††thanks: Version of

# Approximation of skew Brownian motion by snapping-out Brownian motions

Adam Bobrowski  Lublin University of Technology
Nadbystrzycka 38A
20-618 Lublin, Poland<EMAIL_ADDRESS>Elżbieta Ratajczyk  Lublin
University of Technology
Nadbystrzycka 38A
20-618 Lublin, Poland<EMAIL_ADDRESS>
###### Abstract.

We elaborate on the theorem saying that as permeability coefficients of
snapping-out Brownian motions tend to infinity in such a way that their ratio
remains constant, these processes converge to a skew Brownian motion. In
particular, convergence of the related semigroups, cosine families and
projections is discussed.

###### Key words and phrases:

skew and snapping-out Brownian motion, invariant subspaces, projection,
complemented spaces, transmission conditions

###### 1991 Mathematics Subject Classification:

35B06, 46E05, 47D06,
47D07, 47D09

##  1\. Introduction

###  1.1. Skew Brownian motion

As described in [lejayskew] ‘The skew Brownian Motion appeared in the ’70 in
[ito, walsh] as a natural generalization of the Brownian motion: it is a
process that behaves like a Brownian motion except that the sign of each
excursion is chosen using an independent Bernoulli random variable’. This
characteristic property of skew Brownian motion is expressed in the boundary
conditions

$f^{\prime\prime}(0+)=f^{\prime\prime}(0-)\quad\text{ and }\quad\beta
f^{\prime}(0-)=\alpha f^{\prime}(0+),$ (1.1)

where $\alpha$ and $\beta$ are non-negative parameters such that
$\alpha+\beta>0$. More precisely, the skew Brownian motion is an honest Feller
process on $\mathbb{R}$, and as such can be described by means of its
generator, that is, a Laplace operator, say,
$A^{\textnormal{{skew}}}_{\alpha,\beta}$, in the space
$\mathfrak{C}[-\infty,\infty]$ ($[-\infty,\infty]$ is a two-point
compactification of $\mathbb{R}$) of continuous functions on $\mathbb{R}$ that
have finite limits at $\infty$ and $-\infty$. The domain of
$A^{\textnormal{{skew}}}_{\alpha,\beta}$ is composed of functions satisfying
the following three conditions:

  * (a)

$f$ is twice continuously differentiable in both $(-\infty,0]$ and
$[0,\infty)$, separately, with left-hand and right-hand derivatives at $x=0$,
respectively,

  * (b)

both the limits $\lim_{x\to\infty}f^{\prime\prime}(x)$ and
$\lim_{x\to-\infty}f^{\prime\prime}(x)$ exist and are finite (it follows that,
in fact, they have to be equal to $0$), and

  * (c)

conditions (1.1) hold; note that the first of them implies that, although
$f^{\prime}(0)$ need not exist, it is meaningful to speak of
$f^{\prime\prime}(0)$.

Furthermore, we define

$A^{\textnormal{{skew}}}_{\alpha,\beta}f\coloneqq f^{\prime\prime}.$

Even though, for reasons that will become obvious in what follows, it is
convenient to consider all non-negative parameters $\alpha$ and $\beta$ in
(1.1) (such that $\alpha+\beta>0$), their probabilistic meaning is more clear
when they are normalized by dividing both of them by $\alpha+\beta$. Then the
new $\alpha$ is the probability that the sign of excursion is chosen to be
positive, and the new $\beta$ is the probability that the sign is negative.

The paper by A. Lejay cited above discusses a number of constructions of the
skew Brownian motion that appeared in the literature since the process was
first discovered, and various contexts in which this process and its
generalizations are studied; see also [yor97]*p. 107 and [manyor]*pp. 115–117.
It is well-known, for example, that skew Brownian motion can be obtained in
the limit procedure of Friedlin and Wentzell’s averaging principle
[fwbook]—see [fw]*Thm. 5.1, comp. [emergence]*Eq. (3.1). Much more recently,
in [abtk], a link has been provided between skew Brownian motion and kinetic
models of motion of a phonon involving an interface, of the type studied in
[tomekkinetic, tomekkinetic3, tomekkinetic2, tomekkinetic1], and the telegraph
process with elastic boundary at the origin [wlosi, wlosi1].

###  1.2. Snapping-out Brownian motion

The goal of this paper is to use the recent results of [aberman] to strengthen
the limit theorem of [tombatty] (see also [knigaz]*Chapter 11) saying that the
skew Brownian motion can be obtained as a limit of _snapping out Brownian
motions_.

The snapping out Brownian motion is a diffusion on two half-lines,
$(-\infty,0)$ and $(0,\infty)$, separated by a semi-permeable membrane located
at $0$. It can be described by a Feller semigroup of operators in the space

$\mathfrak{C}(\mathbb{R}_{\sharp})$

of continuous functions on

$\mathbb{R}_{\sharp}\coloneqq[-\infty,0-]\cup[0+,\infty]$

where $0-$ and $0+$ are two distinct points, representing positions to the
immediate left and to the immediate right of the membrane; alternatively,
members of $\mathfrak{C}(\mathbb{R}_{\sharp})$ can be seen as continuous
functions on $(-\infty,0)\cup(0,\infty)$ that have finite limits at
$\pm\infty$ and one-sided finite limits at $0$. The membrane, in turn, is
characterized by two non-negative parameters, say, $\alpha$ and $\beta$,
describing its permeability for a particle diffusing from the left to the
right and from the right to the left, respectively. More precisely, given such
$\alpha$ and $\beta$, we define the generator of the snapping out Brownian
motion as follows: it is the operator $A_{\alpha,\beta}^{\textnormal{s-o}}$ in
$\mathfrak{C}(\mathbb{R}_{\sharp})$ given by

$A_{\alpha,\beta}^{\textnormal{s-o}}f\coloneqq f^{\prime\prime},$

on the domain composed of twice continuously differentiable
$f\in\mathfrak{C}(\mathbb{R}_{\sharp})$ such that
$f^{\prime},f^{\prime\prime}\in\mathfrak{C}(\mathbb{R}_{\sharp})$ and

$\displaystyle\begin{split}f^{\prime}(0-)&=\alpha(f(0+)-f(0-)),\\\
f^{\prime}(0+)&=\beta(f(0+)-f(0-)).\end{split}$ (1.2)

These _transmission conditions_ are well-known in the literature and can be
plausibly interpreted: according to Newton’s Law of Cooling, the temperature
at $x=0$ changes at a rate proportional to the difference of temperatures on
either sides of the membrane, see [crank, p. 9]. In this context, J. Crank
uses the term _radiation boundary condition_. In the context of passing or
diffusing through membranes, analogous transmission conditions were introduced
by J.E. Tanner [tanner, eq. (7)], who studied diffusion of particles through a
sequence of permeable barriers (see also Powles et al. [powles, eq. (1.4)],
for a continuation of the subject). In [andrews] (see e.g. eq. (4) there)
similar conditions are used in describing absorption and desorption phenomena.
We refer also to [fireman], where a compartment model with permeable walls
(representing e.g., cells, and axons in the white matter of the brain in
particular) is analyzed, and to eq. [42] there. Among more recent literature
involving relatives of (1.2), one should mention the model of inhibitory
synaptic receptor dynamics of P. Bressloff [bressloff2023]; see also
[bressloff2022] and references given in these papers.

In [bobmor] (see also [knigaz]*Chapters 4 and 11), relations (1.2) were put in
the context of Feller–Wentzel boundary conditions, and this allowed describing
the related process by means of the _Lévy local time_. Later, in [lejayn], a
stochastic construction was provided, and the term _snapping out Brownian
motion_ was coined. To summarize these findings, we note that conditions (1.2)
are akin to the elastic barrier condition: An elastic Brownian motion on
$\mathbb{R}^{+}:=[0,\infty)$ (see e.g. [ito, karatzas]) is the process with
generator $Gf=\frac{1}{2}f^{\prime\prime}$ defined on the domain composed of
$f\in\mathfrak{C}^{2}[0,\infty]$ satisfying the _Robin boundary condition_
(known also as _elastic barrier condition_):

$f^{\prime}(0+)=\beta f(0+).$

In this process, the state-space is $\mathbb{R}^{+}$, and each particle
performs a standard Brownian motion while away from the barrier $x=0.$ Each
time a particle touches the barrier it is reflected, but its time spent at the
boundary, the local time mentioned above, is measured and after an exponential
time with parameter $\beta$ with respect to the local time, the particle is
killed and no longer observed. The second condition in (1.2) expresses the
fact that in the snapping out Brownian motion, each particle diffusing on the
right half-axis, instead of being killed after the time described, is
transferred to the other side of the membrane $x=0$. In other words, the
particle after some time spent ‘at’ the right side of the boundary, permeates
to the the left side. Similarly, a particle on the left side of the boundary
permeates through the membrane after a random time characterized by the
coefficient $\alpha$. In particular, we see that $\alpha$ and $\beta$ are
permeability coefficients: the larger they are the shorter is the time for
particles to diffuse through the membrane.

###  1.3. Approximation of skew B.M. by snapping out B.M.

Since $\alpha$ and $\beta$ in (1.2) are permeability coefficients, it may be
thought reasonable that letting them to be infinite should lead to the case in
which the membrane is completely permeable, that is, to the standard Brownian
motion. As it turns out, however, the matter is not so simple: in the limit
some kind of asymmetry in the way particles pass to the left and to the right
through the membrane remains.

To explain this phenomenon in more datail, let us replace $\alpha$ and $\beta$
in the transmission conditions (1.2) by $n\alpha$ and $n\beta$, respectively,
and let $n\to\infty$. Heuristically, it is then clear that the limit
conditions should read

$f(0+)=f(0-)\quad\text{ and }\quad\beta f^{\prime}(0-)=\alpha f^{\prime}(0+).$

The first of these relations tells us that we should work with
$f\in\mathfrak{C}(\mathbb{R}_{\sharp})$ that are continuous at $x=0$, that is
with $f\in\mathfrak{C}[-\infty,\infty]$, and the other is precisely the second
of conditions that characterize the skew Brownian motion, see (1.1). Hence, we
anticipate a theorem saying that a skew Brownian motion is a limit of snapping
out Brownian motions, provided that permeability coefficients $\alpha$ and
$\beta$ of the semi-permeable membrane converge to infinity whereas their
ratio remains constant.

As we know from [tombatty], these intuitions can be transformed into a formal
theorem saying that for any $s>0$,

$\displaystyle\lim_{n\to\infty}\sup_{t\in[0,s]}\big{\|}\mathrm{e}^{tA_{n\alpha,n\beta}^{\textnormal{s-o}}}f-\mathrm{e}^{tA^{\textnormal{{skew}}}_{\alpha,\beta}}f\big{\|}=0,\qquad
f\in\mathfrak{C}[-\infty,\infty],$ (1.3)

where $\mathfrak{C}[-\infty,\infty]$ is seen as the subspace of
$\mathfrak{C}(\mathbb{R}_{\sharp})$ composed of functions $f$ such that
$f(0-)=f(0+)$, $\\{\mathrm{e}^{tA_{\alpha,\beta}^{\textnormal{s-o}}},\,t\geq
0\\}$ is the Feller semigroup describing snapping-out Brownian motion, and
$\\{\mathrm{e}^{tA^{\textnormal{{skew}}}_{\alpha,\beta}},\,t\geq 0\\}$ is that
describing the skew Brownian motion. In view of the
Trotter–Sova–Kurtz–Mackevic̆ius theorem [kallenbergnew], this result can be
seen as expressing a convergence of the random processes involved.

###  1.4. The goal of the paper

The goal of the paper is to strengthen the result of [tombatty] by exhibiting
a number of phenomena accompanying convergence (1.3). These can be summarized
as follows.

  * (A)

_Convergence of cosine families._ As proved in [tombatty], each
$A_{\alpha,\beta}^{\textnormal{s-o}}$ generates not only a semigroup but also
a cosine family, and the latter is in a sense more fundamental than the
semigroup. We will show that on $\mathfrak{C}[-\infty,\infty]$ (but not
outside of this space) not only the semigroups generated by
$A_{n\alpha,n\beta}^{\textnormal{s-o}}$ converge; so do also the related
cosine families. Moreover, the limit is in fact uniform with respect to
$t\in\mathbb{R}$, and for Lipschitz continuous functions the rate of
convergence can be estimated—see Section 2, and Theorem 2.1 (b) and Remarks 1
and 3 in particular.

