# A sharp Harnack bound for a nonlocal heat equation

Mateusz Dembny  Faculty of Mathematics, Mechanics and Informatics
University of Warsaw
ul. Banacha 2, 02-097 Warsaw, Poland<EMAIL_ADDRESS>and  Mikołaj
Sierżęga  Faculty of Mathematics, Mechanics and Informatics
University of Warsaw
ul. Banacha 2, 02-097 Warsaw, Poland Department of Mathematics Cornell
University, 583 Malott Hall Ithaca, NY 14853 USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

A sharp double-sided Harnack bound is derived for positive solutions of a
fractional order heat equation.

###### Key words and phrases:

Harnack inequality, Widder uniqueness theorem, fractional heat equation

###### 2020 Mathematics Subject Classification:

Primary 35R11; Secondary 35K08

##  1\. Introduction

Consider the classical linear heat equation,
$\partial_{t}w-\partial_{xx}^{2}w=0$, posed in an infinite strip
$S_{T}={\mathbb{R}}\times(0,T)$. If we restrict our attention to smooth
positive solutions, then the following important lower bound may be deduced:

(1.1) $\partial_{t}\ln w-|\partial_{x}\ln w|^{2}\geq-\frac{1}{2t}\quad\mbox{
in }\quad S_{T}.$

This result is an instance of a family of estimates derived by Aronson and
Bénilan to tackle the problem of regularity of solutions of the porous medium
equation [1]. A highly consequential generalisation to the setting of
Riemannian manifolds, due to Li and Yau [9], resulted in (1.1) being commonly
associated with their names in the literature.

Note, that inequality (1.1) does not mention the initial moment and applies to
_all positive smooth solutions_ regardless of their origin. In particular, no
further assumptions on the asymptotic behaviour of solutions are required. On
the contrary, it is this remarkable generality of the Aronson-Bénilan-Li-Yau
(ABLY) bound that imposes limitations on the spatial growth of solutions and
the nature of the initial object. It should be stressed that (1.1) is _sharp_
\- a rare and desirable property in the field of analysis of partial
differential equations. The bound is identically satisfied when evaluated on
the fundamental solution,

$g(x-x_{*},t):=\frac{1}{\sqrt{4\pi t}}{\rm e}^{-\frac{|x-x_{*}|^{2}}{4t}},$

_irrespective_ of where the location $x_{*}$ of the source is at the initial
instant. Estimate (1.1) is also called a _differential Harnack bound_ for the
linear heat flow. In line with this terminology, (1.1) may be appropriately
integrated (see [7, 9]), to reveal the well-known classical parabolic Harnack
estimate, due independently to Hadamard and Pini [6, 12],

(1.2) $\frac{w(x_{2},t_{2})}{w(x_{1},t_{1})}\geq\sqrt{\frac{t_{1}}{t_{2}}}{\rm
e}^{-\frac{|x_{2}-x_{1}|^{2}}{4(t_{2}-t_{1})}},$

where $x_{1}$ and $x_{2}$ are arbitrary and $0<t_{1}<t_{2}<T\leq\infty$.

The differential bound (1.1) is _sharper_ than the integrated one (1.2), as
the latter reduces to identity only for $u(x,t)=g(x-x_{*},t)$, with the
initial mass concentrated at a particular point, related to $(x_{i},t_{i})$
through

$x_{*}=\frac{x_{1}t_{2}-x_{2}t_{1}}{t_{2}-t_{1}}.$

Observe moreover, that for the bound to be useful, there needs to be a nonzero
gap between $t_{1}$ and $t_{2}$, i.e., we cannot compare the solution at two
different points in space at the same instant of time. This is a
characteristic feature of parabolic Harnack bounds, see [11].

The following note is inspired by a natural problem, put forward in [5], of
finding an appropriate extension for the ABLY inequality to the context of
nonlocal diffusion and in particular to the canonical model of the fractional
heat equation

$\partial_{t}u+(-\Delta)^{\frac{\alpha}{2}}u=0\quad\mbox{on}\quad S_{T}$

with $0<\alpha<2$. There are numerous definitions of the fractional power of
the Laplace operator, see e.g. [8], none of which will be employed in any of
the considerations below. However, to fix ideas, we may opt for the standard
potential-theoretic definition:

$(-\Delta)^{\frac{\alpha}{2}}f(x):=C(n,\alpha)P.V.\int_{{\mathbb{R}}}\frac{f(x)-f(y)}{|x-y|^{n+\alpha}}{\,\rm
d}y,$

where

$C(n,\alpha):=\frac{2^{\alpha}\Gamma\left(\frac{n+\alpha}{2}\right)}{\pi^{\frac{n}{2}}\left|\Gamma\left(-\frac{\alpha}{2}\right)\right|}.$

For the purpose of this note we will restrict our attention to one space
dimension and $\alpha=1$, i.e.,the _half-laplacian_ , whereby

(1.3) $\partial_{t}u+(-\Delta)^{\frac{1}{2}}u=0\quad\mbox{ on }\quad S_{T},$

with

$(-\Delta)^{\frac{1}{2}}f(x)=-\frac{1}{\pi}P.V.\int_{{\mathbb{R}}}\frac{f(x)-f(y)}{|x-y|^{2}}{\,\rm
d}y.$

Before we begin, let us define the notion of a strong solution, used in the
considerations that follow. Following [2], we will say that $u(x,t)$ is a
_strong solution_ of the fractional heat equation (1.3) in the strip $S_{T}$
if

  * •

$\partial_{t}u\in C({\mathbb{R}}\times(0,T))$,

  * •

$u\in C({\mathbb{R}}\times[0,T))$,

  * •

the equation (1.3) is satisfied pointwise for every $(x,t)\in S_{T}$.

The same definition, with obvious modifications, applies to the notion of the
strong solution for the classical heat equation.

One promising line of inquiry is to find a fractional counterpart of the ABLY
inequality. Note, that (1.1) is equivalent to

$\partial_{xx}^{2}\ln w\geq-\frac{1}{2t}$

in $S_{T}$. This form of the inequality has been generalised to the fractional
setting in [15] where, among other things, the following elegant inequality
has been been proven.

######  Theorem 1.1 (Thm. 3.2 + Prop. 3.3 in [15]).

Let $u:S_{T}\mapsto(0,\infty)$ be a strong solution to the fractional heat
equation (1.3). Then, the Li-Yau type inequality

$-(-\Delta)^{\frac{1}{2}}\ln u\geq-\frac{1}{2t}$

holds in $S_{T}$.

This inequality may be then employed to derive a Harnack bound.

######  Theorem 1.2 (Thm. 5.2 in [15]).

Let $0<t_{1}<t_{2}<\infty$ and $x_{1},x_{2}\in{\mathbb{R}}$. If $u$ is a
strong positive solution of (1.3) on ${\mathbb{R}}\times[0,\infty)$, then

(1.4) $\frac{u(x_{2},t_{2})}{u(x_{1},t_{1})}\geq\sqrt{\frac{t_{1}}{t_{2}}}{\rm
e}^{-C\left[1+\frac{|x_{2}-x_{1}|^{2}}{(t_{2}-t_{1})^{2}}\right]},$

for some positive constant $C$.

For a more general statement and further results consult [15]. In that work,
the authors explain how their bound differs from the Hadamard-Pini estimate.
In particular, (1.4) does not reduce to identity when applied to the
fractional heat kernel with an appropriate choice of $(x_{i},t_{i})$.
Moreover, due to the polynomial behaviour of the heat kernel, it is expected
that the sharp Harnack bound would display a polynomial rather than
exponential decay. Lastly, it would be desirable not to assume continuity of
the solution at the initial time, thus allowing for generalised initial
conditions.

Another result in this direction is provided in [4]. There the authors
consider a weaker class of solutions, the very weak solutions. If we assume,
that the solution is also a strong one, then we obtain the following result in
our setting.

######  Theorem 1.3 (Thm. 8.2 in [4]).

Let $u$ be a positive strong solution of (1.3) for $t>0$. Suppose moreover,
that the initial condition is dominated by the fractional heat kernel away
from the origin, in the following sense:

$0\leq u_{0}(x)\leq\frac{1}{1+|x|^{2}},$

for $|x|\geq R_{0}\geq 0$. Then, for all $x_{1},x_{2}\in{\mathbb{R}}$ and
$t_{1},t_{2}>0$, we have

(1.5)
$\left(\frac{t_{1}}{t_{2}}\right)C_{*}\leq\frac{u(x_{2},t_{2})}{u(x_{1},t_{1})}\leq\left(\frac{t_{1}}{t_{2}}\right)C^{*},$

where

$C_{*}:=C(R_{0})\left(\frac{t_{2}}{t_{1}}\right)^{2}\left[1+\frac{\sqrt{|t_{2}-t_{1}|}+\left||x_{1}|^{2}-|x_{2}|^{2}\right|}{\sqrt{t_{1}}+|x_{1}|^{2}}\right]$

and

$C^{*}:=C(R_{0})\left(\frac{t_{2}}{t_{1}}\right)^{2}\left[1+\frac{\sqrt{|t_{2}-t_{1}|}+\left||x_{1}|^{2}-|x_{2}|^{2}\right|}{\sqrt{t_{2}}+|x_{2}|^{2}}\right],$

for some constant $C$ dependent on $R_{0}$.

