{d/2}\Gamma(\alpha/2)}\frac{(z^{(1)}-x^{(1)})^{\alpha/2}}{|x-z|^{d}}{\rm
d}z,$ (27) $\displaystyle=\int_{\\{x^{(1)}<z^{(1)}\\}}g(z)U_{x}^{+}({\rm
d}z).$

###### Proof.

By the last exit formula (12) we note that

$\displaystyle\mathbb{E}_{x}\left[\int_{0}^{\tau_{r}^{\vee}}g(X_{s}){\rm
d}s\right]$
$\displaystyle=\mathbb{E}_{x}\left[\sum_{g\in\mathtt{G}}\mathbf{1}_{\\{\underline{X}_{g}^{(1)}>r\\}}\int_{g}^{d}g(\underline{X}_{g}^{(1)}+\epsilon^{(1)}_{g}(s),\epsilon^{(2:d)}_{g}(s))\mathrm{d}s\right]$
$\displaystyle=\mathbb{E}_{x}\left[\int_{0}^{\infty}\mathbf{1}_{\\{\underline{X}_{s}^{(1)}>r\\}}\mathbb{N}_{X^{(2:d)}_{s}}\left(\int_{0}^{\zeta}g(y+\epsilon^{(1)}(u),\epsilon^{(2:d)}(u))\mathrm{d}u\right)\Big{|}_{y=X^{(1)}_{s}}{\rm
d}L_{s}\right]$
$\displaystyle=\mathbb{E}_{x}\left[\int_{0}^{\infty}\mathbf{1}_{\\{H^{(1)-}_{s}>r\\}}\mathbb{N}_{H^{(2:d)-}_{s}}\left(\int_{0}^{\zeta}g(y+\epsilon^{(1)}(u),\epsilon^{(2:d)}(u))\mathrm{d}u\right)\Big{|}_{y=H^{(1)-}_{s}}{\rm
d}s\right]$ $\displaystyle=\int_{\\{r<z^{(1)}<x^{(1)}\\}}U_{x}^{-}({\rm
d}z)\mathbb{N}_{z^{(2:d)}}\left(\int_{0}^{\zeta}g(z^{(1)}+\epsilon^{(1)}(u),\epsilon^{(2:d)}(u)){\rm
d}u\right)$ (28)
$\displaystyle=\int_{\\{r<z^{(1)}<x^{(1)}\\}}\tilde{U}_{x}^{-}({\rm
d}z)(z^{(1)})^{\alpha/2}\mathbb{N}_{z^{(2:d)}}\left(\int_{0}^{\zeta}g(z^{(1)}+\epsilon^{(1)}(u),\epsilon^{(2:d)}(u)){\rm
d}u\right).$

Next, we want to make use of an argument very similar to (17) replacing the
role of $f$ there by the function

$\kappa(z):=(z^{(1)})^{\alpha/2}\mathbb{N}_{z^{(2:d)}}\left(\int_{0}^{\zeta}g(z^{(1)}+\epsilon(u),\epsilon^{(2:d)}(u))\right).$

Note that we cannot use (17) directly as the function $f$ there cannot take
the form of $\kappa$. Nonetheless, the result in (17) still holds when we make
the aforesaid exchange of functions, providing we can demonstrate that
$\kappa(z)$ is bounded and continuous in $z$.

Boundedness of $\kappa$ is ensured by the compact support property of $g$.
Continuity of $\kappa$ is ensured by dominated convergence, using the compact
embedding of the support of $g$, the boundedness and continuity of $g$ and the
fact that

$\mathbb{N}_{z^{(2:d)}}\left(\int_{0}^{\zeta}g(z^{(1)}+\epsilon(u),\epsilon^{(2:d)}(u))\right)=\mathbb{N}_{0}\left(\int_{0}^{\zeta}g(z^{(1)}+\epsilon(u),z^{(2:d)}+\epsilon^{(2:d)}(u))\right).$

Hence using $\kappa$ in place of $f$ in (17) we have on the one hand that

(29) $\displaystyle(x^{(1)})^{\alpha/2}\mathbb{N}_{x^{(2:d)}}$
$\displaystyle\left(\int_{0}^{\zeta}g(x^{(1)}+\epsilon(u),\epsilon^{(2:d)}(u)){\rm
d}u\right)=\lim_{r\uparrow
x^{(1)}}\frac{\mathbb{E}_{x}\left[\int_{0}^{\tau_{r}^{\vee}}g(X_{s}){\rm
d}s\right]}{\int_{\\{r<z^{(1)}<x^{(1)}\\}}\tilde{U}_{x}^{-}({\rm d}z)}.$

