# Fixed Points of $K$-Fibonacci Sequences

Brennan Benfield and Oliver Lippard  Brennan Benfield: Department of
Mathematics and Statistics, University of North Carolina at Charlotte, 9201
University City Blvd., Charlotte, NC 28223, USA  Oliver Lippard: Department of
Mathematics and Statistics, University of North Carolina at Charlotte, 9201
University City Blvd., Charlotte, NC 28223, USA

###### Abstract.

A $K$-Fibonacci sequence is a binary recurrence sequence where $F_{0}=0$,
$F_{1}=1$, and $F_{n}=K\cdot F_{n-1}+F_{n-2}$. These sequences are known to be
periodic modulo every positive integer greater than $1$. If the length of one
shortest period of a $K$-Fibonacci sequence modulo a positive integer is equal
to the modulus, then that positive integer is called a fixed point. This paper
determines the fixed points of $K$-Fibonacci sequences according to the
factorization of $K^{2}+4$ and concludes that if this process is iterated,
then every modulus greater than 3 eventually terminates at a fixed point.

Mathematics Subject Classifications. 11B39, 11B50
Keywords. Fibonacci sequence, Pisano period, fixed points

##  1\. Introduction

Perhaps the most popular sequence of all, the Fibonacci sequence is a binary
recurrence defined by $F_{0}=0$, $F_{1}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ and
has been studied so thoroughly that discovering new properties at all is
almost as surprising as the property discovered. The focus of this paper is a
particular property that was first recognized in 1877 by Lagrange [10]: the
terms in the Fibonacci sequence modulo $10$ (i.e. the one’s place digits)
repeat every $60$ terms. Inquiry regarding the periodicity of the sequence
modulo a positive integer has continued and in 2004 it was proven by Everest &
Shparlinski [4] that every binary recurrence sequence is periodic modulo a
positive integer $m>1$.

Define the Pisano period as the length of one (shortest) period of the
Fibonacci sequence modulo $m$. The Pisano period is denoted $\pi(m)$ and can
be recursively applied: denote $\pi(\pi(m))~{}=~{}\pi^{2}(m)$ and
$\pi(\pi^{i}(m))=\pi^{i+1}(m)$ for $n>2$. The trajectory of an integer $m$ is
the set of integers given by recursively applying the Pisano period to $m$.

######  Example.

Trajectories of the integers in the Fibonacci sequence for $m\leq 24$.

$\displaystyle\pi(2)=3\rightarrow\pi(3)=8\rightarrow\pi(8)=12\rightarrow\pi(12)=24\qquad$
$\displaystyle\pi(13)=28\rightarrow\pi(28)=48\rightarrow\pi(48)=24$
$\displaystyle\pi(3)=8\rightarrow\pi(8)=12\rightarrow\pi(12)=24\qquad$
$\displaystyle\pi(14)=48\rightarrow\pi(48)=24$
$\displaystyle\pi(4)=6\rightarrow\pi(6)=24\qquad$
$\displaystyle\pi(15)=40\rightarrow\pi(40)=60\rightarrow\pi(60)=120$
$\displaystyle\pi(5)=20\rightarrow\pi(20)=60\rightarrow\pi(60)=120\qquad$
$\displaystyle\pi(16)=24$ $\displaystyle\pi(6)=24\qquad$
$\displaystyle\pi(17)=36\rightarrow\pi(36)=24$
$\displaystyle\pi(7)=16\rightarrow\pi(16)=24\qquad$ $\displaystyle\pi(18)=24$
$\displaystyle\pi(8)=12\rightarrow\pi(12)=24\qquad$
$\displaystyle\pi(19)=18\rightarrow\pi(18)=24$ $\displaystyle\pi(9)=24\qquad$
$\displaystyle\pi(20)=60\rightarrow\pi(60)=120$
$\displaystyle\pi(10)=60\rightarrow\pi(60)=120\qquad$
$\displaystyle\pi(21)=16\rightarrow\pi(16)=24$
$\displaystyle\pi(11)=10\rightarrow\pi(10)=60\rightarrow\pi(60)=120\qquad$
$\displaystyle\pi(22)=30\rightarrow\pi(30)=120$
$\displaystyle\pi(12)=24\qquad$ $\displaystyle\pi(23)=48\rightarrow\pi(48)=24$
$\displaystyle\pi(24)=24$

A fixed point is a positive integer $m$ such that $\pi(m)=m$. A $k$-periodic
point is a $k$-tuple of positive integers such that
$\pi(m)=\pi^{2}(m)=\ldots=\pi^{k}(m)=m$. In 1969, Fulton & Morris [6]
determined the fixed points of the Fibonacci sequence and proved that the
trajectory of every integer will eventually terminate at a fixed point. In
2020, Trojovská [14] showed that there are no $k$-periodic points in the
Fibonacci sequence.

######  Theorem 1.1 (Fixed Point Theorem, Fulton & Morris [6]).

For and integer $m>1$, $\pi(m)=m$ if and only if $m=24\times 5^{k-1}$.

######  Theorem 1.2.

(Iteration Theorem, Fulton & Morris [6]) For every integer $m>1$, there exists
a least integer $j$ such that $\pi^{j+1}(m)=\pi^{j}(m)$.

Closely related are $K$-Fibonacci sequences defined by $F_{K,0}=0$,
$F_{K,1}=1$, and $F_{K,n}=KF_{n-1}+F_{n-2}$. The Pisano period of these
sequences is commonly denoted $\pi_{K}(m)$. What are the fixed points of
$K$-Fibonacci sequences? Some fixed points have already been calculated:

######  Theorem 1.3 (Falcon and Plaza [5]).

If $K\geq 2$ is even, then $\pi_{K}(2K)=4$. If $K\geq 3$ is odd, then
$\pi_{K}(2K)=6$.

It follows that if $K=2$, then $\pi_{K}(4)=4$ and if $K=3$, then
$\pi_{K}(6)=6$. The main result of this paper extends the results of Fulton &
Morris [6], establishing that for each $K$, fixed points may be classified
according to one of four categories of $K$. Essentially, for every $K$ the
trajectory of every $m>1$ terminates at a fixed point (or is a $k$-periodic
point).

######  Theorem 1.4 ($K$-Fixed Point Theorem).

Let $p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{t}^{e_{t}}$ be the prime
factorization of $K^{2}+4$ for a positive integer $K\geq 1$, then for every
modulus $m>1$ and for $j_{i}=0,1,2,\ldots$, $\pi_{K}(m)=m$ if and only if:

  1. (i)

$K\equiv\pm 1\pmod{6}$ and $m=24\times p_{1}^{j_{1}}p_{2}^{j_{2}}\cdots
p_{t}^{j_{t}}$.

  2. (ii)

$K\equiv 2\pmod{4}$ and $m=2^{j_{1}}$ or $m=2^{j_{1}+1}p_{2}^{j_{2}}\cdots
p_{t}^{j_{t}}$ where $p_{1}=2$.

  3. (iii)

$K\equiv 3\pmod{6}$ and $m=6$ or $m=12\times p_{3}^{j_{3}}p_{4}^{j_{4}}\cdots
p_{t}^{j_{t}}$.

  4. (iv)

$K\equiv 0\pmod{4}$ and $m=2$ or $m=4\times
p_{2}^{j_{2}+1}p_{3}^{j_{3}+1}\cdots p_{t}^{j_{t}+1}$ where
$p_{1}^{e_{1}}=2^{2}$.

with the only exception when $K\equiv 3\pmod{6}$, which has the $2$-periodic
points $\pi(2)=3$ and $\pi(3)=2$.

######  Theorem 1.5 ($K$-Iteration Theorem).

For every $m>1$, there exists a positive integer $N$ such that
$\pi_{K}^{N}(m)=m$, with the only exception when $K\equiv 3\pmod{6}$, which
has the $2$-periodic points $\pi(2)=3$ and $\pi(3)=2$.

Note that every positive integer $K\geq 1$ belongs to exactly one of the four
categories in Theorem 1.4. Perhaps it is surprising that the prime factors of
$K^{2}+4$ are important in finding fixed points. Famously, the limit of the
ratio between successive Fibonacci numbers is the number $\phi$ where Binet’s
formula gives the $n$th Fibonacci number:

$F_{n}=\frac{\phi^{n}-\phi^{-n}}{\sqrt{5}}\qquad\text{where}\
\qquad\phi=\frac{1+\sqrt{5}}{2}.$

Something analogous occurs for $K$-Fibonacci sequences (see [5, 11]). Note the
similarity when $K=1$:

$F_{K,n}=\frac{\sigma^{n}-\sigma^{-n}}{\sqrt{K^{2}+4}}\qquad\text{where}\
\qquad\sigma=\frac{K+\sqrt{K^{2}+4}}{2}.$

$K$ $K\equiv\pm 1\pmod{6}$ $K\equiv 2\pmod{4}$ $K\equiv 3\pmod{6}$ $K\equiv
0\pmod{4}$ $K=1$ $24\times 5^{j_{1}}$ $K=2$ $2^{j_{1}}$ $K=3$ $6$ or $12\times
13^{j_{1}}$ $K=4$ $2$ or $4\times 5^{j_{1}+1}$ $K=5$ $24\times 29^{j_{1}}$
$K=6$ $2^{j_{1}}$ or $2^{j_{1}+1}\times 5^{j_{2}}$ $K=7$ $24\times 53^{j_{1}}$
$K=8$ $2$ or $4\times 17^{j_{1}+1}$ $K=9$ $6$ or $12\times 5^{j_{1}}\times
17^{j_{2}}$ $K=10$ $2^{j_{1}}$ or $2^{j_{1}+1}\times 13^{j_{2}}$ $K=11$
$24\times 5^{j_{1}}$ $K=12$ $2$ or $4\times 37^{j_{1}+1}$ $K=13$ $24\times
173^{j_{1}}$ $K=14$ $2^{j_{1}}$ or $2^{j_{1}+1}\times 5^{j_{2}}$ $K=15$ $6$ or
$12\times 229^{j_{1}}$ $K=16$ $2$ or $4\times 5^{j_{1}+1}\times 17^{j_{2}+1}$
$K=17$ $24\times 293^{j_{1}}$ $K=18$ $2^{j_{1}}$ or $2^{j_{1}+1}\times
41^{j_{2}}$ $K=19$ $24\times 5^{j_{1}}\times 73^{j_{2}}$ $K=20$ $2$ or
$4\times 101^{j_{1}+1}$ $K=21$ $6$ or $12\times 5^{j_{1}}\times 89^{j_{2}}$
$K=22$ $2^{j_{1}}$ or $2^{j_{1}+1}\times 61^{j_{2}}$ $K=23$ $24\times
13^{j_{1}}\times 41^{j_{2}}$ $K=24$ $2$ or $4\times 5^{j_{1}+1}\times
29^{j_{2}+1}$

Table 1. Fixed points of $(K,1,0,1)$-sequences for $k=1,2,\ldots,24$ and
$j_{i}=0,1,2,\ldots$

##  2\. Preliminaries

Used in the proof of Theorems 1.4 and 1.5 are select results for $K$-Fibonacci
sequences. Renault [11] generalizes a result of Wall [15] and establishes a
particular type of multiplicativity for $\pi_{K}(m)$ among the prime factors
of $m$.

######  Theorem 2.1 (Renault [11]).

Let $m=p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{r}^{e_{r}}$ be the prime
factorization of $m$, then

$\pi_{K}(m)=\operatorname{lcm}\left[\pi_{K}(p_{1}^{e_{1}}),\pi_{K}(p_{2}^{e_{2}}),\ldots,\pi_{K}(p_{r}^{e_{r}})\right].$

Another result generalized by Renault [11] concerns Wall-Sun-Sun primes \-
conjectured primes $p$ where $\pi(p)=\pi(p^{2})$. For the Fibonacci sequence,
these primes were originally investigated by Wall [15] in 1960 but became
exceedingly popular in the 1990s when Sun & Sun [13] demonstrated that the
first case of Fermat’s Last Theorem is false for an exponent $p$ only if $p$
is a Wall-Sun-Sun prime. No Wall-Sun-Sun prime has been found, though
infinitely many are conjectured to exist. For $K$-Fibonacci sequences, any
prime $p$ such that $\pi_{K}(p)=\pi_{K}(p^{2})$ is a $K$-Wall-Sun-Sun prime.
These are much more common: if $K=2$, then $\pi_{2}(13)=\pi_{2}(169)$ and
$\pi_{2}(31)=\pi_{2}(961)$.

######  Theorem 2.2 (Renault [11]).

For all $K$ and all primes $p$, there exists a maximal $e$ such that
$\pi_{K}(p^{e})=\pi_{K}(p)$. For all $x\geq e\geq 1$,
$\pi_{K}(p^{x})=p^{x-e}\cdot\pi_{K}(p)$.

######  Theorem 2.3 (Renault [11]).

If $\pi_{K}(p^{e})\neq\pi_{K}(p^{e+1})$, then
$\pi_{K}(p^{e+1})\neq\pi_{K}(p^{e+2})$.

For $K$-Fibonacci sequences, Bouazzaoui [1, 2] showed that a prime $p$ that
does not divide $K^{2}+4$ is a $K$-Wall-Sun-Sun prime if and only if
$\mathbb{Q}\left(\sqrt{K^{2}+4}\right)$ is not $p$-rational. Results from
Harrington & Jones [7, 8, 9] establish necessary and sufficient conditions for
precisely which primes that do divide $K^{2}+4$ are $K$-Wall-Sun-Sun primes.
In particular, Harrington & Jones establish that no odd prime divisor of
$K^{2}+4$ is ever a $K$-Wall-Sun-Sun prime.

######  Theorem 2.4 (Harrington & Jones [7]).

Let $p$ be a prime divisor of $K^{2}+4$. Then $p$ is a $K$-Wall-Sun-Sun prime
if and only if $p=2$ and $K\equiv 0\pmod{4}$.

In an effort to characterize the behavior of $\pi_{K}(p)$ for a prime $p$,
Renault [11] develops a trichotomy depending on the quadratic residues of
$K^{2}+4$ modulo $p$.

######  Theorem 2.5 (Renault [11]).

Let $p$ be an odd prime.

  * •

If $K^{2}+4$ is a nonzero quadratic residue modulo $p$, then
$\pi_{K}(p)\mid(p-1)$.

  * •

If $K^{2}+4$ is a quadratic non-residue modulo $p$, then $\pi_{K}(p)\mid
2(p+1)$.

  * •

If $p\mid(K^{2}+4)$, then $\pi_{K}(p)=p\cdot\text{ord}_{p}(2^{-1}K)$.

In light of Theorem 2.5, it is possible to completely determine $\pi_{K}(p)$
for primes that divide $K^{2}+4$.

######  Theorem 2.6.

If $p$ is an odd prime and $p\mid(K^{2}+4)$, then, $\text{ord}_{p}(2^{-1}K)=4$
and $\pi_{K}(p)=4p$.

###### Proof.

Notice that $K^{4}=(K^{2}+4)(K^{2}-4)+16$, hence $K^{4}\equiv
16\pmod{K^{2}+4}$. It follows that $(2^{-1}K)^{4}=\frac{K^{4}}{16}\equiv
1\pmod{K^{2}+4}$. Similarly, if a prime $p\mid(K^{2}+4)$, then
$(2^{-1}K)^{4}\equiv 1\pmod{p}$. Thus, $\text{ord}_{p}(2^{-1}K)\mid 4$. Notice
that $K^{2}\equiv-4\pmod{K^{2}+4}$. It follows that
$(2^{-1}K)^{2}=\frac{K^{2}}{4}\equiv-1\pmod{K^{2}+4}$. Hence
$(2^{-1}K)^{2}\equiv-1\pmod{p}$ for all $p\mid(K^{2}+4)$. The condition that
$(2^{-1}K)^{2}\equiv-1\equiv 1\pmod{p}$ is equivalent to the condition that
$p=2$, contradicting the assumption that $p$ is an odd prime. Thus, if $p\neq
2$, then $\text{ord}_{p}(2^{-1}K)=4$. ∎

Harrington & Jones [7] discovered the value of $\pi_{K}(2)$ when $K$ is an
even integer.

######  Theorem 2.7 (Harrington & Jones [7]).

If $K$ is even, (i.e. if $2\mid(K^{2}+4)$), then $\pi_{K}(2)=2$.

For the proof of the main result, it is important to also understand the
nature of $\pi_{K}(2)$ when $K$ is odd.

######  Theorem 2.8.

If $K$ is odd, then $\pi_{K}(2)=3$.

###### Proof.

Consider the $K$-Fibonacci sequence modulo $2$. Since $K\equiv 1\pmod{2}$, the
sequence becomes $0,1,1,\circlearrowleft$, with a period of $3$. ∎

######  Theorem 2.9.

If $3\mid K$, then $\pi_{K}(3)=2$. Otherwise, $\pi_{K}(3)=8$.

###### Proof.

Consider the $K$-Fibonacci sequence modulo $3$. If $3\mid K$, the sequence
becomes $0,1,\circlearrowleft$, with a period of $2$. If $K\equiv 1\pmod{3}$,
the sequence is $0,1,1,2,0,2,2,1,0,1,\circlearrowleft$ with a period of $8$.
If $K\equiv 2\pmod{3}$, the sequence is $0,1,2,2,0,2,1,1,0,1,\circlearrowleft$
again with a period of $8$. ∎

######  Theorem 2.10.

For all odd primes $p$, if $q$ is a prime such that $q\mid\pi_{K}(p)$, then
$q\leq p$, with equality only if $p\mid(K^{2}+4)$.

