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## J Projective and affine Hjelmslev spaces and spaces over rings

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## K Projective lines and circle geometries over rings

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## L Ring geometries and buildings

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## M Ring geometries and coding theory

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## N Ring geometries in quantum information theory

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# Tour de force in Quantum Chromodynamics:
A first next-to-next-to-leading order study of three-jet production at the LHC

Michał Czakon  Institut für Theoretische Teilchenphysik und Kosmologie, RWTH
Aachen University, D-52056 Aachen, Germany Alexander Mitov  Cavendish
Laboratory, University of Cambridge, Cambridge CB3 0HE, UK Rene Poncelet
Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK

###### Abstract

Multi-jet rates at hadron colliders provide a unique possibility for probing
Quantum Chromodynamics (QCD), the theory of strong interactions. By comparing
theory predictions with collider data, one can directly test perturbative QCD,
extract fundamental parameters like the strong coupling $\alpha_{s}$ and
search for physics beyond the Standard Model. In this work we calculate, for
the first time, the next-to-next-to-leading (NNLO) QCD corrections to typical
three-jet observables and to differential three-to-two jet ratios. We
demonstrate that the inclusion of the NNLO corrections significantly reduces
the dependence of those observables on the factorization and renormalization
scales. Besides its phenomenological value, this proof-of-principle
computation represents a milestone in perturbative QCD.

††preprint: Cavendish-HEP-21/09, P3H-21-043, TTK-21-20

##  I Introduction

The production of highly energetic sprays of particles, also known as jets, is
a dominant process at hadron colliders. At high energies, where perturbation
theory is expected to hold, this process offers the possibility for studying
QCD in great detail. The theory–data comparison of differential multi-jet
rates provides essential information about perturbative QCD and the modeling
of jet production. The precision of these predictions is typically limited by
their dependence on unphysical parameters – such as the renormalization and
factorization scales – but it can be systematically increased by including
higher-order corrections.

Three-jet production at the Large Hadron Collider (LHC) has been studied in
great detail by experimental collaborations, see for example ref. Aad:2011tqa
; Aad:2014rma ; CMS:2014mna ; ATLAS:2015yaa ; Aaboud:2018hie ; Aad:2020fch .
Typical observables are jet transverse momenta, angular correlations and, more
generally, event-shape observables. A particular type of observable suited for
perturbative QCD are ratios $R_{3/2}$ of three-to-two jet rates
Chatrchyan:2013txa . These ratios are directly sensitive to parton splittings
and are, therefore, proportional to the strong coupling constant $\alpha_{s}$.
This provides an opportunity for measuring the value of $\alpha_{s}$ at the
LHC. Cross section ratios have the additional advantage that some systematic
uncertainties of experimental and theoretical nature cancel out. A prime
example is the dependence on parton distribution functions (pdf).

There is extensive literature on theoretical predictions for multi-jet
production through NLO in perturbative QCD Ellis:1992en ; Giele:1993dj ;
Nagy:2001fj ; Nagy:2003tz ; Bern:2011ep ; Badger:2013yda , including NLO EW
corrections Dittmaier:2012kx ; Frederix:2016ost ; Reyer:2019obz . NLO
computations have also been matched to parton-showers Alioli:2010xa ;
Hoeche:2012fm  and are generally present in multi-purpose event generators
Gleisberg:2008ta ; Alwall:2014hca ; Frederix:2018nkq . Higher-order
predictions for two-jet and single-inclusive jet production have seen
extensive development in the past decade and are implemented through NNLO in
QCD Currie:2016bfm ; Currie:2017eqf ; Czakon:2019tmo ; AbdulKhalek:2020jut .
The feasibility of NNLO QCD predictions for higher jet multiplicity is limited
by the availability of two-loop virtual amplitudes and by the efficient
treatment of real radiation contributions. The three-jet two-loop amplitudes
have recently been made public in the leading-color approximation
Chicherin:2020oor ; Abreu:2021fuk , leaving the real radiation as the last
obstacle predictions accurate at second order in $\alpha_{S}$.

The aim of this article is twofold. Firstly, it presents NNLO QCD predictions
for the production of three jets and $R_{3/2}$ ratios at the LHC at 13 TeV.
Secondly, it demonstrates the technical ability to treat the NNLO real
radiation contributions for processes with five colored partons at the Born
level. The completion of the second order corrections to three jet production
is a milestone in perturbative QCD computations since, judging by its infra-
red structure, it is among the most complicated two-to-three processes at the
LHC.

This paper is organized as follows: in section II we discuss the technical
details of our computation. Section III contains the phenomenological results
and their analysis. We conclude with a summary and outlook on future
applications in section IV.

##  II Calculation details

The computation has been performed within the sector-improved residue
subtraction scheme formalism Czakon:2010td ; Czakon:2014oma  which has already
been successfully applied to single inclusive jet production Czakon:2019tmo
and many other hadron collider processes, see refs. Czakon:2015owf ;
Chawdhry:2019bji ; Chawdhry:2021hkp . We work in five-flavour massless QCD
without the top quark. Tree-level matrix elements have been taken from the AvH
library avhlib ; Bury:2015dla  while all one-loop matrix elements have been
implemented with the OpenLoops library Buccioni:2019sur . The double virtual
matrix elements are not yet available beyond the leading-colour approximation.
For this reason we approximate the finite two-loop contribution

$\displaystyle\mathcal{R}^{(2)}(\mu_{R}^{2})$ $\displaystyle=$ $\displaystyle
2\real\left[\mathcal{M}^{\dagger(0)}\mathcal{F}^{(2)}\right](\mu_{R}^{2})+\big{|}\mathcal{F}^{(1)}\big{|}^{2}(\mu_{R}^{2})$
(1) $\displaystyle\equiv$ $\displaystyle{\cal
R}^{(2)}(s_{12})+\sum_{i=1}^{4}c_{i}\ln^{i}\left({\mu_{R}^{2}\over
s_{12}}\right)\,,$

in the following way

${\cal R}^{(2)}(s_{12})\approx{\cal R}^{(2)l.c.}(s_{12})\,,$ (2)

where ${\cal R}^{(2)l.c.}(s_{12})$ denotes its leading-colour approximation.
It is taken from the C++ implementation provided in ref. Abreu:2021fuk .

Eq. (2) above is the only approximation made in the present computation. We
have checked that the overall contribution of ${\cal R}^{(2)l.c.}(s_{12})$ is
about $\order{2\%}$ and we expect the missing pure virtual contributions
beyond the leading-colour approximations to be further suppressed.

We consider production of two and three jets at the LHC with a center of mass
energy of 13 TeV with jet requirements adapted from experimental phase space
definitions like, for example, ref. Aad:2020fch . Jets are clustered using the
anti-$k_{T}$ algorithm Cacciari:2008gp  with a radius of $R=0.4$ and required
to have transverse momentum $p_{T}(j)$ of at least $60$ GeV and rapidity
$y(j)$ fulfilling $|y(j)|<4.4$. All jets passing this requirement are sorted
and labeled according to their $p_{T}$ from largest to smallest. Among those
jets we require the two leading jets to fulfill
$p_{T}(j_{1})+p_{T}(j_{2})>250$ GeV in order to avoid large higher-order
corrections in two-jet production close to the phase space boundary. We denote
by $\differential\sigma$ the differential cross section for at least $n$ jets
fulfilling the above criteria. Its expansion in $\alpha_{S}$ reads

$\displaystyle\differential\sigma_{n}=\differential\sigma_{n}^{(0)}+\differential\sigma_{n}^{(1)}+\differential\sigma_{n}^{(2)}+\order{\alpha_{S}^{n+3}}\;$
$\displaystyle\differential\sigma_{n}^{\rm
LO}=\differential\sigma_{n}^{(0)}\;,$
$\displaystyle\differential\sigma_{n}^{\rm
NLO}=\differential\sigma_{n}^{(0)}+\differential\sigma_{n}^{(1)}\;,$
$\displaystyle\differential\sigma_{n}^{\rm
NNLO}=\differential\sigma_{n}^{(0)}+\differential\sigma_{n}^{(1)}+\differential\sigma_{n}^{(2)}\;.$
(3)

We quantify the size of (N)NLO corrections with the help of the following
ratios of differential cross sections

$\displaystyle K^{\rm NNLO}=\frac{\differential\sigma^{\rm
NNLO}}{\differential\sigma^{\rm NLO}}\;\quad\text{and}\quad K^{\rm
NLO}=\frac{\differential\sigma^{\rm NLO}}{\differential\sigma^{\rm LO}}\;.$
(4)

The pdf set NNPDF31_nnlo_as_0118 is used for all perturbative orders. The
renormalization $\mu_{R}$ and factorization $\mu_{F}$ scales are set equal
$\mu_{R}=\mu_{F}=\mu_{0}$. The central scale $\mu_{0}$ is chosen as
$\hat{H}_{T}/n$ for $n=1,2$, where

$\hat{H}_{T}=\sum_{i\in\text{partons}}p_{T,i}\,.$ (5)

The sum in the above equation is over all final state partons, irrespective of
the jet requirements. Previous studies of perturbative convergence in jet
production support this event-based dynamic scale Currie:2018xkj ;
Bellm:2019yyh . Unless stated otherwise, uncertainties from missing higher
orders in perturbation theory are estimated by independent variation of
$\mu_{F}$ and $\mu_{R}$ by a factor of $2$ around the central scale $\mu_{0}$,
subject to the constraint $1/2\leq\mu_{R}/\mu_{F}\leq 2$.

