# Asymptotic stability of evolution systems of probability measures for
nonautonomous stochastic systems: Theoretical results and applications

Renhai Wang  Institute of Applied Physics and Computational Mathematics,
Beijing 100088, China<EMAIL_ADDRESS>,  Tomás Caraballo  Departamento
de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas,
Universidad de Sevilla, C/ Tarfia s/n, 41012-Sevilla, Spain<EMAIL_ADDRESS>and
Nguyen Huy Tuan∗ Division of Applied Mathematics, Science and Technology
Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam,
Faculty of Technology, Van Lang University, Ho Chi Minh City, Vietnam
<EMAIL_ADDRESS>
###### Abstract.

The limiting stability of invariant probability measures of time homogeneous
transition semigroups for autonomous stochastic systems has been extensively
discussed in the literature. In this paper we initially initiate a program to
study the asymptotic stability of evolution systems of probability measures of
time inhomogeneous transition operators for nonautonomous stochastic systems.
A general theoretical result on this topic is established in a Polish space by
establishing some sufficient conditions which can be verified in applications.
Our abstract results are applied to a stochastic lattice reaction-diffusion
equation driven by a time-dependent nonlinear noise. A time-average method and
a mild condition on the time-dependent diffusion function are used to prove
that the limit of every evolution system of probability measures must be an
evolution system of probability measures of the limiting equation. The
theoretical results are expected to be applied to various stochastic lattice
systems/ODEs/PDEs in the future.

###### Key words and phrases:

Invariant measure; tightness; Limit measure; nonlinear noise; lattice system.

###### 2010 Mathematics Subject Classification:

37L30,37L40,37L55, 35B40,35B41,60H10

∗Corresponding author: Nguyen Huy Tuan (E-mail<EMAIL_ADDRESS>
##  1\. Introduction

###  1.1. Statement of problems

A basic approach to look at the asymptotic stability of stochastic systems
with noise perturbations is to consider the limiting stability of their
invariant probability measures with respect to the noise intensity. Recently,
this kind of the limiting stability of invariant probability measures of time
homogeneous transition semigroups was discussed in the literature for
_autonomous_ stochastic lattice systems/ODEs/PDEs, see e.g., [5, 6, 13, 14].
As far as the authors can find, currently, there are no results reported in
the literature on the limiting stability of an _evolution system of
probability measures_ (an extension of invariant measures from autonomous to
nonautonomous developed by Da Prato and Röckner [9, 10]) of time inhomogeneous
transition operators for _nonautonomous_ stochastic systems.

###  1.2. General framework and theoretical results

The goal of the present work is to initiate a program of studying limiting
stability of evolution systems of probability measures of time inhomogeneous
transition operators. We will establish a general setting on the limiting
stability of evolution system of probability measures for an abstract time
inhomogeneous transition operator on a Polish space. More specifically, we
will show, under certain conditions, that the limit of every evolution system
of probability measures (if exists) must be an evolution system of probability
measures of the limiting transition operator.

Let $(\mathbb{X},\|\cdot\|_{\mathbb{X}})$ be a Polish space,
$\mathcal{P}(\mathbb{X})$ be the set of all probability measures on
$\mathbb{X}$, $\mathscr{B}(\mathbb{X})$ be the Borel $O$-algebra on
$\mathbb{X}$, and $B_{b}(\mathbb{X})$ ($C_{b}(\mathbb{X})$) the space of all
bounded Borel (continuous) functions on $\mathbb{X}$. Assume
$\\{X^{\epsilon}(t,\tau,x),t\geqslant\tau\in\mathbb{R}\\}$ on $\mathbb{X}$ is
a unique noise driven stochastic process with value $x\in\mathbb{X}$ and the
noise intensity $\epsilon\in[a,b]$ for $0\leqslant a<b<\infty$. For
$\varphi\in B_{b}(\mathbb{X})$, $\Lambda\in\mathscr{B}(\mathbb{X})$ and
$\eta\in\mathcal{P}(\mathbb{X})$, we define a transition operator
$(P^{\epsilon}_{\tau,t})_{t\geqslant\tau}$ by
$(P^{\epsilon}_{\tau,t}\varphi)(x)=\mathbb{E}[\varphi(X^{\epsilon}(t,\tau,x))]$,
a transition probability function
$P^{\epsilon}_{\tau,t}(u_{0},\Lambda)=\mathbb{P}\\{\omega\in\Omega:u^{\epsilon}(t,\tau,u_{0})\in\Lambda\\}$
and the adjoint operator $(Q^{\epsilon}_{\tau,t})_{t\geqslant\tau}$ of
$(P^{\epsilon}_{\tau,t})_{t\geqslant\tau}$ by
$Q^{\epsilon}_{\tau,t}\eta(\Lambda)=\int_{\mathbb{X}}P^{\epsilon}_{\tau,t}(x,\Lambda)\eta(dx)$.
As in Da Prato and Röckner [9, 10], we say
$\\{\eta^{\epsilon}_{t}\\}_{t\in\mathbb{R}}\subseteq\mathcal{P}(\mathbb{X})$
is an _evolution system of probability measures_ of
$(P^{\epsilon}_{\tau,t})_{t\geqslant\tau}$ if
$Q^{\epsilon}_{\tau,t}\eta_{\tau}=\eta_{t}$ for all
$t\geqslant\tau\in\mathbb{R}$.

For some technical reasons, the following assumption is needed when we discuss
the limiting stability of evolution systems of probability measures of time
inhomogeneous transition operators.

CIP(Convergence in Probability). For every compact set
$\mathbf{K}\subseteq\mathbb{X}$, $\tau\in\mathbb{R}$, $t\geqslant\tau$,
$\epsilon_{0}\in[a,b]$ and $\delta>0$,

$\displaystyle\lim_{\epsilon\rightarrow\epsilon_{0}}\sup_{x\in\mathbf{K}}\mathbb{P}\bigg{(}\Big{\\{}\omega\in\Omega:\|X^{\epsilon}(t,\tau,x)-X^{\epsilon_{0}}(t,\tau,x)\|_{\mathbb{X}}\geqslant\delta\Big{\\}}\bigg{)}=0.$
(1.1)

######  Theorem 1.1.

(Theoretical results) Assume CIP holds. Let
$\\{\eta^{\epsilon_{0}}_{t}\\}_{t\in\mathbb{R}}$ be a family of probability
measures on $\mathbb{X}$ for $\epsilon_{0}\in[a,b]$, and
$\\{\eta^{\epsilon_{n}}_{t}\\}_{t\in\mathbb{R}}$ be an evolution system of
probability measures of $(P^{\epsilon_{n}}_{\tau,t})_{t\geqslant\tau}$ on
$\mathbb{X}$ for $n\in\mathbb{N}$. If
$(P^{\epsilon_{0}}_{\tau,t})_{t\geqslant\tau}$ is Feller,
$\epsilon_{n}\rightarrow\epsilon_{0}$, and
$\eta^{\epsilon_{n}}_{t}\rightarrow\eta^{\epsilon_{0}}_{t}$ weakly for each
$t\in\mathbb{R}$, then $\\{\eta^{\epsilon_{0}}_{t}\\}_{t\in\mathbb{R}}$ must
be an evolution system of probability measures of
$(P^{\epsilon_{0}}_{\tau,t})_{t\geqslant\tau}$.

######  Remark 1.2.

(i) We only require $(P^{\epsilon}_{\tau,t})_{t\geqslant\tau}$ is Feller at
$\epsilon=\epsilon_{0}$. (ii) The convergence in probability in (1.1) is not
necessary to be uniform for $t$. But for most stochastic systems, (1.1) can be
proved uniformly for $t$:

$\displaystyle\lim_{\epsilon\rightarrow\epsilon_{0}}\sup_{x\in\mathbf{K}}\mathbb{P}\bigg{(}\Big{\\{}\omega\in\Omega:\sup_{t\in[\tau,\tau+T]}\|X^{\epsilon}(t,\tau,x)-X^{\epsilon_{0}}(t,\tau,x)\|_{\mathbb{X}}\geqslant\delta\Big{\\}}\bigg{)}=0.$

Theorem 1.1 can be viewed as an extension version of limiting stability of
invariant measures of time homogeneous transition operators from the
autonomous to the nonautonomous framework.

