# Law of the SLE tip

Oleg Butkovsky Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, FRG.
Email<EMAIL_ADDRESS>Vlad Margarint University of North Carolina
at Charlotte. Email<EMAIL_ADDRESS>Yizheng Yuan Technische Universität
Berlin, Straße des 17. Juni 136, 10623 Berlin. Email<EMAIL_ADDRESS>
###### Abstract

We analyse the law of the SLE tip at a fixed time in capacity parametrization.
We describe it as the stationary law of a suitable diffusion process, and show
that it has a density which is the unique solution (up to a multiplicative
constant) of a certain PDE. Moreover, we identify the phases in which the even
negative moments of the imaginary value are finite. For the negative second
and negative fourth moments we provide closed-form expressions.

##  1 Introduction

The Schramm-Loewner evolution (SLEκ) is a family of random planar fractal
curves indexed by the real parameter $\kappa\geq 0$, introduced by Schramm in
[Sch00]. These random fractal curves are proved to describe scaling limits of
a number of discrete models that are of great interest in planar statistical
physics. For instance, it was proved in [LSW04] that the scaling limit of
loop-erased random walk (with the loops erased in a chronological order)
converges in the scaling limit to SLEκ with $\kappa=2\,.$ Moreover, other two-
dimensional discrete models from statistical mechanics including Ising model
cluster boundaries, Gaussian free field interfaces, percolation on the
triangular lattice at critical probability, and uniform spanning tree Peano
curves were proved to converge in the scaling limit to SLEκ for values of
$\kappa=3,$ $\kappa=4,$ $\kappa=6$ and $\kappa=8$ respectively in the series
of works [Smi10], [SS09], [Smi01] and [LSW04]. There are also other models of
statistical physics in 2D that are conjectured to have SLEκ, for some value of
$\kappa$, as a scaling limit, among which is the two-dimensional self-avoiding
walk which is conjectured to converge in the scaling limit to SLE8/3. For a
detailed exposure and pedagogical introduction to SLE theory, we refer the
reader to [RS05], [Law05], and [Kem17].

Questions concerning the behaviour of the SLE trace at the tip can be found in
the existing body of SLE literature, for example in [JVL12] where the almost
sure multi-fractal spectrum of the SLE trace near its tip is computed, and in
[Zha16] in which the ergodic properties of the harmonic measure near the tip
of the SLE trace are studied.

However, to the best of our knowledge, the law of the SLE tip at fixed
capacity time has not been studied in the SLE literature until very recently.
One of the first papers in this direction is [LMN19] where a method based on
stopping times was applied in order to try to deduce information about the law
of the SLE tip.

In this article, we develop an approach that allows for an in-depth study of
this fundamental quantity. More precisely, we derive a PDE whose unique
solution is the density of the SLE tip. This allows us to obtain explicit
values for the negative second and negative fourth moment of the imaginary
value of the SLE tip. We deduce that they are finite only for $\kappa<8$ resp.
$\kappa<8/3$. For further negative moments, we identify the values of $\kappa$
where the moments are finite.

To obtain these results we combine PDE techniques with certain tools from the
theory of stochastic stability of stochastic differential equations (SDEs).
Namely, we work with an SDE obtained from the backward Loewner differential
equation. By a scaling argument, we derive a two-dimensional diffusion process
that converges in law to the SLE tip. Using tools from ergodic theory (in the
spirit of [MSH02]), we prove that this diffusion process has a unique
invariant measure. This allows us to show that the density of the SLE tip
solves the Fokker-Planck-Kolmogorov (FPK) equation associated with the
process.

Showing that the density of SLE tip is the unique solution of the FPK equation
requires further tools. Note that while there is a vast literature on FPK
equations (see e.g. [BKRS15]), usually only the case of elliptic operators are
considered, while our FPK is hypoelliptic. Therefore, to show uniqueness of
solutions to this equation and derive the support of the solution we utilise
the generalized Ambrosio-Figalli-Trevisan superposition principle obtained
recently in [BRS21] as well as more standard methods such as Lyapunov
functions and Harnack inequalities.

This paper is organised in three sections, the first one being the
introduction. In the second section we state the main results. In the last
section which is further divided in two subsections we give their proofs.

Convention on constants. Throughout the paper $C$ denotes a positive constant
whose value may change from line to line.

Acknowledgements. The authors are deeply indebted to Stas Shaposhnikov for his
help, patience, detailed explanations of some parts of the theory of FPK
equations and for suggesting some useful ideas for the proofs. We are very
grateful to Paolo Pigato and Peter Friz for fruitful discussions. We also
would like to express our deep gratitude to the referee for thoroughly reading
the paper and for offering very valuable suggestions. OB has received funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation program (grant agreement No. 683164), from the
DFG Research Unit FOR 2402, and is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy — The Berlin Mathematics Research Center MATH+
(EXC-2046/1, project ID: 390685689, sub-project EF1-22). YY acknowledges
partial support from ERC through Consolidator Grant 683164 (PI: Peter Friz).

##  2 Main results

First, let us introduce the basic notation. For a domain
$D\subset\mathbb{R}^{k}$, $k\geqslant 1$, let
$\mathcal{C}^{\infty}(D,\mathbb{R})$ be a set of functions $D\to\mathbb{R}$
which have derivatives of all orders. The set of functions from
$\mathcal{C}^{\infty}(D,\mathbb{R})$ which are bounded and have bounded
derivatives of all orders will be denoted by
$\mathcal{C}_{b}^{\infty}(D,\mathbb{R})$. As usual, for a function $f\colon
D\to\mathbb{R}^{d}$, $d\geqslant 1$, we will denote its supremum norm by
$\|f\|_{\infty}:=\sup_{x\in D}|f(x)|$. Let $\mathbb{H}$ be the open complex
upper half-plane $\\{\operatorname{Im}(z)>0\\}$.

Until the end of the paper we fix $\kappa\in(0,\infty)$. Let $g_{t}\colon
H_{t}\to\mathbb{H}$, $t\geqslant 0$, be the forward SLE flow, that is the
solution to the Loewner ODE

$\partial_{t}g_{t}(z)=\frac{2}{g_{t}(z)-\sqrt{\kappa}B_{t}},\quad
g_{0}(z)=z,\quad t\geqslant 0,\,z\in\mathbb{H},$

where $B$ is a standard Brownian motion, $H_{t}=\\{z\in\mathbb{H}\mid
T_{z}>t\\}$, and $T_{z}$ is the time until which the ODE is solvable. Let
$(\gamma_{t})_{t\geqslant 0}$ be the SLEκ path associated with this flow. It
is well-known [RS05, Theorem 3.6], [LSW04, Theorem 4.7] that $\mathsf{P}$-a.s.
for any $t\geqslant 0$

$\gamma(t)=\lim_{u\to 0+}g_{t}^{-1}(\sqrt{\kappa}B_{t}+iu).$ (2.1)

Throughout the paper we use the notations $\gamma(t)$ and $\gamma_{t}$
interchangeably.

Our main result is the following statement.

######  Theorem 2.1.

The random vector
$(\operatorname{Re}(\gamma_{1}),\operatorname{Im}(\gamma_{1}))$ has a density
$\psi\in\mathcal{C}^{\infty}(\mathbb{R}\times(0,\infty),\mathbb{R})$ which is
the unique solution in the class of probability densities (non-negative
functions that integrate to $1$ over the whole space) of the following PDE:

$\frac{\kappa}{2}\partial^{2}_{xx}\psi+\left(\frac{1}{2}x+\frac{2x}{x^{2}+y^{2}}\right)\partial_{x}\psi+\left(\frac{1}{2}y-\frac{2y}{x^{2}+y^{2}}\right)\partial_{y}\psi+\left(1+\frac{4(y^{2}-x^{2})}{(x^{2}+y^{2})^{2}}\right)\psi=0,$
(2.2)

where $x\in\mathbb{R}$, $y\in(0,\infty)$.

Furthermore, $\psi$ is strictly positive in $\mathbb{R}\times(0,2)$,
$\psi\equiv 0$ on $\mathbb{R}\times[2,\infty)$, and $\psi(x,y)=\psi(-x,y)$ for
$x\in\mathbb{R}$, $y>0$.

We have attached in Figure 1 numerical simulations of $\gamma(1)$ with various
values of $\kappa$. There we have chosen the coordinates $(\alpha,y)$ where
$\alpha=\arg\gamma(1)$ and $y=\operatorname{Im}\gamma(1)$ so that they fit
well in the plot.

Figure 1: Simulation of $\gamma(1)$ with $20000$ samples each. Plotted are the
coordinates $(\alpha,y)$ where $\alpha=\arg\gamma(1)$ and
$y=\operatorname{Im}\gamma(1)$.

As an application of Theorem 2.1, we show that the following quantities can be
explicitly calculated.

######  Theorem 2.2.

The following holds:

  1. (i)

For any measurable set $\Lambda\subset\overline{\mathbb{H}}$ one has

$\hskip 0.6458pt\mathsf{E}\hskip
0.43057pt\int_{0}^{\infty}\mathbbm{1}(\gamma(t)\in\Lambda)\,dt=\frac{\Gamma(1+\frac{4}{\kappa})}{2\sqrt{\pi}\Gamma(\frac{1}{2}+\frac{4}{\kappa})}\int_{\Lambda}\left(1+\frac{x^{2}}{y^{2}}\right)^{-4/\kappa}\,dx\,dy.$
(2.3)

  2. (ii)

For any $n\in\mathbb{N}$ we have

$\hskip 0.6458pt\mathsf{E}\hskip
0.43057pt(\operatorname{Im}\gamma_{1})^{-2n}<\infty\,\,\text{if and only if
$\kappa<8/(2n-1)$}.$ (2.4)

Further,

$\displaystyle\hskip 0.6458pt\mathsf{E}\hskip
0.43057pt(\operatorname{Im}\gamma_{1})^{-2}=\frac{2}{8-\kappa}\quad\text{for
$\kappa<8$}$ (2.5) $\displaystyle\hskip 0.6458pt\mathsf{E}\hskip
0.43057pt(\operatorname{Im}\gamma_{1})^{-4}=\frac{16(3-\kappa)}{(12-\kappa)(8-\kappa)(8-3\kappa)}\quad\text{for
$\kappa<8/3$}.$ (2.6)

######  Remark 2.3.

Note that the left-hand side of (2.3) is an average amount of time SLE spends
in a set $\Lambda$. A version of this identity has previously appeared in
[Zha19, Corollary 5.3]. However, in that paper the constant in front of the
integral has been implicitly specified as $1/C_{\kappa,1}$ with

$C_{\kappa,1}=\int_{\mathbb{H}}\left(M_{0}(z)-\hskip 0.6458pt\mathsf{E}\hskip
0.43057pt[M_{1}(z)\mathbbm{1}_{T_{z}>1}]\right)\,dx\,dy$

and $M_{t}(z)=\lvert
g_{t}^{\prime}(z)\rvert^{2}\left(\frac{\operatorname{Im}g_{t}(z)}{\lvert
g_{t}(z)-\sqrt{\kappa}B_{t}\rvert}\right)^{8/\kappa}$. In particular, our
result implies

$C_{\kappa,1}=\frac{2\sqrt{\pi}\Gamma(\frac{1}{2}+\frac{4}{\kappa})}{\Gamma(1+\frac{4}{\kappa})}$

As we will point out in Section 3.2, our Theorem 2.2 may seem like a simple
consequence of Theorem 2.1 that can be heuristically deduced from integration
by parts arguments. However, it is surprisingly tricky to control the boundary
behaviour of $\psi$ and its derivatives. Therefore it requires more work to
rigorously prove Theorem 2.2.

One of our initial motivations was to know more about the marginal law of
$\alpha=\arg\gamma(1)$. We believe that the marginal density should behave
like $\alpha^{8/\kappa}$ as $\alpha\searrow 0$. We did not succeed in proving
this; instead, we prove the following in Section 3.2. Denote
$(\alpha,y)=(\arg\gamma(1),\operatorname{Im}\gamma(1))$ and let
$q(\alpha,y)=\psi(y\cot\alpha,y)\frac{y}{\sin^{2}\alpha}$ the density in these
coordinates. Then for $n\geqslant 1$ we have
$\int_{0}^{2}y^{-2n}q(\alpha,y)\,dy\approx\alpha^{8/\kappa-2n}$ as
$\alpha\searrow 0$.

######  Remark 2.4.

The support of the density is quite natural since the half-plane capacity of
$\gamma[0,t]$ is always at least $\frac{1}{2}\operatorname{Im}\gamma(t)^{2}$,
and hence we always have
$\operatorname{Im}\gamma(t)\leqslant\sqrt{2\operatorname{hcap}(\gamma[0,t])}=2\sqrt{t}$.
Note also that $\operatorname{Im}\gamma(t)=2\sqrt{t}$ is only attained by
SLE0, i.e. $\gamma(t)=i2\sqrt{t}$ which is driven by the constant driving
function.

To obtain these results we establish the following lemma which links the law
of SLEκ with invariant measure of a certain diffusion process. Introduce the
reverse SLE flow

$\partial_{t}h_{t}(z)=\frac{-2}{h_{t}(z)-\sqrt{\kappa}\widetilde{B}_{t}},\quad
h_{0}(z)=z,\quad t\geqslant 0,\,z\in\mathbb{H};$ (2.7)

where $\widetilde{B}$ is the time-reversed Brownian motion, that is,

$\widetilde{B}_{t}:=B_{1-t}-B_{1}\quad\text{for $t\leqslant
1$;}\quad\widetilde{B}_{t}:=B^{\prime}_{t-1}-B_{1}\quad\text{for $t\geqslant
1$},$ (2.8)

where $B^{\prime}$ is a Brownian motion independent of $B$. It is obvious that
$\widetilde{B}$ is a Brownian motion.

######  Lemma 2.5.

We have

$\frac{1}{\sqrt{t}}(h_{t}(i)-\sqrt{\kappa}\widetilde{B}_{t})\to\gamma(1)\,\,\text{in
law as $t\to\infty$}.$

##  3 Proofs

###  3.1 Proofs of Lemmas 2.5 and 2.1

We begin with the proof of Lemma 2.5.

###### Proof of Lemma 2.5.

Introduce $\widehat{f}_{t}(z):=g_{t}^{-1}(\sqrt{\kappa}B_{t}+z)$,
$z\in\mathbb{H}$, $t\geqslant 0$. We claim that

$\widehat{f}_{1}(z)=h_{1}(z)+\sqrt{\kappa}B_{1}.$ (3.1)

Indeed, it follows from (2.7) that for $z\in\mathbb{H}$, $t\in[0,1]$

$\partial_{t}(h_{1-t}(z)+\sqrt{\kappa}B_{1})=\frac{2}{h_{1-t}(z)-\sqrt{\kappa}\widetilde{B}_{1-t}}=\frac{2}{(h_{1-t}(z)+\sqrt{\kappa}B_{1})-\sqrt{\kappa}B_{t}},$

which implies
$h_{1-t}(z)+\sqrt{\kappa}B_{1}=g_{t}(h_{1}(z)+\sqrt{\kappa}B_{1})$. Recalling
the definition of $\widehat{f}$ and taking $t=1$, we obtain (3.1).

