# A nonlinear Strassen Law for singular SPDEs

Shalin Parekh<EMAIL_ADDRESS>
###### Abstract.

A result of Arcones [Arc95] implies that if a measure-preserving linear
operator $S$ on an abstract Wiener space $(X,H,\mu)$ is strongly mixing, then
the set of limit points of the random sequence $((2\log
n)^{-1/2}S^{n}x)_{n\in\mathbb{N}}$ equals the closed unit ball of $H$ for a.e.
$x\in X$, which may be seen as a generalization of the classical Strassen’s
law of the iterated logarithm. We extend this result to the case of a
continuous parameter $n$ and higher Gaussian chaoses, and we also prove a
contraction principle for Strassen laws of such chaoses under continuous maps.
We then use these extensions to recover or prove Strassen-type laws for a
broad collection of processes derived from a Gaussian measure, including
“nonlinear” Strassen laws for singular SPDEs such as the KPZ equation.

###### Key words and phrases:

Large deviations of Wiener chaos, Singular SPDEs, Law of the iterated
logarithm, Ergodicity

###### Contents

  1. 1 Introduction
  2. 2 Proofs of main theorems
    1. 2.1 Strassen law for strongly mixing linear operators
    2. 2.2 Strassen law for higher Gaussian chaos
    3. 2.3 Contraction principle for Strassen Laws
  3. 3 Examples and applications to SPDEs
    1. 3.1 First examples
    2. 3.2 Strassen law for the KPZ equation
  4. 4 Further generalizations
    1. 4.1 Noise-smoothing and non-measure-preserving systems
    2. 4.2 Iterated processes
  5. A Ergodic properties of measure-preserving linear operators
  6. B Gaussian measures on spaces of distributions

##  1\. Introduction

The law of the iterated logarithm for Brownian motion states that if $B$ is a
standard Brownian motion then $\limsup_{t\to
0}(2t\log\log(1/t))^{-1/2}B_{t}=1$. Strassen in a seminal work [Str64]
generalized this statement to show the functional form of this statement,
namely that if we let $B^{\epsilon}(t)=\epsilon^{-1/2}B(\epsilon t)$ then the
set of limit points as $\epsilon\to 0$ in $C[0,1]$ of the sequence
$\\{(2\log\log(1/\epsilon))^{-1/2}B^{\epsilon}\\}_{\epsilon}$ is almost surely
equal to the closed unit ball of its Cameron-Martin space. Since Strassen’s
original work, there has been a tremendous effort resulting in a large
literature expanding the scope of the theorem into many different settings
including Banach space-valued processes [KL73, Kue77, GKZ81, Ale89], Gaussian
processes and higher chaoses [Ood72, GK91, GK93, Arc95], iterated processes
[Bur93, Arc95, CCFR95, Neu98], stronger topologies [BBAK92], sharper envelopes
[R7́9], as well as more complicated stochastic processes driven by
multiparameter fields and fractional processes [Par75, DU99, LWC04, OM05,
Fat21], etc. See [LT11, Chapter 8] for surveys of classical topics.

The goal of the present paper is to extend Strassen’s law in yet another
general direction, related to many of the aforementioned extensions,
eventually culminating with a compact limit set theorem for the small-noise
regime of subcritical singular SPDEs such as the KPZ equation, which have been
of popular interest in the probability literature recently, see e.g. [Cor12,
CW15, CS20], etc. One difference from the aforementioned results is that we
take a dynamical-systems and semigroup-theory perspective on Strassen’s law,
rather than considering an arbitrary sequence of random variables in a Banach
space. We are uncertain if this perspective is novel, or if it merely provides
a convenient notational framework to summarize proof methods that may already
be well-known to experts in the area. Nonetheless the dynamical systems
perspective does allow us to concisely recover and generalize some of the more
classical results cited above, in particular our main results (see Theorem 1.3
and its generalizations Theorems 4.5, and 4.6) recover results from [KL73,
Par75, BBAK92, Bur93, GK93, CCFR95, Arc95, Neu98, DU99, LWC04], see Section 3.

First we establish some notation. If $H$ is any real Hilbert space and $S:H\to
H$ is any bounded operator, we denote by $S^{*}$ its adjoint operator with
respect to the inner product of $H$. Throughout this work we will use the
notion of abstract Wiener spaces introduced by Leonard Gross [Gro67]. These
are formal triples $(X,H,\mu)$ where $X$ is a Banach space, $\mu$ is a
centered Gaussian measure on $X$, and $H\hookrightarrow X$ is the compactly
embedded Cameron-Martin space. In this paper we will consider measure-
preserving a.e. linear maps on $X$. These are Borel-measurable linear maps
$S:E\to X$ where $E\subset X$ is a Borel-measurable linear subspace of full
measure in $X$, such that $S$ preserves the measure $\mu$. It is known that
any such map necessarily maps $H$ to itself and must satisfy $SS^{*}=I$ there,
see Lemma A.2.

Conversely, given any bounded linear map $S:H\to H$ satisfying $SS^{*}=I$,
there exists a Borel-measurable linear subspace $E\subset X$ of full measure
and a linear extension $\hat{S}:E\to X$ of $S$ which is unique up to a.e.
equivalence (this is despite the fact that $H$ may have measure zero).
Moreover the condition $SS^{*}=I$ guarantees that this extension is measure-
preserving. We refer to Lemma A.2 for a proof of these statements. Letting
$E_{2}:=\\{x\in E:\hat{S}x\in E\\}$, we see that $E_{2}$ is a measurable
linear subspace and we can sensibly define $\hat{S}^{2}x:=\hat{S}(\hat{S}x)$
for $x\in E_{2}$. Moreover, the measure-preserving property implies that
$\mu(E_{2})=1$. Continuing iteratively we may define $E_{k+1}=\\{x:\hat{S}x\in
E_{k}\\},$ which will have full measure and admits a sensible definition of
$\hat{S}^{k+1}$. Then $E_{\infty}:=\bigcap E_{k}$ will have full measure and
admits a simultaneous definition of all $\hat{S}^{n}$.

In the sequel we will not distinguish between $\hat{S}$ and $S$ and simply
write $\hat{S}^{n}=:S_{n}$ without explicitly specifying these formalities of
its definition on a linear subspace of full measure. The forward implication
of the following result may be seen as a special case of [Arc95, Theorem 2.1]
(albeit formulated differently) and the main focus of the present paper will
be to extend it to continuous-parameter settings and higher Gaussian chaoses,
ultimately culminating in a nonlinear version of Strassen’s Law for some
singular SPDEs. The reverse implication has not appeared in the literature as
far as we know.

######  Proposition 1.1.

Let $(X,H,\mu)$ be an abstract Wiener space. Let $S:H\to H$ be a bounded
linear operator and write $S_{N}:=S^{N}$ as explained above. Suppose that
$SS^{*}=I$, i.e., $S$ is measure-preserving.

If $\langle S_{N}x,y\rangle_{H}\to 0$ as $N\to\infty$ for all $x,y\in H$
(i.e., $S$ is strongly mixing) then for $\mu$-almost every $x\in X$, the set
of limit points of $\\{\frac{S_{N}x}{\sqrt{2\log N}}:N\in\mathbb{N}\\}$ is
equal to the closed unit ball of $H.$

Conversely, if the set of limit points of $\\{\frac{S_{N}x}{\sqrt{2\log
N}}:N\in\mathbb{N}\\}$ is a.s. equal to the closed unit ball of $H,$ then one
necessarily has $\frac{1}{N}\sum_{k=1}^{N}|\langle S_{k}x,y\rangle_{H}|\to 0$
for all $x,y\in H$ (i.e., $S$ is weakly mixing).

This is proved as Proposition 2.7 in the main body. Just to be clear, we are
referring to limit points with respect to the topology of $X$, and $S_{N}x$ is
meant to be understood in terms of the unique measurable linear extensions
(explained above) if $x\in X$. The equivalence of strong mixing or weak mixing
to the stated conditions on the inner products $\langle S_{k}x,y\rangle$ is
proved in Proposition A.4 below. It turns out that in many cases of interest,
an even stronger mixing condition $\|S_{N}x\|_{H}\to 0$ is satisfied for all
$x\in H$, and by Lemma A.3 this can be equivalently stated as
$\bigcap_{N\in\mathbb{N}}\text{Im}(S_{N}^{*})=\\{0\\}$, where $\text{Im}(S)$
denotes the image in $H$ of $S$. This is easier to check in certain instances.
Lemma B.10 provides another efficient way to check that the mixing condition
is satisfied in many examples of interest.

Now we will formulate a continuous-time version of Proposition 1.1. Recall
that a strongly continuous semigroup on a Banach space $X$ is a semigroup
$(S_{t})_{t\geq 0}$ of bounded operators from $X\to X$ such that
$\|S_{t}x-x\|_{X}\to 0$ as $t\to 0$ for all $x\in X$. Moreover, if $X$ is a
Banach space and if $\gamma:[0,\infty)\to X$ is any function, then we call
$x\in X$ a cluster point at infinity of $\gamma$ if there exists a sequence
$t_{n}\uparrow\infty$ such that $\|\gamma(t_{n})-x\|_{X}\to 0$. Similarly if
$\gamma:(0,1]\to X$ then we will call $x\in X$ a cluster point at zero of
$\gamma$ if there exists a sequence $\epsilon_{n}\downarrow 0$ such that
$\|\gamma(\epsilon_{n})-x\|_{X}\to 0$.

######  Proposition 1.2.

Let $(X,H,\mu)$ be an abstract Wiener space. Suppose that $(S_{t})_{t\geq 0}$
is a family of bounded operators from $H\to H$ satisfying the following four
properties:

  1. (1)

$S_{t+u}=S_{t}S_{u}$ for all $t,u\geq 0$.

  2. (2)

$S_{t}S_{t}^{*}=I$.

  3. (3)

$\langle S_{t}x,y\rangle_{H}\to 0$ as $t\to\infty$ for all $x,y\in H$.

  4. (4)

$(S_{t})_{t\geq 0}$ extends to a strongly continuous semigroup on the larger
space $X.$

Then for $\mu$-almost every $x\in X$, the set of cluster points at infinity of
$\\{\frac{S_{t}x}{\sqrt{2\log t}}:t\geq 0\\}$ is equal to the closed unit ball
of $H.$

This will be proved by combining the previous proposition with Lemma 2.8 in
the main body. Now let us relate this to the usual law of the iterated
logarithm. We will say that a strongly continuous multiplicative semigroup is
a family of bounded operators $(R_{\epsilon})_{\epsilon\in(0,1]}$ satisfying
$R_{\epsilon}R_{\delta}=R_{\epsilon\delta}$ and moreover
$\|R_{\epsilon}x-x\|_{X}\to 0$ as $\epsilon\uparrow 1$ for all $x\in X$. Then
by setting $S_{t}:=R_{e^{-t}}$, Proposition 1.2 can be restated in terms of
multiplicative semigroups as follows.

Let $(X,H,\mu)$ be an abstract Wiener space. Suppose that
$(R_{\epsilon})_{\epsilon\in(0,1]}$ is a family of bounded operators from
$H\to H$ satisfying the following four properties:

  1. (1)

$R_{\epsilon}R_{\delta}=R_{\epsilon\delta}$ for all $\epsilon,\delta\in(0,1]$.

  2. (2)

$R_{\epsilon}R_{\epsilon}^{*}=I$.

  3. (3)

$\langle R_{\epsilon}x,y\rangle_{H}\to 0$ as $\epsilon\to 0$ for all $x,y\in
H$.

  4. (4)

$(R_{\epsilon})_{\epsilon\in(0,1]}$ extends to a strongly continuous
multiplicative semigroup on the larger space $X.$

Then for $\mu$-almost every $x\in X$, the set of cluster points at zero of
$\\{\frac{R_{\epsilon}x}{\sqrt{2\log\log(1/\epsilon)}}:\epsilon\in(0,1]\\}$ is
equal to the closed unit ball of $H.$

The quintessential example of such a setup is the classical Wiener space
$X=C[0,1],H=H_{0}^{1}[0,1]$, and $\mu$ is the law of a standard Brownian
motion. Then one sets $R_{\epsilon}f(x)=\epsilon^{-1/2}f(\epsilon x).$ One
easily checks that $\|R_{\epsilon}f\|_{H_{0}^{1}}\to 0$ as $\epsilon\to 0$ for
all $f\in H_{0}^{1}[0,1],$ and thus the above statement recovers the classical
version of Strassen’s law. Another important example in the context of
stochastic PDEs is the additive-noise stochastic heat equation $h$ driven by
space-time white noise, started either from two-sided Brownian motion or from
zero initial condition, in which one obtains a compact limit set theorem for
the $\epsilon$-indexed family of functions
$\epsilon^{-1/2}h(\epsilon^{2}t,\epsilon x)$, see Example 3.2 below. We will
give a self-contained proof of Propositions 1.1 and 1.2 in Subsection 2.1
below.

Next let us discuss our main result of the paper, the generalization to higher
chaos and the contraction principle. If $(X,H,\mu)$ is an abstract Wiener
space, then the $n^{th}$ homogeneous Wiener chaos, denoted by
$\mathcal{H}^{k}(X,\mu)$ is defined to be the closure in $L^{2}(X,\mu)$ of the
linear span of $H_{k}\circ g$ as $g$ varies through all elements of the
continuous dual space $X^{*},$ where $H_{k}$ denotes the $k^{th}$ Hermite
polynomial (normalized so that $H_{k}(Z)$ has unit variance for a standard
normal $Z$). Letting $Y$ be another separable Banach space, a Borel-measurable
map $T:X\to Y$ is called a chaos of order $k$ over $X$ if $f\circ
T\in\mathcal{H}^{k}(X,\mu)$ for all $f\in Y^{*}$. For such a chaos we may
define (following [Led96, HW15]) its “homogeneous form” $T_{\mathrm{hom}}:H\to
Y_{i}$ by the Bochner integral formula

$T_{\mathrm{hom}}(h):=\int_{X}T(x+h)\mu(dx)=\frac{1}{k!}\int_{X}T(x)\langle
x,h\rangle^{k}\mu(dx),$

which may be shown to converge, see Proposition 2.10 below which is based on
work of [AG93].

For an abstract Wiener space $(X,H,\mu)$ we always let $B(H)$ denote the
closed unit ball of the Cameron-Martin space $H$. Our main result in this work
is the following.