###### Proof.

If $K^{2}+4$ is a nonzero quadratic residue modulo $p$, then $q\mid(p-1)$ and
$q<p$. If $K^{2}+4$ is a quadratic non-residue, then $q\mid 2(p+1)$. Since $p$
is an odd prime, $p+1$ is composite, and is therefore composed of prime
factors smaller than $p$. Finally, if $p\mid(K^{2}+4)$, then by Theorem 2.5
$\pi_{K}(p)=p\cdot\text{ord}_{p}(2^{-1}K)$. It is clear that
$p\mid\pi_{K}(p)$. Since the integers modulo $p$ form a cyclic group of order
at most $p-1$, it follows that $\text{ord}_{p}(2^{-1}K)\mid(p-1)$. Hence,
$q\leq p$. ∎

##  3\. Proof of the $K$-Fixed Point Theorem

######  Theorem 3.1.

If $K\equiv\pm 1\pmod{6}$ and $m=24\cdot p_{1}^{j_{1}}\cdots p_{r}^{j_{r}}$
for $j_{i}=0,1,2,\ldots$, where $p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the
prime factorization of $K^{2}+4$, then $\pi_{K}(m)=m$.

###### Proof.

Suppose $K\equiv\pm 1\pmod{6}$ and $m=24\cdot p_{1}^{j_{1}}\cdots
p_{r}^{j_{r}}$ for $j_{i}=0,1,2,\ldots$, where $p_{1}^{a_{1}}\cdots
p_{r}^{a_{r}}$ is the prime factorization of $K^{2}+4$. By Theorem 2.1,

$\displaystyle\pi_{K}(m)=\operatorname{lcm}\left[\pi_{K}(2^{3}),\pi_{K}(3),\pi_{K}(p_{1}^{j_{1}}),\ldots,\pi_{K}(p_{r}^{j_{r}})\right]$
(1)

By Theorem 2.4, neither $2$, $3$, nor any $p_{i}$ is a $K$-Wall-Sun-Sun prime.
By Theorems 2.2 and 2.8, $\pi_{K}(2^{3})=2^{3-1}\pi_{K}(2)=2^{2}\cdot 3=12$.
By Theorem 2.9, $\pi_{K}(3)=8$. By Theorem 2.2,
$\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})$ for $i=1,2,\ldots r$.
By Theorem 2.5, $\pi_{K}(p_{i})=p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)$ and by
Theorem 2.6, $\text{ord}_{p_{i}}(2^{-1}K)=4$ since $p_{i}$ is odd and divides
$K^{2}+4$. It follows that

$\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})=p_{i}^{j_{i}-1}\cdot
p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)=4p_{i}^{j_{i}}$

These may be substituted into equation (1) to obtain:

$\pi_{K}(m)=\operatorname{lcm}\left[12,8,4p_{1},\ldots,4p_{r}\right]=24\cdot
p_{1}^{j_{1}}\cdots p_{r}^{j_{r}}=m.$

∎

######  Theorem 3.2.

If $K\equiv\pm 1\pmod{6}$ and $\pi_{K}(m)=m$, then $m=24\cdot
p_{1}^{j_{1}}\cdots p_{r}^{j_{r}}$ for $j_{i}=0,1,2,\ldots$, where
$p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime factorization of $K^{2}+4$.

###### Proof.

Suppose $K\equiv\pm 1\pmod{6}$ and $\pi_{K}(m)=m$. Let $m=2^{j_{1}}\cdot
3^{j_{2}}\cdot p_{1}^{j_{1}}\cdots p_{r}^{j_{r}}\cdot q_{1}^{\theta_{1}}\cdots
q_{s}^{\theta_{s}}$ where $p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime
factorization of $K^{2}+4$. Note that if $\theta_{\lambda}\neq 0$ then
$q_{\lambda}\mid m$. By Theorem 2.1,

$\displaystyle
m=\pi_{K}(m)=\operatorname{lcm}\left[\pi_{K}(2^{j_{1}}),\pi_{K}(3^{j_{2}}),\pi_{K}(p_{1}^{j_{1}}),\ldots,\pi_{K}(p_{r}^{j_{r}}),\pi_{K}(q_{1}^{\theta_{1}}),\ldots,\pi_{K}(q_{s}^{\theta_{s}})\right]$
(2)

By Theorem 2.10, if $\rho\mid\pi_{K}(q_{\lambda}^{\theta_{\lambda}})$, then
$\rho<q_{\lambda}$ for all $1\leq i\leq s$. By Theorem 2.2,
$\pi_{K}(q_{\lambda}^{\theta_{\lambda}})=q_{\lambda}^{\theta_{\lambda}-1}\pi_{K}(q_{\lambda})$
and by Theorem 2.5, $\pi_{K}(q_{\lambda})\mid(q_{\lambda}-1)$ if $K^{2}+4$ is
a quadratic residue, or $\pi_{K}(q_{\lambda})\mid 2(q_{\lambda}+1)$ if
$K^{2}+4$ is not a quadratic residue modulo $q_{\lambda}$. Either way, it
follows that $q_{\lambda}\not\mid\pi_{K}(q_{\lambda})$ and
$q_{\lambda}^{\theta_{\lambda}-1}$ is the largest factor of $q_{\lambda}$ that
divides $\pi_{K}(q_{\lambda}^{\theta_{\lambda}})$, that is,
$q_{\lambda}^{\theta_{\lambda}-1}\mid\mid\pi_{K}(q_{\lambda}^{\theta_{\lambda}})$.

By Theorem 2.4, neither $2$, $3$, $p_{i}$, nor $q_{\lambda}$ are $K$-Wall-Sun-
Sun primes. By Theorems 2.2 and 2.8,
$\pi_{K}(2^{j_{1}})=2^{j_{1}-1}\pi_{K}(2)=2^{j_{1}-1}\cdot 3$. By Theorems 2.2
and 2.9, $\pi_{K}(3^{j_{2}})=3^{j_{2}-1}\pi_{K}(3)=3^{j_{2}-1}\cdot 8$. By
Theorem 2.2, $\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})$ for
$i=1,2,\ldots r$. By Theorem 2.5,
$\pi_{K}(p_{i})=p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)$ and by Theorem 2.6,
$\text{ord}_{p_{i}}(2^{-1}K)=4$ since $p_{i}$ is odd and divides $K^{2}+4$. It
follows that

$\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})=p_{i}^{j_{i}-1}\cdot
p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)=4p_{i}^{j_{i}}$

Note that $q_{\lambda}\not\mid\pi_{K}(2^{j_{1}})=2^{j_{1}-1}\cdot 3$ and
$q_{\lambda}\not\mid\pi_{K}(3^{j_{2}})=3^{j_{2}-1}\cdot 2$, and
$q_{\lambda}\not\mid\pi_{K}(p_{i}^{j_{i}})=4p_{i}^{j_{i}}$. Thus,
$q_{\lambda}^{\theta_{\lambda}-1}\mid\mid m$. But this implies that
$q_{\lambda}^{\theta_{\lambda}}\not\mid m$, which is a contradiction. Equation
(7) is reduced to:

$\displaystyle m=\pi_{K}(m)=\operatorname{lcm}\left[2^{j_{1}-1}\cdot
3,3^{j_{2}-1}\cdot 8,4p_{1}^{j_{1}},\ldots,4p_{r}^{j_{r}}\right]$ (3)

Recall that by hypothesis, $m=2^{j_{1}}\cdot 3^{j_{2}}\cdot
p_{3}^{j_{3}}\cdots p_{r}^{j_{r}}\cdot q_{1}^{\theta_{1}}\cdots
q_{s}^{\theta_{s}}$. Suppose $j_{1}\geq 4$, then $2^{j_{1}-1}$ is the largest
power of $2$ in (8), implying $2^{j_{1}-1}\mid\mid m$. However, this
contradicts the hypothesis $2^{j_{1}}\mid m$. Suppose $j_{2}\geq 2$, then
$3^{j_{2}-1}$ is the largest power of $3$ in (13), implying
$3^{j_{2}-1}\mid\mid m$. However, this contradicts the hypothesis
$3^{j_{2}}\mid m$. Thus, $j_{2}=1$, and equation (8) is reduced to:

$m=\pi_{k}(m)=\operatorname{lcm}\left[2^{j_{1}-1}\cdot
3,24,4p_{1}^{j_{1}},\ldots,4p_{r}^{j_{r}}\right]=24\cdot p_{1}^{j_{1}}\cdots
p_{r}^{j_{r}}$

where $p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime factorization of
$K^{2}+4$. ∎

######  Theorem 3.3.

If $K\equiv 2\pmod{4}$ and $m=2^{j_{1}}$ or $m=2^{j_{1}+1}p_{2}^{j_{2}}\cdots
p_{r}^{j_{r}}$ for $j_{i}=0,1,2,\ldots$, where $p_{1}^{a_{1}}\cdots
p_{r}^{a_{r}}$ is the prime factorization of $K^{2}+4$ and $p_{1}=2$, then
$\pi_{K}(m)=m$.

###### Proof.

Suppose $K\equiv 2\pmod{4}$ and let $m=2^{j_{1}}$. By Theorem 2.4, neither $2$
nor any $p_{i}$ is a $K$-Wall-Sun-Sun prime. By Theorems 2.2 and 2.7,
$\pi_{K}(2^{j_{1}})=2^{j_{1}-1}\cdot\pi_{K}(2)=2^{j_{1}-1}\cdot 2=2^{j_{1}}$.

Suppose $m=2^{j_{1}+1}\cdot p_{2}^{j_{2}}\cdots p_{r}^{j_{r}}$ for
$j_{i}=0,1,2,\ldots$, where $p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime
factorization of $K^{2}+4$. By Theorem 2.1,

$\displaystyle\pi_{K}(m)=\operatorname{lcm}\left[\pi_{K}(2^{j_{1}+1}),\pi_{K}(p_{2}^{j_{2}}),\ldots,\pi_{K}(p_{r}^{j_{r}})\right]$
(4)

By Theorem 2.4, neither $2$ nor any $p_{i}$ is a $K$-Wall-Sun-Sun prime. By
Theorem 2.2, $\pi_{K}(2^{j_{1}}+1)=2^{j_{1}}\cdot\pi_{K}(2)$ and
$\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})$ for $i=2,3,\ldots r$.
By Theorem 2.7, $\pi_{K}(2)=2$, implying that
$2^{j_{1}}\cdot\pi_{K}(2)=2^{j_{1}}\cdot 2=2^{j_{1}+1}$. By Theorem 2.5,
$\pi_{K}(p_{i})=p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)$ and by Theorem 2.6,
$\text{ord}_{p_{i}}(2^{-1}K)=4$ since $p_{i}$ is odd and divides $K^{2}+4$. It
follows that

$\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})=p_{i}^{j_{i}-1}\cdot
p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)=4p_{i}^{j_{i}}$

These may be substituted into equation (4) to obtain:

$\pi(m)=\operatorname{lcm}\left[2^{j_{1}+1},4p_{2}^{j_{2}},\ldots,4p_{r}^{j_{r}}\right]=2^{j_{1}+1}\cdot
p_{2}^{j_{2}}\cdots p_{r}^{j_{r}}=m.$

∎

######  Theorem 3.4.

If $K\equiv 2\pmod{4}$ and $\pi_{K}(m)=m$, then $m=2^{j_{1}+1}$ or
$m=2^{j_{1}+2}p_{2}^{j_{2}}\cdots p_{r}^{j_{r}}$ for $j_{i}=0,1,2,\ldots$,
where $p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime factorization of
$K^{2}+4$ and $p_{1}=2$.

###### Proof.

Suppose $K\equiv 2\pmod{4}$ and $\pi_{K}(m)=m$. Let $m=2^{j_{1}}\cdot
p_{2}^{j_{2}}\cdots p_{r}^{j_{r}}\cdot q_{1}^{\theta_{1}}\cdots
q_{s}^{\theta_{s}}$ where $p_{1}^{j_{1}}\cdots p_{r}^{j_{r}}$ is the prime
factorization of $K^{2}+4$ and $p_{1}=2$. Note that if $\theta_{\lambda}\neq
0$ then $q_{\lambda}\mid m$. By Theorem 2.1,

$\displaystyle
m=\pi(m)=\operatorname{lcm}\left[\pi_{K}(2^{j_{1}}),\pi_{K}(p_{2}^{j_{2}}),\ldots,\pi_{K}(p_{r}^{j_{r}}),\pi_{K}(q_{1}^{\theta_{1}}),\ldots,\pi_{K}(q_{s}^{\theta_{s}})\right]$
(5)

By Theorem 2.10, if $\rho\mid\pi_{K}(q_{\lambda}^{\theta_{\lambda}})$, then
$\rho<q_{\lambda}$ for all $1\leq i\leq s$. By Theorem 2.2,
$\pi_{K}(q_{\lambda}^{\theta_{\lambda}})=q_{\lambda}^{\theta_{\lambda}-1}\pi_{K}(q_{\lambda})$
and by 2.5, $\pi_{K}(q_{\lambda})\mid(q_{\lambda}-1)$ if $K^{2}+4$ is a
quadratic residue and $\pi_{K}(q_{\lambda})\mid 2(q_{\lambda}+1)$ if $K^{2}+4$
is not a quadratic residue modulo $q_{\lambda}$. Either way, it follows that
$q_{\lambda}^{\theta_{\lambda}-1}$ is the largest factor of $q_{\lambda}$ that
divides $\pi_{K}(q_{\lambda}^{\theta_{\lambda}})$, that is,
$q_{\lambda}^{\theta_{\lambda}-1}\mid\mid\pi_{K}(q_{\lambda}^{\theta_{\lambda}})$.

By Theorem 2.4, neither $2$, $p_{i}$, nor $q_{\lambda}$ are $K$-Wall-Sun-Sun
primes. By Theorems 2.2 and 2.7,
$\pi_{K}(2^{j_{1}})=2^{j_{1}-1}\pi_{K}(2)=2^{j_{1}-1}\cdot 2=2^{j_{1}}$. By
Theorem 2.2, $\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})$ for
$i=1,2,\ldots r$. By Theorem 2.5,
$\pi_{K}(p_{i})=p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)$ and by Theorem 2.6,
$\text{ord}_{p_{i}}(2^{-1}K)=4$ since $p_{i}$ is odd and divides $K^{2}+4$. It
follows that

$\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})=p_{i}^{j_{i}-1}\cdot
p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)=4p_{i}^{j_{i}}$

Note that $q_{\lambda}\not\mid\pi_{K}(2^{j_{1}})=2^{j_{1}}$ and
$q_{\lambda}\not\mid\pi_{K}(3^{j_{2}})=3^{j_{2}-1}\cdot 8$, and
$q_{\lambda}\not\mid\pi_{K}(p_{i}^{j_{i}})=4p_{i}^{j_{i}}$. Thus,
$q_{\lambda}^{\theta_{\lambda}-1}\mid\mid m$. But this implies that
$q_{\lambda}^{\theta_{\lambda}}\not\mid m$, which is a contradiction. Equation
(5) is reduced to:

$m=\pi_{K}(m)=\operatorname{lcm}\left[2^{j_{1}},4p_{3}^{j_{3}},\ldots,4p_{r}^{j_{r}}\right]=2^{j_{1}}\
\text{or}\ 2^{j_{1}+1}p_{2}^{j_{2}}\cdots p_{r}^{j_{r}}$

where $p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime factorization of
$K^{2}+4$ and $p_{1}=2$. ∎

######  Theorem 3.5.

If $K\equiv 3\pmod{6}$ and $m=6$ or $m=12\cdot p_{3}^{j_{3}}\cdots
p_{r}^{j_{r}}$ for $j_{i}=0,1,2,\ldots$, where $p_{1}^{a_{1}}\cdots
p_{r}^{a_{r}}$ is the prime factorization of $K^{2}+4$ and
$p_{1}^{a_{1}}=2^{2}$ and $p_{2}^{a_{2}}=3$, then $\pi_{K}(m)=m$.

###### Proof.

Suppose $K\equiv 3\pmod{6}$ and $m=6$. Then by Theorem 2.1,
$\pi_{K}(m)=\operatorname{lcm}\left[\pi_{K}(2),\pi_{K}(3)\right]$. By Theorem
2.8, $\pi_{K}(2)=3$. By Theorem 2.9, $\pi_{K}(3)=2$. Hence,
$\pi_{K}(6)=\operatorname{lcm}\left[3,2\right]=6$. Note, this also establishes
the $2$-periodic point $\pi_{K}(2)=3\rightarrow\pi_{K}(3)=2\circlearrowleft$.

Suppose $m=12\cdot p_{3}^{j_{3}}\cdots p_{r}^{j_{r}}$ for
$j_{i}=0,1,2,\ldots$, where $p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime
factorization of $K^{2}+4$ and $p_{1}^{a_{1}}=2^{2}$ and $p_{2}^{a_{2}}=3$. By
Theorem 2.4, neither $2$, $3$, nor any $p_{i}$ is a $K$-Wall-Sun-Sun prime. By
Theorem 2.1,

$\displaystyle\pi_{K}(m)=\operatorname{lcm}\left[\pi_{K}(2^{2}),\pi_{K}(3),\pi_{K}(p_{3}^{j_{3}}),\ldots,\pi_{K}(p_{r}^{j_{r})}\right]$
(6)

By Theorems 2.2 and 2.8, $\pi_{K}(2^{2})=2^{2-1}\pi_{K}(2)=2\cdot 3=6$. By
Theorem 2.9, $\pi_{K}(3)=2$. By Theorem 2.2,
$\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})$ for $i=1,2,\ldots r$.
By Theorem 2.5, $\pi_{K}(p_{i})=p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)$ and by
Theorem 2.6, $\text{ord}_{p_{i}}(2^{-1}K)=4$ since $p_{i}$ is odd and divides
$K^{2}+4$. It follows that

$\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})=p_{i}^{j_{i}-1}\cdot
p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)=4p_{i}^{j_{i}}$

These may be substituted into equation (6) to obtain:

$\pi_{K}(m)=\operatorname{lcm}\left[6,2,4p_{3},\ldots,4p_{r}\right]=12\cdot
p_{3}^{j_{3}}\cdots p_{r}^{j_{r}}=m.$

∎

######  Theorem 3.6.

If $K\equiv 3\pmod{6}$ and $\pi_{K}(m)=m$, then $m=6$ or $m=12\cdot
p_{3}^{j_{3}}\cdots p_{r}^{j_{r}}$ for $j_{i}=0,1,2,\ldots$, where
$p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime factorization of $K^{2}+4$
and $p_{1}^{a_{1}}=2^{2}$ and $p_{2}^{a_{2}}=3$.