Next, we note that the following scaling property holds: for any $c>0$

$\operatorname{Law}(\frac{1}{c}\sqrt{\kappa}\widetilde{B}_{c^{2}},\frac{1}{c}h_{c^{2}}(cz))=\operatorname{Law}(\sqrt{\kappa}\widetilde{B}_{1},h_{1}(z))).$
(3.2)

Indeed, using again the definition of $h$ in (2.7), we see that for any
$t\geqslant 0$

$\partial_{t}(\frac{1}{c}h_{c^{2}t}(cz))=\frac{-2c}{h_{c^{2}t}(cz)-\sqrt{\kappa}\widetilde{B}_{c^{2}t}}=\frac{-2}{\frac{1}{c}h_{c^{2}t}(cz)-\frac{1}{c}\sqrt{\kappa}\widetilde{B}_{c^{2}t}}.$

Since the process
$(\frac{1}{c}\sqrt{\kappa}\widetilde{B}_{c^{2}t})_{t\geqslant 0}$ has the same
law as $(\sqrt{\kappa}\widetilde{B}_{t})_{t\geq 0}$ and the solution of the
Loewner differential equation is a deterministic function of the driver, we
see that (3.2) holds.

Fix $u>0$. Applying (3.1) with $z=iu$ and (3.2) with $z=iu$, $c=1/u$, we
deduce

$\operatorname{Law}(\widehat{f}_{1}(iu))=\operatorname{Law}(uh_{\frac{1}{u^{2}}}(i)-u\sqrt{\kappa}\widetilde{B}_{\frac{1}{u^{2}}}).$

where we have also used the fact that $B_{1}=-\widetilde{B}_{1}$. Since, by
(2.1), we have $\gamma(1)=\lim_{u\searrow 0}\widehat{f}_{1}(iu)$, it follows
that $u(h_{1/u^{2}}(i)-\sqrt{\kappa}\widetilde{B}_{1/u^{2}})$ converges in law
to $\gamma(1)$ as $u\searrow 0$. This implies the statement of the lemma. ∎

Recall the definition of the reverse SLE flow $h$ in (2.7) and the reversed
Brownian Motion $\widetilde{B}$ in (2.8). Lemma 2.5 implies the following
result.

######  Corollary 3.1.

Let $(\widehat{X}_{t},\widehat{Z}_{t})_{t\geqslant 0}$ be the stochastic
process that satisfies the following equation

$\displaystyle d\widehat{X}_{t}$
$\displaystyle=\bigl{(}-\frac{1}{2}\widehat{X}_{t}-\frac{2\widehat{X}_{t}}{\widehat{X}_{t}^{2}+e^{2\widehat{Z}_{t}}}\bigr{)}\,dt+\sqrt{\kappa}\,d\widehat{B}_{t},$
(3.3) $\displaystyle d\widehat{Z}_{t}$
$\displaystyle=\bigl{(}-\frac{1}{2}+\frac{2}{\widehat{X}_{t}^{2}+e^{2\widehat{Z}_{t}}}\bigr{)}\,dt,$
(3.4)

with the initial data
$\widehat{X}_{0}=\operatorname{Re}(h_{1}(i))-\sqrt{\kappa}\widetilde{B}_{1}$,
$\widehat{Z}_{0}=\log(\operatorname{Im}(h_{1}(i)))$; here
$\widehat{B}_{t}:=-\int_{0}^{t}e^{-s/2}\,d\widetilde{B}_{e^{s}}$ and the
filtration
$\widehat{\mathcal{F}}_{t}:=\sigma(\widetilde{B}_{r},r\in[0,e^{t}])$. Then

$(\widehat{X}_{t},\widehat{Z}_{t})\to(\operatorname{Re}(\gamma_{1}),\log(\operatorname{Im}(\gamma_{1})))\quad\text{in
law as $t\to\infty$}.$ (3.5)

Note that the initial value of the process $(\widehat{X},\widehat{Z})$ is
random but measurable with respect to $\widehat{\mathcal{F}}_{0}$.

###### Proof.

Put
$X_{t}+iY_{t}\mathrel{\mathop{:}}=h_{t}(i)-\sqrt{\kappa}\widetilde{B}_{t}$.
Then, it follows from (2.7) that

$\displaystyle dX_{t}$
$\displaystyle=\frac{-2X_{t}}{X_{t}^{2}+Y_{t}^{2}}\,dt-\sqrt{\kappa}\,d\widetilde{B}_{t},$
$\displaystyle dY_{t}$ $\displaystyle=\frac{2Y_{t}}{X_{t}^{2}+Y_{t}^{2}}\,dt,$
(3.6)

$X_{0}=0$, $Y_{0}=1$. For $t\geqslant 0$, let
$\widehat{X}_{t}:=e^{-t/2}X_{e^{t}}$ and $\widehat{Y}_{t}:=e^{-t/2}Y_{e^{t}}$.
We apply Itô’s formula to derive

$\displaystyle d\widehat{X}_{t}$
$\displaystyle=\bigl{(}-\frac{1}{2}\widehat{X}_{t}-\frac{2\widehat{X}_{t}}{\widehat{X}_{t}^{2}+\widehat{Y}_{t}^{2}}\bigr{)}\,dt+\sqrt{\kappa}\,d\widehat{B}_{t},$
(3.7) $\displaystyle d\widehat{Y}_{t}$
$\displaystyle=\bigl{(}-\frac{1}{2}\widehat{Y}_{t}+\frac{2\widehat{Y}_{t}}{\widehat{X}_{t}^{2}+\widehat{Y}_{t}^{2}}\bigr{)}\,dt.$
(3.8)

Clearly, $\widehat{B}$ is a standard Brownian motion with respect to the
filtration $\widehat{\mathcal{F}}_{t}$. By definition, we also have
$\widehat{X}_{0}=X_{1}$, $\widehat{Y}_{0}=Y_{1}$. The change of variables
$\widehat{Z}_{t}:=\log\widehat{Y}_{t}$ and another application of Itô’s
formula implies that the process $(\widehat{X},\widehat{Z})_{t\geqslant 0}$
satisfies SDE (3.3)–(3.4) with the initial conditions
$\widehat{X}_{0}=X_{1}=\operatorname{Re}(h_{1}(i))-\sqrt{\kappa}\widetilde{B}_{1}$,
$\widehat{Z}_{0}=\log(Y_{1})=\log(\operatorname{Im}(h_{1}(i)))$. Note that by
(3.6), $Y_{1}\geqslant Y_{0}=1$, therefore $|\widehat{Z}_{0}|<\infty$.

Furthermore,

$e^{-t/2}(h_{e^{t}}(i)-\sqrt{\kappa}\widetilde{B}_{e^{t}})=e^{-t/2}(X_{e^{t}}+iY_{e^{t}})=\widehat{X}_{t}+i\widehat{Y}_{t}.$

Thus, by Lemma 2.5, we have

$(\widehat{X}_{t},\widehat{Y}_{t})\to(\operatorname{Re}(\gamma(1)),\operatorname{Im}(\gamma(1)))\,\,\text{in
law as $t\to\infty$}.$ (3.9)

Note that

$\mathsf{P}(\operatorname{Im}(\gamma(1))=0)=0.$ (3.10)

Indeed, the trace of a Loewner chain a.s. spends zero capacity time at the
boundary, i.e., $\lambda(\\{t\mid\operatorname{Im}\gamma(t)=0\\})=0$ a.s.,
where $\lambda$ is the Lebesgue measure (cf. [Yua22, Proposition 1.7]; the
case for SLEκ appeared already in [Zha19, Corollary 5.3]). Therefore, by
Fubini’s theorem, $\mathsf{P}(\operatorname{Im}(\gamma(t))=0)=0$ Lebesgue
a.e.. By scale invariance, this implies (3.10).

Now, combining (3.9) and (3.10), we get (3.5). ∎

It follows from Corollary 3.1 that to prove Theorem 2.1 one needs to study
invariant measures of (3.3)–(3.4). PDE (2.2) is then the Fokker-Planck-
Kolmogorov equation for this process. However, since the coefficients have a
singularity at $0$, a bit of care is needed to make the statements rigorous.

First, we show that this SDE is well-posed and is a Markov process. We will
need the following notation. For a vector field
$U\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ denote its derivative matrix by
$(DU)_{i,j}:=\partial_{x_{j}}U_{i}$. The Lie bracket between two vector fields
$U,V\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ is given by

$[U,V](x):=DV(x)U(x)-DU(x)V(x),\quad x\in\mathbb{R}^{2}.$

It is immediate to see that if $U=\left(\begin{smallmatrix}1\\\
0\end{smallmatrix}\right)$, then

$[U,V]=\begin{pmatrix}\partial_{x_{1}}V_{1}\\\
\partial_{x_{1}}V_{2}\end{pmatrix},\quad[U,[U,V]\,]=\begin{pmatrix}\partial^{2}_{x_{1}x_{1}}V_{1}\\\
\partial^{2}_{x_{1}x_{1}}V_{2}\end{pmatrix}.\quad$ (3.11)

We begin with the following technical statement.

Let $W$ be a standard Brownian motion. For $\varepsilon>0$, let
$g_{\varepsilon}\colon\mathbb{R}\to[\varepsilon/2,+\infty)$ be a
$\mathcal{C}^{\infty}(\mathbb{R})$ function with bounded derivatives of all
orders such that

$\begin{cases}g_{\varepsilon}(x)=x,\quad&x\geqslant\varepsilon;\\\
\varepsilon/2\leqslant
g_{\varepsilon}(x)\leqslant\varepsilon,\quad&-\infty<x<\varepsilon.\end{cases}$

######  Lemma 3.2.

Fix $\varepsilon>0$ and consider stochastic differential equation

$\displaystyle dX^{\varepsilon}_{t}$
$\displaystyle=\bigl{(}-\frac{1}{2}X^{\varepsilon}_{t}-\frac{2X^{\varepsilon}_{t}}{(X^{\varepsilon}_{t})^{2}+g_{\varepsilon}(e^{2Z_{t}^{\varepsilon}})}\bigr{)}\,dt+\sqrt{\kappa}\,dW_{t},$
(3.12) $\displaystyle dZ^{\varepsilon}_{t}$
$\displaystyle=\bigl{(}-\frac{1}{2}+\frac{2}{(X^{\varepsilon}_{t})^{2}+g_{\varepsilon}(e^{2Z_{t}^{\varepsilon}})}\bigr{)}\,dt,$
(3.13)

where
$(X^{\varepsilon}_{0},Z^{\varepsilon}_{0})=(x_{0},z_{0})\in\mathbb{R}^{2}$.
Then for any initial condition $(x_{0},z_{0})\in\mathbb{R}^{2}$ SDE
(3.12)–(3.13) has a unique strong solution. This solution is a strong Feller
Markov process.

###### Proof.

Since the drift and diffusion of (3.12)–(3.13) are uniformly Lipschitz
continuous functions, it is immediate that SDE (3.12)–(3.13) has a unique
strong solution and this solution is a Markov process. To show that
$(X_{t}^{\varepsilon},Z_{t}^{\varepsilon})$ is a strong Feller process we use
(parabolic) Hörmander’s theorem.

Denote

$b^{\varepsilon}(x,z):=\begin{pmatrix}b^{1,\varepsilon}(x,z)\\\
b^{2,\varepsilon}(x,z)\end{pmatrix}:=\begin{pmatrix}-\frac{1}{2}x-\frac{2x}{x^{2}+g_{\varepsilon}(e^{2z})}\\\\[6.45831pt]
-\frac{1}{2}+\frac{2}{x^{2}+g_{\varepsilon}(e^{2z})}\end{pmatrix},\quad
x,z\in\mathbb{R};\qquad\sigma:=\begin{pmatrix}\sqrt{\kappa}\\\
0\end{pmatrix}.$ (3.14)

Then we can rewrite (3.12)–(3.13) as

$d\xi_{t}^{\varepsilon}=b^{\varepsilon}(\xi^{\varepsilon}_{t})dt+\sigma
dW_{t},$ (3.15)

where we put $\xi^{\varepsilon}:=\begin{pmatrix}X^{\varepsilon}\\\
Z^{\varepsilon}.\end{pmatrix}$ Let us verify that SDE (3.15) satisfies all
conditions of Hörmander’s theorem [Hai11, Theorem 1.3] (see also [Pav14,
Theorem 6.1]).

We see that the drift $b^{\varepsilon}$ is in $\mathcal{C}^{\infty}$ and all
its derivatives are bounded. Furthermore, using (3.11), we see that for $x\neq
0$, $z\in\mathbb{R}$ we have
$\operatorname{span}(\sigma,\bigl{[}\sigma,b^{\varepsilon}(x,z)\bigr{]})=\mathbb{R}^{2}$,
and for $x=0$, $z\in\mathbb{R}$ we have
$\operatorname{span}(\sigma,\Bigl{[}\sigma,\bigl{[}\sigma,b^{\varepsilon}(x,z)\bigr{]}\Bigr{]})=\mathbb{R}^{2}$.
Thus, the parabolic Hörmander condition holds. Hence, all the conditions of
the Hörmander theorem are met and [Hai11, Theorem 1.3] implies that
$(X^{\varepsilon},Z^{\varepsilon})$ is strong Feller. ∎

Now we can show well-posedness of (3.3)–(3.4).

######  Lemma 3.3.

For any random vector $(\widehat{x}_{0},\widehat{z}_{0})$ independent of
$\widehat{B}$ the stochastic differential equation (3.3)–(3.4) has a unique
strong solution with
$(\widehat{X}_{0},\widehat{Z}_{0})=(\widehat{x}_{0},\widehat{z}_{0})$. This
solution is a Markov process in the state space $\mathbb{R}^{2}$ and its
transition kernel $P_{t}$ is strong Feller for any $t>0$.

###### Proof.

First, we consider the case when the initial data
$(\widehat{x}_{0},\widehat{z}_{0})$ is deterministic. Then it is immediate to
see that for any $T>0$ a solution to (3.3)–(3.4) satisfies

$\widehat{Z}_{t}\geqslant\widehat{z}_{0}-T/2,$ (3.16)

$t\in[0,T]$. Hence, on time interval $[0,T]$, any solution to (3.3)–(3.4)
solves SDE (3.12)–(3.13) with
$(X_{0}^{\varepsilon},Z_{0}^{\varepsilon})=(\widehat{x}_{0},\widehat{z}_{0})$,
$\varepsilon=\exp(2\widehat{z}_{0}-T)$, $W=\widehat{B}$ and vice versa. Since,
by Lemma 3.2, the latter equation has a unique strong solution, we see that
SDE (3.3)-(3.4) has a unique strong solution on $[0,T]$ and

$(\widehat{X}_{t},\widehat{Z}_{t})=(X^{\varepsilon}_{t},Z^{\varepsilon}_{t}),\quad
t\in[0,T].$ (3.17)

Since $T$ is arbitrary, it follows that SDE (3.3)-(3.4) has a unique strong
solution on $[0,\infty)$.