######  Theorem 1.3.

Let $(X,H,\mu)$ be an abstract Wiener space, and let
$(R_{\epsilon})_{\epsilon\in(0,1]}$ be a family of Borel-measurable a.e.
linear maps from $X\to X$ which are measure-preserving and satisfy $\langle
R_{\epsilon}x,y\rangle_{H}\to 0$ as $\epsilon\to 0$ for all $x,y\in H$ (i.e.,
each $R_{\epsilon}$ is strongly mixing). Let $T^{i}:X\to Y_{i}$ be a chaos of
degree $k_{i}$ over $X$ for $1\leq i\leq m$, where $Y_{i}$ may be any
separable Banach spaces. Suppose that there exist strongly continuous
semigroups $(Q^{i}_{\epsilon})_{\epsilon\in(0,1]}$ of operators from $Y_{i}\to
Y_{i}$ for $1\leq i\leq m$ such that

$T^{i}\circ R_{\epsilon}=Q^{i}_{\epsilon}\circ
T^{i},\;\;\;\;\;\;\;\;\mu\text{-a.e.}\;\;\;\;\;\;\text{ for all
}\epsilon\in(0,1]\text{ and }1\leq i\leq m.$ (1.1)

Let $\mathcal{M}\subset Y_{1}\times\cdots Y_{m}$ be a closed subset, such that
for all $\delta>0$

$\mu(\\{x\in
X:(\delta^{k_{1}}T^{1}(x),...,\delta^{k_{m}}T^{m}(x))\in\mathcal{M}\\})=1.$

Then the compact set
$K:=\\{(T^{1}_{\mathrm{hom}}(h),...,T^{m}_{\mathrm{hom}}(h)):h\in B(H)\\}$ is
contained in $\mathcal{M}$. Moreover for any Banach space $Z$ and any
continuous map $\Phi:\mathcal{M}\to Z$, the set of cluster points at zero of

$\big{\\{}\Phi\big{(}(2\log\log(1/\epsilon))^{-k_{1}/2}Q_{\epsilon}^{1}T^{1}(x),\;.\;.\;.\;,(2\log\log(1/\epsilon))^{-k_{m}/2}Q_{\epsilon}^{m}T^{m}(x)\big{)}:\epsilon\in(0,1]\\}$

is almost surely equal to $\Phi(K).$

This result builds upon a body of literature on sample paths of ergodic self-
similar processes derived from a Gaussian measure and their chaoses, which
started with works such as [MO86, Tak89, Alb90]. Besides the more abstract
setup, a novelty of our result is that there is no additional difficulty in
allowing the topology on the spaces $Y_{i}$ and $Z$ to be taken as strong as
one desires (e.g. Hölder spaces as opposed to just continuous functions).

Theorem 1.3 is proved as Corollary 2.18 in the main body of the paper. Note
that one recovers Proposition 1.2 by setting
$m=1,k_{1}=1,Y_{1}=X=Z=\mathcal{M}$, and making $\Phi$ the identity map.
Classical and new examples using this general result are illustrated in
Sections 3 and 4. One particular example of interest which seems new, and
which sparked our interest in this problem, is the following.

######  Theorem 1.4.

For $\epsilon\in(0,1/e)$ let $C_{\epsilon}:=(\log\log(1/\epsilon))^{1/2}$, and
let $h^{\epsilon}$ denote the Hopf-Cole solution to the KPZ equation on
$\mathbb{R}_{+}\times\mathbb{R}$ driven by standard Gaussian space-time white
noise:

$\partial_{t}h^{\epsilon}=\partial_{x}^{2}h^{\epsilon}+C_{\epsilon}(\partial_{x}h^{\epsilon})^{2}+\xi,\;\;\;\;\;\;\;\;\;\;t\geq
0,x\in\mathbb{R},$

with initial data $h(0,x)=0$. Then for any $s,y>0$ the set of cluster points
as $\epsilon\downarrow 0$ in $\mathcal{C}_{s,y}:=C([0,s]\times[-y,y])$ of the
sequence of functions
$C_{\epsilon}^{-1}\epsilon^{-1/2}h^{\epsilon}(\epsilon^{2}t,\epsilon x)$ is
a.s. equal to the compact set $K_{\mathrm{Zero}}$ given by the closure in
$\mathcal{C}_{s,y}$ of the set of smooth functions $h$ satisfying

$h(0,x)=0,\;\;\;\;\;\big{\|}\partial_{t}h-\partial_{x}^{2}h-(\partial_{x}h)^{2}\big{\|}_{L^{2}([0,s]\times[-y,y])}\leq
1.$

Moreover, the same compact limit set result holds for the same sequence of
functions in stronger topologies given by parabolic Hölder norms up to but
excluding exponent 1/2 (see (3.9)).

This will be proved in Section 3.2. We remark that the compact limit set is
precisely the compact set on which the large deviation rate function of small-
noise KPZ is less than or equal to 1, see for instance [HW15, LT21].

The result of Theorem 1.3 can also be used to prove variants of Theorem 1.4.
For instance if we instead let $h^{\epsilon}(0,x)$ be a two sided Brownian
motion (fixed for different values of $\epsilon)$ then one could show that a
similar result holds for
$C_{\epsilon}^{-1}\epsilon^{-1/2}h^{\epsilon}(\epsilon^{2}t,\epsilon x)$, but
with compact limit set $K_{Br}$ given by the closure of smooth functions
$h\in\mathcal{C}_{s,y}$ satisfying

$h(0,0)=0,\;\;\;\;\;\|\partial_{x}h(0,\cdot)\|_{L^{2}[-y,y]}^{2}+\big{\|}\partial_{t}h-\partial_{x}^{2}h-(\partial_{x}h)^{2}\big{\|}^{2}_{L^{2}([0,s]\times[-y,y])}\leq
1.$

If we likewise define $k^{\epsilon}$ to be the solution of

$\partial_{t}k^{\epsilon}=\partial_{x}^{2}k^{\epsilon}+C_{\epsilon}^{-1}(\partial_{x}k^{\epsilon})^{2}+\xi,$

then one can further show that the same compact limit set results hold for
$C_{\epsilon}^{-1}\epsilon^{1/2}k^{\epsilon}(\epsilon^{-2}t,\epsilon^{-1}x)$,
in other words the large-time regime. For brevity we will not write down the
proofs of these variants.

Note that in Theorem 1.4, the nonlinearity in the equation describing
$h^{\epsilon}$ must be scaled along with the parameter $\epsilon$, so that it
blows up in the $\epsilon\to 0$ limit. If we did not do this, then the
limiting compact set would simply agree with that of the linearized equation
$\partial_{t}h_{\mathrm{Linear}}=\partial_{x}^{2}h_{\mathrm{Linear}}+\xi$,
namely

$K_{\mathrm{Linear}}:=\\{h\in\mathcal{C}_{s,y}:h(0,x)=0,\big{\|}\partial_{t}h-\partial_{x}^{2}h\big{\|}_{L^{2}([0,s]\times[-y,y])}\leq
1\\}.$

Indeed this can be proved by decomposing
$h_{\mathrm{KPZ}}=h_{\mathrm{Linear}}+v$ where $h_{\mathrm{KPZ}}$ is the Hopf-
Cole solution to KPZ with initial data zero and $v$ is a remainder term which
has better regularity than $h_{\mathrm{Linear}}$ (see e.g. Theorem 3.19 of
[PR19]). Then under the scaling necessary to obtain Strassen’s law, it is easy
to check that the remainder term converges a.s. to zero in the topology of
$C([0,s]\times[-y,y])$ and the set of limit points for the part corresponding
to $h_{\mathrm{Linear}}$ can be shown to be $K_{\mathrm{Linear}}$ by applying
Proposition 1.2 above (see Example 3.2 below). Likewise in the result stated
above for the family $k^{\epsilon}$, the nonlinearity must be scaled along
with the parameter $\epsilon$ so that it vanishes in the $\epsilon\to 0$
limit. If we did not do this, then the asymptotics would be wrong entirely and
instead one would need to apply a scaling that respects the tail behavior of
the so-called KPZ fixed point [MQR21], namely $(\log\log(1/\epsilon))^{2/3}$,
see [DG21]. The functional form of the latter result seems to be a very
interesting open problem, see [QT22] for some progress on large deviations
under the KPZ fixed point scaling.

The manner in which we prove Theorem 1.4 uses results from the theory of
regularity structures [Hai13] and is robust enough to prove similar theorems
for other rough equations on a full space that admit a factorization of the
solution map in terms of a finite number of Gaussian chaoses and a continuous
part. In the case of $\Phi_{2}^{4}$ and $\Phi_{3}^{4}$, such a methodology
would only be applicable once one has developed such a solution theory for the
latter equation on the full space $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$ using
regularity structures. To show the generality of our method, we also prove a
similar result for mollified smooth versions of the noise, specifically we
show that even though mollifying is not a measure-preserving operation on the
noise, it is enough to be “asymptotically rapidly measure-preserving,” see
Theorem 4.2 and its corollary Theorem 4.4 for the KPZ equation.

See Theorem 4.5 and the subsequent discussion for the most general formulation
of the main theorem above, in which we discuss how one could potentially use
our results to give Strassen laws even for discrete or non-Gaussian systems,
assuming that a strong enough form of the “almost sure invariance principle”
holds for the non-Gaussian driving noise as well as for its higher chaoses.

In Section 4.2 we consider yet another generalization of the main result that
may be applied to obtain Strassen laws for so-called “iterated processes,”
recovering and generalizing [Bur93, CCFR95, Neu98].

Organization. In Section 2 we prove the main result Theorem 1.3, first
expositing on the known results before moving onto the newer higher chaos
results. Section 3 is devoted to giving many examples culminating in the proof
of Theorem 1.4. Section 4 gives further generalizations of Theorem 1.3.
Appendix A studies the ergodic properties of measure-preserving linear
operators on an abstract Wiener space. Appendix B contains functional-analytic
lemmas that are useful in checking the conditions of Theorem 1.3 in cases of
interest.

Notation and Conventions. Throughout this paper we use $(X,H,\mu)$ to denote
an abstract Wiener space, and we use $x$ or $\xi$ to denote a general element
of the space $X$. We generally use $S_{N},G_{t},R_{\epsilon},Q_{\epsilon}$ for
measure-preserving semigroups of operators on these spaces. For a Gaussian
chaos we typically use $T$ or sometimes $\psi$, and for its homogeneous form
we always use the subscript “$\mathrm{hom}$.”

Acknowledgements. We thank Yier Lin for many fruitful discussions on the topic
of large deviations of Gaussian chaos. This research was partially supported
by Ivan Corwin’s NSF grant DMS-1811143, and the Fernholz Foundation’s “Summer
Minerva Fellows” program.

##  2\. Proofs of main theorems

###  2.1. Strassen law for strongly mixing linear operators

In this subsection we prove Propositions 1.1 and 1.2. Although most of these
results for first-order chaos are already known (as we explained in the
introduction), we give a short proof here because we use some of the lemmas
later for the higher chaos results, and because it allows the exposition to
flow more smoothly in later sections. First we need some preliminary lemmas
and then we will formulate the result as Theorem 2.7.

######  Lemma 2.1.

Let $(X_{i})_{i\geq 0}$ be a stationary sequence of real-valued jointly
Gaussian random variables. Suppose that $\mathrm{var}(X_{0})=1$ and
$\mathrm{cov}(X_{0},X_{n})\to 0$ as $n\to\infty$. Then

$\limsup_{n\to\infty}\frac{X_{n}}{\sqrt{2\log n}}=1,\;\;\;\;\;\;\;a.s..$

This lemma is classical and is a special case of [Arc95, Lemma 2.1], which in
turn is an improvement of classical results from [Pic67, Nis67, Lai74]. The
proof may be found in those works, or proved directly as a straightforward
corollary of Slepian’s Lemma (using a comparison with the stationary Gaussian
process $\xi_{j}$ with unit variance, whose covariance is given for $i<j$ by
$\mathrm{cov}(\xi_{i},\xi_{j})=\epsilon$ for some fixed small
$\epsilon\in(0,1)$).

We remark that the above lemma is the only place where the mixing condition is
used, and is also the reason why we are not able to extend the result to the
ergodic case (i.e., it seems difficult to prove or find a counterexample to
the above fact under the weaker assumption
$\frac{1}{n}\sum_{1}^{n}|\mathrm{cov}(X_{0},X_{n})|\to 0$). Next we generalize
the above lemma to $\mathbb{R}^{N}$. We denote the unit sphere of
$\mathbb{R}^{N}$ to be the set of all points $(a_{1},...,a_{N})$ with
$\sum_{1}^{N}a_{i}^{2}=1.$

######  Corollary 2.2.

Let $(\vec{X}_{n})_{n\geq 1}$ be a stationary sequence of jointly Gaussian
random variables in $\mathbb{R}^{N}$, say
$\vec{X}_{n}=(X_{n}^{1},X_{n}^{2},...,X_{n}^{N}).$ Suppose that
$\mathrm{cov}(X_{0}^{i},X_{0}^{j})=\delta_{ij}$ and that
$\mathrm{cov}(X_{0}^{i},X_{n}^{j})\to 0$ as $n\to\infty$ for all $1\leq
i,j\leq N$. Then the unit sphere of $\mathbb{R}^{N}$ is contained in the set
of limit points of the random sequence $((2\log n)^{-1/2}\vec{X}_{n})_{n\geq
2}.$

###### Proof.

By Lemma 2.1, if we restrict our attention to only the first coordinate, then
the set of limit points of the random sequence $((2\log
n)^{-1/2}\vec{X}_{n})_{n\geq 1}$ must contain a point on the set
$A:=\\{(a_{1},...,a_{N}):a_{1}=1\\}$. On the other hand, by Borel-Cantelli and
$\mathrm{cov}(X_{0}^{i},X_{0}^{j})=\delta_{ij}$, it is clear that the set of
limit points must be contained in the set
$B:=\\{(a_{1},...,a_{N}):\sum_{i=1}^{N}a_{i}^{2}\leq 1\\}.$