###### Proof.

Suppose $K\equiv 3\pmod{6}$ and $\pi_{K}(m)=m$. Let $m=2^{j_{1}}\cdot
3^{j_{2}}\cdot p_{3}^{j_{3}}\cdots p_{r}^{j_{r}}\cdot q_{1}^{\theta_{1}}\cdots
q_{s}^{\theta_{s}}$ where $2^{2}\cdot 3\cdot p_{3}^{j_{3}}\cdots
p_{r}^{j_{r}}$ is the prime factorization of $K^{2}+4$. Note that if
$\theta_{\lambda}\neq 0$ then $q_{\lambda}\mid m$. By Theorem 2.1,

$\displaystyle
m=\pi(m)=\operatorname{lcm}\left[\pi_{K}(2^{j_{1}})\,\pi_{K}(3^{j_{2}})\,\pi_{K}(p_{3}^{j_{3}}),\ldots,\pi_{K}(p_{r}^{j_{r}}),\pi_{K}(q_{1}^{\theta_{1}}),\ldots,\pi_{K}(q_{s}^{\theta_{s}})\right]$
(7)

By Theorem 2.10, if $\rho\mid\pi_{K}(q_{\lambda}^{\theta_{\lambda}})$, then
$\rho<q_{\lambda}$ for all $1\leq i\leq s$. By Theorem 2.2,
$\pi_{K}(q_{\lambda}^{\theta_{\lambda}})=q_{\lambda}^{\theta_{\lambda}-1}\pi_{K}(q_{\lambda})$
and by 2.5, $\pi_{K}(q_{\lambda})\mid(q_{\lambda}-1)$ if $K^{2}+4$ is a
quadratic residue and $\pi_{K}(q_{\lambda})\mid 2(q_{\lambda}+1)$ if $K^{2}+4$
is not a quadratic residue modulo $q_{\lambda}$. Either way, it follows that
$q_{\lambda}^{\theta_{\lambda}-1}$ is the largest factor of $q_{\lambda}$ that
divides $\pi_{K}(q_{\lambda}^{\theta_{\lambda}})$, that is,
$q_{\lambda}^{\theta_{\lambda}-1}\mid\mid\pi_{K}(q_{\lambda}^{\theta_{\lambda}})$.

By Theorem 2.4, neither $2$, $3$, $p_{i}$, nor $q_{\lambda}$ are $K$-Wall-Sun-
Sun primes. By Theorems 2.2 and 2.8,
$\pi_{K}(2^{j_{1}})=2^{j_{1}-1}\pi_{K}(2)=2^{j_{1}-1}\cdot 3$. By Theorems 2.2
and 2.9, $\pi_{K}(3^{j_{2}})=3^{j_{2}-1}\pi_{K}(3)=3^{j_{2}-1}\cdot 2$. By
Theorem 2.2, $\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})$ for
$i=1,2,\ldots r$. By Theorem 2.5,
$\pi_{K}(p_{i})=p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)$ and by Theorem 2.6,
$\text{ord}_{p_{i}}(2^{-1}K)=4$ since $p_{i}$ is odd and divides $K^{2}+4$. It
follows that

$\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})=p_{i}^{j_{i}-1}\cdot
p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)=4p_{i}^{j_{i}}$

Note that $q_{\lambda}\not\mid\pi_{K}(2^{j_{1}})=2^{j_{1}-1}\cdot 3$ and
$q_{\lambda}\not\mid\pi_{K}(3^{j_{2}})=3^{j_{2}-1}\cdot 2$, and
$q_{\lambda}\not\mid\pi_{K}(p_{i}^{j_{i}})=4p_{i}^{j_{i}}$. Thus,
$q_{\lambda}^{\theta_{\lambda}-1}\mid\mid m$. But this implies that
$q_{\lambda}^{\theta_{\lambda}}\not\mid m$, which is a contradiction. Equation
(7) is reduced to:

$\displaystyle m=\pi_{K}(m)=\operatorname{lcm}\left[2^{j_{1}-1}\cdot
3,3^{j_{2}-1}\cdot 2,4p_{3}^{j_{3}},\ldots,4p_{r}^{j_{r}}\right]$ (8)

Recall that by hypothesis, $m=2^{j_{1}}\cdot 3^{j_{2}}\cdot
p_{3}^{j_{3}}\cdots p_{r}^{j_{r}}\cdot q_{1}^{\theta_{1}}\cdots
q_{s}^{\theta_{s}}$. Suppose $j_{1}\geq 3$, then $2^{j_{1}-1}$ is the largest
power of $2$ in (8), implying $2^{j_{1}-1}\mid\mid m$. However, this
contradicts the hypothesis $2^{j_{1}}\mid m$. Thus, $j_{1}=1$ or $2$. Suppose
$j_{2}\geq 2$, then $3^{j_{2}-1}$ is the largest power of $3$ in (8), implying
$3^{j_{2}-1}\mid\mid m$. However, this contradicts the hypothesis
$3^{j_{2}}\mid m$. Thus, $j_{2}=1$, and equation (8) is reduced to:

$m=\pi_{k}(m)=\operatorname{lcm}\left[(6\ \text{or}\
12),6,4p_{3}^{j_{3}},\ldots,4p_{r}^{j_{r}}\right]=12\cdot p_{3}^{j_{3}}\cdots
p_{r}^{j_{r}}\ \text{or}\ 6$

where $p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime factorization of
$K^{2}+4$ and $p_{1}^{a_{1}}=2^{2}$ and $p_{2}^{a_{2}}=3$. ∎

######  Theorem 3.7.

If $K\equiv 0\pmod{4}$ and $m=2$ or $m=4\cdot p_{2}^{j_{2}}\cdots
p_{r}^{j_{r}}$ for $j_{i}=0,1,2,\ldots$, where $p_{1}^{a_{1}}\cdots
p_{r}^{a_{r}}$ is the prime factorization of $K^{2}+4$ and
$p_{1}^{a_{1}}=2^{2}$, then $\pi_{K}(m)=m$.

###### Proof.

Suppose $K\equiv 0\pmod{4}$ and $m=2$. By Theorem 2.7, $\pi_{K}(2)=2$. Suppose
$m=4\cdot p_{2}^{j_{2}}\cdots p_{r}^{j_{r}}$ for $j_{i}=0,1,2,\ldots$, where
$p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime factorization of $K^{2}+4$
and $p_{1}^{j_{1}}=2^{2}$. By Theorem 2.1,

$\displaystyle\pi_{K}(m)=\operatorname{lcm}\left[\pi_{K}(2^{2}),\pi_{K}(p_{2}^{j_{2}}),\ldots,\pi_{K}(p_{r}^{j_{r}})\right].$
(9)

By Theorem 2.4, $2$ is a $K$-Wall-Sun-Sun prime, so
$\pi_{K}(2^{2})=\pi_{K}(2)=2$. Also by Theorem 2.4, no other $p_{i}$ is a
$K$-Wall-Sun-Sun prime. By Theorem 2.2,
$\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})$ for $i=1,2,\ldots r$.
By Theorem 2.5, $\pi_{K}(p_{i})=p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)$ and by
Theorem 2.6, $\text{ord}_{p_{i}}(2^{-1}K)=4$ since $p_{i}$ is odd and divides
$K^{2}+4$. It follows that

$\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})=p_{i}^{j_{i}-1}\cdot
p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)=4p_{i}^{j_{i}}$

These may be substituted into equation (9) to obtain:

$\pi_{K}(m)=\operatorname{lcm}\left[2,4p_{2},\ldots,4p_{r}\right]=4\cdot
p_{2}^{j_{2}}\cdots p_{r}^{j_{r}}=m.$

∎

######  Theorem 3.8.

If $\pi_{K}(m)=m$ and $K\equiv 0\pmod{4}$, then $m=2$ or $m=4\cdot
p_{2}^{j_{2}}\cdots p_{r}^{j_{r}}$ for $j_{i}=0,1,2,\ldots$, where
$p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime factorization of $K^{2}+4$
and $p_{1}^{a_{1}}=2^{2}$.

###### Proof.

Suppose $K\equiv 0\pmod{4}$ and $\pi_{K}(m)=m$. Let $m=2^{j_{1}}\cdot
3^{j_{2}}\cdot p_{1}^{j_{1}}\cdots p_{r}^{j_{r}}\cdot q_{1}^{\theta_{1}}\cdots
q_{s}^{\theta_{s}}$ where $p_{1}^{j_{1}}\cdots p_{r}^{j_{r}}$ is the prime
factorization of $K^{2}+4$. Note that if $\theta_{\lambda}\neq 0$ then
$q_{\lambda}\mid m$. By Theorem 2.1,

$\displaystyle
m=\pi(m)=\operatorname{lcm}\left[\pi_{K}(2^{j_{1}})\,\pi_{K}(3^{j_{2}})\,\pi_{K}(p_{1}^{j_{1}}),\ldots,\pi_{K}(p_{r}^{j_{r}}),\pi_{K}(q_{1}^{\theta_{1}}),\ldots,\pi_{K}(q_{s}^{\theta_{s}})\right]$
(10)

By Theorem 2.10, if $\rho\mid\pi_{K}(q_{\lambda}^{\theta_{\lambda}})$, then
$\rho<q_{\lambda}$ for all $1\leq i\leq s$. By Theorem 2.2,
$\pi_{K}(q_{\lambda}^{\theta_{\lambda}})=q_{\lambda}^{\theta_{\lambda}-1}\pi_{K}(q_{\lambda})$
and by Theorem 2.5, $\pi_{K}(q_{\lambda})\mid(q_{\lambda}-1)$ if $K^{2}+4$ is
a quadratic residue, or $\pi_{K}(q_{\lambda})\mid 2(q_{\lambda}+1)$ if
$K^{2}+4$ is not a quadratic residue modulo $q_{\lambda}$. Either way, it
follows that $q_{\lambda}^{\theta_{\lambda}-1}$ is the largest factor of
$q_{\lambda}$ that divides $\pi_{K}(q_{\lambda}^{\theta_{\lambda}})$, that is,
$q_{\lambda}^{\theta_{\lambda}-1}\mid\mid\pi_{K}(q_{\lambda}^{\theta_{\lambda}})$.

By Theorem 2.4, $2$ is a $K$-Wall-Sun-Sun prime where
$\pi_{K}(2)=\pi_{K}(2^{2})=2$ but neither $3$, $p_{i}$, nor $q_{\lambda}$ are
$K$-Wall-Sun-Sun primes. By Theorems 2.2 and 2.7, for $j_{1}\geq
2,\pi_{K}(2^{j_{1}})=2^{j_{1}-2}\pi_{K}(2)=2^{j_{1}-2}\cdot 2=2^{j_{1}-1}$. If
$j_{1}=1,\pi_{K}(2)=2$, thus forming a fixed point. By Theorems 2.2 and 2.9,
$\pi_{K}(3^{j_{2}})=3^{j_{2}-1}\pi_{K}(3)=3^{j_{2}-1}\cdot 2$ if $3\mid K$ and
$\pi_{K}(3^{j_{2}})=3^{j_{2}-1}\pi_{K}(3)=3^{j_{2}-1}\cdot 8$ if $3\not\mid
K$. By Theorem 2.2, $\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})$ for
$i=1,2,\ldots r$. By Theorem 2.5,
$\pi_{K}(p_{i})=p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)$ and by Theorem 2.6,
$\text{ord}_{p_{i}}(2^{-1}K)=4$ since $p_{i}$ is odd and divides $K^{2}+4$. It
follows that

$\pi_{K}(p_{i}^{j_{i}})=p_{i}^{j_{i}-1}\pi_{K}(p_{i})=p_{i}^{j_{i}-1}\cdot
p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)=4p_{i}^{j_{i}}$

Note that $q_{\lambda}\not\mid\pi_{K}(2^{j_{1}-1})=2^{j_{1}}$ and
$q_{\lambda}\not\mid\pi_{K}(3^{j_{2}})=3^{j_{2}-1}\cdot 2\ \text{or}\ 8$, and
$q_{\lambda}\not\mid\pi_{K}(p_{i}^{j_{i}})=4p_{i}^{j_{i}}$. Thus,
$q_{\lambda}^{\theta_{\lambda}-1}\mid\mid m$. But this implies that
$q_{\lambda}^{\theta_{\lambda}}\not\mid m$, which is a contradiction. Equation
(10) is reduced to:

$\displaystyle
m=\pi_{K}(m)=\operatorname{lcm}\left[2^{j_{1}-1},3^{j_{2}-1}\cdot(2\
\text{or}\ 8),4p_{2}^{j_{1}},\ldots,4p_{r}^{j_{r}}\right]$ (11)

Recall that by hypothesis, if $4\mid m$, then $m=2^{j_{1}-1}\cdot
p_{2}^{j_{2}}\cdots p_{r}^{j_{r}}\cdot q_{1}^{\theta_{1}}\cdots
q_{s}^{\theta_{s}}$. Suppose $j_{1}\geq 3$, then $2^{j_{1}-1}$ is the largest
power of $2$ in (8), implying $2^{j_{1}-1}\mid\mid m$. However, this
contradicts the hypothesis $2^{j_{1}}\mid m$. Thus, equation (11) is reduced
to:

$m=\pi_{k}(m)=\operatorname{lcm}\left[2^{j_{1}-1},4p_{2}^{j_{2}},\ldots,4p_{r}^{j_{r}}\right]=4\cdot
p_{2}^{j_{2}}\cdots p_{r}^{j_{r}}$

where $p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime factorization of
$K^{2}+4$ and $p_{1}^{a_{1}}=2^{2}$. ∎

##  4\. Proof of the $K$-Iteration Theorem

######  Theorem 4.1.

(Cerda-Morales [3]). Let $U(K)=\left(\begin{matrix}K&1\\\
1&0\end{matrix}\right)$, then for any integer $n\geq 1$,

$U^{n}(K)=\left(\begin{matrix}F_{K,n+1}&F_{K,n}\\\
F_{K,n}&F_{K,n-1}\end{matrix}\right).$

######  Theorem 4.2.

For all $m>2$ and $K\geq 1$, $\pi_{K}(m)$ is even.

###### Proof.

Define $U(K)$ as in Theorem 4.1, and note that $\det U(K)=-1$. Let $m>2$ and
let $N=\pi_{K}(m)+1$. Then, $F_{K,N+1}=F_{K,2}\equiv K\pmod{m}$,
$F_{K,N}=F_{K,1}\equiv 1\pmod{m}$, and $F_{K,N-1}=F_{K,0}\equiv 0\pmod{m}$.
The determinant of $U^{N}(K)$ is
$F_{K,N+1}F_{K,N-1}-F_{K,N}^{2}\equiv-1\pmod{m}$. Hence, $\det
U^{N}(K)=\left(\det U(K)\right)^{N}=(-1)^{N}$. If $m>2,-1\not\equiv
1\pmod{m}$, and it follows that $N$ is odd, so $\pi_{K}(m)=N-1$ is even. ∎

######  Theorem 4.3.

If $K$ is odd, then $2$ is not a $K$-Wall-Sun-Sun prime.

###### Proof.

Note that by Theorem 2.8 that $\pi_{K}(2)=3$ for odd $K$. It remains to show
that $\pi_{K}(4)\neq 3$ for odd $K$. If $K\equiv 1\pmod{4}$, then the
$K$-Fibonacci sequence (modulo 4) becomes 0, 1, 1, 2, 3, 1, 0, 1,
$\circlearrowleft$, with period 6. If $K\equiv 3\pmod{4}$, then the
$K$-Fibonacci sequence (modulo 4) becomes 0, 1, 3, 2, 1, 1, 0, 1,
$\circlearrowleft$, again with period 6. Hence, $\pi_{K}(2)\neq\pi_{K}(4)$ for
all odd $K$. ∎

######  Theorem 4.4.

If $p\mid(K^{2}+4)$, then for all $N$, $p\mid\pi_{K}^{N}(p^{j})$.

###### Proof.

Suppose $p\mid(K^{2}+4)$. By Theorem 2.4, $p$ is a $K$-Wall-Sun-Sun prime if
and only if $p=2$ and $K\equiv 0\pmod{4}$. In this case, by Theorem 2.7
$\pi_{K}(2)=\pi_{K}(4)=2$. Hence, $2\mid\pi_{K}^{N}(p)$ for all $N$. If $p=2$
and $K\equiv 2\pmod{4}$, then $\pi_{K}^{N}(p^{j})=2^{j}$ for all $j$, because
this is precisely one family of $K$-fixed points by Theorem 3.4. On the other
hand, suppose that $p$ is an odd prime. By Theorems 2.2 and 2.5,
$\pi_{K}(p^{j})=p^{j-1}\pi_{K}(p_{i})=4p$. Recursively applying the Pisano
period yields
$\pi_{K}^{2}(p)=\pi_{K}\left(4p\right)=\operatorname{lcm}\left[\pi_{K}(2^{2}),\pi_{K}(p)\right]=\operatorname{lcm}\left[\pi_{K}(2^{2}),4p\right]$.
Iterating the Pisano period always preserves a factor of $p$ by induction.
Thus, for all $N$, $p\mid\pi_{K}^{N}(p)$. ∎

######  Theorem 4.5.

If $q\not\mid(K^{2}+4)$, then there exists a positive integer $N$ such that
for all $n\geq N$, $q\not\mid\pi_{K}^{n}(q^{\theta})$ for any $\theta\geq 1$.