###  1.3. Application of theoretical results

Our abstract result in Theorem 1.1 is expected to be applied to various
stochastic ODEs/PDEs/lattice systems with noise perturbations. In particular,
we apply this abstract result to a stochastic lattice reaction-diffusion
equation driven by a _nonlinear_ noise on $\mathbb{Z}$ for $t>\tau$ with
$\tau\in\mathbb{R}$:

$\displaystyle du^{\epsilon}_{i}(t)+\lambda
u^{\epsilon}_{i}(t)dt-\nu\big{(}u^{\epsilon}_{i}(t)-2u^{\epsilon}_{i-1}(t)+u^{\epsilon}_{i+1}(t)\big{)}dt+|u^{\epsilon}_{i}(t)|^{p-2}|u^{\epsilon}_{i}(t)|dt=\epsilon\sigma(t,u^{\epsilon}_{i}(t))dW(t),$
(1.2)

with initial condition $u^{\epsilon}_{i}(\tau)=u_{0,i}$, where
$\lambda,\nu>0$, $p>2$, $\epsilon>0$, $W$ is a two-sided, real-valued Wiener
process on
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\in\mathbb{R}},\mathbb{P})$ the
complete filtered probability space, and the nonlinear function
$\sigma(t,\cdot):\mathbb{R}\rightarrow\mathbb{R}$ is locally Lipschitz such
that $|\sigma(t,s)|\leqslant\delta|s|+g(t)$ for a constant $\delta>0$ and a
time-dependent deterministic function $g$ satisfying certain conditions. We
say the noise in (1.2) is a time-dependent nonlinear noise just because the
diffusion function $\sigma$ depends on time $t$, and is nonlinear in the
unknown function $u^{\epsilon}_{i}$.

Note that lattice systems can be regarded as space discretization versions of
PDEs that have many applications in the real world such as electric circuits,
pattern formation, propagation of nerve pulse, chemical reaction and others.
The existence and limiting stability of invariant probability measures for
_autonomous_ stochastic lattice systems has been considered recently in [5, 6,
13, 14, 15]. The reader is referred to [1, 2, 3, 4, 11, 12, 17, 18] for other
mathematical topics such as random attractors for stochastic lattice systems.
In this paper we study existence and limiting stability of evolution systems
of probability measures for the _nonautonomous_ stochastic lattice system
(1.2). To our knowledge, it seems that this is the first time to study
evolution systems of probability measures for stochastic lattice systems.

######  Theorem 1.3.

(Existence of evolution system of probability measures) If
$\int_{-\infty}^{0}e^{\lambda r}\|g(r)\|^{2}dr<\infty$, then the transition
operator $(P^{\epsilon}_{\tau,t})_{t\geqslant\tau}$ for (1.2) has an evolution
system of probability measures $\\{\mu^{\epsilon}_{t}\\}_{t\in\mathbb{R}}$ on
$\ell^{2}$ for any
$\epsilon\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}$.

Let
$\mathcal{E}^{\epsilon}_{t}=\big{\\{}\mu^{\epsilon}_{t}:\\{\mu^{\epsilon}_{t}\\}_{t\in\mathbb{R}}\
\mbox{is an evolution system of probability measures of
$(P^{\epsilon}_{\tau,t})_{t\geqslant\tau}$ for \eqref{AA1}}\big{\\}}$ for
$t\in\mathbb{R}$ and
$\epsilon\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}$. Then we discuss
the tightness of
$\bigcup_{\epsilon\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}}\mathcal{E}^{\epsilon}_{t}$
and limiting stability of any sequences of $\mathcal{E}^{\epsilon}_{t}$.

######  Theorem 1.4.

(Application of theoretical results) If $\int_{-\infty}^{t}e^{\lambda
r}\|g(r)\|^{2}dr<\infty$ for each $t\in\mathbb{R}$, then we have the following
two results.

(i) The union
$\bigcup_{\epsilon\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}}\mathcal{E}_{t}^{\epsilon}$
is tight on $\ell^{2}$ for each $t\in\mathbb{R}$.

(ii)If $\mu^{\epsilon_{n}}_{t}\in\mathcal{E}_{t}^{\epsilon_{n}}$ with
$\epsilon_{n}\rightarrow\epsilon_{0}\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}$,
then there exists $\mu^{\epsilon_{0}}_{t}\in\mathcal{E}^{\epsilon_{0}}_{t}$
and a subsequence (not relabeled) such that
$\mu^{\epsilon_{n}}_{t}\rightarrow\mu^{\epsilon_{0}}_{t}$ weakly for each
$t\in\mathbb{R}$.

##  2\. Proof of Theorem 1.1

###### Proof.

Proof of Theorem 1.1. Given $\tau\in\mathbb{R}$, it is sufficient to show that
for any $\varphi\in C_{b}(\mathbb{X})$,

$\displaystyle\int_{\mathbb{X}}\varphi(x)(Q^{\epsilon_{0}}_{\tau,t}\eta^{\epsilon_{0}}_{\tau})(dx)=\int_{\mathbb{X}}(P^{\epsilon_{0}}_{\tau,t}\varphi)(x)\eta^{\epsilon_{0}}_{\tau}(dx)=\int_{\mathbb{X}}\varphi(x)\eta^{\epsilon_{0}}_{t}(dx),\
\ \forall\ t\geqslant\tau.$ (2.1)

The first equality is obvious. To prove the second one in (2.1) we consider
the following equality:

$\displaystyle\int_{\mathbb{X}}(P^{\epsilon_{0}}_{\tau,t}\varphi)(x)\eta^{\epsilon_{0}}_{\tau}(dx)-\int_{\mathbb{X}}\varphi(x)\eta^{\epsilon_{0}}_{t}(dx)$
$\displaystyle=\int_{\mathbb{X}}(P^{\epsilon_{0}}_{\tau,t}\varphi)(x)\eta^{\epsilon_{0}}_{\tau}(dx)-\int_{\mathbb{X}}(P^{\epsilon_{0}}_{\tau,t}\varphi)(x)\eta^{\epsilon_{n}}_{\tau}(dx)$
$\displaystyle\ \
+\int_{\mathbb{X}}(P^{\epsilon_{0}}_{\tau,t}\varphi)(x)\eta^{\epsilon_{n}}_{\tau}(dx)-\int_{\mathbb{X}}\varphi(x)\eta^{\epsilon_{n}}_{t}(dx)$
$\displaystyle\ \
+\int_{\mathbb{X}}\varphi(x)\eta^{\epsilon_{n}}_{t}(dx)-\int_{\mathbb{X}}\varphi(x)\eta^{\epsilon_{0}}_{t}(dx).$
(2.2)

For the last line on the right-hand side of (2), since
$\eta^{\epsilon_{n}}_{t}\rightarrow\eta^{\epsilon_{0}}_{t}$ weakly, we know
that for every $\gamma$, there exists $N_{1}\in\mathbb{N}$ such that

$\displaystyle\int_{\mathbb{X}}\varphi(x)\eta^{\epsilon_{n}}_{t}(dx)-\int_{\mathbb{X}}\varphi(x)\eta^{\epsilon_{0}}_{t}(dx)\leqslant\gamma\
\ \mbox{for all $n\geqslant N_{1}$}.$ (2.3)

For the first line on the right-hand side of (2), since
$(P^{\epsilon_{0}}_{\tau,t})_{t\geqslant\tau}$ is Feller and
$\eta^{\epsilon_{n}}_{\tau}\rightarrow\eta^{\epsilon_{0}}_{\tau}$ weakly, we
know that for every $\gamma$, there exists $N_{2}\in\mathbb{N}$ such that

$\displaystyle\int_{\mathbb{X}}(P^{\epsilon_{0}}_{\tau,t}\varphi)(x)\eta^{\epsilon_{0}}_{\tau}(dx)-\int_{\mathbb{X}}(P^{\epsilon_{0}}_{\tau,t}\varphi)(x)\eta^{\epsilon_{n}}_{\tau}(dx)\leqslant\gamma\
\ \mbox{for all $n\geqslant N_{2}$}.$ (2.4)

It remains to consider the second line on the right-hand side of (2). By
assumption CIP we know that for every $\gamma>0$ and $\delta>0$, there exists
$N_{3}\in\mathbb{N}$ such that