Since $A\cap B=\\{(1,0,...,0)\\}$ it follows that $(1,0,...,0)$ is a.s. a
limit point of the random sequence $((2\log n)^{-1/2}\vec{X}_{n})_{n\geq 1}.$
The fact that any point on the unit sphere is a limit point then follows from
rotational invariance of the conditions that
$\mathrm{cov}(X_{0}^{i},X_{0}^{j})=\delta_{ij}$ and
$\mathrm{cov}(X_{0}^{i},X_{n}^{j})\to 0$ (i.e., these conditions remain true
if we replace $(\vec{X}_{n})_{n}$ by $(U(\vec{X}_{n}))_{n}$ for some
orthogonal $N\times N$ matrix $U$). ∎

######  Definition 2.3.

Let $H$ be a real Hilbert space. We will henceforth denote $S(H):=\\{h\in
H:\|h\|_{H}=1\\}$ and $B(H):=\\{h\in H:\|h\|_{H}\leq 1\\}.$

######  Lemma 2.4.

Let $(X,H,\mu)$ be an abstract Wiener space of infinite dimension. Then the
closure in $X$ of $S(H)$ is $B(H).$

###### Proof.

$B(H)$ is a compact (hence closed) subset of $X$ which contains $S(H)$, so the
set of limit points of $S(H)$ must be contained in $B(H)$. Choose an
orthonormal basis $\\{e_{n}\\}_{n}$ for $H$. Since $e_{n}\in B(H)$, since
$B(H)$ is compact in $X$ [Bog98], and since $e_{n}\to 0$ weakly in $H$, it
follows that $\|e_{n}\|_{X}\to 0$ as $n\to\infty$. Now fix $h\in H$ with
$\|h\|_{H}\leq 1$. Choose $c_{n}\in\mathbb{R}$ such that
$\|h+c_{n}e_{n}\|_{H}=1$. It is clear that $|c_{n}|\leq 2$ (otherwise
$1=\|c_{n}e_{n}+h\|_{H}\geq|c_{n}|\|e_{n}\|-\|h\|>2-1=1$), and therefore
$\|c_{n}e_{n}\|_{X}\to 0$. Thus $h+c_{n}e_{n}$ is a sequence in $S(H)$
converging to $h\in B(H)$ with respect to the topology of $X.$ ∎

At this point we invite the reader to review the main results of Appendix A.
Given an abstract Wiener space $(X,H,\mu)$, the set of measure-preserving a.e.
linear maps from $X\to X$ (modulo a.e. equivalence) is in bijection with the
set of bounded linear maps from $H\to H$ satisfying $SS^{*}=I$. Moreover, the
bijection is given by simply restricting $S$ to $H$. This is shown in the
appendix, see Lemma A.2. In the sequel we will not notationally distinguish
between such a map $S$ on $H$ and its a.e.-unique and a.e.-linear measure-
preserving extension on $X$.

######  Lemma 2.5.

Let $(X,H,\mu)$ be an abstract Wiener space. Suppose that $S:H\to H$ is a
bounded linear operator satisfying $SS^{*}=I$. Then for any $a\notin B(H)$
there exists $\epsilon>0$ such that

$\mu\bigg{(}\bigg{\\{}x\in X:\frac{S_{k}x}{\sqrt{\log k}}\in
B(a,\epsilon)\;\;\text{infinitely often}\bigg{\\}}\bigg{)}=0.$

###### Proof.

By definition of the Cameron-Martin space, for every $x\in X$, we have

$\sup\\{\ell(x):\ell\in X^{*},\int\ell^{2}d\mu=1\\}=\|x\|_{H}$

(we set $\|x\|_{H}=\infty$ when $x\notin H$). For any $x\in X\backslash B(H)$,
there exists $\ell\in X^{*}$ such that $\ell(x)>\sqrt{2}$ and
$\int_{X}\ell^{2}d\mu=1$. For each $n\in\mathbb{N}$, under $\mu$,
$\ell(S_{n}x)$ is a standard Gaussian random variables. Using Borel-Cantelli
lemma and Gaussian tail bound, together with $\ell(x)>\sqrt{2}$, there exist
small enough $\epsilon_{x}>0$ such that

$\mathbb{P}\Big{(}\big{\\{}\frac{\ell(S_{n}\xi)}{\sqrt{\log
n}}\big{\\}}_{n=1}^{\infty}\in B(\ell(x),\epsilon_{x})\;\;\text{infinitely
often}\Big{)}=0$

By making $\epsilon_{x}$ smaller, we have

$\mathbb{P}\Big{(}\big{\\{}\frac{S_{n}\xi}{\sqrt{\log
n}}\big{\\}}_{n=1}^{\infty}\in B(x,\epsilon_{x})\;\;\text{infinitely
often}\Big{)}=0.\qed$

######  Lemma 2.6.

[Led96, Equation (4.4)] Let $\mu$ be a centered Gaussian measure on a
separable Banach space, then for all $a>0$

$\mu(\\{x\in X:\|x\|_{X}>a+\int_{X}\|u\|_{X}\mu(du)\\})\leq
e^{-a^{2}/(2\sigma^{2})},$

where $\sigma:=\sup_{\|f\|_{X^{*}}\leq
1}\int_{X}f(x)^{2}\mu(dx)=\sup_{\|h\|_{H}\leq 1}\|h\|_{X}.$

Finally we are ready to prove the main result of the section, a restatement of
Proposition 1.1 in addition to a partial converse.

######  Proposition 2.7.

Let $(X,H,\mu)$ be an abstract Wiener space. Let $E\subset X$ be a Borel
measurable linear subspace of measure 1, and suppose $S:E\to X$ is linear and
measure-preserving. If $S$ is strongly mixing, then the set of limit points of
the random sequence $((2\log n)^{-1/2}S^{n}x)_{n\in\mathbb{N}}$ equals the
closed unit ball of $H$ for a.e. $x\in X$.

Conversely, if $S$ is any measure-preserving linear operator such that the set
of limit points of the random sequence $((2\log
n)^{-1/2}S^{n}x)_{n\in\mathbb{N}}$ equals the closed unit ball of $H$ for a.e.
$x\in X$, then $S$ must be ergodic (equivalently weakly mixing by Proposition
A.4).

###### Proof.

Fix $h\in H$ with $\|h\|_{H}=1$, and let $\epsilon>0$. We wish to show that
$\|(2\log n)^{-1/2}S_{n}x-h\|_{H}<\epsilon$ infinitely often. To this end, let
$\\{e_{i}\\}_{i}$ be an orthonormal basis of $H$ with $e_{1}=h$. Let $P_{k}$
denote the orthogonal projection onto the subspace $M_{k}$ which is spanned by
$\\{e_{i}\\}_{i=1}^{k}.$

Choose some $k\in\mathbb{N}$ so that
$\int_{X}\|x-P_{k}x\|_{X}^{2}\mu(dx)<(\epsilon/5)^{2}.$ Then for all $n$ we
have that

$\mathbb{P}(\|S_{n}\xi-P_{k}(S_{n}\xi)\|_{X}>a)=\mathbb{P}(\|\xi-
P_{k}(\xi)\|_{X}>a)\leq e^{-Ma^{2}}\mathbb{E}[e^{M\|\xi-
P_{k}(\xi)\|_{X}^{2}}]$ (2.1)

for any $a>0$ and for any $M>0$ such that the expectation on the right side is
finite. By Lemma 2.6, it is true that $\int_{X}e^{\alpha\|x\|^{2}}\nu(dx)$ is
finite whenever $\alpha<(2\int_{X}\|x\|^{2}\nu(dx))^{-1},$ for any centered
Gaussian measure $\nu$ on $X.$ Consequently in (2.1) we can take $M$ to be
$(5/\epsilon)^{2}$ and we can take $a$ to be $\epsilon\sqrt{2\log n}/4$ and we
find

$\sum_{n>1}\mathbb{P}\bigg{(}\frac{\|S_{n}\xi-
P_{k}(S_{n}\xi)\|_{X}}{\sqrt{2\log n}}>\epsilon/4\bigg{)}<\infty,$

where $\xi$ is distributed according to $\mu.$ Then by the Borel-Cantelli
lemma, we find that

$(2\log n)^{-1/2}\|S_{n}\xi-P_{k}(S_{n}\xi)\|_{X}<\epsilon/4$

for all but finitely many $n$, a.s.. Thus we just need to show that $\|(2\log
n)^{-1/2}P_{k}(S_{n}x)-h\|_{X}<\epsilon/4$ infinitely often. But this is just
a finite dimensional statement which immediately follows from Corollary 2.2.
Indeed, by assumption $h$ lies on the unit sphere of $M_{k}$, and the joint
covariances tend to zero precisely because of the condition that $S$ is mixing
(via Item (2) in Proposition A.4).

In the notation of Lemma 2.4, we have shown that any point on $S(H)$ is almost
surely a limit point of $(2\log n)^{-1/2}S_{n}x$. By that same lemma, any
point of $B(H)$ is also a limit point. It is clear from Lemma 2.5 that no
point outside of $B(H)$ can be a limit point of $(2\log n)^{-1/2}S_{n}x$. That
lemma is still valid in this case, since it only relies on the measure-
preserving property of $S_{n}$ and nothing else. This completes the proof of
the first statement.

Next we prove that $S$ must be ergodic if the set of limit points of $((2\log
n)^{-1/2}S^{n}x)_{n\in\mathbb{N}}$ equals the closed unit ball of $H$ for a.e.
$x\in X$. Indeed, if $S$ is not ergodic, then by Item (3d) in Proposition A.4,
there is a two-dimensional invariant subspace $M$ on which $S$ acts by a
rotation matrix. We claim that no nonzero point which lies in $M$ can be
visited infinitely often, since $(2\log n)^{-1/2}S_{n}x$ converges to zero
a.s. for any $x\in M$. Indeed, let $P$ be the projection onto $M$ and let
$Q=I-P$ denote the projection onto $M^{\perp}.$ Let $\xi$ be sampled from
$\mu$, so that $Q\xi$ has Cameron-Martin space $M^{\perp}$. Since $M$ and
$M^{\perp}$ are invariant under $S$, it follows that $PS_{n}=S_{n}P$ and
$QS_{n}=S_{n}Q$. By Lemma 2.5 it is clear that the set of limit points of
$(2\log n)^{-1/2}QS_{n}\xi=(2\log n)^{-1/2}S_{n}(Q\xi)$ must be contained in
the closed unit ball of the Cameron-Martin space of $Q\xi$, namely $M^{\perp}$
(this lemma only relies on the measure-preserving property of $S_{n}$ and
nothing else). Furthermore, since $(2\log n)^{-1/2}S_{n}x$ converges to zero
a.s. for any $x\in M$, it follows that $(2\log n)^{-1/2}PS_{n}\xi=(2\log
n)^{-1/2}S_{n}(P\xi)$ converges to zero a.s. Consequently the set of limit
points of $(2\log n)^{-1/2}S_{n}\xi=(2\log n)^{-1/2}PS_{n}\xi+(2\log
n)^{-1/2}QS_{n}\xi$ must also be contained in $M^{\perp}$, and hence contain
no nonzero points of $M$, proving the claim. ∎

This already proves Proposition 1.1, and to prove Proposition 1.2 one only
needs the following.

######  Lemma 2.8.

Let $(X,H,\mu)$ be an abstract Wiener space. Suppose that $S_{t}:H\to H$ is
some family of operators satisfying $S_{t}S_{t}^{*}=I$. Moreover assume that
$(S_{t})$ extends to a strongly continuous semigroup on $X$. Then there exists
a deterministic function $C:[0,1]\to\mathbb{R}_{+}$ such that $C(\rho)\to 0$
as $\rho\downarrow 0$ and such that

$\mu\bigg{(}\bigg{\\{}x\in
X:\limsup_{k\to\infty}\frac{\sup_{t\in[k\rho,(k+1)\rho]}\|S_{t}x-S_{k\rho}x\|_{X}}{\sqrt{\log
k}}\leq C(\rho)\bigg{\\}}\bigg{)}=1$

for all $\rho\in[0,1]$. In particular $(S_{t}x)_{t\geq 0}$ has the same set of
cluster points as $(S_{N}x)_{N\in\mathbb{N}}$ for $x$ in a set of full
$\mu$-measure.

###### Proof.

Let $\xi$ denote a random variable in $X$ with law $\mu$, defined on some
probability space $(\Omega,\mathcal{F},\mathbb{P})$. Then
$(S_{t}\xi)_{t\in[0,1]}$ is continuous in $t$, thus it can be viewed as a
Gaussian random variable taking values in the Banach space $C([0,1],X)$ of
continuous paths in $X$, endowed with norm
$\|F\|_{C([0,1],X)}:=\sup_{t\in[0,1]}\|F(t)\|_{X}.$ By Fernique’s theorem,
$\mathbb{E}[\sup_{t\in[0,1]}\|S_{t}\xi-\xi\|_{X}^{p}]<\infty$ for all $p\geq
1$. By continuity, we also know that
$\sup_{t\in[0,\rho]}\|S_{t}\xi-\xi\|_{X}\to 0$ as $\rho\downarrow 0$. Letting

$C(\rho):=\mathbb{E}[\sup_{t\in[0,\rho]}\|S_{t}\xi-\xi\|_{X}^{2}],$

we thus have by uniform integrability that $\lim_{\rho\downarrow 0}C(\rho)=0.$
Letting

$A(k,\rho):=\sup_{t\in[k\rho,(k+1)\rho]}\|S_{t}x-S_{k\rho}x\|_{X},$

the stationarity of the process $(S_{t}\xi)_{t\geq 0}$ implies that
$A(k,\rho)\stackrel{{\scriptstyle d}}{{=}}A(0,\rho)$ for all $k\in\mathbb{N}$.
We thus find that

$\mathbb{P}(A(k,\rho)>\mathbb{E}[A(0,\rho)]+u)=\mathbb{P}(A(0,\rho)>\mathbb{E}[A(0,\rho)]+u)\leq
e^{-u^{2}/(2C(\rho))},$

where we used Lemma 2.6 in the last inequality. But
$\mathbb{E}[A(0,\rho)]\leq\mathbb{E}[A(0,\rho)^{2}]^{1/2}=C(\rho)^{1/2}$, and
therefore

$\mathbb{P}(A(k,\rho)>u)\leq e^{-(u-C(\rho)^{1/2})^{2}/(2C(\rho))}\leq
e^{1/2}\cdot e^{-u^{2}/(4C(\rho))}$

where we used $(a-b)^{2}\geq\frac{1}{2}a^{2}-b^{2}$ in the last inequality.
Thus

$\mathbb{P}(A(k,\rho)>4C(\rho)\sqrt{\log k})\lesssim k^{-4},$

so by the Borel-Cantelli lemma we find that

$\limsup_{k\to\infty}\frac{A(k,\rho)}{\sqrt{\log k}}\leq
4C(\rho),\;\;\;\;a.s.$

As we already observed eariler $C(\rho)\to 0$ as $\rho\downarrow 0$,
completing the proof. ∎

To summarize the main results of this section, let $(X,H,\mu)$ be an abstract
Wiener space, let $S:X\to X$ be measure-preserving and a.e. linear, and
consider the following statements:

  1. (1)

$\bigcap_{n}\sigma(S_{n})$ is a trivial sigma algebra.