  * (B)

_Convergence of semigroups outside of $\mathfrak{C}[-\infty,\infty]$._ As a
consequence of (A), for $f\not\in\mathfrak{C}[-\infty,\infty]$ the limit
$\lim_{n\to\infty}\mathrm{e}^{tA_{n\alpha,n\beta}^{\textnormal{s-o}}}f$ still
exists for all $t>0$, but is merely uniform with respect to $t$ in compact
subintervals of $(0,\infty)$—see Theorem 2.1 (c).

  * (C)

_Asymptotic behavior._ We will show that the cosine families generated by
$A_{\alpha,\beta}^{\textnormal{s-o}}$ have the same time averages as those
generated by $A^{\textnormal{{skew}}}_{\alpha,\beta}$—see Section 3.

  * (D)

_Convergence of projections._ It is also proved in [aberman] that each
transmission condition (1.2) shapes unequivocally an extension of each member
of $\mathfrak{C}(\mathbb{R}_{\sharp})$ to a pair of members of
$\mathfrak{C}[-\infty,\infty]$, that is, to a member of the Cartesian product

$\mathcal{C}\coloneqq(\mathfrak{C}[-\infty,\infty])^{2}.$

The subspace $\mathcal{C}_{\alpha,\beta}^{\textnormal{s-o}}\subset\mathcal{C}$
of such extensions, besides being invariant under the Cartesian product basic
cosine family in $\mathcal{C}$ (see (2.7)), is _complemented_. Therefore,
there are related projections
$P_{\alpha,\beta}^{\textnormal{s-o}}\in\mathcal{L}(\mathcal{C})$ on
$\mathcal{C}_{\alpha,\beta}^{\textnormal{s-o}}$. We prove that
$\lim_{n\to\infty}P_{n\alpha,n\beta}^{\textnormal{s-o}}$ exists and is a
projection on the subspace of extensions unequivocally shaped by the skew
Brownian motion boundary condition (1.1). See Section 4.

  * (E)

_Convergence of complementary cosine families and semigroups._ As also proved
in [aberman], the complementary subspace of
$\mathcal{C}_{\alpha,\beta}^{\textnormal{s-o}}$, say,
$\mathcal{C}_{\alpha,\beta}^{\textnormal{o-s}}$, in $\mathcal{C}$ is
unequivocally shaped by certain transmission conditions too. As a result there
are complementary cosine families and semigroups to those generated by
$A_{\alpha,\beta}^{\textnormal{s-o}}$. We prove that these cosine families and
semigroups converge similarly as those generated by
$A_{\alpha,\beta}^{\textnormal{s-o}}$. Surprisingly, the limit semigroup and
the limit cosine family turn out to be isomorphic copies of the semigroup and
cosine family related to another skew Brownian motion, with the role of
coefficients $\alpha$ and $\beta$ reversed—see Section 5.

Additional results completing the paper are presented in Section 6.

###  1.5. Commonly used notation

####  1.5.1. Constants $\alpha$ and $\beta$

Throughout the paper we assume that $\alpha$ and $\beta$ are fixed non-
negative constants. Also, to shorten formulae, we write

$\displaystyle\gamma\coloneqq\sqrt{2(\alpha^{2}+\beta^{2})}.$

Since the case of $\alpha=\beta=0$ is not interesting, in what follows we
assume that ${\alpha+\beta}>0$, implying that $\gamma>0$ as well.

####  1.5.2. Limits at infinities, and transformations of functions

For $f\in\mathfrak{C}(\mathbb{R}_{\sharp})$, we write

$f(\pm\infty)\coloneqq\lim_{x\to\pm\infty}f(x).$

Moreover, we define $f^{\textnormal{T}},f^{e}$ and $f^{o}$, also belonging to
$\mathfrak{C}(\mathbb{R}_{\sharp})$, by

$\displaystyle f^{\textnormal{T}}(x)$ $\displaystyle\coloneqq f(-x),\qquad
x\in\mathbb{R}{\setminus\\{0\\}},$

and

$\displaystyle f^{e}$
$\displaystyle\coloneqq{\textstyle\frac{1}{2}}(f+f^{\textnormal{T}}),\qquad
f^{o}\coloneqq{\textstyle\frac{1}{2}}(f-f^{\textnormal{T}}).$ (1.4)

####  1.5.3. Convolution with exponential function

For $a>0$ we introduce

$e_{a}(x)\coloneqq\mathrm{e}^{-ax},\qquad x\in\mathbb{R}^{+},$

and for $f\in\mathfrak{C}(\mathbb{R}_{\sharp})$, we write

$e_{a}*f(x)\coloneqq\int_{0}^{x}\mathrm{e}^{-a(x-y)}f(y)\,\mathrm{d}y,\qquad
x\in\mathbb{R}^{+}.$

####  1.5.4. The restriction operator

The following restriction operator, denoted $R$ and mapping $\mathcal{C}$ onto
the space $\mathfrak{C}(\mathbb{R}_{\sharp})$ of Section 1.2, will be of key
importance in the entire paper. By definition $R$ assigns to a pair
$(f_{1},f_{2})\in\mathcal{C}$ the member $f$ of
$\mathfrak{C}(\mathbb{R}_{\sharp})$ given by

$f(x)=f_{1}(x),\text{ for }x<0\quad\text{ and }\quad f(x)=f_{2}(x),\text{ for
}x>0.$

##  2\. Convergence of cosine families and semigroups describing snapping-out
Brownian motions

Before our main theorem in this section is presented we need to explain the
nature of the result it generalizes, i.e., to put (1.3) in the context of
general theory of convergence of semigroups. Also, in Section 2.2 we recall
the notion of cosine family.

###  2.1. Theory of convergence of semigroups of operators

The main idea of the Trotter–Kato–Neveu convergence theorem [abhn, engel,
goldstein, pazy], a cornerstone of the theory of convergence of semigroups
[knigaz, bobrud], is that convergence of resolvents of _equibounded_
semigroups in a Banach space $\mathsf{F}$, gives an insight into convergence
of the semigroups themselves. Hence, in studying the limit of semigroups, say,
$\\{\mathrm{e}^{tB_{n}},t\geq 0\\},n\geq 1$, generated by the operators
$B_{n},n\geq 1$ we should first establish the existence of the strong limit

$R_{\lambda}\coloneqq\lim_{n\to\infty}\big{(}\lambda-B_{n}\big{)}^{-1}.$

The general theory of convergence (see [knigaz]*Chapter 8) covers also the
case in which, unlike in the classical version of the Trotter–Kato–Neveu
theorem, the (common) range of the operators $R_{\lambda},\lambda>0$ so-
obtained is _not_ dense in $\mathsf{F}$, and stresses the role of the so-
called _regularity space_ , defined as the closure of the range of
$R_{\lambda}$:

$\mathsf{F}_{\textnormal{reg}}\coloneqq
cl(RangeR_{\lambda})\subset\mathsf{F}.$ (2.1)

Namely, $\mathsf{F}_{\textnormal{reg}}$ turns out to be composed of
$f\in\mathsf{F}$ such the limit
$T(t)f\coloneqq\lim_{n\to\infty}\mathrm{e}^{tB_{n}}f$ exists and is uniform
with respect to $t$ in compact subintervals of $[0,\infty)$; then
$\\{T(t),\,t\geq 0\\}$ is a strongly continuous semigroup in
$\mathsf{F}_{\textnormal{reg}}$.

Condition (1.3) (more precisely, the theorem proved in [tombatty]) is thus a
typical result of convergence theory: it characterizes the regularity space
for $B_{n}\coloneqq A_{n\alpha,n\beta}^{\textnormal{s-o}}$ as equal to
$\mathfrak{C}[-\infty,\infty]\subset\mathfrak{C}(\mathbb{R}_{\sharp})$, and
identifies $\\{T(t),\,t\geq 0\\}$ as
$\\{\mathrm{e}^{A^{\textnormal{{skew}}}_{\alpha,\beta}},t\geq 0\\}$. It should
be stressed, though, that this statement does not exclude the possibility of
the existence of $f\not\in\mathfrak{C}[-\infty,\infty]$ such that
$\big{(}\mathrm{e}^{tA_{n\alpha,n\beta}^{\textnormal{s-o}}}f\big{)}_{n\geq 1}$
converges for all $t>0$. Such _irregular_ convergence of semigroups, which is
known to be always uniform with respect to $t$ in compact subsets of
$(0,\infty)$ — see [note] or [knigaz]*Thm 28.4 — is not so uncommon,
especially in the context of singular perturbations [banmika, banalachokniga,
knigaz], but needs to be established by different means. We will prove that
the limit
$\lim_{n\to\infty}\mathrm{e}^{tA_{n\alpha,n\beta}^{\textnormal{s-o}}}f$ exists
for all $t>0$ and $f\in\mathfrak{C}(\mathbb{R}_{\sharp})$, by examining the
related cosine families.

###  2.2. Cosine families

A strongly continuous family $\\{C(t),t\in\mathbb{R}\\}$ of operators in a
Banach space $\mathsf{F}$ is said to be a cosine family iff $C(0)$ is the
identity operator and

$2C(t)C(s)=C(s+t)+C(t-s),\qquad t,s\in\mathbb{R}.$

The generator of such a family is defined by

$Af=\lim_{t\to 0}2t^{-2}{(C(t)f-f)}$

for all $f\in\mathsf{F}$ such the limit on the right-hand side exists. For
example, in $\mathfrak{C}[-\infty,\infty]$ there is the _basic cosine family_
given by

$C(t)f(x)={\textstyle\frac{1}{2}}[f(x+t)+f(x-t)],\qquad
x\in\mathbb{R},t\in\mathbb{R}.$ (2.2)

Its generator is the one-dimensional Laplace operator $f\mapsto
f^{\prime\prime}$ with domain composed of twice continuously differentiable
functions on $\mathbb{R}$ such that
$f^{\prime\prime}\in\mathfrak{C}[-\infty,\infty]$.

Each cosine family generator, say, $A$, is automatically the generator of a
strongly continuous semigroup (but not vice versa). The semigroup such $A$
generates is given by the _Weierstrass formula_ (see e.g. [abhn]*p. 219)

$T(0)f=f\quad\text{and}\quad T(t)f={\textstyle\frac{1}{2\sqrt{\pi
t}}}\int_{-\infty}^{\infty}\mathrm{e}^{-\frac{s^{2}}{4t}}C(s)f\,\mathrm{d}s,\qquad
t>0,f\in\mathsf{F}.$ (2.3)

This formula expresses the fact that the cosine family is then a more
fundamental object than the semigroup, and properties of the semigroup can be
hidden in those of the cosine family.

In our main theorem of this section, we will show that (1.3) can be
strengthened by proving that not only the semigroups generated by
$A_{\alpha,\beta}^{\textnormal{s-o}}$ converge but so also do the related
cosine families. The fact that each $A_{\alpha,\beta}^{\textnormal{s-o}}$ is a
generator of a cosine family was proved in [tombatty], but from the analysis
presented there it is not clear that the cosine families generated by
$A_{\alpha,\beta}^{\textnormal{s-o}}$’s are equibounded or that they do
converge (see Remark 6.1 in that paper).

###  2.3. Convergence theorem for solution families

We are finally ready to present our main theorem in this section.

######  Theorem 2.1.

  * (a)

The cosine families
$\\{\operatorname{Cos_{\alpha,\beta}^{\textnormal{s-o}}}(t),\,t\in\mathbb{R}\\}$
generated by $A_{\alpha,\beta}^{\textnormal{s-o}}$ are equibounded:

$\|\operatorname{Cos_{\alpha,\beta}^{\textnormal{s-o}}}(t)\|\leq
5,\qquad\alpha,\beta\geq 0,t\in\mathbb{R}.$ (2.4)

  * (b)

We have

$\lim_{n\to\infty}\sup_{t\in\mathbb{R}}\big{\|}\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{s-o}}}(t)f-\operatorname{Cos_{\alpha,\beta}^{\textnormal{skew}}}(t)f\big{\|}=0,\qquad
f\in\mathfrak{C}[-\infty,\infty],$ (2.5)

where
$\\{\operatorname{Cos_{\alpha,\beta}^{\textnormal{skew}}}(t),\,t\in\mathbb{R}\\}$
is the cosine family generated by $A^{\textnormal{{skew}}}_{\alpha,\beta}$.
For $f\not\in\mathfrak{C}[-\infty,\infty]$, this limit exists for no
$t\not=0$.