$\displaystyle\sup_{x\in\mathbf{K}}\mathbb{P}\bigg{(}\Big{\\{}\omega\in\Omega:\|X^{\epsilon_{n}}(t,\tau,x)-X^{\epsilon_{0}}(t,\tau,x)\|_{\mathbb{X}}\geqslant\delta\Big{\\}}\bigg{)}\leqslant\gamma\
\ \mbox{for all $n\geqslant N_{3}$}.$ (2.5)

Since $\eta^{\epsilon_{n}}_{\tau}\rightarrow\eta^{\epsilon_{0}}_{\tau}$
weakly, by Prohorov theorem (see [7, 8]) we know
$\\{\eta^{\epsilon_{n}}_{\tau}\\}_{n=1}^{\infty}$ is tight on $\mathbb{X}$.
This means that for every $\gamma>0$, there exists a compact set
$\mathbf{K}(\gamma,\tau)\subseteq\mathbb{X}$ such that

$\displaystyle\eta^{\epsilon_{n}}_{\tau}(\mathbb{X}/\mathbf{K}(\gamma,\tau))<\gamma\
\ \mbox{for all $n\in\mathbb{N}$}.$ (2.6)

By $\varphi\in C_{b}(\mathbb{X})$ and the compactness of $\mathbf{K}$ we know
that for every $\gamma>0$, there exists $\eta>0$ such that

$\displaystyle|\varphi(y)-\varphi(z)|<\gamma\ \ \mbox{for all
$y,z\in\mathbf{K}$\ with \ $\|y-z\|_{\mathbb{X}}\leqslant\eta$}.$ (2.7)

Indeed, if (2.7) is false, then there exist $\gamma_{0}>0$,
$y_{n},z_{n}\in\mathbf{K}$ with $\|y_{n}-z_{n}\|_{\mathbb{X}}\leqslant 1/n$
such that $|\varphi(y_{n})-\varphi(z_{n})|\geqslant\gamma_{0}$. Since
$\mathbf{K}$ is a compact subset of $\mathbb{X}$, there exist $x\in\mathbf{K}$
and a subsequence $\\{n_{k}\\}_{k=1}^{\infty}\subseteq\\{n\\}_{n=1}^{\infty}$
such that $\lim_{k\rightarrow\infty}\|y_{n_{k}}-x\|_{\mathbb{X}}=0$ . Then one
can verify $\lim_{k\rightarrow\infty}\|z_{n_{k}}-x\|_{\mathbb{X}}=0$. Since
$\varphi$ is continuous, letting $k\rightarrow\infty$ in
$\gamma_{0}\leqslant|\varphi(y_{n_{k}})-\varphi(z_{n_{k}})|\leqslant|\varphi(y_{n_{k}})-\varphi(x)|+|\varphi(z_{n_{k}})-\varphi(x)|$,
we find a contradiction $\gamma_{0}\leqslant 0$.

Since $\\{\eta^{\epsilon_{n}}_{t}\\}_{t\in\mathbb{R}}$ is an evolution system
of probability measures of $(P^{\epsilon_{n}}_{\tau,t})_{t\geqslant\tau}$ on
$\mathbb{X}$, by (2.5)-(2.7) we find that for all $n\geqslant
N:=\max\\{N_{1},N_{2},N_{3}\\}$,

$\displaystyle\int_{\mathbb{X}}(P^{\epsilon_{0}}_{\tau,t}\varphi)(x)\eta^{\epsilon_{n}}_{\tau}(dx)-\int_{\mathbb{X}}\varphi(x)\eta^{\epsilon_{n}}_{t}(dx)$
$\displaystyle=\int_{\mathbb{X}}\mathbb{E}\big{[}\varphi(X^{\epsilon_{0}}(t,\tau,x))\big{]}\eta^{\epsilon_{n}}_{\tau}(dx)-\int_{\mathbb{X}}\mathbb{E}\big{[}\varphi(X^{\epsilon_{n}}(t,\tau,x))\big{]}\eta^{\epsilon_{n}}_{\tau}(dx)$
$\displaystyle\leqslant\int_{\mathbb{X}}\mathbb{E}\big{[}|\varphi(X^{\epsilon_{n}}(t,\tau,x))-\varphi(X^{\epsilon_{0}}(t,\tau,x))|\big{]}\eta^{\epsilon_{n}}_{\tau}(dx)$
$\displaystyle\leqslant\int_{\mathbb{\mathbf{K}(\gamma,\tau)}}\mathbb{E}\big{[}|\varphi(X^{\epsilon_{n}}(t,\tau,x))-\varphi(X^{\epsilon_{0}}(t,\tau,x))|\big{]}\eta^{\epsilon_{n}}_{\tau}(dx)+2\sup_{x\in\mathbb{X}}|\varphi(x)|\eta^{\epsilon_{n}}_{\tau}(\mathbb{X}/\mathbf{K}(\gamma,\tau))$
$\displaystyle\leqslant\int_{\mathbb{\mathbf{K}(\gamma,\tau)}}\mathbb{E}\big{[}|\varphi(X^{\epsilon_{n}}(t,\tau,x))-\varphi(X^{\epsilon_{0}}(t,\tau,x))|\big{]}\eta^{\epsilon_{n}}_{\tau}(dx)+2\gamma\sup_{x\in\mathbb{X}}|\varphi(x)|$
$\displaystyle\leqslant\int_{\mathbb{\mathbf{K}(\gamma,\tau)}}\int_{\\{\omega\in\Omega:\|X^{\epsilon_{n}}(t,\tau,x)-X^{\epsilon_{0}}(t,\tau,x)\|_{\mathbb{X}}\geqslant\eta\\}}\big{|}\varphi(X^{\epsilon_{n}}(t,\tau,x))-\varphi(X^{\epsilon_{0}}(t,\tau,x))\big{|}\mathbb{P}(d\omega)\eta^{\epsilon_{n}}_{\tau}(dx)$
$\displaystyle\ \
+\int_{\mathbb{\mathbf{K}(\gamma,\tau)}}\int_{\\{\omega\in\Omega:\|X^{\epsilon_{n}}(t,\tau,x)-X^{\epsilon_{0}}(t,\tau,x)\|_{\mathbb{X}}<\eta\\}}\big{|}\varphi(X^{\epsilon_{n}}(t,\tau,x))-\varphi(X^{\epsilon_{0}}(t,\tau,x))\big{|}\mathbb{P}(d\omega)\eta^{\epsilon_{n}}_{\tau}(dx)$
$\displaystyle\ \ +2\gamma\sup_{x\in\mathbb{X}}|\varphi(x)|$
$\displaystyle\leqslant
2\sup_{x\in\mathbb{X}}|\varphi(x)|\eta^{\epsilon_{n}}_{\tau}(\mathbf{K}(\gamma,\tau))\mathbb{P}\bigg{(}\Big{\\{}\omega\in\Omega:\|u^{\epsilon}(t,\tau,x)-u^{\epsilon_{0}}(t,\tau,x)\|_{\mathbb{X}}\geqslant\eta\Big{\\}}\bigg{)}$
$\displaystyle+\gamma\eta^{\epsilon_{n}}_{\tau}(\mathbf{K}(\gamma,\tau))\mathbb{P}\bigg{(}\Big{\\{}\omega\in\Omega:\|u^{\epsilon}(t,\tau,x)-u^{\epsilon_{0}}(t,\tau,x)\|_{\mathbb{X}}<\eta\Big{\\}}\bigg{)}+2\gamma\sup_{x\in\mathbb{X}}|\varphi(x)|$
$\displaystyle\leqslant\gamma(1+4\sup_{x\in\mathbb{X}}|\varphi(x)|).$ (2.8)

Thus, as a result of (2.3)-(2.4) and (2) we have

$\displaystyle\int_{\mathbb{X}}(P^{\epsilon_{0}}_{\tau,t}\varphi)(x)\eta^{\epsilon_{0}}_{\tau}(dx)-\int_{\mathbb{X}}\varphi(x)\eta^{\epsilon_{0}}_{t}(dx)\leqslant\gamma(3+4\sup_{x\in\mathbb{X}}|\varphi(x)|).$