  2. (2)

$S$ is strongly mixing.

  3. (3)

The set of limit points of the random sequence $((2\log
n)^{-1/2}S^{n}x)_{n\in\mathbb{N}}$ equals the closed unit ball of $H$ for a.e.
$x\in X$.

  4. (4)

$S$ is ergodic/weakly mixing.

Appendix A shows that (1) implies (2) (which is actually true for any
dynamical system, by e.g. the reverse martingale convergence theorem), that
(2) implies (3), and that (3) implies (4). From Proposition A.4, it is clear
that (2) does not imply (1) in general, and that (4) does not imply (2) in
general.

We do not know if (3) implies (2) or if (4) implies (3), although Proposition
A.4 implies that both cannot simultaneously be true in general. To show that
(4) implies (3), one would need to prove Lemma 2.1 with the condition
$\mathrm{cov}(X_{0},X_{n})\to 0$ replaced by the condition
$\frac{1}{n}\sum_{j=1}^{n}|\mathrm{cov}(X_{0},X_{j})|\to 0.$ We do not know
how to prove this, nor are we certain that it is even true. We have reason to
suspect that “(4) implies (3)” may actually be false, and a counterexample
might be given by an operator whose spectral measure is atomless but highly
singular with respect to Lebesgue measure. For instance, in the case that the
spectral measure looks like the usual two-thirds Cantor measure, we have some
reason to suspect that the set of limit points of the random sequence $((2\log
n)^{-1/2}S^{n}x)_{n\in\mathbb{N}}$ equals the ball in $H$ of radius
$\sqrt{\log_{3}2}$ for a.e. $x\in X$, rather than the closed unit ball.
However, we do not have a proof of this.

###  2.2. Strassen law for higher Gaussian chaos

The main goal of this subsection is to prove a version of Strassen’s law for
higher Gaussian chaoses, which reduces to the previous version for first-order
chaos.

If $(X,H,\mu)$ is an abstract Wiener space, then the $k^{th}$ homogeneous
Wiener chaos, denoted by $\mathcal{H}^{k}(X,\mu)$ is defined to be the closure
in $L^{2}(X,\mu)$ of the linear span of $H_{k}\circ g$ as $g$ varies through
all elements of the continuous dual space $X^{*},$ where $H_{k}$ denotes the
$k^{th}$ Hermite polynomial

$H_{k}(x):=(-1)^{k}e^{\frac{x^{2}}{2}}\frac{d^{k}}{dx^{k}}e^{-\frac{x^{2}}{2}}.$

Equivalently $\mathcal{H}^{k}$ can be described as the closure in
$L^{2}(X,\mu)$ of the closure of the linear span of
$H_{k}(\langle\cdot,v\rangle)$ as $v$ ranges through all elements of $H$ and
satisfies $\|v\|_{H}=1$. One always has the orthonormal decomposition
$L^{2}(X,\mu)=\bigoplus_{k\geq 0}\mathcal{H}^{k}(X,\mu),$ see e.g. [Nua06,
Section 1.1]. Sometimes the $k^{th}$ chaos is also described slightly
differently as linear combinations of products of Hermite polynomials of
degree adding up to $k$, for instance in [HW15, Section 3], but our
formulation is equivalent by the umbral identity for Hermite polynomials (see
e.g. [Hai16, Corollary 2.3]) together with the fact that for any vector space
$H$ the “diagonal” elements of the form $h^{\otimes_{s}k}\;\;\;(h\in H)$ span
span the entire symmetric tensor product $H^{\otimes_{s}k}$.

######  Definition 2.9.

Let $(X,H,\mu)$ be an abstract Wiener space, and let $Y$ be another separable
Banach space. A Borel-measurable map $T:X\to Y$ is called a homogeneous chaos
of order $k$ if $f\circ T\in\mathcal{H}^{k}(X,\mu)$ for all $f\in Y^{*}$.

Before formulating the main result of this section, we collect a few important
results about homogeneous variables. In [AG93] the authors show that the norm
of any Gaussian chaos has moments of all orders. Moreover, [AG93, equation
(4.1)] gives a hypercontractive bound for any Gaussian chaos $T$ of order $k$:

$\int_{X}\|T(x)\|_{Y}^{2p}\mu(dx)\leq(2p-1)^{pk}\bigg{[}\int_{X}\|T(x)\|_{Y}^{2}\mu(dx)\bigg{]}^{p}.$
(2.2)

Here $p\geq 1$ is arbitrary. Immediately this gives the following strong
integrability result.

######  Proposition 2.10.

Let $(X,H,\mu)$ be an abstract Wiener space and let $T:X\to Y$ be a chaos of
order $k.$ Let $\|T\|_{L^{2}(X,\mu;Y)}^{2}:=\int_{X}\|T(x)\|_{Y}^{2}\mu(dx).$
Then $\|T\|_{L^{2}(X,\mu;Y)}<\infty$ and moreover

$\mu(\\{x\in
X:\|T(x)\|_{Y}>u\\})<C\exp\big{[}-\alpha\big{(}u/\|T\|_{L^{2}(X,\mu;Y)}\big{)}^{2/k}\big{]},$
(2.3)

where $C,\alpha>0$ depend on $k$ but are independent of the choice of
$X,H,\mu,Y,T,$ and $u>0$.

Let $(X,H,\mu)$ be an abstract Wiener space. Choose an orthonormal basis
$\\{e_{i}\\}_{i}$ for $H,$ and for $x\in H$ let

$P_{N}x:=\sum_{j=1}^{N}\langle x,e_{j}\rangle
e_{j},\;\;\;\;\;\;\;\;\;\;\;\;\;\;Q_{N}x:=\sum_{j=N+1}^{\infty}\langle
x,e_{j}\rangle e_{j}=x-P_{N}x.$

where the infinite sum converges in the topology of $H$. Note that if $x$ is
sampled from $\mu$ then $P_{N}x,Q_{N}x$ still make sense (in a set of measure
1) and moreover they are independent since $\\{\langle
x,e_{j}\rangle\\}_{j\geq 1}$ are i.i.d. standard Gaussian random variables
under $\mu$. Therefore if $(x,y)$ is sampled from $\mu^{\otimes 2}$, then
$P_{N}x+Q_{N}y$ is distributed as $\mu$. If $T:X\to Y$ is a chaos of order
$k$, we thus define a sequence of “finite-rank Cameron-Martin projections” for
$T$ by the formula

$T_{N}(x):=\int_{X}T(P_{N}x+Q_{N}y)\mu(dy).$ (2.4)

This is a well-defined Bochner integral for $\mu$ a.e. $x\in X$. Indeed, since

$\int_{X}\int_{X}\|T(P_{N}x+Q_{N}y)\|_{Y}\mu(dy)\mu(dx)=\int_{X}\|T(u)\|_{Y}\mu(du)<\infty,$

it follows that $\int_{X}\|T(P_{N}x+Q_{N}y)\|_{Y}\mu(dy)$ is finite for a.e.
$x$.

######  Corollary 2.11.

Let $(X,H,\mu)$ be an abstract Wiener space, and let $T:X\to Y$ be a chaos of
order $k$. If $T_{N}$ is defined as in (2.4), then $T_{N}$ is also a chaos of
order $k$, and moreover $\|T_{N}-T\|_{Y}\to 0$ a.e. and in every
$L^{p}(X,\mu)$ as $N\to\infty$. In fact, one has the following super-
polynomial convergence bound:

$\mu(\\{x\in X:\|T_{N}(x)-T(x)\|_{Y}>u\\})\leq
C\exp\big{[}-\alpha\big{(}u/\|T_{N}-T\|_{L^{2}(X,\mu;Y)}^{2}\big{)}^{2/k}\big{]}$
(2.5)

where $C,\alpha$ are independent of $X,H,\mu,Y,T,u,N$ and the choice of basis
$\\{e_{i}\\}_{i}$, but may depend on the homogeneity $k$.

###### Proof.

That $T_{N}$ is a chaos of order $k$ is clear. The a.e. and $L^{2}$
convergence follows immediately from the martingale convergence theorem for
Banach-valued random variables [Cha68], since
$T_{N}=\mathbb{E}_{\mu}[T|\mathcal{F}_{N}]$ where $\mathcal{F}_{N}$ is the
$\sigma$-algebra generated by the i.i.d. variables $\langle x,e_{i}\rangle$
for $1\leq i\leq N$. The super-polynomial convergence bound then follows
immediately from Proposition 2.10 applied to the chaos $T-T_{N}.$ ∎

If $(X,H,\mu)$ is an abstract Wiener space and $T:X\to Y$ is a chaos of order
$k$, then we define the associated map $T_{\mathrm{hom}}:H\to Y$ by

$T_{\mathrm{hom}}(h):=\int_{X}T(x+h)\mu(dx)=\frac{1}{k!}\int_{X}T(x)\langle
x,h\rangle^{k}\mu(dx).$ (2.6)

The latter equality follows by applying the Cameron-Martin formula to first
replace $T(x+h)$ by $T(x)e^{\langle x,h\rangle-\frac{1}{2}\|h\|_{H}^{2}}$,
then expanding the exponential as
$\sum_{n=0}^{\infty}\frac{\|h\|^{n}}{n!}H_{n}(\langle x,h/\|h\|\rangle)$ and
using the fact that $T$ is orthogonal to all $H_{n}\circ g$ when $n\neq k$ and
$g\in X^{*}$.

######  Lemma 2.12.

Let $(X,H,\mu)$ be an abstract Wiener space, and let $T:X\to Y$ be a chaos of
order $k$. Choose an orthonormal basis $\\{e_{i}\\}$ of $H$ and let $T_{N}$ be
the finite-rank approximation given in (2.4). Then we have the uniform
convergence

$\lim_{N\to\infty}\sup_{\|h\|_{H}\leq
1}\big{\|}(T_{N})_{hom}(h)-T_{\mathrm{hom}}(h)\big{\|}_{Y}=0.$

Letting $B(H)$ denote the closed unit ball of $H$, it follows that
$T_{\mathrm{hom}}$ is continuous from $B(H)\to Y$, where $B(H)$ is given the
topology of $X$ (not of $H$).

This is proved in [HW15, (3.8)]. Since continuous functions map compact sets
to compact sets, we immediately obtain the following.

######  Corollary 2.13.

Let $(X,H,\mu)$ be an abstract Wiener space, and let $T^{i}:X\to Y_{i}$ be a
chaos of order $k_{i}$ for $1\leq i\leq m$, where $m\in\mathbb{N}$. Then the
set

$\\{(T^{1}_{\mathrm{hom}}(h),...,T^{m}_{\mathrm{hom}}(h)):h\in B(H)\\}$

is a compact subset of $Y_{1}\times\cdots Y_{m}$.

With all of these preliminaries in place, we are ready to formulate the main
theorem of this subsection, which is a generalization of the main theorem to
homogeneous variables of order $k$.

######  Theorem 2.14.

Let $(X,H,\mu)$ be an abstract Wiener space, fix $m\in\mathbb{N}$. Let
$T^{i}:X\to Y_{i}$ be chaoses of order $k_{i}$ respectively, for $1\leq i\leq
m$. Let $S:X\to X$ be a.e. linear, measure-preserving, and strongly mixing.
Then almost surely, the set of limit points of the random set

$\big{\\{}\big{(}(2\log n)^{-k_{1}/2}T^{1}(S_{n}x),\;.\;.\;.\;,(2\log
n)^{-k_{m}/2}T^{m}(S_{n}x)\big{)}:n\in\mathbb{N}\\}$

is equal to the compact set
$K:=\\{(T^{1}_{\mathrm{hom}}(h),...,T^{m}_{\mathrm{hom}}(h)):h\in B(H)\\}.$

###### Proof.

Recall that $S(H)=\\{h\in H:\|h\|_{H}=1\\}.$ Note by Lemma 2.4 that $S(H)$ is
dense in $B(H)$ with respect to the topology of $X$. Therefore, the set

$D:=\\{(T^{1}_{\mathrm{hom}}(h),...,T^{m}_{\mathrm{hom}}(h)):h\in S(H)\\}$

is dense in $K$. Thus it suffices to show that any point in $D$ is a limit
point of the given sequence. So fix a point
$(T^{1}_{\mathrm{hom}}(h),...,T^{m}_{\mathrm{hom}}(h))\in D$, where
$\|h\|_{H}=1$, and let $\epsilon>0$. We wish to show that

$\sum_{i=1}^{m}\big{\|}(2\log
n)^{-k_{i}/2}T^{i}(S_{n}x)-T^{i}_{hom}(h)\big{\|}_{Y_{i}}<\epsilon$ (2.7)

infinitely often. Fix an orthonormal basis $\\{e_{i}\\}$ for $H$, with
$e_{1}=h$, and let $T^{i}_{N}$ be the associated finite rank Cameron-Martin
projections as in (2.4). We claim that it is enough to prove (2.7) with each
$T^{i}(S_{n}x)$ replaced by $T^{i}_{N}(S_{n}x)$ and $T^{i}_{hom}(h)$ replaced
by $(T_{N}^{i})_{hom}(h)$ (and also replacing $\epsilon$ by $\epsilon/2$), for
some large enough $N$.