###### Proof.

Suppose $q\not\mid(K^{2}+4)$. By Theorem 2.2, there is a maximal $e$ such that
$\pi_{K}(q^{e})=\pi_{K}(q)$ and for $x\geq e\geq 1$,
$\pi_{K}(q^{x})=q^{x-e}\pi_{K}(q)$. Applying the Pisano period again yields

$\displaystyle\pi_{K}^{2}(q^{\theta})$
$\displaystyle=\pi_{K}\left(q^{x-e}\pi_{K}(q)\right)=\operatorname{lcm}\left[\pi_{K}(q^{x-e}),\pi_{K}^{2}(q)\right]=\operatorname{lcm}\left[q^{x-e-1}\pi_{k}(q),\pi_{K}^{2}(q)\right].$

By Theorem 2.5, since $q_{i}\not\mid(K^{2}+4)$, it follows that
$q^{\theta}\not\mid\pi_{K}(q)$ for any positive integer $\theta$. As the
Pisano period is recursively applied, the highest power of $q$ that carries to
the next iteration is strictly decreasing. Applying the Pisano period
precisely $x-e$ times ensures that the highest power of $q$ that emerges is
$q^{x-e-(x-e)}$. Hence, there exists a positive integer $N$ such that for all
$n\geq N$, $q\not\mid\pi_{K}^{n}(q^{\theta})$. ∎

######  Theorem 4.6.

If $K\equiv\pm 1\pmod{6}$ and $m>1$, then there exists an $N$ such that for
all $n\geq N$, $\pi_{K}^{n}(m)=24\cdot p_{1}^{j_{1}}\cdots p_{r}^{j_{r}}$ for
$j_{i}=0,1,2,\ldots$, where $p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime
factorization of $K^{2}+4$.

###### Proof.

Suppose $K\equiv\pm 1\pmod{6}$ and $m=2^{s_{1}}3^{s_{2}}p_{1}^{j_{1}}\cdots
p_{r}^{j_{r}}q_{1}^{\lambda_{t}}\cdots q_{r}^{\lambda_{t}}$ for
$s_{i},j_{i}\geq 0$ and $\lambda_{i}\geq 1$, where $p_{1}^{a_{1}}\cdots
p_{r}^{a_{r}}$ is the prime factorization of $K^{2}+4$. By Theorem 2.1,

$\displaystyle\pi_{K}(m)=\operatorname{lcm}\left[\pi_{K}(2^{j_{1}}),\pi_{K}(3^{j_{2}}),\pi_{K}(p_{1}^{j_{1}}),\ldots,\pi_{K}(p_{r}^{j_{r}}),\pi_{K}(q_{1}^{\lambda_{1}}),\ldots,\pi_{K}(q_{s}^{\lambda_{2}})\right]$
(12)

By Theorem 4.2, $2\mid\pi_{K}(m)$. By Theorem 2.8, $\pi_{K}(2)=3$ implying
that $3\mid\pi_{K}^{2}(m)$. By Theorem 2.9, $\pi_{K}(3)=8$, implying that
$2^{3}\mid\pi_{K}^{3}(m)$. By Theorems 2.2 and 2.7,
$\pi_{K}(2^{3})=2^{3-1}\pi_{K}(2)=4\cdot 3=12$, implying
$12\mid\pi_{K}^{4}(m)$. By Theorems 2.1, 2.7, and 2.9,
$\pi_{K}(12)=\operatorname{lcm}\left[\pi_{K}(2^{2}),\pi_{K}(3)\right]=\operatorname{lcm}\left[6,8\right]=24$,
implying $24\mid\pi_{K}^{5}(m)$. By Theorem 4.4, if any $j_{i}\geq 1$, then
the corresponding $p_{i}$ is preserved for all $N$, and $p_{i}\mid\pi_{K}(m)$.
By Theorem 4.5, for any $q_{i}\not\mid(K^{2}+4)$, $q_{i}$ vanishes for some
sufficiently large $N$, and $q_{i}\not\mid\pi_{K}(m)$. Hence, for some $N$,
$\pi_{K}^{n}(m)=24\cdot p_{1}^{j_{1}}\cdots p_{r}^{j_{r}}$ where
$p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime factorization of $K^{2}+4$. ∎

######  Theorem 4.7.

If $K\equiv 3\pmod{6}$ and $m>3$, then there exists an $N$ such that for all
$n\geq N$, $\pi_{K}^{n}(m)=p_{1}^{j_{1}}\cdots p_{r}^{j_{r}}$ for
$j_{i}=0,1,2,\ldots$, where $p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime
factorization of $K^{2}+4$. And if $m=2$ or $m=3$, then $\pi_{K}(2)=3$ and
$\pi_{K}(3)=2$.

###### Proof.

Suppose $K\equiv 3\pmod{6}$ and $m=p_{1}^{j_{1}}\cdots
p_{r}^{j_{r}}q_{1}^{\lambda_{t}}\cdots q_{r}^{\lambda_{t}}>3$ for $j_{i}\geq
0$ and $\lambda_{i}\geq 1$, where $p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the
prime factorization of $K^{2}+4$. By Theorem 2.1,

$\displaystyle\pi_{K}(m)=\operatorname{lcm}\left[\pi_{K}(p_{1}^{j_{1}}),\ldots,\pi_{K}(p_{r}^{j_{r}}),\pi_{K}(q_{1}^{\lambda_{1}}),\ldots,\pi_{K}(q_{s}^{\lambda_{2}})\right]$
(13)

By Theorem 4.2, $2\mid\pi_{K}(m)$. If $\pi_{K}(m)=2$, then the $K$-Fibonacci
sequence must reduce to $0,1,K\equiv 0\pmod{m},\circlearrowleft$, so $m\mid
K$. In this case, the iterative Pisano period cycles at the 2-periodic point
$\pi(2)=3,\pi(3)=2,\circlearrowleft$. Otherwise, either
$\pi_{K}(m)=2^{\alpha}\cdot\beta$, where $\alpha\geq 1$ and $\beta$ is an odd
number greater than 1, or $\pi_{K}(m)=2^{\gamma}$, where $\gamma\geq 2$. By
Theorem 2.8, $\pi_{K}(2)=3$. If $\pi_{K}(m)=2^{\alpha}\cdot\beta$, then
$\pi_{K}(m)=\operatorname{lcm}\left[\pi_{K}(2^{\alpha}),\pi_{K}(\beta)\right]$.
Since $3\mid\pi_{K}(2^{\alpha})$ and $2\mid\pi_{K}(\beta)$,
$6\mid\pi_{K}^{2}(m)$. Else, if $\pi_{K}(m)=2^{\gamma}$, and since 2 is not a
$K$-Wall-Sun-Sun prime by Theorem 4.3, then $\pi_{K}^{2}(m)=3\cdot
2^{\gamma-1}$, so $6\mid\pi_{K}^{2}(m)$. By Theorem 4.4, if any $j_{i}\geq 1$,
then the corresponding $p_{i}$ is preserved for all $N$, and
$p_{i}\mid\pi_{K}(m)$. By Theorem 2.5,
$\pi_{K}(p_{i})=p_{i}\cdot\text{ord}_{p_{i}}(2^{-1}K)$ and by Theorem 2.6,
$\text{ord}_{p_{i}}(2^{-1}K)=4$ since $p_{i}$ is odd and divides $K^{2}+4$. By
Theorem 4.5, for any $q_{i}\not\mid(K^{2}+4)$, $q_{i}$ vanishes for some
sufficiently large $N$, and $q_{i}\not\mid\pi_{K}(m)$. Hence, for some $N$,
$\pi_{K}^{n}(m)=\operatorname{lcm}\left[6,4p_{1}^{j_{1}},\ldots,4p_{r}^{j_{r}}\right]$.
If there is at least one such $p_{i}$, then $\pi_{K}^{n}(m)=12\cdot
p_{1}^{j_{1}}\cdots p_{r}^{j_{r}}$ where $p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$
is the prime factorization of $K^{2}+4$; otherwise, $\pi_{K}^{n}(m)=6$.
Finally, since $K\equiv 3\pmod{6}$, there is a 2-periodic point:
$\pi_{K}(2)=3\rightarrow\pi_{K}(3)=2\circlearrowleft$. ∎

######  Theorem 4.8.

If $K\equiv 2\ \text{or}\ 0\pmod{4}$ and $m>1$, then there exists an $N$ such
that for all $n\geq N$, $\pi_{K}^{n}(m)=p_{1}^{j_{1}}\cdots p_{r}^{j_{r}}$ for
$j_{i}=0,1,2,\ldots$, where $p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime
factorization of $K^{2}+4$.

###### Proof.

Suppose $K\equiv 2\ \text{or}\ 0\pmod{4}$ and suppose $m=p_{1}^{j_{1}}\cdots
p_{r}^{j_{r}}q_{1}^{\lambda_{t}}\cdots q_{t}^{\lambda_{t}}>1$ for $j_{i}\geq
0$ and $\lambda_{i}\geq 1$ (and $m>3$ if $K\equiv 3\pmod{6}$), where
$p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime factorization of $K^{2}+4$.
By Theorem 2.1,

$\displaystyle\pi_{K}(m)=\operatorname{lcm}\left[\pi_{K}(p_{1}^{j_{1}}),\ldots,\pi_{K}(p_{r}^{j_{r}}),\pi_{K}(q_{1}^{\lambda_{1}}),\ldots,\pi_{K}(q_{s}^{\lambda_{2}})\right]$
(14)

By Theorem 4.4, if any $j_{i}\geq 1$, then the corresponding $p_{i}$ is
preserved for all $N$, and $p_{i}\mid\pi_{K}^{N}(m)$. By Theorem 4.5, for any
$q_{i}\not\mid(K^{2}+4)$, $q_{i}$ vanishes for some sufficiently large $N$,
and $q_{i}\not\mid\pi_{K}^{N}(m)$. Hence, there is a positive integer $N$ such
that for all $n\geq N$, $\pi_{K}^{n}(m)=p_{1}^{j_{1}}\cdots p_{r}^{j_{r}}$
where $p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ is the prime factorization of
$K^{2}+4$. ∎

##  5\. Final Thoughts

A general binary recurrence sequence $\mathcal{U}_{n}$ is determined by the
four parameters $(a,b,c,d)$ where

$\mathcal{U}_{0}=c,\qquad\mathcal{U}_{1}=d,\qquad\mathcal{U}_{n}=a\times\mathcal{U}_{n-1}+b\times\mathcal{U}_{n-2}.$

For example, the Fibonacci sequence is determined by $(1,1,0,1)$, the Lucas
sequence by $(1,1,2,1)$, the Pell sequence by $(2,1,0,1)$, the Jacobsthal
sequence by $(1,2,0,1)$, etc. In all of these cases, the binary recurrence
sequence is periodic modulo a positive integer $m>1$ [4]. Naturally, the
Pisano period has been considered for many binary recurrence sequences. Is it
possible to determine the fixed points for general binary recurrence
sequences?

Sequence | Pisano Periods for $m=2,3,\ldots,24$
---|---
Fibonacci | 3,8,6,20,24,16,12,24,60,10,24,28,48,40,24,36,24,18,60,16,30,48,24,…
Lucas | 3,8,6,4,24,16,12,24,12,10,24,28,48,8,24,36,24,18,12,16,30,48,24,…
Pell |  2,8,4,12,8,6,8,24,12,24,8,28,6,24,16,16,24,40,12,24,24,22,8…
Jacobsthal | 1,6,2,4,6,6,2,18,4,10,6,12,6,12,2,8,18,18,4,6,10,22,6,24…

It is well-established that the Pisano period of the Lucas sequence differs
from the Pisano period of the Fibonacci sequence only when $m$ is a multiple
of $5$ [12]. And note that the Pell sequence is simply the $K$-Fibonacci
sequence when $K=2$.

######  Corollary 5.1.

In the Lucas sequence, $m>1$ is a fixed point if and only if $m=24$.

######  Corollary 5.2.

In the Pell sequence, $m>1$ is a fixed point if and only if $m=2^{k}$ for
$k=1,2,\ldots$

######  Conjecture 5.3.

In the Jacobsthal sequence, $m>1$ is a fixed point if and only if $m=2\cdot
3^{k}$ for $k=1,2,\ldots$

######  Question 5.4.

Given positive integers $a,b,c,$ and $d$, what are the fixed points of general
binary recurrence sequences $\mathcal{U}_{n}$?

######  Question 5.5.

Does the trajectory of every element in a general binary recurrence sequence
terminate at a fixed point (or is a $k$-periodic point)? Are there any
unbounded trajectories?

## References

  * [1] Zakariae Bouazzaoui.  Fibonacci numbers and real quadratic p-rational fields.  Periodica Mathematica Hungarica, 81(1):123–133, 2020.
  * [2] Zakariae Bouazzaoui.  On periods of Fibonacci sequences and real quadratic p-rational fields.  Fibonacci Quarterly, 58(5):103–110, 2020.
  * [3] Gamaliel Cerda-Morales.  On generalized Fibonacci and Lucas numbers by matrix methods.  Hacettepe Journal of Mathematics, 42(2):173–179, 2013.
  * [4] G. Everest, A. van der Poorten, I. Sharplinski, and T. Ward.  Recurrence Sequences.  Rhode Island, USA: American Mathematical Society, 2004.
  * [5] Sergio Falcon and Ángel Plaza.  k-Fibonacci sequences modulo m.  Chaos, Solitons & Fractals, 41(1):497–504, 2009.
  * [6] John Fulton and William Morris.  On arithmetical functions related to the Fibonacci numbers.  Acta Arithmetica, 2(16):105–110, 1969.
  * [7] Joshua Harrington and Lenny Jones.  A note on generalised Wall–Sun–Sun primes.  Bulletin of the Australian Mathematical Society, 108(3):373–378, 2023.
  * [8] Lenny Jones.  Generalized Wall-Sun-Sun primes and monogenic power-compositional trinomials.  Albanian Journal of Mathematics, 17(2), 2023.
  * [9] Lenny Jones.  A new condition for $k$-Wall–Sun–Sun primes.  Taiwanese Journal of Mathematics, 28(1):17–28, 2024.
  * [10] J. L. Lagrange.  Oeuvres de LaGrange.  Gauthier Villars, Paris, 7:5–182, 1877.
  * [11] Marc Renault.  The period, rank, and order of the (a, b)-Fibonacci sequence mod m.  Mathematics Magazine, 86(5):372–380, 2013.
  * [12] Neil J. A. Sloane and The OEIS Foundation Inc.  The on-line encyclopedia of integer sequences A106291.  http://oeis.org/A106291, 2021.
  * [13] Zhi-Wei Sun.  Fibonacci numbers and Fermat’s last theorem.  Acta Arithmetica, 60(4):371–388, 1992.
  * [14] Eva Trojovská.  On periodic points of the order of appearance in the Fibonacci sequence.  Mathematics, 8(5):773, 2020.
  * [15] Donald Dines Wall.  Fibonacci series modulo m.  The American Mathematical Monthly, 67(6):525–532, 1960.


# An integrable bound for rough stochastic partial differential equations

M. Ghani Varzaneh  Mazyar Ghani Varzaneh
Fakultät für Mathematik und Informatik, FernUniversität in Hagen, Hagen,
Germany<EMAIL_ADDRESS>and  S. Riedel  Sebastian
Riedel
Fakultät für Mathematik und Informatik, FernUniversität in Hagen, Hagen,
Germany<EMAIL_ADDRESS>
###### Abstract.

We study semilinear rough stochastic partial differential equations as
introduced in [Gerasimovičs, Hairer; EJP 2019] and provide an a priori bound.
When replacing the driving signal by a suitable Gaussian process, this leads
to an integrable bound and implies $\mathcal{L}^{p}(\Omega)$-integrability of
the solution.

###### Key words and phrases:

Rough stochastic partial differential equations, Gaussian processes

###### 1991 Mathematics Subject Classification:

Primary 60L50; Secondary 60G15.

## Introduction

In [GH19], Gerasimovičs and Hairer introduced a new solution concept to study
parabolic semilinear stochastic partial differential equations (SPDEs) driven
by a finite-dimensional noise. One important property of this theory is that
it is completely _pathwise_ since no stochastic integration theory is used to
define the solution to the equation. Instead, the paper uses key ideas of
Lyons’ rough paths theory [Lyo98], meaning that the stochastic integral is
replaced by a pathwise defined _rough_ integral. A nice feature of this
concept of a _rough partial differential equation_ (RPDE) is that it is fully
compatible with classical rough path theory. In particular, one can easily
study RPDEs driven by Gaussian noises, e.g. fractional Brownian motions, as
introduced in [CQ02, FV10a, FGGR16]. Let us mention that this framework of
RPDEs was later generalized to study non-autonomous SPDEs, too [GHN21]. The
theory was particularly applied to study dynamical properties of solutions to
SPDEs using the concept of random dynamical systems [Arn98] where a pathwise
theory is crucial, cf. [KN23, YLZ23].