Although we have not calculated an expression for the numerator on the right-
hand side of (29), it is easily derived from Lemma 1. Indeed, it is easy to
derive from the underlying Lévy system of $X$ that, for
$x,y\in\mathbb{H}_{r\uparrow}$ and $z\in\mathbb{H}_{r\downarrow}$,

$\displaystyle\Xi_{x}(\bullet,y,z)\mathrm{d}y\mathrm{d}z=\mathbb{E}_{x}\left[\int_{0}^{\tau_{r}^{\vee}}\mathbf{1}_{\\{X_{s}\in\mathrm{d}y\\}}{\rm
d}s\right]\Pi(\mathrm{d}z-y)$

As such, we have, for $x,y\in\mathbb{H}_{r\uparrow}$ and
$z\in\mathbb{H}_{r\downarrow}$,

$\displaystyle h_{r}^{\vee}(x,y)$
$\displaystyle:=\mathbb{E}_{x}\left[\int_{0}^{\tau_{r}^{\vee}}\mathbf{1}_{\\{X_{s}\in\mathrm{d}y\\}}{\rm
d}s\right]$ (30)
$\displaystyle=E_{\alpha,d}|x-y|^{\alpha-d}\int_{0}^{\zeta_{r}^{\vee}(x,y)}(u+1)^{-d/2}u^{\alpha/2-1}{\rm
d}u,$

where $E_{\alpha,d}=\Gamma(d/2)/2^{\alpha}\pi^{d/2}\Gamma(\alpha/2)^{2}$ and
we recall

$\zeta_{r}^{\vee}(x,y):=\frac{4(x^{(1)}-r)(y^{(1)}-r)}{|x-y|^{2}}.$

Therefore, on the other hand we have that

$\displaystyle\lim_{r\uparrow
x^{(1)}}\frac{\mathbb{E}_{x}\left[\int_{0}^{\tau_{r}^{\vee}}g(X_{s}){\rm
d}s\right]}{\int_{\\{r<z^{(1)}<x^{(1)}\\}}\tilde{U}_{x}^{-}({\rm d}z)}$
$\displaystyle=\lim_{r\uparrow
x^{(1)}}\frac{\int_{\\{x^{(1)}<z^{(1)}\\}}\mathbf{1}_{\\{r<z^{(1)}\\}}g(z)h_{r}^{\vee}(x,z){\rm
d}z}{\int_{\\{r<z^{(1)}<x^{(1)}\\}}(z^{(1)})^{-\alpha/2}U_{x}^{(1)-}({\rm
d}z^{(1)})}$ $\displaystyle=\lim_{r\uparrow
x^{(1)}}\frac{\int_{\\{x^{(1)}<z^{(1)}\\}}\mathbf{1}_{\\{r<z^{(1)}\\}}g(z)E_{\alpha,d}|x-z|^{\alpha-d}\int_{0}^{\zeta_{r}^{\vee}(x,z)}(u+1)^{-d/2}u^{\alpha/2-1}{\rm
d}u{\rm
d}z}{\Gamma(\alpha/2)^{-1}\int_{r}^{x^{(1)}}u^{-\alpha/2}(x^{(1)}-u)^{\alpha/2-1}{\rm
d}u}$
$\displaystyle=E_{\alpha,d}\Gamma(\alpha/2)\int_{\\{x^{(1)}<z^{(1)}\\}}g(z)|x-z|^{\alpha-d}\lim_{r\uparrow
x^{(1)}}\frac{\int_{0}^{\zeta_{r}^{\vee}(x,z)}(u+1)^{-d/2}u^{\alpha/2-1}{\rm
d}u}{\int_{r}^{x^{(1)}}u^{-\alpha/2}(x^{(1)}-u)^{\alpha/2-1}{\rm d}u}{\rm d}z$
$\displaystyle=E_{\alpha,d}\Gamma(\alpha/2)(x^{(1)})^{\alpha/2}\int_{\\{x^{(1)}<z^{(1)}\\}}g(z)|x-z|^{\alpha-d}\left(\frac{4(z^{(1)}-x^{(1)})}{|x-z|^{2}}\right)^{\alpha/2}{\rm
d}z$ (31)
$\displaystyle=2^{\alpha}E_{\alpha,d}\Gamma(\alpha/2)(x^{(1)})^{\alpha/2}\int_{\\{x^{(1)}<z^{(1)}\\}}g(z)\frac{(z^{(1)}-x^{(1)})^{\alpha/2}}{|x-z|^{d}}{\rm
d}z.$

where we have used (20) and, in the last equality, we used L’Hôspital rule.
Equating (29) with (31) and noting that