Indeed, by Corollary 2.11 we can choose $N$ so large that
$\int_{X}\|T^{i}_{N}-T^{i}\|_{Y_{i}}^{2}d\mu<\epsilon/(2m).$ Then by (2.5) and
the fact that $S_{n}$ is measure preserving, it is clear that

$\sum_{n\geq 2}\mu(\\{x:(2\log
n)^{-k_{i}/2}\|T^{i}_{N}(S_{n}x)-T^{i}(S_{n}x)\|_{Y}>\epsilon/m\\})<\infty,$
(2.8)

so by Borel-Cantelli lemma, $(2\log
n)^{-k_{i}/2}\|T^{i}_{N}(S_{n}x)-T^{i}(S_{n}x)\|_{Y}<\epsilon/m$ for all but
finitely many $n\in\mathbb{N}$ almost surely. Furthermore, by Lemma 2.12 we
can (by making $N$ larger) ensure that
$\|(T^{i}_{N})_{hom}(h)-T^{i}_{hom}(h)\|_{Y}<\epsilon/m.$

Thus we just need to show that

$\sum_{i=1}^{m}\big{\|}(2\log
n)^{-k_{i}/2}T^{i}_{N}(S_{n}x)-(T^{i}_{N})_{hom}(h)\big{\|}_{Y_{i}}<\epsilon/2$
(2.9)

infinitely often. Note that each $T^{i}$ is a chaos of order $k$ and
measurable with respect to a finite collection
$\\{\langle\cdot,e_{i}\rangle\\}_{i=1}^{N}$ of i.i.d. standard Gaussian random
variables, therefore one can verify that each $T^{i}_{N}$ can be written as

$T^{i}_{N}(x)=\sum_{j=1}^{M_{i}}y_{j}^{i}H_{k_{i}}(\langle
v^{i}_{j},x\rangle),$ (2.10)

for some finite collection of vectors
$\\{v^{i}_{j}\\}\subset\mathrm{span}(\\{e_{i}\\}_{i=1}^{N})$,
$\\{y^{i}_{j}\\}\subset Y$, and $M_{i}\in\mathbb{N}$. Here $H_{k}$ is the
$k^{th}$ Hermite polynomial. Using Cameron-Martin theorem, one can see that

$(T^{i}_{N})_{hom}(h)=\sum_{j=1}^{M_{i}}y_{j}^{i}\langle
v^{i}_{j},h\rangle^{k_{i}}.$ (2.11)

Note that

$\big{|}\langle v^{i}_{j},(2\log n)^{-1/2}S_{n}x\rangle^{k_{i}}-(2\log
n)^{-k_{i}/2}H_{k_{i}}(\langle v^{i}_{j},S_{n}x\rangle)\big{|}\to
0\;\;\;\;\;\;\;a.s.,$

by Borel-Cantelli lemma and the fact that $x\mapsto\langle
v^{i}_{j},S_{n}x\rangle$ are standard Gaussian random variables under $\mu$.
Thus it suffices to show that

$\sum_{\begin{subarray}{c}1\leq i\leq m\\\ 1\leq j\leq
M_{i}\end{subarray}}\|y^{i}_{j}\|_{Y}\big{|}\langle v^{i}_{j},(2\log
n)^{-1/2}S_{n}x\rangle^{k_{i}}-\langle
v^{i}_{j},h\rangle^{k_{i}}\big{|}<\epsilon/2$

happens infinitely often. Letting
$P_{N}:H\to\mathrm{span}\\{e_{i}\\}_{i=1}^{N}$ denote the orthogonal
projection, it is clear that $v^{i}_{j}=P_{N}(v^{i}_{j})$, thus by exploiting
self-adjointness of $P_{N}$, the previous expression is equivalent to showing
that

$\sum_{\begin{subarray}{c}1\leq i\leq m\\\ 1\leq j\leq
M_{i}\end{subarray}}\|y^{i}_{j}\|_{Y}\big{|}\langle v^{i}_{j},(2\log
n)^{-1/2}P_{N}(S_{n}x)\rangle^{k_{i}}-\langle
v^{i}_{j},h\rangle^{k_{i}}\big{|}<\epsilon/2$

infinitely often. Since $h=e_{1}$, we know from Corollary 2.2 that $\|(\log
n)^{-1/2}P_{N}(S_{n}x)-h\|_{H}<\delta$ infinitely often, for arbitrary
$\delta>0$. By choosing $\delta$ small enough and noting that $\langle
v^{i}_{j},\cdot\rangle$ is a continuous function on
$\text{span}\\{e_{i}\\}_{i=1}^{N}$, the claim (2.9) immediately follows, and
thus (2.7) is proved.

Now the only thing left to show is that the set of limit points of the given
sequence cannot contain points outside of the set $K.$ This will be done in
the following lemma. ∎

######  Lemma 2.15.

In the setting of Theorem 2.14, let

$a_{n}(x):=\big{(}(2\log n)^{-k_{1}/2}T^{1}(S_{n}x),\;.\;.\;.\;,(2\log
n)^{-k_{m}/2}T^{m}(S_{n}x)\big{)}.$

For a.e $x\in X$ one has that $\mathrm{dist}(a_{n}(x),K)<\epsilon$ for all but
finitely many $n$.

###### Proof.

Note that if our abstract Wiener space $(X,H,\mu)$ is finite dimensional, then
the statement is straightforward, since $T$ and $T_{\mathrm{hom}}$ are of the
form (2.10) and (2.11) respectively, and since all of the relevant quantities
are continuous functions.

Now we move to the infinite-dimensional case. Suppose for contradiction the
claim was false. Let $U$ denote a neighborhood of size $\epsilon$ around $K$.
Then since $S$ is strongly mixing (hence ergodic) and since the event
“$\mathrm{dist}(U^{c},a_{n}(x))<\epsilon/5$ infinitely often” is shift
invariant, it follows that it actually occurs with probability 1. By the same
argument used in deriving (2.8), we can choose $N$ so large that
$\sum_{n=1}^{m}(2\log
n)^{-k_{i}/2}\|T^{i}(S_{n}x)-T^{i}_{N}(S_{n}x)\|_{Y_{i}}<\epsilon/5$ for all
but finitely many $n$ almost surely, and moreover by Lemma 2.12 we can ensure
(by making $N$ possibly larger) that $\sum_{n=1}^{m}\sup_{\|h\|\leq
1}\|T^{i}_{hom}(h)-(T^{i}_{N})_{hom}(h)\|_{Y_{i}}<\epsilon/5.$ By the latter
bound and the definition of $U$ it is clear that

$dist\bigg{(}\;U^{c}\;,\;\big{(}(T^{1}_{\mathrm{hom}})_{N}(h),...,(T^{m}_{N})_{hom}(h)\big{)}\;\bigg{)}>4\epsilon/5$

for any $h$ such that $\|h\|_{H}\leq 1$. On the other hand the former bound
and our shift-invariant event of full probability guarantees that
$\mathrm{dist}(U^{c},a^{N}_{n}(x))<2\epsilon/5$ infinitely often (for a.e.
$x$), where

$a_{n}^{N}(x):=\big{(}(2\log n)^{-k_{1}/2}T^{1}_{N}(S_{n}x),...,(2\log
n)^{-k_{m}/2}T^{m}_{N}(S_{n}x)\big{)}.$

The preceding two sentences imply (by finite dimensionality) that the sequence
$(a_{n}^{N}(x))_{n\geq 1}$ contains a limit point outside of the set

$\\{((T^{1}_{N})_{hom}(h),...,(T^{m}_{N})_{hom}(h)):\|h\|_{H}\leq 1\\},$

since the distance of $a_{n}^{N}(x)$ to that set must be greater than
$2\epsilon/5$ infinitely often. This contradicts the finite dimensional
version of the statement that the set of limit points must be contained in
$K$, which is impossible as noted earlier. ∎

Next we formulate a continuous-time version of the above results. If $Y$ is a
Banach space, we denote by $C([0,1],Y)$ the space of continuous maps from
$[0,1]\to Y,$ equipped with the Banach space norm
$\|\gamma\|_{C([0,1],Y)}:=\sup_{t\in[0,1]}\|\gamma(t)\|_{Y}.$ For $t\in[0,1]$
we define $\pi_{t}:C([0,1],Y)\to Y$ by sending $\gamma\mapsto\gamma(t).$

######  Theorem 2.16.

Let $(X,H,\mu)$ be an abstract Wiener space, let $(S_{t})_{t\geq 0}$ be a
family of Borel-measurable a.e. linear maps from $X\to X$ which are measure-
preserving and strongly mixing, and let $T^{i}:X\to Y_{i}$ be homogeneous of
degree $k_{i}$ for $1\leq i\leq m$. Suppose that there exist strongly
continuous semigroups $(G^{i}_{t})_{t\geq 0}$ of operators from $Y_{i}\to
Y_{i}$ for $1\leq i\leq m$ with the property that

$T^{i}\circ S_{t}=G^{i}_{t}\circ
T^{i},\;\;\;\;\;\;\;\;\mu\text{-}\mathrm{a.e.}\;\;\;\;\;\;\text{ for all
}t\geq 0.$ (2.12)

Then almost surely, the set of cluster points at infinity of the random set

$\big{\\{}\big{(}(2\log t)^{-k_{1}/2}G_{t}^{1}T^{1}(x),\;.\;.\;.\;,(2\log
t)^{-k_{m}/2}G_{t}^{m}T^{m}(x)\big{)}:t\in[0,\infty)\\}$

is equal to the compact set
$K:=\\{(T^{1}_{\mathrm{hom}}(h),...,T^{m}_{\mathrm{hom}}(h)):h\in B(H)\\}.$

Note that we impose no semigroup condition on $(S_{t})$ itself. This is
because we do not need to, though in practice $(S_{t})$ will usually be a
strongly continuous semigroup on $X$. Note that this theorem implies
Proposition 1.2: set $k=1$, $m=1$, $Y=X$, let $T^{1}$ be the identity on $X$,
and let $G_{t}^{1}=S_{t}$.

###### Proof.

First we claim that that $(G_{t}^{i}(T^{i}(\xi)))_{t\in[0,1]}$ is homogeneous
variable of order $k_{i}$ taking values in the space $C([0,1],Y_{i}).$ To
prove this, note that if $Y,Z$ are Banach spaces, if $T:X\to Y$ is a chaos of
order $k$, and if $A:Y\to Z$ is a bounded linear map, then $A\circ T$ is also
a chaos of order $k$. We simply apply this to the case where $Y=Y_{i}$,
$Z=C([0,1],Y_{i})$ and $A:Y_{i}\to C([0,1],Y_{i})$ sends a point $y$ to
$(G^{i}_{t}y)_{t\in[0,1]}$. This linear map is bounded by the uniform
boundedness principle.

Next, note that the set of cluster points must contain $K$ by Theorem 2.14.
Thus we just need to show it contains no other points. Consider the random set

$\big{\\{}\big{(}(2\log t)^{-k_{1}/2}G_{t}^{1}(T^{1}(x)),\;.\;.\;.\;,(2\log
t)^{-k_{m}/2}G_{t}^{m}(T^{m}(x))\big{)}:t\in[0,\infty)\\}.$

The argument that this set contains no cluster points outside of $K$ is very
similar to that of Lemma 2.8. More precisely, we show that there exists
deterministic functions $C_{i}:[0,1]\to\mathbb{R}_{+}$ such that
$C_{i}(\rho)\to 0$ as $\rho\downarrow 0$ and such that

$\mu\bigg{(}\bigg{\\{}x\in
X:\limsup_{n\to\infty}\frac{\sup_{t\in[n\rho,(n+1)\rho]}\|G_{t}^{i}(T^{i}(x))-G_{n\rho}^{i}(T^{i}(x))\|_{Y_{i}}}{(\log
n)^{k_{i}/2}}\leq C_{i}(\rho)\bigg{\\}}\bigg{)}=1$

for all $\rho\in[0,1]$ and $1\leq i\leq m$. In particular
$(G_{t}^{i}(T^{i}(x)))_{t\geq 0}$ has the same set of cluster points as
$(G_{N}(T^{i}(x)))_{N\in\mathbb{N}}$ for $x$ in a set of full $\mu$-measure.

To prove the above claim, one uses precisely the same arguments as we did in
the proof of Lemma 2.8. Namely one defines
$C_{i}(\rho):=C\mathbb{E}[\sup_{t\in[0,\rho]}\|G_{t}^{i}(T^{i}(\xi))-T^{i}(\xi)\|_{Y_{i}}^{2}]$,
where $\xi$ is sampled from $\mu$ and $C>0$ is to be determined later. Then
one uses the fact that $(G_{t}^{i}(T^{i}(\xi)))_{t\in[0,1]}$ is homogeneous
variable of order $k_{i}$ taking values in the space $C([0,1],Y_{i}),$ by the
discussion above. Finally one uses the associated tail bounds for such
homogeneous variables as given in Proposition 2.10, and concludes using the
Borel-Cantelli lemma. ∎

###  2.3. Contraction principle for Strassen Laws

We derive a corollary that will be used in deriving Strassen’s Law for
singular semilinear SPDEs later. The following can be viewed as a sort of
contraction principle for Strassen’s law under continuous maps which may be
nonlinear.

######  Corollary 2.17.

Let $(X,H,\mu)$ be an abstract Wiener space, and let $T^{i}:X\to Y_{i}$ be a
chaos of order $k_{i}$ for $1\leq i\leq m$, where $m\in\mathbb{N}$. Let $Z$ be
a Banach space, and let $\mathcal{M}\subset Y_{1}\times\cdots Y_{m}$ be a
closed subset, such that for all $\delta>0$ one has

$\mu(\\{x\in
X:(\delta^{k_{1}}T^{1}(x),...,\delta^{k_{m}}T^{m}(x))\in\mathcal{M}\\})=1.$
(2.13)

Let $\Phi:\mathcal{M}\to Z$ be continuous. Then the compact set
$K:=\\{(T^{1}_{\mathrm{hom}}(h),...,T^{m}_{\mathrm{hom}}(h)):h\in B(H)\\}$ is
necessarily contained in $\mathcal{M}$, and moreover the set of cluster points
at infinity of the random set

$\big{\\{}\Phi\big{(}(2\log n)^{-k_{1}/2}T^{1}(S_{n}x),\;.\;.\;.\;,(2\log
n)^{-k_{m}/2}T^{m}(S_{n}x)\big{)}:n\in\mathbb{N}\big{\\}}$

is almost surely equal to $\Phi(K).$

###### Proof.

Note that if $(a_{1},...,a_{m})\in K$ then by Theorem 2.14 there exists a
subset $E\subset X$ of full measure such that for each $x\in E$, the sequence
$\big{(}(2\log n)^{-k_{1}/2}T^{1}(S_{n}x),\;.\;.\;.\;,(2\log
n)^{-k_{m}/2}T^{m}(S_{n}x)\big{)}$ converges to $(a_{1},...,a_{m})$ along some
subsequence as $n\to\infty$. Since $S_{n}$ are measure-preserving it holds by
(2.13) that

$\big{(}(2\log n)^{-k_{1}/2}T^{1}(S_{n}x),\;.\;.\;.\;,(2\log
n)^{-k_{m}/2}T^{m}(S_{n}x)\big{)}\in\mathcal{M}$

for a.e. $x\in X$. Consequently $(a_{1},...,a_{m})$ is a limit point of the
closed set $\mathcal{M}$ and thus belongs to $\mathcal{M}$. This implies that
$K$ is contained in $\mathcal{M}$.