Strong existence for the case of arbitrary initial data follows now from
[Kal96, Theorem 1], and strong uniqueness from [IW89, Remark IV.1.4].
Moreover, [KS91, Theorem 5.4.20] shows that
$(\widehat{X}_{t},\widehat{Z}_{t})_{t\geqslant 0}$ is a Markov process with
the state space $\mathbb{R}^{2}$ equipped with the Borel topology.

Now let us show that $(P_{t})_{t\geqslant 0}$ is strong Feller. Let $f$ be an
arbitrary bounded measurable function $\mathbb{R}^{2}\to\mathbb{R}$, let
$(x_{0},z_{0})\in\mathbb{R}^{2}$. Let
$(x_{0}^{n},z_{0}^{n})\in\mathbb{R}^{2}$, $n\in\mathbb{Z}_{+}$ be a sequence
converging to $(x_{0},z_{0})$ as $n\to\infty$. Without loss of generality we
can assume that $z_{0}^{n}\geqslant-2|z_{0}|$ for all $n\in\mathbb{Z}_{+}$.
Fix $t>0$. Then, denoting by $(P^{\varepsilon}_{t})_{t\geqslant 0}$ the
transition kernel associated with SDE (3.12)–(3.13), we derive

$P_{t}f(x^{n}_{0},z^{n}_{0})=\hskip 0.6458pt\mathsf{E}\hskip
0.43057pt_{(x^{n}_{0},z^{n}_{0})}f(\widehat{X}_{t},\widehat{Z}_{t})=\hskip
0.6458pt\mathsf{E}\hskip
0.43057pt_{(x^{n}_{0},z^{n}_{0})}f(X^{\varepsilon}_{t},Z^{\varepsilon}_{t})=P_{t}^{\varepsilon}f(x^{n}_{0},z^{n}_{0}),$
(3.18)

where $\varepsilon:=\exp(-4|z_{0}|-t)$ and we used here (3.16) and (3.17). By
Lemma 3.2, we have

$P_{t}^{\varepsilon}f(x^{n}_{0},z^{n}_{0})\to
P_{t}^{\varepsilon}f(x_{0},z_{0})=P_{t}f(x_{0},z_{0}),\quad\text{as
$n\to\infty$},$ (3.19)

here we used once again (3.16) and (3.17). Combining (3.18) and (3.19), we see
that $P_{t}$ is strong Feller. ∎

To show uniqueness of the invariant measure of $(P_{t})$, we will need the
following support theorem. For $\delta>0$, $v\in\mathbb{R}^{2}$ let
$B_{\delta,v}$ be the ball of radius $\delta$ centred at $v$.

######  Lemma 3.4.

For any $(x_{0},z_{0})\in\mathbb{R}^{2}$, $\delta>0$, there exists $T>0$ such
that

$P_{T}((x_{0},z_{0}),B_{\delta,(0,\log 2)})>0.$

###### Proof.

Fix $(x_{0},z_{0})\in\mathbb{R}^{2}$. Consider the following deterministic
control problem associated with (3.3)–(3.4):

$\displaystyle\frac{d}{dt}x_{t}$
$\displaystyle=\bigl{(}-\frac{1}{2}x_{t}-\frac{2x_{t}}{x_{t}^{2}+e^{2z_{t}}}\bigr{)}+\sqrt{\kappa}\frac{d}{dt}U_{t},$
(3.20) $\displaystyle\frac{d}{dt}z_{t}$
$\displaystyle=\bigl{(}-\frac{1}{2}+\frac{2}{x_{t}^{2}+e^{2z_{t}}}\bigr{)},$
(3.21)

where $x(0)=x_{0}$, $z(0)=z_{0}$ and $U\in\mathcal{C}^{1}([0,T];\mathbb{R})$
is a non-random function with $U_{0}=0$. We claim that we can find $T>0$ and
$U$ such that $x_{T}=0$ and $|z_{T}-\log 2|<\delta/2$.

First, we take a $\mathcal{C}^{1}$ path $x\colon[0,1]\to\mathbb{R}$ such that
$x(0)=x_{0}$, $x(1)=0$, $\frac{d}{dt}x(t)\Bigr{|}_{t=1}=0$. Let $z_{t}$,
$t\in[0,1]$, be a solution to (3.21) with the initial condition $z_{0}$ (for
$x$ constructed above).

Consider now the equation

$\frac{d}{dt}z_{t}=\bigl{(}-\frac{1}{2}+\frac{2}{e^{2z_{t}}}\bigr{)},\quad
t\geqslant 1$

with the initial condition $z_{1}$ constructed above. It is easy to see that
there exists $T=T(x_{0},z_{0})>1$ such that $|z_{T}-\log 2|<\delta/2$. Set
$x_{t}=0$ for $t\in[1,T]$.

Finally, let $U_{t}$, $t\in[0,T]$, be a $\mathcal{C}^{1}$ path such that
(3.20) holds for $x,z$ constructed above and $U_{0}=0$. The desired control
$U$ has been constructed.

Now for arbitrary $\varepsilon>0$, consider the event

$A_{\varepsilon}:=\\{\sup_{t\in[0,T]}|W_{t}-U_{t}|<\varepsilon\\}.$

It is well-known (see, e.g., [Fre71, Theorem 38]) that
$\mathsf{P}(A_{\varepsilon})>0$. Let
$(\widehat{X}_{t},\widehat{Z}_{t})_{t\in[0,T]}$ be the solution of (3.3)-(3.4)
with the initial condition $(x_{0},z_{0})$. Then

$z_{t}\geqslant z_{0}-T/2,\quad\widehat{Z}_{t}\geqslant
z_{0}-T/2,\qquad\text{for all $t\in[0,T]$}.$ (3.22)

Therefore, for any $t\in[0,T]$ we have on $A_{\varepsilon}$

$|\widehat{X}_{t}-x_{t}|+|\widehat{Z}_{t}-z_{t}|\leqslant
C\int_{0}^{t}(|\widehat{X}_{s}-x_{s}|+|\widehat{Z}_{s}-z_{s}|)\,ds+\sqrt{\kappa}\varepsilon,$
(3.23)

where we used (3.22) and the fact that the Lipschitz constant of the drift of
SDE (3.3)-(3.4) is bounded on the set $\mathbb{R}\times[z_{0}-T/2,+\infty)$.
By the Gronwall inequality and (3.23), we have on $A_{\varepsilon}$

$|\widehat{X}_{T}-x_{T}|+|\widehat{Z}_{T}-z_{T}|\leqslant
C(T)\sqrt{\kappa}\varepsilon.$

Choose now $\varepsilon$ small enough, such that the right-hand side of the
above inequality is less than $\delta/2$. Then recalling that $x_{T}=0$ and
$|z_{T}-\log 2|<\delta/2$, we finally deduce

$P_{T}((x_{0},z_{0}),B_{\delta,(0,\log
2)})\geqslant\mathsf{P}(A_{\varepsilon})>0.\qed$

######  Lemma 3.5.

The measure
$\pi:=\operatorname{Law}\bigl{(}\operatorname{Re}(\gamma_{1}),\log(\operatorname{Im}(\gamma_{1}))\bigr{)}$
is the unique invariant measure for the process (3.3)–(3.4).

###### Proof.

The fact that the measure $\pi$ is invariant follows by a standard argument.
Denote, as usual, for a measurable bounded function
$f\colon\mathbb{R}^{2}\to\mathbb{R}$ and a measure $\nu$ on $\mathbb{R}^{2}$

$\displaystyle
P_{t}f(x):=\int_{\mathbb{R}^{2}}f(y)P_{t}(x,dy),\,\,x\in\mathbb{R}^{2};\qquad
P_{t}\nu(A):=\int_{\mathbb{R}^{2}}P_{t}(y,A)\,\nu(dy),\,\,A\in\mathcal{B}(\mathbb{R}^{2}).$

Consider the measure
$\mu:=\operatorname{Law}\Bigl{(}\operatorname{Re}(h_{1}(i))-\sqrt{\kappa}\widetilde{B}_{1},\log(\operatorname{Im}(h_{1}(i)))\Bigr{)}$.
Rewriting (3.5), we see that

$P_{t}\mu\to\pi\,\,\text{weakly as $t\to\infty$}.$ (3.24)

Fix any $s\geqslant 0$. Let us show that $P_{s}\pi=\pi$. Indeed, let
$f\colon\mathbb{R}^{2}\to\mathbb{R}$ be an arbitrary continuous bounded
function. Then

$\displaystyle\int_{\mathbb{R}^{2}}f(x)\,P_{s}\pi(dx)$
$\displaystyle=\int_{\mathbb{R}^{2}}P_{s}f(x)\,\pi(dx)=\lim_{t\to\infty}\int_{\mathbb{R}^{2}}P_{s}f(x)\,P_{t}\mu(dx)$
$\displaystyle=\lim_{t\to\infty}\int_{\mathbb{R}^{2}}f(x)\,P_{t+s}\mu(dx)=\int_{\mathbb{R}^{2}}f(x)\,\pi(dx),$

where the second identity follows from (3.24) and the fact that $P_{s}f$ is a
bounded continuous function (this is guaranteed by the Feller property of
$P$). Since $f$ was arbitrary bounded continuous function, we see that
$P_{s}\pi=\pi$ for any $s\geqslant 0$. Thus, the measure $\pi$ is invariant
for SDE (3.3)-(3.4).

Now let us show that SDE (3.3)-(3.4) have a unique invariant measure. Assume
the contrary. Then SDE (3.3)-(3.4) must have two different ergodic invariant
measures $\nu$, $\widetilde{\nu}$ ([Hai08, Lemma 7.1], [VO16, Theorem
5.1.3(iv)]). By Lemma 3.3 the semigroup $(P_{t})$ is strong Feller. Therefore,
by [DP06, Proposition 7.8]

$\operatorname{supp}(\nu)\cap\operatorname{supp}(\widetilde{\nu})=\emptyset.$
(3.25)

We claim now that the point $(0,\log 2)$ belongs to the support of both of
these measures.

Indeed, fix arbitrary $\delta>0$. Take any
$(x_{0},z_{0})\in\operatorname{supp}(\nu)$. Then, by Lemma 3.4, there exists
$T>0$, $\varepsilon>0$ such that $P_{T}((x_{0},z_{0}),B_{\delta,(0,\log
2)})>\varepsilon$. By the strong Feller property of $P_{T}$, the function
$(x,z)\mapsto P_{T}((x,z),B_{\delta,(0,\log 2)})$ is continuous. Therefore,
there exists $\delta^{\prime}>0$ such that

$P_{T}((x,z),B_{\delta,(0,\log 2)})>\varepsilon/2,\quad\text{for any $(x,z)\in
B_{\delta^{\prime},(x_{0},z_{0})}$}.$

This implies that

$\nu(B_{\delta,(0,\log
2)})\geqslant\int_{B_{\delta^{\prime},(x_{0},z_{0})}}\nu(x,z)P_{T}((x,z),B_{\delta,(0,\log
2)})\,dxdz\geqslant\frac{\varepsilon}{2}\nu(B_{\delta^{\prime},(x_{0},z_{0})})>0$

where the last inequality follows from the fact that
$(x_{0},z_{0})\in\operatorname{supp}(\nu)$. Since $\delta$ was arbitrary, we
see that $(0,\log 2)\in\operatorname{supp}(\nu)$. Similarly, $(0,\log
2)\in\operatorname{supp}(\widetilde{\nu})$, which contradicts (3.25).
Therefore, SDE (3.3)-(3.4) has a unique invariant measure. ∎

Let $L$ be the generator of the semigroup $P$

$Lf:=\frac{1}{2}\kappa\partial^{2}_{xx}f+\bigl{(}-\frac{1}{2}x-\frac{2x}{x^{2}+e^{2z}}\bigr{)}\partial_{x}f+\bigl{(}-\frac{1}{2}+\frac{2}{x^{2}+e^{2z}}\bigr{)}\partial_{z}f,$

where $f\in\mathcal{C}^{\infty}(\mathbb{R}^{2})$. As usual, the adjoint of $L$
will be denoted by $L^{*}$.

######  Lemma 3.6.

The measure
$\pi:=\operatorname{Law}\bigl{(}\operatorname{Re}(\gamma_{1}),\log(\operatorname{Im}(\gamma_{1}))\bigr{)}$
has a smooth density $p$ with respect to the Lebesgue measure. Further, $p$ is
the unique solution in the class of densities of the Fokker-Planck-Kolmogorov
equation

$L^{*}p=0.$ (3.26)

Finally, $p(x,z)=0$ for $x\in\mathbb{R}$, $z\geqslant\log 2$, and $p(x,z)>0$
for $x\in\mathbb{R}$, $z<\log 2$.

###### Proof.

Since the measure $\pi$ is invariant for $P$, we have (in the weak sense)

$L^{*}\pi=0.$ (3.27)

Let us now check that $L^{*}$ satisfies the (standard) Hörmander condition.

Denote by $b$ the drift of (3.3)-(3.4)

$b(x,z):=\begin{pmatrix}b^{1}(x,z)\\\
b^{2}(x,z)\end{pmatrix}:=\begin{pmatrix}-\frac{1}{2}x-\frac{2x}{x^{2}+e^{2z}}\\\\[6.45831pt]
-\frac{1}{2}+\frac{2}{x^{2}+e^{2z}}\end{pmatrix},\quad x,z\in\mathbb{R}.$
(3.28)

and recall the notation for $\sigma$ (3.14). Using (3.11), we see that for
$x\neq 0$, $z\in\mathbb{R}$ we have
$\operatorname{span}(\sigma,\bigl{[}\sigma,b(x,z)\bigr{]})=\mathbb{R}^{2}$,
and for $x=0$, $z\in\mathbb{R}$ we have
$\operatorname{span}(\sigma,\Bigl{[}\sigma,\bigl{[}\sigma,b(x,z)\bigr{]}\bigr{]})=\mathbb{R}^{2}$.
Thus, the Hörmander condition holds and by Hörmander’s theorem [Str08, Theorem
7.4.3], $L^{*}$ is hypoelliptic.444In the proof of Lemma 3.2, we use [Hai11,
Theorem 1.3] which is a probabilistic version of Hörmander’s theorem, and it
imposes global assumptions on boundedness of derivatives of the drift. Here we
use [Str08, Theorem 7.4.3] which is a purely PDE result and it does not
require any global assumptions. Therefore we do not have to smoothen the drift
$b$ here. Therefore, (3.27) implies that the Schwarz distribution
$\pi\in\mathcal{C}^{\infty}(\mathbb{R}^{2})$. Thus, the measure $\pi$ has a
$\mathcal{C}^{\infty}$ density $p$ with respect to the Lebesgue measure and
(3.26) holds.

