# Arithmetic properties of equivariant birational types

Andrew Kresch  Institut für Mathematik, Universität Zürich,
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
<EMAIL_ADDRESS>and  Yuri Tschinkel  Courant Institute, 251 Mercer
Street, New York, NY 10012, USA<EMAIL_ADDRESS>Simons Foundation
160 Fifth Avenue
New York, NY 10010
USA

(Date: December 7, 2020)

###### Abstract.

We study arithmetic properties of equivariant birational types introduced by
Kontsevich, Pestun, and the second author.

##  1\. Introduction

Let $G$ be a finite abelian group and $k$ an algebraically closed field of
characteristic zero. Investigations of obstructions to $G$-equivariant
birationality over $k$ led to the definition, in [2], of new invariants of
actions of $G$ on algebraic varieties $X$ defined over $k$. These invariants
were further developed in [4], where specialization maps were defined,
generalizing the ones from the non-equivariant setting [3].

The invariants from [2] are computed on a suitable smooth projective model
$X$, where $G$ acts regularly. To such an action one associates a class

$[X\righttoleftarrow G]:=\sum_{\alpha}\beta_{\alpha},$ (1.1)

where the sum is over components of the fixed point locus $F_{\alpha}\subset
X^{G}$, and $\beta_{\alpha}$ are the characters of $G$ appearing in the
tangent bundle to a point $x_{\alpha}\in F_{\alpha}$. Equivariant birational
maps can be factored into sequences of blowups (and blowdowns) of smooth
$G$-stable subvarieties, thanks to Equivariant Weak Factorization. To obtain
an invariant, one imposes relations on the formal sums in (1.1), of the type

$[\tilde{X}\righttoleftarrow G]-[X\righttoleftarrow G]=0,$

for every equivariant blowup $\tilde{X}\to X$.

This construction motivated the introduction of two closely related quotients
of the free abelian group ${\mathcal{S}}_{n}(G)$, generated by symbols

$\beta=[a_{1},\ldots,a_{n}]=[a_{\sigma(1)},\ldots,a_{\sigma(n)}],\quad\forall\sigma\in\mathfrak{S}_{n},$
(1.2)

where $\beta$ is an $n$-dimensional faithful representation of $G$ over $k$,
i.e., a collection of characters $a_{1},\ldots,a_{n}$ of $G$, up to
permutation, spanning the character group of $G$.

We have a diagram

$\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
15.27858pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{S}}_{n}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
23.23691pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\mathsf{b}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 40.32024pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
40.32024pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{B}}_{n}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
56.36272pt\raise-20.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\mu}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 56.36272pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{S}}_{n}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
22.26468pt\raise-35.49306pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\mathsf{m}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 39.27858pt\raise-40.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
39.27858pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{M}}_{n}(G)}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(1.3)

Here, the projection $\mathsf{b}$ is the quotient by the relation:

  * ($\mathrm{\bf B}$)

Blow-up: for all $[a_{1},a_{2},b_{1},\ldots,b_{n-2}]\in{\mathcal{S}}_{n}(G)$
one has

$\displaystyle\begin{split}[&a_{1},a_{2},b_{1},\ldots,b_{n-2}]=\\\
&\begin{cases}[a_{1},a_{2}-a_{1},b_{1},\ldots,b_{n-2}]+[a_{1}-a_{2},a_{2},b_{1},\ldots,b_{n-2}],&a_{1}\neq
a_{2},\\\ [0,a_{1},b_{1},\ldots,b_{n-2}],&a_{1}=a_{2}.\end{cases}\end{split}$
(1.4)

One of the main results in [2] is the following

######  Theorem 1.1.

Let $X$ be a smooth projective algebraic variety of dimension $n$ over $k$,
with a regular action of $G$. The class

$[X\righttoleftarrow G]\in{\mathcal{B}}_{n}(G)$

is a well-defined $G$-equivariant birational invariant.

Numerical experiments revealed an interesting structure of the other quotient
of ${\mathcal{S}}_{n}(G)$, namely, via the projection $\mathsf{m}$ in (1.3),
which is defined as quotient by the relation:

  * ($\mathrm{\bf M}$)

Modular blow-up: for all
$[a_{1},a_{2},b_{1},\ldots,b_{n-2}]\in{\mathcal{S}}_{n}(G)$ one has

$\displaystyle\begin{split}[&a_{1},a_{2},b_{1},\ldots,b_{n-2}]=\\\
&\,\,\,\,\,\,[a_{1},a_{2}-a_{1},b_{1},\ldots,b_{n-2}]+[a_{1}-a_{2},a_{2},b_{1},\ldots,b_{n-2}].\end{split}$
(1.5)

To distinguish, we write

$[a_{1},\ldots,a_{n}],\quad\text{ respectively, }\quad\langle
a_{1},\ldots,a_{n}\rangle,$

for the image of a generator in ${\mathcal{B}}_{n}(G)$, respectively, the
image of a generator in ${\mathcal{M}}_{n}(G)$.

When $a_{1}\neq a_{2}$, the relations are identical; the only difference is

$\displaystyle[a_{1},a_{1},\ldots,a_{n}]$
$\displaystyle=[a_{1},0,\ldots,a_{n}]\,\,\,\,\,\in{\mathcal{B}}_{n}(G)$
$\displaystyle\langle a_{1},a_{1},\ldots,a_{n}\rangle$ $\displaystyle=2\langle
a_{1},0,\ldots,a_{n}\rangle\in{\mathcal{M}}_{n}(G).$

The homomorphism in (1.3)

$\mu:{\mathcal{B}}_{n}(G)\to{\mathcal{M}}_{n}(G),\quad n\geq 2,$ (1.6)

is defined on symbols by:

$\mu([a_{1},\ldots,a_{n}]):=\begin{cases}\langle
a_{1},\ldots,a_{n}\rangle&\text{ if all }a_{i}\neq 0,\\\ 2\langle
a_{1},\ldots,a_{n}\rangle&\text{ if exactly one }a_{i}=0,\\\ 0&\text{
otherwise}.\end{cases}$

In [2] it was shown that this map on symbols is compatible with relations.

### Antisymmetry

We have a diagram of homomorphisms

$\textstyle{{\mathcal{B}}_{n}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{{\mathcal{B}}^{-}_{n}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{-}}$$\textstyle{{\mathcal{M}}_{n}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{M}}_{n}^{-}(G)}$

where the horizontal maps are projections to the corresponding quotients by
the additional relation

$[-a_{1},\ldots,a_{n}]=-[a_{1},\ldots,a_{n}],$

defined only for nontrivial $G$. On symbols, the map $\mu^{-}$ is the same as
$\mu$; its compatibility with defining relations is obvious.

In this note, we prove a comparison, left open in [2, Conjecture 8]:

######  Theorem 1.2.

Both homomorphisms $\mu$ and $\mu^{-}$ are isomorphisms, after tensoring with
${\mathbb{Q}}$.

This implies that the main constructions connected with
${\mathcal{M}}_{n}(G)$, from Sections 4,5,6, and 9 of [2], also apply to
${\mathcal{B}}_{n}(G)\otimes{\mathbb{Q}}$. We briefly sketch these structures:

  * •

Lattices and cones: elements $\langle
a_{1},\ldots,a_{n}\rangle\in{\mathcal{M}}_{n}(G)$ can be identified with
isomorphism classes of triples

$(\mathbf{L},\chi,\Lambda),$ (1.7)

where $\mathbf{L}={\mathbb{Z}}^{n}$ is a lattice, $\chi\in\mathbf{L}\otimes
A$, and $\Lambda\subset\mathbf{L}\otimes{\mathbb{R}}$ is a basic simplicial
cone. Here $A$ denotes the character group of $G$, and by a basic simplicial
cone we mean one that is spanned by a basis of $\mathbf{L}$. Concretely,
choosing a basis $e_{1},\ldots,e_{n}$ of lattice vectors spanning $\Lambda$,
one can write

$\chi=\sum_{i=1}^{n}\,e_{i}\otimes a_{i},\quad a_{i}\in A,$

and put

$(\mathbf{L},\chi,\Lambda)\mapsto\langle a_{1},\ldots,a_{n}\rangle.$

Changing the basis spanning $\Lambda$ permutes the entries
$a_{1},\ldots,a_{n}$, and relation $(\mathbf{M})$ arises from decompositions
of a simplicial cone into simplicial subcones. We will discuss this in more
detail in Section 3.