The bound (1.5) does reflect the decay rate of the fractional heat kernel.
Moreover, this bound, unlike the Hadamard-Pini estimate, is double-sided. This
interesting feature is a result of nonlocality of the fractional flow. Here
however, a constraint is placed on the initial condition. Since one of the
uses of Harnack bounds is to obtain constraints on the initial data [7], it is
desirable to have bounds, derivation of which avoids introducing restrictions
on the initial datum.

Our contribution in this note concerns obtaining an _unconditional_ bound,
which is to say, that apart from the solution being classical and positive, we
do not impose any further restrictions on the spatial growth of solutions and
of the initial data.

##  2\. The Harnack bound

In order to demonstrate, that no additional requirement is needed, we will
refer to the fractional counterpart of the classical uniqueness theorem of
Widder [2, 16]. The approach presented herein applies both to the classical
heat equation and its fractional counterpart.

Let us first reprove (1.2) through the convolution formula for the heat
equation. To begin with, suppose that $w$ is a nonnegative strong solution on
${\mathbb{R}}\times[0,T)$, with initial condition $w_{0}$. By Widder’s
representation and uniqueness theorems [16], the solution is unique and
expressed by the integral

$w(x,t)=\int_{{\mathbb{R}}}g(x-y,t)w_{0}(y){\,\rm d}y,$

with the kernel

$g(x,t)=\frac{1}{\sqrt{4\pi t}}{\rm e}^{-\frac{|x|^{2}}{4t}}.$

Since this kernel is strictly positive, nothing prevents us from performing an
elementary estimate

$\displaystyle w(x_{2},t_{2})$
$\displaystyle=\int_{\mathbb{R}}\left[\frac{g(x_{2}-y,t_{2})}{g(x_{1}-y,t_{1})}\right]g(x_{1}-y,t_{1})w_{0}(y){\,\rm
d}y$
$\displaystyle\geq\inf_{y\in{\mathbb{R}}}\left[\frac{g(x_{2}-y,t_{2})}{g(x_{1}-y,t_{1})}\right]\int_{\mathbb{R}}g(x_{1}-y,t_{1})w_{0}(y){\,\rm
d}y$
$\displaystyle=w(x_{1},t_{1})\inf_{y\in{\mathbb{R}}}\left[\frac{g(x_{2}-y,t_{2})}{g(x_{1}-y,t_{1})}\right].$

It is easy to convince oneself, that whenever
$0<\sigma_{1}<\sigma_{2}<+\infty$, we have

$\frac{|x_{1}-y|^{2}}{\sigma_{1}}-\frac{|x_{2}-y|^{2}}{\sigma_{2}}\geq-\frac{|x_{2}-x_{1}|^{2}}{\sigma_{2}-\sigma_{1}},$

which in turn, when applied to the heat kernel, yields

${\rm e}^{-\frac{|x_{2}|^{2}}{4\sigma_{2}}}\bigg{/}{\rm
e}^{-\frac{|x_{1}|^{2}}{4\sigma_{1}}}\geq{\rm
e}^{-\frac{|x_{2}-x_{1}|^{2}}{4(\sigma_{2}-\sigma_{1})}},$

i.e.,

(2.1)
$\sqrt{\frac{\sigma_{2}}{\sigma_{1}}}\frac{g(x_{2}-y,\sigma_{2})}{g(x_{1}-y,\sigma_{1})}\geq\sqrt{4\pi(\sigma_{2}-\sigma_{1})}g(x_{2}-x_{1},\sigma_{2}-\sigma_{1}).$

This inequality is sharp and resolves into identity for $y=x_{*}$, with

$x_{*}:=\frac{x_{1}\sigma_{2}-x_{2}\sigma_{1}}{\sigma_{2}-\sigma_{1}}.$

Hence, with $t_{i}=\sigma_{i}$, we find

$w(x_{2},t_{2})\geq
w(x_{1},t_{1})\sqrt{\frac{t_{1}}{t_{2}}}\sqrt{4\pi(t_{2}-t_{1})}g(x_{2}-x_{1},t_{2}-t_{1}),$

which is (1.2).

Thus, we arrive at the desired classical estimate, with this difference
however, that the initial data (and so the solution class covered) is
restricted, as compared to the scope of the technique resting on the ABLY
inequality.

We will now lift this restriction. Suppose $w$ is a smooth positive solution
in ${\mathbb{R}}\times(0,T)$. Choose $0<\tau<t_{1}$ and consider $w^{\tau}$,
defined as the restriction of $w$ to the subdomain
${\mathbb{R}}\times[\tau,T)$. Clearly, $w^{\tau}$ is a strong solution on its
domain and by the Widder representation and uniqueness theorems we have

$w^{\tau}(x,t)=\int_{{\mathbb{R}}^{n}}g(x-y,t-\tau)w(y,\tau){\,\rm d}y$

$w^{\tau}=w$ in ${\mathbb{R}}\times(\tau,T)$. We can now perform the same
estimate as before but with $\sigma_{i}=t_{i}-\tau$ in place of $t_{i}$. In
effect, we find

$\frac{w(x_{2},t_{2})}{w(x_{1},t_{1})}=\frac{w^{\tau}(x_{2},t_{2})}{w^{\tau}(x_{1},t_{1})}\geq\sqrt{\frac{t_{1}-\tau}{t_{2}-\tau}}\sqrt{4\pi(t_{2}-t_{1})}g(x_{2}-x_{1},t_{2}-t_{1}).$

We are free to apply this procedure with any choice of $\tau\in(0,t_{1})$ and
so

$\displaystyle\frac{w(x_{2},t_{2})}{w(x_{1},t_{1})}$
$\displaystyle\geq\sqrt{4\pi(t_{2}-t_{1})}g(x_{2}-x_{1},t_{2}-t_{1})\sup_{0<\tau<t_{1}}\sqrt{\frac{t_{1}-\tau}{t_{2}-\tau}}$
$\displaystyle=\sqrt{\frac{t_{1}}{t_{2}}}\sqrt{4\pi(t_{2}-t_{1})}g(x_{2}-x_{1},t_{2}-t_{1}).$

Thus, provided the solution we work with is smooth and positive in
${\mathbb{R}}\times(0,T)$, we do not need to impose additional constraints on
the nature of the initial condition.

We see then, that it is the Widder representation and uniqueness theorems that
ensure, that our straightforward estimation catches the proper class of
solutions, without unnecessary additional restrictions. To apply a similar
reasoning to the fractional heat flow we need an appropriate generalisation of
these theorems. Indeed, such a result is available [2] and below we cite a
version tailored to our needs.

######  Theorem 2.1 (Thm. 1.4 + Thm. 2.1 in [2]).

Let $v$ be a nonnegative strong solution of the problem

$\partial_{t}v+(-\Delta)^{\frac{1}{2}}v=0\quad\mbox{in}\quad{\mathbb{R}}\times[\tau,T),$

with $v(\cdot,\tau)=\nu\in C({\mathbb{R}})$. Then, $v$ is unique and admits
the representation

$v(x,t)=\int_{{\mathbb{R}}}k(x-y,t-\tau)\nu(y){\,\rm d}y,$

with

$k(x,t):=\frac{1}{\pi t}\left(\frac{1}{1+\frac{|x|^{2}}{t^{2}}}\right).$

In particular, since the kernel is a smooth function, the solution needs to be
smooth in ${\mathbb{R}}\times(\tau,T)$. Actually, in this one-dimensional
context and for the square root of the Laplace operator, the above
representation has been known to Widder even before his celebrated
representation theorem for the classical heat flow [10].

First, we will derive a simple counterpart of (1.2), that relies on a lemma
inspired by the appealing inequality (2.1).