Now let us show that (3.26) does not have any other solutions. We have already
seen that semigroup $(P_{t})$ has a unique invariant measure (this has been
established in Lemma 3.5). In general, without extra conditions, this does not
immediately imply uniqueness of solutions to (3.26) in the class of
probability measures, see [BKRS15, hint to exercise 9.8.48]. This is because
not every probability solution to the Fokker–Planck–Kolmogorov equation
corresponds to a solution of the martingale problem; we refer to [BRS21, p.
719] for further discussion.

Thus, we assume the contrary and suppose that $p^{\prime}$ is another
probability density which solves (3.26). Let $\pi^{\prime}$ be the measure
with density $p^{\prime}$. We claim that $\pi^{\prime}$ is another invariant
measure for $(P_{t})$.

Consider a Lyapunov function $V$ (the suggestion to take this specific
function is due to Stas Shaposhnikov)

$V(x,z):=x^{2}+\log(1+z^{2}),\quad(x,z)\in\mathbb{R}^{2}.$

Then

$\displaystyle LV(x,z)$
$\displaystyle=\kappa-x^{2}-\frac{4x^{2}}{x^{2}+e^{2z}}-\frac{z}{1+z^{2}}+\frac{4z}{(x^{2}+e^{2z})(1+z^{2})}$
$\displaystyle\leqslant\kappa+3-\Bigl{(}x^{2}+\frac{4|z|\mathbbm{1}(z\leqslant
0)}{(x^{2}+e^{2z})(1+z^{2})}\Bigr{)}.$

By [BKRS15, Theorem 2.3.2 and inequality (2.3.2)], this implies (note that $V$
is obviously quasi-compact in the sense of [BKRS15, Definition 2.3.1])

$\int_{\mathbb{R}^{2}}\bigl{(}x^{2}+\frac{4|z|\mathbbm{1}(z\leqslant
0)}{(x^{2}+e^{2z})(1+z^{2})}\bigr{)}p^{\prime}(x,z)\,dxdz<\infty.$ (3.29)

Then recalling (3.28) we have

$\frac{1+|b^{1}(x,z)x|+|b^{2}(x,z)z|}{1+x^{2}+z^{2}}\leqslant
5+\frac{2|z|\mathbbm{1}(z\leqslant 0)}{(x^{2}+e^{2z})(1+z^{2})}.$

Combining this with (3.29), we see that for any $T>0$

$\int_{0}^{T}\int_{\mathbb{R}^{2}}\frac{1+|b^{1}(x,z)x|+|b^{2}(x,z)z|}{1+x^{2}+z^{2}}p^{\prime}(x,z)\,dxdzdt<\infty.$

Therefore, by the generalized Ambrosio-Figalli-Trevisan superposition
principle [BRS21, Theorem 1.1] and the standard equivalence between weak
solutions of SDE and the martingale problems, see, e.g., [KS91, Proposition
5.4.11], there exists a weak solution to SDE (3.3)–(3.4) on the interval
$[0,T]$ such that for any $t\geqslant 0$ we have
$\operatorname{Law}(\widehat{X}_{t},\widehat{Z}_{t})=\pi^{\prime}$. Thus, the
measure $\pi^{\prime}$ is also invariant for the semigroup $(P_{t})$. However,
this contradicts Lemma 3.5. Therefore, (3.26) has a unique solution in the
class of probability densities.

Finally, let us prove the results concerning the support of $p$. Note that if
$\widehat{Z}_{0}(\omega)>\log 2$, then
$\widehat{Z}_{0}(\omega)>\widehat{Z}_{1}(\omega)$. Let
$f\colon\mathbb{R}\to[0,\infty)$ be an increasing function such that $f(x)=0$
for $x\leqslant\log 2$ and $f(x)>0$ for $x>\log 2$. Then
$f(\widehat{Z}_{0})-f(\widehat{Z}_{1})\geqslant 0$. On the other hand, by
invariance

$\hskip 0.6458pt\mathsf{E}\hskip
0.43057pt_{\pi}(f(\widehat{Z}_{0})-f(\widehat{Z}_{1}))=0.$

This implies that $\mathsf{P}_{\pi}$ a.s. we have
$f(\widehat{Z}_{0})=f(\widehat{Z}_{1})$. By the definition of $f$ this implies
that $\mathsf{P}_{\pi}(\widehat{Z}_{0}>\log 2)=0$ and thus
$\pi(\mathbb{R}\times(\log 2,\infty))=0$. Since the density $p$ is continuous
we have

$p(x,z)=0,\quad x\in\mathbb{R},z\geqslant\log 2.$ (3.30)

Figure 2: Support of the density $p$ (yellow and red regions). The process
$\widehat{Z}_{t}$ is increasing when $(\widehat{X}_{t},\widehat{Z}_{t})$ is in
the red region, and decreasing whenever $(\widehat{X}_{t},\widehat{Z}_{t})$ is
in the yellow region. The dashed line, which touches the red region, is
$z=\log 2$.

Now let us show that $p(x,z)>0$ for any $z<\log 2$. The idea of this part of
the proof is due to Stas Shaposhnikov. Suppose the contrary that for some
$x_{0}\in\mathbb{R}$, $z_{0}<\log 2$ we have $p(x_{0},z_{0})=0$. We claim that
this implies that $p\equiv 0$. Note that the set $\\{z=z_{0}\\}$ is the set of
elliptic connectivity for operator $L^{*}$ in the sense of [OR73, Chapter
III.1] (see also [Hil70, Section 2]). Therefore, the maximum principle for
degenerate elliptic equations [OR73, Theorem 3.1.2] (see also [Hil70, Theorem
1], [Ale58, Theorem 4]) implies that $p(x,z_{0})=0$ for any $x\in\mathbb{R}$.

Note that in the domain

$D:=\\{x^{2}+\exp(2z)<4\\}$

PDE (3.26) becomes a parabolic equation in $(z,x)$ and on its complement
(3.26) is a backward parabolic equation. This corresponds to the fact that the
process $\widehat{Z}_{t}$ is increasing on $D$ and decreasing on
$\mathbb{R}^{2}\setminus D$, see Figure 2.

Fix now small $\delta$ such that $\delta^{2}+\exp(2z_{0})<4$ (this is possible
since $z_{0}<\log 2$). Consider now the domain
$D^{\prime}:=[-\delta,\delta]\times(-\infty,z_{0})\subset D$. In this domain
(3.26) is a parabolic equation

$\partial_{z}p-a(x,z)\partial_{xx}p+b(x,z)\partial_{x}p+c(x,z)p=0,$ (3.31)

for certain smooth functions $a,b,c$ and

$a(x,z)=\frac{\kappa}{\frac{4}{x^{2}+e^{2z}}-1}>0,\quad(x,z)\in D^{\prime},$

since $\frac{4}{x^{2}+e^{2z}}>\frac{4}{\delta^{2}+e^{2z_{0}}}>1$ on
$D^{\prime}$. Therefore, by the Harnack inequality for parabolic equations
(see, e.g., [Eva98, Section 7.1, Theorem 10]), we get for arbitrary
$z_{1}\leqslant z_{0}$, and $C>0$

$\sup_{x\in(-\delta,\delta)}p(x,z_{1})\leqslant
C\inf_{x\in(-\delta,\delta)}p(x,z_{0})=0.$

Using again the maximum principle for degenerate elliptic equations, we deduce
from this that $p(x,z_{1})=0$ for any $x\in\mathbb{R}$. Since $z_{1}\leqslant
z_{0}$ was arbitrary we have that $p\equiv 0$ on
$\mathbb{R}\times(-\infty;z_{0}]$.

We use a similar argument to treat the case $z\geqslant z_{0}$. Consider now
the domain
$D^{\prime\prime}:=[3,4]\times(z_{0},\infty)\subset\mathbb{R}^{2}\setminus D$.
In this domain (3.26) is a backward parabolic equation (3.31) and

$a(x,z)=\frac{\kappa}{\frac{4}{x^{2}+e^{2z}}-1}<0,\quad(x,z)\in
D^{\prime\prime}$

since $\frac{4}{x^{2}+e^{2z}}<\frac{4}{9}<1$ on $D^{\prime\prime}$. The
Harnack inequality for parabolic equations implies now for arbitrary
$z_{1}\geqslant z_{0}$, and $C>0$

$\sup_{x\in(3,4)}p(x,z_{1})\leqslant C\inf_{x\in(3,4)}p(x,z_{0})=0.$

and thus, as above, the maximum principle implies that $p\equiv 0$ on
$\mathbb{R}\times[z_{0},\infty)$.

Therefore the function $p$ is identically $0$ which is not possible since $p$
is a density. This contradiction shows that $p(x,z)>0$ for any
$x\in\mathbb{R}$, $z<\log 2$. Together with (3.30) this concludes the proof of
the theorem. ∎

###### Proof of Theorem 2.1.

By Lemma 3.6, the measure
$\operatorname{Law}\bigl{(}\operatorname{Re}(\gamma_{1}),\log(\operatorname{Im}(\gamma_{1}))\bigr{)}$
has a smooth density $p$ with respect to the Lebesgue measure, which solves
(3.26). Therefore, the measure
$\operatorname{Law}\bigl{(}\operatorname{Re}(\gamma_{1}),\operatorname{Im}(\gamma_{1})\bigr{)}$
has a density

$\psi(x,y):=\frac{1}{y}p(x,\log y),\quad x\in\mathbb{R},y>0.$

Now, by change of variables, it is easy to see that $\psi$ is the unique
solution of (2.2) in the class of probability densities. Since $p(x,z)$ is
positive whenever $z<\log 2$, we see that $\psi(x,y)$ is positive whenever
$y\in(0,2)$. Finally, it is immediate that the function
$\bar{\psi}(x,y):=\psi(-x,y)$ also solves (2.2). By uniqueness, this implies
that $\psi(x,y)=\psi(-x,y)$. ∎

###  3.2 Proof of Theorem 2.2

To establish Theorem 2.2, it will be convenient to work in the coordinates
$(A,U)$, where

$A:=\arg\gamma_{1}=\cot^{-1}(\operatorname{Re}\gamma_{1}/\operatorname{Im}\gamma_{1}),\quad
U:=(\operatorname{Im}\gamma_{1})^{2}.$

Denoting the density of $(A,U)$ by $\varphi$, we note that

$\psi(x,y)=\frac{2y^{2}}{x^{2}+y^{2}}\varphi(\cot^{-1}(x/y),y^{2}),\quad
x\in\mathbb{R},\ y>0.$

It follows from Theorem 2.1 that the density $\varphi$ is the unique solution
to the corresponding Fokker-Planck-Kolmogorov equation, which in the new
coordinates is given by

$\displaystyle\frac{\kappa}{2u}\sin^{4}\alpha\,\partial^{2}_{\alpha\alpha}\varphi+\frac{3\kappa-4}{u}\sin^{3}\alpha\cos\alpha\,\partial_{\alpha}\varphi+(u-4\sin^{2}\alpha)\,\partial_{u}\varphi$
$\displaystyle\qquad+\frac{\kappa-4}{u}(3\sin^{2}\alpha\cos^{2}\alpha-\sin^{4}\alpha)\,\varphi+\varphi=0,\quad(\alpha,u)\in(0,\pi)\times(0,4].$
(3.32)

Recall that we can consider this equation on a larger domain
$(0,\pi)\times(0,\infty)$, but since $\psi(x,y)=0$ for $y\geqslant 2$, we have
$\varphi(\alpha,u)=0$ for $u\geqslant 4$.

Note that this PDE can be rewritten as

$\partial_{u}((u-4\sin^{2}\alpha)\,\varphi)+\frac{\kappa-4}{u}\partial_{\alpha}(\sin^{3}\alpha\cos\alpha\,\varphi)+\frac{\kappa}{2u}\partial_{\alpha}(\sin^{4}\alpha\,\partial_{\alpha}\varphi)=0.$
(3.33)

The crucial statement on the way to prove Theorem 2.2 is the following lemma.

######  Lemma 3.7.

For any $\alpha\in(0,\pi)$ we have

$\int_{0}^{4}\frac{1}{u}\varphi(\alpha,u)\,du=\frac{\Gamma(1+\frac{4}{\kappa})}{4\sqrt{\pi}\Gamma(\frac{1}{2}+\frac{4}{\kappa})}(\sin\alpha)^{8/\kappa-2}.$
(3.34)

Before we go into the technical details, let us outline heuristically the main
idea of the proof. If we assume $\varphi(\alpha,0+)=\varphi(\alpha,4)=0$, then
integrating (3.33) in $u$ yields

$\partial_{\alpha}\Bigl{(}\int_{0}^{4}(\frac{\kappa-4}{u}\sin^{3}\alpha\cos\alpha\,\varphi(\alpha,u)+\frac{\kappa}{2u}\sin^{4}\alpha\,\partial_{\alpha}\varphi(\alpha,u))\,du\Bigr{)}=0.$

Hence the expression
$J(\alpha):=\int_{0}^{4}(\frac{\kappa-4}{u}\sin^{3}\alpha\cos\alpha\,\varphi+\frac{\kappa}{2u}\sin^{4}\alpha\,\partial_{\alpha}\varphi)\,du$
does not depend on $\alpha$. Moreover, let us suppose that
$\alpha^{4}|\partial_{\alpha}\varphi|$ and $\alpha^{3}\varphi$ monotonically
go to $0$ as $\alpha\to 0$ for any $u\in(0,4]$. Then $J(0+)=0$ and thus
$J(\alpha)=0$ for any $\alpha\in(0,\pi)$. Therefore,

$0=J(\alpha)\sin^{-8/\kappa-2}\alpha=\int_{0}^{4}\frac{\kappa}{2u}\partial_{\alpha}((\sin\alpha)^{2-8/\kappa}\,\varphi(\alpha,u))\,du.$

This yields that
$\int_{0}^{4}\frac{1}{u}(\sin\alpha)^{2-8/\kappa}\,\varphi(\alpha,u)\,du$ is
constant in $\alpha$, which gives

$\int_{0}^{4}\frac{1}{u}\,\varphi(\alpha,u)\,du=c(\sin\alpha)^{8/\kappa-2}$

for some $c>0$, which is almost the statement of Lemma 3.7.

However, since the boundary behavior of $\varphi$ as $\alpha$ approaches $0$
is not clear, we developed an alternative approach which avoids these steps.
Instead of integrating all the way to $0$, we will integrate only up to
$\varepsilon>0$ and obtain approximate identities. Then we would like to let
$\varepsilon\searrow 0$. For this, we would need the following technical
results about approximating ODEs.