Avoiding stochastic integration when defining solutions to SPDEs has many
advantages, but also leads to new challenges since probabilistic properties of
the solution are often not so easy to obtain. This is particularly true when
the driving signal is not a Brownian motion, but a general Gaussian process.
One of these problems concerns the $\mathcal{L}^{p}(\Omega)$-integrability of
the solution to a random RPDE. For rough differential equations (RDEs), this
problem was a major obstacle when generalizing Hörmander theory to rough
differential equations driven by Gaussian processes where obtaining the
$\mathcal{L}^{p}(\Omega)$-integrability of the Jacobian of an RDE is a crucial
step [CF10]. The confusing problem was that the known a priori bounds for
solutions to deterministic RDEs, formulated in terms of the rough path norm,
were optimal [FV10b], but not integrable if the noise was replaced by a
Gaussian process. This problem was solved in the seminal paper [CLL13] where
the known a priori bounds were modified in such a way that probabilistic
properties of the Gaussian rough paths could be applied. Later, the results in
[CLL13] were slightly simplified and extended to study a larger class of rough
differential equations, too [FR13]. For RPDEs in the sense of [GH19], the
problem of finding an integrable a priori bound remained unsolved up to now.
In fact, it is stated in [GH19, page 51] that The [integrable] moment bounds
for the rough path norms of solution and Jacobian (…) [for the RPDE] might not
be easy to obtain in general and require a closer look as a separate problem
on its own. We decide to postpone the study of such moments but refer the
reader to [FR13] where this question was answered for the rough SDE case. In
the present paper, we provide exactly these bounds, cf. Theorem 1.12 which is
our main result. We believe that our conclusions will have many implications
and important consequences e.g. when extending non-Markovian Hörmander theory
to RPDEs as initiated in [GH19] or in the study of random invariant manifolds
as it was started in [KN23]. We will leave these important questions for
future work.

### Review on Rough stochastic partial differential equations

We are interested in the solution of the following form of rough SPDEs

(0.1)
$\displaystyle\mathrm{d}Z_{t}=AZ_{t}\,\mathrm{d}t+F(Z_{t})\,\mathrm{d}t+G(Z_{t})\circ\mathrm{d}\mathbf{X}_{t},\
\ \ \ Z_{0}=z_{0}\in\mathcal{B}_{\alpha}$

where $\mathbf{X}$ is a rough path. This family of SPDEs is studied in [GH19]
and [GHN21]. Let us first quickly review some basic definitions and notations
for setting up the solution for this family of equations. For more details,
the reader can refer to [GHN21]. The following definition is taken from
[GHN21, Definition 2.1].

######  Definition 0.1.

We call a family of indexed separable Banach spaces
$\\{(\mathcal{B}_{\beta},|\ |_{\beta})\\}_{\beta\in\mathbb{R}}$ a _monotone
family of interpolation spaces_ if

  * (i)

For every $\alpha\leq\beta$: $\mathcal{B}_{\beta}$ is a dense subset of
$\mathcal{B}_{\alpha}$ and the identity map
$id:\mathcal{B}_{\beta}\rightarrow\mathcal{B}_{\alpha}$ is continuous.

  * (ii)

For every $\alpha\leq\beta\leq\theta$ and
$x\in\mathcal{B}_{\alpha}\cap\mathcal{B}_{\theta}:\
|x|_{\beta}\lesssim|x|_{\alpha}^{\frac{\theta-\beta}{\theta-\alpha}}|x|_{\theta}^{\frac{\beta-\alpha}{\theta-\alpha}}$.

We also assume :

######  Assumption 0.2.

Let $\frac{1}{3}<\gamma\leq\frac{1}{2}$ and
$\mathbf{X}=(X,\mathbb{X})\in\mathscr{C}^{\gamma}((0,\infty),\mathbb{R}^{n})$
be $\gamma$-_rough path_ (cf. [FH20] for the precise definition). We will
further assume that $0\leq\sigma<1$, $0\leq\delta<\gamma$ and
$\theta\in\\{0,\gamma,\gamma+\delta,2\gamma\\}$. Also

  * •

$F:\mathcal{B}_{\alpha}\rightarrow\mathcal{B}_{\alpha-\sigma}$ is a Lipschitz
continuous function.

  * •

$G:\mathcal{B}_{\alpha-\theta}\rightarrow\mathcal{B}_{\alpha-\theta-\delta}$
is a bounded Fréchet differentiable function up to three times with bounded
derivatives.

  * •

$A$ generates a continuous semigroup $(S_{t})_{t\geq 0}$ such that for every
$\beta\in[\min\\{\alpha-2\gamma-\delta,\alpha-\sigma\\},\alpha]$,
$S_{t}\in\mathcal{L}(\mathcal{B}_{\beta})$. Also for every
${\sigma}_{1}\in[0,1)$ with $\beta+{\sigma}_{1}\leq\alpha$,

(0.2) $\displaystyle\begin{split}|S_{t}x|_{\beta+{\sigma}_{1}}&\lesssim
t^{-{\sigma}_{1}}|x|_{{\beta}},\\\ |(I-S_{t})x|_{{\beta}}&\lesssim
t^{{\sigma}_{1}}|x|_{\beta+{\sigma}_{1}}.\end{split}$

######  Remark 0.3.

Note that as a direct result of (0.2) for
$\beta\in[\min\\{\alpha-2\gamma-\delta,\alpha-\sigma\\},\alpha]$ and
$\sigma_{1},\sigma_{2}\in[0,1)$ such that
$\beta-\sigma_{1}+\sigma_{2}\in[\min\\{\alpha-2\gamma-\delta,\alpha-\sigma\\},\alpha]$,
we have

(0.3)
$\displaystyle|S_{t-u}(I-S_{u-v})x|_{\beta}\lesssim(t-u)^{-\sigma_{1}}(u-v)^{\sigma_{2}}|x|_{\beta-\sigma_{1}+\sigma_{2}}.\
\ $

Let us quickly review the required framework for solving this family of
equations and also some preliminary definitions mostly taken from [GHN21]. For
the interval $I\subset\mathbb{R}$ and
$\tilde{\alpha}\in\\{\alpha-\sigma,\alpha\\}$, set

$\displaystyle\mathcal{E}^{0,\gamma}_{\tilde{\alpha}-\gamma;I}:=C(I;\mathcal{B}_{\tilde{\alpha}-\gamma})\cap
C^{\gamma}(I;\mathcal{B}_{\tilde{\alpha}-2\gamma}),\ \ \
\mathcal{E}^{\gamma,2\gamma}_{\tilde{\alpha};I}:=C^{\gamma}_{2}(I;\mathcal{B}_{\tilde{\alpha}-\gamma})\cap
C^{2\gamma}_{2}(I;\mathcal{B}_{\tilde{\alpha}-2\gamma}).$

We write $Z\in\mathcal{D}_{\mathbf{X},\tilde{\alpha}}^{\gamma}(I)$ if for
$s,t\in I$, $Z_{s,t}=Z^{\prime}_{s}(\delta X)_{s,t}+Z^{\\#}_{s,t}$ such that

$\displaystyle\|Z\|_{\mathcal{D}_{\mathbf{X},\tilde{\alpha}}^{\gamma}(I)}:=\|Z\|_{C(I;\mathcal{B}_{\tilde{\alpha}})}+\|Z^{\prime}\|_{\mathcal{E}^{0,\gamma}_{\tilde{\alpha}-\gamma;I}}+\|Z^{\\#}\|_{\mathcal{E}^{\gamma,2\gamma}_{\tilde{\alpha};I}}<\infty$

where

$\displaystyle\|Z\|_{C(I;\mathcal{B}_{\tilde{\alpha}})}:=\sup_{\tau\in
I}|Z_{\tau}|_{\tilde{\alpha}},\ \
\|Z^{\prime}\|_{\mathcal{E}^{0,\gamma}_{\tilde{\alpha}-\gamma;I}}:=\max\\{\sup_{\tau\in
I}|Z^{\prime}_{\tau}|_{\tilde{\alpha}-\gamma},\sup_{\begin{subarray}{c}\tau,\nu\in
I,\\\ \tau<\nu\end{subarray}}\frac{|(\delta
Z^{\prime})_{\tau,\nu}\|_{\tilde{\alpha}-2\gamma}}{(\nu-\tau)^{\gamma}}\\}\ \
\text{and}$
$\displaystyle\quad\|Z^{\\#}|_{\mathcal{E}^{\gamma,2\gamma}_{\tilde{\alpha};I}}:=\max\\{\sup_{\begin{subarray}{c}\tau,\nu\in
I,\\\
\tau<\nu\end{subarray}}\frac{|Z^{\\#}_{\tau,\nu}|_{\tilde{\alpha}-\gamma}}{(\nu-\tau)^{\gamma}},\sup_{\begin{subarray}{c}\tau,\nu\in
I,\\\
\tau<\nu\end{subarray}}\frac{|Z^{\\#}_{\tau,\nu}|_{\tilde{\alpha}-2\gamma}}{(\nu-\tau)^{2\gamma}}\\}.$

We recall that for $Y\in\mathcal{D}_{\mathbf{X},\tilde{\alpha}}^{\gamma}(I)$,
the following limit exists:

(0.4)
$\displaystyle\int_{s}^{t}S_{t-\tau}Y_{\tau}\circ\mathrm{d}\mathbf{X}_{\tau}:=\lim_{\begin{subarray}{c}|\pi|\rightarrow
0,\\\ \pi=\\{\tau_{0}=s,\tau_{1},...\tau_{m}=t\\}\end{subarray}}\sum_{0\leq
j<m}\big{[}S_{t-\tau_{j}}Y_{\tau_{j}}\circ(\delta
X)_{\tau_{j},\tau_{j+1}}+S_{t-\tau_{j}}Y^{\prime}_{\tau_{j}}\circ\mathbb{X}_{\tau_{j},\tau_{j+1}}\big{]}.$

Let $Y\in\mathcal{D}_{\mathbf{X},\tilde{\alpha}}^{\gamma}(I)$. Then, it can
easily be shown that
$G(Y)\in\mathcal{D}_{\mathbf{X},\tilde{\alpha}}^{\gamma}(I)$. Finally, we can
define the mild solution to 0.1:

######  Definition 0.4.

We say that $Z\in\mathcal{D}_{\mathbf{X},{\alpha}}^{\gamma}(I)$ solves
equation (0.1) if and only if$Z$ satisfies the identity

(0.5) $\displaystyle(\delta
Z)_{s,t}=S_{t-s}Z_{s}+\int_{s}^{t}S_{t-\tau}F(Z_{\tau})\mathrm{d}\tau+\int_{s}^{t}S_{t-\tau}G(Z_{\tau})\circ\mathrm{d}\mathbf{X}_{\tau},\
\ Z_{0}=z_{0},\ s,t\in\mathbb{R},$

where the second integral is understood as (0.4).

######  Remark 0.5.

Existence and uniqueness of the solution for this type of equation are
discussed in several articles. For example, in [HN22], the authors prove that
the mild solutions of the equation under Assumption 0.2 exists, is unique and
globally defined.

##  1\. Main result

We aim to prove that the solution to (0.1) has an integrable bound. The main
obstacle we face here is the presence of the linear part: if we just apply
Grönwall’s lemma naively, we will only get an exponential bound in terms of
the noise which is not integrable. To overcome this problem, we employ a
modified version of the greedy points technique and modify the Sewing lemma.

As we stated earlier, the well-posedness and global existence of the solution
for this family of equations is well understood. However, the a priori bound
that is provided is not optimal and fails to be integrable for Gaussian
noises. Therefore, we need some more careful bound for the solution. Let us
first start with the following lemma:

######  Lemma 1.1.

For $0\leq\delta_{1}<\gamma$, set

(1.1)
$\displaystyle\begin{split}&W_{\mathbf{X},\gamma,\delta_{1}}:=\Delta_{T}=\\{(s,t):0\leq
s\leq t\leq T\\}\rightarrow\mathbb{R},\\\
&W_{\mathbf{X},\gamma,\delta_{1}}(s,t):=\sup_{\begin{subarray}{c}\pi,\\\
\pi=\\{\kappa_{0}=s,\kappa_{1},...\kappa_{m}=t\\}\end{subarray}}\big{\\{}\sum_{j}(\kappa_{j+1}-\kappa_{j})^{\frac{-\delta_{1}}{\gamma-\delta_{1}}}\big{[}|(\delta
X)_{\kappa_{j},\kappa_{j+1}}|^{\frac{1}{\gamma-\delta_{1}}}+\|\mathbb{X}_{\kappa_{j},\kappa_{j+1}}\|^{\frac{1}{2(\gamma-\delta_{1})}}\big{]}\big{\\}}.\end{split}$

Then $W_{\mathbf{X},\gamma,\delta_{1}}$ is a continuous control function, i.e.

(1.2) $\displaystyle
W_{\mathbf{X},\gamma,\delta_{1}}(s,u)+W_{\mathbf{X},\gamma,\delta_{1}}(u,t)\leq
W_{\mathbf{X},\gamma,\delta_{1}}(s,t),\ \ \ s\leq u\leq t.$

###### Proof.

Follows from our assumption on $\mathbf{X}$. ∎

The next Lemma is basic.

######  Lemma 1.2.

Assume $Z\in\mathcal{D}_{\mathbf{X},{\alpha}}^{\gamma}(I)$ is a mild solution
of (0.1). Then

$\displaystyle Z^{\prime}_{s}=G(Z_{s})\quad\text{and}\quad
Z^{\\#}_{s,t}=Z_{s,t}-G(Z_{s})\circ(\delta X)_{s,t}.$

Moreover, $[G(Z)]^{\prime}_{s}=D_{Z_{s}}G[G(Z_{s})]$ and

(1.3)
$\displaystyle\begin{split}[G(Z)]^{\\#}_{s,t}&=G(Z_{t})-G(Z_{s})-D_{Z_{s}}G[G(Z_{s})\circ(\delta
X)_{s,t}]\\\ &=\int_{0}^{1}\int_{0}^{1}\sigma D^{2}_{Z_{s}+\sigma u(\delta
Z)_{s,t}}G[G(Z_{s})\circ(\delta X)_{s,t},G(Z_{s})\circ(\delta
X)_{s,t}+Z_{s,t}^{\\#}]\mathrm{d}u\ \mathrm{d}\sigma\\\
&\quad+\int_{0}^{1}D_{Z_{s}+\sigma(\delta
Z)_{s,t}}G[Z^{\\#}_{s,t}]\mathrm{d}\sigma.\end{split}$

Let $s\leq u\leq v\leq w\leq t$ and

$\displaystyle\tilde{\Xi}^{u,v}_{s,t}\coloneqq S_{t-u}G(Z_{u})\circ(\delta
X)_{u,v}+S_{t-u}D_{Z_{u}}G[G(Z_{u})]\circ\mathbb{X}_{u,v}.$

Then

(1.4)
$\displaystyle\begin{split}&\tilde{\Xi}^{u,v}_{s,t}+\tilde{\Xi}^{v,w}_{s,t}-\tilde{\Xi}^{u,w}_{s,t}\\\
&\ =S_{t-u}([G(Z)]^{\\#}_{u,v})\circ(\delta
X)_{v,w}-S_{t-v}(S_{v-u}-I)G(Z_{v})\circ(\delta
X)_{v,w}-S_{t-v}(S_{v-u}-I)D_{Z_{u}}G[G(Z_{u})]\circ\mathbb{X}_{v,w}\\\
&\quad+S_{t-v}\big{(}\int_{0}^{1}D^{2}_{Z_{u}+\sigma(\delta
z)_{u,v}}G\big{[}G(Z_{u})\circ(\delta
X)_{u,v}+Z^{\\#}_{u,v},G(Z_{u})\big{]}\mathrm{d}\sigma\big{)}\circ\mathbb{X}_{v,w}\\\
&\quad+S_{t-v}\big{(}D_{Z_{v}}G\big{[}\int_{0}^{1}D_{Z_{u}+\sigma(\delta
Z)_{u,v}}G[G(Z_{u})\circ(\delta
X)_{u,v}+Z^{\\#}_{u,v}]\mathrm{d}\sigma\big{]}\big{)}\circ\mathbb{X}_{v,w}.\end{split}$

###### Proof.

Follows from definition of the mild solutions and our assumption. ∎

######  Lemma 1.3.

Let us fix $s<t$ and set $\tau^{n}_{m}\coloneqq s+\frac{n}{2^{m}}(t-s)$ where
$0\leq n<2^{m}-1$. We also define

(1.5) $\displaystyle\Xi^{n,m}_{s,t}\coloneqq
S_{t-\tau^{n}_{m}}G(Z_{\tau^{n}_{m}})\circ(\delta
X)_{\tau^{n}_{m},\tau^{n+1}_{m}}+S_{t-\tau^{n}_{m}}D_{Z_{\tau^{n}_{m}}}G[G(Z_{\tau^{n}_{m}})]\circ\mathbb{X}_{\tau^{n}_{m},\tau^{n+1}_{m}}.$

Then for $A^{n}_{m}:=G(Z_{\tau^{2n}_{m+1}})\circ(\delta
X)_{{\tau^{2n}_{m+1}},{\tau^{2n+1}_{m+1}}}$ we have

(1.6)
$\displaystyle\begin{split}&\Xi^{2n,m+1}_{s,t}+\Xi^{2n+1,m+1}_{s,t}-\Xi^{2n+2,m+1}_{s,t}\\\
&\ =S_{t-\tau^{2n}_{m+1}}\big{(}\int_{0}^{1}\int_{0}^{1}\sigma
D^{2}_{Z_{\tau^{2n}_{m+1}}+\sigma u(\delta
Z)_{\tau^{2n}_{m+1},\tau^{2n+1}_{m+1}}}G[A^{n}_{m};A^{n}_{m}+Z^{\\#}_{{\tau^{2n}_{m+1}},{\tau^{2n+1}_{m+1}}}]\mathrm{d}u\
\mathrm{d}\sigma\big{)}\circ(\delta
X)_{\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1}}\\\
&\quad+S_{t-\tau^{2n}_{m+1}}\big{(}\int_{0}^{1}D_{Z_{\tau^{2n}_{m+1}}+\sigma(\delta
Z)_{\tau^{2n}_{m+1},\tau^{2n+1}_{m+1}}}G[Z^{\\#}_{{\tau^{2n}_{m+1}},\tau^{2n+1}_{m+1}}]\mathrm{d}\sigma\big{)}\circ(\delta
X)_{\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1}}\\\
&\quad+S_{t-\tau^{2n}_{m+1}}\big{(}\int_{0}^{1}D^{2}_{Z_{\tau^{2n}_{m+1}}+\sigma(\delta
Z)_{\tau^{2n}_{m+1},\tau^{2n+1}_{m+1}}}G[A^{n}_{m}+Z^{\\#}_{\tau^{2n}_{m+1},\tau^{2n+1}_{m+1}};G(Z_{\tau^{2n}_{m+1}})]\mathrm{d}\sigma\big{)}\circ\mathbb{X}_{\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1}}\\\
&\quad+S_{t-\tau^{2n+1}_{m+1}}\big{(}D_{Z_{\tau^{2n+1}_{m+1}}}G[\int_{0}^{1}D_{Z_{\tau^{2n}_{m+1}}+\sigma(\delta
Z)_{\tau^{2n}_{m+1},\tau^{2n+1}_{m+1}}}G[A^{n}_{m}+Z^{\\#}_{\tau^{2n}_{m+1},\tau^{2n+1}_{m+1}}]\mathrm{d}\sigma]\big{)}\circ\mathbb{X}_{\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1}}\\\
&\quad-
S_{t-\tau^{2n+1}_{m+1}}\big{(}S_{\tau^{2n+1}_{m+1}-\tau^{2n}_{m+1}}-I\big{)}\big{(}G(Z_{\tau^{2n+1}_{m+1}})\circ(\delta
X)_{\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1}}+D_{Z_{\tau^{2n}_{m+1}}}G[G(Z_{\tau^{2n}_{m+1}})]\circ\mathbb{X}_{\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1}}\big{)}.\end{split}$

###### Proof.

Follows from (1.3) and (1.4). ∎

In the next proposition, we obtain an upper bound over the latter formula in
terms of our control function defined in Lemma 1.1.