  * (c)

The limit

$\lim_{n\to\infty}\mathrm{e}^{tA_{n\alpha,n\beta}^{\textnormal{s-o}}}f$ (2.6)

exists for all $t>0$ and $f\in\mathfrak{C}(\mathbb{R}_{\sharp}),$ and for
$f\not\in\mathfrak{C}[-\infty,\infty]$ it is uniform with respect to $t$ in
compact subintervals of $(0,\infty)$.

Information on the rate of convergence will be given in Remark 3 further down.

######  Remark 1.

Thesis (b) and the Weierstrass formula imply that the limit in (1.3) is in
fact uniform in $t\geq 0$. To see this, let $f\in\mathfrak{C}[-\infty,\infty]$
and $\varepsilon>0$ be fixed and let $n_{0}$ be so large that

$\big{\|}\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{s-o}}}(s)f-\operatorname{Cos_{\alpha,\beta}^{\textnormal{skew}}}(s)f\big{\|}\leq\varepsilon$

for $n\geq n_{0}$ and $s\in\mathbb{R}$. Then, for any $t>0$,

$\displaystyle\big{\|}\mathrm{e}^{tA_{n\alpha,n\beta}^{\textnormal{s-o}}}f-\mathrm{e}^{tA^{\textnormal{{skew}}}_{\alpha,\beta}}f\big{\|}$
$\displaystyle\leq{\textstyle\frac{1}{2\sqrt{\pi
t}}}\int_{-\infty}^{\infty}\mathrm{e}^{-\frac{s^{2}}{4t}}\big{\|}\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{s-o}}}(s)f-\operatorname{Cos_{\alpha,\beta}^{\textnormal{skew}}}(s)f\big{\|}\,\mathrm{d}s$
$\displaystyle={\textstyle\frac{1}{2\sqrt{\pi
t}}}\int_{-\infty}^{\infty}\mathrm{e}^{-\frac{s^{2}}{4t}}\varepsilon\,\mathrm{d}s=\varepsilon,$

(and for $t=0$ the left hand side is $0$). Hence, (b) improves (1.3).

Our proof of Theorem 2.1 hinges on the representation of cosine families
$\\{\operatorname{Cos_{\alpha,\beta}^{\textnormal{s-o}}}(t),t\in\mathbb{R}\\}$
and
$\\{\operatorname{Cos_{\alpha,\beta}^{\textnormal{skew}}}(t),t\in\mathbb{R}\\}$
by means of the _basic Cartesian product cosine family_

$\\{C_{\textnormal{D}}(t),\,t\in\mathbb{R}\\}$

(‘D’ for ‘Descartes’) defined in $\mathcal{C}$ by the formula

$C_{\textnormal{D}}(t)(f_{1},f_{2})=(C(t)f_{1},C(t)f_{2}),\qquad
t\in\mathbb{R},f_{1},f_{2}\in\mathfrak{C}[-\infty,\infty],$ (2.7)

where $C(t)\in\mathcal{L}(\mathfrak{C}[-\infty,\infty])$ are introduced in
(2.2). The following two propositions were obtained in [aberman]*Section 2 and
[abtk]*Section 6, respectively. Extension operators featured in these
propositions are illustrated in our Figure 1.

$\widetilde{f_{\ell}}$$\widetilde{f_{\textnormal{r}}}$ Figure 1. Two
extensions of a single function $f\in\mathfrak{C}(\mathbb{R}_{\sharp})$ (solid
lines): extension $\widetilde{f_{\ell}}$ of its left part, and extension
$\widetilde{f_{\textnormal{r}}}$ of its right part.

######  Proposition 2.2.

The cosine family
$\\{\operatorname{Cos_{\alpha,\beta}^{\textnormal{s-o}}}(t),t\in\mathbb{R}\\}$
in $\mathfrak{C}(\mathbb{R}_{\sharp})$ generated by
$A_{\alpha,\beta}^{\textnormal{s-o}}$ is given by the abstract Kelvin formula

$\operatorname{Cos_{\alpha,\beta}^{\textnormal{s-o}}}(t)=RC_{\textnormal{D}}(t)E_{\alpha,\beta}^{\textnormal{s-o}},\qquad
t\in\mathbb{R}.$ (2.8)

Here, (a) $R$ is the restriction operator of Section 1.5.4, and (b) the
extension operator
$E_{\alpha,\beta}^{\textnormal{s-o}}\colon\mathfrak{C}(\mathbb{R}_{\sharp})\to\mathcal{C}$
maps an $f\in\mathfrak{C}(\mathbb{R}_{\sharp})$ to the pair
$(\widetilde{f_{\ell}},\widetilde{f_{\textnormal{r}}})\in\mathcal{C}$ of
extensions of its left and right parts determined by

$\displaystyle\widetilde{f_{\ell}}(x)$
$\displaystyle=\begin{cases*}f(x),&$x<0$,\\\ f(-x)+2\alpha
e_{{\alpha+\beta}}*[f-f^{\textnormal{T}}](x),&$x>0,$\end{cases*}$ (2.9)

and

$\displaystyle\widetilde{f_{\textnormal{r}}}(x)$
$\displaystyle=\begin{cases*}f(-x)-2\beta
e_{\alpha+\beta}*[f-f^{\textnormal{T}}](-x),&$x<0$,\\\
f(x),&$x>0$.\end{cases*}$ (2.10)

In particular,

$\|E_{\alpha,\beta}^{\textnormal{s-o}}\|\leq
1+\textstyle{\frac{4}{\alpha+\beta}}\max(\alpha,\beta)\leq 5.$ (2.11)

######  Proposition 2.3.

The cosine family
$\\{\operatorname{Cos_{\alpha,\beta}^{\textnormal{skew}}}(t),t\in\mathbb{R}\\}$
in $\mathfrak{C}[-\infty,\infty]$ generated by
$A^{\textnormal{{skew}}}_{\alpha,\beta}$ is given by the abstract Kelvin
formula:

$\operatorname{Cos_{\alpha,\beta}^{\textnormal{skew}}}(t)=RC_{\textnormal{D}}(t)E_{\alpha,\beta}^{\textnormal{skew}},\qquad
t\in\mathbb{R},$ (2.12)

in which (a) $R$ is the restriction operator of Section 1.5.4, and (b) the
extension operator
$E_{\alpha,\beta}^{\textnormal{skew}}\colon\mathfrak{C}[-\infty,\infty]\to\mathcal{C}$
maps an $f\in\mathfrak{C}[-\infty,\infty]$ to the pair
$(\widetilde{f_{\ell}},\widetilde{f_{\textnormal{r}}})\in\mathcal{C}$ of
extensions of its left and right parts defined by

$\displaystyle\widetilde{f_{\ell}}(x)$
$\displaystyle=\begin{cases*}f(x),&$x\leq 0$,\\\
\frac{\beta-\alpha}{{\alpha+\beta}}f(-x)+\frac{2\alpha}{{\alpha+\beta}}f(x),&$x>0,$\end{cases*}$
(2.13)

and

$\displaystyle\widetilde{f_{\textnormal{r}}}(x)$
$\displaystyle=\begin{cases*}\frac{2\beta}{{\alpha+\beta}}f(x)+\frac{\alpha-\beta}{{\alpha+\beta}}f(-x),&$x<0$,\\\
f(x),&$x\geq 0$.\end{cases*}$ (2.14)

######  Remark 2.

The cartesian product basic cosine family leaves the image of
$\mathfrak{C}[-\infty,\infty]$ via $E_{\alpha,\beta}^{\textnormal{skew}}$
invariant. On the other hand, (2.13) and (2.14) make it clear that
$\widetilde{f_{\ell}}(0)=f(0)=\widetilde{f_{\textnormal{r}}}(0)$. As a result
$RC_{\textnormal{D}}(t)E_{\alpha,\beta}^{\textnormal{skew}}f$ is a member of
$\mathfrak{C}[-\infty,\infty]$ for $f\in\mathfrak{C}[-\infty,\infty]$.

Figure 2. Top: example of $f\in\mathfrak{C}[-\infty,\infty]$. Bottom:
$E_{\alpha,\beta}^{\textnormal{skew}}f$ (black) and
$E_{n\alpha,n\beta}^{\textnormal{s-o}}f$ for $n=1,2,5,15$ (pink, green,
purple, blue, resp.) and $\alpha=0.2,\beta=0.1$. Graphs of $f_{\ell}$’s are on
the left and of $f_{\textnormal{r}}$’s on the right.

###  2.4. Proof of Theorem 2.1

_Proof of (a)_ In view of (2.8), estimate (2.4) is a direct consequence of
(2.11) because the operator norms of $R$ and $C_{\textnormal{D}}(t)$ are equal
to $1.$

_Proof of (b)_ By (2.8) and (2.12), to establish (2.5), it suffices to prove
that

$\lim_{n\to\infty}E_{n\alpha,n\beta}^{\textnormal{s-o}}f=E_{\alpha,\beta}^{\textnormal{skew}}f,\qquad
f\in\mathfrak{C}[-\infty,\infty].$ (2.15)

This, in turn, is an immediate consequence of the fact that distributions of
exponential random variables with large parameters converge to Dirac measure
at $0$, as expressed in Lemma 7.1 presented in Appendix. For, Lemma 7.1 (c)
implies that for $f\in\mathfrak{C}[-\infty,\infty]$, the second term in the
second line of the definition (2.9) of $\widetilde{f_{\ell}}$ with $\alpha$
and $\beta$ replaced by $n\alpha$ and $n\beta$, respectively, converges, as
$n\to\infty$, to $\frac{\alpha}{\alpha+\beta}(f(x)-f(-x))$ uniformly in $x\geq
0.$ This is because the function $\phi$ defined by $\phi(x)=f(x)-f(-x),x\geq
0$ belongs to $\mathfrak{C}[0,\infty]$, and we have $\phi(0)=0$. It follows
that the first coordinate of $E_{n\alpha,n\beta}^{\textnormal{s-o}}f$
converges (in the norm of $\mathfrak{C}[-\infty,\infty]$) to
$\widetilde{f_{\ell}}$ defined by (2.13)—see Fig. 2. Since, similarly, its
second coordinate converges to $\widetilde{f_{\textnormal{r}}}$ from (2.14),
the proof of (2.15) is completed, and so is the proof of (2.5).

Before completing the proof of (b), we recall the general theorem shown in
[zwojtkiem] (see also [knigaz]*Chapter 61) saying that outside of the
regularity space (i.e., outside of the subspace defined in (2.1)—the
regularity space of a sequence of cosine families is by definition the
regularity space of the sequence of corresponding semigroups) cosine families
cannot converge. Since in the case of cosine families
$\\{\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{s-o}}}(t),\,t\in\mathbb{R}\\},n\geq
1$ the regularity space equals $\mathfrak{C}[-\infty,\infty]$, outside of
$\mathfrak{C}[-\infty,\infty]$ the limit
$\lim_{n\to\infty}\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{s-o}}}(t)f$
does not exist for at least one $t\in\mathbb{R}\setminus\\{0\\}$.

However, the abstract Kelvin formula (2.8) provides additional insight into
convergence under study. Namely, for $f$ belonging to
$\mathfrak{C}(\mathbb{R}_{\sharp})$, but not to
$\mathfrak{C}[-\infty,\infty]$,
$\lim_{n\to\infty}E_{n\alpha,n\beta}^{\textnormal{s-o}}f$ still exits and is
given by (2.13) and (2.14), but the convergence is no longer uniform in $x$
and in the limit we obtain a pair of functions which are continuous everywhere
except for $x=0$—see Fig. 3. Hence, if $t\neq 0$, the pointwise
$\lim_{n\to\infty}\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{s-o}}}(t)f$
is not an element of $\mathfrak{C}(\mathbb{R}_{\sharp})$, being discontinuous
at $x=\pm t$. In particular, for $f\not\in\mathfrak{C}[-\infty,\infty]$, the
strong limit exists for no $t\not=0$.