  * •

Operations: Given an exact sequence of groups

$0\rightarrow G^{\prime}\rightarrow G\rightarrow G^{\prime\prime}\rightarrow
0$

there is a ${\mathbb{Z}}$-bilinear multiplication homomorphism

$\nabla:{\mathcal{M}}_{n^{\prime}}(G^{\prime})\otimes{\mathcal{M}}_{n^{\prime\prime}}(G^{\prime\prime})\rightarrow{\mathcal{M}}_{n^{\prime}+n^{\prime\prime}}(G),\quad
n^{\prime},n^{\prime\prime}\geq 1,$

which descends to the antisymmetric versions, as well as a co-multiplication
homomorphism

$\Delta:{\mathcal{M}}_{n^{\prime}+n^{\prime\prime}}(G)\rightarrow{\mathcal{M}}_{n^{\prime}}(G^{\prime})\otimes{\mathcal{M}}_{n^{\prime\prime}}^{-}(G^{\prime\prime}),$

(the minus on the second factor is not an error), which also comes with an
antisymmetric version

$\Delta^{-}:{\mathcal{M}}_{n^{\prime}+n^{\prime\prime}}^{-}(G)\rightarrow{\mathcal{M}}_{n^{\prime}}^{-}(G^{\prime})\otimes{\mathcal{M}}_{n^{\prime\prime}}^{-}(G^{\prime\prime}).$

These homomorphisms allow to decompose ${\mathcal{M}}_{n}(G)$ into primitive
pieces, and reveal a rich internal structure.

  * •

Hecke operators: The lattice-theoretic interpretation of
${\mathcal{M}}_{n}(G)$ leads to the definition of commuting operators

$T_{\ell,r}:{\mathcal{M}}_{n}(G)\otimes{\mathbb{Q}}\to{\mathcal{M}}_{n}(G)$

for all $1\leq r\leq n-1$ and primes $\ell$ not dividing the order of $G$. By
Theorem 1.2, the groups ${\mathcal{B}}_{n}(G)\otimes{\mathbb{Q}}$ also carry
Hecke operators.

  * •

Cohomology of arithmetic groups: Let

$\Gamma(G,n)\subset\mathrm{GL}_{n}({\mathbb{Z}})$

be the stabilizer of $\chi$ in (1.7). Let

    * –

$\mathcal{F}_{n}$ be the ${\mathbb{Q}}$-vector space generated by
characteristic functions of convex finitely generated rational polyhedral
cones $\Lambda\subset{\mathbb{R}}^{n}$, modulo those of dimension $\leq n-1$,

    * –

$\mathrm{St}_{n}$ be the Steinberg-module, and

    * –

$\mathrm{or}_{n}$ be the sign of the determinant module.

By [2, Prop. 22] and Theorem 1.2, we have a commutative diagram

$\textstyle{\mathcal{M}_{n}(G)\otimes{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\mathcal{M}_{n}^{-}(G)\otimes{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{H_{0}(\Gamma(G,n),\mathcal{F}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{0}(\Gamma(G,n),\mathrm{St}_{n}\otimes\mathrm{or}_{n})}$$\textstyle{\mathcal{B}_{n}(G)\otimes{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\mathcal{B}_{n}^{-}(G)\otimes{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\simeq}$

The connection between the groups ${\mathcal{B}}_{n}(G)\otimes{\mathbb{Q}}$,
encoding invariants of abelian actions on algebraic varieties, and the theory
of automorphic forms, via cohomology of congruence subgroups, seems intriguing
to us. However, given the link between ${\mathcal{B}}_{n}(G)$ and
${\mathcal{M}}_{n}(G)$ it is natural to seek a lattice theoretic
interpretation of ${\mathcal{B}}_{n}(G)$ as well. This is done in Section 3.
One of the byproducts is the definition of Hecke operators

$T_{\ell,r}:{\mathcal{B}}_{n}(G)\to{\mathcal{B}}_{n}(G),$

where $\ell$ is a prime not dividing the order of $G$ and $1\leq r\leq n-1$,
over the integers.

Acknowledgments: We are very grateful to Brendan Hassett for his interest and
help on this and related projects. The first author was partially supported by
the Swiss National Science Foundation. The second author was partially
supported by NSF grant 2000099.

##  2\. Comparison

We continue to assume that $G$ is a finite abelian group. This section is
closely related to [2, Sections 3, 5, and 11]. In particular, we settle
Conjecture 8 from ibid, asserting that

${\mathcal{B}}_{n}(G)\otimes{\mathbb{Q}}\simeq{\mathcal{M}}_{n}(G)\otimes{\mathbb{Q}}.$

Our first result is a refinement of [1, Prop. 3.2].

######  Theorem 2.1.

Let $n\geq 2$.
$\mathrm{(i)}$ Let $p$ be a prime and
$a\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}$. The class

$[a,0,\ldots]+[-a,0,\ldots]\in{\mathcal{B}}_{n}({\mathbb{Z}}/p{\mathbb{Z}})$

is zero when $p\leq 5$, and is annihilated by $(p^{2}-1)/24$ when $p\geq 7$.
$\mathrm{(ii)}$ Let $N>1$ be an integer and
$a\in({\mathbb{Z}}/N{\mathbb{Z}})^{\times}$. Then

$[a,0,\ldots]+[-a,0,\ldots]\in{\mathcal{B}}_{n}({\mathbb{Z}}/N{\mathbb{Z}})_{\rm
tors},$

the subgroup of torsion elements.

We start with a sequence of technical lemmas.

######  Lemma 2.2.

For $a$, $b\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}$ we have

$[a,b]+[a,-b]=[a,0]\in{\mathcal{B}}_{2}({\mathbb{Z}}/p{\mathbb{Z}}).$

###### Proof.

We write $b=ma$ with $m\in\\{1,\dots,p-1\\}$ and proceed by induction on $m$.
The base case $m=1$ is clear, since

$[a,a]=[a,0]\qquad\text{and}\qquad[a,-a]=0.$

The induction hypothesis, in combination with

$[a,(m+1)a]=[a,ma]+[(m+1)a,-ma]$

and

$[a,-ma]=[a,-(m+1)a]+[(m+1)a,-ma],$

gives the inductive step. ∎

######  Lemma 2.3.

For $a$, $b\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}$ we have

$[a,0]+[-a,0]=[a,b]+[a,-b]+[-a,b]+[-a,-b]$

in ${\mathcal{B}}_{2}({\mathbb{Z}}/p{\mathbb{Z}})$, and this element is
independent of $a$ and $b$.

###### Proof.

The equality holds by Lemma 2.2. The right-hand side is symmetric in $a$ and
$b$ and, by the equality, is independent of $b$. Hence it is also independent
of $a$. ∎

######  Lemma 2.4.

For $a$, $b\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}$ with $a+b\neq 0$, we have

$[a,0]=[a,b]+[-b,a+b]+[-a-b,a]\in{\mathcal{B}}_{2}({\mathbb{Z}}/p{\mathbb{Z}}).$

###### Proof.

This follows from

$[a,-b]=[-b,a+b]+[-a-b,a]$

and $[a,b]+[a,-b]=[a,0]$. ∎

Lemma 2.3 tells us that

$\delta:=[a,0]+[-a,0]\in{\mathcal{B}}_{2}({\mathbb{Z}}/p{\mathbb{Z}})$ (2.1)

is independent of $a\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}$.

######  Lemma 2.5.

For $a$, $b\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}$ with $a+b\neq 0$, we have

$\displaystyle\delta$ $\displaystyle=[a,b]+[-b,a+b]+[-a-b,a]$
$\displaystyle\,\,\,\,\,\,\,\,+[-a,-b]+[b,-a-b]+[a+b,-a]\in{\mathcal{B}}_{2}({\mathbb{Z}}/p{\mathbb{Z}}).$

###### Proof.

We add together

$[a,0]=[a,b]+[-b,a+b]+[-a-b,a]$

and

$[-a,0]=[-a,-b]+[b,-a-b]+[a+b,-a],$

and recognize $\delta$ on the left-hand side. ∎

######  Lemma 2.6.

We have in ${\mathcal{B}}_{2}({\mathbb{Z}}/p{\mathbb{Z}})$:

$\displaystyle\sum_{a\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}}[a,a]$
$\displaystyle=\frac{p-1}{2}\delta,$
$\displaystyle\sum_{a\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}}[a,-2a]$
$\displaystyle=0.$

###### Proof.

We pair summands indexed by $a$ and $-a$ and use $[a,a]=[a,0]$ and the
definition of $\delta$ to get the first equality. Also, from

$[a,0]=[a,a]+[-a,2a]+[-2a,a]$

follows the vanishing of pairs of summands in the second equality. ∎

######  Lemma 2.7.

Let $\beta$, $\beta^{\prime}$,
$\beta^{\prime\prime}\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}\setminus\\{-1\\}$
with

$\beta^{\prime}=-\beta^{-1}-1\qquad\text{and}\qquad\beta^{\prime\prime}=-(\beta+1)^{-1}.$

Then

$\sum_{a\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}}[a,\beta
a]+[a,\beta^{\prime}a]+[a,\beta^{\prime\prime}a]=\frac{p-1}{2}\delta.$

Furthermore, if $\beta=\beta^{\prime}=\beta^{\prime\prime}$ then

$\sum_{a\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}}[a,\beta
a]=\frac{p-1}{6}\delta.$

###### Proof.

We identify pairs of summands in the first expression with $\delta$. We have
$\beta=\beta^{\prime}=\beta^{\prime\prime}$ if and only if $\beta$ is a
primitive cube root of unity. Then we may identify $6$-tuples of summands with
$\delta$ to get the second equality. ∎