##  III Results

Figure 1: The three panels show the $i$th leading jet transverse momentum
$p_{T}(j_{i})$ for $i=1,2,3$ for the production of (at least) three jets. LO
(green), NLO (blue) and NNLO (red) are shown for the central scale (solid
line). 7-point scale variation is shown as a coloured band. The grey band
corresponds to the uncertainty from Monte Carlo integration.

We begin by discussing typical jet observables at hadron colliders. In fig. 1
we show differential cross sections for three-jet production with respect to
the transverse momentum $p_{T}(j_{i})$ of the $i$th leading jet. In all
histograms the outer bins do not include over- or under-flow events.

The NNLO $K$-factor of the $p_{T}(j_{1})$ distribution is not flat: at small
$p_{T}(j_{1})$ one observes negative NNLO corrections of about $-10\%$, while
at large $p_{T}(j_{1})$ the corrections tend to be small and positive. The
change in scale dependence for this observable when going from NLO to NNLO is
also dependent on $p_{T}(j_{1})$. One observes a rather significant reduction
at large $p_{T}(j_{1})$ (from about $7\%$ at NLO to about $2\%$ at NNLO) while
at small $p_{T}(j_{1})$, where the $K$-factor is largest, the scale dependence
slightly increases (from about $4\%$ at NLO to about $5\%$ at NNLO).
Interestingly, the scale dependence at NLO and NNLO behaves rather
differently: at NLO it steadily increases with $p_{T}(j_{1})$ while at NNLO it
decreases with $p_{T}(j_{1})$. Throughout this work we define the scale
dependence as one half of the width of the scale uncertainty band. This is
relevant for cases where the scale variation is asymmetric, as for example is
the case of $p_{T}(j_{1})$ at NLO.

The $p_{T}(j_{2})$ distribution has a similar pattern of NNLO corrections:
relative to NLO they are negative, about $-20\%$ at low $p_{T}(j_{2})$, and
steadily increase towards larger $p_{T}(j_{2})$ values. At both NLO and NNLO
the scale dependence of $p_{T}(j_{2})$ is similar to that of $p_{T}(j_{1})$.
On the technical side, the convergence of the numerical integration for the
$p_{T}(j_{2})$ spectrum has been significantly slower than for the other
$p_{T}$ observables, which results in increased Monte Carlo uncertainty. To
compensate for this, a larger bin size has been used for $p_{T}(j_{2})$.
Independently of its slower numerical convergence, the $p_{T}(j_{2})$ spectrum
shows good perturbative convergence. Such a behavior is in contrast to the
two-jet case where the sub-leading $p_{T}$ spectrum is known to get large
perturbative corrections due to the strict back-to-back tree-level kinematics
Currie:2018xkj .

The $p_{T}(j_{3})$ distribution is well-behaved: it has a flat $K$-factor and
fairly symmetric uncertainty band at both NLO and NNLO. The scale variation is
almost independent of $p_{T}(j_{3})$, about the $5\%$ at NNLO, which is only
slightly smaller than in the NLO case.

The fact that this observable shows such good convergence and perturbative
stability is somewhat remarkable. Naively, one may suspect that the scale used
here may not perform very well for this distribution because the scale is
based on the kinematics of the full event which, in turn, is dominated by the
leading jet(s). Since this finding may be of relevance for the extraction of
the strong coupling constant from three-jet events, it may be worth
investigating this behavior in more detail. This is outside the scope of the
present work.

Figure 2: The observable $H_{T}$ in two-jet production for two different
central scale choices. Scale variation corresponds to 3-point variation. The
colours are the same as in fig. 1.

Next we discuss the observable $H_{T}$, defined as

$H_{T}=\sum_{i\in\text{jets}}p_{T}(j_{i})\;,$ (6)

where the sum is over all jets that pass the jet requirements. We show this
observable in fig. 2 for the two-jet process and in fig. 3 for the three-jet
process. Both figures are subdivided in two panels showing the same observable
but for a different central scale choice: the upper panels for
$\mu_{0}=\hat{H}_{T}$ and the lower panels for $\mu_{0}=\hat{H}_{T}/2$. Note
that here the colored bands show the scale dependence derived from a 3-point
variation. Turning to the two-jet case we see that both the perturbative
convergence and the scale dependence improve if the central scale choice is
lowered. For $\mu_{0}=\hat{H}_{T}$ the inclusion of the NNLO QCD corrections
does not reduce significantly the scale dependence with respect to NLO and
both bands barely overlap. However, $K^{\rm NNLO}\approx 1.2$ is much smaller
than $K^{\rm NLO}\approx 2$ indicating the stabilization of higher-order
corrections beyond NNLO. For the production of three jets we find that the two
central scale choices $\mu_{0}=\hat{H}_{T}$ and $\mu_{0}=\hat{H}_{T}/2$
produce comparable results, albeit $\mu_{0}=\hat{H}_{T}/2$ has smaller scale
variation. The scale dependence is small compared to the two-jet case and
$K^{\rm NNLO}$ is closer to one. These findings indicate that a central scale
of $\mu_{0}=\hat{H}/2$ leads to slightly better perturbative convergence and,
thus, better approximates the actual energy scale relevant for this
observable. We have checked that even lower scales, like
$\mu_{0}=\hat{H}_{T}/4$, spoil perturbative convergence.

Figure 3: As in fig. 2 but for three-jet production.

As a first application of the NNLO–accurate three-jet rates computed in this
work we consider ratios between three-jet and two-jet rates. The ratios are
defined as

$R_{3/2}(X,\mu_{R},\mu_{F})=\frac{\differential\sigma_{3}(\mu_{R},\mu_{F})/\differential
X}{\differential\sigma_{2}(\mu_{R},\mu_{F})/\differential X}\;,$ (7)

where $X$ is some observable of interest. The (N)NLO ratio is defined in such
a way that the numerator and denominator on the right hand side are evaluated
at the matching order. The scale dependence of the differential cross sections
is shown explicitly to emphasize that the scale choices in the numerator and
denominator are correlated.

In the upper two panels of fig. 4 we show the ratio $R_{3/2}(p_{T}(j_{1}))$.
The ratio changes drastically when going from LO to NLO mostly due to the
change in the two-jet cross section. The NNLO correction stabilizes the ratio
and leads to a very small scale dependence. The $K^{\rm NNLO}$ factor slightly
decreases for large momenta, however, it is always fully contained within the
NLO scale band. An important observation is that the NNLO scale band is very
small in comparison to NLO, reducing it from about $10\%$ down to $3\%$.

Next we consider the lower two panels in fig. 4, where we show the ratio
$R_{3/2}(H_{T})$ for a central scale $\mu_{0}=H_{T}/2$. This observable
behaves similarly to $R_{3/2}(p_{T}(j_{1}))$ albeit with a slightly larger
scale dependence. The reduction in the scale uncertainty when going from NLO
to NNLO is of particular importance since this observable is used
experimentally for measurements of $\alpha_{S}$ Aaboud:2018hie . The leading
source of perturbative uncertainty in this data–theory comparison is the scale
dependence. The pdf dependence, which is not computed in this work, is
expected to largely cancel out in the ratio.

Figure 4: The top two panels show $R_{3/2}(p_{T}(j_{1}))$ (in absolute and as
ratio to NLO) and the bottom two panels $R_{3/2}(H_{T})$. The colours are the
same as in fig. 1.