Figure 3. Top: example of $f\in\mathfrak{C}(\mathbb{R}_{\sharp})$. Bottom:
$E_{n\alpha,n\beta}^{\textnormal{s-o}}f$, $n=1,2,5,15$ (pink, green, purple,
blue, resp.) and $\lim_{n\to\infty}E_{n\alpha,n\beta}^{\textnormal{s-o}}f$
(black) for $\alpha=0.2,\beta=0.1$. Graphs of $f_{\ell}$’s are on the left and
of $f_{\textnormal{r}}$’s on the right.

_Proof of (c)_ We recall that, as an application of his algebraic version of
the Hille–Yosida Theorem, J. Kisyński has proved the following result
accompanying Trotter–Kato–Neveu Theorem (see Corollary 5.8 in [prof10]): if
$B_{n},n\geq 1$ are generators of equibounded semigroups, then the existence
of the strong limit $\lim_{n\to\infty}\big{(}\lambda-B_{n}\big{)}^{-1}$ is
equivalent to the existence of

$\lim_{n\to\infty}\int_{0}^{\infty}\Psi(t)\mathrm{e}^{tB_{n}}\,\mathrm{d}t$

for any $\Psi$ that is absolutely integrable on $\mathbb{R}^{+}$ (see also
[hn]; the fact that the existence of the first limit implies the existence of
the second for $\Psi=1_{[0,\tau]}$ with $\tau>0$, has been noted already in
the 1970 paper of T. G. Kurtz [kurtz2]).

An analogue of this result for cosine families was found in [odessa]. In our
case it says that the existence of the strong limit
$\lim_{n\to\infty}\big{(}\lambda-
A_{n\alpha,n\beta}^{\textnormal{s-o}}\big{)}^{-1}$ (which is established in
[tombatty]) implies the existence of the strong limit

$\lim_{n\to\infty}\int_{-\infty}^{\infty}\Psi(s)\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{s-o}}}(s)\,\mathrm{d}s$
(2.16)

for any $\Psi$ that is absolutely integrable on the entire $\mathbb{R}$ and
even (as long as we have an estimate of the type (2.4)). Since, by the
Weierstrass formula (2.3),

$\mathrm{e}^{tA_{n\alpha,n\beta}^{\textnormal{s-o}}}f={\textstyle\frac{1}{2\sqrt{\pi
t}}}\int_{-\infty}^{\infty}\mathrm{e}^{-\frac{s^{2}}{4t}}\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{s-o}}}(s)f\,\mathrm{d}s,\qquad
f\in\mathfrak{C}(\mathbb{R}_{\sharp}),t>0$

the limit in (2.6) is a particular case of (2.16) for
$\Psi(s)=\Psi_{t}(s)\coloneqq{\textstyle\frac{1}{2\sqrt{\pi
t}}}\mathrm{e}^{-\frac{s^{2}}{4t}},$ $s\in\mathbb{R},t>0$. The fact that the
limit is uniform with respect to $t$ in compact subintervals of $(0,\infty)$
is a consequence of the general result discussed in [note] and [knigaz]*Thm
28.4.

######  Remark 3.

The rate of convergence in Theorem 2.1 (b) depends apparently on the modulus
of continuity of $f\in\mathfrak{C}[-\infty,\infty]$. We can be more specific
about this rate if we restrict ourselves to a dense subset of
$\mathfrak{C}[-\infty,\infty]$: for $\phi\in C^{1}[-\infty,\infty]$, or, more
generally, for $\phi$ that are Lipschitz continuous, the difference between
$\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{s-o}}}(t)f$
and $\operatorname{Cos_{\alpha,\beta}^{\textnormal{skew}}}(t)f$ is of the
order of $n^{-1}$. Indeed, if $|f(x)-f(y)|\leq L|x-y|,x,y\in\mathbb{R}$ then
$\phi\in\mathfrak{C}[0,\infty]$ defined by $\phi(x)\coloneqq f(x)-f(-x),x\geq
0$ is Lipschitz continuous on $\mathbb{R}^{+}$ with the same Lipschitz
constant as $f$, and the the second line in (2.9) with $\alpha$ and $\beta$
replaced by $n\alpha$ and $n\beta$, respectively, differs from the second line
in (2.13) by

$\frac{2\alpha}{\alpha+\beta}[n(\alpha+\beta)e_{n(\alpha+\beta)}*\phi(x)-\phi(x)]$

and this in absolute value does not exceed $\frac{2\alpha
L}{n(\alpha+\beta)^{2}}$ by Remark 7. Since a similar reasoning shows that the
absolute value of the difference between the first line in (2.10) with
$\alpha$ and $\beta$ replaced by $n\alpha$ and $n\beta$, respectively, and the
first line in (2.14) does not exceed $\frac{2\beta L}{n(\alpha+\beta)^{2}}$,
we conclude that

$\sup_{t\in\mathbb{R}}\|\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{s-o}}}(t)f-\operatorname{Cos_{\alpha,\beta}^{\textnormal{skew}}}(t)f\|\leq\|E_{n\alpha,n\beta}^{\textnormal{s-o}}f-E_{\alpha,\beta}^{\textnormal{skew}}f\|\leq{\textstyle\frac{K}{n}},$

where $K\coloneqq\frac{2\max(\alpha,\beta)L}{(\alpha+\beta)^{2}}$. This, in
turn, as in Remark 1, implies

$\sup_{t\geq
0}\big{\|}\mathrm{e}^{tA_{n\alpha,n\beta}^{\textnormal{s-o}}}f-\mathrm{e}^{tA^{\textnormal{{skew}}}_{\alpha,\beta}}f\big{\|}\leq{\textstyle\frac{K}{n}}.$

##  3\. Asymptotic behavior

It is well-known (see [zwojtkiem]*Corollary 1 which extends [abhn]*Prop.
3.14.6) that, if $\\{C(t),t\in\mathbb{R}\\}$ is a strongly continuous family
in a Banach space $\mathsf{F}$, then the limit $\lim_{t\to\infty}C(t)f$ exist
for an $f\in\mathsf{F}$ only in the trivial case, that is, iff
$C(t)f=f,t\in\mathbb{R}$. On the other hand, for rather non-trivial cosine
families it often happens that the mean

$Mf\coloneqq\lim_{t\to\infty}t^{-1}\int_{0}^{t}C(s)f\,\mathrm{d}s$

exists for all $f\in\mathsf{F}$. For example, for the basic cosine family this
limit exists in the bounded pointwise topology and we have

$Mf={\textstyle\frac{1}{2}}(f(-\infty)+f(\infty)),\qquad
f\in\mathfrak{C}[-\infty,\infty].$ (3.1)

(Here, and in the rest of this section, for simplicity of notations, we do not
distinguish between a real number, like
${\textstyle\frac{1}{2}}(f(-\infty)+f(\infty))$, and a function that is
everywhere equal to this number.) In this section we show that the mean
operator $M$ for $\operatorname{Cos_{\alpha,\beta}^{\textnormal{s-o}}}$
remains the same when $\alpha$ and $\beta$ are replaced by $n\alpha$ and
$n\beta$ respectively, and in fact coincides with the mean operator for
$\operatorname{Cos_{\alpha,\beta}^{\textnormal{skew}}}$. As a result, in view
of Theorem 2.1 (b), for $f\in\mathfrak{C}[-\infty,\infty]$ we can write

$\displaystyle\lim_{t\to\infty}t^{-1}\int_{0}^{t}\lim_{n\to\infty}\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{s-o}}}(s)f\,\mathrm{d}s$
$\displaystyle=\lim_{t\to\infty}t^{-1}\int_{0}^{t}\operatorname{Cos_{\alpha,\beta}^{\textnormal{skew}}}(s)f\,\mathrm{d}s$
$\displaystyle=\lim_{n\to\infty}\lim_{t\to\infty}t^{-1}\int_{0}^{t}\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{s-o}}}(s)f\,\mathrm{d}s.$

######  Proposition 3.1.

We have

$\displaystyle\lim_{t\to\infty}t^{-1}\int_{0}^{t}\operatorname{Cos_{\alpha,\beta}^{\textnormal{s-o}}}(s)f\,\mathrm{d}s$
$\displaystyle=M_{\alpha,\beta}f,\qquad
f\in\mathfrak{C}(\mathbb{R}_{\sharp}),$
$\displaystyle\lim_{t\to\infty}t^{-1}\int_{0}^{t}\operatorname{Cos_{\alpha,\beta}^{\textnormal{skew}}}(s)f\,\mathrm{d}s$
$\displaystyle=M_{\alpha,\beta}f,\quad f\in\mathfrak{C}[-\infty,\infty],$

where

$M_{\alpha,\beta}f\coloneqq{\textstyle\frac{\beta}{\alpha+\beta}}f(-\infty)+{\textstyle\frac{\alpha}{\alpha+\beta}}f(\infty),\qquad
f\in\mathfrak{C}(\mathbb{R}_{\sharp}).$

###### Proof.

Because of (3.1), for $\widetilde{f_{\ell}}$ of (2.13), we have

$\lim_{t\to\infty}t^{-1}\int_{0}^{t}C(s)\widetilde{f_{\ell}}\,\mathrm{d}s={\textstyle\frac{1}{2}}(f(-\infty)+{\textstyle\frac{\beta-\alpha}{\alpha+\beta}}f(-\infty)+\textstyle{\frac{2\alpha}{\alpha+\beta}}f(\infty))=M_{\alpha,\beta}f.$

and a similar calculation based on (2.14) shows that
$\lim_{t\to\infty}t^{-1}\int_{0}^{t}C(s)\widetilde{f_{\textnormal{r}}}\,\mathrm{d}s$
$=M_{\alpha,\beta}f$ too. Hence,
$\lim_{t\to\infty}t^{-1}\int_{0}^{t}C_{\textnormal{D}}(s)E_{\alpha,\beta}^{\textnormal{skew}}f\,\mathrm{d}s=(M_{\alpha,\beta}f,M_{\alpha,\beta}f)\in\mathcal{C}$,
and thus

$\lim_{t\to\infty}t^{-1}\int_{0}^{t}\operatorname{Cos_{\alpha,\beta}^{\textnormal{skew}}}(s)f\,\mathrm{d}s=R(M_{\alpha,\beta}f,M_{\alpha,\beta}f)=M_{\alpha,\beta}f,$

showing the second relation to be proved.

The proof of the first of these relations is rather similar. For
$\widetilde{f_{\ell}}$ of (2.9)

$\lim_{t\to\infty}t^{-1}\int_{0}^{t}C(s)\widetilde{f_{\ell}}\,\mathrm{d}s=f(-\infty)+{\textstyle\frac{\alpha}{\alpha+\beta}}(f(\infty)-f(-\infty))=M_{\alpha,\beta}f,$

because
$\lim_{x\to\infty}e_{\alpha+\beta}*[f-f^{\textnormal{T}}](x)={\textstyle\frac{1}{\alpha+\beta}}(f(\infty)-f(-\infty))$.
Since the calculation for $\widetilde{f_{\textnormal{r}}}$ gives the same
result, we argue as above to complete the proof. ∎

##  4\. Convergence of projections

The abstract Kelvin formula (2.8) works well because the extension operator
$E_{\alpha,\beta}^{\textnormal{s-o}}$ is shaped by transmission conditions
(1.2) in a very specific manner. In particular, the space

$\mathcal{C}_{\alpha,\beta}^{\textnormal{s-o}}\subset\mathcal{C}$

of extensions of all elements of $\mathfrak{C}(\mathbb{R}_{\sharp})$ is
invariant under the basic cartesian product cosine family
$\\{C_{\textnormal{D}}(t),t\in\mathbb{R}\\}$,

It is one of the main findings of [aberman] that
$\mathcal{C}_{\alpha,\beta}^{\textnormal{s-o}}$ is _complemented_. In fact,
Section 3 in [aberman] provides an explicit form of a natural projection of
$\mathcal{C}$ on this subspace. A ‘nearly orthogonal’ projection, say,
$P_{\alpha,\beta}^{\textnormal{s-o}}$ on
$\mathcal{C}_{\alpha,\beta}^{\textnormal{s-o}}$ is given by