######  Lemma 2.2.

Let $0<\sigma_{1}<\sigma_{2}<+\infty$ and $x_{1},x_{2}\in{\mathbb{R}}$, then

$\left(\frac{\sigma_{2}}{\sigma_{1}}\right)\frac{k(x_{2}-y,\sigma_{2})}{k(x_{1}-y,\sigma_{1})}\geq\pi(\sigma_{2}-\sigma_{1})k(x_{2}-x_{1},\sigma_{2}-\sigma_{1}).$

###### Proof.

Set $\lambda=\frac{|x_{2}-x_{1}|^{2}}{(\sigma_{2}-\sigma_{1})^{2}}$. We need
to show, that

$\frac{1+\frac{|x_{1}-y|^{2}}{\sigma_{1}^{2}}}{1+\frac{|x_{2}-y|^{2}}{\sigma_{2}^{2}}}\geq\frac{1}{1+\frac{|x_{2}-x_{1}|^{2}}{(\sigma_{2}-\sigma_{1})^{2}}},$

which is equivalent to

$\sigma_{2}^{2}|x_{1}-y|^{2}(1+\lambda)-\sigma_{1}^{2}|x_{2}-y|^{2}+\lambda\sigma_{1}^{2}\sigma_{2}^{2}\geq
0.$

This in turn, may be rephrased as a quadratic inequality in $y$:

$Ay^{2}+By+C\geq 0$

with

$\begin{cases}A=(1+\lambda)\sigma_{2}^{2}-\sigma_{1}^{2},\\\
B=2\sigma_{1}^{2}x_{2}-2(1+\lambda)\sigma_{2}^{2}x_{1},\\\
C=(1+\lambda)\sigma_{2}^{2}x_{1}^{2}-\sigma_{1}^{2}x_{2}^{2}+\lambda\sigma_{1}^{2}\sigma_{2}^{2}.\end{cases}$

Since $\lambda\geq 0$ and $\sigma_{2}>\sigma_{1}$, we have $A>0$. Hence, it
suffices to check, that the discriminant is not positive. Thus, after a brief
calculation, this amounts to the requirement that:

$4\sigma_{1}^{2}\sigma_{2}^{2}(1+\lambda)\left|x_{2}-x_{1}\right|^{2}-4\lambda\sigma_{1}^{2}\sigma_{2}^{2}\left[\left(1+\lambda\right)\sigma_{2}^{2}-\sigma_{1}^{2}\right]\leq
0.$

Now, $\sigma_{i}\neq 0$ and by definition
$|x_{2}-x_{1}|^{2}=\lambda(\sigma_{2}-\sigma_{1})^{2}$. Hence, the above
simplifies further to

$\lambda\left[(1+\lambda)(2\sigma_{1}\sigma_{2}-\sigma_{1}^{2})-\sigma_{1}^{2}\right]\geq
0,$

which is true, since $\lambda\geq 0$ and $\sigma_{2}>\sigma_{1}$. ∎

######  Theorem 2.3.

Let $u$ be a smooth positive solution of the fractional heat equation (1.3) in
${\mathbb{R}}\times(0,T)$. Then, given $0<t_{1}<t_{2}<T$ and
$x_{1},x_{2}\in{\mathbb{R}}$, we have

$\frac{u(x_{2},t_{2})}{u(x_{1},t_{1})}\geq\left(\frac{t_{1}}{t_{2}}\right)\frac{1}{1+\frac{|x_{2}-x_{1}|^{2}}{(t_{2}-t_{1})^{2}}}.$

###### Proof.

Choose $0<\tau<t_{1}$ and restrict $u$ to the ${\mathbb{R}}\times[\tau,T)$
subdomain, where it becomes a strong solution. By the Widder-type
representation and uniqueness theorem [2], we may write

$u(x,t)=\int_{{\mathbb{R}}}k(x-y,t-\tau)u(y,\tau){\,\rm d}y.$

Further, due to Lemma 2.2 with $\sigma_{i}=t_{i}-\tau$, we have

$\displaystyle u(x_{2},t_{2})$
$\displaystyle=\int_{{\mathbb{R}}}\left[\frac{k(x_{2}-y,t_{2}-\tau)}{k(x_{1}-y,t_{1}-\tau)}\right]k(x_{1}-y,t_{1}-\tau)u(y,\tau){\,\rm
d}y$
$\displaystyle\geq\left(\frac{t_{1}-\tau}{t_{2}-\tau}\right)\frac{1}{1+\frac{|x_{2}-x_{1}|^{2}}{(t_{2}-t_{1})^{2}}}u(x_{1},t_{1}).$

The above inequality is valid for all $\tau\in(0,t_{1})$ and so we may
optimise by taking $\tau$ arbitrarily small, to the effect that

$u(x_{2},t_{2})\geq\left(\frac{t_{1}}{t_{2}}\right)\frac{1}{1+\frac{|x_{2}-x_{1}|^{2}}{(t_{2}-t_{1})^{2}}}u(x_{1},t_{1}),$

as required. ∎

The above theorem satisfies some of the expected properties of a Harnack bound
for (1.3) in that it is related to the fractional heat kernel and its decay
properties. However, it is one-sided when a double-sided bound is expected and
it does not reduce to identity when the solution is given by the fractional
heat kernel, originating at some specific location related to $(x_{i},t_{i})$.

Next, we move on to our main result, i.e., an optimal fractional counterpart
of the Hadamard-Pini bound.

######  Theorem 2.4.

Let $u$ be a positive classical solution of (1.3) on
${\mathbb{R}}\times(0,T)$. Given $0<t_{1},t_{2}<T$ and
$x_{1},x_{2}\in{\mathbb{R}}$, we have

(2.2)
$\left(\frac{t_{1}}{t_{2}}\right)C_{*}\leq\frac{u(x_{2},t_{2})}{u(x_{1},t_{1})}\leq\left(\frac{t_{1}}{t_{2}}\right)C^{*},$

with

$C_{*}=\frac{\sqrt{\kappa_{0}}-(t_{2}-t_{1})(t_{2}+t_{1})-|x_{2}-x_{1}|^{2}}{\sqrt{\kappa_{0}}-(t_{2}-t_{1})(t_{2}+t_{1})+|x_{2}-x_{1}|^{2}},$
$C^{*}=\frac{\sqrt{\kappa_{0}}+(t_{2}-t_{1})(t_{2}+t_{1})+|x_{2}-x_{1}|^{2}}{\sqrt{\kappa_{0}}+(t_{2}-t_{1})(t_{2}+t_{1})-|x_{2}-x_{1}|^{2}},$

where

$\kappa_{0}=\Big{(}\big{|}x_{2}-x_{1}\big{|}^{2}+\big{|}t_{2}-t_{1}\big{|}^{2}\Big{)}\Big{(}\big{|}x_{2}-x_{1}\big{|}^{2}+\big{|}t_{2}+t_{1}\big{|}^{2}\Big{)}.$

Moreover,

$C_{*}=\frac{x_{1}-x_{*}}{x_{2}-x_{*}}\quad\mbox{ and }\quad
C^{*}=\frac{x_{1}-x^{*}}{x_{2}-x^{*}},$

where $x_{*},x^{*}\in{\mathbb{R}}$ satisfy

$\left(\frac{t_{1}}{t_{2}}\right)C_{*}=\frac{k(x_{2}-x_{*},t_{2})}{k(x_{1}-x_{*},t_{1})}\quad\mbox{
and
}\quad\frac{k(x_{2}-x^{*},t_{2})}{k(x_{1}-x^{*},t_{1})}=\left(\frac{t_{1}}{t_{2}}\right)C^{*}.$

Before laying out the proof, it is worth stressing that, like in the estimate
found in [4], the times $t_{1}$ and $t_{2}$ _need not_ be ordered or
different. The formulae for $M_{*}$ and $M^{*}$ are presented in a possibly
simple form that emphasises the spatial $|x_{2}-x_{1}|$ and temporal
$|t_{2}-t_{1}|$ distance between the points $(x_{1},t_{1})$ and
$(x_{2},t_{2})$. In the Hadamard-Pini bound (1.2), the counterparts of $C_{*}$
and $C^{*}$ depend on the points $(x_{i},t_{i})$ through the distances
$|x_{2}-x_{1}|$ and $|t_{2}-t_{1}|$. In the fractional case however our
estimate is also sensitive to the life-span of the solution prior to the
instants $t_{1}$ and $t_{2}$, which manifests itself through appearance of the
term $(t_{2}+t_{1})$ alongside $(t_{2}-t_{1})$.

###### Proof.

Take $0<\tau<\min\\{t_{1},t_{2}\\}$. As before, when considered on
${\mathbb{R}}\times[\tau,T)$, the solution $u$ is strong and due to the
representation formula on $C({\mathbb{R}}\times[\tau,T))$ we may write

$\displaystyle u(x_{2},t_{2})$
$\displaystyle=\int_{{\mathbb{R}}}k(x_{2}-y,t_{2}-\tau)u(y,\tau){\,\rm d}y$
$\displaystyle=\int_{\mathbb{R}}\left[\frac{k(x_{2}-y,t_{2}-\tau)}{k(x_{1}-y,t_{1}-\tau)}\right]k(x_{1}-y,t_{1}-\tau)u(y,\tau){\,\rm
d}y.$

Since $u$ is positive

$m_{*}(\tau)u(x_{1},t_{1})\leq u(x_{2},t_{2})\leq m^{*}(\tau)u(x_{1},t_{1}),$

where

$m_{*}(\tau):=\inf_{y\in{\mathbb{R}}}\,\frac{k(x_{2}-y,t_{2}-\tau)}{k(x_{1}-y,t_{1}-\tau)}\quad\mbox{and}\quad
m^{*}(\tau):=\sup_{y\in{\mathbb{R}}}\,\frac{k(x_{2}-y,t_{2}-\tau)}{k(x_{1}-y,t_{1}-\tau)}.$

If we now put

$M_{*}:=\sup_{0<\tau<\min\\{t_{1},t_{2}\\}}m_{*}(\tau)\quad\mbox{ and }\quad
M^{*}:=\inf_{0<\tau<\min\\{t_{1},t_{2}\\}}m^{*}(\tau)$

then

$M_{*}\leq\frac{u(x_{2},t_{2})}{u(x_{1},t_{1})}\leq M^{*},$

i.e., (2.2) with

$C_{*}:=\left(\frac{t_{1}}{t_{2}}\right)M_{*}\quad\mbox{ and }\quad
C^{*}=\left(\frac{t_{1}}{t_{2}}\right)M^{*}.$

It remains to show, that these bounds are finite and given as in the statement
of the theorem.