Jet rates are typically measured in slices of jet rapidity. To demonstrate how
our calculation performs in this situation, we divide the phase space in
slices of the rapidity difference between the two leading jets

$y^{*}=|y(j_{1})-y(j_{2})|/2\;,$ (8)

and define the ratio of the two- and three-jet rates as

$R_{3/2}(H_{T},y^{*})=\frac{\differential^{2}\sigma_{3}/\differential
H_{T}/\differential y^{*}}{\differential^{2}\sigma_{2}/\differential
H_{T}/\differential y^{*}}\;.$ (9)

The NNLO prediction for this cross section ratio can be found in fig. 5 . The
prediction is shown relative to the NLO one. The NNLO correction is negative
across the full kinematic range and, overall, behaves very similarly to the
one for the rapidity-inclusive ratio $R_{3/2}(H_{T})$. This remains the case
as $y^{*}$ increases, at least in the range of rapidities considered here.

Figure 5: The three panels show $R_{3/2}(H_{T},y^{*})$, in each panel a
different slice in $y^{*}$ as ratio to NLO. The colours are the same as in
fig. 1.

##  IV Conclusions

In this work we present for the first time NNLO-accurate predictions for
three-jet rates at the LHC. We compute differential distributions for typical
jet observables like $H_{T}$ and the transverse momentum of the $i$th leading
jet, $i=1,2,3$, as well as differential three-to-two jet ratios. Scale
dependence is the main source of theoretical uncertainty for this process at
NLO, and it gets significantly reduced after the inclusion of the NNLO QCD
corrections. Notably, the three-to-two jet ratios stabilize once the second-
order QCD corrections are accounted for.

A central goal of the present work is to demonstrate the feasibility of three-
jet hadron collider computations with NNLO precision. With this proof-of-
principle goal attained, one can now turn one’s attention to the broad
landscape of phenomenological applications for three-jet production at the
LHC. Examples include studies of event-shapes Banfi:2004nk ; Banfi:2010xy ;
Aad:2020fch , determination of the running of the strong coupling constant
$\alpha_{s}$ through TeV scales and resolving the question of scale setting in
multi-jet production. Another major benefit from having NNLO–accurate
predictions is the reliability of the theory uncertainty estimates.

On the technical side, the enormous computational cost of the present
calculation makes it clear that further refinements in the handling of real
radiation contributions to NNLO calculations are desirable.

###### Acknowledgements.

We would like to thank Manuel Estevez, Javier Llorente and Jennifer Roloff for
helpful discussions about ATLAS jet measurements. The work of M.C. was
supported by the Deutsche Forschungsgemeinschaft under grant 396021762 - TRR
257\. The research of A.M. and R.P. has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 Research and
Innovation Programme (grant agreement no. 683211). A.M. was also supported by
the UK STFC grants ST/L002760/1 and ST/K004883/1. A.M. acknowledges the use of
the DiRAC Cumulus HPC facility under Grant No. PPSP226. Simulations were
performed with computing resources granted by RWTH Aachen University under
project rwth0414.

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# Smoothing of Boundary Behaviour in Stochastic Planar Evolutions

Atul Shekhar Lindstedtsvägen 25, Royal Institute of Technology, Stockholm,
Sweden. Email<EMAIL_ADDRESS>
###### Abstract

Motivated by the study of trace for Schramm-Loewner evolutions, we consider
evolutions of planar domains governed by ordinary differential equations with
holomorphic vector fields $F$ defined on the upper half plane $\mathbb{H}$. We
show a smoothing effect of the presence of noise on the boundary behaviour of
associated conformal maps. More precisely, if $F$ is Hölder, we show that
evolving domains vary continuously in uniform topology and their boundaries
are continuously differentiable Jordan arcs. This is in contrast with examples
from deterministic setting where the corner points on the boundary of domain
$F(\mathbb{H})$ may give rise to corner points on the boundaries of
corresponding evolving domains.

Keywords: Regularization by noise, Stochastic evolutions, trace of Loewner
chains.

AMS 2010 Subject Classification: 60H10, 30C35.

##  1 Introduction and Results.

In this article we consider evolutions of planar domains governed by ordinary
differential equations (ODEs) with holomorphic vector fields. Let
$\mathbb{H}:=\\{z\in\mathbb{C}|Im(z)>0\\}$ be the upper half plane and
$F:\mathbb{H}\to\mathbb{H}$ be a holomorphic function. Let
$U:[0,\infty)\to\mathbb{R}$ be a continuous real valued function with
$U_{0}=0$. One can construct a family of simply connected planar domains by
letting $\mathbb{H}$ evolve under the flow of the equation

$dZ_{t}=F(Z_{t})dt+dU_{t},\hskip 5.69054ptZ_{s}=z\in\mathbb{H}.$ (1.1)

For $(s,t)\in\Delta:=\\{(s,t)|0\leq s\leq t<\infty\\}$, let $\varphi(s,t,z)$
denote the value of solution $Z$ to equation (1.1) at time $t$ solved with the
initial condition $Z_{s}=z\in\mathbb{H}$. Note that since $F$ is holomorphic
on $\mathbb{H}$, viewing (1.1) as a time dependent ODE, solution $Z$ is
uniquely well defined till the time the solution escapes $\mathbb{H}$. Also,
since $Im(F(z))>0$ for $z\in\mathbb{H}$ and $U$ is real valued, $Im(Z_{t})$ is
increasing in time $t$ and thus $Z_{t}$ stays in $\mathbb{H}$. Assuming that
the solution $Z$ doesn’t blow to infinity in finite time, $\varphi(s,t,z)$ is
well defined for all $(s,t)\in\Delta$ and $z\in\mathbb{H}$. We will call
$\varphi:\Delta\times\mathbb{H}\to\mathbb{H}$ as the flow associated to
equation (1.1). Further note that from the uniqueness of solution to ODEs, it
follows easily that for each fixed $(s,t)\in\Delta$, $\varphi(s,t,.)$ is an
injective holomorphic map defined on $\mathbb{H}$. The planar evolution
associated to $(F,U)$ is then defined to be the family
$\mathcal{H}=\\{H_{s,t}\\}_{(s,t)\in\Delta}$ of simply connected domains given
by $H_{s,t}:=\varphi(s,t,\mathbb{H})$. Note that $H_{s,t}\subset
H_{u,t}\subset\mathbb{H}$ for all $s\leq u\leq t$. It however doesn’t satisfy
any inclusion property in the $t$ variable. We are interested in understanding
qualitative properties of domains in $\mathcal{H}$. Following questions
arising naturally in this context:

  1. (a)

Are the boundaries of domains in family $\mathcal{H}$ locally connected, i.e.
can it be parameterized by a continuous function? If yes, are these boundaries
also Jordan arcs?

  2. (b)

How regular are the boundaries of domains in $\mathcal{H}$?

  3. (c)

Do boundaries of domains in $\mathcal{H}$ vary continuously in uniform
topology as $s,t$ vary?

Motivation for considering above questions come from the study of trace in the
theory of Loewner chains. When $F$ is given by the inversion of the upper half
plane, i.e. $F(z)=\frac{-2}{z}$, equation (1.1) is known as reverse time
Loewner differential equation driven by driving function $U$, also see
interesting related works [9],[10] which considers more general vector fields
$F$. Since our main result doesn’t apply to Loewner chains and is related to
it only on a heuristic level, we postpone an introduction to the theory of
Loewner chains and its relation to (1.1) till section 5. The main goal of this
paper is to exhibit a clear distinction in the properties of planar evolution
$\mathcal{H}$ when $U$ is a deterministic and regular driver versus when $U$
is a random and irregular driver. We will show that random and irregular
drivers have some inbuilt smoothing effects which can not be expected when $U$
is a generic deterministic regular driver. We thus argue that the fundamental
reason behind the existence of trace for deterministic and random Loewner
chains are in fact different, see section 5.1.