$P_{\alpha,\beta}^{\textnormal{s-o}}(f_{1},f_{2})=(g_{1},g_{2})$

where

$\displaystyle\begin{split}g_{1}(x)&=f_{1}^{e}(x)+{\textstyle\frac{1}{2}}(\gamma+2\alpha)\int_{-\infty}^{x}\big{[}\textstyle{\frac{1}{\gamma}}k_{1}(y)-{\textstyle\frac{1}{2}}k_{2}(y)\big{]}\mathrm{e}^{-\gamma(x-y)}\,\mathrm{d}y\\\
&\phantom{=}-{\textstyle\frac{1}{2}}(\gamma-2\alpha)\int_{x}^{\infty}\big{[}\textstyle{\frac{1}{\gamma}}k_{1}(y)+{\textstyle\frac{1}{2}}k_{2}(y)\big{]}\mathrm{e}^{\gamma(x-y)}\,\mathrm{d}y,\\\
g_{2}(x)&=f_{2}^{e}(x)+{\textstyle\frac{1}{2}}(\gamma+2\beta)\int_{x}^{\infty}\big{[}\textstyle{\frac{1}{\gamma}}k_{1}(y)+{\textstyle\frac{1}{2}}k_{2}(y)\big{]}\mathrm{e}^{\gamma(x-y)}\,\mathrm{d}y\\\
&\phantom{=}-{\textstyle\frac{1}{2}}(\gamma-2\beta)\int_{-\infty}^{x}\big{[}\textstyle{\frac{1}{\gamma}}k_{1}(y)-{\textstyle\frac{1}{2}}k_{2}(y)\big{]}\mathrm{e}^{-\gamma(x-y)}\,\mathrm{d}y,\qquad
x\in\mathbb{R},\end{split}$ (4.1)

and

$\displaystyle k_{1}\coloneqq\alpha f_{1}^{o}+\beta f_{2}^{o},\qquad
k_{2}\coloneqq f_{1}^{e}-f_{2}^{e}.$

We will show that convergence of solution families described in the previous
section is accompanied by convergence of the related projections. We will show
namely that projections $P_{n\alpha,n\beta}^{\textnormal{s-o}}$ converge
strongly to a projection on the space of extensions related to the skew
Brownian motion.

######  Theorem 4.1.

  * (a)

For every $(f_{1},f_{2})\in\mathcal{C}$, we have

$\displaystyle\lim_{n\to\infty}P_{n\alpha,n\beta}^{\textnormal{s-o}}(f_{1},f_{2})=P_{\alpha,\beta}^{\textnormal{skew}}(f_{1},f_{2}),$

where

$P_{\alpha,\beta}^{\textnormal{skew}}(f_{1},f_{2})\coloneqq\big{(}f_{1}^{e}+\textstyle{\frac{2\alpha}{\gamma^{2}}}k_{1}-{\textstyle\frac{1}{2}}k_{2},f_{2}^{e}+\textstyle{\frac{2\beta}{\gamma^{2}}}k_{1}+{\textstyle\frac{1}{2}}k_{2}\big{)}.$

  * (b)

The map $P_{\alpha,\beta}^{\textnormal{skew}}$ is a projection on the space

$\mathcal{C}_{\alpha,\beta}^{\textnormal{skew}}\coloneqq\\{(g_{1},g_{2})\in\mathcal{C}\colon
g_{1}^{e}=g_{2}^{e},\beta g_{1}^{o}=\alpha g_{2}^{o}\\}.$

  * (c)

The space $\mathcal{C}_{\alpha,\beta}^{\textnormal{skew}}$ is precisely the
subspace of extensions related to the skew Brownian motion generated by
$A^{\textnormal{{skew}}}_{\alpha,\beta}$.

###### Proof.

(a) From the first formula in (4.1) it follows that the first coordinate of
$P_{n\alpha,n\beta}^{\textnormal{s-o}}(f_{1},f_{2})$ is

$\displaystyle g_{1,n}(x)$
$\displaystyle=f_{1}^{e}(x)+{\textstyle\frac{1}{2}}(\gamma+2\alpha)n\int_{-\infty}^{x}\big{[}\textstyle{\frac{1}{\gamma}}k_{1}(y)-{\textstyle\frac{1}{2}}k_{2}(y)\big{]}\mathrm{e}^{-n\gamma(x-y)}\,\mathrm{d}y$
$\displaystyle\phantom{=}-{\textstyle\frac{1}{2}}(\gamma-2\alpha)n\int_{x}^{\infty}\big{[}\textstyle{\frac{1}{\gamma}}k_{1}(y)+{\textstyle\frac{1}{2}}k_{2}(y)\big{]}\mathrm{e}^{n\gamma(x-y)}\,\mathrm{d}y,\qquad
x\in\mathbb{R}.$

Lemma 7.1 in Appendix implies now that $g_{1,n}$ converges in the norm of
$\mathfrak{C}[-\infty,\infty]$, as $n\to\infty$, to
$f_{1}^{e}+\frac{2\alpha}{\gamma^{2}}k_{1}-\frac{1}{2}k_{2}$, as desired.

For the proof of convergence of the second coordinate of
$P_{n\alpha,n\beta}^{\textnormal{s-o}}(f_{1},f_{2})$, denoted $g_{2,n}$, we
proceed analogously. Namely, by (4.1), we have

$\displaystyle g_{2,n}(x)$
$\displaystyle=f_{2}^{e}(x)+{\textstyle\frac{1}{2}}(\gamma+2\beta)n\int_{x}^{\infty}\big{[}\textstyle{\frac{1}{\gamma}}k_{1}(y)+{\textstyle\frac{1}{2}}k_{2}(y)\big{]}\mathrm{e}^{n\gamma(x-y)}\,\mathrm{d}y$
$\displaystyle\phantom{=}-{\textstyle\frac{1}{2}}(\gamma-2\beta)n\int_{-\infty}^{x}\big{[}\textstyle{\frac{1}{\gamma}}k_{1}(y)-{\textstyle\frac{1}{2}}k_{2}(y)\big{]}\mathrm{e}^{-n\gamma(x-y)}\,\mathrm{d}y,\qquad
x\in\mathbb{R}.$

Hence, by Lemma 7.1, $g_{2,n}$ converges in the norm of
$\mathfrak{C}[-\infty,\infty]$ to
$f_{2}^{e}+\frac{2\beta}{\gamma^{2}}k_{1}+\frac{1}{2}k_{2}$.

(b) As a strong limit of projections, $P_{\alpha,\beta}^{\textnormal{skew}}$
is a projection also. To characterize its range we consider
$(f_{1},f_{2})\in\mathcal{C}$ and
$(g_{1},g_{2})=P_{\alpha,\beta}^{\textnormal{skew}}(f_{1},f_{2})$. Since
$f_{1}^{e},f_{2}^{e},$ $k_{1}$ and $k_{2}$ all belong to
$\mathfrak{C}[-\infty,\infty]$, it follows that so do $g_{1}$ and $g_{2}$.
Furthermore,
$g_{1}^{e}=f_{1}^{e}-{\textstyle\frac{1}{2}}k_{2}={\textstyle\frac{1}{2}}(f_{1}^{e}+f_{2}^{e})=f_{2}^{e}+{\textstyle\frac{1}{2}}k_{2}=g_{2}^{e}$,
because $k_{1}$ is odd and $k_{2}$ is even. Similarly, we see that
$g_{1}^{o}=\frac{2\alpha}{\gamma^{2}}k_{1}$ and
$g_{2}^{o}=\frac{2\beta}{\gamma^{2}}k_{1}$. This shows that
$(g_{1},g_{2})\in\mathcal{C}_{\alpha,\beta}^{\textnormal{skew}}$, that is,
that the range of $P_{\alpha,\beta}^{\textnormal{skew}}$ is contained in
$\mathcal{C}_{\alpha,\beta}^{\textnormal{skew}}$.

To prove the other inclusion, we assume that $\beta f_{1}^{o}=\alpha
f_{2}^{o}$ and $f_{1}^{e}=f_{2}^{e}.$ Then, $2\alpha
k_{1}=\gamma^{2}f_{1}^{o}$, $2\beta k_{1}=\gamma^{2}f_{2}^{o}$ and $k_{2}=0$.
Hence the definition of $g_{1}$ and $g_{2}$ simplifies to
$g_{1}=f_{1}^{e}+f_{1}^{o}=f_{1}$ and $g_{2}=f_{2}^{e}+f_{2}^{o}=f_{2}$. This
means that for
$(f_{1},f_{2})\in\mathcal{C}_{\alpha,\beta}^{\textnormal{skew}}$,
$P_{\alpha,\beta}^{\textnormal{skew}}(f_{1},f_{2})=(f_{1},f_{2})$. Therefore,
the range of $P_{\alpha,\beta}^{\textnormal{skew}}$ contains
$\mathcal{C}_{\alpha,\beta}^{\textnormal{skew}}$ and the proof of (b) is
complete.

(c) Relations (2.13) and (2.14) show that the subspace of extensions related
to the skew Brownian motion can be characterized as the space of
$(g_{1},g_{2})\in\mathcal{C}$ satisfying
$g_{1}(x)=\frac{\beta-\alpha}{{\alpha+\beta}}g_{1}(-x)+\frac{2\alpha}{{\alpha+\beta}}g_{2}(x)$
and
$g_{2}(-x)=\frac{2\beta}{{\alpha+\beta}}g_{1}(-x)+\frac{\alpha-\beta}{{\alpha+\beta}}g_{2}(x)$
for $x\geq 0$. Multiplying by $\alpha+\beta$ and rearranging (using (1.4)) we
obtain the following equivalent form of these equations:

$\displaystyle\beta g_{1}^{o}$
$\displaystyle=\alpha(g_{2}-g_{1}^{e}),\qquad\alpha
g_{2}^{o}=\beta(g_{2}^{e}-g_{1}^{\textnormal{T}})\quad\textrm{on
}\mathbb{R}^{+}.$ (4.2)

These, in turn, are equivalent to

$\displaystyle g_{1}^{e}$ $\displaystyle=g_{2}^{e},\qquad\beta
g_{1}^{o}=\alpha g_{2}^{o}\quad\textrm{on }\mathbb{R}^{+}.$ (4.3)

Indeed, summing sides of (4.2) and rearranging once again yields the first
relation in (4.3), and then the first relation in (4.2) renders the second in
(4.3). Conversely, a straightforward argument leads from (4.3) to (4.2).
Finally, equations featured in (4.3) hold on $\mathbb{R}^{+}$ whenever they
hold on $\mathbb{R}$. Hence, the space of extensions equals
$\mathcal{C}_{\alpha,\beta}^{\textnormal{skew}}$. ∎

######  Remark 4.

The operator $P_{\alpha,\beta}^{\textnormal{skew}}$ mimics orthogonal
projection in the Hilbert space of square integrable functions on
$\mathbb{R}$. More precisely, for $(f_{1},f_{2})\in\mathcal{C}$, the pair
$P_{\alpha,\beta}^{\textnormal{skew}}(f_{1},f_{2})$ minimizes the value of

$L(g_{1},g_{2})\coloneqq\int_{-y}^{y}(g_{1}(x)-f_{1}(x))^{2}+(g_{2}(x)-f_{2}(x))^{2}\,\mathrm{d}x$

over all $(g_{1},g_{2})\in\mathcal{C}_{\alpha,\beta}^{\textnormal{skew}}$ for
any positive $y$. To prove this claim, we assume for a moment that
$\alpha\neq\beta$. Summing the sides of the first equation in (4.3) multiplied
by $\alpha$ with the sides of the second, we check that
$g_{1}(-x)=\frac{{\alpha+\beta}}{\beta-\alpha}g_{1}(x)-\frac{2\alpha}{\beta-\alpha}g_{2}(x)$
for $(g_{1},g_{2})\in\mathcal{C}_{\alpha,\beta}^{\textnormal{skew}}$ and
$x\geq 0$. Also, repeating these calculations with $\alpha$ replaced by
$\beta$ yields
$g_{2}(-x)=\frac{2\beta}{\beta-\alpha}g_{1}(x)-\frac{{\alpha+\beta}}{\beta-\alpha}g_{2}(x)$
for $x\geq 0$. It follows that

$\displaystyle L(g_{1},g_{2})=$
$\displaystyle\int_{0}^{y}(g_{1}(x)-f_{1}(x))^{2}+\Big{(}{\textstyle\frac{{\alpha+\beta}}{\beta-\alpha}}g_{1}(x)-{\textstyle\frac{2\alpha}{\beta-\alpha}}g_{2}(x)-f_{1}(-x)\Big{)}^{2}$
$\displaystyle+(g_{2}(x)-f_{2}(x))^{2}+\Big{(}{\textstyle\frac{2\beta}{\beta-\alpha}}g_{1}(x)-{\textstyle\frac{{\alpha+\beta}}{\beta-\alpha}}g_{2}(x)-f_{2}(-x)\Big{)}^{2}\,\mathrm{d}x.$

We can minimize the integrand for each $x\in(0,y)$ by finding the minimum of a
quadratic function in two variables ($f_{1}(\pm x)$ and $f_{2}(\pm x)$ are
treated as fixed parameters). Direct calculations verify that such solution
coincides with $P_{\alpha,\beta}^{\textnormal{skew}}(f_{1},f_{2})$ on the
interval $(-y,y)$.