###### Proof of Theorem 2.1.

For $\mathrm{(i)}$, let $p\geq 5$. We partition
$({\mathbb{Z}}/p{\mathbb{Z}})^{\times}\setminus\\{1,-1\\}$ into
$\\{-2,-1/2\\}$, the primitive cube roots of unity (which exist only when
$p\equiv 1$ mod $3$), and $6$-element sets

$\\{\beta,\beta^{\prime},\beta^{\prime\prime},\beta^{-1},\beta^{\prime-1},\beta^{\prime\prime-1}\\},$

with distinct $\beta$, $\beta^{\prime}$, $\beta^{\prime\prime}$ as above. We
take a subset

$I\subset({\mathbb{Z}}/p{\mathbb{Z}})^{\times}\setminus\\{1,-1\\},$

to consist of one of $-2$, $-1/2$, one primitive cube root of unity if it
exists, and $\beta$, $\beta^{\prime}$, $\beta^{\prime\prime}$ from every
$6$-element set as above. Then $\delta$ from (2.1) satisfies

$\displaystyle\frac{(p-1)(p-2)}{6}\delta$
$\displaystyle=\sum_{\beta=1}^{p-3}\sum_{a\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}}[a,\beta
a]$ $\displaystyle=\sum_{\beta\in
I}\sum_{a\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}}[a,(\beta-1)a]+[a,(\beta^{-1}-1)a]$
$\displaystyle=\sum_{\beta\in
I}\sum_{a\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}}[a,\beta a]$
$\displaystyle=\frac{(p-1)(p-5)}{12}\delta.$

It follows that $\delta$ is annihilated by $(p^{2}-1)/12$. Next we prove
annihilation by $(p^{2}-1)/8$, and thus by $(p^{2}-1)/24$ as claimed. This is
an adaptation of the previous argument: take

$J\subset({\mathbb{Z}}/p{\mathbb{Z}})^{\times}\setminus\\{1,-1\\}$

to consist of one square root of $-1$ when $p\equiv 1$ mod $4$ as well as
$\beta$ and $-\beta$ from every $4$-element set

$\\{\beta,-\beta,\beta^{-1},-\beta^{-1}\\}.$

From

$\displaystyle\frac{(p-1)^{2}}{4}\delta$
$\displaystyle=\sum_{\beta=1}^{p-3}\sum_{a\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}}[a,\beta
a]$ $\displaystyle=\sum_{\beta\in
J}\sum_{a\in({\mathbb{Z}}/p{\mathbb{Z}})^{\times}}[a,\beta
a]=\frac{(p-1)(p-3)}{8}\delta$

we get the desired conclusion.

For $\mathrm{(ii)}$, we treat composite $N$, as in the proof of [1, Prop.
3.2]: We recall that, for $a,b$ with $\gcd(a,b,N)=1$, we have

$\langle a,b\rangle=\begin{cases}[a,b]&\text{ when both }a,b\neq 0,\\\
\frac{1}{2}[a,0]&\text{ when }b=0.\end{cases}$

In this case, we work with

$\delta(a,b):=\langle a,b\rangle+\langle-a,b\rangle+\langle
a,-b\rangle+\langle-a,-b\rangle\in{\mathcal{B}}_{2}({\mathbb{Z}}/N{\mathbb{Z}}).$

We observe that $\delta(a,b)$ satisfies the blow-up relation ($\mathbf{M}$),
thus

$S:=\sum_{a,b}\delta(a,b)=2S.$

It follows that $S=0$ in ${\mathcal{B}}_{2}({\mathbb{Z}}/N{\mathbb{Z}})$. On
the other hand, $\delta(a,b)$ is seen to be invariant under
$\mathrm{SL}_{2}({\mathbb{Z}}/N{\mathbb{Z}})$. This implies that $\delta(a,b)$
is torsion in ${\mathcal{B}}_{2}({\mathbb{Z}}/N{\mathbb{Z}})$ (annihilated by
the number of summands in $S$). Substituting $b=0$, we $[a,0]+[-a,0]=0$ in
${\mathcal{B}}_{2}({\mathbb{Z}}/N{\mathbb{Z}})\otimes{\mathbb{Q}}$. ∎

The following theorem settles Conjectures 8 and 9 of [2]:

######  Theorem 2.8.

Let $n\geq 3$.
$\mathrm{(i)}$ Let $p$ be a prime. Then

$[0,0,1,\dots]\in{\mathcal{B}}_{n}({\mathbb{Z}}/p{\mathbb{Z}})$

is zero when $p\leq 5$, and is annihilated by $(p^{2}-1)/24$ when $p\geq 7$.
$\mathrm{(ii)}$ Let $G$ be a finite abelian group. Any element of the form

$[0,0,\dots]\in{\mathcal{B}}_{n}(G)$

is a torsion element.

###### Proof.

For $\mathrm{(ii)}$ it suffices to consider cyclic
$G={\mathbb{Z}}/N{\mathbb{Z}}$. Theorem 2.1 $\mathrm{(ii)}$ gives, for
$a\in({\mathbb{Z}}/N{\mathbb{Z}})^{\times}$, that

$[a,0,c\ldots]+[-a,0,c,\ldots]$

is torsion in ${\mathcal{B}}_{n}({\mathbb{Z}}/N{\mathbb{Z}})$. Substituting
$c=a$, and using that

$[a,0,a,\dots]=[0,0,a,\dots]\in{\mathcal{B}}_{n}({\mathbb{Z}}/N{\mathbb{Z}})$

and

$[-a,0,a,\ldots]=0\in{\mathcal{B}}_{n}({\mathbb{Z}}/N{\mathbb{Z}})$

we obtain the result. We obtain $\mathrm{(i)}$ similarly, from Theorem 2.1
$\mathrm{(i)}$. ∎

###### Proof of Theorem 1.2.

The assertion for $\mu^{-}$ follows immediately from the vanishing of all
$[0,a_{2},\ldots,a_{n}]$, respectively $\langle 0,a_{2},\ldots,a_{n}\rangle$
in ${\mathcal{B}}_{n}^{-}(G)\otimes{\mathbb{Q}}$, respectively
${\mathcal{M}}_{n}^{-}(G)\otimes{\mathbb{Q}}$.

To obtain the assertion for $\mu$, we combine Theorem 2.8 $\mathrm{(ii)}$ with
analogous relations in ${\mathcal{M}}_{n}(G)$, stated at the beginning of
Section 3 of [2], to show directly that $\mu$ induces an isomorphism after
tensoring with ${\mathbb{Q}}$. ∎

##  3\. Interpretation via lattices

As before, $G$ is a finite abelian group $G$; we denote by $A$ the character
group of $G$. Our starting point is the free abelian group on triples

$(\mathbf{L},\chi,\Lambda),$

where

  * •

$\mathbf{L}\simeq{\mathbb{Z}}^{n}$ is an $n$-dimensional lattice,

  * •

$\chi\in\mathbf{L}\otimes A$ is an element inducing, by duality, a surjection
$\mathbf{L}^{\vee}\to A$,

  * •

$\Lambda$ is a basic cone, i.e., a simplicial cone spanned by a basis of
$\mathbf{L}$.

Let $\mathbf{T}$ be the quotient of this group by the equivalence relation:
two triples are equivalent if they differ by the action of
$\mathrm{GL}_{n}({\mathbb{Z}})$. There is a natural map

$\begin{array}[]{ccc}\mathbf{T}&\to&{\mathcal{S}}_{n}(G),\\\
(\mathbf{L},\chi,\Lambda)&\mapsto&[a_{1},\ldots,a_{n}],\end{array}$

defined by decomposing

$\chi=\sum_{i=1}^{n}e_{i}\otimes a_{i},\quad a_{i}\in A,$ (3.1)

where $\\{e_{1},\ldots,e_{n}\\}$ is a basis of $\Lambda$. The symmetry
property (1.2) is precisely the ambiguity in the order of generating elements
of $\Lambda$. Imposing scissor-type relations [2, (4.4)] on $\mathbf{T}$, we
obtain a diagram

---
$\textstyle{\mathbf{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{\mathsf{s}}$$\textstyle{\mathcal{M}_{n}(G)}$$\textstyle{\mathbf{T}/(\text{scissor-
type
relations})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$

We propose a similar group $\widetilde{\mathbf{T}}$, based on triples

$(\mathbf{L},\chi,\Lambda^{\prime}),$

where now $\Lambda^{\prime}$ is a smooth cone of _arbitrary_ dimension (i.e.,
one spanned by part of a basis of $\mathbf{L}$), such that when we let
$\mathbf{L}^{\prime}$ denote the sublattice of $\mathbf{L}$ spanned by
$\Lambda^{\prime}$, we have

$\chi\in\mathrm{Im}(\mathbf{L}^{\prime}\otimes A\to\mathbf{L}\otimes A).$
(3.2)

Again, we impose the relations coming from the evident
$\mathrm{GL}_{n}({\mathbb{Z}})$-action. There is a natural map