We will restrict ourselves to Hölder continuous functions $F$. Recall that for
$\alpha\in(0,1]$, $F$ is locally $\alpha$-Hölder if for all bounded subsets
$A\subset\mathbb{H}$,

$||F||_{\alpha,A}:=\sup_{z,w\in A}\frac{|F(z)-F(w)|}{|z-w|^{\alpha}}<\infty.$

$F$ is called globally $\alpha$-Hölder if
$||F||_{\alpha}:=\sup_{A\subset\mathbb{H}}||F||_{\alpha,A}<\infty$. Hölder
holomorphic functions typically arise as Riemann uniformizing maps of simply
connected domains whose boundary may have corners but has no cusps. Such
domains are called Hölder domains, examples include domains bounded by various
fractal curves such as von Koch snowflake. Riemann uniformizing maps of
domains $H_{s,t}$ is by definition given by conformal maps $\varphi(s,t,.)$.
Boundaries of domains $H_{s,t}$ can be thus understood by studying the
boundary behaviour of maps $\varphi(s,t,.)$ which in turn is intricately
connected to the boundary behaviour of $F$. Our main result is the following
theorem answering questions $(a)$-$(c)$ when $F$ is Hölder and $U$ is a sample
path of standard Brownian motion $B$ defined on a filtered probability space
$(\Omega,\\{\mathcal{F}_{t}\\}_{t\geq 0},\mathcal{F},\mathbb{P})$. Domains in
the corresponding stochastic evolution $\mathcal{H}$ will in fact have
continuously differentiable boundaries even if boundary of $F(\mathbb{H})$ may
have corners (we however do not require $F$ to be injective). We denote the
complex derivative of a holomorphic function $G(z)$ by $G^{\prime}(z)$.
$\overline{\mathbb{H}}=\mathbb{H}\cup\mathbb{R}$ is the closed upper half
plane.

######  Theorem 1.

Let $F$ be an uniformly bounded and globally $\alpha$-Hölder holomorphic
function. Then almost surely for all $\omega\in\Omega$ and $U=B(\omega)$,

  1. (i)

$\varphi$ extends continuously to $\Delta\times\overline{\mathbb{H}}$. In
fact, $\varphi$ is jointly $\eta$-Hölder on compact subsets of
$\Delta\times\overline{\mathbb{H}}$ for all $\eta<\frac{1}{2}$. Further, for
all $s\leq t$, $\varphi(s,t,.)$ is injective on $\overline{\mathbb{H}}$. In
particular, boundaries of domains in $\mathcal{H}$ are continuously varying
Jordan arcs.

  2. (ii)

$\varphi^{\prime}$ extends continuously to $\Delta\times\overline{\mathbb{H}}$
and for each $s\leq t$, $\varphi^{\prime}(s,t,.)$ is $\eta$-Hölder on compact
subsets of $\overline{\mathbb{H}}$ for all $\eta<\alpha$.

######  Remark 1.1.

We have assumed $F$ to be uniformly bounded only to guarantee that the
solution to (1.1) doesn’t explode in finite time and the flow $\varphi$ is
globally well defined. This assumption can be replaced by Osgood-type
conditions on $F$, e.g. $F(z)=O(|z|)$ as $z\to\infty$ which implies non-
explosion of solution in finite time. The global Hölder condition on $F$ can
also be replaced by local Hölder condition by using a localisation argument.
For brevity, we will work with the global boundedness and global Hölder
assumptions on $F$ and leave the details of the general case to interested
readers.

######  Remark 1.2.

A classical result known as Kellogg-Warschawski Theorem states that Riemann
uniformizing map of a Jordan domain has derivative which is $\eta$-Hölder on
$\overline{\mathbb{H}}$ for some $\eta\in(0,1)$ if and only if the boundary of
such a domain can be parametrized by a $C^{1+\eta}$ (i.e. differentiable with
$\eta$-Hölder derivative) function defined on $[0,1]$. Thus, Theorem 1-(ii)
can be equivalently stated by saying that boundary of $H_{s,t}$ is of class
$C^{1+\eta}$. Such an equivalence is no longer true for Hölder domains, i.e.
for class $C^{\alpha}$, $\alpha\in(0,1)$. It is not enough for a domain with
the boundary which can be parametrized by a Hölder function to be a Hölder
domain and one requires the boundary to be Hölder in its conformal
parametrization, see Privalov-Hardy-Littlewood Theorem in [5, Chapter $2$] and
[6, Chapter $3$].

######  Example 1.1.

To emphasize on the smoothing effect of noise term $B$ in Theorem 1, consider
the case of $F(z)=z^{\alpha}$ for $\alpha\in(0,1)$. It is easy to check that
$F$ is $\alpha$-Hölder. The function $F$ maps $\mathbb{H}$ to the infinite
sector formed by rays $\\{y=\tan(\pi\alpha)x,x\geq 0\\}$ and $\\{x\geq 0\\}$
meeting at origin and $F(\mathbb{H})$ has a corner at origin. If we consider
the equation $dZ_{t}=F(Z_{t})dt$ without the noise term $B$, the flow
$\varphi$ can be explicitly solved and is given by

$\varphi(s,t,z)=\\{(1-\alpha)(t-s)+z^{1-\alpha}\\}^{\frac{1}{1-\alpha}}.$

Clearly, $\varphi^{\prime}$ doesn’t extend continuously to $0$ for all $s<t$
and domains $\varphi(s,t,\mathbb{H})$ will in fact have a corner at the point
$\varphi(s,t,0)$. Theorem 1 shows that in the presence of noise, such corners
get smoothed out.

######  Example 1.2.

Various examples of random simply connected domains can be constructed with
different choices of holomorphic functions $F:\mathbb{H}\to\mathbb{H}$. Such
functions are also called Nevanlinna-Herglotz functions and are completely
characterized by the representation formula

$F(z)=C+Dz+\int_{\mathbb{R}}\biggl{(}\frac{1}{x-z}-\frac{x}{1+x^{2}}\biggr{)}d\mu(x),$

where $C,D$ are reals with $D\geq 0$ and $\mu$ is a Borel measure on
$\mathbb{R}$ satisfying $\int_{\mathbb{R}}\frac{d\mu(x)}{1+x^{2}}<\infty$.

More examples can be produced by iterating the above scheme. Denote
$\varphi=\varphi^{F}$ to show its dependence on $F$. Starting with any
function $F$, one can construct a sequence of holomorphic vector fields
$\\{F_{n}\\}_{n\geq 0}$ with $F_{0}=F$ and
$F_{n+1}(z)=\varphi^{F_{n}}(0,1,z)$. In the special case when
$F(z)=\frac{-2/\kappa}{z}$, $\kappa>0$, $F_{1}(\mathbb{H})$ is the complement
of Schramm-Loewner-Evolution SLEκ curve in $\mathbb{H}$ which clearly has a
fractal boundary. However, when $\kappa\neq 4$, it was proven in [7] that
$F_{1}(\mathbb{H})$ is a Hölder domain and Theorem 1 implies that the second
iteration $F_{2}(\mathbb{H})$ will in fact have continuously differentiable
boundary.

Theorem 1 has its root in earlier works on the phenomena of regularization by
noise, see [11],[12],[13],[14] and references there in. A similar result as
Theorem 1 was proven in [12, Theorem $5$] using partial differential equation
(PDE) techniques. A key difference between [12, Theorem $5$] and Theorem 1 is
that though the equation (1.1) is a two dimensional system of real valued
equations, a one dimensional noise term $B$ is enough to produce smoothing
effects. The fact that $F$ is complex differentiable on $\mathbb{H}$ is very
crucial in our case so that the set where $F$ is irregular is indeed one
dimensional. We however believe that PDE techniques from [12] can be used to
give a different proof of Theorem 1, see section 5.2 for a brief sketch of
their approach and how to adapt it to our case. Our proof of Theorem 1 is
purely based on stochastic analysis and it will even allow one to consider
non-Hölder functions $F$ satisfying certain conditions. For such conditions on
$F$, we refer the interested reader to [16] where we have used the same
technique as in this paper but in a slightly different context.

To prove Theorem 1, we will consider (1.1) starting from a point on
$\mathbb{R}=\partial\mathbb{H}$ interpreted as a stochastic differential
equation (SDE). We will establish the existence and uniqueness of strong
solution to such SDEs. But since unique strong solutions are only well defined
upto a set of $\mathbb{P}$ measure $0$, we require some additional arguments
to prove the almost sure joint continuity of $\varphi$. Continuity of
$\varphi^{\prime}$ will be established by obtaining an exponential identity
for difference of solutions to (1.1) starting from two different points, see
Proposition 4.1 for details.

The organization of this article is as follows. In section 2, we establish
some preparatory lemmas used in the proof of Theorem 1. In section 3, we prove
the strong uniqueness of solution to (1.1) starting on $\mathbb{R}$. Section 4
contains the proof of Theorem 1. We close the article with some concluding
remarks in section 5.

Acknowledgements: I would like to thank Siva Athreya, Suprio Bhar, Peter Friz,
Fredrik Viklund and Yilin Wang for various discussions and their fruitful
comments. I acknowledge the financial support in the form of KTH visiting
researcher scholarship through research grant of Fredrik Viklund from Göran
Gustafsson foundation.