######  Lemma 3.8.

Let $S,T\in\mathbb{R}$, $S\leqslant T$. Suppose
$x_{k}\colon[S,T]\to\mathbb{R}^{d}$, $k\in\mathbb{N}$, are continuous
functions that solve the integral equation

$x_{k}(t)-x_{k}(s)=\int_{s}^{t}\big{(}F(r,x_{k}(r))+g_{k}(r)\big{)}\,dr+h_{k}(s,t),\quad
s,t\in[S,T],$ (3.35)

where

  * $\bullet$

$F$ is a continuous function $[S,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ and
there exists $C>0$ such that $|F(t,x)|\leqslant C(1+|x|)$ for $t\in[S,T]$,
$x\in\mathbb{R}^{d}$;

  * $\bullet$

$g$ and $g_{k}$, $k\in\mathbb{Z}_{+}$, are integrable functions
$[S,T]\to\mathbb{R}^{d}$, $g_{k}\to g$ pointwise as $k\to\infty$, and
$\sup_{k}\lVert g_{k}\rVert_{\infty}<\infty$;

  * $\bullet$

$h_{k}$, $k\in\mathbb{Z}_{+}$, are functions $[S,T]^{2}\to\mathbb{R}^{d}$, and
$\lVert h_{k}\rVert_{\infty}\to 0$ as $k\to\infty$.

Moreover, suppose that there exist $t_{k}\in[S,T]$ such that
$\sup_{k}|x_{k}(t_{k})|<\infty$.

Then there exists a continuous function $x\colon[S,T]\to\mathbb{R}^{d}$ such
that along some subsequence $(k_{j})_{j\in\mathbb{Z}_{+}}$ we have
$x_{k_{j}}\to x$ uniformly as $j\to\infty$ and

$x(t)-x(s)=\int_{s}^{t}\big{(}F(r,x(r))+g(r)\big{)}\,dr,\quad s,t\in[S,T].$
(3.36)

###### Proof.

First, we show that $x_{k}$ are uniformly bounded. Indeed, by our assumptions
we have for any $t\in[S,T]$

$\begin{split}\lvert x_{k}(t)\rvert&\leqslant\lvert
x_{k}(t_{k})\rvert+\int_{t_{k}}^{t}\lvert
F(r,x_{k}(r))+g_{k}(r)\rvert\,dr+\lVert h_{k}\rVert_{\infty}\\\ &\leqslant
C+C\int_{t_{k}}^{t}(1+\lvert x_{k}(r)\rvert)\,dr,\end{split}$

and an application of Grönwall’s inequality implies $x_{k}$ are uniformly
bounded.

Consequently, we can assume $F$ to be bounded. It follows that the family
$(x_{k})_{k\in\mathbb{Z}_{+}}$ is equicontinuous. Indeed, for $\varepsilon>0$
let $k_{\varepsilon}$ large enough such that $\lVert
h_{k}\rVert_{\infty}<\varepsilon$ for $k\geqslant k_{\varepsilon}$. Then, for
$k\geqslant k_{\varepsilon}$, we have

$\begin{split}\lvert x_{k}(t)-x_{k}(s)\rvert&\leqslant\int_{s}^{t}\lvert
F(r,x_{k}(r))+g_{k}(r)\rvert\,dr+\varepsilon\\\ &\leqslant C\lvert
t-s\rvert+\varepsilon\end{split}$

which is smaller than $2\varepsilon$ whenever $\lvert
t-s\rvert<\varepsilon/C$. For $k<k_{\varepsilon}$, by continuity of $x_{k}$ we
can find $\delta_{\varepsilon}>0$ such that $\lvert
x_{k}(t)-x_{k}(s)\rvert<\varepsilon$ whenever $\lvert
t-s\rvert<\delta_{\varepsilon}$.

Hence, by the Arzelà-Ascoli theorem, we have $x_{k_{j}}\to x$ uniformly along
some subsequence. Equation (3.36) follows now from (3.35) by taking limits. ∎

We will later also frequently use integation by parts arguments. In order to
control the boundary terms that appear, the following lemma will be useful.

######  Lemma 3.9.

Let $T>0$, and $f\colon(0,T]\to\mathbb{R}$ be a differentiable function such
that $\int_{\varepsilon}^{T}f(s)\,ds$ neither diverges to $+\infty$ nor
$-\infty$ as $\varepsilon\searrow 0$. Let $h\colon{(0,T]}\to{(0,\infty)}$ be a
non-increasing differentiable function such that
$\int_{0}^{T}h(s)\,ds=+\infty$. Then there exists a sequence $t_{k}\searrow 0$
such that

$\lvert f(t_{k})\rvert\leqslant h(t_{k})\quad\text{and}\quad
f^{\prime}(t_{k})\geqslant h^{\prime}(t_{k}),\quad k\in\mathbb{Z}_{+}.$

###### Proof.

First we note that there must exist a sequence $s_{k}\searrow 0$ such that
$\lvert f(s_{k})\rvert<h(s_{k})$ for all $k\in\mathbb{Z}_{+}$, otherwise we
would have $\int_{0}^{T}f(s)\,ds=+\infty$ or $\int_{0}^{T}f(s)\,ds=-\infty$.
To control $f^{\prime}$, we distinguish two cases.

Case 1: We have $\lvert f(t)\rvert\leqslant h(t)$ for all small $t$. In that
case, consider $g(t):=h(t)-f(t)$. The function $g$ cannot be always increasing
for small $t$, otherwise we would have $\int_{0}^{T}f(s)\,ds=\infty$.
Consequently there must be a sequence $t_{k}\searrow 0$ such that
$g^{\prime}(t_{k})\leqslant 0$.

Case 2: We have $\lvert f(r_{k})\rvert>h(r_{k})$ along a sequence
$r_{k}\searrow 0$. We can pick the sequence such that either
$f(r_{k})>h(r_{k})$ for all $k$ or $f(r_{k})<-h(r_{k})$ for all $k$. In the
former case $f(r_{k})>h(r_{k})$ for all $k$, let $t_{k}=\sup\\{t<r_{k}\mid
f(t)\leqslant h(t)\\}$ (this set is non-empty due to the existence of a
sequence $s_{k}$ with $\lvert f(s_{k})\rvert<h(s_{k})$). Then
$f(t_{k})=h(t_{k})$ and $f^{\prime}(t_{k})\geqslant h^{\prime}(t_{k})$ as
desired. In the latter case $f(r_{k})<-h(r_{k})$ for all $k$, let
$t_{k}=\inf\\{t>r_{k}\mid f(t)\geqslant-h(t)\\}$ (again, $(t_{k})$ is well-
defined and tends to $0$ due to the existence of $(s_{k})$ as above). Then
$f(t_{k})=h(t_{k})$ and $f^{\prime}(t_{k})\geqslant-h^{\prime}(t_{k})\geqslant
h^{\prime}(t_{k})$ since $h$ is non-increasing. ∎

######  Corollary 3.10.

Consider the same setup as Lemma 3.9, and suppose additionally that
$f\geqslant 0$. Then there exists a sequence $t_{k}\searrow 0$ such that

$f(t_{k})\leqslant h(t_{k})\quad\text{and}\quad\lvert
f^{\prime}(t_{k})\rvert\leqslant\lvert h^{\prime}(t_{k})\rvert,\quad
k\in\mathbb{Z}_{+}.$ (3.37)

###### Proof.

Let $(t_{k})_{k\in\mathbb{Z}_{+}}$ be a sequence as in Lemma 3.9. Fix now
$k\in\mathbb{Z}_{+}$. If $f^{\prime}(t_{k})\leqslant\lvert
h^{\prime}(t_{k})\rvert$, then, by Lemma 3.9 we have
$f^{\prime}(t_{k})\geqslant-\lvert h^{\prime}(t_{k})\rvert$ and
$f(t_{k})\leqslant h(t_{k})$. Hence the point $t_{k}$ satisfies (3.37).

Otherwise, if $f^{\prime}(t_{k})>\lvert h^{\prime}(t_{k})\rvert$, define
$s_{k}=\sup\\{t\leqslant t_{k}\mid f^{\prime}(t)\leqslant\lvert
h^{\prime}(t)\rvert\\}$ (this set is non-empty, otherwise we would have
$f(t)\to-\infty$ as $t\searrow 0$). By definition, we have
$f^{\prime}(t)>\lvert h^{\prime}(t)\rvert$ for $t\in{]s_{k},t_{k}]}$, and
hence also $f(s_{k})<f(t_{k})$. Moreover, we find some $r_{k}\leqslant s_{k}$
close to $s_{k}$ with $f^{\prime}(r_{k})\leqslant\lvert
h^{\prime}(r_{k})\rvert$. By continuity, we still have $f(t)<f(t_{k})$ for
$t\in[r_{k},t_{k}[$. Since the derivative of any differentiable function
satisfies the intermediate value theorem, we find some
$\widetilde{t}_{k}\in[r_{k},t_{k}]$ such that
$f^{\prime}(\widetilde{t}_{k})=\lvert h^{\prime}(\widetilde{t}_{k})\rvert$.
Then we also have $f(\widetilde{t}_{k})<f(t_{k})\leqslant h(t_{k})\leqslant
h(\widetilde{t}_{k})$ as desired. ∎

We now proceed to the main part of our proof. In the following, we denote for
$n\in\mathbb{Z}_{+}$, $\alpha\in(0,\pi)$, and $\varepsilon>0$

$I_{n}(\alpha):=\int_{0}^{4}u^{-n}\varphi(\alpha,u)\,du;\qquad
I_{n}^{\varepsilon}(\alpha):=\int_{\varepsilon}^{4}u^{-n}\varphi(\alpha,u)\,du.$

From the equation (3.2), we will deduce a recursive system of ODEs that are
satisfied for the functions $I_{n}$. In fact, the relation is satisfied for
general $n\in\mathbb{R}$ but we will use it only with $n\in\mathbb{Z}_{+}$.

######  Lemma 3.11.

Let $n\in\mathbb{Z}_{+}$ be fixed. Suppose that either $n=0$ or $I_{n}$ is
continuous (and finite) on $(0,\pi)$. Assume that for any $\delta>0$ there
exists a sequence $(\varepsilon_{k})_{k\in\mathbb{Z}_{+}}$ converging to $0$
such that

$\varepsilon_{k}^{-n}\int_{\delta}^{\pi-\delta}\varphi(\alpha,\varepsilon_{k})\,d\alpha\to
0$ (3.38)

Then either $I_{n+1}=\infty$ everywhere on $(0,\pi)$ or $I_{n+1}$ is twice
differentiable on $(0,\pi)$ and satisfies the following ODE

$0=nI_{n}-4n\sin^{2}\alpha\,I_{n+1}+(\kappa-4)(3\sin^{2}\alpha\cos^{2}\alpha-\sin^{4}\alpha)\,I_{n+1}\\\
+(3\kappa-4)\sin^{3}\alpha\cos\alpha\,I_{n+1}^{\prime}+\frac{\kappa}{2}\sin^{4}\alpha\,I_{n+1}^{\prime\prime}.$
(3.39)

###### Proof.

Fix $n\in\mathbb{Z}_{+}$. Let $\varepsilon>0$. Multiplying (3.2) by $u^{-n}$
and integrating in $u\in[\varepsilon,4]$ yields

$\displaystyle 0={}$
$\displaystyle\int_{\varepsilon}^{4}u^{-n}\partial_{u}((u-4\sin^{2}\alpha)\,\varphi)\,du$
$\displaystyle+(\kappa-4)(3\sin^{2}\alpha\cos^{2}\alpha-\sin^{4}\alpha)\int_{\varepsilon}^{4}u^{-n-1}\,\varphi\,du$
$\displaystyle+(3\kappa-4)\sin^{3}\alpha\cos\alpha\int_{\varepsilon}^{4}u^{-n-1}\,\partial_{\alpha}\varphi\,du+\frac{\kappa}{2}\sin^{4}\alpha\int_{\varepsilon}^{4}u^{-n-1}\,\partial^{2}_{\alpha\alpha}\varphi\,du$
$\displaystyle={}$
$\displaystyle-\varepsilon^{-n}(\varepsilon-4\sin^{2}\alpha)\,\varphi(\alpha,\varepsilon)+nI_{n}^{\varepsilon}-4n\sin^{2}\alpha\,I_{n+1}^{\varepsilon}$
$\displaystyle+(\kappa-4)(3\sin^{2}\alpha\cos^{2}\alpha-\sin^{4}\alpha)\,I_{n+1}^{\varepsilon}$
$\displaystyle+(3\kappa-4)\sin^{3}\alpha\cos\alpha\,(I_{n+1}^{\varepsilon})^{\prime}+\frac{\kappa}{2}\sin^{4}\alpha\,(I_{n+1}^{\varepsilon})^{\prime\prime}.$
(3.40)

We would like to apply Lemma 3.8 to pass to the limit as $\varepsilon\to 0$ in
the above ODE. Suppose now that $I_{n+1}$ is not infinite everywhere, i.e.
$I_{n+1}(\alpha_{0})<\infty$ for some $\alpha_{0}\in(0,\pi)$. Fix arbitrary
$\delta>0$ small enough such that $\alpha_{0}\in(\delta,\pi-\delta)$ and set
$S\mathrel{\mathop{:}}=\delta$, $T\mathrel{\mathop{:}}=\pi-\delta$,

$\displaystyle x_{k}$
$\displaystyle\mathrel{\mathop{:}}=\begin{pmatrix}I_{n+1}^{(\varepsilon_{k})}\\\
\partial_{\alpha}I_{n+1}^{(\varepsilon_{k})}\end{pmatrix},$ $\displaystyle
F(\alpha,x^{(1)},x^{(2)})$
$\displaystyle\mathrel{\mathop{:}}=\begin{pmatrix}x^{(2)}\\\\[4.30554pt]
\frac{2}{\kappa\sin^{2}\alpha}\left(4nx^{(1)}-x^{(1)}(\kappa-4)(3\cos^{2}\alpha-\sin^{2}\alpha)-(3\kappa-4)x^{(2)}\sin\alpha\cos\alpha\right)\end{pmatrix},$
$\displaystyle g_{k}(\alpha)$
$\displaystyle\mathrel{\mathop{:}}=\begin{pmatrix}0\\\
-\frac{2}{\kappa\sin^{4}\alpha}nI_{n}^{\varepsilon_{k}}(\alpha)\end{pmatrix},$
$\displaystyle h_{k}(\alpha_{1},\alpha_{2})$
$\displaystyle\mathrel{\mathop{:}}=\begin{pmatrix}0\\\
\int_{\alpha_{1}}^{\alpha_{2}}\frac{2}{\kappa\sin^{4}\alpha}\varepsilon_{k}^{-n}(\varepsilon_{k}-4\sin^{2}\alpha)\,\varphi(\alpha,\varepsilon_{k})\,d\alpha\end{pmatrix}$

where $\varepsilon_{k}$ are the same as in the condition (3.38). It is obvious
that on $[\delta,\pi-\delta]$ the function $F$ is continuous and has linear
growth in $x^{(1)},x^{(2)}$. Moreover, $g_{k}(\alpha)\to
g(\alpha)=\begin{pmatrix}0&-\frac{2nI_{n}(\alpha)}{\kappa\sin^{4}\alpha}\end{pmatrix}^{T}$
monotonically by the assumptions of the Lemma. Finally, thanks to (3.38), we
have $\lVert h_{k}\rVert_{\infty}\to 0$ on $[\delta,\pi-\delta]^{2}$.