$\begin{array}[]{ccl}\widetilde{\mathbf{T}}&\to&{\mathcal{S}}_{n}(G),\\\
(\mathbf{L},\chi,\Lambda^{\prime})&\mapsto&[a_{1},\ldots,a_{n}].\end{array}$

We introduce Subdivision relations on $\widetilde{\mathbf{T}}$:

  * ($\mathrm{\bf S}$)

for a face $\Lambda^{\prime\prime}$ of $\Lambda^{\prime}$ of dimension at
least $2$,

$\Lambda^{\prime\prime}={\mathbb{R}}_{\geq 0}\langle
e_{1},\dots,e_{r}\rangle\subset\Lambda^{\prime}={\mathbb{R}}_{\geq 0}\langle
e_{1},\dots,e_{s}\rangle,$

consider the star subdivision
$\Sigma^{*}_{\Lambda^{\prime}}(\Lambda^{\prime\prime})$, consisting of the
$2^{r}-1$ cones spanned by $e_{1}+\dots+e_{r}$, $e_{r+1}$, $\dots$, $e_{s}$,
and all proper subsets of $\\{e_{1},\dots,e_{r}\\}$. Then

$\displaystyle(\mathbf{L},\chi,\Lambda^{\prime})$
$\displaystyle=\sum_{\begin{subarray}{c}\widetilde{\Lambda}^{\prime}\in\Sigma^{*}_{\Lambda^{\prime}}(\Lambda^{\prime\prime})\\\
\chi\in\mathrm{Im}(\widetilde{\mathbf{L}}^{\prime}\otimes
A\to\mathbf{L}\otimes
A)\end{subarray}}(-1)^{\dim(\Lambda^{\prime})-\dim(\widetilde{\Lambda}^{\prime})}(\mathbf{L},\chi,\widetilde{\Lambda}^{\prime}),$
(3.3) $\displaystyle(\mathbf{L},\chi,\Lambda^{\prime})$
$\displaystyle=(\mathbf{L},\chi,\Lambda),$ (3.4)

for a basic cone $\Lambda$, having $\Lambda^{\prime}$ as a face.

We have:

$\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
58.09732pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\widetilde{\mathbf{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
44.62372pt\raise-4.69444pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\tilde{\psi}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
82.09732pt\raise-18.9295pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\tilde{\mathsf{s}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-36.11111pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\lower-3.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
82.09732pt\raise-22.61111pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{B}}_{n}(G)}$}}}}}}}{\hbox{\kern-58.09732pt\raise-46.61111pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\widetilde{\mathbf{T}}/(\text{subdivision
relations})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
41.95065pt\raise-30.32706pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.28406pt\hbox{$\scriptstyle{\sim}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
82.09732pt\raise-26.52773pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}\ignorespaces\end{split}$
(3.5)

######  Lemma 3.1.

The subdivision relations are generated by (3.3) for $r=2$, and (3.4).

###### Proof.

As in the proof of [1, Prop. 2.1], we show inductively that the relations
(3.3) for given $r>2$ are generated by (3.3) with smaller values of $r$. ∎

In (3.5) we have an obvious map from ${\mathcal{B}}_{n}(G)$ to the quotient of
$\widetilde{\mathbf{T}}$ by the subdivision relations, sending
$[a_{1},\dots,a_{n}]$ to a triple $(\mathbf{L},\chi,\Lambda)$ with $\Lambda$ a
basic cone and $\chi$ given by the formula (3.1). It is readily verified that
this respects the relation (1.4), and that the bottom map in (3.5) is an
isomorphism.

As in [2, Section 4] we extend the definition of
$\tilde{\psi}(\mathbf{L},\chi,\Lambda^{\prime})$ to the case of a simplicial
cone $\Lambda^{\prime}$, satisfying (3.2) with
$\mathbf{L}^{\prime}=\mathbf{L}\cap\Lambda^{\prime}\otimes{\mathbb{R}}$. We
choose a subdivision by smooth cones and sum, with signs, the contributions
from the cones, not contained in any proper face of $\Lambda^{\prime}$. Here,
as in (3.3), the signs are given by codimension, and contributions are only
taken from summands satisfying the analogous condition to (3.2).

Now we can define Hecke operators

$T_{\ell,r}:{\mathcal{B}}_{n}(G)\to{\mathcal{B}}_{n}(G),$

where $\ell$ is a prime not dividing the order of $G$ and $1\leq r\leq n-1$,
following the construction in [2, Section 6], as a sum over certain
overlattices:

$T_{\ell,r}(\tilde{\psi}(\mathbf{L},\chi,\Lambda^{\prime})):=\sum_{\begin{subarray}{c}\mathbf{L}\subset\widehat{\mathbf{L}}\subset\mathbf{L}\otimes{\mathbb{Q}}\\\
\widehat{\mathbf{L}}/\mathbf{L}\simeq({\mathbb{Z}}/\ell{\mathbb{Z}})^{r}\end{subarray}}\tilde{\psi}(\widehat{\mathbf{L}},\chi,\Lambda^{\prime}).$

## References

  * [1] B. Hassett, A. Kresch, and Yu. Tschinkel.  Symbols and equivariant birational geometry in small dimensions, 2020\.  arXiv:2010.08902.
  * [2] M. Kontsevich, V. Pestun, and Yu. Tschinkel.  Equivariant birational geometry and modular symbols, 2019.  arXiv:1902.09894, to appear in J. Eur. Math. Soc.
  * [3] M. Kontsevich and Yu. Tschinkel.  Specialization of birational types.  Invent. Math., 217(2):415–432, 2019.
  * [4] A. Kresch and Yu. Tschinkel.  Equivariant birational types and Burnside volume, 2020.  arXiv:2007.12538.


Grotunits=360

# RANDOM ATTRACTORS OF STOCHASTIC LATTICE DYNAMICAL SYSTEMS DRIVEN BY
FRACTIONAL BROWNIAN MOTIONS

ANHUI GU  Department of Mathematics, Shanghai Normal University, Shanghai
200234, PR China
and College of Science, Guilin University of Technology,
Guangxi, Guilin 541004, PR China
<EMAIL_ADDRESS>
(January 8, 2012; March 25, 2012)

###### Abstract

This paper is devoted to considering the stochastic lattice dynamical systems
(SLDS) driven by fractional Brownian motions with Hurst parameter bigger than
$1/2$. Under usual dissipativity conditions these SLDS are shown to generate a
random dynamical system for which the existence and unique of a random
attractor is established. Furthermore, the random attractor is in fact a
singleton sets random attractor.

###### keywords:

Stochastic lattice dynamical system; fractional Brownian motion; random
dynamical systems; random attractor.

;

##  1 Introduction

The purpose of this paper is to investigate the long-term behavior for the
following SLDS

$\displaystyle\frac{du_{i}(t)}{dt}$ $\displaystyle=$
$\displaystyle\kappa(u_{i-1}-2u_{i}+u_{i+1})-\lambda u_{i}+f_{i}(u_{i})$ (1)
$\displaystyle+g_{i}+\sigma_{i}\frac{d\beta_{i}^{H}(t)}{dt},~{}i\in\mathbb{Z},$

where $\mathbb{Z}$ denotes the integer set,
$u=(u_{i})_{i\in\mathbb{Z}}\in\ell^{2}$, $\kappa$ and $\lambda$ are positive
constants, $f_{i}$ are smooth functions satisfying some dissipative
conditions, $g=(g_{i})_{i\in\mathbb{Z}}\in\ell^{2}$,
$\sigma=(\sigma_{i})_{i\in\mathbb{Z}}\in\ell^{2}$, and
$\\{\beta_{i}^{H}:i\in\mathbb{Z}\\}$ are independent two-sided fractional
Brownian motions (fBms) with Hurst parameter $H\in(1/2,1)$.

Recently, the dynamics of infinite lattice dynamical systems have drawn much
attention of mathematicians and physicists, see e.g. Bates et al. [2001,
2006]; Zhou [2002, 2003, 2004]; Zhou &Shi [2006]; Wang [2006]; Lv & Sun
[2006a, b]; Huang [2007]; Wang et al. [2008]; Caraballo & Lu [2008]; Zhao &
Zhou [2009]; Wang et al. [2010]; Han et al. [2011]; Han [2011a, b] and the
references therein. Since most of the realistic systems involve noises which
may play an important role as intrinsic phenomena rather than just
compensation of defects in deterministic models, SLDS then arise naturally
while these random influences or uncertainties are taken into account.