Now we prove that the limit set is necessarily $\Phi(K)$. The fact that any
point $z\in\Phi(K)$ is a limit point is due to the fact that $\Phi$ is
continuous and any point of $K$ is a limit point of the sequence

$\big{(}(2\log n)^{-k_{1}/2}T^{1}(S_{n}x),\;.\;.\;.\;,(2\log
n)^{-k_{m}/2}T^{m}(S_{n}x)\big{)}$

as $n\to\infty$ (see Theorem 2.14).

Now we need to prove that points outside of $\Phi(K)$ are not limit points.
Suppose $z\notin\Phi(K)$. The latter set is closed so we may choose
$\epsilon>0$ such that $\|z-b\|_{Z}>\epsilon$ for all $b\in\Phi(K)$. Choose
$\delta>0$ so that $\mathrm{dist}(\Phi(a),\Phi(K))<\epsilon$ whenever
$\mathrm{dist}(a,K)<\delta$ (this $\delta$ exists by compactness of $K$). We
choose points $a_{1},...,a_{N}$ so that $B(a_{i},\epsilon)$ form an open cover
of $\Phi(K)$, then consider the open cover
$U_{i}:=\Phi^{-1}(B(a_{i},\epsilon))$ of $K$, then let $U$ denote the union of
the $U_{i}$, and take $\delta:=\min_{x\in K}\mathrm{dist}(x,U^{c})>0$).
Letting $a_{n}(x):=\big{(}(2\log n)^{-k_{1}/2}T^{1}(S_{n}x),\;.\;.\;.\;,(2\log
n)^{-k_{m}/2}T^{m}(S_{n}x)\big{)}$, by Lemma 2.15 we know that for a.e $x\in
X$ that $\mathrm{dist}(a_{n}(x),K)<\delta$ for all but finitely many $n$ and
therefore $z$ is not a limit point of $\Phi(a_{n}(x))$. ∎

######  Corollary 2.18.

Let $(X,H,\mu)$ be an abstract Wiener space, let $(S_{t})_{t\geq 0}$ be a
family of Borel-measurable a.e. linear maps from $X\to X$ which are measure-
preserving and strongly mixing, and let $T^{i}:X\to Y_{i}$ be homogeneous of
degree $k_{i}$ for $1\leq i\leq m$. Suppose that there exist strongly
continuous semigroups $(G^{i}_{t})_{t\geq 0}$ of operators from $Y_{i}\to
Y_{i}$ for $1\leq i\leq m$ with the property that

$T^{i}\circ S_{t}=G^{i}_{t}\circ
T^{i},\;\;\;\;\;\;\;\;\mu\text{-a.e.}\;\;\;\;\;\;\text{ for all }t\geq 0.$

Let $Z$ be a Banach space, and let $\mathcal{M}\subset Y_{1}\times\cdots
Y_{m}$ be a closed subset, such that for all $\delta>0$

$\mu(\\{x\in
X:(\delta^{k_{1}}T^{1}(x),...,\delta^{k_{m}}T^{m}(x))\in\mathcal{M}\\})=1.$

Let $\Phi:\mathcal{M}\to Z$ be continuous. Then the compact set
$K:=\\{(T^{1}_{\mathrm{hom}}(h),...,T^{m}_{\mathrm{hom}}(h)):h\in B(H)\\}$ is
necessarily contained in $\mathcal{M}$, and moreover the set of cluster points
at infinity of

$\big{\\{}\Phi\big{(}(2\log t)^{-k_{1}/2}G_{t}^{1}T^{1}(x),\;.\;.\;.\;,(2\log
t)^{-k_{m}/2}G_{t}^{m}T^{m}(x)\big{)}:t\in[0,\infty)\\}$

is almost surely equal to $\Phi(K).$

###### Proof.

The argument is similar to the previous corollary but applying Theorem 2.16
rather than Theorem 2.14. ∎

Note that the previous two corollaries are the most general version of the
Strassen’s Law that we have stated so far (e.g. set $Z=Y_{1}\times\cdots
Y_{m}=\mathcal{M}$ and let $\Phi$ be the identity). Note also that Corollary
2.18 will be used in multiplicative form later (see Theorem 1.3), not the
additive form stated above. Even further generalizations will be given in
Theorems 4.2, 4.5, and 4.6 below.

##  3\. Examples and applications to SPDEs

With the above results in place, we now move on to the main result of the
paper. Although we want to study the application of the above results to
singular SPDEs such as KPZ, we start with a series of simpler examples just to
illustrate the methods.

###  3.1. First examples

The first three examples will be done with just the first chaos (in other
words just a Gaussian measure) so that only Proposition 1.2 is needed as
opposed to the full generality of Theorem 1.3. Then the next two examples will
be done with higher chaos and contraction by continuous maps. A few of these
examples will concisely recover classical results.

######  Example 3.1 (Brownian and fractional Brownian motion).

Consider the fractional Brownian motion $(B^{H}(t))_{t\in[0,1]}$ with Hurst
parameter $H\in(0,1)$. Let $\mathcal{F}$ denote the space of smooth real-
valued functions on $[0,1]$ which vanish at $0$. We can let $X$ be the closure
of $\mathcal{F}$ in the Hölder space $C^{\kappa}[0,1]$ for $\kappa\in(0,H)$.
In fact, one may consider the closure of $\mathcal{F}$ with respect to an even
stronger norm

$\|f\|_{X}=\sup_{0<|t-s|<1/3}\frac{|f(x)-f(y)|}{\varphi(|t-s|)}$

where $\varphi$ may be any increasing function vanishing at $0$ whose growth
near $0$ is strictly larger than the modulus of continuity of $B^{H}$ (for
instance $\varphi(a)=a^{H}(\log(1/a))^{\frac{1}{2}+\gamma}$ and $\gamma>0$ is
small). One may apply Theorems 2.52 and 8.27 in [Wea18] to see that the law of
$B^{H}$ is supported on the closure of smooth functions in such a norm (in
fact, the closure contains any function with a strictly better modulus of
continuity than $\varphi$).

One defines the measure-preserving operator
$R_{\epsilon}f(t):=\epsilon^{-H}f(\epsilon t)$. Since $\varphi$ is increasing
we see that
$\sup_{\epsilon\in[e^{-1},1]}\sup_{a\in(0,1/4]}\epsilon^{-H}\varphi(\epsilon
a)/\varphi(a)\leq e^{H}<\infty$, therefore Lemma B.9 implies that
$(R_{\epsilon})$ is strongly continuous on this Banach space. Now to prove the
Strassen law, we need to verify a strong mixing condition on the appropriate
space. Since this mixing condition is the crucial input needed to yield the
Strassen law, and since one of the main contributions of this work is to
reduce the proof of the Strassen law to this deterministic mixing calculation,
we give several different proofs of the mixing condition.

  1. (1)

A result of [DU99, (6)] shows that the Cameron-Martin space $\mathcal{H}$ of
$B^{H}$ consists of functions of the form
$h(t)=\int_{0}^{t}K_{H}(t,s)\ell(s)ds$ for some $\ell\in L^{2}[0,1]$, with the
isometric identity $\|h\|_{\mathcal{H}}=\|\ell\|_{L^{2}}$, where $K_{H}$ is an
explicit kernel that satisfies $K_{H}(\epsilon t,\epsilon
s)=\epsilon^{H-\frac{1}{2}}K_{H}(t,s)$. Consequently one has
$\|R_{\epsilon}h\|_{\mathcal{H}}=\|\tilde{R}_{\epsilon}\ell\|_{L^{2}[0,1]}\to
0$ as $\epsilon\to 0$, where
$\tilde{R}_{\epsilon}\ell(t)=\epsilon^{1/2}\ell(\epsilon t).$

  2. (2)

Again using [DU99, (6)], one may write
$B^{H}(t)=\int_{0}^{t}K_{H}(s,t)dB^{\frac{1}{2}}(s)$ for a standard Brownian
motion $B^{\frac{1}{2}}$ on $[0,1]$. If we let $\tilde{X}:=C[0,1]$,
$\tilde{\mathcal{H}}=H_{0}^{1}[0,1]$ and $\mu$ the law on $X$ of $C[0,1]$,
then we may then view $B^{H}$ as an $X$-valued chaos of order $1$ over
$(\tilde{X},\tilde{\mathcal{H}},\mu)$. More precisely we have
$B^{H}=\psi(B^{1/2})$ where $\psi(B)=\int_{0}^{\bullet}K_{H}(s,\bullet)dB(s).$
Note that $\psi_{hom}(\ell)=\int_{0}^{\bullet}K_{H}(s,\bullet)\ell(s)ds$ for
$\ell\in L^{2}[0,1].$ Moreover, we clearly have the relation
$R_{\epsilon}\psi=\psi\tilde{R}_{\epsilon}$ where $\tilde{R}_{\epsilon}$ is
the same semigroup on $L^{2}[0,1]$ as in the proof (1) above. Thus we may
conclude the Strassen law from Theorem 1.3. While this perspective is
equivalent to the proof used in (1) just above, the paradigm will be more
useful when we deal with higher chaos.

This concludes the proof for the Strassen law of $B^{H}$ on the unit interval
$[0,1]$. Rather than considering the Strassen law for $B^{H}$ restricted to
the unit interval, we can instead consider it on the full real line
$\mathbb{R}$ as well. Indeed, the Banach space $X$ can be taken to be
$X:=\\{f\in C[0,\infty):\lim_{t\to\infty}\frac{|f(t)|}{t}=0\\}$ with norm
given by $\sup_{t\geq 0}\frac{|f(t)|}{1+t}$. This is a separable Banach space
which supports $B^{H}$ (for all time, not just the unit interval) almost
surely. Alternatively one can also use Holder spaces with suitable weight
functions.

In this full-line setting, we can give alternative and more self-contained
proofs of the calculation of the Cameron-Martin space and the mixing property
of the operators $R_{\epsilon}$ (defined by the same formula) without using
[DU99], and instead using results from Appendix B. Letting
$C(s,t):=|t|^{2H}+|s|^{2H}-|t-s|^{2H}$, we can define a bounded operator on
$L^{2}[0,1]$ by $Qf(t)=\frac{1}{2}\int_{0}^{1}C(s,t)f(s)ds.$ In the sense of
tempered distributions the Fourier transform of $a(t)=|t|^{2H}$ is given by
(the finite part of) $\hat{a}(\xi)=c_{H}|\xi|^{-2H-1}$ for some universal
$c_{H}>0$ (this is immediate from a scaling argument). This implies that if
$\phi\in C_{c}^{\infty}(\mathbb{R})$, then

$(Q\phi^{\prime},\phi^{\prime})_{L^{2}[0,1]}=-\frac{1}{2}\int_{\mathbb{R}}\int_{\mathbb{R}}|t-s|^{2H}\phi^{\prime}(s)\phi^{\prime}(t)dsdt=\frac{c_{H}}{2}\int_{\mathbb{R}}|\hat{\phi}(\xi)|^{2}|\xi|^{1-2H}d\xi,$
(3.1)

where $\phi^{\prime}$ is the usual derivative, and the last equality can be
deduced using Plancherel’s identity and the convolution-to-multiplication
property of the Fourier transform. Since $|\xi|^{1-2H}$ has polynomial growth
or decay at infinity, this identity is enough to show that the completion of
$L^{2}(\mathbb{R})$ with respect to the norm $(Q\phi,\phi)^{1/2}$ is contained
in the space $\mathcal{S}^{\prime}(\mathbb{R})$ of tempered distributions.
Using Lemma B.8 we see that the image of $Q^{1/2}$ contains
$\mathcal{S}(\mathbb{R})$ and, regardless of the choice of norm on the larger
space $X$, the Cameron-Martin space of $B^{H}$ is given by the completion of
$\mathcal{S}(\mathbb{R})$ under
$\|\phi\|_{\mathcal{H}}=\|Q^{-1/2}\phi\|_{L^{2}}$. Now to verify the mixing
condition, note that if $f,g$ are any smooth functions such that
$f^{\prime},g^{\prime}$ have compact support, then

$\langle R_{\epsilon}f,g\rangle_{H}=(\epsilon\cdot
R_{\epsilon}(f^{\prime}),Q_{\mathrm{der}}^{-1}g^{\prime})_{L^{2}[0,1]}=\frac{2}{c_{H}}\int_{\mathbb{R}}|\xi|^{2H-1}\epsilon^{-H-1}\cdot
i\xi\hat{f}(\epsilon^{-1}\xi)\cdot\overline{i\xi\hat{g}(\xi)}d\xi,$ (3.2)

where $Q_{\mathrm{der}}:=-\frac{d}{dt}Q\frac{d}{dt}$ whose Fourier symbol is
$\frac{c_{H}}{2}|\xi|^{1-2H}$ by (3.1). Note that by setting $\epsilon=1$,
this shows that $\langle
f,g\rangle_{H}=\frac{2}{c_{H}}\int_{\mathbb{R}}|\xi|^{2H+1}\hat{f}(\xi)\overline{\hat{g}(\xi)}d\xi.$

Using the fact that $\epsilon^{-1}\hat{f}(\epsilon^{-1}\xi)$ is an approximate
delta function, the right side of (3.2) can be bounded above by
$C\epsilon^{-1-H}\cdot\epsilon\cdot\epsilon^{2H+1}=C\epsilon^{1+H}$ where $C$
may depend on $f,g$ but not on $\epsilon\in(0,1]$. We may now use Lemma B.10,
noting that such $f,g$ are dense in $\mathcal{S}(\mathbb{R})$. Then Strassen’s
law for $(\log\log(1/\epsilon))^{-1/2}R_{\epsilon}B^{H}$ thus follows from the
multiplicative form of Proposition 1.2. This concludes the alternative direct
proof of the Strassen’s law (on the full line).