Since $\gamma>0$ is arbitrary, we thus complete the proof. ∎

##  3\. Proof of Theorem 1.3

Let us consider the Banach space
$\ell^{q}=\big{\\{}u=(u_{i})_{i\in\mathbb{Z}}:\sum_{i\in\mathbb{Z}}|u_{i}|^{q}<+\infty\big{\\}}$
endowed with the norm

$\|u\|_{q}=\left\\{\begin{array}[]{ll}\big{(}\sum_{i\in\mathbb{Z}}|u_{i}|^{q}\big{)}^{1/q},&\hbox{if
$q\in[1,\infty)$,}\\\\[6.0pt] \ \sup_{i\in\mathbb{Z}}|u_{i}|,&\hbox{if
$q=\infty$.}\end{array}\right.$

In particular, we write $\|\cdot\|=\|\cdot\|_{2}$. It is known from Wang [15]
that for each $\tau\in\mathbb{R}$ and $u_{0}\in
L^{2}(\Omega,\mathcal{F}_{\tau};\ell^{2})$, equation (1.2) has a unique
solution which is continuous $\ell^{2}$-valued $\mathcal{F}_{t}$-adapted
Markov process $u(\cdot,\tau,u_{0})\in
L^{2}(\Omega,C([\tau,\infty),\ell^{2}))\cap
L^{p}(\Omega,L^{p}(\tau,\infty);\ell^{p}))$. Then we can prove that the
transition operator $(P^{\epsilon}_{\tau,t})_{t\geqslant\tau}$ for
$u(t,\tau,u_{0})$ is Feller, and the process laws hold:
$P^{\epsilon}_{\tau,t}=P^{\epsilon}_{\tau,r}P^{\epsilon}_{r,t}$ and
$Q^{\epsilon}_{\tau,t}=Q^{\epsilon}_{r,t}Q_{\tau,r}^{\epsilon}$,
$-\infty<\tau\leqslant r\leqslant t<+\infty$.

Next, we derive different uniform estimates of time average of the solutions.

######  Lemma 3.1.

If $\int_{-\infty}^{t}e^{\lambda r}\|g(r)\|^{2}dr<\infty$ for each
$t\in\mathbb{R}$, then we have the following results.

(i) For each $t\in\mathbb{R}$ and $\mathbb{N}\ni k>-t$,

$\displaystyle\sup_{\epsilon\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}}\frac{1}{k+t}\int_{-k}^{t}\mathbb{E}[\|u^{\epsilon}(t,\tau,u_{0})\|^{2}]d\tau+\sup_{\epsilon\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}}\frac{1}{k+t}\int_{-k}^{t}\int_{\tau}^{t}e^{\lambda(r-t)}\mathbb{E}\left[\|u^{\epsilon}(r,\tau,u_{0})\|^{2}\right]drd\tau$
$\displaystyle\leqslant\frac{c}{(k+t)}\mathbb{E}\big{[}\|u_{0}\|^{2}\big{]}+c\int_{-\infty}^{t}e^{\lambda(r-t)}\|g(r)\|^{2}dr,$
(3.1)

where $c>0$ is a constant independent of $\epsilon$, $t$, $k$ and $u_{0}$.

(ii) For each $t\in\mathbb{R}$ and compact set $\mathbf{K}$ of $\ell^{2}$, the
solutions satisfy

$\displaystyle\lim_{n\rightarrow\infty}\sup_{\epsilon\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}}\sup_{\mathbb{N}\ni
k>-t}\sup_{u_{0}\in\mathbf{K}}\frac{1}{k+t}\int_{-k}^{t}\sum_{|i|\geqslant
n}\mathbb{E}[|u^{\epsilon}(t,\tau,u_{0})|^{2}]d\tau=0.$

(iii) For each $t\in\mathbb{R}$ and bounded set $\mathbf{B}$ of $\ell^{2}$,
the solutions satisfy

$\displaystyle\lim_{n,k\rightarrow\infty}\sup_{\epsilon\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}}\sup_{u_{0}\in\mathbf{B}}\frac{1}{k+t}\int_{-k}^{t}\sum_{|i|\geqslant
n}\mathbb{E}[|u^{\epsilon}(t,\tau,u_{0})|^{2}]d\tau=0.$

###### Proof.

(i) Applying Itô’s formula to (1.2), we infer that for any
$\epsilon\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}$,

$\displaystyle\frac{d}{dt}\mathbb{E}\big{[}\|u(t)\|^{2}\big{]}+\frac{3}{2}\lambda\mathbb{E}\big{[}\|u(t)\|^{2}\big{]}\leqslant
c\|g(t)\|^{2},$ (3.2)

where $c>0$ is a constant. Multiplying (3.2) by $e^{\lambda t}$ and
integrating over $(\tau,t)$, we deduce

$\displaystyle\mathbb{E}\big{[}\|u^{\epsilon}(t,\tau,u_{0})\|^{2}\big{]}$
$\displaystyle+\frac{\lambda}{2}\int_{\tau}^{t}e^{\lambda(r-t)}\mathbb{E}\left[\|u^{\epsilon}(r,\tau,u_{0})\|^{2}\right]dr$
$\displaystyle\leqslant
e^{\lambda(\tau-t)}\mathbb{E}\big{[}\|u_{0}\|^{2}\big{]}+c\int_{-\infty}^{t}e^{\lambda(s-t)}\|g(s)\|^{2}ds.$
(3.3)

Integrating (3) with respect to $\tau$ over $(-k,t)$, we obtain (3.1).

(ii)-(iii) By a cut-off technique as used by Wang [15, Lemma 4.2] (see also
[16]) one can derive, for any
$\epsilon\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}$,

$\displaystyle\mathbb{E}\Big{[}\sum_{|i|\geqslant
2n}|u_{i}(t,\tau,u_{0})|^{2}\Big{]}$ $\displaystyle\leqslant
e^{\lambda(\tau-t)}\sum_{|i|\geqslant
n}|u_{0,i}|^{2}+c\int_{\tau}^{t}e^{\lambda(r-t)}\sum_{|i|\geqslant
n}|g_{i}(r)|^{2}dr$ $\displaystyle\ \
+\frac{c}{n}\int_{\tau}^{t}e^{\lambda(r-t)}\mathbb{E}\left[\|u^{\epsilon}(r,\tau,u_{0})\|^{2}\right]dr$
(3.4)

Integrating (3) with respect to $\tau$ over $(-k,t)$, we have

$\displaystyle\frac{1}{k+t}\int_{-k}^{t}\mathbb{E}\Big{[}\sum_{|i|\geqslant
2n}|u_{i}(t,\tau,u_{0})|^{2}\Big{]}d\tau$
$\displaystyle\leqslant\frac{1}{\lambda(k+t)}\sum_{|i|\geqslant
n}|u_{0,i}|^{2}+c\int_{-\infty}^{t}e^{\lambda(r-t)}\sum_{|i|\geqslant
n}|g_{i}(r)|^{2}dr$ $\displaystyle\ \
+\frac{c}{n(k+t)}\int_{-k}^{t}\int_{\tau}^{t}e^{\lambda(r-t)}\mathbb{E}\left[\|u^{\epsilon}(r,\tau,u_{0})\|^{2}\right]drd\tau.$

This along with $\int_{-\infty}^{t}e^{\lambda r}\|g(r)\|^{2}dr<\infty$ and
(3.1) completes the proof of (ii)-(iii). ∎

###### Proof.

Proof of Theorem 1.3. The proof is completed by an extended Krylov-Bogolyubov
method proposed by Da Prato and Röckner [10]. For $n\in\mathbb{N}$, we let
$\chi_{[-n,n]}$ be the characteristic function of $[-n,n]$. Then
$u^{\epsilon}(t,\tau,u_{0})=\tilde{u}^{\epsilon}_{n}(t,\tau,u_{0})+\hat{u}^{\epsilon}_{n}(t,\tau,u_{0})$
with
$\tilde{u}^{\epsilon}_{n}(t,\tau,u_{0})=\big{(}\chi_{[-n,n]}(|i|)u^{\epsilon}_{i}(t,\tau,u_{0})\big{)}_{i\in\mathbb{Z}}$
and
$\hat{u}^{\epsilon}_{n}(t,\tau,u_{0})=\big{(}(1-\chi_{[-n,n]}(|i|))u^{\epsilon}_{i}(t,\tau,u_{0})\big{)}_{i\in\mathbb{Z}}$.
Given $\delta>0$, $l\in\mathbb{N}$ and $m,k\in\mathbb{N}$ with $m\leqslant k$,
by (i)-(ii) of Lemma 3.1 there exist $n_{l}^{\delta}\in\mathbb{N}$ and $C>0$
independent on $\epsilon$, $m$ and $k$ such that