Since Bates et al. Bates et al. [2006] initiated the study of SLDS, many works
have been done regarding the existence of global random attractors for SLDS
with white noises on lattices $\mathbb{Z}$ (see e.g. Lv & Sun [2006a, b];
Huang [2007]; Wang et al. [2008]; Caraballo & Lu [2008]; Zhao & Zhou [2009];
Wang et al. [2010]). Later, the existence of global random attractors have
been extended to other SLDS with additive white noises, for example, first-
order SLDS on $\mathbb{Z}^{k}$ Lv & Sun [2006a], stochastic Ginzburg-Landau
lattice equations Lv & Sun [2006b], stochastic FitzHugh-Nagumo lattice
equations Huang [2007]; Wang et al. [2008], second-order stochastic lattice
systems Wang et al. [2010] and the first (or second)-order SLDS with a
multiplicative white noise Caraballo & Lu [2008]; Han [2011a]. Zhao and Zhou
Zhao & Zhou [2009] gave some sufficient conditions for the existence of a
global random attractor for general SLDS in the non-weighted space
$\mathbb{R}$ of infinite sequences and provided an application to damped sine-
Gordon lattice system with additive noises. Very recently, Han, et al. Han et
al. [2011] provided some sufficient conditions for the existence of global
compact random attractors for general random dynamical systems in weighted
space $\ell_{\rho}^{p}$ $(p\geqslant 1)$ of infinite sequences, and their
results are applied to second-order SLDS in Han [2011b].

However, as can be seen that all the works above are considered in the
frameworks of the classical It$\ddot{\rm o}$ theory of Brownian motion. There
are no results on these systems when they are perturbed by a fractional
Brownian motion (fBm) to the best of our knowledge. FBm appears naturally in
the modeling of many complex phenomena in applications when the systems are
subject to “rough” external forcing. An fBm is a stochastic process which
differs significantly from the standard Brownian motion and semi-martingales,
and other classically used in the theory of stochastic process. As a centered
Gaussian process, it is characterized by the stationarity of its increments
and a medium- or long-memory property. It also exhibits power scaling with
exponent $H$. Its paths are H$\ddot{\rm o}$lder continuous of any order
$H^{\prime}\in(0,H)$. An fBm is not a semi-martingale nor a Markov process. So
an fBm is the good candidate to model random long term influences in climate
systems, hydrology, medicine and physical phenomena. For more details on fBm,
we can refer to the books Biagini et al. [2008]; Mishura [2008] for its
further development.

The goal of this article is to establish the existence of a random attractor
for SLDS with the nonlinearity $f$ under some dissipative conditions and
driven by fBms with Hurst parameter $H\in(1/2,1)$. By borrowing the main ideas
of the proofs in Garrido-Atienza et al. [2009], we firstly define a random
dynamical system by using the possibility of a pathwise interpretation of the
stochastic integral with respect to the fBms. This method is based on the fact
that a stochastic integral with respect to an fBm with Hurst parameter
$H\in(1/2,1)$ can be defined by a generalized pathwise Riemann-Stieltjes
integral (see e.g. Zahle [1998]; Decreusefond & Ustunel [1999]; Nualart &
Rascanu [2002]; Tindel et al. [2003]). And then we show the existence of a
pullback absorbing set for the random dynamical system which achieved by means
of a fractional Ornstein-Uhlenbeck transformation and Gronwall lemma. Since
every trajectory of the solutions of system (1) cannot be differentiated, we
have to consider the difference between any two solutions among them, which is
pathwise differentiable (see Garrido-Atienza et al. [2009]). Due to the
stationarity of the fractional Ornstein-Uhlenbeck solution, we get the random
attractor finally. All solutions converge pathwise to each other, so the
random attractor is proven to be a singleton sets random attractor.

The paper is organized as follows. In Sec. 2, we recall some basic concepts on
random dynamical systems. In Sec.3, we give a unique solution to system (1)
and make sure that the solution generates a random dynamical system. We
establish the main result, that is, the random attractor generated by equation
(1) turns out to be a singleton sets random attractor in Sec.4.

##  2 Random dynamical systems and Random attractor

In this section, we introduce some basic concepts related to random dynamical
systems and random attractor, which are taken from Crauel et al. [1997];
Arnold [1998].

Let $(\mathbb{E},\|\cdot\|_{\mathbb{E}})$ be a separable Hilbert space and
$(\Omega,\mathcal{F},\mathbb{P})$ be a probability space.

######  Definition 2.1.

A metric dynamical system $(\Omega,\mathcal{F},\mathbb{P},\theta)$ with two-
sided $\mathbb{R}$ consists of a measurable flow

$\theta:(\mathbb{R}\times\Omega,\mathcal{B}(\mathbb{R})\otimes\mathcal{F})\rightarrow(\Omega,\mathcal{F}),$

where the flow property for the mapping $\theta$ holds for the partial
mappings $\theta_{t}=\theta(t,\cdot)$:

$\theta_{t}\circ\theta_{s}=\theta_{t}\theta_{s}=\theta_{t+s},\ \
\theta_{0}={\rm id}_{\Omega}$

for all $s,t\in\mathbb{R}$, and $\theta\mathbb{P}=\mathbb{P}$ for all
$t\in\mathbb{R}$.

######  Definition 2.2.

A continuous random dynamical system (RDS) $\varphi$ on $\mathbb{E}$ over
$(\Omega,\mathcal{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})$ is a
$(\mathcal{B}(\mathbb{R}^{+})\times\mathcal{F}\times\mathcal{B}(\mathbb{E}),\mathcal{B}(\mathbb{E}))$-measurable
mapping and satisfies

(i) $\varphi(0,\omega)$ is the identity on $\mathbb{E}$;

(ii) $\varphi(t+s,\omega)=\varphi(t,\theta_{s}\omega)\circ\varphi(s,\omega)$
for all $s,$ $t\in\mathbb{R}^{+}$, $\omega\in\Omega$;

(iii) $\varphi(t,\omega)$ is continuous on $\mathbb{E}$ for all
$(t,\omega)\in\mathbb{R}^{+}\times\Omega$.

A universe $\mathcal{D}=\\{D(\omega),\omega\in\Omega\\}$ is a collection of
nonempty subsets $D(\omega)$ of $\mathbb{E}$ satisfying the following
inclusion property: if $D\in\mathcal{D}$ and $D^{\prime}(\omega)\subset
D(\omega)$ for all $\omega\in\Omega$, then $D^{\prime}\in\mathcal{D}$.

######  Definition 2.3.

A family of $\mathcal{A}=\\{A(\omega),\omega\in\Omega\\}$ of nonempty
measurable compact subsets $\mathcal{A}(\omega)$ of $\mathbb{E}$ is called
$\varphi$\- invariant if
$\varphi(t,\omega,\mathcal{A}(\omega))=\mathcal{A}(\theta_{t}\omega)$ for all
$t\in\mathbb{R^{+}}$ and is called a random attractor if in addition it is
pathwise pullback attracting in the sense that

$H_{d}^{*}(\varphi(t,\theta_{-t}\omega,D(\theta_{-t}\omega)),\mathcal{A}(\omega))\rightarrow
0\ \ \mbox{as}\ \ t\rightarrow\infty$

for all $D\in\mathcal{D}$. Here $H_{d}^{*}$ is the Hausdorff semi-distance on
$\mathbb{E}$.

######  Definition 2.4.

A random variable $r:\Omega\rightarrow\mathbb{R}$ is called tempered if

$\lim_{t\rightarrow\pm\infty}\frac{\log|r(\theta_{t}\omega)|}{|t|}=0\ \
\mathbb{P}-a.s.$

and a random set $\\{D(\omega),\omega\in\Omega\\}$ with
$D(\omega)\in\mathbb{E}$ is called tempered if it is contained in the ball
$\\{x\in\mathbb{R}:|x|\leq r(\omega)\\}$, where $r$ is a tempered random
variable.

Here we will always work with the attracting universe given by the tempered
random sets.

######  Definition 2.5.

A family $\hat{B}=\\{B\mathcal{(\omega)},\omega\in\Omega\\}$ is said to be
pullback absorbing if for every $D(\omega)\in\mathcal{D}$, there exists
$T_{D}(\omega)\geq 0$ such that

$\varphi(t,\theta_{-t}\omega,D(\theta_{-t}\omega))\subset B(\omega)\ \ \forall
t\geq T_{D}(\omega).$ (2)

######  Theorem 2.6.

(See Schmalfuß [1992]; Crauel et al. [1997]; Arnold [1998].) Let
$(\theta,\varphi)$ be a continuous RDS on $\Omega\times\mathbb{E}$. If there
exists a pullback absorbing family
$\hat{B}=\\{B\mathcal{(\omega)},\omega\in\Omega\\}$ such that, for every
$\omega\in\Omega$, $B(\omega)$ is compact and $B(\omega)\in\mathcal{D}$, then
the RDS $(\theta,\varphi)$ has a random attractor

$\mathcal{A}(\omega)=\bigcap_{\tau>0}\overline{\bigcup_{t\geqslant\tau}\varphi(t,\theta_{-t}\omega)B(\theta_{-t}\omega)}.$

Note that if the random attractor consists of singleton sets, i.e.
$\mathcal{A}(\omega)=\\{u^{*}(\omega)\\}$ for some random variable $u^{*}$,
then $u^{*}(t)(\omega)=u^{*}(t)(\theta_{t}\omega)$ is a stationary stochastic
process.