$2^{\alpha}E_{\alpha,d}\Gamma(\alpha/2)=\frac{\Gamma(d/2)}{\pi^{d/2}\Gamma(\alpha/2)}$

completes the proof. ∎

######  Remark 4.

We can remove the assumption that $g$ is compactly supported in the previous
Lemma, leaving only the requirement that $g$ is continuous and bounded by
appealing to a sequence of increasing functions $g_{n}\uparrow g$ and monotone
convergence across the equality (27) applied to the sequence $g_{n}$.

###### Proof of Theorem 2.

Using the fact that $X^{(1)}$ does not creep downwards, from the underlying
Lévy system we get that

(32)
$\displaystyle\mathbb{E}_{w}\left[f(X_{G(\tau_{r}^{\vee})})g(X_{\tau_{r}^{\vee}-})h(X_{\tau_{r}^{\vee}})\right]=\mathbb{E}_{w}\left[\int_{0}^{\tau_{r}^{\vee}}f(X_{G(t)})g(X_{t})k(X_{t}){\rm
d}t\right],$

where $k(y)=\int_{\\{y^{(1)}+u^{(1)}<r\\}}\Pi({\rm d}u)h(y+u)$.

Putting together identity (32) with Lemma 4, and rearranging the integrals we
obtain

$\displaystyle\mathbb{E}_{w}\left[\int_{0}^{\tau_{r}^{\vee}}f(X_{G(t)})g(X_{t})k(X_{t}){\rm
d}t\right]$
$\displaystyle\quad=\int_{\\{r<x^{(1)}<w^{(1)}\\}}f(x)U_{w}^{-}({\rm
d}x)\mathbb{N}_{x^{(2:d)}}^{-}\left(\int_{0}^{\infty}(g\cdot
k)(x^{(1)}+\epsilon^{(1)}(s),\epsilon^{(2:d)}(s))\mathrm{d}s\right)$ (33)
$\displaystyle\quad=\int_{\\{r<x^{(1)}<w^{(1)}\\}}f(x)U_{w}^{-}({\rm
d}x)\int_{\\{x^{(1)}<y^{(1)}\\}}g(y)U_{x}^{+}({\rm
d}y)\int_{\\{y^{(1)}+u^{(1)}<r\\}}\Pi({\rm d}u)h(y+u)$
$\displaystyle\quad=\int_{\\{r<x^{(1)}<w^{(1)}\\}}\int_{\\{x^{(1)}<y^{(1)}\\}}\int_{\\{z^{(1)}<r\\}}f(x)g(y)h(z)U_{w}^{-}({\rm
d}x)U_{x}^{+}({\rm d}y)K_{\alpha,d}|z-y|^{-(\alpha+d)}.$

where
$K_{\alpha,d}=2^{\alpha}\pi^{-d/2}\Gamma((d+\alpha)/2)/|\Gamma(-\alpha/2)|$
and we have used (1), from which the result follows. ∎

######  Remark 5.

The equality given by (33) agrees with a general formula given by [13], albeit
in the latter, the factors where not clearly introduced as the potential
measures of the ascending and descending ladder processes.

##  8 Concluding remarks

Many of the arguments we have given here are adapted from [9], which took a
perspective on the stable process via the Lamperti–Kiu transform. Many of the
calculations in [9] demonstrate that the nature of the underlying Markov
Additive Process (MAP) captures the behaviour of the stable process via a
generalised spherical polar coordinate system. The reason why we have been
able to secure similarly explicit identities in relation to the hyperplane in
this paper is because we have decomposed the stable process in an orthogonal
coordinate system, that is, $X=(X^{(1)},X^{(2:d)})$. Moreover, the pair
$(X^{(1)},X^{(2:d)})$ may also be considered as a MAP and, although there is
no time change involved as with the Lamperti–Kiu transform, the underlying
principles of MAP fluctuation theory is still what drives our proofs to
conclusion here, just as in [9].