First, we will obtain an explicit expression for $m_{*}(\tau)$ and
$m^{*}(\tau)$ and further show, that $M_{*}=m_{*}(0)$ and $M^{*}=m^{*}(0)$. In
the calculations below, it will be expedient to introduce new coordinates:

$\omega(y):=\frac{1}{2}\left(\frac{x_{2}-y}{t_{2}-\tau}+\frac{x_{1}-y}{t_{1}-\tau}\right)\quad\mbox{and}\quad\rho(y):=\frac{1}{2}\left(\frac{x_{2}-y}{t_{2}-\tau}-\frac{x_{1}-y}{t_{1}-\tau}\right).$

We are interested in extrema of the function

$\frac{k(x_{2}-y,t_{2}-\tau)}{k(x_{1}-y,t_{1}-\tau)}=\left(\frac{t_{1}-\tau}{t_{2}-\tau}\right)\frac{k\left(\frac{x_{2}-y}{t_{2}-\tau},1\right)}{k\left(\frac{x_{1}-y}{t_{1}-\tau},1\right)}=\left(\frac{t_{1}-\tau}{t_{2}-\tau}\right)\frac{1+|\omega-\rho|^{2}}{1+|\omega+\rho|^{2}},$

with respect to $y$. This function is smooth, not constant and approaches
$\frac{t_{2}-\tau}{t_{1}-\tau}$ as $|y|\to+\infty$. Moreover, it achieves
values both above and below $\frac{t_{2}-\tau}{t_{1}-\tau}$. Hence, there is a
global global minimum and a global maximum and we will compute them directly.
It is computationally more convenient to work with the function

$H(y):=\ln\left(\frac{k(x_{2}-y,t_{2}-\tau)}{k(x_{1}-y,t_{1}-\tau)}\right).$

We find that

$H^{\prime}(y)=2\frac{\omega-\rho}{1+|\omega-\rho|^{2}}(\omega^{\prime}-\rho^{\prime})-2\frac{\omega+\rho}{1+|\omega+\rho|^{2}}(\omega^{\prime}+\rho^{\prime})=0,$

when

(2.3)
$2(\rho^{\prime}\omega-\rho\omega^{\prime})(\omega-\rho)(\omega+\rho)+2(\rho\omega)^{\prime}=0.$

Now

$\omega-\rho=\frac{x_{1}-y}{t_{1}-\tau},\quad\omega+\rho=\frac{x_{2}-y}{t_{2}-\tau}\quad\mbox{and}\quad
4\rho\omega=\frac{|x_{2}-y|^{2}}{(t_{2}-\tau)^{2}}-\frac{|x_{1}-y|^{2}}{(t_{1}-\tau)^{2}}.$

Further,

$2(\rho^{\prime}\omega-\rho\omega^{\prime})=\frac{x_{2}-x_{1}}{(t_{2}-\tau)(t_{1}-\tau)}$

and

$2(\rho\omega)^{\prime}=\frac{x_{1}-y}{(t_{1}-\tau)^{2}}-\frac{x_{2}-y}{(t_{2}-\tau)^{2}}.$

Hence, (2.3) may be rephrased as

$\frac{(x_{2}-y)(x_{1}-y)(x_{2}-x_{1})}{(t_{2}-\tau)^{2}(t_{1}-\tau)^{2}}+\frac{x_{1}-y}{(t_{1}-\tau)^{2}}-\frac{x_{2}-y}{(t_{2}-\tau)^{2}}=0,$

or

$(x_{2}-y)(x_{1}-y)(x_{2}-x_{1})+(x_{1}-y)(t_{2}-\tau)^{2}-(x_{2}-y)(t_{1}-\tau)^{2}=0.$

This, of course, is simply a quadratic equation in $y$:

(2.4) $Ay^{2}+By+C=0,$

with

$A=x_{2}-x_{1},\quad
B=-\left[x_{2}^{2}-x_{1}^{2}+(t_{2}-\tau)^{2}-(t_{1}-\tau)^{2}\right]$

and

$C=x_{1}x_{2}(x_{2}-x_{1})+\left(t_{2}-{\tau}\right)^{2}x_{1}-\left(t_{1}-{\tau}\right)^{2}x_{2}.$

We see, that the discriminant,

$\kappa(\tau)=\Big{(}\big{|}x_{2}-x_{1}\big{|}^{2}+\big{|}(t_{2}-\tau)-(t_{1}-\tau)\big{|}^{2}\Big{)}\Big{(}\big{|}x_{2}-x_{1}\big{|}^{2}+\big{|}(t_{2}-\tau)+(t_{1}-{\tau})\big{|}^{2}\Big{)},$

is nonnegative and vanishes, if and only if both $x_{1}=x_{2}$ and
$t_{1}=t_{2}$. If $x_{1}=x_{2}$, we simply get

$m_{*}(\tau)=m^{*}(\tau)=\frac{k(0,t_{2}-\tau)}{k(0,t_{1}-\tau)}=\frac{t_{1}-\tau}{t_{2}-\tau}.$

Otherwise, let $x_{*}(\tau)$ and $x^{*}(\tau)$ be the roots of (2.4):

$x_{*}(\tau):=\frac{x_{2}^{2}-x_{1}^{2}+(t_{2}-\tau)^{2}-(t_{1}-\tau)^{2}-\sqrt{\kappa(\tau)}}{2(x_{2}-x_{1})},$
$x^{*}(\tau):=\frac{x_{2}^{2}-x_{1}^{2}+(t_{2}-\tau)^{2}-(t_{1}-\tau)^{2}+\sqrt{\kappa(\tau)}}{2(x_{2}-x_{1})}.$

Then, from (2.3) we infer, that if $\hat{y}(\tau)$ is either of them, we have

$\displaystyle\frac{1+|w-\rho|^{2}}{1+|w+\rho|^{2}}$
$\displaystyle=\frac{(w-\rho)(w^{\prime}-\rho^{\prime})}{(w+\rho)(w^{\prime}+\rho^{\prime})}=\frac{\big{(}(w-\rho)^{2}\big{)}^{\prime}}{\big{(}|w+\rho|^{2}\big{)}^{\prime}}$
$\displaystyle=\frac{\left(t_{2}-{\tau}\right)^{2}}{\left(t_{1}-{\tau}\right)^{2}}\left(\frac{x_{1}-\hat{y}(\tau)}{x_{2}-\hat{y}(\tau)}\right).$

Further,

$\displaystyle\frac{x_{1}-x_{*}(\tau)}{x_{2}-x_{*}(\tau)}$
$\displaystyle=\frac{\sqrt{\kappa(\tau)}-\big{[}(t_{2}-\tau)-(t_{1}-\tau)\big{]}\big{[}(t_{2}-\tau)+(t_{1}-\tau)\big{]}-|x_{2}-x_{1}|^{2}}{\sqrt{\kappa(\tau)}-\big{[}(t_{2}-\tau)-(t_{1}-\tau)\big{]}\big{[}(t_{2}-\tau)+(t_{1}-\tau)\big{]}+|x_{2}-x_{1}|^{2}}$

and

$\displaystyle\frac{x_{1}-x^{*}(\tau)}{x_{2}-x^{*}(\tau)}$
$\displaystyle=\frac{\sqrt{\kappa(\tau)}+\big{[}(t_{2}-\tau)-(t_{1}-\tau)\big{]}\big{[}(t_{2}-\tau)+(t_{1}-\tau)\big{]}+|x_{2}-x_{1}|^{2}}{\sqrt{\kappa(\tau)}+\big{[}(t_{2}-\tau)-(t_{1}-\tau)\big{]}\big{[}(t_{2}-\tau)+(t_{1}-\tau)\big{]}-|x_{2}-x_{1}|^{2}}.$

Now, we may set

$m_{*}(\tau)=\left(\frac{t_{1}-\tau}{t_{2}-\tau}\right)\frac{x_{1}-x_{*}(\tau)}{x_{2}-x_{*}(\tau)}\quad\mbox{and}\quad
m^{*}(\tau)=\left(\frac{t_{1}-\tau}{t_{2}-\tau}\right)\frac{x_{1}-x^{*}(\tau)}{x_{2}-x^{*}(\tau)}.$

In order to see, that irrespective of $\tau$, we have $m_{*}(\tau)\leq
m^{*}(\tau)$, we note, that given $\alpha$ and $\gamma$ nonnegative and
$\beta\in{\mathbb{R}}$, such that $\alpha>|\beta|+\gamma$, we have

(2.5)
$\frac{\alpha+\beta+\gamma}{\alpha+\beta-\gamma}\geq\frac{\alpha-\beta-\gamma}{\alpha-\beta+\gamma}.$

This follows, since the assumptions on $\alpha,\beta$ and $\gamma$ guarantee,
that both denominators in (2.5) are positive and so equivalently

$(\alpha+\beta+\gamma)(\alpha-\beta+\gamma)-(\alpha-\beta-\gamma)(\alpha+\beta-\gamma)\geq
0,$

which, when expanded, reduces to $4\alpha\gamma\geq 0$.