##  2 Some Preliminary Lemmas.

###  2.1 Itô formula in complex $1$-dimension.

We will use Itô formula as a fundamental tool in our proof. We recall it here
in a non-standard form. We will be interested in $\overline{\mathbb{H}}$
valued semimartingales of form $W_{t}=L_{t}+B_{t}$ where $L$ is a
$\overline{\mathbb{H}}$ valued almost surely continuous and bounded variation
process and $B$ is real valued standard Brownian motion both adapted to the
filtration $\\{\mathcal{F}_{t}\\}_{t\geq 0}$. $W$ is actually a two
dimensional semimartingale with the real valued martingale component, but we
will view $W$ as a complex $1$-dimensional process and use the complex field
multiplication to define quadratic variation process $[W]$ of $W$ in the usual
sense, i.e.

$[W]_{t}:=\lim\limits_{|\mathcal{P}|\to
0}\sum_{[u,v]\in\mathcal{P}}(W_{v}-W_{u})^{2},$

where $\mathcal{P}$ denote partitions of $[0,t]$ and the limit is taken in
probability. Clearly, since $L$ is of bounded variation and $B$ is real
valued, $[W]_{t}=t$. Itô formula gives the semimartingale decomposition for
processes of form $G(B_{t})$ where $G$ is a continuously twice real
differentiable function. Since we are working with complex field operations,
we naturally consider complex differentiable or holomorphic functions $G$ and
compute the semimartingale decomposition of process $G(W_{t})$ as sum of a
$\mathbb{C}$ valued martingale and a bounded variation process.

######  Lemma 2.1.

Let $G:\overline{\mathbb{H}}\to\mathbb{C}$ be a holomorphic function, i.e. $G$
is holomorphic in an open set containing $\overline{\mathbb{H}}$. Then almost
surely,

$G(W_{t})=G(W_{0})+\int_{0}^{t}G^{\prime}(W_{r})dW_{r}+\frac{1}{2}\int_{0}^{t}G^{\prime\prime}(W_{r})d[W]_{r}.$

###### Proof.

The proof follows easily using the power series expansion of holomorphic
functions locally around each point in $\overline{\mathbb{H}}$ and details are
left to the reader. ∎

###  2.2 Linear differential equations in the complex plane.

We will use linear differential equations of form $dS_{t}=S_{t}dV_{t}$ as
another basic tool in our proof. We first recall some standard facts about
Riemann-Stieltjes integrals. Given continuous functions
$M,N:[0,T]\to\mathbb{C}$ such that $M$ is of bounded variation, the Riemann-
Stieltjes integral

$\int_{0}^{t}M_{r}dN_{r}:=\lim\limits_{|\mathcal{P}|\to
0}\sum_{\xi\in[u,v]\in\mathcal{P}}M_{\xi}(N_{v}-N_{u})$

is well defined. Further, the integration by parts formula holds:

$\int_{0}^{t}M_{r}dN_{r}+\int_{0}^{t}N_{r}dM_{r}=M_{t}N_{t}-M_{0}N_{0}.$ (2.1)

In particular, if $\\{M^{k}\\},\\{N^{k}\\}$ are sequences of functions such
that $M^{k}\to M$ in bounded variation norm and $N^{k}\to N$ in supremum norm
as $k\to\infty$, then

$\int_{0}^{.}M_{r}^{k}dN_{r}^{k}\to\int_{0}^{.}M_{r}dN_{r}\hskip
5.69054pt\mbox{in supremum norm as $k\to\infty$.}$

Next, given a continuous function $V:[0,T]\to\mathbb{C}$, consider equations
of form

$S_{t}=S_{0}+\int_{0}^{t}S_{r}dV_{r},$ (2.2)

where we will a priori assume that $S:[0,T]\to\mathbb{C}$ is a continuous
bounded variation function and the right hand side of equation (2.2) is well
defined as a Riemann-Stieltjes integral. We will crucially use the following
lemma.

######  Lemma 2.2.

Let S be a continuous bounded variation function and $V$ be a continuous
function satisfying (2.2). Then,

  1. (i)

if $S_{0}=0$, then $S\equiv 0$.

  2. (ii)

if $S_{0}\neq 0$, then $V$ is of bounded variation and
$S_{t}=S_{0}\exp[V_{t}-V_{0}]$.

###### Proof.

For the part $(i)$, let $Z(S)=\\{t|S_{t}=0\\}$ denote the zero set of $S$
which contains $t=0$ by assumption. If $S$ is not identically zero, then
$[0,T]\setminus Z(S)$ is a non empty open set and it can be written as
countable union of disjoint open intervals. Thus, there exists an interval
$(a,b)$ such that $S_{a}=0$ and $S_{t}\neq 0$ for all $t\in(a,b)$. Then it
easily follows that for all $\epsilon>0$ small enough and
$t\in(a+\epsilon,b)$,

$\int_{a+\epsilon}^{t}\frac{dS_{r}}{S_{r}}=V_{t}-V_{a+\epsilon}.$

Evaluating the real part both sides gives
$\log|S_{t}|-\log|S_{a+\epsilon}|=Re(V_{t})-Re(V_{a+\epsilon})$. Taking
$\epsilon\to 0+$ gives a contradiction. Thus $S$ is identically zero.

For the part $(ii)$, if $S_{0}\neq 0$, it follows from the proof of part $(i)$
that $S_{t}\neq 0$ for all $t$ and

$V_{t}=V_{0}+\int_{0}^{t}\frac{dS_{r}}{S_{r}}.$

Thus $V$ is of bounded variation. Define
$\tilde{S}_{t}=S_{0}\exp[V_{t}-V_{0}]$. Then $\tilde{S}$ is of bounded
variation and solves $d\tilde{S}_{t}=\tilde{S}_{t}dV_{t}$. So,
$d\\{S-\tilde{S}\\}_{t}=\\{S-\tilde{S}\\}_{t}dV_{t}$ with
$S_{0}-\tilde{S}_{0}=0$. It then follows from part $(i)$ that
$S_{t}=\tilde{S}_{t}$ which finishes the proof. ∎

##  3 Uniqueness of strong solution to (1.1) starting on $\mathbb{R}$.

In this section we fix a $x\in\mathbb{R}$ ($x$ can also be a $\mathcal{F}_{0}$
measurable random variable) and consider the SDE

$X_{t}=x+\int_{0}^{t}F(X_{r})dr+B_{t},$ (3.1)

where $F:\mathbb{H}\to\mathbb{H}$ is bounded and $\alpha$-Hölder and solution
$X$ is assumed to be $\overline{\mathbb{H}}$ valued. Since $F$ is Hölder, it
extends continuously to $F:\overline{\mathbb{H}}\to\overline{\mathbb{H}}$ and
the above equation is well defined. We next establish the existence and
uniqueness of $\overline{\mathbb{H}}$ valued strong solution, also see [17]
for a related work.

######  Proposition 3.1.

There exists a unique (upto indistinguishability) continuous process $X$
adapted to the filtration $\\{\mathcal{F}_{t}\\}_{t\geq 0}$, taking values in
$\overline{\mathbb{H}}$ and satisfying equation (3.1).

###### Proof.

We first establish the uniqueness of solution $X$. Let $Y$ be another such
solution. For $\theta\in[0,1]$, define $W^{\theta}_{t}=\theta
X_{t}+(1-\theta)Y_{t}$. Note that $W^{\theta}$ is a $\overline{\mathbb{H}}$
valued semimartingale of the same form as considered in section 2.1. Denote
$S_{t}=X_{t}-Y_{t}$. Then,

$S_{t}=\int_{0}^{t}\\{F(X_{r})-F(Y_{r})\\}dr.$ (3.2)

Note that $S$ is of bounded variation. Since $\mathbb{H}$ is simply connected
and $F$ is holomorphic on $\mathbb{H}$, there exists a holomorphic function
$G$ (unique upto constant) defined on $\mathbb{H}$ such that
$G^{\prime}(z)=F(z)$. Since $F$ is continuous on $\overline{\mathbb{H}}$, $G$
also extends continuously to $\overline{\mathbb{H}}$. Let

$V_{t}=2\int_{0}^{1}\biggl{\\{}G(W_{t}^{\theta})-G(W_{0}^{\theta})-\int_{0}^{t}F(W_{r}^{\theta})dW_{r}^{\theta}\biggr{\\}}d\theta.$

Note that $V$ is almost surely a continuous process. We claim that almost
surely,