We remark that this full-line proof generalizes to yield Strassen’s law for
the multiparameter fractional Brownian sheet studied in [LWC04], in every
Hölder space. Rather than considering the small-time version of Strassen’s law
for fractional Brownian motion, we can also directly prove the large-time
version. Indeed one similarly checks that the family of operators
$\tilde{R}_{\epsilon}f(x)=\epsilon^{H}f(\epsilon^{-1}x)$ is strongly mixing
with respect to this norm, using the result of Lemma B.10. Consequently one
obtains the Strassen’s Law for
$\epsilon^{H}B(\epsilon^{-1}t)/\sqrt{2\log\log(1/\epsilon)}$ with respect to
the norm of $X$.

######  Example 3.2.

Consider the stochastic heat equation

$\partial_{t}h(t,x)=\partial^{2}_{x}h(t,x)+\xi(t,x),\quad h(0,x)=0,\quad t\geq
0,x\in\mathbb{R}^{d},$ (3.3)

where $\xi$ is a Gaussian space-time white noise on
$\mathbb{R}_{+}\times\mathbb{R}$. The solution is defined by $h=K*\xi$, where
$K(t,x)=\frac{1}{\sqrt{2\pi t}}{e^{-x^{2}/2t}}1_{\\{t\geq 0\\}},$ where $*$
denotes convolution in both space and time, and the convolution may be
interpreted as a stochastic integral against $\xi$.

In the notation of Appendix B, let $U=\mathbb{(}0,\infty)\times\mathbb{R}^{d}$
and we define the Frechet space $\mathcal{F}(U)$ to be the set of all smooth
functions of rapid decay on $\mathbb{R}_{+}\times\mathbb{R}^{d}$ such that
$f(0,x)=0$ for all $x\in\mathbb{R}$. For $d=1$ one may show that the solution
$h$ is supported on the space $X$ given by the completion of $\mathcal{F}(U)$
with respect to $\|\phi\|_{X}:=\sup_{t,x}\frac{\phi(t,x)}{1+t+|x|}.$ For $d>1$
the equation cannot be realized as a continuous function, thus one has to use
a space of generalized functions for $X$. An example would be the closure of
Schwartz functions in a weighted parabolic Besov-Hölder space of negative
exponent $\alpha:=\frac{1}{2}-\frac{d}{2}-\kappa$ where $\kappa>0$ is
arbitrary. More specifically, one takes the closure of $\mathcal{F}(U)$ under

$\|\phi\|_{X}:=\sup_{\|\psi\|_{C^{r}\leq
1}}\sup_{\lambda\in(0,1],(t,x)\in\mathbb{R}_{+}\times\mathbb{R}}\frac{\lambda^{-\alpha}|(\phi,\psi^{\lambda}_{t,x})|}{w(t,x)},$
(3.4)

where $r=-\lfloor\alpha\rfloor$,
$\psi^{\lambda}_{t,x}(s,y)=\lambda^{-d-2}\phi(\lambda^{-2}(t-s),\lambda^{-1}(x-y))$.
Here $w(t,x)$ may be taken to be $(1+t+|x|)^{\sigma}$ for sufficiently large
$\sigma>0$. See [MW17, Section 5 - Lemma 9] for a proof of these support
statements.

If we define
$R_{\epsilon}h(x)=\epsilon^{\frac{d}{2}-1}h(\epsilon^{2}t,\epsilon x)$ (to be
interpreted by integration against a test function if $X$ consists of a space
of distributions), then one verifies that $R_{\epsilon}$ sends $X$ boundedly
to itself and satisfies all of the conditions of Lemma B.9 for either choice
of norm above. Next we need to verify the mixing condition on an appropriate
space. Again we give two separate paradigms for doing this.

  1. (1)

Using e.g. Lemma B.7 and Remark B.8, one verifies that the solution of the
equation has Cameron-Martin space given by the completion of $\mathcal{F}(U)$
with respect to the norm
$\|h\|_{H}:=\|\partial_{t}h-\partial_{x}^{2}h\|_{L^{2}(\mathbb{R}_{+}\times\mathbb{R}^{d})}$.
Indeed, note that
$\mathbb{E}[(h,\phi)(h,\psi)]=\langle\phi,K*K^{s}*\psi\rangle_{L^{2}(\mathbb{R}_{+}\times\mathbb{R})}$
where $K^{s}(t,x)=\frac{1}{\sqrt{2\pi|t|}}{e^{-x^{2}/2|t|}}1_{\\{t\leq 0\\}}$.
Since $K*K^{s}$ is the kernel for the inverse operator of
$(\partial_{t}-\partial_{x}^{2})(\partial_{t}+\partial_{x}^{2})$, the claim is
immediate from Lemma B.8. Consequently by Proposition 1.2 (and the discussion
afterwards) one finds that the set
$\\{(\log\log(1/\epsilon))^{-1/2}R_{\epsilon}h\\}_{\epsilon\in(0,1]}$ is
precompact and that its set of cluster points in $X$ as $\epsilon\to 0$ equals
the closed unit ball of $H$.

  2. (2)

Rather than viewing $h$ as an intrinsic object we can view it as a chaos of
order 1 over $\xi$, specifically $h=\psi(\xi):=K*\xi$ so that
$\psi_{hom}(h)=K*h$. We may view $\xi$ as taking values in the Banach space
$\tilde{X}$ given (for example) by the closure of $\mathcal{F}(U)$ under the
norm $\|K*f\|_{X}$. If we define
$\tilde{R}_{\epsilon}f(t,x):=\epsilon^{3/2}f(\epsilon^{2}t,\epsilon x)$ then
one uses Lemma B.9 to verify the mixing condition required of
$\tilde{R}_{\epsilon}$ (considering functions with compact support contained
in $U$). Furthermore one has the relation
$R_{\epsilon}\psi=\psi\tilde{R}_{\epsilon},$ thanks to the scaling property of
the heat kernel. Thus one applies Theorem 1.3 to obtain the Strassen law.

This concludes the examples with first-order chaos, and now we move on to
examples with higher chaos. Henceforth we shall not verify the calculation of
the Cameron-Martin norm and the mixing condition as explicitly as we have done
above, as it should be understood that this may be done similarly to above
using e.g. results from Appendix B.

######  Example 3.3.

We give a simple example in the second chaos. Let $X=C([0,1],\mathbb{R}^{2})$,
take $\mu$ to be the law on $X$ of two-dimensional standard Brownian motion,
which has Cameron-Martin space $H=\\{(f_{1},f_{2})\in
X:\int_{0}^{1}(f_{1}^{\prime}(t)^{2}+f_{2}^{\prime}(t)^{2})dt<\infty\\}.$ Let
$Y=C[0,1]$ and consider the (discontinuous) map $\psi:X\to Y$

$\psi(B_{1},B_{2})=\int_{0}^{\bullet}B_{1}(t)dB_{2}(t).$

It is known that $\psi$ is a homogeneous chaos of order two, see e.g. [Nua06,
Chapter 1]. For $f\in Y$ and $(f_{1},f_{2})\in X$ let us define

$\displaystyle Q_{\epsilon}f(t)$ $\displaystyle:=\epsilon^{-1}f(\epsilon t),$
$\displaystyle R_{\epsilon}(f_{1},f_{2})(t)$
$\displaystyle:=(\epsilon^{-1/2}f_{1}(\epsilon
t),\epsilon^{-1/2}f_{2}(\epsilon t)).$

Then it is clear that $\psi\circ R_{\epsilon}=Q_{\epsilon}\circ\psi$ a.s., and
thus one can obtain the Strassen law (as $\epsilon\to 0$) for the family of
processes

$\bigg{\\{}\bigg{(}(2\epsilon\log\log(1/\epsilon))^{-1}\int_{0}^{\epsilon
t}B_{1}(s)dB_{2}(s)\bigg{)}_{t\in[0,1]}\bigg{\\}}_{0<\epsilon<1/e}.$

The compact limit set is checked to be
$\\{\int_{0}^{\bullet}f_{1}(s)f_{2}^{\prime}(s)ds:\int_{0}^{1}f_{1}^{\prime}(s)^{2}+f_{2}^{\prime}(s)^{2}\leq
1\\}.$ Note that one may strengthen the topology of $Y$ to the closure of
smooth functions with respect to the Hölder norm of any exponent less than
1/2. See e.g. [FH20, Proposition 3.4] to see that the path has this Hölder
regularity, and note that Lemma B.7 ensures that the compact limit set remains
unchanged by changing the Banach spaces $X$ and $Y$ chosen to contain
$(B_{1},B_{2})$ or $\psi(B_{1},B_{2})$ respectively.

Note that there is no additional difficulty in carrying out this same
procedure for $k$-fold iterated integrals for any $k\in\mathbb{N}$, which
recovers and generalizes [MO86].

######  Example 3.4.

Consider a standard space-time white noise $\xi$ on
$\mathbb{R}_{+}\times\mathbb{R}$ and consider a $k$-fold stochastic integral
of the form

$F(t,x):=\int_{t_{1}<...<t_{k}}\int_{\mathbb{R}^{k}}\prod_{j=1}^{k}p_{t_{i}-t_{i-1}}(x_{i}-x_{i-1})\xi(dt_{k},dx_{k})\cdots\xi(dt_{1},dx_{1}),$
(3.5)

where $t_{0}=t$, $x_{0}=x,$ and $p_{t}(x)=(2\pi t)^{-1/2}e^{-x^{2}/2t}$ is the
standard heat kernel. Such an expression appears when considering chaos
expansions for the Hopf-Cole transform of the KPZ equation, see e.g. [Wal86].
If we consider $\psi(\xi)=F$ as a homogeneous chaos of order $k$ [Nua06,
Chapter 1], then it turns out [Wal86] that $\psi$ can take values in the space
$\mathcal{C}^{\alpha}$ given by the closure of smooth functions in the Banach
space of Hölder continuous functions on $B:=[0,1]\times[-1,1]$ (for instance),
with norm

$\|h\|_{\mathcal{C}^{\alpha}}:=\sup_{(t,x)\in
B}|h(t,x)|+\sup_{(t,x)\neq(t^{\prime},x)\in
B}\frac{|h(t,x)-h(t^{\prime},x^{\prime})|}{|t-t^{\prime}|^{\alpha/2}+|x-x^{\prime}|^{\alpha}}.$

Furthermore, if one defines
$(R_{\epsilon}\xi)(t,x):=\epsilon^{3/2}\xi(\epsilon^{2}t,\epsilon x)$, then by
using the identity $cp_{c^{2}t}(cx)=p_{t}(x)$ for $c>0$, one verifies that
$\psi R_{\epsilon}=Q_{\epsilon}\psi$ where
$Q_{\epsilon}F(t,x)=\epsilon^{-k/2}F(\epsilon^{2}t,\epsilon x)$. Consequently,
with $F$ given by (3.5), the set of limit points of
$\\{(\log\log(1/\epsilon))^{-k/2}\epsilon^{-k/2}F(\epsilon^{2}t,\epsilon
x)\\}$ is equal to the compact set of functions in $\mathcal{C}^{\alpha}$
which are of the form

$\int_{t_{1}<...<t_{k}}\int_{\mathbb{R}^{k}}\prod_{j=1}^{k}p_{t_{i}-t_{i-1}}(x_{i}-x_{i-1})f(t_{k},x_{k})\cdots
f(t_{1},x_{1})dx_{k}\cdots dx_{1}dt_{k}\cdots dt_{1},$

where $\|f\|_{L^{2}(\mathbb{R}_{+}\times\mathbb{R})}\leq 1.$

######  Example 3.5.

Let $b:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be smooth and globally Lipchitz, and
let $\sigma:\mathbb{R}^{d}\to\mathbb{R}^{d\times d}$ be smooth and globally
Lipchitz. Let $B=(B_{t})_{t\geq 0}$ be a standard Brownian motion. For
$\delta>0$ we consider the Itô SDE in $\mathbb{R}^{d}$ given by

$dX^{\delta}=b(X^{\delta})dt+\delta\cdot\sigma(X^{\delta})dB,$ (3.6)

with $X^{\delta}(0)=0$. Then a unique solution exists which is adapted to the
filtration of $B$ [RY99, Chapter XI]. It is known for $d\geq 2$ that the
measurable map $B\mapsto X^{\delta}$ is discontinuous in general.

However we can factor it into a continuous map and a second-order chaos as
follows. We fix $\alpha\in(1/3,1/2)$. We let
$C^{\alpha}([0,1],\mathbb{R}^{d})$ be the closure of smooth paths with respect
to the $\alpha$-Hölder norm on paths $[0,1]\to\mathbb{R}^{d}$, and we let
$C^{\alpha}([0,1]^{2},\mathbb{R}^{d}\otimes\mathbb{R}^{d})$ be the Banach
space consisting of those functions
$(s,t)\mapsto\mathbb{F}_{s,t}\in\mathbb{R}^{d}\otimes\mathbb{R}^{d}$ which lie
in the closure of the set of smooth functions vanishing along the diagonal,
with respect to the norm given by

$\sup_{s\neq t}\frac{|\mathbb{F}_{s,t}|}{|t-s|^{2\alpha}},$

where $|\cdot|$ can be any norm on $\mathbb{R}^{d}\otimes\mathbb{R}^{d}$. We
then let $\mathcal{M}\subset C^{\alpha}([0,1],\mathbb{R}^{d})\times
C^{\alpha}([0,1]^{2},\mathbb{R}^{d}\otimes\mathbb{R}^{d})$ be the space of
rough paths in the sense of [FH20, Definition 2.1], that is all those pairs
$(F,\mathbb{F})\in C^{\alpha}([0,1],\mathbb{R}^{d})\times
C^{\alpha}([0,1]^{2},\mathbb{R}^{d}\otimes\mathbb{R}^{d})$ satisfying

$\mathbb{F}_{s,t}=\mathbb{F}_{s,u}+\mathbb{F}_{u,t}+(F_{u}-F_{s})\otimes(F_{s}-F_{t}),\;\;\;\;\;\;\;\;\;\;\;0\leq
s,t,u\leq 1.$

Then there exists a continuous map $\Phi:\mathcal{M}\to
C([0,1],\mathbb{R}^{d})$ such that the solution of (3.6) is given by

$X^{\delta}=\Phi\bigg{(}\delta
B,\delta^{2}\int_{\bullet}^{-}(B_{r}-B_{\bullet})\otimes dB_{r}\bigg{)},$
(3.7)

for all $\delta>0$, where explicitly the latter stochastic integral should be
understood as

$\bigg{(}\int_{\bullet}^{-}(B_{r}-B_{\bullet})\otimes
dB_{r}\bigg{)}_{s,t}:=\sum_{i,j=1}^{d}\bigg{(}\int_{s}^{t}(B^{i}_{r}-B^{i}_{s})dB^{j}_{r}\bigg{)}e_{i}\otimes
e_{j}$

where $\\{e_{i}\\}$ is the standard basis of $\mathbb{R}^{d}$. The fact that
this second-order chaos is a.e. defined (albeit discontinuous as a function of
$B$) and takes values in the Banach space
$C^{\alpha}([0,1]^{2},\mathbb{R}^{d}\otimes\mathbb{R}^{d})$ for
$\alpha\in(1/3,1/2)$ is proved in [FH20, Proposition 3.4]. This map $\Phi$ is
called the Itô-Lyons map and its construction and structural properties are
described in great detail in [FH20, Chapter 8]. More specifically, the
continuity of $\Phi$ is proved as [FH20, Theorem 8.5] and the identity (3.7)
is proved as [FH20, Theorem 9.1].