Set

$\alpha=\sqrt{\kappa(\tau)},\quad\beta=\big{[}(t_{2}-\tau)-(t_{1}-\tau)\big{]}\big{[}(t_{2}-\tau)+(t_{1}-\tau)\big{]}$

and

$\gamma=|x_{2}-x_{1}|^{2}.$

We will now establish, that $\alpha>|\beta|+\gamma$. It suffices to show, that

$\alpha^{2}-\left(|\beta|+\gamma\right)^{2}=\left(\alpha-|\beta|-\gamma\right)\left(\alpha+|\beta|+\gamma\right)>0,$

which, in our case, translates to

$\displaystyle\alpha^{2}$
$\displaystyle=\Big{(}\gamma+\big{|}(t_{2}-\tau)-(t_{1}-\tau)\big{|}^{2}\Big{)}\Big{(}\gamma+\big{|}(t_{2}-\tau)+(t_{1}-{\tau})\big{|}^{2}\Big{)}$
$\displaystyle=\gamma^{2}+\gamma\left[|(t_{2}-\tau)-(t_{1}-\tau)|^{2}+|(t_{2}-\tau)+(t_{1}-{\tau})|^{2}\right]+\beta^{2}$
$\displaystyle=\gamma^{2}+2\gamma\left[(t_{2}-\tau)^{2}+(t_{1}-\tau)^{2}\right]+|\beta|^{2}.$

We have

$(t_{2}-\tau)^{2}+(t_{1}-\tau)^{2}=\left|(t_{2}-\tau)^{2}-(t_{1}-\tau)^{2}\right|+2\min\left\\{(t_{2}-\tau)^{2},(t_{1}-\tau)^{2}\right\\}$

and so

$\alpha^{2}=\left(\gamma+|\beta|\right)^{2}+2\min\left\\{(t_{2}-\tau)^{2},(t_{1}-\tau)^{2}\right\\}>0.$

In effect, we conclude that

$m_{*}(\tau)\leq\frac{k(x_{2}-y,t_{2}-\tau)}{k(x_{1}-y,t_{1}-\tau)}\leq
m^{*}(\tau)$

for all $y\in{\mathbb{R}}$. It is a matter of a tedious but elementary
calculation to show that

$\frac{{\,\rm d}}{{\,\rm
d}\tau}m_{*}(\tau)=-\frac{\sqrt{\left|x_{2}-x_{1}\right|^{2}+\left(t_{2}-t_{1}\right)^{2}}\left((t_{2}-\tau)+(t_{1}-\tau)\right)}{\left(t_{2}-{\tau}\right)\left(t_{1}-{\tau}\right)\sqrt{\left|x_{2}-x_{1}\right|^{2}+\left((t_{2}-\tau)+(t_{1}-\tau)\right)^{2}}}\leq
0$

and likewise

$\frac{{\,\rm d}}{{\,\rm
d}\tau}m^{*}(\tau)=\frac{\sqrt{\left|x_{2}-x_{1}\right|^{2}+\left(t_{2}-t_{1}\right)^{2}}\left((t_{2}-\tau)+(t_{1}-\tau)\right)}{\left(t_{2}-{\tau}\right)\left(t_{1}-{\tau}\right)\sqrt{\left|x_{2}-x_{1}\right|^{2}+\left((t_{2}-\tau)+(t_{1}-\tau)\right)^{2}}}\geq
0.$

From this, we infer that

$M_{*}=\sup_{0<\tau<\min\\{t_{1},t_{2}\\}}m_{*}(\tau)=\left(\frac{t_{1}}{t_{2}}\right)\frac{\sqrt{\kappa_{0}}-(t_{2}-t_{1})(t_{2}+t_{1})-|x_{2}-x_{1}|^{2}}{\sqrt{\kappa_{0}}-(t_{2}-t_{1})(t_{2}+t_{1})+|x_{2}-x_{1}|^{2}}$

and

$M^{*}=\inf_{0<\tau<\min\\{t_{1},t_{2}\\}}m^{*}(\tau)=\left(\frac{t_{1}}{t_{2}}\right)\frac{\sqrt{\kappa_{0}}+(t_{2}-t_{1})(t_{2}+t_{1})+|x_{2}-x_{1}|^{2}}{\sqrt{\kappa_{0}}+(t_{2}-t_{1})(t_{2}+t_{1})-|x_{2}-x_{1}|^{2}}$

with

$\kappa_{0}=\Big{(}\big{|}x_{2}-x_{1}\big{|}^{2}+\big{|}t_{2}-t_{1}\big{|}^{2}\Big{)}\Big{(}\big{|}x_{2}-x_{1}\big{|}^{2}+\big{|}t_{2}+t_{1}\big{|}^{2}\Big{)}$

as required. ∎

##  3\. Discussion

One striking difference between the Hadamard-Pini bound (1.2) and its
fractional counterpart (2.2) is the double-sidedness of the latter. When
perceived as a model of diffusion, that - morally speaking - should share some
broad characteristics with the standard heat flow, it may appear surprising,
that the fractional flow admits an upper bound. Upon closer inspection we see
however, that replacing the Laplace operator with its fractional power
radically constraints the space of initial data, or – if we choose not to
refer to the initial condition – the admissible growth of solutions in the
spatial direction. This limitation is a necessary prerequisite of using
$(-\Delta)^{\frac{1}{2}}$ in the first place. No such _a priori_ constraint is
found in the local case. The relatively narrow domain of the fractional
operator means, that some of the intended applications of the Harnack bound
become irrelevant. For example the Aronson-Bénilan-Li-Yau inequality may be
used to characterise those smooth positive solutions of the heat equation,
that exist on a strip ${\mathbb{R}}\times(0,T)$, leading in effect to Tikhonov
type conditions, see [7]. The necessity of such constraints stems from the
fact that smooth positive solutions of the heat equation may blow-up in finite
time, in case there is so much heat “tucked away at space-infinity”, that the
averaging process commanded by the diffusion operator cannot redistribute it
efficiently enough and there comes a time when the solution becomes unbounded
everywhere. Following this line of thought, we may use the Hadamard-Pini bound
to characterise those initial conditions, that give rise to global-in-time
solutions. Considerations of this type are unnecessary for the fractional heat
equation (1.3), since _all smooth positive solutions are global_ , see [2,
14]. This circumstance is not limited to the particular instance of the
fractional heat flow considered here but applies more broadly to other
exponents and nonlocal operators.

Existence of a double-sided bound adds another layer of difficulty to the
problem of finding a proper extension of differential Harnack bounds of
Aronson-Bénilan-Li-Yau type. In the case of the standard heat flow, the
transition from the differential Harnack bound to the Hadamard-Pini estimate
involves integration along a straight line segment (a geodesic segment in the
setting of Riemannian manifolds) connecting the space-time points being
considered. Adjusting this argument to accommodate two paths – one optimal for
the lower bound and one optimal for the upper bound – is not obvious,
especially given the fundamental nature of the geodesic path used in the
original argument.

In this note we addressed the diffusion process driven by the “half-Laplace”
operator and we took advantage of the explicit form of the heat kernel. The
aim was to obtain, by a direct computation, a sharp Harnack bound, free of the
artifacts brought about by the standard estimation techniques. Even though
there is no such representation of fractional heat kernels for other powers of
the Laplace operator, precise asymptotic bounds have been available for a long
time [3, 13]. The exact form of the Harnack bound (2.2) sheds light on the way
in which the relative position of the two space-time points manifests itself
in the estimate. A generalisation of this result to other powers of the
operator and arbitrary spatial dimension will be addressed in a separate
paper.

### Acknowledgements

This research was partially supported by the Polish National Science Center
grant SONATA BIS no. 2020/38/E/ST1/00596.

The second author also acknowledges the support of the NAWA Bekker Scholarship
Programme BPN/BEK/2021/1/00277.