The case $\alpha=\beta$ is straightforward because it comes down to the fact
that
$(g_{1},g_{2})=({\textstyle\frac{1}{2}}(f_{1}+f_{2}),{\textstyle\frac{1}{2}}(f_{1}+f_{2}))=P^{\textnormal{skew}}_{\alpha,\alpha}(f_{1},f_{2})$
miminizes
$L(g_{1},g_{2})=\int_{-y}^{y}(g_{1}(x)-f_{1}(x))^{2}+(g_{1}(x)-f_{2}(x))^{2}\,\mathrm{d}x$
over all $(g_{1},g_{2})\in\mathcal{C}$ satisfying $g_{1}=g_{2}$.

######  Remark 5.

Arguing as in Remark 3, we can estimate the rate of convergence in Theorem 4.1
(a); we omit the details.

##  5\. Convergence of ‘complementary’ families

###  5.1. Definition of ‘complementary’ cosine families

The fact that $P_{\alpha,\beta}^{\textnormal{s-o}}$ of Section 4 is a
projection on
$\mathcal{C}_{\alpha,\beta}^{\textnormal{s-o}}\subset\mathcal{C}$ implies the
existence of the subspace $\mathcal{C}_{\alpha,\beta}^{\textnormal{o-s}}$ that
complements $\mathcal{C}_{\alpha,\beta}^{\textnormal{s-o}}$ to the entire
$\mathcal{C}$:

$\mathcal{C}=\mathcal{C}_{\alpha,\beta}^{\textnormal{s-o}}\oplus\mathcal{C}_{\alpha,\beta}^{\textnormal{o-s}};$

$\mathcal{C}_{\alpha,\beta}^{\textnormal{o-s}}$ is defined as the kernel of
$P_{\alpha,\beta}^{\textnormal{s-o}}$ or, equivalently, as the image of
$\mathcal{C}$ via

$P_{\alpha,\beta}^{\textnormal{o-s}}\coloneqq
I_{\mathcal{C}}-P_{\alpha,\beta}^{\textnormal{s-o}}.$ (5.1)

Interestingly, $\mathcal{C}_{\alpha,\beta}^{\textnormal{o-s}}$ turns out to be
invariant under the cartesian product basic cosine family also, and in fact is
unequivocally shaped by certain boundary conditions.

Elaborating on this succinct statement, we first define the space

$\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp})$

as the subspace of functions $f\in\mathfrak{C}(\mathbb{R}_{\sharp})$
satisfying $f(0+)=-f(0-)$ (‘ov’ stands for ‘opposite values’). As it
transpires, each member $f$ of
$\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp})$ can be identified with
the pair

$E_{\alpha,\beta}^{\textnormal{o-s}}f=(\widetilde{f_{\ell}},\widetilde{f_{\textnormal{r}}})\in\mathcal{C}_{\alpha,\beta}^{\textnormal{o-s}}$

determined by

$\displaystyle\widetilde{f_{\ell}}(x)$
$\displaystyle=\begin{cases*}f(x),&$x<0$,\\\
-f(-x)-2f(0+)e_{\alpha+\beta}(x)-2e_{\alpha+\beta}*[\beta f-\alpha
f^{\textnormal{T}}](x),&$x>0,$\end{cases*}$

and

$\displaystyle\widetilde{f_{\textnormal{r}}}(x)$
$\displaystyle=\begin{cases*}-f(-x)+2f(0+)e_{\alpha+\beta}(-x)+2e_{\alpha+\beta}*[\beta
f-\alpha f^{\textnormal{T}}](-x),&$x<0$,\\\ f(x),&$x>0$.\end{cases*}$

As a result, the formula

$\operatorname{Cos_{\alpha,\beta}^{\textnormal{o-s}}}(t)\coloneqq
RC_{D}(t)E_{\alpha,\beta}^{\textnormal{o-s}},\qquad t\in\mathbb{R},$ (5.2)

defines a cosine family in
$\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp})$. Since
$\|E_{\alpha,\beta}^{\textnormal{o-s}}\|\leq 5$, we have

$\|\operatorname{Cos_{\alpha,\beta}^{\textnormal{o-s}}}(t)\|\leq
5,\qquad\alpha,\beta\geq 0,t\in\mathbb{R}.$

It is also proved in [aberman] that the generator, say,
$A^{\textnormal{o-s}}_{\alpha,\beta}$, of this cosine family can be
characterized as follows. Its domain $D(A^{\textnormal{o-s}}_{\alpha,\beta})$
consists of functions
$f\in\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp})$ satisfying the
following conditions:

  * (a)

$f$ is twice continuously differentiable in both $(-\infty,0-]$ and
$[0+,\infty)$, separately, with left-hand and right-hand derivatives at
$x=0\mp$, respectively,

  * (b)

both the limits $\lim_{x\to\infty}f^{\prime\prime}(x)$ and
$\lim_{x\to-\infty}f^{\prime\prime}(x)$ exist and are finite, and

  * (c)

$f^{\prime\prime}(0+)=\alpha f^{\prime}(0-)+\beta f^{\prime}(0+)\quad\text{
and }\quad f^{\prime\prime}(0+)=-f^{\prime\prime}(0-).$ (5.3)

Moreover, for such $f$ we have

$A^{\textnormal{o-s}}_{\alpha,\beta}f=f^{\prime\prime};$

we note in passing that the second condition in (5.3) is necessary for
$f^{\prime\prime}$ to belong to
$\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp})$.

It is worth stressing two points here. First of all, because of the way
$\operatorname{Cos_{\alpha,\beta}^{\textnormal{o-s}}}$ was defined, it can be
seen as a ‘complementary’ cosine family to
$\operatorname{Cos_{\alpha,\beta}^{\textnormal{s-o}}}$. Moreover, it can be
shown that if (5.2) is to define the cosine family generated by
$A^{\textnormal{o-s}}_{\alpha,\beta}$, the extension operator
$E_{\alpha,\beta}^{\textnormal{o-s}}$ has to be defined as above. In this
sense, $E_{\alpha,\beta}^{\textnormal{o-s}}$, and thus
$\mathcal{C}_{\alpha,\beta}^{\textnormal{o-s}}$ also, is shaped by the
boundary conditions (5.3).

###  5.2. Convergence of ‘complementary’ solution families

We know from Theorem 2.1 that cosine families
$\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{s-o}}}$
converge to the cosine families
$\operatorname{Cos_{\alpha,\beta}^{\textnormal{skew}}}$ as $n\to\infty$, and
that implies convergence of the related semigroups. In this section, we want
to complement the aforementioned theorem with the result concerning
convergence of
$\\{\operatorname{Cos_{\alpha,\beta}^{\textnormal{o-s}}},\,t\in\mathbb{R}\\}$
introduced in Section 5.1. We will show that (see Theorem 5.1), in contrast to
$\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{s-o}}}$
which converge only on a subspace of $\mathfrak{C}(\mathbb{R}_{\sharp})$,
$\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{o-s}}}$
converge, as $n\to\infty$, on the entire
$\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp})$. Moreover, perhaps more
surprisingly, the limit cosine family is a mirror image of that related to
skew Brownian motion (see Proposition 5.2).

######  Theorem 5.1.

We have

$\lim_{n\to\infty}\sup_{t\in\mathbb{R}}\|\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{o-s}}}(t)f-\operatorname{Cos_{\alpha,\beta}^{\textnormal{weks}}}(t)f\|=0,\qquad
f\in\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp}),$ (5.4)

where $\operatorname{Cos_{\alpha,\beta}^{\textnormal{weks}}}(t)\coloneqq
RC_{\textnormal{D}}(t)E_{\alpha,\beta}^{\textnormal{weks}}$ and
$E_{\alpha,\beta}^{\textnormal{weks}}f=(\widetilde{f_{\ell}},\widetilde{f_{\textnormal{r}}})$
is defined by

$\displaystyle\widetilde{f_{\ell}}(x)$
$\displaystyle=\begin{cases*}f(x),&$x<0$,\\\
\frac{\alpha-\beta}{{\alpha+\beta}}f(-x)-\frac{2\beta}{{\alpha+\beta}}f(x),&$x>0,$\end{cases*}$
(5.5)

and

$\displaystyle\widetilde{f_{\textnormal{r}}}(x)$
$\displaystyle=\begin{cases*}-\frac{2\alpha}{{\alpha+\beta}}f(x)+\frac{\beta-\alpha}{{\alpha+\beta}}f(-x),&$x<0$,\\\
f(x),&$x>0$.\end{cases*}$ (5.6)

As a consequence,

$\lim_{n\to\infty}\sup_{t\geq
0}\big{\|}\mathrm{e}^{tA^{\textnormal{o-s}}_{n\alpha,n\beta}}-T(t)f\big{\|}=0,\qquad
f\in\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp}),$

where $T(0)f=f$ and

$T(t)f={\textstyle\frac{1}{2\sqrt{\pi
t}}}\int_{-\infty}^{\infty}\mathrm{e}^{-\frac{s^{2}}{4t}}\operatorname{Cos_{\alpha,\beta}^{\textnormal{weks}}}(s)f\,\mathrm{d}s,\qquad
t>0,f\in\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp}).$

###### Proof.

It suffices to show (5.4), because the rest then can be proved as in Remark 1.
Indeed, $A^{\textnormal{o-s}}_{n\alpha,n\beta}$ is the generator of
$\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{o-s}}}$,
and $\operatorname{Cos_{\alpha,\beta}^{\textnormal{weks}}}$ is a cosine family
as a limit of cosine families. But, to establish (5.4) it suffices to check
that

$\lim_{n\to\infty}E_{n\alpha,n\beta}^{\textnormal{o-s}}f=E_{\alpha,\beta}^{\textnormal{weks}}f,\qquad
f\in\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp})$ (5.7)

for $E_{\alpha,\beta}^{\textnormal{o-s}}$ defined in the preceding section.

To this end, we note that, for
$f\in\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp})$, the continuous
function $\phi$ on $\mathbb{R}^{+}$ determined by $\phi(x)=\beta f(x)-\alpha
f(-x),x>0$, (in particular $\phi(0)=(\alpha+\beta)f(0+)$) belongs to
$\mathfrak{C}[0,\infty]$. Therefore, by Lemma 7.1 (c),

$f(0+)e_{n(\alpha+\beta)}(x)+ne_{n({\alpha+\beta})}*[\beta f-\alpha
f^{\textnormal{T}}](x)$

converges, as $n\to\infty$, to $(\alpha+\beta)^{-1}[\beta f(x)-\alpha
f^{\textnormal{T}}(x)]$ uniformly in $x\geq 0.$ It follows that the first
coordinate of $E_{n\alpha,n\beta}^{\textnormal{o-s}}f$ converges in the norm
of $\mathfrak{C}[-\infty,\infty]$ to $\widetilde{f_{\ell}}$ defined by (5.5).
Since, similarly, the second coordinate of
$E_{n\alpha,n\beta}^{\textnormal{o-s}}f$ converges to
$\widetilde{f_{\textnormal{r}}}$ of (5.6), the proof is complete. ∎

The space $\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp})$ in which
$\operatorname{Cos_{\alpha,\beta}^{\textnormal{weks}}}$ is defined is somewhat
unusual, and it is perhaps the main reason why
$\operatorname{Cos_{\alpha,\beta}^{\textnormal{weks}}}$ remains a bit
mysterious even if a direct characterization of its generator is provided
(that characterization can be given, but does not seem to be very
informative). Fortunately,
$\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp})$ is isometrically
isomorphic to the space $\mathfrak{C}[-\infty,\infty]$ where a number of
semigroups and cosine families can be interpreted probabilistically, and it
turns out that the image of
$\operatorname{Cos_{\alpha,\beta}^{\textnormal{weks}}}$ in
$\mathfrak{C}[-\infty,\infty]$ coincides with
$\operatorname{Cos_{\beta,\alpha}^{\textnormal{skew}}}$—that is, describes the
skew Brownian motion with the roles of $\alpha$ and $\beta$ exchanged.