It remains to find a sequence $\alpha_{k}$ such that
$I_{n+1}^{(\varepsilon_{k})}(\alpha_{k})$ and
$(I_{n+1}^{(\varepsilon_{k})})^{\prime}(\alpha_{k})$ are bounded.

First assume that there exists
$\alpha^{\prime},\alpha^{\prime\prime}\in[\delta,\pi-\delta]$ such that
$\alpha^{\prime}<\alpha_{0}<\alpha^{\prime\prime}$ and
$I_{n+1}(\alpha_{0})<I_{n+1}(\alpha^{\prime})$,
$I_{n+1}(\alpha_{0})<I_{n+1}(\alpha^{\prime\prime})$ (at this point, we allow
$I_{n+1}(\alpha^{\prime})$ or $I_{n+1}(\alpha^{\prime\prime})$ to be
infinite). Then for all large enough $k$ we have
$I_{n+1}^{\varepsilon_{k}}(\alpha_{0})<I_{n+1}^{\varepsilon_{k}}(\alpha^{\prime})$,
$I_{n+1}^{\varepsilon_{k}}(\alpha_{0})<I_{n+1}^{\varepsilon_{k}}(\alpha^{\prime\prime})$.
Pick some
$\alpha_{k}\in\operatorname{argmin}_{[\alpha^{\prime},\alpha^{\prime\prime}]}I_{n+1}^{\varepsilon_{k}}$.
By above, $\alpha_{k}\in(\alpha^{\prime},\alpha^{\prime\prime})$ and hence
$(I_{n+1}^{\varepsilon_{k}})^{\prime}(\alpha_{k})=0$. Moreover,
$I_{n+1}^{\varepsilon_{k}}(\alpha_{k})\leqslant
I_{n+1}^{\varepsilon_{k}}(\alpha_{0})\leqslant I_{n+1}(\alpha_{0})$. Thus the
sequence
$\bigl{(}I_{n+1}^{\varepsilon_{k}}(\alpha_{k}),(I_{n+1}^{\varepsilon_{k}})^{\prime}(\alpha_{k})\bigr{)}_{k\in\mathbb{Z}_{+}}$
is bounded.

If $I_{n+1}(\alpha_{0})\geqslant I_{n+1}(\alpha)$ for all
$\alpha\in[\delta,\alpha_{0}]$, then

$\sup_{\alpha\in[\delta,\alpha_{0}]}I_{n+1}^{\varepsilon_{k}}(\alpha)\leqslant\sup_{\alpha\in[\delta,\alpha_{0}]}I_{n+1}(\alpha)\leqslant
I_{n+1}(\alpha_{0}).$ (3.41)

Hence for each $k$ there exists $\alpha_{k}\in[\delta,\alpha_{0}]$ such that
$\lvert(I_{n+1}^{\varepsilon_{k}})^{\prime}(\alpha_{k})\rvert\leqslant
I_{n+1}(\alpha_{0})/(\alpha_{0}-\delta)$. Combining this with (3.41) we see
again that the sequence
$\bigl{(}I_{n+1}^{\varepsilon_{k}}(\alpha_{k}),(I_{n+1}^{\varepsilon_{k}})^{\prime}(\alpha_{k})\bigr{)}_{k\in\mathbb{Z}_{+}}$
is bounded.

The case when $I_{n+1}(\alpha_{0})\geqslant I_{n+1}(\alpha)$ for all
$\alpha\in[\alpha_{0},\pi-\delta]$ is treated in a similar way.

Thus we see that all the conditions of Lemma 3.8 are satisfied. By passing to
the limit as $\varepsilon\to 0$ in (3.40) and using continuity of $g$, we get
(3.39). ∎

As we mentioned before, we are planning to apply Lemma 3.11 recursively
starting with $n=0$. To verify condition (3.38) we will use the following
result.

######  Lemma 3.12.

For any $n\geqslant 0$ we have

$\int_{0}^{4}\int_{0}^{\pi}u^{-n-1}\sin^{2}\alpha\,\varphi(\alpha,u)\,d\alpha
du=\frac{1}{4}\int_{0}^{4}\int_{0}^{\pi}u^{-n}\varphi(\alpha,u)\,d\alpha du.$
(3.42)

In case both sides of this identity are finite, for any $\delta>0$ there
exists a sequence $(\varepsilon_{k})_{k\in\mathbb{Z}_{+}}\searrow 0$ such that

$\varepsilon_{k}^{-n}\int_{\delta}^{\pi-\delta}\varphi(\alpha,\varepsilon_{k})\,d\alpha\to
0\quad\text{as $k\to\infty$}.$ (3.43)

Note that (3.42) can be rewritten as

$\int_{0}^{\pi}I_{n+1}(\alpha)\sin^{2}\alpha\,d\alpha=\frac{1}{4}\int_{0}^{\pi}I_{n}(\alpha)d\alpha.$

###### Proof.

Fix arbitrary $\delta>0$. Integrating (3.33) in $\alpha$ from $\delta$ to
$\pi/2$ yields for any $u\in(0,4]$

$\partial_{u}\int_{\delta}^{\pi/2}(u-4\sin^{2}\alpha)\varphi(\alpha,u)\,d\alpha=\frac{(\kappa-4)}{u}\varphi(\delta,u)\sin^{3}\delta\cos\delta+\frac{\kappa}{2u}\sin^{4}\delta\,\,\partial_{\alpha}\varphi(\alpha,u)\bigg{\rvert}_{\alpha=\delta}.$
(3.44)

By Theorem 2.1, $\varphi(\alpha,4)=0$ for any $\alpha\in(0,\pi/2)$. Fix now
arbitrary $u_{0}\in(0,4]$ and denote

$J(\alpha)\mathrel{\mathop{:}}=I_{1}^{u_{0}}(\alpha)=\int_{u_{0}}^{4}u^{-1}\varphi(\alpha,u)du,\quad\alpha\in(0,\pi).$

Integrating (3.44) in $u$ from $u_{0}$ to $4$ we get

$\Bigl{|}\int_{\delta}^{\pi/2}(u_{0}-4\sin^{2}\alpha)\varphi(\alpha,u_{0})\,d\alpha\Bigr{|}\leqslant
CJ(\delta)\delta^{3}+C\delta^{4}|J^{\prime}(\delta)|.$ (3.45)

Let us pass to the limit in (3.45) as $\delta\to 0$. Note that
$A:=\int_{0}^{1}J(\alpha)\,d\alpha$ is obviously finite. Hence, we can apply
Corollary 3.10 with $f=J$, $h(t)=1/t$. Then, there exists a sequence
$(\delta_{k})_{k\in\mathbb{Z}_{+}}$, such that

$\delta_{k}\downarrow 0,\,\,\,\delta_{k}J(\delta_{k})\leqslant
1,\,\,\,\delta_{k}^{2}|J^{\prime}(\delta_{k})|\leqslant 1$

for all $k\in\mathbb{Z}_{+}$. Applying now (3.45) with $\delta=\delta_{k}$ and
passing to the limit as $k\to\infty$, we get

$\int_{0}^{\pi/2}(u_{0}-4\sin^{2}\alpha)\varphi(\alpha,u_{0})\,d\alpha=0,$

which by symmetry of $\varphi$ implies

$\int_{0}^{\pi}(u_{0}-4\sin^{2}\alpha)\varphi(\alpha,u_{0})\,d\alpha=0$

for any $u_{0}\in(0,4]$. Dividing now this identity by $u_{0}^{n+1}$ and
integrating in $u_{0}$, we get (3.42).

To show (3.43), fix $\delta>0$. Assuming the left-hand side of (3.42) to be
finite, we get

$\int_{0}^{4}\int_{\delta}^{\pi-\delta}u^{-n-1}\varphi(\alpha,u)\,d\alpha\,du\leqslant\frac{1}{\sin^{2}\delta}\int_{0}^{4}\int_{\delta}^{\pi-\delta}u^{-n-1}\sin^{2}\alpha\,\varphi(\alpha,u)\,d\alpha\,du<\infty.$

Therefore there must exist a sequence of $\varepsilon_{k}\searrow 0$
satisfying (3.43) because otherwise the left-hand side of the above inequality
would be infinite. ∎

######  Remark 3.13.

Lemma 3.12 can be deduced from a general PDE argument [BKRS15, Theorem 2.3.2
and inequality (2.3.2)]. Indeed, note that PDE (3.2) can be written as

$\mathcal{L}^{*}\varphi=0,$

where
$\mathcal{L}=\frac{\kappa}{2u}\sin^{4}\alpha\,\partial^{2}_{\alpha\alpha}+\frac{4+\kappa}{u}\cos\alpha\sin^{3}\alpha\,\partial_{\alpha}+(4\sin^{2}\alpha-u)\,\partial_{u}$,
$u>0$, $\alpha\in(0,\pi)$. If $n=1$, take a Lyapunov function
$V(\alpha,u):=-\log u$; otherwise set $V(\alpha,u):=\frac{1}{n-1}u^{-n+1}$.
Then

$\mathcal{L}V(\alpha,u)=(u-4\sin^{2}\alpha)u^{-n}.$

Note however that even though $V$ does not satisfy all the conditions of
[BKRS15, Theorem 2.3.2], a standard mollification argument and [BKRS15,
inequality (2.3.2)] yield (3.42). However, writing up rigorously all the
technical details gets a bit complicated, so we found it simpler to give a
direct proof.

We are now able to prove (3.34) rigorously.

###### Proof of Lemma 3.7.

Let us apply Lemma 3.11 with $n=0$. We see that the right-hand side of (3.42)
is finite for $n=0$. Hence Lemma 3.12 implies that(3.43) holds for $n=0$.
Therefore, condition (3.38) is satisfied for $n=0$.

Note now that if $I_{1}=\infty$ for all $\alpha\in(0,\pi)$, then the left-hand
side of (3.42) with $n=0$ is infinite. However this is not the case. Thus, by
Lemma 3.11, the function $I_{1}$ is twice differentiable and solves

$(\kappa-4)(3\sin^{2}\alpha\cos^{2}\alpha-\sin^{4}\alpha)\,I_{1}+(3\kappa-4)\sin^{3}\alpha\cos\alpha\,I_{1}^{\prime}+\frac{\kappa}{2}\sin^{4}\alpha\,I_{1}^{\prime\prime}=0.$

This can be rewritten as

$\partial_{\alpha}\Bigl{(}(\kappa-4)\sin^{3}\alpha\cos\alpha\,I_{1}+\frac{\kappa}{2}\sin^{4}\alpha\,I_{1}^{\prime}\Bigr{)}=0.$
(3.46)

Let $\alpha\in(0,\pi)$. Then integrating (3.46) in
$\alpha^{\prime}\in[\alpha,\pi-\alpha]$, we get

$(\kappa-4)\sin^{3}\alpha\cos\alpha(I_{1}(\alpha)+I_{1}(\pi-\alpha))+\frac{\kappa}{2}\sin^{4}\alpha(I_{1}^{\prime}(\alpha)-I_{1}^{\prime}(\pi-\alpha))=0$
(3.47)

Recall that by Theorem 2.1 we have that the density $\psi$ is symmetric,
$\psi(x,y)=\psi(-x,y)$ for $x\in\mathbb{R}$, $y>0$. This implies that
$\varphi$ is also symmetric and $\varphi(\alpha,u)=\varphi(\pi-\alpha,u)$,
$\partial_{\alpha}\varphi(\alpha,u)=-\partial_{\alpha}\varphi(\pi-\alpha,u)$
for $\alpha\in(0,\pi)$, $u>0$. Hence $I_{1}(\alpha)=I_{1}(\pi-\alpha)$,
$I_{1}^{\prime}(\alpha)=-I_{1}^{\prime}(\pi-\alpha)$ and (3.47) yields

$(\kappa-4)\sin^{3}\alpha\cos\alpha\,I_{1}(\alpha)+\frac{\kappa}{2}\sin^{4}\alpha\,I_{1}^{\prime}(\alpha)=0.$
(3.48)

Therefore,

$\partial_{\alpha}(\sin^{2-8/\kappa}\alpha\,I_{1}(\alpha))=0,$

and we finally get

$I_{1}(\alpha)=c\sin^{8/\kappa-2}\alpha,\quad\alpha\in(0,\pi),$

for some $c>0$. The precise value of $c$ follows from (3.42):

$\frac{1}{4}=\int_{0}^{4}\int_{0}^{\pi}\frac{1}{u}\sin^{2}\alpha\,\varphi(\alpha,u)\,d\alpha\,du=c\int_{0}^{\pi}\sin^{8/\kappa}\alpha\,d\alpha=c\sqrt{\pi}\frac{\Gamma(\frac{1}{2}+\frac{4}{\kappa})}{\Gamma(1+\frac{4}{\kappa})},$

which gives (3.34). ∎

###### Proof of Theorem 2.2(i).

Note that the
$\operatorname{Law}(\gamma(t))=\operatorname{Law}(t^{1/2}\gamma(1))$.
Therefore

$\hskip 0.6458pt\mathsf{E}\hskip
0.43057pt\int_{0}^{\infty}\mathbbm{1}_{\gamma(t)\in\Lambda}\,dt=\int_{0}^{\infty}\mathsf{P}(\gamma(t)\in\Lambda)\,dt=\int_{0}^{\infty}\mathsf{P}(\gamma(1)\in
t^{-1/2}\Lambda)\,dt.$ (3.49)

Fix $0<a<b$, $0<\alpha<\beta<\pi$. First consider sets $\Lambda$ of the form

$\Lambda=\\{x+iy\mid\cot^{-1}(x/y)\in[\alpha,\beta],\ y^{2}\in[a,b]\\}.$
(3.50)

Then, writing $\gamma(1)=\sqrt{U}(\cot A+i)$, we continue (3.49) in the
following way

$\displaystyle\hskip 0.6458pt\mathsf{E}\hskip
0.43057pt\int_{0}^{\infty}\mathbbm{1}_{\gamma(t)\in\Lambda}\,dt$
$\displaystyle=\int_{0}^{\infty}\mathsf{P}(A\in[\alpha,\beta],U\in[a/t,b/t])\,dt$
$\displaystyle=\int_{0}^{\infty}\int_{\alpha}^{\beta}\int_{a/t}^{b/t}\varphi(\alpha^{\prime},u)\,dud\alpha^{\prime}dt$
$\displaystyle=\int_{0}^{\infty}\int_{\alpha}^{\beta}\frac{b-a}{u}\varphi(\alpha^{\prime},u)\,d\alpha^{\prime}du$
$\displaystyle=(b-a)\frac{\Gamma(1+\frac{4}{\kappa})}{4\sqrt{\pi}\Gamma(\frac{1}{2}+\frac{4}{\kappa})}\int_{\alpha}^{\beta}(\sin\alpha^{\prime})^{8/\kappa-2}\,d\alpha^{\prime},$

where the last identity follows from Lemma 3.7. Since

$\displaystyle\int_{\Lambda}\bigl{(}1+\frac{x^{2}}{y^{2}}\bigr{)}^{-4/\kappa}\,dx\,dy$
$\displaystyle=\frac{1}{2}\int_{\alpha}^{\beta}\int_{a}^{b}(\sin\alpha^{\prime})^{8/\kappa-2}\,dud\alpha^{\prime}$
$\displaystyle=\frac{b-a}{2}\int_{\alpha}^{\beta}(\sin\alpha^{\prime})^{8/\kappa-2}\,d\alpha^{\prime},$

we see that

$\hskip 0.6458pt\mathsf{E}\hskip
0.43057pt\int_{0}^{\infty}\mathbbm{1}_{\gamma(t)\in\Lambda}\,dt=\frac{\Gamma(1+\frac{4}{\kappa})}{2\sqrt{\pi}\Gamma(\frac{1}{2}+\frac{4}{\kappa})}\int_{\Lambda}\bigl{(}1+\frac{x^{2}}{y^{2}}\bigr{)}^{-4/\kappa}\,dx\,dy.$

Clearly, sets $\Lambda$ of the form (3.50) generate the Borel $\sigma$-algebra
on $\mathbb{H}$. This implies (2.3). ∎

To prove Theorem 2.2(ii), we need the following key result.