The assimilation of our ‘orthogonal’ path decomposition in this article with
what we know for ‘spherical polar’ path decompositions of the stable process
goes further than what we have have demonstrated in the previous sections.
Below are but two examples of what more can be done. We leave them as
exercises to the reader, with full proofs available in [12].

### Orthogonal Wiener–Hopf factorisation of the stable process

The Wiener–Hopf factorisation for one-dimensional Lévy processes has
undisputedly played a crucial role in understanding its fluctuations. The
factorisation offers an actual decomposition of the Lévy–Khintchine formula
into two factors which correspond to the exponents of the ascending and
descending ladder processes. In essence the factorisation is the result of
splitting the path of the Lévy process over an independent exponentially
distributed time at its minimum (or maximum) and applying duality to the post
minimum (or maximum).

In higher dimensions it is less clear what such a factorisation should look
like, however with an appropriate coordinate system, such as spherical polar
or orthogonal coordinates one may still seek to split the path at a minimum
(or maximum) in one of the coordinate directions.

Let us introduce the following potential operators, for $f$ a continuous
function with compact support on $\mathbb{R}^{d-1}$, $\theta\in\mathbb{R}$,
and $x^{(2:d)}\in\mathbb{R}^{d-1}$

$\mathbf{R}_{\theta}[f](x):=\mathbb{E}_{0,x^{(2:d)}}\left[\int_{0}^{\infty}\mathbf{e}^{{\rm
i}\theta X_{t}^{(1)}}f(X_{t}^{(2:d)}){\rm d}t\right].$

Now define the potentials of the descending and ascending ladder processes,
$(H^{(1)-},H^{(2:d)-})$ and $(H^{(1)+},H^{(2:d)+})$ respectively (see the
discussion at the start of Section 7 for the latter), by

$\hat{\bm{\rho}}_{\theta}[f](x):=\mathbb{E}_{0,x^{(2:d)}}\left[\int_{0}^{\infty}\mathbf{e}^{{\rm
i}\theta H_{t}^{(1)-}}f(H_{t}^{(2:d)-}){\rm d}t\right],$

and

$\bm{\rho}_{\theta}[f](x):=\mathbb{E}_{0,x^{(2:d)}}\left[\int_{0}^{\infty}\mathbf{e}^{{\rm
i}\theta H_{t}^{(1)+}}f(H_{t}^{(2:d)+}){\rm d}t\right].$

The density of the potential of a $d$-dimensional stable process is a well
known object (cf. Chapter 3 of [7]), and allows us to write the operator
$\mathbf{R}_{\theta}[f](x)$ as follows,

$\mathbf{R}_{\theta}[f](x)=2^{-\alpha}\pi^{-d/2}\frac{\Gamma((d-\alpha)/2)}{\Gamma(\alpha/2)}\int_{\mathbb{R}^{d}}\mathbf{e}^{{\rm
i}\theta y^{(1)}}f(y^{(2:d)})|x-y|^{\alpha-d}{\rm d}y.$

######  Theorem 4 (Orthogonal Wiener-Hopf factorisation).

For a continuous compactly supported function
$f:\mathbb{R}^{d-1}\to\mathbb{R}$, $\theta\in\mathbb{R}$ and
$x=(0,x^{(2:d)}),\,x^{(2:d)}\in\mathbb{R}^{d-1}$ the potential operator
$\mathbf{R}$ corresponding to the isotropic $\alpha$-stable can be represented
as the composition of ascending and descending potential operators as follows

(34)
$\mathbf{R}_{\theta}[f](x)=\hat{\bm{\rho}}_{\theta}\left[\bm{\rho}_{\theta}[f]\right](x).$

Moreover, we have

$\bm{\rho}_{\theta}[f](x)=\int_{\\{y^{(1)}>0\\}}\mathbf{e}^{{\rm i}\theta
y^{(1)}}f(y^{(2:d)})\frac{\Gamma(d/2)}{\pi^{d/2}\Gamma(\alpha/2)}\frac{(y^{(1)})^{\alpha/2}}{|x-y|^{d}}\mathrm{d}y$

and

$\hat{\bm{\rho}}_{\theta}[f](x)=\bm{\rho}_{-\theta}[f](x).$

### Conditioning to hit plane continuously from one side

The following results are an extension of those in [6], where the law of the
point of closest reach to the origin is used to condition the isotropic
$\alpha$-stable Lévy process $(X,\mathbb{P})$ to continuously approach a patch
on the surface of the unit sphere from outside of the sphere. In the current
setting, we can use the law of $X_{G(\tau_{0}^{\vee})}$ to condition
$(X,\mathbb{P})$ to continuously approach a patch on the hyperplane
$\mathbb{H}_{0}=\\{x\in\mathbb{R}^{d}:x^{(1)}=0\\}$ from one side.