To make these remarks more precise, we introduce the isometric isomorphism

$J\colon\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp})\to\mathfrak{C}[-\infty,\infty]$

of $\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp})$ and
$\mathfrak{C}[-\infty,\infty]$, given by $Jf(x)=-f(x),x<0$, $Jf(x)=f(x),x>0$
and $Jf(0)=f(0+)$. In terms of $J$ our result takes the following form.

######  Proposition 5.2.

For all $t\in\mathbb{R}$,

$J\operatorname{Cos_{\alpha,\beta}^{\textnormal{weks}}}(t)J^{-1}=\operatorname{Cos_{\beta,\alpha}^{\textnormal{skew}}}(t).$

###### Proof.

Let $\mathcal{J}$ be the isometric automorphism of $\mathcal{C}$ given by
$\mathcal{J}(f_{1},f_{2})=(-f_{1},f_{2})$. It is clear that $\mathcal{J}$ is
its own inverse. We note the following three identities that can be
established by a straightforward calculation.

  * (i)

$E_{\alpha,\beta}^{\textnormal{weks}}J^{-1}=\mathcal{J}E_{\beta,\alpha}^{\textnormal{skew}}$.

  * (ii)

$\mathcal{J}C_{\textnormal{D}}(t)\mathcal{J}=C_{\textnormal{D}}(t),t\in\mathbb{R}$.

  * (iii)

$JR(f_{1},f_{2})=R\mathcal{J}(f_{1},f_{2}),$ provided that
$f_{1}(0)=-f_{2}(0)$.

Hence,

$\displaystyle
J\operatorname{Cos_{\alpha,\beta}^{\textnormal{weks}}}(t)J^{-1}$
$\displaystyle=(JR)C_{\textnormal{D}}(t)(E_{\alpha,\beta}^{\textnormal{weks}}J^{-1})=(R\mathcal{J})C_{\textnormal{D}}(t)(\mathcal{J}E_{\beta,\alpha}^{\textnormal{skew}})$
$\displaystyle=RC_{\textnormal{D}}(t)E_{\beta,\alpha}^{\textnormal{skew}}=\operatorname{Cos_{\beta,\alpha}^{\textnormal{skew}}}(t),$

as desired. ∎

We have established that, as $n\to\infty$, along with
$\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{s-o}}}$
converge also complementary families
$\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{o-s}}}$;
the first of these families converge to
$\operatorname{Cos_{\alpha,\beta}^{\textnormal{skew}}}$, the second to a
mirror image of $\operatorname{Cos_{\beta,\alpha}^{\textnormal{skew}}}$.

######  Remark 6.

In Theorem 5.1 (a) the rate of convergence can also be estimated. Arguing as
in Remark 3 we can prove that, provided that $Jf$ is Lipschitz continuous with
Lipschitz constant $L$,
$\sup_{t\in\mathbb{R}}\|\operatorname{Cos_{\textrm{$n$}\alpha,\textrm{$n$}\beta}^{\textnormal{o-s}}}(t)f-\operatorname{Cos_{\alpha,\beta}^{\textnormal{weks}}}(t)f\|\leq\frac{K}{n}$
for $K\coloneqq\frac{2L}{\alpha+\beta}$.

##  6\. Additional results

To complete the paper, we need to answer two natural questions. First of all,
as a direct consequence of Theorem 4.1 (a),

$\lim_{n\to\infty}P_{n\alpha,n\beta}^{\textnormal{o-s}}(f_{1},f_{2})=P_{\alpha,\beta}^{\textnormal{weks}}(f_{1},f_{2})$

where $P_{\alpha,\beta}^{\textnormal{o-s}}$ was defined in (5.1) whereas

$P_{\alpha,\beta}^{\textnormal{weks}}(f_{1},f_{2})\coloneqq\big{(}f_{1}^{o}-\textstyle{\frac{2\alpha}{\gamma^{2}}}k_{1}+{\textstyle\frac{1}{2}}k_{2},f_{2}^{o}-\textstyle{\frac{2\beta}{\gamma^{2}}}k_{1}-{\textstyle\frac{1}{2}}k_{2}\big{)}.$

It is natural to ask whether $P_{\alpha,\beta}^{\textnormal{weks}}$ has
anything to do with $\operatorname{Cos_{\alpha,\beta}^{\textnormal{weks}}}$.
As the following result shows, the answer is in the affirmative:
$P_{\alpha,\beta}^{\textnormal{weks}}$ projects on the invariant space of
extensions related to $\operatorname{Cos_{\alpha,\beta}^{\textnormal{weks}}}$,
that is, on the range of the operator $E_{\alpha,\beta}^{\textnormal{weks}}$.

######  Proposition 6.1.

The map $P_{\alpha,\beta}^{\textnormal{weks}}$ is a projection on the space

$\mathcal{C}_{\alpha,\beta}^{\textnormal{weks}}\coloneqq\\{(g_{1},g_{2})\in\mathcal{C}\colon
g_{1}^{e}=-g_{2}^{e},\alpha g_{1}^{o}=-\beta g_{2}^{o}\\}.$

Moreover, $\mathcal{C}_{\alpha,\beta}^{\textnormal{weks}}$ is the range of
$E_{\alpha,\beta}^{\textnormal{weks}}$.

###### Proof.

We start by noting that $P_{\alpha,\beta}^{\textnormal{weks}}$ is a
projection, being the limit of projections. Hence, to prove the first
statement it suffices to find the range of
$P_{\alpha,\beta}^{\textnormal{weks}}$. To this end, for
$(f_{1},f_{2})\in\mathcal{C}$, let
$(g_{1},g_{2})=P_{\alpha,\beta}^{\textnormal{weks}}(f_{1},f_{2})$, that is,
$g_{1}=f_{1}^{o}-\textstyle{\frac{2\alpha}{\gamma^{2}}}k_{1}+{\textstyle\frac{1}{2}}k_{2}$
and
$g_{2}=f_{2}^{o}-\textstyle{\frac{2\beta}{\gamma^{2}}}k_{1}-{\textstyle\frac{1}{2}}k_{2}$.
The fact that $g_{1},g_{2}\in\mathfrak{C}[-\infty,\infty]$ follows from point
(b) in Theorem 4.1. Also, as in the proof of that result, we obtain
$g_{1}^{e}={\textstyle\frac{1}{2}}(f_{1}^{e}-f_{2}^{e})=-g_{2}^{e}$ and
$\alpha g_{1}^{o}=\frac{2\alpha\beta}{\gamma^{2}}(\beta f_{1}^{o}-\alpha
f_{2}^{o})=-\beta g_{2}^{o}$ implying that
$(g_{1},g_{2})\in\mathcal{C}_{\alpha,\beta}^{\textnormal{weks}}.$ Moreover, it
is easy to check that, if
$(f_{1},f_{2})\in\mathcal{C}_{\alpha,\beta}^{\textnormal{weks}}$, then
$g_{1}=f_{1}$ and $g_{2}=f_{2}$. This proves the first statement in the
proposition.

Identity (i) presented in the proof of Theorem 5.2 can be rewritten as
$E_{\alpha,\beta}^{\textnormal{weks}}=\mathcal{J}E_{\beta,\alpha}^{\textnormal{skew}}J$.
Since the range of $J$ is the entire $\mathfrak{C}[-\infty,\infty]$, the range
of $E_{\alpha,\beta}^{\textnormal{weks}}$ is the image of the range of
$E_{\beta,\alpha}^{\textnormal{skew}}$ under $\mathcal{J}$. However, the range
of $E_{\beta,\alpha}^{\textnormal{skew}}$ is characterized in Theorem 4.1 (b).
This characterization reveals that the image of the range of
$E_{\beta,\alpha}^{\textnormal{skew}}$ is precisely
$\mathcal{C}_{\alpha,\beta}^{\textnormal{weks}}$ as defined above. ∎

######  Corollary 6.2.

The space $\mathcal{C}$ is a direct sum of two spaces that are invariant under
the basic Cartesian product cosine family:

$\displaystyle\mathcal{C}=\mathcal{C}_{\alpha,\beta}^{\textnormal{skew}}\oplus\mathcal{C}_{\alpha,\beta}^{\textnormal{weks}}.$

The second, and final, question one need to answer in this paper reads: is
there a counterpart of Proposition 3.1 pertaining to the complementary cosine
families $\operatorname{Cos_{\alpha,\beta}^{\textnormal{o-s}}}$? The answer is
given below; we omit the proof of this statement, since it is very similar to
that of Proposition 3.1.

######  Proposition 6.3.

For all $f\in\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp})$,

$\displaystyle\lim_{t\to\infty}t^{-1}\int_{0}^{t}\operatorname{Cos_{\alpha,\beta}^{\textnormal{o-s}}}(s)f\,\mathrm{d}s=\lim_{t\to\infty}t^{-1}\int_{0}^{t}\operatorname{Cos_{\alpha,\beta}^{\textnormal{weks}}}(s)f\,\mathrm{d}s$
$\displaystyle=N_{\alpha,\beta}f,$

where
$N_{\alpha,\beta}f\in\mathfrak{C}_{\textnormal{ov}}(\mathbb{R}_{\sharp})$ is
defined as follows

$N_{\alpha,\beta}f(x)\coloneqq\begin{cases}{\textstyle\frac{\alpha}{\alpha+\beta}}f(-\infty)-{\textstyle\frac{\beta}{\alpha+\beta}}f(\infty),&x<0,\\\
{\textstyle\frac{\beta}{\alpha+\beta}}f(\infty)-{\textstyle\frac{\alpha}{\alpha+\beta}}f(-\infty),&x>0.\end{cases}$

We note that this result agrees nicely with Proposition 3.1 and Proposition
5.2, for we have

$\displaystyle\lim_{t\to\infty}t^{-1}\int_{0}^{t}\operatorname{Cos_{\beta,\alpha}^{\textnormal{skew}}}(s)f\,\mathrm{d}s$
$\displaystyle=J\lim_{t\to\infty}t^{-1}\int_{0}^{t}\operatorname{Cos_{\alpha,\beta}^{\textnormal{weks}}}(t)J^{-1}f\,\mathrm{d}s$
$\displaystyle=JN_{\alpha,\beta}J^{-1}f=M_{\beta,\alpha}f.$

##  7\. Appendix

The following lemma is indispensable in proving our convergence theorems.

######  Lemma 7.1.

Let $a>0$, $\phi\in\mathfrak{C}[-\infty,\infty]$ and
$\varphi\in\mathfrak{C}[0,\infty]$ be given. Then,

  * (a)

$\displaystyle\lim_{n\to\infty}na\int_{-\infty}^{x}\mathrm{e}^{-na(x-y)}\phi(y)\,\mathrm{d}y=\phi(x),$

  * (b)

$\displaystyle\lim_{n\to\infty}na\int_{x}^{\infty}\mathrm{e}^{na(x-y)}\phi(y)\,\mathrm{d}y=\phi(x)$,

  * (c)

$\displaystyle\lim_{n\to\infty}\left[na\int_{0}^{x}\mathrm{e}^{-na(x-y)}\varphi(y)\,\mathrm{d}y+\mathrm{e}^{-nax}\varphi(0)\right]=\varphi(x).$

The first two limits are uniform with respect to $x\in\mathbb{R}$, the third
is uniform with respect to $x\geq 0.$

###### Proof.

To prove (a), we apply a typical argument involving so-called Dirac sequences
(see e.g. [lang]*pp. 227–235 or [feller]*pp. 219–220). Namely, given
$\varepsilon>0$ we can find a $\delta$ such that
$|\phi(x)-\phi(y)|<\frac{\varepsilon}{2}$ as long as $|x-y|<\delta$; this is
because members of $\mathfrak{C}[-\infty,\infty]$ are uniformly continuous.
Having chosen such a $\delta$ we can also find an $n_{0}$ so large that
$2\mathrm{e}^{-na\delta}\|\phi\|<\frac{\varepsilon}{2}$, provided that
$n>n_{0}$, where $\|\phi\|\coloneqq\|\phi\|_{\mathfrak{C}[-\infty,\infty]}$.