$\displaystyle\sup_{\epsilon\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}}\frac{1}{k-m}\int_{-k}^{-m}\mathbb{E}\big{[}\|u(-m,\tau,0)\|^{2}\big{]}d\tau\leqslant
C\ $

and

$\displaystyle\
\sup_{\epsilon\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}}\frac{1}{k-m}\int_{-k}^{-m}\mathbb{E}\big{[}\|\hat{u}^{\epsilon}_{n_{l}^{\delta}}(-m,\tau,0)\|^{2}\big{]}d\tau<\frac{\delta}{2^{4l}}.$

Define

$\mathcal{Y}^{\delta}_{l}=\bigg{\\{}u\in\ell^{2}:u_{i}=0\ \mbox{for
}|i|>n_{l}^{\delta}\ \mbox{and}\
\|u\|\leqslant\frac{2^{l}\sqrt{C}}{\sqrt{\delta}}\bigg{\\}}$

and

$\mathcal{Z}^{\delta}_{l}=\Big{\\{}u\in\ell^{2}:\|u-v\|\leqslant\frac{1}{2^{l}}\
\mbox{for some $v\in\mathcal{Y}^{\delta}_{l}$}\Big{\\}}.$

Define a probability measure
$\eta^{\epsilon}_{k,m}=\frac{1}{k-m}\int_{-k}^{-m}\mathbb{P}\Big{\\{}\omega\in\Omega:u^{\epsilon}(-m,\tau,0)\in\cdot\Big{\\}}d\tau$
on $\ell^{2}$. Then by Chebychev’s inequality one can verify

$\displaystyle\eta^{\epsilon}_{k,m}(\ell^{2}\setminus\mathcal{Z}^{\delta}_{l})$
$\displaystyle\leqslant\frac{\delta}{2^{2l}C}\frac{1}{k-m}\int_{-k}^{-m}\mathbb{E}\big{(}\|u^{\epsilon}(-m,\tau,0)\|^{2}\big{)}d\tau$
$\displaystyle\ \
+\frac{2^{2l}}{k-m}\int_{-k}^{-m}\mathbb{E}\big{[}\|\hat{u}^{\epsilon}_{n^{\delta}_{l}}(-m,\tau,0)\|^{2}\big{]}d\tau\leqslant\frac{\delta}{2^{2l-1}}.$
(3.5)

Let $\mathcal{Z}^{\delta}=\bigcap_{l=1}^{\infty}\mathcal{Z}^{\delta}_{l}$.
Since one can verify that $\mathcal{Z}^{\delta}$ is closed and totally bounded
in $\ell^{2}$, it is compact in $\ell^{2}$. By (3) we know
$\eta^{\epsilon}_{k,m}(\ell^{2}\setminus\mathcal{Z}^{\delta})\leqslant\sum_{l=1}^{\infty}\eta^{\epsilon}_{k,m}(\ell^{2}\setminus\mathcal{Z}^{\delta}_{l})<\delta$.
Then $\\{\eta^{\epsilon}_{k,m}\\}_{\mathbb{N}\ni k\geqslant m}$ is tight on
$\ell^{2}$ for each fixed $m\in\mathbb{N}$. Then
$\\{\eta^{\epsilon}_{k,m}\\}_{\mathbb{N}\ni k\geqslant m}$ is tight on
$\ell^{2}$ for each $m\in\mathbb{N}$. Then there exists
$\eta^{\epsilon}_{m}\in\mathcal{P}(\ell^{2})$ and a subsequence (not
relabeled) such that $\eta^{\epsilon}_{k,m}\rightarrow\eta^{\epsilon}_{m}$
weakly as $k\rightarrow\infty$. For each fixed $t\in\mathbb{R}$, we choose
$m\in\mathbb{N}$ such that $-m\leqslant t$, and define
$\mu^{\epsilon}_{t}:=Q^{\epsilon}_{-m,t}\eta^{\epsilon}_{m}$. One can verify
that this definition is independent of the choice of $m$. Then for every fixed
$t\in\mathbb{R}$ and for any $\mathbb{N}\ni m\geqslant-\tau\geqslant-t$, we
have
$Q^{\epsilon}_{\tau,t}\mu^{\epsilon}_{\tau}=Q^{\epsilon}_{\tau,t}Q^{\epsilon}_{-m,\tau}\eta^{\epsilon}_{-m}=(P^{\epsilon}_{-m,\tau}P^{\epsilon}_{\tau,t})^{*}\eta^{\epsilon}_{-m}=Q^{\epsilon}_{-m,t}\eta_{-m}=\mu^{\epsilon}_{t}$.
This complete the proof. ∎

##  4\. Proof of Theorem 1.4

###### Proof.

Proof of Theorem 1.4. (i) For every $\delta>0$, $t\in\mathbb{R}$,
$\epsilon\in[a,b]$ and
$\mu_{t}^{\epsilon}\in\bigcup_{\epsilon\in[a,b]}\mathcal{E}_{t}^{\epsilon}$,
we shall find a compact set $\mathcal{Z}^{\delta}(t)$ independent of
$\epsilon$ such that $\mu_{t}^{\epsilon}(\mathcal{Z}^{\delta}(t))>1-\delta$.
Given $t\in\mathbb{R}$ and $l\in\mathbb{N}$, by (i) and (iii) of Lemma 3.1
there exists $n^{\delta}_{l}(t)\in\mathbb{N}$,
$K_{l}^{\delta}(t)\in\mathbb{N}$ and $C(t)>0$ independent of $\epsilon$ and
$u_{0}$ such that

$\displaystyle\sup_{k\geqslant
K_{l}^{\delta}(t)}\sup_{\epsilon\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}}\sup_{u_{0}\in\ell^{2}}\frac{1}{k+t}\int_{-k}^{t}\mathbb{E}\big{[}\|u(t,\tau,u_{0})\|^{2}\big{]}d\tau\leqslant
C(t),$ (4.1)
$\displaystyle\sup_{\epsilon\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}}\sup_{u_{0}\in\ell^{2}}\frac{1}{k+t}\int_{-k}^{t}\mathbb{E}\big{[}\|\hat{u}^{\epsilon}_{n^{\delta}_{l}(t)}(t,\tau,u_{0})\|^{2}\big{]}d\tau<\frac{\delta}{2^{4l}}.$
(4.2)

Define $\mathcal{Y}^{\delta}_{l}(t):=\bigg{\\{}u\in\ell^{2}:u_{i}=0\ \mbox{for
}|i|>n^{\delta}_{l}(t)\ \mbox{and}\
\|u\|\leqslant\frac{2^{l}\sqrt{C(t)}}{\sqrt{\delta}}\bigg{\\}}$,
$\mathcal{Z}_{l}^{\delta}(t):=\Big{\\{}u\in\ell^{2}:\|u-v\|\leqslant\frac{1}{2^{l}}\
\mbox{for some $v\in\mathcal{Y}^{\delta}_{l}(t)$}\Big{\\}}$ and
$\mathcal{Z}^{\delta}(t):=\bigcap_{l=1}^{\infty}\mathcal{Z}^{\delta}_{l}(t)$.
Note that $\mathcal{Z}^{\delta}(t)$ is compact in $\ell^{2}$. In what follows
we prove $\mu_{t}^{\epsilon}(\mathcal{Z}^{\delta}(t))>1-\delta$. Denote by
$\mathcal{X}^{\delta}_{n}(t):=\bigcap_{l=1}^{n}\mathcal{Z}^{\delta}_{l}(t)$
for $n\in\mathbb{N}$. Then
$\mu_{t}^{\epsilon}(\mathcal{Z}^{\delta}(t))=\mu_{t}^{\epsilon}\big{(}\bigcap_{n=1}^{\infty}\mathcal{X}^{\delta}_{n}(t)\big{)}=\lim\limits_{n\rightarrow\infty}\mu_{t}^{\epsilon}\big{(}\mathcal{X}^{\delta}_{n}(t)\big{)}$,
and hence there exists $N=N(\delta,t)\in\mathbb{N}$ such that
$0\leqslant\mu_{t}^{\epsilon}(\mathcal{X}^{\delta}_{n}(t))-\mu_{t}^{\epsilon}(\mathcal{Z}^{\delta}(t))\leqslant\delta/3$
for all $n\geqslant N$. By Fubini’s theorem and the definition of
$\\{\mu^{\epsilon}_{t}\\}_{t\in\mathbb{R}}$ of
$(P^{\epsilon}_{\tau,t})_{t\geqslant\tau}$, we find that for all
$\tau\leqslant t\in\mathbb{R}$,