######  Proposition 1.4.

For $i\in\\{0,1,2\\}$ and $\epsilon>0$ chosen such that
$\delta+\epsilon<\gamma$, there exists a constant $M_{\epsilon}$ depending on
$\epsilon$ and $G$ such that

(1.7) $\displaystyle\begin{split}&\sum_{m\geq 1}\sum_{0\leq
n<2^{m}}|S_{t-\tau^{2n}_{m+1}}\big{(}\int_{0}^{1}\int_{0}^{1}\sigma
D^{2}_{Z_{\tau^{2n}_{m+1}}+\sigma u(\delta
Z)_{\tau^{2n}_{m+1},\tau^{2n+1}_{m+1}}}G[A^{n}_{m};Z^{\\#}_{{\tau^{2n}_{m+1}},{\tau^{2n+1}_{m+1}}}]\mathrm{d}u\
\mathrm{d}\sigma\big{)}\circ(\delta
X)_{\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1}}|_{\alpha-i\gamma}\\\
&\quad+|S_{t-\tau^{2n}_{m+1}}\big{(}\int_{0}^{1}D_{Z_{\tau^{2n}_{m+1}}+\sigma(\delta
Z)_{\tau^{2n}_{m+1},\tau^{2n+1}_{m+1}}}G[Z^{\\#}_{{\tau^{2n}_{m+1}},\tau^{2n+1}_{m+1}}]\mathrm{d}\sigma\big{)}\circ(\delta
X)_{\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1}}|_{\alpha-i\gamma}\\\
&\quad+|S_{t-\tau^{2n}_{m+1}}\big{(}\int_{0}^{1}D^{2}_{Z_{\tau^{2n}_{m+1}}+\sigma(\delta
Z)_{\tau^{2n}_{m+1},\tau^{2n+1}_{m+1}}}G[Z^{\\#}_{\tau^{2n}_{m+1},\tau^{2n+1}_{m+1}};G(Z_{\tau^{2n}_{m+1}})]\big{)}\circ\mathbb{X}_{\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1}}|_{\alpha-i\gamma}\\\
&\quad+|S_{t-\tau^{2n+1}_{m+1}}\big{(}D_{Z_{\tau^{2n+1}_{m+1}}}G[\int_{0}^{1}D_{Z_{\tau^{2n}_{m+1}}+\sigma(\delta
Z)_{\tau^{2n}_{m+1},\tau^{2n+1}_{m+1}}}G[Z^{\\#}_{\tau^{2n}_{m+1},\tau^{2n+1}_{m+1}}]\mathrm{d}\sigma]\big{)}\circ\mathbb{X}_{\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1}}|_{\alpha-i\gamma}\\\
&\leq
M_{\epsilon}(t-s)^{i\gamma}\max\big{\\{}(t-s)^{\epsilon}\big{(}W_{\mathbf{X},\gamma,\delta+\epsilon}(s,t)\big{)}^{\gamma-\delta-\epsilon},(t-s)^{2\epsilon}\big{(}W_{\mathbf{X},\gamma,\delta+\epsilon}(s,t)\big{)}^{2(\gamma-\delta-\epsilon)}\big{\\}}|Z^{\\#}|_{\mathcal{E}^{\gamma,2\gamma}_{{\alpha};I}}.\end{split}$

###### Proof.

We have to show that each term on the left hand side of (1.7) can be bounded
up to a constant depending on $\epsilon$ and $G$ by
$(t-s)^{i\gamma+\epsilon}W_{\mathbf{X},\gamma,\delta+\epsilon}(s,t)^{\gamma-\delta-\epsilon}$.
We will show this bound for the second and fourth term whose proofs have some
distinctions. For the remaining terms, our claim can be confirmed by a similar
technique. Remember that
$\|Z^{\\#}\|_{\mathcal{E}^{\gamma,2\gamma}_{{\alpha};I}}<\infty$ and also that
(0.2) holds true. Then

$\displaystyle\sum_{m\geq 1}\sum_{0\leq
n<2^{m}}|S_{t-\tau^{2n}_{m+1}}\big{(}\int_{0}^{1}D_{\sigma
Z_{\tau^{2n}_{m+1},\tau^{2n+1}_{m+1}}+Z_{\tau^{2n}_{m+1}}}G[Z^{\\#}_{{\tau^{2n}_{m+1}},Z{\tau^{2n+1}_{m+1}}}]\mathrm{d}\sigma\big{)}\circ(\delta
X)_{\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1}}|_{\alpha-i\gamma}$
$\displaystyle\lesssim\ $
$\displaystyle|Z^{\\#}|_{\mathcal{E}^{\gamma,2\gamma}_{{\alpha};I}}\sum_{m\geq
1}\sum_{0\leq
n<2^{m}}(t-\tau^{2n}_{m+1})^{\gamma(i-2)-\delta}(t-s)^{2\gamma}(\frac{1}{2^{m+1}})^{2\gamma}|{X}_{\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1}}|$
$\displaystyle\leq\ $ $\displaystyle(t-s)^{\gamma
i+\epsilon}|Z^{\\#}|_{\mathcal{E}^{\gamma,2\gamma}_{{\alpha};I}}\sum_{m\geq
1}\sum_{0\leq
n<2^{m}}(1-\frac{2n}{2^{m+1}})^{\gamma(i-2)-\delta}(\frac{1}{2^{m+1}})^{(2\gamma+\delta+\epsilon)}\big{(}W_{\mathbf{X},\gamma,\delta+\epsilon}(\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1})\big{)}^{(\gamma-\delta-\epsilon)}.$

From the Hölder inequality and (1.2),

$\displaystyle\sum_{0\leq
n<2^{m}}(1-\frac{2n}{2^{m+1}})^{\gamma(i-2)-\delta}(\frac{1}{2^{m+1}})^{(2\gamma+\delta+\epsilon)}\big{(}W_{\mathbf{X},\gamma,\delta+\epsilon}(\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1})\big{)}^{(\gamma-\delta-\epsilon)}$
$\displaystyle\leq\ $ $\displaystyle\big{(}\sum_{0\leq
n<2^{m}}(1-\frac{2n}{2^{m+1}})^{\frac{\gamma(i-2)-\delta}{1-\gamma+\delta+\epsilon}}(\frac{1}{2^{m+1}})^{\frac{2\gamma+\delta+\epsilon)}{1-\gamma+\delta+\epsilon}}\big{)}^{1-\gamma+\delta+\epsilon}\big{(}\sum_{0\leq
n<2^{m}}W_{\mathbf{X},\gamma,\delta+\epsilon}(\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1})\big{)}^{\gamma-\delta-\epsilon}$
$\displaystyle\leq\ $
$\displaystyle\big{(}W_{\mathbf{X},\gamma,\delta+\epsilon}(s,t)\big{)}^{\gamma-\delta-\epsilon}\big{(}\sum_{0\leq
n<2^{m}}(1-\frac{2n}{2^{m+1}})^{\frac{\gamma(i-2)-\delta}{1-\gamma+\delta+\epsilon}}(\frac{1}{2^{m+1}})^{\frac{2\gamma+\delta+\epsilon)}{1-\gamma+\delta+\epsilon}}\big{)}^{1-\gamma+\delta+\epsilon}.$

Note that $\frac{1}{2^{m+1}}\leq 1-\frac{2n}{2^{m+1}}$. Therefore,

$\displaystyle\sum_{0\leq
n<2^{m}}(1-\frac{2n}{2^{m+1}})^{\frac{\gamma(i-2)-\delta}{1-\gamma+\delta+\epsilon}}(\frac{1}{2^{m+1}})^{\frac{2\gamma+\delta+\epsilon}{1-\gamma+\delta+\epsilon}}\leq(\frac{1}{2^{m}})^{\frac{\frac{\epsilon}{2}}{1-\gamma+\delta+\epsilon}}\sum_{0\leq
n<2^{m}}(1-\frac{2n}{2^{m+1}})^{\frac{\gamma(i+1)-1-\delta-\frac{\epsilon}{2}}{1-\gamma+\delta+\epsilon}}\frac{1}{2^{m+1}}.$

Since

$\displaystyle\lim_{m\rightarrow\infty}\sum_{0\leq
n<2^{m}}(1-\frac{2n}{2^{m+1}})^{\frac{\gamma(i+1)-1-\delta-\frac{\epsilon}{2}}{1-\gamma+\delta+\epsilon}}\frac{1}{2^{m+1}}=\frac{1}{2}\int_{0}^{1}(1-x)^{\frac{\gamma(i+1)-\delta-1-\frac{\epsilon}{2}}{1-\gamma+\delta+\epsilon}}\mathrm{d}x,$

We can conclude that for some $\tilde{M}_{\epsilon}<\infty$,

$\displaystyle\sum_{m\geq 1}\sum_{0\leq
n<2^{m}}(1-\frac{2n}{2^{m+1}})^{\gamma(i-2)-\delta}(\frac{1}{2^{m+1}})^{(2\gamma+\delta+\epsilon)}\big{(}W_{\mathbf{X},\gamma,\delta+\epsilon}(\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1})\big{)}^{(\gamma-\delta-\epsilon)}\leq\tilde{M}_{\epsilon}\big{(}W_{\mathbf{X},\gamma,\delta+\epsilon}(s,t)\big{)}^{\gamma-\delta-\epsilon}.$

This proves our claim for the second term. For estimating the fourth term, we
use the same idea with some modifications. First note that in (1.7) from the
interpolation property,

$\displaystyle|Z^{\\#}_{s,t}|_{\alpha-2\gamma+\delta}\lesssim(t-s)^{2\gamma-\delta}|Z^{\\#}|_{\mathcal{E}^{\gamma,2\gamma}_{{\alpha};I}}.$

For

$\displaystyle{B}^{i,\epsilon}_{\mathbf{X}}(s,t)\coloneqq(t-s)^{\gamma
i+2\epsilon}W_{\mathbf{X},\gamma,\delta+\epsilon}(s,t)^{2(\gamma-\delta-\epsilon)},$

we have

$\displaystyle\sum_{m\geq 1}\sum_{0\leq
n<2^{m}}|S_{t-\tau^{2n+1}_{m+1}}\big{(}D_{Z_{\tau^{2n+1}_{m+1}}}G[\int_{0}^{1}D_{\sigma
Z_{\tau^{2n}_{m+1},\tau^{2n+1}_{m+1}}+Z_{\tau^{2n}_{m+1}}}G[Z^{\\#}_{\tau^{2n}_{m+1},\tau^{2n+1}_{m+1}}]\mathrm{d}\sigma]\big{)}\circ\mathbb{X}_{\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1}}|_{\alpha-i\gamma}$
$\displaystyle\lesssim\ $
$\displaystyle|Z^{\\#}|_{\mathcal{E}^{\gamma,2\gamma}_{{\alpha};I}}\sum_{m\geq
1}\sum_{0\leq
n<2^{m}}(t-\tau_{m+1}^{2n+1})^{\gamma(i-2)-\delta}(t-s)^{2\gamma-\delta}(\frac{1}{2^{m+1}})^{2\gamma-\delta}\|\mathbb{X}_{\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1}}\|$
$\displaystyle\leq\ $
$\displaystyle|Z^{\\#}|_{\mathcal{E}^{\gamma,2\gamma}_{{\alpha};I}}\sum_{m\geq
1}\sum_{0\leq
n<2^{m}}(t-\tau^{2n+1}_{m+1})^{\gamma(i-2)-\delta}(t-s)^{2\gamma+\delta+2\epsilon}(\frac{1}{2^{m+1}})^{2\gamma+\delta+2\epsilon}\big{(}W_{\mathbf{X},\gamma,\delta+\epsilon}(\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1})\big{)}^{2(\gamma-\delta-\epsilon)}$
$\displaystyle=\ $ $\displaystyle(t-s)^{\gamma
i+2\epsilon}|Z^{\\#}|_{\mathcal{E}^{\gamma,2\gamma}_{{\alpha};I}}\sum_{m\geq
1}\sum_{0\leq
n<2^{m}}(1-\frac{2n+1}{2^{m+1}})^{\gamma(i-2)-\delta}(\frac{1}{2^{m+1}})^{2\gamma+\delta+2\epsilon}\big{(}W_{\mathbf{X},\gamma,\delta+\epsilon}(\tau^{2n+1}_{m+1},\tau^{2n+2}_{m+1})\big{)}^{2(\gamma-\delta-\epsilon)}$
$\displaystyle\leq\ $
$\displaystyle{B}^{i,\epsilon}_{\mathbf{X}}(s,t)|Z^{\\#}|_{\mathcal{E}^{\gamma,2\gamma}_{{\alpha};I}}\sum_{m\geq
1}\big{(}\sum_{0\leq
n<2^{m}}(1-\frac{2n+1}{2^{m+1}})^{\frac{\gamma(i-2)-\delta}{1-2\gamma+2\delta+2\epsilon}}(\frac{1}{2^{m+1}})^{\frac{2\gamma+\delta+2\epsilon}{1-2\gamma+2\delta+2\epsilon}}\big{)}^{1-2\gamma+2\delta+2\epsilon}$
$\displaystyle\leq\ $
$\displaystyle{B}^{i,\epsilon}_{\mathbf{X}}(s,t)|Z^{\\#}|_{\mathcal{E}^{\gamma,2\gamma}_{{\alpha};I}}\sum_{m\geq
1}(\frac{1}{2^{m+1}})^{\epsilon}\big{(}\sum_{0\leq
n<2^{m}}(1-\frac{2n+1}{2^{m+1}})^{\frac{\gamma(i+2)-2\delta-\epsilon-1}{1-2\gamma+2\delta+2\epsilon}}\frac{1}{2^{m+1}}\big{)}^{1-2\gamma+2\delta+2\epsilon}$
$\displaystyle\leq\ $ $\displaystyle\tilde{\tilde{M}}_{\epsilon}(t-s)^{\gamma
i+2\epsilon}W_{\mathbf{X},\gamma,\delta+\epsilon}(s,t)^{2(\gamma-\delta-\epsilon)}|Z^{\\#}|_{\mathcal{E}^{\gamma,2\gamma}_{{\alpha};I}},$

where $\tilde{{\tilde{M}}}_{\epsilon}<\infty$ only depends on $\epsilon$. ∎

Now we can prove the following lemma

######  Lemma 1.5.

Assume $\delta+\epsilon<\gamma$ and
$Z\in\mathcal{E}^{\gamma,2\gamma}_{{\alpha};I}$ solves the equation (0.1).
Then for $M_{\epsilon}<\infty$ and $i\in\\{0,1,2\\}$,

(1.8)
$\displaystyle\begin{split}&|\int_{s}^{t}S_{t-\tau}G(Z_{\tau})\circ\mathrm{d}\mathbf{X}_{\tau}-S_{t-s}G(Z_{s})\circ(\delta
X)_{s,t}-S_{t-s}D_{Z_{s}}G[G(Z_{s})]\circ\mathbb{X}_{s,t}|_{\alpha-i\gamma}\\\
\leq\
&M_{\epsilon}(t-s)^{i\gamma}\max\big{\\{}(t-s)^{\epsilon}\big{(}W_{\mathbf{X},\gamma,\delta+\epsilon}(s,t)\big{)}^{\gamma-\delta-\epsilon},(t-s)^{2\epsilon}\big{(}W_{\mathbf{X},\gamma,\delta+\epsilon}(s,t)\big{)}^{2(\gamma-\delta-\epsilon)}\big{\\}}|Z^{\\#}|_{\mathcal{E}^{\gamma,2\gamma}_{{\alpha};I}}\\\
&\quad+(t-s)^{i\gamma+3(\gamma-\delta)}\|X\|_{\gamma}[\|X\|_{\gamma}^{2}+\|\mathbb{X}\|_{2\gamma}].\end{split}$

###### Proof.

We use the same notation as in Lemma 1.3. Set
$\Gamma_{s,t}^{m}\coloneqq\sum_{0\leq n\leq 2^{m}}\Xi^{n,m}_{s,t}$. By the
Sewing lemma,

$\displaystyle|\int_{s}^{t}S_{t-\tau}G(Z_{\tau})\circ\mathrm{d}\mathbf{X}_{\tau}-S_{t-s}G(Z_{s})\circ(\delta
X)_{s,t}-S_{t-s}D_{Z_{s}}G[G(Z_{s})]\circ\mathbb{X}_{s,t}|_{\alpha-i\gamma}$
$\displaystyle\leq\ $ $\displaystyle\sum_{m\geq
1}|\Gamma_{s,t}^{m+1}-\Gamma_{s,t}^{m}|_{\alpha-i\gamma}$ $\displaystyle\leq$
$\displaystyle\sum_{m\geq 1}\sum_{0\leq
n<2^{m}}|\Xi^{2n,m+1}_{s,t}+\Xi^{2n+1,m+1}_{s,t}-\Xi^{2n+2,m+1}_{s,t}|_{\alpha-i\gamma}.$

If we plug (1.3) into (1.4), then our claim follows from Proposition 1.4. ∎

The following result is well known:

######  Lemma 1.6.

Assume
$\mathbf{X}=(X,\mathbb{X})\in\mathscr{C}^{\gamma}([s,t],\mathbb{R}^{n})$ be a
$\gamma$-rough path as before. Let $\gamma^{\prime}>0$ with
$\gamma+\gamma^{\prime}>1$, and also for a given path
$h:[s,t]\rightarrow\mathbb{R}^{d}$

(1.9) $\displaystyle\sup_{\begin{subarray}{c}\pi,\\\
\pi=\\{\kappa_{0}=s,\kappa_{1},...\kappa_{m}=t\\}\end{subarray}}\big{\\{}\sum_{j}\big{[}|h_{\kappa_{j},\kappa_{j+1}}|^{\frac{1}{{\gamma}^{\prime}}}\big{]}\big{\\}}<\infty.$

Then this path can be enhanced to a rough path $\mathbf{h}=(h,\int
h\,\mathrm{d}h)$ where $\int h\,\mathrm{d}h$ is defined by Young integration.
In addition, $\int X\mathrm{d}h$ and $\int h\mathrm{d}X$ can also again be
defined as Young integrals.