Let us consider a Borel set $S\subseteq\mathbb{H}_{0}$ (which may be the whole
of $\mathbb{H}_{0}$) such that it has strictly positive Lebesgue surface
measure, written $\ell_{d-1}$, on $\mathbb{H}_{0}$ or it is a point. We aim to
make sense of the law of $X$ conditioned to approach $S$ continuously from the
half-space $\mathbb{H}_{0\uparrow}:=\\{x\in\mathbb{R}^{d}:x^{(1)}>0\\}$.
Assume momentarily that $S$ is not a point, but has strictly positive Lebesgue
measure. Define for $\epsilon>0$ $S_{\epsilon}=(0,\epsilon)\times S$. Consider
the event $C_{\epsilon}^{\vee}=\\{X_{G(\tau_{0}^{\vee})}\in S_{\epsilon}\\}$.
We are interested in the asymptotic conditioning

(35) $\mathbb{P}_{x}^{\vee}(A,t<\zeta):=\lim_{\beta\to 0}\lim_{\epsilon\to
0}\mathbb{P}_{x}(A,t<\tau_{\beta}^{\vee}|C_{\epsilon}^{\vee}),\quad
x\in\mathbb{H}_{0\uparrow},$

where $A\in\mathcal{F}_{t}=\sigma(X_{u},u\leq t)$ and $\zeta$ is the lifetime
of the process on $\mathbb{D}(\mathbb{R}^{+}\times\mathbb{R}^{d})$, which we
recall is the space of càdlàg paths from $\mathbb{R}^{+}$ to $\mathbb{R}^{d}$
with appended cemetery state, which is where the path is sent at its lifetime.
If $S=\\{v\\}\in\mathbb{H}_{0}$, the sets $S_{\epsilon}$ is adapted to
converge to $S$ on $\mathbb{H}_{0\uparrow}$. For instance one could take
$S_{\epsilon}=(0,\epsilon)\times\mathbb{B}^{(d-1)}(v,\epsilon)$ where
$\mathbb{B}^{(d-1)}(v,\epsilon)=\\{y^{(2:d)}\in\mathbb{H}_{0}:|y^{(2:d)}-v|<\epsilon\\}$.

In order to state our main result let us define the following functions for
$x\notin\mathbb{H}_{0}$,

(36)
$H_{S}(x)=\begin{cases}\int_{S}\big{|}x^{(1)}\big{|}^{\alpha/2}\bigg{|}\left(x^{(1)}\right)^{2}+|x^{(2:d)}-y^{(2:d)}|^{2}\bigg{|}^{-d/2}\ell_{d-1}(\mathrm{d}y^{(2:d)})&\text{
if }\ell_{d-1}(S)>0,\\\
\big{|}x^{(1)}\big{|}^{\alpha/2}\bigg{|}\left(x^{(1)}\right)^{2}+|x^{(2:d)}-v^{(2:d)}|^{2}\bigg{|}^{-d/2}&\text{
if }S=\\{v\\}.\end{cases}$

Note that $H_{S}$ is well defined when $S=\mathbb{H}_{0}$.

######  Theorem 5.

Let $S\subseteq\mathbb{H}_{0}$ such that $\ell_{d-1}(S)>0$ or $S=\\{v\\}$ a
point on $\mathbb{H}_{0}$. Then for all points of issue
$x\notin\mathbb{H}_{0}$ we have as a change of measure on
$\mathbb{D}(\mathbb{R}^{+}\times\mathbb{R}^{d})$ such that, for all
$A\in\mathcal{F}_{t}$,

(37)
$\mathbb{P}^{\vee}_{x}(A,\,t<\zeta)=\mathbb{E}_{x}\left[\mathbf{1}_{(A,\,t<\tau_{0}^{\vee})}\frac{H_{S}(X_{t})}{H_{S}(x)}\right],\qquad
x\in\mathbb{H}_{0\uparrow},$

where $H_{S}(x)$ is given in (36).

## Acknowledgements

SM is supported by a scholarship from the EPSRC Centre for Doctoral Training
in Statistical Applied Mathematics at Bath (SAMBa), under the project
EP/S022945/1.

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