$\int_{0}^{t}\\{F(X_{r})-F(Y_{r})\\}dr=\int_{0}^{t}S_{r}dV_{r},$ (3.3)

where the right hand side is a Riemann-Stieltjes integral as discussed in
section 2.2. Equation (3.3) is an instance of famously known Itô-Tanaka trick.
For proving it, introduce functions $F_{y}(z):=F(z+iy),G_{y}(z):=G(z+iy)$ for
$y>0$. Functions $F_{y},G_{y}$ are defined and holomorphic in an open set
(depending on $y$) containing $\overline{\mathbb{H}}$. Using Fubini Theorem
and Lemma 2.1,

$\int_{0}^{t}\\{F_{y}(X_{r})-F_{y}(Y_{r})\\}dr=\int_{0}^{t}\int_{0}^{1}F_{y}^{\prime}(W_{r}^{\theta})S_{r}d\theta
dr=\int_{0}^{t}S_{r}dV_{r}^{y},$ (3.4)

where

$\displaystyle V^{y}_{t}$
$\displaystyle=\int_{0}^{t}\int_{0}^{1}F_{y}^{\prime}(W_{r}^{\theta})d\theta
dr$
$\displaystyle=\int_{0}^{1}\int_{0}^{t}F_{y}^{\prime}(W_{r}^{\theta})drd\theta$
$\displaystyle=2\int_{0}^{1}\biggl{\\{}G_{y}(W_{t}^{\theta})-G_{y}(W_{0}^{\theta})-\int_{0}^{t}F_{y}(W_{r}^{\theta})dW_{r}^{\theta}\biggr{\\}}d\theta.$

Note that $F_{y},G_{y}$ converge uniformly on compacts subsets of
$\overline{\mathbb{H}}$ to $F,G$ respectively as $y\to 0+$. Thus, from
standard facts about Itô stochastic integrals, almost surely $V^{y}$ converge
uniformly to $V$ as $y\to 0+$. Since $S$ is of bounded variation, using
properties of Riemann-Stieltjes integral discussed in section 2.2,

$\int_{0}^{.}S_{r}dV_{r}^{y}\to\int_{0}^{.}S_{r}dV_{r}\hskip
5.69054pt\mbox{uniformly as $y\to 0+$}.$

Thus, taking $y\to 0+$ in equation (3.4) proves equation (3.3). Finally,
combining (3.2),(3.3) and using Lemma 2.2 implies $S\equiv 0$ or $X\equiv Y$
which finishes the proof.
The existence of solution $X$ can be easily established using standard Picard
iteration method and details are left to the reader. ∎

##  4 Proof of Theorem 1.

We complete the proof of Theorem 1 in this section by providing candidates for
continuous extensions of $\varphi$ and $\varphi^{\prime}$ to
$\Delta\times\overline{\mathbb{H}}$. For
$(s,t,z)\in\Delta\times\overline{\mathbb{H}}$, with a slight abuse of
notation, we use the same symbol $\varphi(s,t,z)$ to denote the random
variable defined by the value at time $t$ of the strong solution to (1.1) with
the initial condition $Z_{s}=z$. Note that we have used Proposition 3.1 to
produce the unique strong solution and the joint law of random variables
$\\{\varphi(s,t,z)\\}_{(s,t,z)\in\Delta\times\overline{\mathbb{H}}}$ is
uniquely well defined. It also follows from Proposition 3.1 that for all fixed
$s\leq u\leq t$ and $z\in\overline{\mathbb{H}}$, almost surely

$\varphi(s,t,z)=\varphi(u,t,\varphi(s,u,z))\hskip 5.69054pt\mbox{and}\hskip
5.69054pt\varphi(s,s,z)=z.$ (4.1)

We call equation (4.1) as the Flow property. Define the following symbols
similarly as in section 3. For $\theta\in[0,1]$, $(s,t)\in\Delta$ and
$z,w\in\overline{\mathbb{H}}$,

$W^{\theta}(s,t,z,w):=\theta\varphi(s,t,z)+(1-\theta)\varphi(s,t,w),$

$\displaystyle I^{\theta}(s,t,z,w):=G($ $\displaystyle
W^{\theta}(s,t,z,w))-G(W^{\theta}(s,s,z,w))$
$\displaystyle-\int_{s}^{t}F(W^{\theta}(s,r,z,w))(\theta
F(\varphi(s,r,z))+(1-\theta)F(\varphi(s,r,w)))dr,$

where $G^{\prime}=F$. Further define,

$J^{\theta}(s,t,z,w):=\int_{s}^{t}F(W^{\theta}(s,r,z,w))dB_{r},$
$I(s,t,z,w):=\int_{0}^{1}I^{\theta}(s,t,z,w)d\theta,\hskip
5.69054ptJ(s,t,z,w):=\int_{0}^{1}J^{\theta}(s,t,z,w)d\theta$

and

$V(s,t,z,w):=2\\{I(s,t,z,w)-J(s,t,z,w)\\}.$

The following Proposition is the key to Theorem 1.

######  Proposition 4.1.

Let $F$ be uniformly bounded globally $\alpha$-Hölder function. Then the
following holds:

  1. (i)

For each fixed $s,t,z,w$, almost surely

$\varphi(s,t,z)-\varphi(s,t,w)=(z-w)\exp[V(s,t,z,w)].$

  2. (ii)

For $p\geq 2$, there exists a constant $C=C(p,F,T)$ depending only on $p,$ $F$
and $T$ such that for all $0\leq s\leq t\leq T$, $0\leq u\leq v\leq T$ and
$z,w\in\overline{\mathbb{H}}$,

$\mathbb{E}[|\varphi(s,t,z)-\varphi(u,v,w)|^{p}]\leq
C(|s-u|^{\frac{p}{2}}+|t-v|^{\frac{p}{2}}+|z-w|^{p}).$

  3. (iii)

For $p\geq\frac{2}{\alpha}$, there exists a constant $C=C(p,F,T)$ depending
only on $p$, $F$ and $T$ such that for all $0\leq s\leq t\leq T$, $0\leq u\leq
v\leq T$ and $z,w,\tilde{z},\tilde{w}\in\overline{\mathbb{H}}$,

$\mathbb{E}[|J(s,t,z,w)-J(u,v,\tilde{z},\tilde{w})|^{p}]\leq
C(|s-u|^{\frac{p\alpha}{2}}+|t-v|^{\frac{p\alpha}{2}}+|z-\tilde{z}|^{p\alpha}+|w-\tilde{w}|^{p\alpha}).$

###### Proof.

  1. $(i)$

Let $S_{t}=\varphi(s,t,z)-\varphi(s,t,w)$. Then

$S_{t}=z-w+\int_{s}^{t}\\{F(\varphi(s,r,z))-F(\varphi(s,r,w))\\}dr.$

Similarly as in the proof of Proposition 3.1, it follows that almost surely,

$\int_{s}^{t}\\{F(\varphi(s,r,z))-F(\varphi(s,r,w))\\}dr=\int_{s}^{t}S_{r}dV(s,r,z,w)$

Thus

$S_{t}=z-w+\int_{s}^{t}S_{r}dV(s,r,z,w)$

and Lemma 2.2 proves the claim.

  2. $(ii)$

We will use $\lesssim$ to denote inequalities upto multiplication by constants
$C$ depending only on $p,F,T$ which may change from line to line. W.l.o.g. we
can assume $s\leq u$. If $t\in[s,u]$, we use the bound

$\displaystyle|\varphi(s,t,z)-\varphi(u,v,w)|$ $\displaystyle=$
$\displaystyle\bigl{|}z-w+\int_{s}^{t}F(\varphi(s,r,z))dr-\int_{u}^{v}F(\varphi(u,r,w))dr+B_{t}-B_{s}-(B_{v}-B_{u})\bigr{|}$
$\displaystyle\lesssim$
$\displaystyle|z-w|+|t-s|+|v-u|+|B_{t}-B_{v}|+|B_{s}-B_{u}|.$

Since in this case $|t-s|\leq|s-u|$ and $|v-u|\leq|v-t|$, the claim follows
using standard moment bounds on increments of Brownian motion.
When $t\geq u$, using the flow property,
$\varphi(s,t,z)-\varphi(u,v,w)=\varphi(u,t,\varphi(s,u,z))-\varphi(u,v,w)$.
Since

$|\varphi(u,v,w)-\varphi(u,t,w)|\lesssim|t-v|+|B_{t}-B_{v}|,$

it is enough to get the desired moment estimate on
$|\varphi(u,t,\varphi(s,u,z))-\varphi(u,t,w)|$. Using previous part $(i)$,

$|\varphi(u,t,\varphi(s,u,z))-\varphi(u,t,w)|=|\varphi(s,u,z)-w|\times|\exp[V(u,t,\varphi(s,u,z),w)]|.$