## References

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# Registered Report:
A Laboratory Experiment on Using Different Financial-Incentivization Schemes
in Software-Engineering Experimentation

Jacob Krüger
Ruhr-University Bochum, Germany  Gül Çalıklı
University of Glasgow, UK  Dmitri Bershadskyy, Robert Heyer
Otto-von-Guericke University Magdeburg, Germany  Sarah Zabel, Siegmar Otto
Otto-von-Guericke University Magdeburg, Germany, & University of Hohenheim,
Germany

###### Abstract

Empirical studies in software engineering are often conducted with open-source
developers or in industrial collaborations. Seemingly, this resulted in few
experiments using financial incentives (e.g., money, vouchers) as a strategy
to motivate the participants’ behavior; which is typically done in other
research communities, such as economics or psychology. Even the current
version of the SIGSOFT Empirical Standards does mention payouts for completing
surveys only, but not for mimicking the real-world or motivating realistic
behavior during experiments. So, there is a lack of understanding regarding
whether financial incentives can or cannot be useful for software-engineering
experimentation. To tackle this problem, we plan a survey based on which we
will conduct a controlled laboratory experiment. Precisely, we will use the
survey to elicit incentivization schemes we will employ as (up to) four payoff
functions (i.e., mappings of choices or performance in an experiment to a
monetary payment) during a code-review task in the experiment:  (1) a scheme
that employees prefer,  (2) a scheme that is actually employed,  (3) a scheme
that is performance-independent, and  (4) a scheme that mimics an open-source
scenario.  Using a between-subject design, we aim to explore how the different
schemes impact the participants’ performance. Our contributions help
understand the impact of financial incentives on developers in experiments as
well as real-world scenarios, guiding researchers in designing experiments and
organizations in compensating developers.

##  1 Motivation

Experimentation in software engineering rarely involves financial incentives
to compensate and motivate participants. However, in most real-world
situations it arguably matters whether software developers are compensated,
for instance, in the form of wages or bug-bounties [24, 22] of open-source
communities. Particularly experimental economists use financial incentives
during experiments for two reasons [41]. First, financial incentives improve
the validity of the experiment by mimicking real-world incentivisation schemes
to motivate participants’ realistic behavior and performance. To this end, in
addition to show-up or participation fees, the actual performance of
participants during the experiment is rewarded by defining a _payoff function_
that maps the participants’ performance during the experiment to financial
rewards or penalties. Second, they allow to study different incentives with
respect to their impact on participants’ performance. It is likely that using
financial incentives in empirical software engineering can help improve the
validity by mimicking and staying true to the real world, too.

Interestingly, there are no guidelines or recommendations on using financial
incentives in software-engineering experimentation. Namely, the current
SIGSOFT Empirical Standards111https://github.com/acmsigsoft/EmpiricalStandards
[29] (as of August 4, 2021; commit b046f37) mention incentives solely in the
context of rewarding participation in surveys. Also, to the best of our
knowledge and based on a literature review, financial incentives that reward
participants’ performance during an experiment are not used systematically in
empirical software engineering. Although some studies broadly incentivize
performance (e.g., Sayagh et al. [31] or Shargabi et al. [32]), these do not
aim to improve the validity of the experiment, only participation.
Furthermore, we know from experimental economics [8, 7] that finding a
realistic (and thus externally valid) way to reward performance is challenging
and no simple one-fits-all solution exists. For instance, the performance of
open-source developers depends less on financial rewards than those of
industrial developers [3, 42, 19, 4].

As a step towards understanding and systematizing the potential of using
financial incentives in software-engineering experimentation, we propose a
two-part study comprising a survey and a controlled experiment in the context
of bug detection through code reviews. First, we will conduct a survey with
practitioners to elicit real-world incentivisation schemes on bug finding. In
the survey, we will distinguish between the schemes most participants prefer
and those actually employed. Building on the results, we will define two
corresponding payoff functions for our experiment. To extend our experiment,
we will add two more payoff functions: one that is performance-independent and
one that resembles the motives of open-source developers. We derive the latter
function using the induced-value method established in experimental economics
[35, 41], which induces a controlled willingness of participants to achieve a
desired goal (i.e., identify a bug) or obtain a certain good during an
experiment by mimicking its monetary value (e.g., a reward). Second, we will
employ our actual between-subject experiment to explore to what extent each of
the four payoff functions impacts the participants’ behavior. Overall, we
primarily contribute to improving researchers’ understanding of whether and
how financial incentives can help software-engineering experimentation.
However, our experiment also has the potential to reveal whether different
incentivisation schemes could improve practitioners’ motivation. Our survey
and experimental design artifacts are available for peer-
reviewing.222https://osf.io/mcxed/?view_only=602088776ce5498597c473e74bbe0394

##  2 Related Work

Experiments in software engineering are comparable to “real-effort
experiments” in experimental economics, which involve participants who solve
certain tasks to increase their payoffs. Consequently, we build on experiences
from the field of experimental economics, which involves a large amount of
literature on how and when to use financial incentives in real-effort
experiments [38, 14, 12, 10]. For instance, some findings indicate gender
differences regarding the impact of incentivization schemes, which we have to
consider during our experiment. In detail, research has shown that men choose
more competitive schemes (e.g., tournaments, performance payments). Similarly,
participants with higher social preferences select such competitive schemes
more rarely [27, 9]. We will consider such factors when analyzing the results
of our experiment (e.g., comparing gender differences if the number of
participants allows).

Unfortunately, there is much less research on incentivization schemes in
software-engineering experimentation. Mason and Watts [26] have analyzed the
impact of financial incentives on crowd performance during software projects
using online experiments. The results are similar to those in experimental
economics, but the authors also acknowledge that they did not design the
incentives to mimic the real world or to improve the participants’ motivation.
Other studies have been concerned with the impact of payments on employees’
motivation [33, 37], job satisfaction [21, 36], or job change [6, 16, 13]. For
instance, Baddoo et al. [3] conducted a case study and found that developers
perceived wages and benefits as an important motivator, but they did not
connect payments to objective performance metrics. None of the studies we are
aware of decomposes payments or wages into specific components (e.g.,
performance-dependent versus performance-independent). So, the effectiveness
of different payoff schemes on developers’ performance remains unclear.

Software-engineering researchers have investigated the motivations of open-
source developers to a much greater extent [11, 18, 15, 42, 19]. From the
economics perspective, open-source systems represent a public good [5, 25]:
they are available to everyone and their consumption do not yield
disadvantages to anyone else. A typical problem of public goods is that
individual and group incentives collide, which usually leads to an
insufficient provision of the good. While typical explanations for open-source
development focus on high intrinsic motivation to contribute or learn, this is
not always the case. For instance, Roberts et al. [30] show that financial
incentives can actually improve open-source developers’ motivation (in terms
of contributions). Still, financial incentives are at least not always the
predominant motivators for software developers [4, 33]. As a consequence, we
will use the concept of open-source software as a social good [19] as an
extreme example (i.e., the developers help solve a social problem, but do not
receive a payment) for designing a fourth payoff function in our experiment.

##  3 Study Protocol

As explained previously, our study involves two data-collection processes, a
survey and a laboratory experiment. In Table 1, we provide an overview of our
study based on the PCI RR study design template, which we will explain in more
detail in this section.

Table 1: PCI RR study design template for our study.

question hypothesis sampling plan analysis plan sensitivity rationale
interpretation disproved theory Which payoff functions are applied/preferred
in SE practice? (survey) N/A At least 30 participants (personal contacts and
social media). We will analyze the absolute frequency of the combinations of
payment components. We compute the mean values of the weights for the MA and
MP combinations. N/A If MAIT and MPIT should be identical, we will reduce the
number of treatments from four to three. N/A How do different payoff functions
impact the performance of participants in SE experiments? (experiment) H1:
Participants without performance-based incentivization (NPIT) have on average
a worse performance than those with performance-based incentivization (e.g.,
OSIT, MAIT, MPIT).
H2: The experimental performance of participants under performance-based
incentivization (e.g., OSIT, MAIT, MPIT) differs between treatments. We aim to
recruit at least 80 (20 per treatment) computer-science students of the Otto-
von-Guericke University Magdeburg. Furthermore, we will conduct an a
posteriori power analysis to reason on the power of our tests. If their
assumptions are fulfilled, we will use parametric tests to compare between the
treatments. Otherwise, we will employ non-parametric tests. For H1, we will
pairwise compare the performance-independent treatment to the other
treatments:  • NPIT vs. MPIT • NPIT vs. MAIT • NPIT vs. OSIT For H2, we will
pairwise compare the performance-dependent treatments:  • MPIT vs. MAIT • MAIT
vs. OSIT • OSIT vs. MPIT In total, we will compute six pairwise tests to
compare the four treatments with one another and will correct for multiple
hypotheses testing (Holm-Bonferroni method). We will also conduct regression
analyses using the treatments as categorical variables (NPIT as base) and age,
gender, experience, as well as arousal as exogenous variables Due to our
experimental design, we face the issue of multiple hypotheses testing. We
address this issue by applying the Holm-Bonferroni correction. We find support
for H1, if our participants’ performance in NPIT is worse and if the tests
between any of our experimental treatments are significant with $p<0.05$
(after correcting with the Holm-Bonferroni method). Confirming H1 means that
the performance is better in the specific treatment with performance-based
incentives compared to NPIT. This implies that if performance plays a role in
a software-engineering experiment, performance-based incentivization should be
considered.
We find support for H2, if our participants performance between the treatments
differs and the respective tests are significant with $p<0.05$ (after
correcting with the Holm-Bonferroni method). Confirming H2 means that the
practitioners’ performance differs depending on the type of incentivization.
If we cannot confirm H2, we do not find evidence for OSIT, MAIT, and MPIT to
induce different performances. There is no theory focusing on the role of
incentives in software engineering. Incentivization in software-engineering
experiments is scarcely applied. Our results could improve experimental
designs in software engineering by guiding researchers when and how to use
incentives in their experiments. NPIT: No Performance Incentives Treatment –
OSIT: Open-Source Incentives Treatment – MAIT: Most-Applied Incentives
Treatment – MPIT: Most-Preferred Incentives Treatment

###  3.1 Survey Design

Goal. With our survey, we aim to explore  i) which payment components (e.g.,
wages only, bug bounties) are most applied (MA) in practice and  ii) which
payment components are most preferred (MP) by practitioners.  We display an
overview of these payment components with concrete examples in Table 2. Our
intention is to understand what is actually employed compared to what would be
preferred as a payment schema to guide the design of our experiment.