$\displaystyle\eta_{t}^{\epsilon}(\mathcal{X}^{\delta}_{N}(t))$
$\displaystyle=\int_{\ell^{2}}\mathbb{P}\bigg{(}\Big{\\{}\omega\in\Omega:u^{\epsilon}(t,\tau,x)\in\mathcal{X}^{\delta}_{N}(t)\Big{\\}}\bigg{)}\mu_{\tau}^{\epsilon}(dx)$
$\displaystyle=\int_{\ell^{2}}\mathbb{P}\bigg{(}\Big{\\{}\omega\in\Omega:u^{\epsilon}(t,\tau,x)\in\mathcal{X}^{\delta}_{N}(t)\Big{\\}}\bigg{)}h^{\epsilon}(\tau)\mu^{\epsilon}(dx)$
$\displaystyle=\frac{1}{k+t}\int_{-k}^{t}\int_{\ell^{2}}\mathbb{P}\bigg{(}\Big{\\{}\omega\in\Omega:u^{\epsilon}(t,\tau,x)\in\mathcal{X}^{\delta}_{N}(t)\Big{\\}}\bigg{)}h^{\epsilon}(\tau)\mu^{\epsilon}(dx)d\tau$
$\displaystyle=\int_{\ell^{2}}\frac{1}{k+t}\int_{-k}^{t}\mathbb{P}\bigg{(}\Big{\\{}\omega\in\Omega:u^{\epsilon}(t,\tau,x)\in\mathcal{X}^{\delta}_{N}(t)\Big{\\}}\bigg{)}h^{\epsilon}(\tau)d\tau\mu^{\epsilon}(dx),$
(4.3)

where $\mu^{\epsilon}\in\mathcal{P}(\ell^{2})$ and
$h^{\epsilon}:\mathbb{R}\rightarrow[0,1]$ such that
$\mu_{\tau}^{\epsilon}(\cdot)=h^{\epsilon}(\tau)\mu^{\epsilon}(\cdot)$. As
before, by (4.1), we can verify

$\displaystyle\limsup_{k\rightarrow\infty}\frac{1}{k+t}\int_{-k}^{t}\mathbb{P}\bigg{(}\Big{\\{}\omega\in\Omega:u^{\epsilon}(t,\tau,x)\notin\mathcal{X}^{\delta}_{N}(t)\Big{\\}}\bigg{)}h^{\epsilon}(\tau)d\tau$
$\displaystyle\leqslant\sum_{l=1}^{N}\frac{\delta}{2^{2l}C(t)}\limsup_{k\rightarrow\infty}\frac{1}{k+t}\int_{-k}^{t}\mathbb{E}\big{[}\|\tilde{u}^{\epsilon}_{n^{\delta}_{l}(t)}(t,\tau,x)\|^{2}\big{]}h^{\epsilon}(\tau)d\tau$
$\displaystyle\ \
+\sum_{l=1}^{N}2^{2l}\limsup_{k\rightarrow\infty}\frac{1}{k+t}\int_{-k}^{t}\mathbb{E}\big{[}\|\hat{u}^{\epsilon}_{n^{\delta}_{l}(t)}(t,\tau,x)\|^{2}\big{]}h^{\epsilon}(\tau)d\tau\leqslant\sum_{l=1}^{N}\frac{\delta}{2^{2l-1}}\leqslant\frac{2\delta}{3}.$
(4.4)

Then by Fatou’s lemma, (4) and (4) we find

$\displaystyle\eta_{t}^{\epsilon}(\mathcal{X}^{\delta}_{N}(t))$
$\displaystyle=\liminf_{k\rightarrow\infty}\int_{\ell^{2}}\frac{1}{k+t}\int_{-k}^{t}\mathbb{P}\bigg{(}\Big{\\{}\omega\in\Omega:u^{\epsilon}(t,\tau,x)\in\mathcal{X}^{\delta}_{N}(t)\Big{\\}}\bigg{)}h^{\epsilon}(\tau)d\tau\mu^{\epsilon}(dx)$
$\displaystyle\geqslant
1-\int_{\ell^{2}}\limsup_{k\rightarrow\infty}\bigg{[}\frac{1}{k+t}\int_{-k}^{t}\mathbb{P}\bigg{(}\Big{\\{}\omega\in\Omega:u^{\epsilon}(t,\tau,x)\notin\mathcal{X}^{\delta}_{N}(t)\Big{\\}}\bigg{)}h^{\epsilon}(\tau)d\tau\bigg{]}\mu^{\epsilon}(dx)\geqslant\frac{\delta}{3}.$

This concludes the proof of (i).

(ii) For every $\tau\in\mathbb{R}$, $t\geqslant\tau$ $\delta>0$,
$\epsilon_{0}\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}$ and compact
set $\mathbf{K}$ of $\ell^{2}$, since we can prove that all uniform estimates
of the solutions are uniform for
$\epsilon\in\big{[}0,\frac{\sqrt{\lambda}}{2\delta}\big{]}$, by a stoping time
argument, we can prove

$\displaystyle\lim_{\epsilon\rightarrow\epsilon_{0}}\sup_{u_{0}\in\mathbf{K}}\mathbb{P}\bigg{(}\big{\\{}\omega\in\Omega:\|u^{\epsilon}(t,\tau,u_{0})-u^{\epsilon_{0}}(t,\tau,u_{0})\|\geqslant\delta\big{\\}}\bigg{)}=0.$
(4.5)

The proof of (4.5) is quite similar to the autonomous case as in [5, 6, 13,
14], the details are omitted here. Then by (i), (4.5), Theorem 1.1 and
Prohorov’s theorem, we complete the proof of (ii). ∎

##  5\. Acknowledgements

Renhai Wang was supported by China Postdoctoral Science Foundation under grant
numbers 2020TQ0053 and 2020M680456. Tomás Caraballo was supported by the
Spanish Ministerio de Ciencia, Innovación y Universidades project
PGC2018-096540-B-I00, and Junta de Andalucia (Spain) under projects US-1254251
and P18-FR-4509. Nguyen Huy Tuan was supported by Van Lang University.

## References

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  * [8] G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Notes, 229, Cambridge University Press, 1996.
  * [9] G. Da Prato and M. Röckner, Dissipative stochastic equations in Hilbert space with time dependent coefficients, Rend. Lincei Mat. Appl. 17 (2006) 397-403.
  * [10] G. Da Prato and M. Röckner, A note on evolution systems of measures for time-dependent stochastic differential equations, Progress in Probability, 59 (2009) 115-122.
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rightsretained

# Visualizing the Weird and the Eerie

Matthew Brehmer

Tableau Research<EMAIL_ADDRESS>
(2023)

###### Abstract

In this brief essay, I reflect on how Mark Fisher’s definitions of the weird
and the eerie could be applied in communicative data visualization. I ask how
visualization designers might elicit these two impressions when a viewer is
engaging with multimodal representations of data. I argue that there are
situations in which viewers should feel uncertain or suspicious of unseen
forces that account for the presence or absence of audiovisual patterns.
Finally, I conclude that the ability to appreciate the weird and the eerie in
data is particularly important at this moment in history, one marked by
significant ecological and economic disruption.

###### keywords:

Data visualization; reflection; affect; agency; semantics.

††conference: alt.VIS October 2023, Melbourne, Victoria, Australia

##  1 What is Weird? What is Eerie?

In his 2016 book The Weird and the Eerie [8], the late writer Mark Fisher set
out to define these two constructs as both affects and as modes: “modes of
film and fiction, modes of perception, ultimately, you might even say, modes
of being.” In this collection of essays, he references a selection of
critically notable novels, films, television series, and pieces of music that
evoke these two affects. His writing led to my own realization that data
visualization may also be capable of instilling a sense of the weird and the
eerie.