##  3 SLDS with FBms

We now firstly introduce as an example of metric dynamical system a special
noise that is called fractional Brownian motion. Given $H\in(0,1)$, a
continuous centered Gaussian process $\beta^{H}(t),t\in\mathbb{R}$, with the
covariance function

$\mathbf{E}\beta^{H}(t)\beta^{H}(s)=\frac{1}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}),\
\ t,s\in\mathbb{R}$

is called a two-sided one-dimensional fractional Brownian motion, and $H$ is
the Hurst parameter. For $H=1/2$, $\beta$ is a standard Brownian motion, while
for $H\neq 1/2$, it is neither a semimartingale nor a Markov process.
Moreover,

$\mathbf{E}|\beta^{H}(t)-\beta^{H}(s)|^{2}=|t-s|^{2H},\ \ \mbox{for all}\ \
s,t\in\mathbb{R}.$

Here, we assume that $H\in(1/2,1)$ throughout the paper. When $H\in(0,1/2)$ we
cannot define the stochastic integral by a generalized Stieljes integral and,
therefore, dealing with such values of the Hurst parameter seems to be much
more complicated. It is worth mentioning that when $H=1/2$ the fBm becomes the
standard Wiener process, the random dynamical system generated by SLDS has
been studied in Bates et al. [2006].

Using the definition of $\beta^{H}(t)$, Kolmogorov’s theorem ensures that
$\beta^{H}$ has a continuous version, and almost all the paths are H$\ddot{\rm
o}$lder continuous of any order $H^{\prime}\in(0,H)$ (see Kunita [1990]).
Thus, we can consider the canonical interpretation of an fBm: denote
$\Omega=C_{0}(\mathbb{R},\ell^{2})$, the space of continuous functions on
$\mathbb{R}$ with values in $\ell^{2}$ such that $\omega(0)=0$, equipped with
the compact open topology. Let $\mathcal{F}$ be the associated
Borel-$\sigma$-algebra and $\mathbb{P}$ the distribution of the fBm
$\beta^{H}$, and $\\{\theta_{t}\\}_{t\in\mathbb{R}}$ be the flow of Wiener
shifts such that

$\theta_{t}\omega(\cdot)=\omega(\cdot+t)-\omega(t),\ \ t\in\mathbb{R}.$

Due to Maslowski & Schmalfuß [2004]; Garrido-Atienza et al. [2010], we know
that the quadruple $(\Omega,\mathcal{F},\mathbb{P},\theta)$ is a metric
dynamical system which is ergodic. Furthermore, it holds that

$\displaystyle\beta^{H}(\cdot,\omega)=\omega(\cdot),$
$\displaystyle\beta^{H}(\cdot,\theta_{s}\omega)$ $\displaystyle=$
$\displaystyle\beta^{H}(\cdot+s,\omega)-\beta^{H}(s,\omega)$ (3)
$\displaystyle=$ $\displaystyle\omega(\cdot+s)-\omega(s).$

We consider the SLDS

$\displaystyle\frac{du_{i}(t)}{dt}$ $\displaystyle=$
$\displaystyle\kappa(u_{i-1}-2u_{i}+u_{i+1})-\lambda u_{i}+f_{i}(u_{i})+g_{i}$
(4) $\displaystyle+\sigma_{i}\frac{d\omega_{i}(t)}{dt},~{}i\in\mathbb{Z},$

where $\mathbb{Z}$ denotes the integer set,
$u=(u_{i})_{i\in\mathbb{Z}}\in\ell^{2}$, $\kappa$ and $\lambda$ are positive
constants, $f_{i}$ are smooth functions satisfying some dissipative
conditions, $g=(g_{i})_{i\in\mathbb{Z}}\in\ell^{2}$,
$\sigma=(\sigma_{i})_{i\in\mathbb{Z}}\in\ell^{2}$, and
$\\{\omega_{i}=\beta^{H}_{i}:i\in\mathbb{Z}\\}$ are independent two-sided
fractional Brownian motions with Hurst parameter $H\in(1/2,1)$. Then, (4) can
be understood as the pathwise Riemann-Stieltjes integral equations

$\displaystyle u_{i}(t)$ $\displaystyle=$ $\displaystyle
u_{i}(0)+\int_{0}^{t}(\kappa(u_{i-1}(s)-2u_{i}(s)+u_{i+1}(s))$
$\displaystyle-\lambda
u_{i}(s)+f_{i}(u_{i}(s))+g_{i})ds+\sigma_{i}\omega_{i}(t),~{}i\in\mathbb{Z}.$

Let $e^{i}\in\ell^{2}$ denote the element having 1 at position $i$ and 0 at
other components. Then

$W(t)\equiv W(t,\omega)=\sum_{i\in\mathbb{Z}}\sigma_{i}\omega_{i}(t)e^{i}\ \
\mbox{with}\ \ (\sigma_{i})_{i\in\mathbb{Z}}\in\ell^{2},$ (6)

is the special noise with values in $\ell^{2}$ defined on the probability
space $(\Omega,\mathcal{F},\mathbb{P})$.

Assumptions on the nonlinearity $f_{i}$. Assume
$f_{i}:\mathbb{R}\rightarrow\mathbb{R}$ to be continuously differentiable. Let
$f$ be the Nemytski operator associated with $f_{i}$, i.e. for all
$x=(x_{i})_{i\in\mathbb{Z}}\in\ell^{2}$, let
$f(x)=(f_{i}(x_{i}))_{i\in\mathbb{Z}}$. Then we have $f(x)\in\ell^{2}$ (see
Bates et al. [2006]). Assume $f$ satisfies a one-sided dissipative Lipschitz
condition

$\langle x-y,f(x)-f(y)\rangle\leq-L|x-y|^{2},\ \ \mbox{for all}\ \
x,y\in\ell^{2},$ (7)

where $L>0$, and a polynomial growth condition along with its derivative,
i.e.,

$|f(x)|+|Df(x)|\leq K(1+|x|^{p}),\ \ \mbox{for all}\ \ x\in\ell^{2},$ (8)

$p\geq 1$ and a positive constant $K$. Here, we remark that we could consider
more general dissipativity conditions, which would lead to nontrivial
setvalued random attractors. However, to avoid rather technical details, we
will restrict here to the dissipativity conditions (7) and (8).

For $x=(x_{i})_{i\in\mathbb{Z}}\in\ell^{2}$, define $A,B,B^{*}$ to be linear
operators from $\ell^{2}$ to $\ell^{2}$ as follows:

$\displaystyle(Ax)_{i}$ $\displaystyle=$ $\displaystyle-
x_{i-1}+2x_{i}-x_{i+1},$ $\displaystyle(Bx)_{i}$ $\displaystyle=$
$\displaystyle x_{i+1}-x_{i},\ \ (B^{*}x)_{i}=x_{i-1}-x_{i},\ \
i\in\mathbb{Z}.$

It is easy to show that $A=BB^{*}=B^{*}B$, $\langle
B^{*}x,x^{\prime}\rangle=\langle x,Bx^{\prime}\rangle$ for all
$x,x^{\prime}\in\ell^{2}$, which implies that $\langle Ax,x\rangle\geq 0$.

The system (4) with initial values
$u_{0}=(u_{0,i})_{i\in\mathbb{Z}}\in\ell^{2}$ may be written as an equation in
$\ell^{2}$

$\displaystyle u(t)$ $\displaystyle=$ $\displaystyle
u_{0}+\int_{0}^{t}(-\kappa Au(s)-\lambda u(s)+f(u(s))+g)ds$ (9)
$\displaystyle+W(t)$ $\displaystyle:=$ $\displaystyle
u_{0}+\int_{0}^{t}G(u(s))ds+W(t),$ $\displaystyle~{}t\geq
0,~{}\omega\in\Omega,$

where $G(u(t)):=-\kappa Au(t)-\lambda u(t)+f(u(t))+g$.

######  Proposition 3.1.

Let the above assumptions on $f$ be satisfied and $T>0$. Then system (9) has a
unique pathwise solution $u=(u(t))_{t\geq 0}$. In addition that the solution
satisfies

$\displaystyle\sup_{t\in[0,T]}|u(t)|$ $\displaystyle\leq$ $\displaystyle
M(1+|u_{0}|+\sum_{i\in\mathbb{Z}}\sup_{\tau\in[0,T]}|\beta_{i}^{H}(\tau)|$
(10)
$\displaystyle+\sum_{i\in\mathbb{Z}}\sup_{\tau\in[0,T]}|\beta_{i}^{H}(\tau)|^{p}+|g|),$

for all $T>0$, where $M$ is a positive constant and is independent of $T$.