######  Lemma 3.14.

Let $n\in\mathbb{Z}_{+}$, $n\geqslant 1$. Then
$\displaystyle\int_{0}^{\pi}I_{n}(\alpha)\,d\alpha$ is finite for
$\kappa<8/(2n-1)$ and infinite for $\kappa\geqslant 8/(2n-1)$.

Furthermore, let $\frac{8}{\kappa}>2n-3$. Then the function
$I_{n}\colon(0,\pi)\to\mathbb{R}_{+}$ is continuous and for any $\delta>0$
there exists $\alpha_{0}=\alpha_{0}(n,\delta)\in(0,\pi/2)$ such that for
$\alpha\in(0,\alpha_{0})$

$I_{n}(\alpha)\geqslant\alpha^{8/\kappa-2n}\lvert\log\alpha\rvert^{-\delta}.$
(3.51)

If, additionally, $\frac{8}{\kappa}>(2n-3)$ and $\kappa<\frac{16}{3}$, then
for any $\delta>0$ there exists $\alpha_{0}=\alpha_{0}(n,\delta)\in(0,\pi/2)$
such that for $\alpha\in(0,\alpha_{0})$

$I_{n}(\alpha)\leqslant\alpha^{8/\kappa-2n-\delta}.$ (3.52)

###### Proof.

We will prove this lemma by induction over $n$, with the case $n=1$ already
established in Lemma 3.7. Let us first explain the heuristic idea. Consider
for simplicity the first non-trivial case $n=2$. Then approximating (3.39)
near $\alpha\approx 0$ and knowing that
$I_{1}=c_{0}(\sin\alpha)^{8/\kappa-2}$, the equation reads

$0\approx
c_{0}\alpha^{8/\kappa-2}-4\alpha^{2}\,I_{2}+(\kappa-4)3\alpha^{2}\,I_{2}+(3\kappa-4)\alpha^{3}\,I_{2}^{\prime}+\frac{\kappa}{2}\alpha^{4}\,I_{2}^{\prime\prime}.$

If we naively suppose $I_{2}\approx\alpha^{s}$, $I_{2}^{\prime}\approx
s\alpha^{s-1}$, $I_{2}^{\prime\prime}\approx s(s-1)\alpha^{s-2}$, then we find
that either $s=8/\kappa-4$, cancelling the first term $I_{1}$, or
$s<8/\kappa-4$ in which case the remaining terms need to cancel each other. In
the latter case, the coefficients need to sum to $0$, i.e.

$0=(3\kappa-16)+(3\kappa-4)s+\frac{\kappa}{2}s(s-1).$ (3.53)

Recall also, that by Lemma 3.12 with $n=1$, we have
$\int_{0}^{\pi}I_{2}\sin^{2}\alpha\,d\alpha<\infty$, which implies $s>-3$.
However, on the interval $(-3,8/\kappa-4)$ equation (3.53) has no solutions,
and thus the case $s<8/\kappa-4$ is not possible. Hence, the only remaining
option is $s=8/\kappa-4$.

To make this heuristic precise, we find a suitable subsequence
$\alpha_{k}\searrow 0$ where we can apply a similar argument.

Let us now proceed to the rigorous induction on $n$.

Base case. $n=1$. In this case (3.51), (3.52) and continuity of $I_{1}$ was
already proven in (3.34). The fact that $\int_{0}^{\pi}I_{1}(\alpha)\,d\alpha$
is finite if and only if $\kappa<8$ is immediate.

Inductive step. Suppose that the statement of the lemma is valid for
$n\in\mathbb{Z}_{+}$. Let us prove it for $n+1$.

If $\kappa\geqslant 8/(2n-1)$, then
$\int_{0}^{\pi}I_{n}(\alpha)\,d\alpha=\infty$, and this obviously implies that
$\int_{0}^{\pi}I_{n+1}(\alpha)\,d\alpha=\infty$. Therefore it is sufficient to
consider the case $\kappa<8/(2n-1)$. By the inductive step, for these values
of $\kappa$ we have $\int_{0}^{\pi}I_{n}(\alpha)\,d\alpha<\infty$. Hence,
Lemma 3.12 implies that condition (3.43) holds. This, together with continuity
of $I_{n}$, shows that all the conditions of Lemma 3.11 are met. Note that we
cannot have $I_{n+1}=\infty$ for all $\alpha\in(0,\pi)$. Indeed, in this case
the left-hand side of identity (3.42) would be infinite but the right-hand
side of this identity is finite (because it is equal to
$C\int_{0}^{\pi}I_{n}(\alpha)\,d\alpha$). Thus, Lemma 3.11 implies that

$\text{$I_{n+1}$ is twice differentiable and satisfies \eqref{eq:Iq_ode}}.$
(3.54)

Using this, we now show (3.51) and (3.52). The statement about the finiteness
of $\int I_{n}\,d\alpha$ follows immediately.

Lower bound. We begin with the lower bound (3.51). Denote

$s:=\frac{8}{\kappa}-2n-2.$ (3.55)

Fix $\delta\in(0,1)$ and suppose that the lower bound does not hold, i.e. we
have
$I_{n+1}(\widetilde{\alpha}_{k})<\tilde{\alpha}_{k}^{s}\lvert\log\tilde{\alpha}_{k}\rvert^{-\delta}$
for a sequence of $\widetilde{\alpha}_{k}\searrow 0$. We distinguish two
cases.

Case 1.1. $I_{n+1}(\alpha)\leqslant\alpha^{s}\lvert\log\alpha\rvert^{-\delta}$
for all small $\alpha>0$. We apply Lemma 3.9 with
$f(\alpha):=\partial_{\alpha}(\alpha^{-s}I_{n+1}(\alpha))$,
$h(\alpha)=\alpha^{-1}\lvert\log\alpha\rvert^{-\delta}$. It is easy to see
that all the conditions of the lemma are satisfied, and therefore there exists
a sequence of $\alpha_{k}\searrow 0$ such that (for some $C<\infty$)

$\displaystyle\lvert I_{n+1}(\alpha_{k})\rvert$
$\displaystyle\leqslant\alpha_{k}^{s}\lvert\log\alpha_{k}\rvert^{-\delta},$
$\displaystyle\lvert I_{n+1}^{\prime}(\alpha_{k})\rvert$
$\displaystyle\leqslant
C\alpha_{k}^{s-1}\lvert\log\alpha_{k}\rvert^{-\delta},$ $\displaystyle
I_{n+1}^{\prime\prime}(\alpha_{k})$
$\displaystyle\geqslant-C\alpha_{k}^{s-2}\lvert\log\alpha_{k}\rvert^{-\delta}.$

Plugging this into (3.39) we derive

$\displaystyle 0$
$\displaystyle=nI_{n}(\alpha_{k})+\sin^{2}\alpha_{k}(3\kappa-4n-12+(16-4\kappa)\sin^{2}\alpha_{k})\,I_{n+1}(\alpha_{k})$
$\displaystyle\hphantom{=}+(3\kappa-4)\sin^{3}\alpha_{k}\cos\alpha_{k}\,I_{n+1}^{\prime}(\alpha_{k})+\frac{\kappa}{2}\sin^{4}\alpha_{k}\,I_{n+1}^{\prime\prime}(\alpha_{k})$
$\displaystyle\geqslant
nI_{n}(\alpha_{k})+\sin^{2}\alpha_{k}[(3\kappa-4n-12+(16-4\kappa)\sin^{2}\alpha_{k})\wedge
0]\,\alpha_{k}^{s}\lvert\log\alpha_{k}\rvert^{-\delta}$
$\displaystyle\hphantom{=}-C\sin^{3}\alpha_{k}\cos\alpha_{k}\,\alpha_{k}^{s-1}\lvert\log\alpha_{k}\rvert^{-\delta}-C\sin^{4}\alpha_{k}\,\alpha_{k}^{s-2}\lvert\log\alpha_{k}\rvert^{-\delta}.$
(3.56)

Multiplying (3.56) by $\alpha^{-s-2}\lvert\log\alpha\rvert^{\delta}$ and
passing to the limit as $\alpha_{k}\searrow 0$, we get

$-C+n\liminf_{\alpha\to
0}\frac{I_{n}(\alpha)\lvert\log\alpha\rvert^{\delta}}{\alpha^{s+2}}\leqslant
0$ (3.57)

By induction hypothesis (applied with $\delta/2$ in place of $\delta$),
$I_{n}(\alpha)\alpha^{-s-2}\lvert\log\alpha\rvert^{\delta}\to\infty$. This
contradicts (3.57). Therefore it cannot be that
$I_{n+1}(\alpha)\leqslant\alpha^{s}\lvert\log\alpha\rvert^{-\delta}$ for all
small $\alpha$.

Case 1.2. In the other case one can find two sequences
$\widetilde{\alpha}_{k},\accentset{\approx}{\alpha}_{k}\searrow 0$ such that
$\accentset{\approx}{\alpha}_{k+1}\leqslant\widetilde{\alpha}_{k}\leqslant\accentset{\approx}{\alpha}_{k}$
and
$I_{n+1}(\widetilde{\alpha}_{k})<\tilde{\alpha}_{k}^{s}\lvert\log\tilde{\alpha}_{k}\rvert^{-\delta}$,
$I_{n+1}(\accentset{\approx}{\alpha}_{k})>\accentset{\approx}{\alpha}_{k}^{s}\lvert\log\accentset{\approx}{\alpha}_{k}\rvert^{-\delta}$.
Pick
$\alpha_{k}\in\operatorname*{arg\,min}_{[\accentset{\approx}{\alpha}_{k+1},\accentset{\approx}{\alpha}_{k}]}(I_{n+1}(\alpha)-\alpha^{s}\lvert\log\alpha\rvert^{-\delta}$).
Then
$\alpha_{k}\in(\accentset{\approx}{\alpha}_{k+1},\accentset{\approx}{\alpha}_{k})$
and

$\displaystyle\lvert I_{n+1}(\alpha_{k})\rvert$
$\displaystyle<\alpha_{k}^{s}\lvert\log\alpha_{k}\rvert^{-\delta},$
$\displaystyle I_{n+1}^{\prime}(\alpha_{k})$
$\displaystyle=s\alpha_{k}^{s-1}\lvert\log\alpha_{k}\rvert^{-\delta}+o(\alpha_{k}^{s-1}\lvert\log\alpha_{k}\rvert^{-\delta}),$
$\displaystyle I_{n+1}^{\prime\prime}(\alpha_{k})$ $\displaystyle\geqslant
s(s-1)\alpha_{k}^{s-2}\lvert\log\alpha_{k}\rvert^{-\delta}+o(\alpha_{k}^{s-2}\lvert\log\alpha_{k}\rvert^{-\delta}).$

This implies that (3.56) holds for this sequence $(\alpha_{k})$, which again
leads to a contradiction.

Thus, we have shown that
$I_{n+1}(\alpha)\geqslant\alpha^{s}\lvert\log\alpha\rvert^{-\delta}$ for all
small enough $\alpha$. Recalling the definition of $s$ in (3.55), we see that
this is exactly the desired lower bound in (3.51). This bound implies that for
$\kappa\geqslant 8/(2(n+1)-1)=8/(2n+1)$ we have
$\int_{0}^{\pi}I_{n+1}(\alpha)\,d\alpha=\infty$.

Upper bound. Now we proceed with the upper bound in (3.52). We suppose now
that $\kappa<\frac{8}{2n-1}\wedge\frac{16}{3}$. We use again notation (3.55).
Write $I_{n+1}(\alpha)=:\alpha^{s(\alpha)}$ for $\alpha\in(0,\pi)$. We will
distinguish two cases.

Case 2.1. Suppose that

$\liminf_{\alpha\to 0}s(\alpha)<\limsup_{\alpha\to 0}s(\alpha).$

We show that this is impossible by deriving a contradiction.

Note that by Step 1, $\limsup_{\alpha\to 0}s(\alpha)\leqslant s$. Further,
there exists a sequence $\beta_{k}\searrow 0$, such that $s(\beta_{k})>-3$.
Indeed, otherwise the left-hand side of identity (3.42) would be infinite
whilst the right-hand side of this identity is finite thanks to the induction
hypothesis. Therefore, by continuity of $s(\alpha)$ there exists $r\in[-3,s)$,
and sequences $\widetilde{\alpha}_{k},\accentset{\approx}{\alpha}_{k}\searrow
0$ such that
$\accentset{\approx}{\alpha}_{k+1}\leqslant\widetilde{\alpha}_{k}\leqslant\accentset{\approx}{\alpha}_{k}$
and $s(\widetilde{\alpha}_{k})<r$, $s(\accentset{\approx}{\alpha}_{k})>r$.
Pick now

$\alpha_{k}\in\operatorname*{arg\,max}_{[\accentset{\approx}{\alpha}_{k+1},\accentset{\approx}{\alpha}_{k}]}(I_{n+1}(\alpha)-\alpha^{r}).$

Then
$\alpha_{k}\in(\accentset{\approx}{\alpha}_{k+1},\accentset{\approx}{\alpha}_{k})$
and

$\displaystyle I_{n+1}(\alpha_{k})$ $\displaystyle>\alpha_{k}^{r},$
$\displaystyle I_{n+1}^{\prime}(\alpha_{k})$
$\displaystyle=r\alpha_{k}^{r-1},$ $\displaystyle
I_{n+1}^{\prime\prime}(\alpha_{k})$ $\displaystyle\leqslant
r(r-1)\alpha_{k}^{r-2}.$

Substituting this into (3.39), dividing it by $\alpha_{k}^{r+2}$ and letting
$\alpha_{k}\searrow 0$, we get

$(3\kappa-4n-12)+(3\kappa-4)r+\frac{\kappa}{2}r(r-1)+n\limsup_{\alpha\to
0}\frac{I_{n}(\alpha)}{\alpha^{r+2}}\geqslant 0.$

(Here we have used $\kappa\leqslant 16/3$, implying
$3\kappa-4n-12+(16-4\kappa)\sin^{2}\alpha_{k}<0$).