Fisher argues that the weird and the eerie are distinct from one another, even
though they are both often associated with (and conflated in) the genres of
horror, science fiction, and especially post-apocalyptic speculative fiction.
However, the weird and the eerie are not necessarily horrific or devastating.

According to Fisher, the weird is “that which does not belong,” that is, the
presence of something that seems wrong or at least strange. In this sense, the
weird can be an object of fascination, something to be sought out and enjoyed.
I think of my own enjoyment of certain music or novels that exhibit this sense
of strangeness. Similarly, young children are particularly amused by games of
“one of these things is not like the other.” But the weird goes beyond the
placement of unfamiliar things within a familiar context; the weird is also
invoked when familiar things appear out of place, or when a familiar ordering
of events is seemingly violated. To the extent that events can be weird,
Fisher remarks that an archaic meaning of weird refers to fate, or a “twisted
form of time and causality.”

In contrast to the weird, Fisher defines the eerie as a reaction to situations
“when there is something present when there should be nothing, or when there
is nothing present when there should be something.” In either manifestation of
the eerie, there arises a question of agency, a question of who or what is
responsible for the lack of absence or the lack of presence, and ultimately a
question of what motivated — or continues to motivate — this unseen agent.

##  2 Visualization Design, Affect, and Semantics

This essay is motivated in part by a renewed discussion of affect in
visualization design. There are undoubtedly works of data visualization that
elicit positive affective responses; examples include the playful and humorous
xkcd comic by Randall Munroe [23], Tyler Vigen’s Spurious Correlations [29],
or the oeuvre of Nigel Holmes [15]. Meanwhile, other presentations about data
can evoke profound sadness, such as in Neil Halloran’s multimodal data
documentaries [10]. Beyond the polarities of joy and sadness and simple models
of affect spanning the dimensions of valence and arousal, thoughtful
visualization design can elicit a wider gamut of affective responses, as noted
in a recent paper by Elsie Lee-Robbins and Eytan Adar [21]. The most concrete
affective learning goal in communicative visualization is persuading viewers
to take action [26], or change behavior based on an affective response to a
representation of data. To compel to action, however, first requires
strengthening or changing viewers’ beliefs. Encountering the weird is an
opportunity to change what one believes; Fisher writes that the very existence
of the weird object results in the conclusion that “the weird thing is not
wrong [—] it is our conceptions that must be inadequate.”

Another motivation for this essay stems from my interest in visualization
techniques that integrate semantic cues into encodings of data. For instance,
the cardinal direction ‘up’ might be good in one chart (e.g., profits are
rising), whereas up is bad in another chart (e.g., global average temperatures
are rising); annotations, captions, color palettes, iconography, and other
graphical elements can certainly reinforce these good / bad semantics. The use
of figurative elements and frames [5] can also serve as contextual cues,
suggesting the topic of the visualization upon a cursory glance.

Beyond suggesting semantic associations or that particular visual patterns
exhibited by the data are good or bad, it is less clear how to effectively
communicate that some patterns are simply weird or strange. Similarly, it is
not universally obvious how visualization designers can communicate that a
particular pattern is present or absent, as well as how to express a lack of
an explanation for these presences or absences. The impression of weirdness or
eeriness will dissipate if an explanation is offered, such as when we annotate
erratic spikes and troughs in line charts with contextual information external
to the data [16].

##  3 Related Affects in Experience Design

Before I reflect further on the weird and the eerie in the context of
visualization design, I’ll briefly consider related affects from a broader
human-computer interaction perspective. One particularly memorable alt.CHI
session that I attended in 2011 included Steve Benford and Brendan Walker’s
talk about the design of fearsome interactive experiences [22], i.e., those
that instil a sense of terror that could be implemented in environments such
as virtual reality games, escape rooms, or amusement parks. As mentioned
earlier, the weird and the eerie ere not necessarily terrifying, but I expect
that some of the visual and interaction design patterns that elicit terror
could be applied when visualizing the weird and the eerie. Consider narrative
presentations about data, such as data videos or scrollytelling articles: the
pacing or delivery could establish a sense of tension, delaying or withholding
any revelatory explanation, prompting viewers to imagine or anticipate the
forces responsible for weird and eerie audiovisual patterns. An egocentric
perspective could accentuate this tension; for an example of such a
perspective, I encourage the reader to Ride the Nasdaq [17] (in which the The
Wall Street Journal’s Roger Kenny and Ana Asnes Becker transform a line chart
into a 3D roller coaster ride).

Another related affect, one less visceral than terror, is that of a sense of
uncertainty induced by ambiguity. William Gaver and colleagues [9] proposed a
set of guidelines for eliciting this reaction to an ambiguous interactive
experience, and of these, two are particularly applicable to eliciting a sense
of the weird or the eerie in data. These guidelines are “point[ing] out things
without explaining why” and “cast[ing] doubt on sources to provoke independent
assessment.” In observing these guidelines, attention can be drawn to the
weird without providing the comfort of an explanation. Similarly, a
visualization designer can elicit alternate hypotheses for a failure of
absence or a failure of presence; of course, unintentional human error leading
to inconsistent data collection could be to blame, but there could also an
unseen agent at play, or an agent that is compromising the processes of data
collection and retention.

##  4 Visualizing the Weird

Visualization design consultant Andy Kirk remarked [18] that people “are
naturally drawn to gaps and exceptions and things that don’t really fit in
with the rest.” And yet, designing representations of data that emphasize
these aspects can be “a very fiddly challenge.”

In a paper about scatterplot matrices, Leland Wilkinson and colleagues [30]
observed that real datasets “frequently contain […] outliers, missing data,
and “just plain weird” bivariate distributions” [emphasis added]. A
professional data analyst is trained to spot these weird distributions,
distributions that seem strange or wrong given the context of the data. But
how can the analyst effectively communicate that these distributions are weird
to a lay audience?

I echo the argument of Enrico Bertini and colleagues [3], that despite the
precision and generalizability that scatterplots and their variations offer,
using other forms of representation can promote “serendipitous discovery,
educational impact, hedonic response, or changes in behavior” [emphasis
added]. As with the hedonic enjoyment of weird music or novels, I similarly
enjoy encountering weird representations of data. I admit that part of this
enjoyment can be attributed to what Fisher describes as “seeing the familiar
and the conventional becoming outmoded,” which in my case is a rejection of
dogmatic minimalism and the prioritization of perceptual precision in
visualization design. Beyond this enjoyment is an appreciation that some weird
choices of representation can accentuate weird patterns in data. For instance,
consider the connected scatterplot [11]: it can resemble a line chart, but it
can also exhibit visually prominent line reversals and loops, indicative of
weird events in the relationship between two variables. Absent weird events,
the connected scatterplot is often a poor choice for showing bivariate
temporal relationships [19].

The connected scatterplot is not alone in its capacity to draw attention to
the weird. Visualization consultant Maarten Lambrechts has amassed a catalog
of techniques that he calls xenographics [20], or “weird but (sometimes)
useful charts.” The growing corpus of xenographics suggests that given a weird
observation in data, there may (or should) exist a suitably weird way to
represent it.

Beyond their capacity to herald the presence of weird patterns, xenographics
are also worth discussing for another reason. Several documented xenographics
could be described as hybrids or mash-ups of conventional chart types (e.g., a
stacked area alluvial diagram, a scatterplot of line charts). A hybridization
of form is also characteristic of grotesque art and architecture, in which the
forms of animal and plant, or the organic and inorganic, are combined in ways
that defy the conventions of representational art. According to Fisher, the
grotesque is weird: grotesque works alternately fascinate and repel viewers,
ultimately prompting a reaction that they seem wrong. Hybrid xenographics
could thus be thought of as a grotesque class of data visualization
techniques, designed to communicate what no single chart could do on its
own111Information graphics that meld figurative and abstract elements, such as
Nigel Holmes [15]’s use of images as axes or coordinate spaces (i.e.,
figurative frames [5]) could also be characterized as grotesque..