Again, since $\varphi(s,u,z)$ is $\mathcal{F}_{u}$ measurable and

$|\varphi(s,u,z)-w|\lesssim|z-w|+|s-u|+|B_{s}-B_{u}|,$

it suffices to show that

$\mathbb{E}\bigl{\\{}|\exp[V(u,t,\varphi(s,u,z),w)]|^{p}\bigl{|}\mathcal{F}_{u}\bigr{\\}}\lesssim
1.$ (4.2)

To this end, note that $V=2(I-J)$ and

$|I^{\theta}(u,t,\varphi(s,u,z),w)|\lesssim|t-u|+|B_{t}-B_{u}|.$

Fernique Theorem implies that if $q\lesssim 1$, then
$\mathbb{E}\\{\exp[q|B_{t}-B_{u}|]\\}\lesssim 1$. Thus, using Hölder
inequality, it suffices to prove that for $q\lesssim 1$,

$\mathbb{E}\bigl{\\{}\exp[q|ReJ(u,t,\varphi(s,u,z),w)|]\bigl{|}\mathcal{F}_{u}\bigr{\\}}\lesssim
1.$

Note that $Re(J^{\theta})$ is a martingale with
$[Re(J^{\theta})]_{t}\lesssim(t-u)$. By Dambis-Dubins-Schwarz martingale
embedding theorem, $Re(J_{t}^{\theta})=\tilde{B}_{[Re(J^{\theta})]_{t}}$ for
some another Brownian motion $\tilde{B}$. Using the Fernique Theorem again
gives us the bound (4.2) which proves the claim.

  3. $(iii)$

W.l.o.g. we assume $s\leq u$. If $t\in[s,u]$, then using Fubini Theorem for
stochastic integrals and Burkholder-Davis-Gundy inequality,

$\displaystyle\mathbb{E}[|J(s,t,z,w)|^{p}]$
$\displaystyle=\mathbb{E}\biggl{|}\int_{s}^{t}\int_{0}^{1}F(W^{\theta}(s,r,z,w))d\theta
dB_{r}\biggr{|}^{p}$
$\displaystyle\lesssim\mathbb{E}\biggl{(}\int_{s}^{t}\biggl{\\{}\int_{0}^{1}|F(W^{\theta}(s,r,z,w))|d\theta\biggr{\\}}^{2}dr\biggr{)}^{\frac{p}{2}}$
$\displaystyle\lesssim|t-s|^{\frac{p}{2}}$
$\displaystyle\lesssim|s-u|^{\frac{p}{2}}.$

Similarly,
$\mathbb{E}[|J(u,v,\tilde{z},\tilde{w})|^{p}]\lesssim|t-v|^{\frac{p}{2}}$
which gives

$\mathbb{E}[|J(s,t,z,w)-J(u,v,\tilde{z},\tilde{w})|^{p}]\lesssim|s-u|^{\frac{p}{2}}+|t-v|^{\frac{p}{2}}.$

If $t\geq u$, then

$J(s,t,z,w)=\int_{s}^{u}\int_{0}^{1}F(W^{\theta}(s,r,z,w))d\theta
dB_{r}+\int_{u}^{t}\int_{0}^{1}F(W^{\theta}(s,r,z,w))d\theta dB_{r},$

and

$J(u,v,\tilde{z},\tilde{w})=\int_{t}^{v}\int_{0}^{1}F(W^{\theta}(u,r,\tilde{z},\tilde{w}))d\theta
dB_{r}+\int_{u}^{t}\int_{0}^{1}F(W^{\theta}(u,r,\tilde{z},\tilde{w}))d\theta
dB_{r}.$

Thus using the previous step,

$\displaystyle\mathbb{E}[|J(s,t,z,w)-J(u,v,\tilde{z},\tilde{w})|^{p}]$
$\displaystyle\lesssim|s-u|^{\frac{p}{2}}+|t-v|^{\frac{p}{2}}+\Theta(s,t,z,w,u,v,\tilde{z},\tilde{w}),$

where $\Theta(s,t,z,w,u,v,\tilde{z},\tilde{w})$ is defined by

$\Theta(s,t,z,w,u,v,\tilde{z},\tilde{w})=\mathbb{E}\biggl{[}\biggl{|}\int_{u}^{t}\int_{0}^{1}\\{F(W^{\theta}(s,r,z,w))-F(W^{\theta}(u,r,\tilde{z},\tilde{w}))\\}d\theta
dB_{r}\biggr{|}^{p}\biggr{]}.$

Again using the Burkholder-Davis-Gundy inequality and previous part $(ii)$,

$\displaystyle\Theta(s,t,z,w,u,v,\tilde{z},\tilde{w})$
$\displaystyle\lesssim\int_{u}^{t}\int_{0}^{1}\mathbb{E}[|F(W^{\theta}(s,r,z,w))-F(W^{\theta}(u,r,\tilde{z},\tilde{w}))|^{p}]d\theta
dr$
$\displaystyle\lesssim\int_{0}^{T}\mathbb{E}[|\varphi(s,r,z)-\varphi(u,r,\tilde{z})|^{p\alpha}+|\varphi(s,r,w)-\varphi(u,r,\tilde{w})|^{p\alpha}]dr$
$\displaystyle\lesssim|s-u|^{\frac{p\alpha}{2}}+|z-\tilde{z}|^{p\alpha}+|w-\tilde{w}|^{p\alpha},$

and putting together all the above different cases completes the proof.

∎

###### Proof of Theorem 1.

From the moment estimate obtained in Proposition 4.1-$(ii)$, it follows using
Kolmogorov-Chentsov Theorem that the random field
$\\{\varphi(s,t,z)\\}_{(s,t,z)\in\Delta\times\overline{\mathbb{H}}}$ has a
continuous modification which is also almost surely $\eta$-Hölder on compact
subsets of $\Delta\times\overline{\mathbb{H}}$ for all $\eta<\frac{1}{2}$.
With an abuse of notation, we will use the same symbol $\varphi$ to denote its
continuous version. Similarly, using Proposition 4.1-$(iii)$, the random field
$\\{J(s,t,z,w)\\}_{(s,t,z,w)\in\Delta\times\overline{\mathbb{H}}\times\overline{\mathbb{H}}}$
also has a continuous modification which is almost surely $\eta$-Hölder on
compact subsets of
$\Delta\times\overline{\mathbb{H}}\times\overline{\mathbb{H}}$ for all
$\eta<\frac{\alpha}{2}$. Again, we use the same letter $J$ to denote its
continuous version. It follows that $V=2(I-J)$ is almost surely continuous as
well and Proposition 4.1-$(i)$ implies that almost surely for all
$(s,t,z,w)\in\Delta\times\overline{\mathbb{H}}\times\overline{\mathbb{H}}$,

$\varphi(s,t,z)-\varphi(s,t,w)=(z-w)\exp[V(s,t,z,w)].$

This implies that $\varphi(s,t,.)$ is injective and continuously
differentiable on $\overline{\mathbb{H}}$ with
$\varphi^{\prime}(s,t,z)=\exp[V(s,t,z,z)]$ completing the proof. ∎

##  5 An Informal Discussion.

###  5.1 Trace of Loewner chains.

We first briefly recall some basics about chordal Loewner’s theory in the
upper half plane $\mathbb{H}$, see [1] for details.