Since $r+2<s+2=8/\kappa-2n$, we have $\frac{I_{n}(\alpha)}{\alpha^{r+2}}\to 0$
by the induction hypothesis. Hence,

$(3\kappa-4n-12)+(3\kappa-4)r+\frac{\kappa}{2}r(r-1)\geqslant 0.$ (3.58)

Recall that $r\in[-3,s)$. Note that the left-hand side of the above expression
is strictly negative for $r=-3$ and for $r=s$; in the latter case it equals
$n(\kappa(2n-1)-12)<0$ thanks to our standing assumption $\kappa<8/(2n-1)$.
Hence the left-hand side of (3.58) is strictly negative for any $r\in[-3,s)$
which is a contradiction.

Case 2.2. It follows from above that $\liminf_{\alpha\to
0}s(\alpha)=\limsup_{\alpha\to 0}s(\alpha)=:r$ and $r\in[-3,s]$. We would like
to show $r=s$ which is (3.52).

Suppose $r<s$. Note that (3.54) implies that $s(\alpha)$ is twice
differentiable. Therefore, all the conditions of Lemma 3.9 are satisfied for
the functions $f(\alpha):=s^{\prime}(\alpha)$,
$h(\alpha):=\frac{1}{\alpha\lvert\log\alpha\rvert\log\lvert\log\alpha\rvert}$.
Thus there exists a sequence $\alpha_{k}\searrow 0$ such that
$|s^{\prime}(\alpha_{k})|\leqslant\frac{1}{\alpha_{k}\lvert\log\alpha_{k}\rvert\log\lvert\log\alpha_{k}\rvert}$
and
$s^{\prime\prime}(\alpha_{k})\geqslant-\frac{1}{\alpha_{k}^{2}\lvert\log\alpha_{k}\rvert\log\lvert\log\alpha_{k}\rvert}$.

Recalling that

$\displaystyle I_{n+1}^{\prime}(\alpha)$
$\displaystyle=\left(\frac{s(\alpha)}{\alpha}+s^{\prime}(\alpha)\log\alpha\right)\alpha^{s(\alpha)},$
$\displaystyle I_{n+1}^{\prime\prime}(\alpha)$
$\displaystyle=\left(-\frac{s(\alpha)}{\alpha^{2}}+2\frac{s^{\prime}(\alpha)}{\alpha}+s^{\prime\prime}(\alpha)\log\alpha+\left(\frac{s(\alpha)}{\alpha}+s^{\prime}(\alpha)\log\alpha\right)^{2}\right)\alpha^{s(\alpha)},$

we get

$\displaystyle I_{n+1}(\alpha_{k})$
$\displaystyle=\alpha_{k}^{s(\alpha_{k})},$ $\displaystyle
I_{n+1}^{\prime}(\alpha_{k})$
$\displaystyle=(s(\alpha_{k})+o(1))\alpha_{k}^{s(\alpha_{k})-1},$
$\displaystyle I_{n+1}^{\prime\prime}(\alpha_{k})$
$\displaystyle\leqslant(-s(\alpha_{k})+s(\alpha_{k})^{2}+o(1))\alpha_{k}^{s(\alpha_{k})-2},$

where $o(1)$ denote some sequences that tend to $0$ as $k\to\infty$. Now we
substitute this into (3.39), divide it by $\alpha_{k}^{s(\alpha_{k})+2}$ and
let $\alpha_{k}\searrow 0$. We derive

$(3\kappa-4n-12)+(3\kappa-4)r+\frac{\kappa}{2}r(r-1)+n\limsup_{\alpha\to
0}\frac{I_{n}(\alpha)}{\alpha^{s(\alpha)+2}}\geqslant 0.$ (3.59)

If now $r<s$, then there exists $\delta>0$ such that $s(\alpha)\leqslant
r+\delta<s$ for all $\alpha$ small enough. Hence, thanks to the induction
hypothesis, $\limsup_{\alpha\to
0}\frac{I_{n}(\alpha)}{\alpha^{s(\alpha)+2}}=0$. Therefore inequality (3.58)
holds for a certain $r\in[-3,s)$ which is a contradiction as before.

Thus we have shown that $\liminf_{\alpha\to 0}s(\alpha)=s$. Therefore,
$I_{n+1}(\alpha)=\alpha^{s(\alpha)}\leqslant\alpha^{s-\delta}$ for all
$\alpha$ small enough, so the upper bound (3.52) holds. Hence for
$\kappa<8/(2(n+1)-1)=8/(2n+1)$ we have
$\int_{0}^{\pi}I_{n+1}(\alpha)\,d\alpha<\infty$. ∎

Now we are ready to complete the proof of Theorem 2.2

###### Proof of Theorem 2.2(ii).

Inequality (2.4) follows directly from Lemma 3.14 and the definition of
$I_{n}$. Further, for $\kappa<8$ we have from Lemma 3.7:

$\hskip 0.6458pt\mathsf{E}\hskip
0.43057pt(\operatorname{Im}\gamma_{1})^{-2}=\int_{0}^{\pi}\int_{0}^{4}\frac{1}{u}\varphi(\alpha,u)\,d\alpha
du=\frac{2}{8-\kappa},$

which is (2.5).

To show (2.6), fix $\kappa<8/3$. Note that in this regime by Lemma 3.14, we
have $\int_{0}^{1}I_{2}(\alpha)<\infty$, and thus by Corollary 3.10 with
$f=I_{2}$, $h=1/(\alpha|\log\alpha|)$ there exists a sequence
$\alpha_{k}\searrow 0$ such that

$\displaystyle\lim_{\alpha_{k}\searrow 0}\alpha_{k}I_{2}(\alpha_{k})=0,$
(3.60) $\displaystyle\lim_{\alpha_{k}\searrow
0}\alpha_{k}^{2}I_{2}^{\prime}(\alpha_{k})=0.$ (3.61)

It was shown in the proof of Lemma 3.14, that in this case $I_{2}$ satisfies
(3.39) which can be rewritten as

$\frac{I_{1}}{\sin^{2}\alpha}-4I_{2}+(\kappa-4)(3-4\sin^{2}\alpha)I_{2}+(3\kappa-4)\sin\alpha\cos\alpha\,I_{2}^{\prime}+\frac{\kappa}{2}\sin^{2}\alpha\,I_{2}^{\prime\prime}=0.$
(3.62)

Integrate now the above equation in $\alpha$ from $\alpha_{k}$ to
$\pi-\alpha_{k}$, then integrate by parts. Thanks to (3.60) and (3.61), all
the boundary terms vanish when we send $\alpha_{k}\searrow 0$. Note also that
by Lemma 3.12, we have
$\int_{0}^{\pi}I_{2}(\alpha)\sin^{2}\alpha\,d\alpha=\frac{1}{4}\int_{0}^{\pi}I_{1}(\alpha)\,d\alpha$.
We get

$\int_{0}^{\pi}\frac{I_{1}(\alpha)}{\sin^{2}\alpha}\,d\alpha+(\kappa-12)\int_{0}^{\pi}I_{2}(\alpha)\,d\alpha+2\int_{0}^{\pi}I_{1}(\alpha)\,d\alpha=0.$
(3.63)

Recalling the expression for $I_{1}$ from Lemma 3.7, we deduce

$\hskip 0.6458pt\mathsf{E}\hskip
0.43057pt(\operatorname{Im}\gamma_{1})^{-4}=\int_{0}^{\pi}I_{2}(\alpha)\,d\alpha=\frac{48-16\kappa}{(12-\kappa)(8-\kappa)(8-3\kappa)}.\qed$

######  Remark 3.15.

For general $n$, the identity (3.63) reads

$(4n+8-\kappa)\int_{0}^{\pi}I_{n+1}(\alpha)\,d\alpha=n\int_{0}^{\pi}\frac{I_{n}(\alpha)}{\sin^{2}\alpha}\,d\alpha+2\int_{0}^{\pi}I_{n}(\alpha)\,d\alpha.$

Unfortunately, we do not have an explicit formula for
$\int_{0}^{\pi}\frac{I_{n}(\alpha)}{\sin^{2}\alpha}\,d\alpha$ for $n\geqslant
2$. This prevents us from getting explicit formulas of negative moments of
$\operatorname{Im}(\gamma_{1})$ of higher order.

######  Remark 3.16.

Another possible approach to find explicit formulas for $I_{n}$ would be
through its Fourier coefficients

$a_{0}=\frac{1}{\pi}\int_{0}^{\pi}I_{n}\,d\alpha,\quad
a_{j}=\frac{2}{\pi}\int_{0}^{\pi}I_{n}\cos(2j\alpha)\,d\alpha.$

Formally expanding (3.62), we obtain a (countable) system of linear equations
for $(a_{j})_{j\geqslant 0}$ in terms of the Fourier coefficients
$(b_{j})_{j\geqslant 0}$ of the function $I_{n-1}/\sin^{2}\alpha$. However, it
seems difficult to solve the system of equations explicitly. Only for
$a_{0},a_{1}$ we get a system of two equations in terms of $b_{0},b_{1}$ which
correspond exactly to what we obtain from the proof above.

## References

  * [Ale58] A. D. Aleksandrov.  Investigations on the maximum principle. I.  Izv. Vysš. Učebn. Zaved. Matematika, 1958(5):126–157, 1958\.
  * [BKRS15] Vladimir I. Bogachev, Nicolai V. Krylov, Michael Röckner, and Stanislav V. Shaposhnikov.  Fokker-Planck-Kolmogorov equations, volume 207 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 2015.
  * [BRS21] Vladimir I. Bogachev, Michael Röckner, and Stanislav V. Shaposhnikov.  On the Ambrosio-Figalli-Trevisan superposition principle for probability solutions to Fokker-Planck-Kolmogorov equations.  J. Dynam. Differential Equations, 33(2):715–739, 2021.
  * [DP06] Giuseppe Da Prato.  An introduction to infinite-dimensional analysis.  Universitext. Springer-Verlag, Berlin, 2006.  Revised and extended from the 2001 original by Da Prato.
  * [Eva98] Lawrence C. Evans.  Partial differential equations, volume 19 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 1998.
  * [Fre71] David Freedman.  Brownian motion and diffusion.  Holden-Day, San Francisco, Calif.-Cambridge-Amsterdam, 1971.
  * [Hai08] Martin Hairer.  Ergodic theory for Stochastic PDEs.  Lecture notes, Imperial College London. Available at http://www.hairer.org/notes/Imperial.pdf, 2008.
  * [Hai11] Martin Hairer.  On Malliavin’s proof of Hörmander’s theorem.  Bull. Sci. Math., 135(6-7):650–666, 2011.
  * [Hil70] C. Denson Hill.  A sharp maximum principle for degenerate elliptic-parabolic equations.  Indiana Univ. Math. J., 20:213–229, 1970.
  * [IW89] Nobuyuki Ikeda and Shinzo Watanabe.  Stochastic differential equations and diffusion processes, volume 24 of North-Holland Mathematical Library.  North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, second edition, 1989.
  * [JVL12] Fredrik Johansson Viklund and Gregory F. Lawler.  Almost sure multifractal spectrum for the tip of an SLE curve.  Acta Math., 209(2):265–322, 2012.
  * [Kal96] Olav Kallenberg.  On the existence of universal functional solutions to classical SDE’s.  Ann. Probab., 24(1):196–205, 1996.
  * [KS91] Ioannis Karatzas and Steven E. Shreve.  Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics.  Springer-Verlag, New York, second edition, 1991.
  * [Kem17] Antti Kemppainen.  Schramm-Loewner evolution, volume 24 of SpringerBriefs in Mathematical Physics.  Springer, Cham, 2017.
  * [Law05] Gregory F. Lawler.  Conformally invariant processes in the plane, volume 114 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 2005.
  * [LSW04] Gregory F. Lawler, Oded Schramm, and Wendelin Werner.  Conformal invariance of planar loop-erased random walks and uniform spanning trees.  Ann. Probab., 32(1B):939–995, 2004.
  * [LMN19] Terry J Lyons, Vlad Margarint, and Sina Nejad.  Convergence to closed-form distribution for the backward $SLE_{\kappa}$ at some random times and the phase transition at $\kappa=8$.  arXiv preprint arXiv:1910.05519, 2019.
  * [MSH02] J. C. Mattingly, A. M. Stuart, and D. J. Higham.  Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise.  Stochastic Process. Appl., 101(2):185–232, 2002.
  * [OR73] O. A. Oleinik and E. V. Radkevich.  Second order equations with nonnegative characteristic form.  Plenum Press, New York-London, 1973.
  * [Pav14] Grigorios A. Pavliotis.  Stochastic processes and applications. Diffusion processes, the Fokker-Planck and Langevin equations, volume 60 of Texts in Applied Mathematics.  Springer, New York, 2014.
  * [RS05] Steffen Rohde and Oded Schramm.  Basic properties of SLE.  Ann. of Math. (2), 161(2):883–924, 2005.
  * [Sch00] Oded Schramm.  Scaling limits of loop-erased random walks and uniform spanning trees.  Israel J. Math., 118:221–288, 2000.
  * [SS09] Oded Schramm and Scott Sheffield.  Contour lines of the two-dimensional discrete Gaussian free field.  Acta Math., 202(1):21–137, 2009.
  * [Smi01] Stanislav Smirnov.  Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits.  C. R. Acad. Sci. Paris Sér. I Math., 333(3):239–244, 2001.
  * [Smi10] Stanislav Smirnov.  Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model.  Ann. of Math. (2), 172(2):1435–1467, 2010.
  * [Str08] Daniel W. Stroock.  Partial differential equations for probabilists, volume 112 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2008.
  * [VO16] Marcelo Viana and Krerley Oliveira.  Foundations of ergodic theory, volume 151 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2016.
  * [Yua22] Yizheng Yuan.  Topological characterisations of Loewner traces.  Indiana Univ. Math. J., 71(3):1027–1046, 2022.
  * [Zha16] Dapeng Zhan.  Ergodicity of the tip of an SLE curve.  Probab. Theory Related Fields, 164(1-2):333–360, 2016.
  * [Zha19] Dapeng Zhan.  Decomposition of Schramm-Loewner evolution along its curve.  Stochastic Process. Appl., 129(1):129–152, 2019.