The following estimate will be crucial:

######  Lemma 1.7.

Assume that $I=[s,t]$ is a closed interval and for
$\frac{1}{3}<\gamma<\frac{1}{2}$, $\mathbf{X}=(X,\mathbb{X})$ is a weakly
geometric rough path such that for $\delta_{1}<\gamma$,
$W_{\mathbf{X},\gamma,\delta_{1}}(s,t)<\infty$. In addition, for
$\gamma^{\prime}>0$ with $\gamma+\gamma^{\prime}>1$ and for
$h:I\rightarrow\mathbb{R}^{d}$ being a continuous path satisfying in (1.9), we
assume that $W_{\mathbf{h},\gamma^{\prime},\delta_{1}}(s,t)<\infty$. If
$\gamma+\gamma^{\prime}-2\delta_{1}>1$, then

(1.10) $\displaystyle W_{T_{h}(\mathbf{X}),\gamma,\delta_{1}}(s,t)\leq
C_{\delta_{1}}\big{[}W_{\mathbf{X},\gamma,\delta_{1}}(s,t)+\big{(}W_{\mathbf{h},\gamma^{\prime},\delta_{1}}(s,t)\big{)}^{\frac{\gamma^{\prime}-\delta_{1}}{\gamma-\delta_{1}}}\big{]},$

where $T_{h}(\mathbf{X})=(h+X,\int h\otimes\mathrm{d}h+\int
h\otimes\mathrm{d}X+\int X\otimes\mathrm{d}h+\mathbb{X})$.

###### Proof.

Remember that by Young’s integration theory,

(1.11) $\displaystyle\int_{s}^{t}(\delta
h)_{s,\tau}\otimes\mathrm{d}X_{\tau}:=\lim_{|\pi|\rightarrow
0}\int_{\pi}h_{s,\tau}\circ\mathrm{d}X_{\tau}=\lim_{|\pi|\rightarrow
0}\sum_{0\leq i<m}h_{s,\kappa_{i}}\otimes(\delta
X)_{\kappa_{i},\kappa_{i+1}},$

where $\pi=\\{\kappa_{0}=s,\kappa_{1},...\kappa_{m}=t\\}$ is a partition of
$[s,t]$ and

$\displaystyle\int_{\pi}h_{s,\tau}\circ\mathrm{d}X_{\tau}\coloneqq\sum_{0\leq
i<m}h_{s,\kappa_{i}}\otimes(\delta X)_{\kappa_{i},\kappa_{i+1}}.$

Clearly,

(1.12)
$\displaystyle\begin{split}&\|\int_{\pi}h_{s,\tau}\otimes\mathrm{d}X_{\tau}-\int_{\pi\setminus\\{\kappa_{j}\\}}h_{s,\tau}\otimes\mathrm{d}X_{\tau}\|\leq|h_{\kappa_{j-1},\kappa_{j}}||(\delta
X)_{\kappa_{j},\kappa_{j+1}}|\\\ \leq\
&[W_{\mathbf{h},\gamma^{\prime},\delta_{1}}^{\frac{\gamma^{\prime}-\delta_{1}}{\gamma+{\gamma}^{\prime}-2\delta_{1}}}(\kappa_{j-1},\kappa_{j})W_{\mathbf{X},\gamma,\delta_{1}}^{\frac{\gamma-\delta_{1}}{\gamma+{\gamma}^{\prime}-2\delta_{1}}}(\kappa_{j},\kappa_{j+1})]^{\gamma+{\gamma}^{\prime}-2\delta_{1}}(\kappa_{j}-\kappa_{j-1})^{\delta_{1}}(\kappa_{j+1}-\kappa_{j})^{\delta_{1}}.\end{split}$

Note that since $W_{\mathbf{X},\gamma,\delta_{1}}$ and
$W_{\mathbf{h},\gamma^{\prime},\delta_{1}}$ are control functions (c.f.
(1.2)), consequently we can find $1\leq j<m$, such that

$W_{\mathbf{h},\gamma^{\prime},\delta_{1}}^{\frac{\gamma^{\prime}-\delta_{1}}{\gamma+{\gamma}^{\prime}-2\delta_{1}}}(\kappa_{j-1},\kappa_{j})W_{\mathbf{X},\gamma,\delta_{1}}^{\frac{\gamma-\delta_{1}}{\gamma+{\gamma}^{\prime}-2\delta_{1}}}(\kappa_{j},\kappa_{j+1})<\frac{2}{m-1}W_{\mathbf{h},\gamma^{\prime},\delta_{1}}^{\frac{\gamma^{\prime}-\delta_{1}}{\gamma+{\gamma}^{\prime}-2\delta_{1}}}(s,t)W_{\mathbf{X},\gamma,\delta_{1}}^{\frac{\gamma-\delta_{1}}{\gamma+{\gamma}^{\prime}-2\delta_{1}}}(s,t).$

We can repeat this argument for the new partition
$\pi\setminus\\{\kappa_{j}\\}$, so from (1.11) and (1.12) we conclude

$\displaystyle\|\int_{s}^{t}h_{s,\tau}\otimes\mathrm{d}X_{\tau}\|\leq\sum_{k\geq
1}\frac{2^{\gamma+\gamma^{\prime}-2\delta_{1}}}{k^{\gamma+\gamma^{\prime}-2\delta_{1}}}(t-s)^{2\delta_{1}}W_{\mathbf{h},\gamma^{\prime},\delta_{1}}^{\gamma^{\prime}-\delta_{1}}(s,t)W_{\mathbf{X},\gamma,\delta_{1}}^{\gamma-\delta_{1}}(s,t)\longrightarrow$
$\displaystyle\quad\frac{\|\int_{s}^{t}h_{s,\tau}\otimes\mathrm{d}X_{\tau}\|^{{}^{\frac{1}{2(\gamma-\delta_{1})}}}}{(t-s)^{\frac{\delta_{1}}{\gamma-\delta_{1}}}}\leq\big{[}\sum_{k\geq
1}\frac{2^{\gamma+\gamma^{\prime}-2\delta_{1}}}{k^{\gamma+\gamma^{\prime}-2\delta_{1}}}\big{]}^{\frac{1}{2(\gamma-\delta_{1})}}W_{\mathbf{h},\gamma^{\prime},\delta_{1}}^{\frac{\gamma^{\prime}-\delta_{1}}{2(\gamma-\delta_{1})}}(s,t)W_{\mathbf{X},\gamma,\delta_{1}}^{\frac{1}{2}}(s,t).$

An analogous argument can be run to obtain similar bounds for $\int
X\otimes\mathrm{d}h$ and $\int h\otimes\mathrm{d}h$. This finishes the proof.
∎

We will assume up to the end of this part:

######  Assumption 1.8.

Let $(\mathcal{W},\mathcal{H},\mu)$ be an abstract Wiener space and assume $X$
is a Gaussian process defined on it such that it can be enhanced to a weakly
$\gamma$-Hölder geometric rough path $\mathbf{X}=(X,\mathbb{X})$,
$\frac{1}{3}<\gamma\leq\frac{1}{2}$. For every $h\in\mathcal{H}$, let
Condition (1.9) be fulfilled. In this case, by [CLL13, Lemma 5.4], on a
measurable subset $E\subset\mathcal{W}$ with full measure,

(1.13) $\displaystyle T_{h}\mathbf{X}(\omega)\equiv\mathbf{X}(\omega+h),\ \
\omega\in E\ \ \text{and}\ \forall h\in\mathcal{H}.$

We assume that for every $h\in\mathcal{H}$,

(1.14) $\displaystyle
W_{\mathbf{h},\gamma^{\prime},\delta_{1}}(0,1)\lesssim|h|_{\mathcal{H}}^{\frac{1}{\gamma^{\prime}-\delta_{1}}}.$

In the following, we will show that the lift of a fractional Brownian motion
satisfies Assumption 1.8.

######  Proposition 1.9.

Assume $H\in(\frac{1}{4},\frac{1}{2})$ and let $B^{H}$ be a fractional
Brownian motion with Hurst parameter $H$. Let $\mathcal{H}^{H}$ denote the
associated Cameron–Martin space for this process. Then Assumption 1.8 holds
for every $\frac{1}{2}<\gamma^{\prime}<H+\frac{1}{2}$ and
$\delta_{1}<\gamma^{\prime}-\frac{1}{2}$.

###### Proof.

For $\theta\in(0,1)$ and $q\in(1,\infty)$, for a measurable path
$g\colon[0,1]\rightarrow\mathbb{R}$, define

$\displaystyle|g|_{W^{\theta,q}}\coloneqq\big{(}\int_{[0,1]^{2}}\frac{|g(u)-g(v)|^{q}}{|u-v|^{1+\theta
q}}\mathrm{d}u\mathrm{d}v\big{)}^{\frac{1}{q}}.$

Then $W^{\theta,q}$ is defined as the set of all measurable paths $g$ such
that

$\displaystyle|g|_{W^{\theta,q}}+\big{(}\int_{0}^{1}|g_{u}|^{q}\mathrm{d}u\big{)}^{\frac{1}{q}}<\infty$

and we set

$\displaystyle W^{\theta,q}_{0}\coloneqq\\{g\in W^{\theta,q}:g(0)=0\\}.$

By [FV06, Theorem 3], for $q=2$ and
$\frac{1}{2}<\gamma^{\prime}<H+\frac{1}{2}$, $\mathcal{H}^{H}$ is compactly
embedded in $W^{\gamma^{\prime},2}_{0}$. Therefore, for a constant
$C(\gamma^{\prime})<\infty$,

(1.15)
$\displaystyle\big{(}\int\int_{[0,1]^{2}}\frac{|h(u)-h(v)|^{2}}{|u-v|^{1+2\gamma^{\prime}}}\mathrm{d}u\mathrm{d}v\big{)}^{\frac{1}{2}}\leq
C_{1}(\gamma^{\prime})|h|_{\mathcal{H}^{H}}$

where by $|\cdot|_{\mathcal{H}^{H}}$, we mean the corresponding Hilbert norm
in $\mathcal{H}^{H}$. Also by the Besov–Hölder embedding theorem (cf. [FV10b,
Corollary A.2]), for all $0\leq s<t\leq 1$ and a constant
$C_{2}(\gamma^{\prime})<\infty$,

(1.16) $\displaystyle\begin{split}|h(t)-h(s)|^{2}\leq
C_{2}(\gamma^{\prime})(t-s)^{2\gamma^{\prime}-1}|h|_{W^{\gamma^{\prime},2}}^{2}=C_{2}(\gamma^{\prime})(t-s)^{2\gamma^{\prime}-1}\int\int_{[s,t]^{2}}\frac{|h(u)-h(v)|^{2}}{|u-v|^{1+2\gamma^{\prime}}}\,\mathrm{d}u\,\mathrm{d}v.\end{split}$

Now assume $\pi=\\{\kappa_{0}=0,\kappa_{1},...\kappa_{m}=1\\}$. From (1.16),

$\displaystyle\sum_{0\leq
i<m}\frac{|h(\kappa_{i+1})-h(\kappa_{i})|^{\frac{1}{\gamma^{\prime}-\delta_{1}}}}{(\kappa_{i+1}-\kappa_{i})^{\frac{\delta_{1}}{\gamma^{\prime}-\delta_{1}}}}$
$\displaystyle\leq\ $ $\displaystyle
C_{2}(\theta)^{\frac{1}{2(\gamma^{\prime}-\delta_{1})}}\sum_{0\leq
i<m}(\kappa_{i+1}-\kappa_{i})^{\frac{2\gamma^{\prime}-2\delta_{1}-1}{2(\gamma^{\prime}-\delta_{1})}}\big{(}\int\int_{[\kappa_{i},\kappa_{i+1}]^{2}}\frac{|h(u)-h(v)|^{2}}{|u-v|^{1+2\gamma^{\prime}}}\mathrm{d}u\mathrm{d}v\big{)}^{\frac{1}{2(\gamma^{\prime}-\delta_{1})}}.$

Now for $0<\delta_{1}<\gamma^{\prime}-\frac{1}{2}$, applying the Hölder
inequality and (1.15) yields

$\displaystyle\sum_{0\leq
i<m}\frac{|h(\kappa_{i+1})-h(\kappa_{i})|^{\frac{1}{\gamma^{\prime}-\delta_{1}}}}{(\kappa_{i+1}-\kappa_{i})^{\frac{\delta_{1}}{\gamma^{\prime}-\delta_{1}}}}$
$\displaystyle\leq\ $ $\displaystyle
C_{2}(\gamma^{\prime})^{\frac{1}{2(\gamma^{\prime}-\delta_{1})}}\big{(}\sum_{0\leq
i<m}\int\int_{[\kappa_{i},\kappa_{i+1}]^{2}}\frac{|h(u)-h(v)|^{2}}{|u-v|^{1+2\gamma^{\prime}}}\mathrm{d}u\mathrm{d}v\big{)}^{\frac{1}{2(\gamma^{\prime}-\delta_{1})}}$
$\displaystyle\leq\ $ $\displaystyle
C_{2}(\gamma^{\prime})^{\frac{1}{2(\gamma^{\prime}-\delta_{1})}}\big{(}\int\int_{[0,1]^{2}}\frac{|h(u)-h(v)|^{2}}{|u-v|^{1+2\gamma^{\prime}}}\mathrm{d}u\mathrm{d}v\big{)}^{\frac{1}{2(\gamma^{\prime}-\delta_{1})}}$
$\displaystyle\leq\ $ $\displaystyle
C_{2}(\gamma^{\prime})^{\frac{1}{2(\gamma^{\prime}-\delta_{1})}}C_{1}(\gamma^{\prime})^{\frac{1}{\gamma^{\prime}-\delta_{1}}}|h|_{\mathcal{H}^{H}}^{\frac{1}{\gamma^{\prime}-\delta_{1}}}.$

∎

######  Definition 1.10.

For $I=[a,b]$, the sequence of greedy points denoted with
$\\{\tau^{I}_{n,\delta_{1},\omega}(\chi)\\}_{n\geq 0}$ is defined by setting
$\tau^{I}_{0,\delta_{1},\omega}(\chi)=a$ and recursively

(1.17)
$\displaystyle\begin{split}\tau_{n+1,\omega}^{I}(\chi)\coloneqq\sup\big{\\{}\tau:\
\tau^{I}_{n,\delta_{1},\omega}(\chi)\leq\tau\leq b\ \ \text{and}\ \
W_{\mathbf{X}(\omega),\gamma,\delta_{1}}^{{\gamma-\delta_{1}}}(\tau_{n,\omega}^{I}(\chi),\tau)\leq\chi\big{\\}}.\end{split}$

For $0<\delta_{1}<\gamma$ and $\chi>0$, we set

$\displaystyle N(I,\delta_{1},\chi,\mathbf{X}_{\omega})\coloneqq\inf\\{n>0:\
\tau_{n,\omega}^{I}(\chi)=b\\}.$

The following proposition is analogous to [CLL13, Proposition 6.2].