######  Proof 3.2.

Let $v(t)=u(t)-W(t),t\geq 0$, then system (9) has a solution $u=(u(t))_{t\geq
0}$ for all $\omega\in\Omega$ if and only if the following equation

$\displaystyle v(t)$ $\displaystyle=$ $\displaystyle
u_{0}+\int_{0}^{t}(-\kappa Av(s)-\lambda v(s)+f(v(s)+W(s))$ (11)
$\displaystyle+g-\kappa AW(s)-\lambda W(s))ds.$

has a unique pathwise solution for $t\in[0,T]$. However, since the integrand
is pathwise continuous, the fundamental theorem of calculus says that the left
hand side of (11) is pahtwise differentiable. Thus, for fixed
$\omega\in\Omega$, equation (10) is the pathwise random ordinary differential
equation (RODE)

$\displaystyle\frac{dv(t)}{dt}(\omega)$ $\displaystyle=$ $\displaystyle-\kappa
Av(t)-\lambda v(t)+f(v(t)+W(t))$ (12) $\displaystyle+g-\kappa AW(t)-\lambda
W(t)$ $\displaystyle:=$
$\displaystyle\tilde{G}(t,v(t))(\omega):=\tilde{G}(v(t)+W(t)),$
$\displaystyle~{}~{}~{}~{}\mbox{for}\ \ t\geq 0,\ \
v_{0}(\omega)=u_{0}(\omega).$

Since $\tilde{G}:[0,\infty)\times\ell^{2}\rightarrow\ell^{2}$ is continuous in
$t$ and continuous differentiable in $v$, this RODE has a unique local
solution in a small interval $[0,\tau(\omega)]$, which means that (9) has a
unique local solution in the same small interval $[0,\tau(\omega)]$, see e.g.
Theorem 2.1.4 in Stuart & Humphries [1996].

To see that (9) has a unique solution for every $t\geq 0$, we prove at first
the a priori estimate (10). Suppose that $u=(u(t))_{t\geq 0}$ solves (9) on
the interval $[0,T]$. This implies that the process $v(t)=u(t)-W(t)$ solves
(11) on the interval $[0,T]$. By the one-sided dissipative Lipschitz condition
and H$\ddot{\rm o}$lder inequality, we obtain

$\displaystyle 2|v(t)|\frac{d|v(t)|}{dt}=\frac{d}{dt}|v(t)|^{2}$
$\displaystyle=$ $\displaystyle 2\langle v(t),\tilde{G}(v(t)+W(t))\rangle$
$\displaystyle=$ $\displaystyle 2\langle v(t),-\kappa Av(t)-\lambda
v(t)+f(v(t)+W(t))$ $\displaystyle+g-\kappa AW(t)-\lambda W(t)\rangle$
$\displaystyle\leq$ $\displaystyle-2(\lambda+L)|v(t)|^{2}+2|f(W(t))||v(t)|$
$\displaystyle+2(|g|-|\kappa AW(t)|-|\lambda W(t)|)|v(t)|,$

that is

$\displaystyle\frac{d|v(t)|}{dt}$ $\displaystyle\leq$
$\displaystyle-(\lambda+L)|v(t)|+c_{0}(|g|+\sup_{\tau\in[0,T]}|W(\tau)|$
$\displaystyle+\sup_{\tau\in[0,T]}|W(\tau)|^{p}),\ \ t\in[0,T],$

where $c_{0}$ is a positive constant depends on
$\kappa,\lambda,K,\max_{j\in\mathbb{Z}}|\sigma_{j}|$ and
$\max_{j\in\mathbb{Z}}|g_{j}|$. By Gronwall lemma it yields that

$\displaystyle|v(t)|$ $\displaystyle\leq$ $\displaystyle|u_{0}|e^{-\lambda
t}+\frac{c_{0}}{\lambda}(1-e^{-\lambda t})(|g|+\sup_{\tau\in[0,T]}|W(\tau)|$
$\displaystyle+\sup_{\tau\in[0,T]}|W(\tau)|^{p}),\ \ t\in[0,T].$

Since $\lambda>0$, we have

$\displaystyle|v(t)|$ $\displaystyle\leq$
$\displaystyle|u_{0}|+\frac{c_{0}}{\lambda}(|g|+\sup_{\tau\in[0,T]}|W(\tau)|$
$\displaystyle+\sup_{\tau\in[0,T]}|W(\tau)|^{p}),$

that is

$\displaystyle\sup_{t\in[0,T]}|v(t)|$ $\displaystyle\leq$ $\displaystyle
M(|g|+\sup_{\tau\in[0,T]}|W(\tau)|$
$\displaystyle+\sup_{\tau\in[0,T]}|W(\tau)|^{p}),\ \ t\in[0,T],$

where the constant $M$ depends on
$\kappa,\lambda,K,\max_{j\in\mathbb{Z}}|\sigma_{j}|$ and
$\max_{j\in\mathbb{Z}}|g_{j}|$. The result of (10) follows then by using the
relation $v(t)=u(t)+W(t)$ for $t\in[0,T]$. As a consequence of estimate (10)
the local unique solution to (9) can be extended to a global unique solution
(see e.g. Theorem 2.1.4 in Stuart & Humphries [1996]).

We consider (1) with the linear drift term $G(u(t))=-\lambda u(t)$, that is

$du(t)=-\lambda u(t)dt+dW(t),\ \ \lambda>0,$ (13)

which is called the fractional Ornstein-Uhlenbeck process, where $W(t)$
denotes a one-dimensional fractional Brownian motion defined in (6). It has
the explicit solution

$u(t)=u_{0}e^{-\lambda t}+e^{-\lambda t}\int_{0}^{t}e^{\lambda s}dW(s).$ (14)

Taking the pathwise pullback limit, we get the stochastic stationary solution

$\bar{u}(t)=e^{-\lambda t}\int_{-\infty}^{t}e^{\lambda s}dW(s),\ \
t\in\mathbb{R},$ (15)

which is called the fractional Ornstein-Uhlenbeck solution. Based on Lemma 2.6
in Maslowski & Schmalfuß [2004] and Lemma 1 in Garrido-Atienza et al. [2009],
we have the property

######  Lemma 3.3.

For all $\omega\in\Omega$ the Riemann-Stieltjes integrals

$e^{-\lambda t}\int_{-\infty}^{t}e^{\lambda s}dW(s),\ \ t\in\mathbb{R},$

are well defined in $\ell^{2}$. Moreover, for all $\omega\in\Omega$, we have

$|e^{-\lambda t}\int_{-\infty}^{t}e^{\lambda s}dW(s)|\leq
4\rho(\omega)(1+|t|)^{2},\ \ t\in\mathbb{R},$

where the random constant $\rho(\omega)>0$ and satisfying

$|W(t)|\leq\rho(\omega)(1+|t|^{2}),\ \ \mbox{for}\ \ t\in\mathbb{R},\ \
\omega\in\bar{\Omega}.$ (16)

Here, $\bar{\Omega}\in\mathcal{F}$ is a
$(\theta_{t})_{t\in\mathbb{R}}$-invariant set of full measure.

######  Proof 3.4.

We know that $\lambda>0$, $(\sigma_{i})_{i\in\mathbb{Z}}\in\ell^{2}$. The
proof is similar to Lemma 2.6 in Maslowski & Schmalfuß [2004] and Lemma 1 in
Garrido-Atienza et al. [2009], thus we omit it here.

Since the mapping of $\theta$ on $\bar{\Omega}$ has the same properties as the
original one if we choose the trace $\sigma$-algebra with respect to
$\bar{\Omega}$ to be denoted also by $\mathcal{F}$, we can change our metric
dynamical system with respect to $\bar{\Omega}$, and still denoted by the
symbols $(\Omega,\mathcal{F},\mathbb{P},(\theta_{t})_{t\in\mathbf{R}})$.

Now, we will verify that the solution of system (1) generates a continuous
RDS.

######  Proposition 3.5.

The solution of (1) determinants a continuous random dynamical system
$\varphi:\mathbb{R^{+}}\times\Omega\times\ell^{2}\rightarrow\ell^{2}$, which
is given by

$\displaystyle\varphi(t,\omega,u_{0})$ $\displaystyle=$ $\displaystyle
u_{0}+\int_{0}^{t}(-\kappa Au(s)-\lambda u(s)$ (17)
$\displaystyle+f(u(s))+g)ds+W(t,\omega).$

######  Proof 3.6.

Note that (3) is satisfied for $\omega\in\Omega$ and from the definition of
$(\theta_{t})_{t\in\mathbb{R}}$, we have the property

$W(\tau+t,\omega)=W(\tau,\theta_{t}\omega)+W(t,\omega)\ \ \mbox{for all}\ \
t,\tau\in\mathbb{R}.$ (18)

From Proposition 3.1 we know that $\varphi$ solves (1), thus $\varphi$ is
measurable and satisfies $\varphi(0,\omega,\cdot)=\rm id_{\ell^{2}}$. It
remains to verify the cocycle property in Definition 2.2. Let
$t,\tau\in\mathbb{R^{+}},\omega\in\Omega$ and $u_{0}\in\ell^{2}$, denote
$G(u(t))(\omega):=-\kappa Au(t)-\lambda u(t)+f(u(t))+g$, it yields from (3)
that

$\displaystyle\varphi(t+\tau,\omega,u_{0})$ (19) $\displaystyle=$
$\displaystyle u_{0}+\int_{0}^{t+\tau}G(u(s))(\omega)ds+W(t+\tau,\omega)$
$\displaystyle=$ $\displaystyle
u_{0}+\int_{0}^{t}G(u(s))(\omega)ds+W(t,\omega)$
$\displaystyle+\int_{t}^{t+\tau}G(u(s))(\omega)ds+W(\tau,\theta_{t}\omega)$
$\displaystyle=$ $\displaystyle
u(t)+\int_{0}^{\tau}G(u(s))(\theta_{t}\omega)ds+W(\tau,\theta_{t}\omega)$
$\displaystyle=$
$\displaystyle\varphi(\tau,\theta_{t}\omega,\cdot)\circ\varphi(t,\omega,u_{0}),$

which completes the proof.