Next, for any $x\in\mathbb{R}$ we have

$na\int_{-\infty}^{x}\mathrm{e}^{-na(x-y)}\phi(y)\,\mathrm{d}y-\phi(x)=na\int_{0}^{\infty}\mathrm{e}^{-nay}[\phi(x-y)-\phi(x)]\,\mathrm{d}y.$

It follows that for $n>n_{0}$, the absolute value of the left-hand side above
does not exceed

$na\int_{0}^{\delta}\mathrm{e}^{-nay}{\textstyle\frac{\varepsilon}{2}}\,\mathrm{d}y+na\int_{\delta}^{\infty}\mathrm{e}^{-nay}2\|\phi\|\,\mathrm{d}y={\textstyle\frac{\varepsilon}{2}}(1-\mathrm{e}^{-na\delta})+2\mathrm{e}^{-na\delta}\|\phi\|<\varepsilon.$

This completes the proof, $\varepsilon>0$ being arbitrary.

Condition (b) is (a) in disguise: (a) becomes (b) if $\phi$ and $x$ are
replaced by $\phi^{\textnormal{T}}$ and $-x$, respectively. Likewise, to
obtain (c) we use (a) for $\phi\in\mathfrak{C}[-\infty,\infty]$ defined as
follows: $\phi(x)=\varphi(x),x\geq 0$ and $\phi(x)=\varphi(0),x<0.$ ∎

######  Remark 7.

The proof presented above provides no rate of convergence in (a)–(c), but
suggests that this rate depends on modulus of continuity of $\phi$ and
$\varphi$. If we restrict ourselves to $\phi$ that are Lipschitz continuous,
(a) and (b) can be expressed in a more qualitative form. To wit, if
$|\phi(x)-\phi(y)|\leq L|x-y|$ for all $x,y\in\mathbb{R}$ and certain $L>0$,
we have

$\displaystyle\left|na\int_{-\infty}^{x}\mathrm{e}^{-na(x-y)}\phi(y)\,\mathrm{d}y-\phi(x)\right|$
$\displaystyle=\left|na\int_{0}^{\infty}\mathrm{e}^{-naz}[\phi(x-z)-\phi(x)]\,\mathrm{d}z\right|$
$\displaystyle\leq
Lna\int_{0}^{\infty}\mathrm{e}^{-naz}z\,\mathrm{d}z=\frac{L}{na}.$

Similarly,
$|na\int_{x}^{\infty}\mathrm{e}^{na(x-y)}\phi(y)\,\mathrm{d}y-\phi(x)|\leq\frac{L}{na}$.
Moreover, if $|\varphi(x)-\varphi(y)|\leq L$ for $x,y\geq 0$, then
$|na\int_{0}^{x}\mathrm{e}^{-na(x-y)}\varphi(y)\,\mathrm{d}y+\mathrm{e}^{-nax}\varphi(0)-\varphi(x)|\leq\frac{L}{na}$.

## References


experiments. While stylized, our simulation has the advantage that we can vary
key features of the setting like the dimensionality and distribution of the
data, the size of the labeled and unlabeled datasets, the complexity of the
relationship between the features and the outcome, and so on. To be useful,
however, we must be able to ensure that the key conditions of our method are
met by the data-generating process. To ensure this while also allowing for the
tuneability and flexibility we require, we settle on a hierarchical model
specified by parameterized components that are individually simple but can
serve as building blocks. In particular, the model building blocks consist of:

  * •

Primitive features $Z_{1},...,Z_{m}$

  * •

Conditional probability $b$ of being Black a function of $Z_{1}...Z_{m}$

  * •

Realized status as Black or not $B$ drawn from Bernoulli($b$)

  * •

Downstream features $X_{1},...X_{p}$, a function of $Z_{1},...,Z_{m}$ and $B$

  * •

Score for outcome $P(Y)$, a function of downstream features $X_{1}...X_{p}$

  * •

Outcome $Y$,which is an indicator of $P(Y)$ at threshold $\tau$ with some
noise probability of being flipped $0\leftrightarrow 1$

The primitive features $Z_{1},...,Z_{p}$ intuitively represent the variables
that correspond to proxies in BIFSG, e.g. geographic locations. They serve a
dual role: first, as in BIFSG, they give rise to the probability that an
individual is Black. Second, since the secondary features $X$ are a function
of $Z$, they affect the distribution of these features; thus downstream, they
affect $P(Y)$ and ultimately $Y$, but do not directly enter into $P(Y)$ or $Y$
themselves. This corresponds to how geography and other variables which are
correlated to race may also be correlated to many learning-relevant features,
even when not directly entering causing the outcome of interest themselves.
Note that in addition to primitives affecting $P(Y)$ through each $X$, we
allow for $B$ to affect $P(Y)$. This corresponds to how there may be
associations between group membership and features which affect the outcome of
the interest via the downstream features even if the group status is not
directly relevant tot he outcome of interest.

These relationships are not fully specified by the description in the text
above, of course, so we provide details of the selected functional forms in
Table 8. Figure 11 also summarizes the features and their associative
relationships visually. This visualization, along with the language of
directed acyclic graphs (DAGs), allows us to more easily reason about whether
the covariance conditions are likely to be satisfied in our model, at least
for the underlying outcome.

$Z_{1}$$Z_{2}$…$Z_{p}$$b$$B$$P(Y)$$Y$$X_{1}$$X_{2}$…$X_{k}$ Figure 11: A heuristic depiction of the data generating process for our simulations. Nodes indicate random variables, and edges indicate (causal) relationships between nodes. Importantly, relationships are not necessarily linear. Feature | Interpretation | Functional Form
---|---|---
$Z_{j}$ | Primitive Feature |  $Z_{j}\sim U[0,1]$, $j=1,...m$
$X_{i}$ | Secondary Feature | $X_{i}=\sum_{k=1}^{h_{k}}c_{i}X_{i}^{k},i=1,...p$
$h_{k}$ | Degree | $h_{k}\sim U\\{0,1,2,3\\}$
$c_{i}$ | Coefficients |  $c_{i}\sim U[0,1]$, $i=1,...p$
$b$ | Probability Black |  $b=\max\\{0,\min\\{1,\tilde{b}\\}\\}$,
|  | $\tilde{b}\sim\begin{cases}\mathcal{N}(0.1,.04)&\frac{1}{m}\sum_{j=1}^{m}Z_{j}\leq\tau_{b}\\\ \mathcal{N}(0.9,.04)&\frac{1}{m}\sum_{j=1}^{m}Z_{j}>\tau_{b}\end{cases}$
$\tau_{b}$ | Threshold on $b$ | $\frac{1}{2}+1.2\sqrt{1/(12m)}$
| (based Irwin-Hall distribution) |
$B$ | Indicator for Black | $B\sim\text{Bernoulli}(b)$
$\tilde{P}(Y)$ | Score of Outcome | $\tilde{P}(Y)=\sum_{i}\left[d_{i}X_{i}^{k}+d_{iB}B\right]$
$P(Y)$ | Normalized Score of Outcome | $P(Y)=\frac{\tilde{P}(Y)-\min(\tilde{P}(Y))}{\max(\tilde{P}(Y))-\min(\tilde{P}(Y))}$
$Y$ | Realized Outcome | $Y\sim\begin{cases}\text{Bernoulli}(0.1)&P(Y)\leq\tau\\\ \text{Bernoulli}(0.9)&P(Y)>\tau\\\ \end{cases}$
$d_{i}$ | Coefficients for features $X$ | $d_{i}\sim U[0,1]$
$d_{iB}$ | Coefficients for indicator for Black | $d_{iB}\sim U[0,u_{B}]$
Table 8: Description of several variables we use in our simulation study and
their functional forms. For ease of notation, we omit the index denoting
individuals in the dataset. Unspecified constants were selected by inspection
to match key indicators across scenario and are specified in Table 8. Figure
12: We present a three by three figure showing the test disparity of the our
disparity reduction method when compared with relying on only the labeled
subset to reduce disparity by directly enforcing a constraint on the protected
attribute labels. The rows correspond to datasets of increasing sizes (number
of features from 10 to 50), indicating problems of increasing complexity. The
columns correspond to the size of the overall dataset, ranging from 5,000 to
50,000 samples. The x-axis shows the percentage of the total dataset is
decicated to the labeled subset, and the y-axis denotes the percentage
disparity between the two groups calculated on the test set. The blue graphs
correspond to our method, and the orange to the labeled subset method. The red
dashed line is the desired disparity bound. Figure 13: We present a three by
three figure showing the test accuracy of the models created using our
disparity reduction method when compared with relying on training models only
on the labeled subset and reducing disparity by directly enforcing a
constraint on the protected attribute labels. The rows correspond to datasets
of increasing sizes (number of features from 10 to 50), indicating problems of
increasing complexity. The columns correspond to the size of the overall
dataset, ranging from 5,000 to 50,000 samples. The x-axis shows the percentage
of the total dataset is dedicated to the labeled subset, and the y-axis
denotes the test accuracy of the models. The blue graphs correspond to our
method, and the orange to the labeled subset method.

###  G.2 Experimental Setup

Following the notation above, we have $p$ to be the number of features $X$ in
our data, and let $n$ be the number of datapoints. We run experiments for
$p\in\\{10,20,50\\}$ and $n\in\\{5000,10000,50000\\}$. For each $p$, we fix
the parameters in the data generation process and realize 50,000 datapoints.
Refer to Table 9 for a list of parameter values, which differ slightly for
each $p$ to control demographic disparity on the dataset at around 0.25-0.28.
For experiments $n\in\\{5000,10000\\}$, we simply randomly subsample from the
50,000 dataset.

$p$ | $m$ | $\tau$ | $u_{B}$
---|---|---|---
10 | 4 | 0.4 | 0.05
20 | 5 | 0.4 | 0.1
50 | 10 | 0.425 | 0.2
Table 9: List of parameters in the data generation process for each $p$, the
number of secondary features $X$ in the data. $m$ corresponds to the number of
primitive features $Z$, $\tau$ is the threshold for $P(Y)$, while $u_{B}$ is
the upper bound for the uniform distribution to generate $d_{iB}$, see Table
8.

The last dimension we tune is the size of the labeled subset (measured by the
percentage of $n$), which from hereon we refer to as $e$. For each $n$, we
specified slightly different $e$ as outlined in Table 10. This is to account
for the fact that, for instance, one might need 40% of 5,000 datapoints with
protected attribute labels to learn a predictor that reaches the target
disparity bound. On the other hand, using 20% of 50,000 datapoints might be
more than enough, especially considering the exponentially higher costs to
query thousands of people’s protected attributes.

$n$ | $e$
---|---
5,000 | $\\{2,4,6,8,10,15,20,30,40\\}$
10,000 | $\\{1,2,3,4,5,7,10,20,30\\}$
50,000 | $\\{0.5,1,2,3,4,5,7,10,20\\}$
Table 10: Suite of experiments varying percentage of the data taken as labeled
subset ($e$) by the size of the full dataset ($n$).

We prototype these simulation experiments on demographic parity. For each
experiment, we split the data 80/20 into train/test data, then repeat 10 times
with different seeds. We run both our method and the labeled subset method,
evaluating disparity and accuracy on the test set.

###  G.3 Results

We present our results in Figures 12 and 13. In Figure 12, we see that while
increasing the size of the labeled subset can sometimes lead to a regime where
training on the labeled subset alone can produce a model which comes close to
(or in one case–$n=50,000$, $p=10$, reaches) the desired disparity bound, for
the most part, even with a large labeled subset, the mean of the disparity
over 10 trials is above the desired disparity threshold. Meanwhile, our method
stays below the desired disparity threshold across all nine experiments.

As we can see by looking at the rows from top to bottom, the complex (i.e.,
more features in the data) the problem is, the more data is necessary for the
labeled subset to get close to the desired disparity bound. Thus, our
simulation experiment sheds light on the fact that model applications with
small amounts of labeled data, and more complex data, are particularly well-
suited for our method.