######  Proposition 1.11.

Assume $\gamma+\gamma^{\prime}-2\delta>1$. Choose
$0<\epsilon<\frac{\gamma+\gamma^{\prime}-2\delta-1}{2}$ and set
$\delta_{1}=\delta+\epsilon$. Then for a constant $T(\delta_{1})<\infty$, we
have

(1.18)
$\displaystyle\big{[}N\big{(}I,\delta_{1},2^{\frac{1}{\gamma-\delta_{1}}}C_{\delta_{1}}W_{\mathbf{X}(\omega-h),\gamma,\delta_{1}}(a,b),\mathbf{X}(\omega)\big{)}-1\big{]}W_{\mathbf{X}(\omega-h),\gamma,\delta_{1}}^{{\frac{\gamma-\delta_{1}}{\gamma^{\prime}-\delta_{1}}}}(a,b)\leq
T(\delta_{1})|h|_{\mathcal{H}}^{\frac{1}{\gamma^{\prime}-\delta_{1}}},$

where $C_{\delta_{1}}$ is the constant in (1.10). Also for $\chi>0$, there
exist $M_{1}(\delta_{1},\chi),M_{2}(\delta_{1},\chi)<\infty$ such that

(1.19)
$\displaystyle\mu\big{\\{}\omega:N(I,\delta_{1},\chi,\mathbf{X}(\omega))>n\big{\\}}\leq
M_{1}(\delta_{1},\chi)\exp(-M_{2}(\delta_{1},\chi)n^{2(\gamma^{\prime}-\delta_{1})}).$

###### Proof.

From (1.13) and (1.10)

(1.20)
$\displaystyle\begin{split}&W_{\mathbf{X}(\omega),\gamma,\delta_{1}}^{\gamma-\delta_{1}}(\tau_{n,\omega}^{I}(\chi),\tau_{n+1,\omega}^{I}(\chi))=W_{T_{h}(\mathbf{X}(\omega-h)),\gamma,\delta_{1}}^{\gamma-\delta_{1}}(\tau_{n,\omega}^{I}(\chi),\tau_{n+1,\omega}^{I}(\chi))\\\
\leq\
&C_{\delta_{1}}^{\gamma-\delta_{1}}\big{[}W_{\mathbf{X}(\omega-h),\gamma,\delta_{1}}^{\gamma-\delta_{1}}(\tau_{n,\omega}^{I}(\chi),\tau_{n+1,\omega}^{I}(\chi))+W_{\mathbf{h},\gamma^{\prime},\delta_{1}}^{\gamma^{\prime}-\delta_{1}}(\tau^{I}_{n,\delta_{1},\omega}(\chi),\tau^{I}_{n+1,\delta_{1},\omega}(\chi))\big{]}.\end{split}$

Note that if $\tau_{n+1,\omega}^{I}(\chi)<b$, then the continuity of
$W_{\mathbf{X}(\omega)}$ yields

$W_{\mathbf{X}(\omega),\gamma,\delta_{1}}(\tau_{n,\omega}^{I}(\chi),\tau_{n+1,\omega}^{I}(\chi))=\chi.$

Set $\chi\coloneqq
2^{\frac{1}{\gamma-\delta_{1}}}C_{\delta_{1}}W_{\mathbf{X}(\omega-h),\gamma,\delta_{1}}(a,b)$.
From (1.20),

$\displaystyle
W_{\mathbf{X}(\omega-h),\gamma,\delta_{1}}^{{\frac{\gamma-\delta_{1}}{\gamma^{\prime}-\delta_{1}}}}(a,b)\leq
W_{\mathbf{h},\gamma^{\prime},\delta_{1}}(\tau^{I}_{n,\delta_{1},\omega}(\chi),\tau^{I}_{n+1,\delta_{1},\omega}(\chi))\quad\text{if}\
\tau_{n+1}^{I}(\chi)<b.$

Summing up yields

$\displaystyle\big{[}N\big{(}I,\delta_{1},2^{\frac{1}{\gamma-\delta_{1}}}C_{\delta_{1}}W_{\mathbf{X}(\omega-h),\gamma,\delta_{1}}(a,b),\mathbf{X}(\omega)\big{)}-1\big{]}W_{\mathbf{X}(\omega-h),\gamma,\delta_{1}}^{{\frac{\gamma-\delta_{1}}{\gamma^{\prime}-\delta_{1}}}}(a,b)\leq
W_{\mathbf{h},\gamma^{\prime},\delta_{1}}(a,b).$

Now it is enough to use (1.14) to prove (1.18). For the second claim, let
$\chi>0$. From (1.18)

(1.21)
$\displaystyle\begin{split}&\big{\\{}\omega:N(I,\delta_{1},\chi,\mathbf{X}(\omega))>n\big{\\}}\cap
E\subset\\\
&\quad\Omega\setminus\big{(}\big{\\{}\omega\in\Omega:\frac{\chi}{2^{\frac{1}{\gamma-\delta_{1}}+1}C_{\delta_{1}}}\leq
W_{\mathbf{X}(\omega),\gamma,\delta_{1}}(a,b)\leq\frac{\chi}{2^{\frac{1}{\gamma-\delta_{1}}}C_{\delta_{1}}}\big{\\}}+r_{n}\mathcal{K}\big{)},\end{split}$

where
$r_{n}:=\frac{(n-1)^{\gamma^{\prime}-\delta_{1}}\chi^{\gamma-\delta_{1}}}{{2^{\gamma-\delta_{1}+1}T(\delta_{1})^{\gamma^{\prime}-\delta_{1}}C_{\delta_{1}}^{\gamma-\delta_{1}}}}$
and $\mathcal{K}$ is the unit ball in $\mathcal{H}$. Indeed: Obviously, if
$\chi_{1}<\chi_{2}$, one has $N(I,\delta_{1},\chi_{2},\mathbf{X}(\omega))\leq
N(I,\delta_{1},\chi_{1},\mathbf{X}(\omega))$. Let

$\displaystyle\omega_{1}+h\in\big{(}\big{\\{}\omega\in\Omega:\frac{\chi}{2^{\frac{1}{\gamma-\delta_{1}}+1}C_{\delta_{1}}}\leq
W_{\mathbf{X}(\omega),\gamma,\delta_{1}}(a,b)\leq\frac{\chi}{2^{\frac{1}{\gamma-\delta_{1}}}C_{\delta_{1}}}\big{\\}}+r_{n}\mathcal{K}\big{)}.$

Then from (1.18), setting $\omega=\omega_{1}+h$,

$\displaystyle\big{[}N(I,\delta_{1},\chi,\mathbf{X}(\omega_{1}+h))-1\big{]}(\frac{\chi}{2^{\frac{1}{\gamma-\delta_{1}}+1}C_{\delta_{1}}})^{{\frac{\gamma-\delta_{1}}{\gamma^{\prime}-\delta_{1}}}}\leq(n-1)(\frac{\chi}{2^{\frac{1}{\gamma-\delta_{1}}+1}C_{\delta_{1}}})^{\frac{\gamma-\delta_{1}}{\gamma^{\prime}-\delta_{1}}}.$

Therefore (1.19), follows from (1.21) and the Borell’s inequality (cf.[FV10b,
Theorem D.4]) ∎

We come back now to the initial question of this article. Remember that the
solution $Z$ to (0.1) satisfies

(1.22) $\displaystyle\begin{split}(\delta
Z)_{s,t}&=(S_{t-s}-I)Z_{s}+\int_{s}^{t}S_{t-\tau}F(Z_{\tau})\mathrm{d}\tau+S_{t-s}G(Z_{s})\circ(\delta
X(\omega))_{s,t}+S_{t-s}D_{Z_{s}}G[G(Z_{s})]\circ\mathbb{X}_{s,t}(\omega)\\\
&\quad+\int_{s}^{t}S_{t-\tau}G(Z_{\tau})\circ\mathrm{d}\mathbf{X}_{\tau}(\omega)-S_{t-s}G(Z_{s})\circ(\delta
X)_{s,t}(\omega)-S_{t-s}D_{Z_{s}}G[G(Z_{s})]\circ\mathbb{X}_{s,t}(\omega).\end{split}$

Set $L(x):=\max\\{x,x^{2}\\}$ and assume for $I=[s,t]$ that $t-s\leq 1$. From
Lemma 1.5 and Assumption 0.2, we can conclude for an $M_{\epsilon}>1$ which is
independent from $\mathbf{X}$ it holds that

(1.23)
$\displaystyle\begin{split}\|Z\|_{\mathcal{D}_{\mathbf{X},{\alpha}}^{\gamma}([u,v])}&\leq
M_{\epsilon}\big{[}|Z_{u}|_{\alpha}+(v-u)^{1-\sigma}\|Z\|_{\mathcal{D}_{\mathbf{X}(\omega),{\alpha}}^{\gamma}([u,v])}\\\
&\quad+\|X\|_{\gamma;I}\big{(}\|X(\omega)\|^{2}_{\gamma;I}+\|\mathbb{X}(\omega)\|_{2\gamma;I}\big{)}+\|X(\omega)\|_{\gamma;I}+\|\mathbb{X}(\omega)\|_{2\gamma;I}\\\
&\quad+L\big{(}W_{\mathbf{X}(\omega),\gamma,\delta+\epsilon}(u,v)^{\gamma-\delta-\epsilon}\big{)}\|Z\|_{\mathcal{D}_{\mathbf{X}(\omega),{\alpha}}^{\gamma}([u,v])}\big{]},\end{split}$

where $[u,v]\subset[s,t]$. Let us choose $0<\chi<1$ such that
$M_{\epsilon}\chi^{\gamma-\delta-\epsilon}\leq\frac{1}{4}$. We will assume
further that $M_{\epsilon}(t-s)^{1-\sigma}\leq\frac{1}{4}$ and that
$\\{\tau^{I}_{n,\delta_{1},\omega}(\chi)\\}_{n\geq 0}$ are the greedy points
that are defined in Definition 1.10 with $\delta_{1}=\delta+\epsilon$. From
(1.23),

$\displaystyle\|Z\|_{\mathcal{D}_{\mathbf{X}(\omega),{\alpha}}^{\gamma}([\tau_{n},\tau_{n+1}])}\leq
2M_{\epsilon}|Z_{\tau_{n}}|_{\alpha}+2M_{\epsilon}P(\|X(\omega)\|_{\gamma;I},\|\mathbb{X}(\omega)\|_{2\gamma;I}),$

where
$P(\|X(\omega)\|_{\gamma},\|\mathbb{X}(\omega)\|_{\gamma})=1+\|\mathbb{X}(\omega)\|_{\gamma}+\|X(\omega)\|_{\gamma}(\|X(\omega)\|_{\gamma}^{2}+\|\mathbb{X}(\omega)\|_{2\gamma})$.
Therefore, for $\tilde{M}_{\epsilon}\coloneqq\log(2M_{\epsilon})$,

(1.24)
$\displaystyle\begin{split}\sup_{\tau\in[s,t]}|Z_{\tau}|_{\alpha}&\leq\exp\big{(}N(I,\delta_{1},\chi,\mathbf{X}(\omega))\tilde{M}_{\epsilon}\big{)}|Z_{s}|_{\alpha}\\\
&\quad+\frac{\exp\big{(}N(I,\delta_{1},\chi,\mathbf{X}(\omega))\tilde{M}_{\epsilon}+\tilde{M}_{\epsilon}\big{)}-1}{2M_{\epsilon}-1}P(\|X(\omega)\|_{\gamma;I},\|\mathbb{X}(\omega)\|_{\gamma;I}).\end{split}$

We can now summarize our main result in the following theorem:

######  Theorem 1.12.

Suppose that
$F\colon\mathcal{B}_{\alpha}\rightarrow\mathcal{B}_{\alpha-\sigma}$ is a
Lipschitz continuous function and
$G\colon\mathcal{B}_{\alpha-\theta}\rightarrow\mathcal{B}_{\alpha-\theta-\delta}$
be a bounded Fréchet differentiable function with 3 bounded derivatives.
Assume that $Z$ solves

(1.25)
$\displaystyle\mathrm{d}Z_{t}=AZ_{t}\mathrm{d}t+F(Z_{t})\mathrm{d}t+G(Z_{t})\circ\mathrm{d}\mathbf{X}_{t},\
\ \ \ Z_{0}=\xi\in\mathcal{B}_{\alpha}.$

Then the following holds:

  * (i)

(1.25) admits a unique and global solution $Z$ such that for
$\delta_{1}=\delta+\epsilon$, one has

(1.26)
$\displaystyle\begin{split}\sup_{\tau\in[s,t]}|Z_{\tau}|_{\alpha}&\leq\|Z\|_{\mathcal{D}_{\mathbf{X},{\alpha}}^{\gamma}([s,t])}\\\
&\leq\exp\big{(}N([s,t],\delta_{1},\chi,\mathbf{X})\tilde{M}_{\epsilon}\big{)}|Z_{s}|_{\alpha}\\\
&\quad+\frac{\exp\big{(}N([s,t],\delta_{1},\chi,\mathbf{X})\tilde{M}_{\epsilon}+\tilde{M}_{\epsilon}\big{)}-1}{2M_{\epsilon}-1}P(\|X\|_{\gamma,[s,t]},\|\mathbb{X}\|_{\gamma,[s,t]}),\end{split}$

where $M_{\epsilon}>1$,
$M_{\epsilon}\chi^{\gamma-\sigma-\epsilon}\leq\frac{1}{4},\
M_{\epsilon}(t-s)^{1-\sigma}\leq\frac{1}{4}$ and
$\tilde{M}_{\epsilon}:=\log(2M_{\epsilon})$. In addition
$P(x,y)=1+y+x(x^{2}+y)$.

  * (ii)

Let $(\mathcal{W},\mathcal{H},\mu)$ be an abstract Wiener space and assume
that $X$ is a Gaussian process defined on it that can be enhanced to a weakly
$\gamma$-geometric rough path $\mathbf{X}=(X,\mathbb{X})$,
$\frac{1}{3}<\gamma\leq\frac{1}{2}$. In addition, let $\gamma^{\prime}>0$
satisfy $\gamma+\gamma^{\prime}-2\delta>1$, and for every $h\in\mathcal{H}$,
for $\delta_{1}=\delta+\epsilon$ such that
$0<\epsilon<\frac{\gamma+\gamma^{\prime}-2\delta-1}{2}$, we assume that the
Conditions (1.9) and (1.14) hold. Then

(1.27)
$\displaystyle\exp\big{(}N([s,t],\delta_{1},\chi,\mathbf{X})\tilde{M}_{\epsilon}\big{)}P(\|X\|_{\gamma,[s,t]},\|\mathbb{X}\|_{\gamma,[s,t]})\in\mathcal{L}^{p}(\Omega)$

for every $p>0$.

###### Proof.

The first item is proved in (1.24). For the integrability claim, remember that
from (1.19), we know that

$\displaystyle\mu\big{\\{}\omega:N(I,\delta_{1},\chi,\mathbf{X}(\omega))>n\big{\\}}\leq
M_{1}(\delta_{1},\chi)\exp(-M_{2}(\delta_{1},\chi)n^{2(\gamma^{\prime}-\delta_{1})}).$

Since by assumption $\gamma+\gamma^{\prime}-2\delta_{1}>1$, we can easily
conclude $2(\gamma^{\prime}-\delta_{1})>1$. This proves integrability of
$\exp\big{(}N([s,t],\delta_{1},\chi,\mathbf{X})\tilde{M}_{\epsilon}\big{)}$.
Since $P$ is a polynomial term and $\mathbf{X}$ is a Gaussian process, the
integrability for every moment is clear. Finally, the integrability of the
product of these two terms is a straightforward consequence of Hölder’s
inequality. ∎

## References

  * [Arn98] Ludwig Arnold.  Random dynamical systems.  Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.
  * [CF10] Thomas Cass and Peter K. Friz.  Densities for rough differential equations under Hörmander’s condition.  Ann. of Math. (2), 171(3):2115–2141, 2010.
  * [CLL13] Thomas Cass, Christian Litterer, and Terry J. Lyons.  Integrability and tail estimates for Gaussian rough differential equations.  Ann. Probab., 41(4):3026–3050, 2013.
  * [CQ02] Laure Coutin and Zhongmin Qian.  Stochastic analysis, rough path analysis and fractional Brownian motions.  Probab. Theory Related Fields, 122(1):108–140, 2002.
  * [FGGR16] Peter K. Friz, Benjamin Gess, Archil Gulisashvili, and Sebastian Riedel.  The Jain-Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory.  Ann. Probab., 44(1):684–738, 2016.
  * [FH20] Peter K. Friz and Martin Hairer.  A course on rough paths.  Universitext. Springer, Cham, second edition, [2020] ©2020.  With an introduction to regularity structures.
  * [FR13] Peter K. Friz and Sebastian Riedel.  Integrability of (non-)linear rough differential equations and integrals.  Stoch. Anal. Appl., 31(2):336–358, 2013.
  * [FV06] Peter K. Friz and Nicolas B. Victoir.  A variation embedding theorem and applications.  J. Funct. Anal., 239(2):631–637, 2006.
  * [FV10a] Peter K. Friz and Nicolas B. Victoir.  Differential equations driven by Gaussian signals.  Ann. Inst. Henri Poincaré Probab. Stat., 46(2):369–413, 2010\.
  * [FV10b] Peter K. Friz and Nicolas B. Victoir.  Multidimensional stochastic processes as rough paths, volume 120 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2010.  Theory and applications.
  * [GH19] Andris Gerasimovičs and Martin Hairer.  Hörmander’s theorem for semilinear spdes.  Electron. J. Probab., 24(32):1–56, 2019.
  * [GHN21] Andris Gerasimovičs, Antoine Hocquet, and Torstein Nilssen.  Non-autonomous rough semilinear pdes and the multiplicative sewing lemma.  J. Funct. Anal., 281(10), 2021.
  * [HN22] Robert Hesse and Alexandra Neamtu.  Global solutions for semilinear rough partial differential equations.  Stochastics and Dynamics, 22(02), 2022.
  * [KN23] Christian Kuehn and Alexandra Neamtu.  Center manifolds for rough partial differential equations.  Electron. J. Probab., 2023.
  * [Lyo98] Terry J. Lyons.  Differential equations driven by rough signals.  Rev. Mat. Iberoamericana, 14(2):215–310, 1998.
  * [YLZ23] Qigui Yang, Xiaofang Lin, and Caibin Zeng.  Random attractors for rough stochastic partial differential equations.  Journal of Differential Equations, 371:50–82, 2023.