##  4 Existence of a Random attractor

In this section, we will prove the existence of a random attractor for the RDS
defined in Proposition 3.5. The main proof is based on the first part in
section 4 in Garrido-Atienza et al. [2009].

Let $u,w$ be any two solutions of system (1). Their sample paths are not
differentiable, but the difference satisfies pathwise

$\displaystyle u(t)-w(t)$ $\displaystyle=$ $\displaystyle
u_{0}-w_{0}+\int_{0}^{t}(-\kappa A(u(s)-w(s))$
$\displaystyle-\lambda(u(s)-w(s))$ $\displaystyle+(f(u(s))-f(w(s))))ds,\ \
t\geq 0,$

and again, since the integrand is pathwise continuous, the fundamental theorem
of calculus indicates that the left hand side is pathwise differentiable and
satisfies

$\displaystyle\frac{d}{dt}(u(t)-w(t))$ (20) $\displaystyle=$
$\displaystyle-\kappa A(u(s)-w(s))-\lambda(u(s)-w(s))$
$\displaystyle+(f(u(s))-f(w(s))),\ \ t\geq 0.$

Now we use condition (7) again, we obtain from (20) that

$\displaystyle\frac{d}{dt}|u(t)-w(t)|^{2}$ $\displaystyle=$ $\displaystyle
2\langle u(t)-w(t),-\kappa A(u(s)-w(s))$
$\displaystyle-\lambda(u(s)-w(s))+(f(u(s))-f(w(s)))\rangle$
$\displaystyle\leq$ $\displaystyle-2\lambda|u(t)-w(t)|^{2}.$

Thus pathwise we have

$|u(t)-w(t)|\leq|u_{0}-w_{0}|e^{-\lambda t}\rightarrow 0,\ \ \mbox{as}\ \
t\rightarrow\infty.$

That is to say that all solutions converge pathwise forward to each other in
time.

Now, we consider the difference $u(t)-\bar{u}(t)$. This is also pathwise
differentiable, since the paths are continuous and satisfy the integral
equation

$\displaystyle u(t)-\bar{u}(t)$ $\displaystyle=$ $\displaystyle
u_{0}-\bar{u}_{0}+\int_{0}^{t}(-\kappa Au(s)-\lambda(u(s)$
$\displaystyle-\bar{u}(s))+f(u(s)))ds,$

which is equivalent to the pathwise differential equation

$\displaystyle\frac{d}{dt}(u(t)-\bar{u}(t))$ $\displaystyle=$
$\displaystyle-\kappa Au(s)-\lambda(u(s)-\bar{u}(s))+f(u(s)),t\geq 0.$

By using (7), it follows that

$\displaystyle\frac{d}{dt}|u(t)-\bar{u}(t)|^{2}$ $\displaystyle=$
$\displaystyle 2\langle u(t)-\bar{u}(t),-\kappa Au(s)$
$\displaystyle-\lambda(u(s)-\bar{u}(s))+f(u(s))\rangle$ $\displaystyle\leq$
$\displaystyle-2\lambda|u(t)-\bar{u}(t)|^{2}$
$\displaystyle+2|u(t)-\bar{u}(t)||f(\bar{u}(t))|,$

that is

$\displaystyle\frac{d}{dt}|u(t)-\bar{u}(t)|$ $\displaystyle\leq$
$\displaystyle-\lambda|u(t)-\bar{u}(t)|+|f(\bar{u}(t))|,$

and hence

$\displaystyle|u(\omega)-\bar{u}(\omega)|$ (21) $\displaystyle\leq$
$\displaystyle|u_{0}(\omega)-\bar{u}_{0}(\omega)|e^{-\lambda t}$
$\displaystyle+e^{-\lambda t}\int_{0}^{t}e^{\lambda
s}|f(\bar{u}(s))(\omega)|ds.$

Let us check that the family of balls centered on $\bar{u}_{0}(\omega)$ with
the random radius

$\varrho(\omega):=1+\int_{-\infty}^{0}e^{\lambda s}|f(\bar{u}(s))(\omega)|ds$
(22)

is a pullback absorbing family for the random dynamical system generated by
system (1).

Due to the assumptions on $f$ and Lemma 3.3, the radius defined in (22) is
well defined. Now, by replacing $\omega$ by $\theta_{-t}\omega$ in (21), we
get

$\displaystyle|u(\theta_{-t}\omega)-\bar{u}(\theta_{-t}\omega)|$ (23)
$\displaystyle\leq$
$\displaystyle|u_{0}(\theta_{-t}\omega)-\bar{u}_{0}(\theta_{-t}\omega)|e^{-\lambda
t}$
$\displaystyle+\int_{0}^{t}e^{\lambda(s-t)}|f(\bar{u}(s))(\theta_{-t}\omega)|ds$
$\displaystyle=$
$\displaystyle|u_{0}(\theta_{-t}\omega)-\bar{u}_{0}(\theta_{-t}\omega)|e^{-\lambda
t}$
$\displaystyle+\int_{-t}^{0}e^{\lambda\tau}|f(\bar{u}(\tau))(\omega)|d\tau,\ \
t\geq 0.$

The last term in (23) due to
$\bar{u}(s)(\theta_{-t}\omega)=\bar{u}_{0}(\theta_{s-t}\omega)=\bar{u}(s-t)(\omega)$
which deduced from that $(\bar{u}(t))_{t\in\mathbb{R}}$ is a stationary
process. The conclusion now follows for $t\rightarrow\infty$.

Because of the stationarity and Lemma 3.3, we have $e^{-\lambda
t}|\bar{u}_{0}(\theta_{-t}\omega)|=e^{-\lambda
t}|\bar{u}(-t)(\omega)|\rightarrow 0$ as $t\rightarrow\infty$. Then we have
the pullback absorption

$|u(\theta_{-t}\omega)|\leq|\bar{u}_{0}(\omega)|+\varrho(\omega),\ \ \forall
t\geq T_{\mathcal{D}(\omega)}.$ (24)

As a consequence of Theorem 2.6, system (1) has a random attractor
$\mathcal{A}=\\{A(\omega),\omega\in\Omega\\}$. We have know that all solutions
converge pathwise to each other, so the random attractor sets are in fact
singleton sets $\mathcal{A}=\\{\tilde{u}_{0}(\omega)\\}$, i.e. the random
attractor is formed by a stationary random process
$\tilde{u}(t)(\omega):=\tilde{u}_{0}(\theta_{t}\omega)$, which pathwise
attracts all other solutions in both forward and pullback senses.

Finally, we are in the position to state the result of existence of a random
attractor.

######  Theorem 4.1.

Assume that the conditions on $f$ be satisfied. Then the random dynamical
system $\varphi$ has a unique random attractor. Furthermore, the random
attractor is in fact a singleton sets random attractor.

######  Remark 4.2.

Sometimes, for the need of demonstrating the relations between
$\frac{dv(t)}{dt}(\cdot)$ (or $G(u(t))(\cdot)$, $\tilde{G}(t,v(t))(\cdot)$)
and $\omega$ more explicitly, we will write $\frac{dv(t)}{dt}(\omega)$ (or
$G(u(t))(\omega)$, $\tilde{G}(t,v(t))(\omega)$) instead if necessary.

######  Remark 4.3.

There are several differences between these two kinds of noises: white noise
and fBm for the existence of a random attractor of SLDS. On one hand, in this
work, the SLDS perturbed by additive fBms are considered, which allow to
transform the SLDS into random systems that can be dealt with in a pathwise
way. But for the multiplicative noise, with no transformation into a random
system, the existence of random attractor for stochastic differential
equations was consider in Garrido-Atienza et al. [2010] based on considering a
suitable sequence of stopping times for the fractional Brownian motion. This
method is different from the traditional conjugacy method, which transforms a
stochastic equation into a differential equation with random coefficients but
without white noise. On the other hand, when SLDS perturbed by either additive
white noises (see e.g. Bates et al. [2006]) or a multiplicative white noise
(see e.g. Caraballo & Lu [2008])), the existing random attractor is a
(nontrivial) compact set of tempered random bounded set, but the random
attractor turns out to be a single sets random attractor when the systems
disturbed by additive fBms.

Acknowledgments The author would like to thank the anonymous referees for very
helpful comments and suggestions which largely improve the quality of the
original manuscript. This work has been partially supported by NSF of China
grants 11071165 and Guangxi Provincial Department of Research Project grants
201010LX166.

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