Let $\gamma$ be a continuous injective curve from the compact time interval
$[0,T]$ into $\mathbb{H}\cup\\{0\\}$ with $\gamma(0)=0$. Loewner’s theory
provides a way to encode the curve $\gamma$ via a real valued function $U$
which will be called the driving function or simply the driver of $\gamma$
which is defined as follows. Note that for each $t\geq 0$,
$H_{t}:=\mathbb{H}\setminus\gamma[0,t]$ is a simply connected domain and there
exists a unique conformal map $g_{t}$ from the slit domain $H_{t}$ onto
$\mathbb{H}$ satisfying the so called hydrodynamic normalization given by
$\lim_{z\to\infty}(g_{t}(z)-z)=0$. The map $g_{t}$ is called the mapping-out
function of the set $K_{t}:=\gamma[0,t]$. Laurent series expansion of $g_{t}$
at infinity yields a non-negative constant $b_{t}$ depending on $K_{t}$ such
that

$g_{t}(z)=z+\frac{b_{t}}{z}+O\biggl{(}\frac{1}{|z|^{2}}\biggr{)},\hskip
5.69054pt\mbox{as}\hskip 5.69054ptz\to\infty.$

The constant $b_{t}$ is called the half-plane capacity of $K_{t}$ and denoted
by $b_{t}={\rm hcap}(K_{t})$. It is easy to see that $t\mapsto{\rm
hcap}(K_{t})$ is continuously increasing. Thus it is possible to choose a
parameterization of $\gamma$ so that ${\rm hcap}(K_{t})=2t$ for all
$t\in[0,T]$. The mapping out function $g_{t}$ also admits a continuous
extension to the boundary point $\gamma_{t}$ of the domain $H_{t}$. The driver
$U$ is then defined by $U_{t}:=g_{t}(\gamma_{t})$ which can be easily shown to
be a continuous real valued function. The significance of the driver $U$ comes
from the fact that it describes the evolution of the conformal maps $g_{t}(z)$
in variable $t$ via Loewner differential equation (LDE) given by

$\partial_{t}{g}_{t}(z)=\frac{2}{g_{t}(z)-U_{t}},\hskip 5.69054ptg_{0}(z)=z.$
(5.1)

In fact one can also recover the curve $\gamma$ from $U$ as follows. For each
$z\in\overline{\mathbb{H}}\setminus\\{0\\}$, let $[0,T_{z})$ with
$T_{z}\in(0,\infty]$ denote the maximal interval of existence of the unique
solution to equation (5.1). Also define $T_{0}=0$. Then

$\gamma[0,t]=\\{z\in\overline{\mathbb{H}},\,T(z)\leq t\\}.$

The procedure described above can also be naturally reversed. Given any
continuous real valued function $U$ with $U_{0}=0$, define $g_{t}(z)$ for
$z\in\overline{\mathbb{H}}\setminus\\{0\\}$ to be the solution of (5.1). Let
$T_{z}$ for $z\in\overline{\mathbb{H}}$ be similarly defined as above. Then

$K_{t}:=\\{z\in\overline{\mathbb{H}},\,T(z)\leq t\\}$

defines an increasing family of compact sets in $\overline{\mathbb{H}}$. The
family $K=\\{K_{t}\\}_{t\in[0,T]}$ is called the Loewner chain driven by $U$.
As in the previous case, $H_{t}:=\mathbb{H}\setminus K_{t}$ is simply
connected and $g_{t}$ is the unique conformal map mapping $H_{t}$ to
$\mathbb{H}$ satisfying hydrodynamic normalization. The Loewner chain $K$ also
satisfies ${\rm hcap}(K_{t})=2t$ and the so called conformal local growth
property meaning that the radius of $g_{t}(K_{t+s}\setminus K_{t})$ tends to
$0$ as $s\to 0+$ uniformly with respect to $t$. However, in full generality,
$K_{t}$ may not be locally connected and it cannot always be written as the
image set $\gamma[0,t]$ for a continuous curve $\gamma$, e.g. logarithmic
spirals, see [4]. Even if this is the case, the curve $\gamma$ may be non-
simple and $K_{t}$ has to be described by filling the loops in the image
$\gamma[0,t]$. We say that the Loewner chain $K$ driven by $U$ admits a trace
or synonymously $U$ generates a trace if there exist a continuous curve
$\gamma:[0,T]\to\overline{\mathbb{H}}$ such $\gamma_{0}=0$ and for all
$t\in[0,T]$, $H_{t}$ is the unbounded component of
$\mathbb{H}\setminus\gamma[0,t]$. We then call $\gamma$ is the trace of the
Loewner chain $K$.

Denote $\hat{f}_{t}(z)=g_{t}^{-1}(z+U_{t})$. It can be easily shown that for
each fixed $t\in[0,T]$, $\hat{f}_{t}(z)=h_{t}(z)$, where for $s\in[0,t]$,
$h_{s}(z)$ is the solution to reverse time LDE given by

$h_{s}(z)=z+U_{t}-U_{t-s}+\int_{0}^{s}\frac{-2}{h_{r}(z)}dr.$ (5.2)

Equation (5.2) is a special case of (1.1) driven by time reversal of $U$ with
the vector field $F(z)=-2/z$ and above raised questions $(a)$-$(c)$ translates
into the study of trace of Loewner chains. A natural question to ask is for
which drivers $U$ does the associated Loewner chain $K$ admits a trace? It was
proven in [2],[3] that if $U$ is $1/2$-Hölder with $||U||_{\frac{1}{2}}<4$,
then $U$ generates a simple curve $\gamma$. In the random setting when
$U_{t}=\sqrt{\kappa}B_{t}$, where $\kappa>0$ and $B$ is standard Brownian
motion, the existence of trace $\gamma$ which are called Schramm-Loewner-
Evolution (SLEκ) was shown in [7] (for $\kappa\neq 8$) and [8] (for
$\kappa=8$). Theorem 1 suggests that there is a clear distinction between
existence of trace for $1/2$-Hölder drivers and Brownian drivers because
Brownian drivers produce smoothing effects which is attributed to its high
irregularity. Since Hölder condition on $U$ only measures its regularity by
putting an upper bound on its modulus of continuity, $1/2$-Hölder drivers will
typically not produce any smoothing effects. Heuristically speaking, the
phenomena of existence of trace for Loewner chains is not just dependent on
how regular the driver is but it is also dependent on its irregularity to some
extent. In a recent work Catellier-Gubinelli [15] introduced the notion of
$\rho$-irregularity of functions which captures such smoothing effects from a
deterministic point of view.

###  5.2 A PDE approach to Theorem 1.

We briefly sketch a different approach based on the work of Flandoli-
Gubinelli-Priola [12] to prove Theorem 1. The idea is to find an appropriate
univalent map (i.e. injective and holomorphic) $\Phi:\mathbb{H}\to\mathbb{C}$
and consider the transformation $Y_{t}=\Phi(Z_{t})$, where $Z$ solves (1.1)
with $U=B$. By appropriately choosing $\Phi$, (1.1) can be transformed into a
SDE for $Y$ whose coefficients have more regularity than that of $F$. The
function $\Phi$ is chosen as the unique solution to the following second order
ODE (which is actually a PDE in disguise because of the differentiation w.r.t.
complex variable $z$)

$\frac{1}{2}\Phi^{\prime\prime}(z)+F(z)\Phi^{\prime}(z)=\lambda(\Phi(z)-z),\hskip
5.69054pt\Phi(z)=z+O(1)\hskip 5.69054pt\mbox{as}\hskip 5.69054ptz\to\infty,$
(5.3)

where constant $\lambda$ is appropriately chosen. In fact, $\Phi$ can be
explicitly written as

$\Phi(z)=z+\int_{0}^{\infty}\exp(-\lambda r)\mathbb{E}[F(\varphi(0,r,z))]dr.$
(5.4)

An argument is required to verify that $\Phi$ is indeed an univalent function.
Hadamard’s global inverse function theorem was used in [12] to verify the
injectivity (for large enough $\lambda$) in their case. Testing univalency of
holomorphic functions is more subtle and Hadamard’s theorem will not apply in
this case. However, there are various known criterion such as Becker’s
univalence criteria for such purposes, see [5].

One can easily check using Itô formula that $Y_{t}=\Phi(Z_{t})$ satisfies

$dY_{t}=\tilde{F}(Y_{t})dt+\sigma(Y_{t})dB_{t},\hskip 5.69054ptY_{0}=\Phi(z),$
(5.5)

where $\tilde{F}(z)=\lambda(z-\Phi^{-1}(z))$ and
$\sigma(z)=\Phi^{\prime}(\Phi^{-1}(z))$. It is expected that function $\Phi$
has $C^{2+\alpha}$ regularity on $\mathbb{R}$, i.e. twice differentiable with
$\alpha$-Hölder second derivative. Indeed, in [12] where the authors deal with
multidimensional Brownian motion, the second derivative term in (5.3) is
replaced by the Laplacian and since $F$ is $\alpha$-Hölder, $2+\alpha$
regularity of $\Phi$ is a consequence of Schauder’s estimate from elliptic
regularity theory. Schauder’s estimate do not strictly apply in our case
because of the $B$ is one dimensional and there is no Laplacian term. But,
since $F$ is holomorphic on $\mathbb{H}$ and it is irregular only on
$\mathbb{R}$, it is expected that $\Phi$ will indeed have $2+\alpha$
regularity. Thus, functions $\tilde{F},\sigma$ have $1+\alpha$ regularity and
classical results of Kunita [18] on stochastic flows imply that equation (5.5)
has $C^{1+\alpha}$ regular stochastic flow. Transforming (5.5) back to (1.1)
using $\Phi^{-1}$ gives a different proof of Theorem 1.

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