One final aspect of the weird worth considering here is the impression of not
belonging, that the origin of the weird entity is some other domain. Fisher
discusses how this aspect of the weird is reinforced via visual imagery
signifying an opening between the familiar and the unfamiliar: “The centrality
of doors, thresholds, and portals means that the notion of the between is
crucial to the weird.” For me, this suggests an opportunity to explore these
motifs when revealing weird data observations, with deliberate and
metaphorical animations or interactive affordances: opening a door, parting a
curtain, or traveling through a tunnel.

##  5 Visualizing the Eerie

Fisher’s definition of the eerie as a failure of presence brought to mind
Nicole Hengesbach’s 2021 Information+ presentation [14], which included a
review of the four types of missingness in data, referring to a case study on
the mapping of urban greenery. If an entity (e.g., a tree) exists in reality
and is either completely or partially captured as data, Hengesbach and
colleagues would characterize the representation as complete (or at least
partially complete). However, if the entity is present in reality but
unaccounted for in the data, this is an absence, while the converse is an
emptiness. Lastly, there is nothingness: entities are neither present in
reality or represented in the data. Given these categories or missingness, an
absence or an emptiness could be seen as eerie if there is ambiguity with
respect to the processes of data collection and redaction: who was
responsible, what their motivations were, and if any external forces acted to
bring about absences or emptiness. Nothingness can also be eerie, even when
the processes of data collection are relatively transparent, and particularly
in cases where something or anything was expected222e.g., visualizing the
nothingness of space [31] or ocean depths [2]..

In communicative visualization design, evoking an eerie affect around absences
in data requires establishing what expected patterns look like or how they
have manifested in the past, contrasting presence with absence, and
withholding or deferring possible explanations for this absence. An example of
this formula is Isao Hashimoto’s 1945 – 1998 [12], a multimodal combination of
visualization and sonification, animating every atomic detonation during this
time span on a world map, with different colors and tones indicating the
nation responsible for each detonation. Following several cacophonous decades
with hundreds of detonations and a barrage of color and sound, the last decade
is markedly quieter: a relatively eerie calm has set in. The YouTube video
description includes the statement: “No letter is used for equal messaging to
all viewers without language barrier,” however I maintain that by withholding
any verbal or written explanation in the video itself accentuates an eerie
affect. If it were possible to somehow contain work like this in a time
capsule for a human (or non-human) civilization to discover in the distant
future333e.g., information visualized on Voyager’s Golden Record or on the
moon-bound sapphire discs of the planned Sanctuary project [28]., I expect
that a sense of the eerie would remain even if the viewer cannot readily
interpret its meaning. The viewer may not only question what these audiovisual
patterns represent, but also what or who is responsible for their cessation,
similar how we currently speculate about Stonehenge, the Nazca Lines, crop
circles, and other eerie traces of earlier / alien civilizations.

If a multimodal contrast between hectic noise and calm silence can elicit the
eerie, animation design is another approach though which to reinforce a sense
of the eerie in representations of data. If viewers are to believe that an
unseen force or agent is acting upon what the data points represent, their
disappearance could be animated in such a way that metaphorically suggests the
presence of such forces, such as via simulated physics (e.g., a breeze blows
the data points away) or via biomimicry [7] (e.g., a school / flock of data
points scatter and swim / fly away).

An eerie affect induced by a failure of absence could arguably be easily
triggered given our propensity to apophenia, a bias in which we recognize
patterns and attribute their appearance to an entity (or entities) that have
agency. In most cases, however, there is no agent, and the pattern is
fleeting, illusory, or insignificant. Pareidolia is a form of apophenia that
is specific to visual stimuli, and visualization designers must exercise care
to ensure that their viewers do not succumb to pareidolia unless this is the
intent, such as in Michael Brenner’s humorous Viz in the Wild project [4], in
which spurious visual patterns appearing in photographs are captioned as
charts. However, in select cases, there is an explanation for salient patterns
where none were expected or designed. Dietmar Offenhuber’s notion of
autographic visualization [25] is useful here, as the interpretation of
material traces left by environmental phenomena444e.g., visual patterns of
growth, oscillation, decay, oxidation, saturation, the accumulation of
particulate matter, etc. can often yield meaningful measurements as well as
revelations regarding the process of data generation. Meaningful measurements
depend on several design operations, including the juxtaposition of frames,
annotations, and decoding scales alongside material traces. By undertaking
these operations and measurements, practitioners can arrive at explanations
for the patterns, which may include an attribution of agency, such as air
pollution caused by human industry and transportation. A sense of the eerie is
instantiated when an autographic visualization interpretation either fails to
attribute this agency or fails to explain the motivations of an identifiable
agent. Lastly, it is similarly eerie when an expected material trace vanishes,
and we are bereft of any explanation for this disappearance. I’ll offer a
personal example: for years I regularly visited a Florida beach that was
pockmarked with tiny holes made by sand fleas, a visual pattern distributed
evenly along the shoreline. One year, this pattern had vanished altogether
along one stretch of the beach, and in that moment I could not explain this
lifeless failure of presence555I would later learn that the adjacent resort
hotel had began importing sand from elsewhere, destroying the natural habitat
of the fleas..

##  6 Implications for Visualization Design

Until this point, my essay has been predominantly reflective, an attempt at
connecting the visualization community with ideas that originated from a
critical analysis of popular culture. However, I will nevertheless try to
summarize some prescriptive implications for visualization design, or at least
suggest some creative experiments to undertake:

  * •

Acknowledge situations where it is appropriate for data visualization to make
viewers uncomfortable.

  * •

Employ egocentric perspectives and doorway motifs for distinguishing the
familiar from the weird.

  * •

Avoid manipulating viewers’ impressions by pairing an expected or familiar
data observation with a weird choice of representation, reserving the
grotesque art of xenographics for communicating truly weird patterns in data.

  * •

Withhold or delay revelatory explanations or annotations for failures of
absence and failures of presence.

  * •

Provide subtle indications of agency via animation.

##  7 Visualization for Weird and Eerie Times

We are living in weird times [27]. Around the world, we have experienced weird
weather fluctuations in recent years. Where I live now, the color of the sky
can be a weird dull orange in late summer. Meanwhile, invasive species and
algal blooms have altered ecosystems. While I was growing up, there did not
used to be ticks where I lived, but these weirdly resilient insects and the
threat of Lyme disease can now be found there. In the economic sphere,
sequences of weird events continue to disrupt currencies, real estate markets,
and supply chains. How we represent these and other weird happenings, whenever
we capture them as data, should be commensurately weird, particularly if we
are to communicate just how weird they are. For example, conventions in
meteorological visualization may not capture how weird these extreme weather
events are. New visual idioms like Ed Hawkins’ Warming Stripes [13] are useful
in this regard, though even these elicit questions of how we should visualize
future extreme values: while we could renormalize the color palettes to make
deep blues and deep reds more apparent, renormalization may fail to capture
the severity of the next weird event.

Our world is also increasingly eerie. Jenny Odell recently wrote about the
feeling of living in apocalyptic times [24], which is fitting given how post-
apocalyptic films and novels often evoke a sense of the eerie. Biodiversity
loss has resulted in eerie landscapes and oceans: why are so many species
absent? Human migration has resulted in eerily empty urban centers, empty
rural settlements, and conversely crowded interstitial spaces: why are people
absent in places where they are expected, but present in places where they are
not? Odell’s writing also mentions that the etymology of apocalypse is Greek,
meaning to reveal, and that prior to its modern English usage signifying ‘an
ending’, apocalypse was closer in meaning to insight. Given the old adage “the
purpose of visualization is insight, not pictures” [6], could there be some
utility in the practice of apocalyptic data visualization? By visualizing the
data from the apocalypses unfolding around us, and withholding our attempts to
explain the absences and presences therein, we have the power to invoke the
eerie. From this eerie experience, each viewer can arrive at some degree of
insight, that is, to identify the forces responsible.

For much of what could be constituted as weird or eerie in the capitalocene,
we could (like Fisher) attribute agency to movements of capital, to movements
of an invisible hand. Could we more specific than this? With data
visualization, we have the potential to lend these attributions a semblance of
greater precision, and ultimately, the potential to change what viewers
believe.

##  8 Acknowledgments

Thanks to M. Correll, A. McNutt, and the alt.VIS reviewers.

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