Structure. To achieve our goal, we created an online questionnaire with the
following structure (cf. Table 3). At first, we will welcome our participants,
informing them about the survey’s topic, duration, and their right to withdraw
from our experiment at any point in time without any disadvantages.
Furthermore, we will ask for consent to collect, process, and publish the data
in anonymized from. To allow for questions, we will provide the contact data
of at least one author on this first page. Then, we will ask about each
participant’s background to collect _control variables_ , for instance,
regarding their demographics, role in the organization, the domain they work
in, and experience. Based on their roles, the online survey will show the
questions on the payment structures in an adaptive manner. We designed these
questions as well as their answering options based on established guidelines
and our experiences with empirical studies in software engineering [34, 28,
23].

To explore the payment components (_target variables_), we will display the
ones we summarize in Table 2. We will use a checklist in which a participant
can select the components that are applied in their organization. Each
selected component will have a field in which the participant can enter a
percentage to indicate to what extent that component impacts their payment
(e.g., 80 % wage and 20 % bug bounty). Then, we present the same checklist and
fields again. This time, the participant shall define which of the components
they would prefer to contribute with what share to the payment. While we
present this second list as is to any management role (e.g., project manager,
CEO), we ask software engineers (e.g., developer, tester) to decide upon those
components from the perspective of being the team or organization lead. To
prevent sequence effects, we will randomize the order in which the two
treatment questions occur (applied and preferred). Finally, we ask each
participant to indicate how many hours per week they work unpaid
overtime—which represents a type of performance penalty for our payoff
functions—and allow them to enter any additional comments on the survey.

Table 2: List of components of payment we will ask about in our survey to
design payoff functions for the experiment. Note that the term _check_ refers
to participants selecting or deselecting a line of code during our experiment
(i.e., marking them as buggy or correct as can be seen in Figure 1).

payment component example variable _not performance-based_ hourly wage payment
for hours spent on code review $wage$ payment per task fixed payment for
conducting a code review $payment_{fix}$ others specified by participants
_performance-based_ reward for completing review bonus for finding all bugs
$reward_{complete}$ reward for quality bonus for correctly found bug (e.g.,
bug bounty) $reward_{quality}$ reward for time bonus for performing reviews
fast $reward_{time}$ reward for organization’s performance bonus provided
based on the organization’s profits $reward_{share}$ penalty for low quality
penalty for mistakes within a certain period (e.g., missed bugs)
$penalty_{quality}$ penalty for checks penalty for marking lines of code in
the experiment $penalty_{check}$ penalty for required overtime penalty for not
completing within working hours $penalty_{time}$ others specified by
participants

Sampling Participants. We will invite personal contacts and collaborators from
different organizations, involving software developers, project managers, and
company managers. Note that we exclude self-employed or freelancer developers
who typically ask for a fixed payment for a specific task or project. In
addition, we will distribute a second version (to distinguish both
populations) of our survey through our social media networks. Our goal is to
acquire at least 30 responses to obtain a reasonable understanding of applied
and preferred payments. Note that we will not pay incentives for participating
in the survey. We expect that the survey will take 10 minutes at most, but
will verify the required time and understandability of the survey through test
runs with three PhD students from our work groups.

Analysis Plan. To specify the payoff functions for our experiment, we will
analyze the absolute frequency of combinations of different payment components
to identify the most-preferred and most-applied combinations. For these two
combinations, we will compute the mean values for their weights. As an
example, assume that most of our participants would state to prefer the
combination of fixed wages (with a weight of 75 % on average) and bug bounties
(25 % on average). Then, we would define a cost function as
$0.75*payment_{fix}+0.25*(bugs_{correct}*reward_{quality})$.

Threats to Validity. Our survey relies mostly on our personal contacts, which
may bias its outcomes. We can mitigate this threat, since we have a broad set
of collaborators in different countries and organizations. Moreover, defining
the “ideal” payoff function for practitioners may pressure the participants,
is hard to define (e.g., considering different countries, open-source
communities, or expectations), and challenging to measure (e.g., what is
preferred or efficient). However, this is due to the nature of our experiment
and the property we study: financial incentives. Consequently, these threats
remain and we have to discuss their potential impact, which can only be
mitigated with an appropriately large sample population.

Table 3: List of variables we will check in our survey.

variable description operationalization _control variables_ demographics age,
gender, living country, highest level of education nominal (single-choice
list) role participant’s role in their organization nominal (single-choice
list) experience years of experience in software development and code
reviewing 6-level Likert scale (<1 – >15) frequency current involvement in
software development and code reviewing 5-level Likert scale (none at all –
daily) domain domain of the participant’s organization nominal (single-choice
list) size of organization number of employees 5-level Likert scale (<21 –
>200) size of team number of members in participant’s team (if applicable)
6-level Likert scale (1 – >50) development process whether agile or
traditional development processes are employed $\circ$ agile $\circ$ non-agile
_target variables_ MA/MP incentives list of payment components that can be
selected (cf. Table 2) nominal (checklist) MA/MP percentage percentage to
weigh the payment components chosen before continuous (0–100 %) working hours
per week weekly working hours according to the participant’s contract
continuous unpaid overtime potential unpaid overtime of employees in
proportion to working hours per week ratio MA: most applied; MP: most
preferred

###  3.2 Laboratory Experiment

Goal. After eliciting which payoff functions are used and preferred in
practice, we will conduct our actual experiment to measure the impact of
different payoff functions in software-engineering experiments. We focus on
code reviews and bug identification in this experiment, since these are
typical tasks in software engineering that also involve different types of
incentives. So, we aim to support software-engineering researchers by
identifying which payoff functions can be used to improve the validity of
experiments.

Treatments. As motivated, we aim to compare four treatments to reflect
different payoff functions that stem from our survey and established research.
While we are able to define the payoff functions for the “No Performance
Incentives Treatment” (NPIT) and “Open-Source Incentives Treatment” (OSIT) in
advance, we need data from our survey to proceed with the “MP Incentives
Treatment” (MPIT) and “MA Incentives Treatment” (MAIT). However, we can a
priori describe the method we will use to derive the payoff functions for
those treatments. Note that some treatments may yield the same payoff function
(i.e., NPIT, MAIT, and MPIT). It is unlikely that all three payoff functions
will be identical, but we will merge those that are and reduce the number of
treatments accordingly (see Table 2 for the variable names):

No Performance Incentives Treatment (NPIT):

For NPIT, we provide a fixed payment (i.e., 10 €) that will be payed out at
the end of an experimental session. So, this treatment mimics a participation
fee for experiments or fixed wages for the real world. Consequently, the
payoff is independent of a participant’s actual performance. Overall, the
payoff function ($PF$) for this treatment is:

$PF_{NPIT}=payment_{fix}$

Open-Source Incentives Treatment (OSIT):

Again, this treatment does not depend on our survey results, but builds on
findings from the literature on the motivation of open-source developers [11,
18, 15, 42, 19]. We remark that we focus particularly on those developers that
do not receive payments (e.g., as wages or bug bounties), but work for free.
In a simplified perspective, such developers still act within a conceptual
cost-benefit framework (i.e., they perceive to obtain a benefit from working
on the software). Besides a participation fee, we will involve a performance-
based reward for correctly identifying all bugs to resemble goal-oriented
incentives (e.g., personal fulfillment of achieving a goal or supporting open-
source projects). Furthermore, we consider the opportunity costs of working on
open-source software (i.e., less time for other projects and additional effort
for performing a number of checks). Overall, the payoff function ($PF$) for
this treatment is:

$PF_{NPIT}=payment_{fix}+reward_{complete}-time*penalty_{time}-checks*penalty_{checks}$

MA Incentives Treatment (MAIT):

Using our survey results, we will be able the identify a payoff function that
represents what is mostly applied in practice. We will then derive a payoff
function as explained in Section 3.1.

MP Incentives Treatment (MPIT):

We will use the same method we used for MAIT to define a payoff function for
MPIT.

Note that these payoff functions cannot be perfect, but they are mimicking
real-world scenarios, and thus are feasible to achieve our goals.

We use the same code-review example for all treatments to keep the complexity
of the problem constant. For all treatments, we will calibrate the payoff
function so that the expected payoff for each participant in and between
treatments is approximately the same (i.e., around 10 €). Implementing similar
expected payoffs avoids unfairness between treatments, and ensures that
performance differences are caused by different incentive schemes and not the
total size of the payoff. After the treatment, we will gather demographic data
from the participants (e.g., age, gender) and ask for any concerns or
feedback. We estimate that each sessions of the experiment will take 45
minutes.

Code Example. We selected and adapted three different Java code examples
(i.e., limited calculator, sorting and searching, a Stack), the first from the
learning platform LeetCode333https://leetcode.com and the other two from the
“The Algorithms” GitHub repository.444https://github.com/TheAlgorithms/Java To
create buggy examples, we injected three bugs into each code example by using
mutation operators [20]. Note that we partly reworked the examples to make
them more suitable for our experiment (e.g., combining searching and sorting),
added comments at the top of each example explaining its general purpose, and
kept other comments (potentially adapted) as well as identifier names to
improve the realism. We aimed to limit the time of the experiment to avoid
fatigue and actually allow for a laboratory setting, and thus decided to use
only one example. To select the most suitable subject system for our
experiment, we performed a pilot study in which we measured the time and
performance of the participants. In detail, we asked one M.Sc. student from
the University of Glasgow who has worked as a software practitioner in
industry and four PhD students from the University of Zurich to perform the
code reviews on the buggy examples. Overall, each example was reviewed by
three of these participants. Our results indicate that the sorting and
searching example would be most feasible (i.e., $\approx$12 min, 4/9 bugs
correctly identified, 5 false positives), considering that the task should
neither be too easy nor to hard, the background of the pilot’s participants
and the potential participants for our experiment, as well as the examples’
quality. The other two examples seemed too large or complicated (i.e.,
$\approx$14 min, 2/9 bugs; 4 false positives; $\approx$8 min, 5/9 bugs, 8
false positives), which is why we decided to use the sorting and searching
example (available in our artifacts).2 We remark that none of the participants
from this pilot study will be involved in our actual experiment. In Figure 1,
we display a screenshot of the sorting and searching code example as we will
show it to participants in the lab.

Fig. 1: Screenshot of the code example as it will be shown to the
participants. The checkboxes in front of each line allow the participants to
check buggy lines of code. Note that we will not show the comments indicating
the implemented bugs (i.e., in lines 16, 21, and 38).

Sampling Participants. We aim to recruit a minimum of 80 participants (20 per
treatment) by inviting students and faculty members of the Faculty for
Computer Science of the Otto-von-Guericke University Magdeburg, Germany. In
2019, 1,676 Bachelor and Master students as well as roughly 200 PhD students
had been enrolled at the faculty, and 193 (former) members of the faculty are
listed in the participant pool of the MaXLab555https://maxlab.ovgu.de/en/ at
which we will conduct the laboratory experiment. We will focus on recruiting
participants who passed the faculty courses on Java and algorithms (first two
semester) or equivalent courses to ensure that our participants have the
fundamental knowledge required for understanding our sorting and searching
example. If possible (e.g., considering finances), we will invite further
participants (potentially from industry and other faculties) to strengthen the
validity of our results. Note that we focus on the Otto-von-Guericke
University, since the MaXLab is located there. Regarding the Covid pandemic,
it is currently possible to conduct sessions with reduced numbers of
participants (i.e., 10 instead of 20).

Hypotheses. Reflecting on findings in software engineering as well as other
domains, we define two hypotheses (H) we want to study in our experiment:

  1. H1

Participants without performance-based incentivization (NPIT) have on average
a worse performance (lower value in the F1-score, explained shortly) than
those with performance-based incentivization (e.g., OSIT, MAIT, MPIT).

  2. H2

The experimental performance of participants under performance-based
incentivization (e.g., OSIT, MAIT, MPIT) differs between treatments.

Besides analyzing these hypotheses, we will also compare the behavior (e.g.
risk taking) and performance between all groups to understand which incentives
have what impact. Moreover, we will use eye trackers to explore fixation
counts, fixation lengths, and return fixations. This will allows us to obtain
a deeper understanding of the search and evaluation processes during code
reviews. Also, it enables us to investigate potential differences in eye
movements depending on the incentivization. More precisely, we intend to
follow similar studies from software engineering [1] to explore how our
participants read the source code, for instance, do they focus on the actually
buggy code, what lines are they reading more often, or which code elements do
they focus on to explore bugs?

Metrics. The performance of our participants is primarily depending on their
correctness in identifying bugs during the code review. Since this can be
expressed as confusion matrices, we decided to implement the F1-score. Note
that our participants will not be aware of this metric (they will only know
about the payoff function) to avoid biases, and any decision based on the
payoff function will be reflected by the F1-score (e.g., taking more risks due
to missing penalties under NPIT). So, this metric allows us to compare the
performances of our participants between treatments considering that they
motivate different behaviors, which allows us to test our hypotheses.

Experimental Design. We will allocate participants to their treatment at
random, without anyone repeating the experiment in another treatment. On-site,
we can execute the experiment at the experimental laboratory MaXLab of the
Otto-von-Guericke University using a standardized experimental environment. We
will employ a between-subject design measuring the participants’ performance
and measure the eye movement of four participants (restricted by number of
devices) in each session using eye trackers (60 Hz Tobii Pro Nano H). Note
that we will identify any impact wearing eye-trackers may have on our
participants during our analysis. However, it is not likely that they will
have an impact, because this type of eye trackers is mounted to the screen and
barely noticeable, not a helmet the participants have to wear. In case there
is an impact, we will scale up the number of participants to allow us to
ignore this additional variable. The procedure for each session is as follows:

Welcome and Experimental Instructions:

After the participants of a session enter the laboratory, they are randomly
allocated to working stations with the experimental environment installed.
Moreover, four of them are randomly selected for using eye trackers. To this
end, we will already state in the invitation that eye tracking is involved in
the experiment. If a participant nonetheless disagrees to participate using
eye trackers, we will exclude them from the experiment to avoid selection
bias. Once all participants are at their places, the experimenter begins the
experiment. The participants receive general information about the experiment
(e.g., welcoming text), information about the task at hand (code review),
explanation on how to enter data (e.g., check box), and the definition of
their payoff function for the experiment (with some examples).

Review Task:

All participants receive the code example with the task to identify any bugs
within it. Note that the participants will not be aware of the precise number
of bugs in the code. Instead, a message will explain that the code does not
behave as expected when it is executed. At the end of the task, we can
incorporate unpaid overtime as a payment component by asking participants to
stay for five more minutes to work on the task.

Post Experimental Questionnaire:

After the experiment, the participants will receive a set of demographic
questions (cf. Table 3). We will further apply the distress subscale of the
Short Stress State Questionnaire [17] to measure arousal and stress of the
participants. Eliciting such data on demographics and arousal will enable us
to identify potential confounding parameters.

Payoff Procedure:

After we have collected all the data, we will provide information about their
performance and payoff to the participants by displaying them on their screen.
We will pay out these earnings privately in a separate room in cash
immediately afterwards.

Analysis Plan. To analyze our data, we will employ the following steps:

Data Cleaning:

The experimental environment stores raw data in CSV files. We do not plan to
remove any outliers or data unless we identify a specific reason for which we
believe the data would be invalid.

Descriptive Statistics:

We will present descriptive statistics for the demographic, dependent, and
independent variables for each treatment by reporting means and standard
deviations of the respective variables.

Observational Descriptions:

Since sole statistical testing is often subject to misinterpretation and not
recommended [39, 40, 2], we will focus on describing our observations. For
this purpose, we will start with reporting the results we obtained, plotting
suitable visualizations, and identifying patterns within these. The
statistical tests will help us to improve our confidence in these
observations.

Inferential Statistics:

For our analysis, we will focus on performance (i.e., F1 score). We will first
check whether the assumptions required for parametric tests are fulfilled, and
if not proceed with non-parametric tests. Since we are interested in all
possible differences between the four treatments, we have to conduct all
pairwise treatment tests. In total, this leads to 6 tests, or to 3 tests if
our survey indicates that two treatments are mostly identical. For the
significance analyses, we will apply a confidence interval of $p<0.05$ and
correct for multiple hypotheses testing using the Holm-Bonferroni method.
Though the share of participants who will use eye trackers will be constant
among all treatments, and thus should not affect treatment effects, we will
further check whether the presence of eye trackers affected performance. To
increase the statistical robustness, we will also conduct a regression
analysis using the treatments as categorical variables and NPIT as base. As
exogenous variables, we include: age, gender, experience, and arousal of the
participants.

Based on these steps, we will obtain a detailed understanding of how different
incetivization schemes impact the performance of software developers during
code review.

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