Now let us formulate a version of Strassen’s law for this SDE. If we let
$R_{\epsilon}f:C^{\alpha}([0,1],\mathbb{R}^{d})\to
C^{\alpha}([0,1],\mathbb{R}^{d})$ by
$R_{\epsilon}f(t)=\epsilon^{-1}f(\epsilon^{2}t)$ then we can consider the
sequence $X^{\epsilon}$ of Itô solutions to

$dX^{\epsilon}=b(X^{\epsilon})dt+(\log\log(1/\epsilon))^{-1/2}\sigma(X^{\epsilon})d\big{(}R_{\epsilon}B\big{)}.$

Moreover, we have the second-order chaos
$\psi:C^{\alpha}([0,1],\mathbb{R}^{d})\to
C^{\alpha}([0,1]^{2},\mathbb{R}^{d}\otimes\mathbb{R}^{d})$ given by

$\psi(B)=\int_{\bullet}^{-}(B_{r}-B_{\bullet})\otimes dB_{r}.$

We can then define
$Q^{1}_{\epsilon}:=R_{\epsilon}:C^{\alpha}([0,1],\mathbb{R}^{d})\to
C^{\alpha}([0,1],\mathbb{R}^{d})$ and we can also define
$Q^{2}_{\epsilon}:C^{\alpha}([0,1]^{2},\mathbb{R}^{d}\otimes\mathbb{R}^{d})\to
C^{\alpha}([0,1],\mathbb{R}^{d}\otimes\mathbb{R}^{d})$ by
$Q^{2}_{\epsilon}g(s,t)=\epsilon^{-2}g(\epsilon^{2}s,\epsilon^{2}t),$ then one
verifies that the commutation assumption of Theorem 1.3 holds, namely that
$(R_{\epsilon}B,\psi(R_{\epsilon}B))=(Q^{1}_{\epsilon}B,Q^{2}_{\epsilon}\psi(B)).$
The $Q^{i}$ are checked to be strongly continuous using Lemma B.9. We already
remarked in an earlier example that $R_{\epsilon}$ is strongly mixing, so by
continuity of the map $\Phi$, we may then apply Theorem 1.3, as well as the
result of Example 3.3 to obtain that the set of limit points of
$C([0,1],\mathbb{R}^{d})$ of $\\{X^{\epsilon}\\}_{\epsilon\in(0,1]}$ as
$\epsilon\to 0$ is equal to the compact set $K$ of functions of the form

$\Phi\bigg{(}f,\int_{\bullet}^{-}(f(r)-f(\bullet))\otimes
f^{\prime}(r)dr\bigg{)},$

as $f$ ranges throughout the closed unit ball of the Cameron-Martin space of
$B$, namely $\\{f\in
C([0,1],\mathbb{R}^{d}):\int_{0}^{1}\|f^{\prime}(s)\|_{Eu}^{2}ds\leq 1\\},$
where $\|x\|_{Eu}$ denotes the Euclidean norm of $x\in\mathbb{R}^{d}$. As
above $(\bullet,-)$ denotes the argument of a general two-variable function
here.

Now when $f$ is a smooth function from $[0,1]\to\mathbb{R}^{d}$, then the last
expression inside of $\Phi$ is a smooth geometric rough path, and the Itô-
Lyons map $\Phi$ acts on such a path by sending it to the solution of the
classical ODE $\dot{x}=b(x)+\sigma(x)f$ as explained in [FH20, Remark 8.4]. We
may then conclude that the compact limit set $K$ of $\\{X^{\epsilon}\\}$ is
equal to the closure in $C([0,1],\mathbb{R}^{d})$ of the set of functions $x$
where $x(0)=0$ and $\dot{x}=b(x)+\sigma(x)f$ for some smooth path $f\in
C([0,1],\mathbb{R}^{d})$ such that $\int_{0}^{1}\|f^{\prime}(s)\|_{Eu}^{2}\leq
1.$

######  Example 3.6 (A counterexample).

We conclude with an example where the semigroup $(R_{\epsilon})$ does not
extend to the full space $X$ in a strongly continuous fashion. Let
$H=L^{2}(\mathbb{T})$ where $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. If we view
$\mathbb{T}=[0,1]$ with the endpoints identified, we may sensibly define
$(R_{\epsilon}\xi)(x)=\epsilon^{1/2}\xi(\epsilon x)$ for $\xi\in H$. It is
clear that this defines a bounded linear map on $H$ satisfying
$R_{\epsilon}R_{\epsilon}^{*}=I$. Let $X=C^{-\alpha}(\mathbb{T})$ where
$\alpha>1/2$, which is defined to be the closure of smooth functions
$f:\mathbb{T}\to\mathbb{R}$ under the norm given by

$\|f\|:=\sup_{\lambda\in(0,1]}\sup_{x\in\mathbb{T}}\sup_{\|\varphi\|_{C^{1}}\leq
1}\lambda^{\alpha}(f,\varphi^{\lambda}_{x})_{L^{2}(\mathbb{T})}$

where $\varphi^{\lambda}_{x}(y)=\lambda^{-1}\varphi(\lambda^{-1}(y-x))$. Then
the law of $\xi$ is supported on $X$ as shown in e.g. [HL18, Section 2], thus
$R_{\epsilon}$ extends from $H$ to an a.e. linear measure-preserving operator
on $X$ for each $\epsilon>0$ by Lemma A.2. Nonetheless $R_{\epsilon}$ does not
extend to a bounded operator on the full space $X$ as it would violate the
periodic boundary conditions. On one hand, by Theorem 1.1 one still has a
“discrete-time” Strassen law along any subsequence of the form
$\epsilon_{N}=\alpha^{N}$ where $\alpha\in(0,1)$, but to even sensibly talk
about the continuous version one would first need to simultaneously define the
uncountable collection $(R_{\epsilon})_{\epsilon\in(0,1]}$ on a subset of full
measure, which seems nontrivial. This illustrates that the assumptions in
Theorem 1.3 are necessary.

###  3.2. Strassen law for the KPZ equation

In this subsection we prove Theorem 1.4. Although we focus on KPZ, the method
described here is general and may apply to any SPDE whose solution map can be
factored into a finite number of chaoses and a continuous part

$\xi\mapsto(\psi_{1}(\xi),...,\psi_{m}(\xi))\stackrel{{\scriptstyle\text{continuous}}}{{\mapsto}}\text{solution
of SPDE},$

which is always the case when using the theory of regularity structures as we
do here. The only caveat is that it is difficult to come up with interesting
examples of semigroups satisfying the mixing conditions unless the solution
(in addition to admitting a factorization as above) is globally well-posed on
the full space $\mathbb{R}^{d}$, for which KPZ (or rather its Hopf-Cole
transform) seems to be the only example [HL18, PR19]. For example, in [MW17]
the $\Phi_{2}^{4}$ equation is shown to be globally well-posed on
$\mathbb{R}^{2}$ but it lacks the explicit construction of a continuous part
of the solution map on the full space, which is needed in our context.

Let $\xi$ be a standard space-time white noise on
$\mathbb{R}_{+}\times\mathbb{R}$ and define the following distribution-valued
chaoses for $z\in\mathbb{R}_{+}\times\mathbb{R}$:

$\displaystyle\Pi_{z}^{\xi}\Xi(\psi)$
$\displaystyle:=\int_{\mathbb{R}_{+}\times\mathbb{R}}\psi(w)\xi(dw),$
$\displaystyle\Pi_{z}^{\xi}\big{[}\Xi\mathcal{I}[\Xi]\big{]}(\psi)$
$\displaystyle:=\int_{(\mathbb{R}_{+}\times\mathbb{R})^{2}}\psi(w)(K(w-a)-K(z-a))\xi(da)\xi(dw),$
$\displaystyle\Pi_{z}^{\xi}\big{[}\Xi\mathcal{I}[\Xi\mathcal{I}[\Xi]]\big{]}(\psi)$
$\displaystyle:=\int_{(\mathbb{R}_{+}\times\mathbb{R})^{3}}\psi(w)(K(w-a)-K(z-a))(K(a-b)-K(z-b))\xi(db)\xi(da)\xi(dw),$
$\displaystyle\Pi_{z}^{\xi}\big{[}\Xi\mathcal{I}[\Xi$
$\displaystyle\mathcal{I}[\Xi\mathcal{I}[\Xi]]]\big{]}(\psi):=\int_{(\mathbb{R}_{+}\times\mathbb{R})^{4}}\psi(w)(K(w-a)-K(z-a))(K(a-b)-K(z-b))$
$\displaystyle\cdot\big{(}K(b-c)-K(z-c)-(t_{z}-t_{c})\partial_{t}K(z-c)-(x_{z}-x_{c})\partial_{x}K(z-c)\big{)}\xi(dc)\xi(db)\xi(da)\xi(dw),$
$\Pi_{z}^{\xi}\big{[}\Xi\mathcal{I}[X_{1}\Xi]\big{]}(\psi):=\int_{(\mathbb{R}_{+}\times\mathbb{R})^{2}}\psi(w)K(w-a)(x_{z}-x_{a})\xi(da)\xi(dw)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$

where $z=(t_{z},x_{z}),a=(t_{a},x_{a}),b=(t_{b},x_{b}),$ and
$c=(t_{c},x_{c})\in\mathbb{R}_{+}\times\mathbb{R}.$ We refer to e.g. [CW15,
Appendix A] for a definition of the iterated stochastic integrals against
space-time white noise on the full space. The requirement for these stochastic
integrals to exist is that the integrands are $L^{2}$ functions jointly in all
of the integration variables, which may be checked using the decay properties
of the heat kernel. The usage of the abstract symbols on the left side will be
explained shortly, but for now one should simply take the stochastic integrals
on the right side as the definition of the objects on the left side, which are
indexed by $\psi$ and $z$. For

$\tau\in\mathcal{W}:=\\{\Xi,\Xi\mathcal{I}[\Xi],\Xi\mathcal{I}[\Xi\mathcal{I}[\Xi]],\Xi\mathcal{I}[\Xi\mathcal{I}[\Xi\mathcal{I}[\Xi]]],\Xi\mathcal{I}[X_{1}\Xi]\\},$
(3.8)

we define the order $|\tau|$ to be
$-3/2-\kappa,-1-2\kappa,-1/2-3\kappa,-4\kappa,-2\kappa$ respectively, where
$\kappa\in(0,1/8)$ is arbitrary but fixed.

Now we want to find a Banach space for the above chaoses to be supported. To
do this we follow a method used by [HW15, Section 4]. Clearly each
$\Pi_{z}^{\xi}\tau$ is linear in $\psi$ as a map from
$L^{2}(\mathbb{R}_{+}\times\mathbb{R})\to L^{2}(\Omega,\mathcal{F},P)$ where
the latter denotes the underlying probability space of the noise $\xi.$ Thus
one can expect to be supported on a space of distributions in space-time. In
fact we will do this jointly in the $(z,\psi)$ coordinates, so that the
$\Pi^{\xi}$ are viewed as a generalized function of two space-time coordinates
$(z,z^{\prime})$.

######  Definition 3.7.

Fix a time horizon $T>0$ and set $\Lambda_{T}:=[0,T]\times\mathbb{R}$. We thus
identify $\Lambda_{T}$ as our space-time, so that the space
$\Lambda_{T}^{2}=\Lambda_{T}\times\Lambda_{T}$ consists of pairs of space-time
points $(z,z^{\prime})$. For $j=1,2$ we say that a function
$\Lambda_{T}^{j}\to\mathbb{R}$ is Schwartz if it is the restriction to
$\Lambda_{T}^{j}$ of some Schwartz function from
$\mathbb{R}^{2j}\to\mathbb{R}$. Given a Schwartz function
$\Pi:\Lambda_{T}^{2}\to\mathbb{R}$ and $\alpha<0$ one may define a norm

$\|\Pi\|:=\sup_{\lambda\in(0,1]}\sup_{\varphi\in
B_{r}}\sup_{z\in\Lambda_{T}}w(z)^{-1}\lambda^{-\alpha}\int_{\Lambda_{T}}\Pi(z,z^{\prime})\mathcal{S}^{\lambda}_{z}\varphi(z^{\prime})dz^{\prime},$

where if $z=(s,y)$ and $z^{\prime}=(t,x)$ then
$\mathcal{S}^{\lambda}_{z}\varphi(z^{\prime})=\lambda^{-3}\varphi(\lambda^{-2}(t-s),\lambda^{-1}(x-y))$,
where $w$ is a weight function, and where $B_{r}$ is the set of smooth
functions $\mathbb{R}^{2}\to\mathbb{R}$ supported on the closed unit ball with
$C^{r}$ norm less than $1$ where $r:=-\lfloor\alpha\rfloor$. One may then
define $E_{w}^{\alpha}$ to be the closure of Schwartz functions $\Pi$ under
this norm (thus $E_{w}^{\alpha}$ implicitly depends on the time horizon $T$
which we have chosen).

Note that if $\Pi\in E^{\alpha}_{w}$, then for all $z\in\Lambda_{T}$ we can
sensibly define $\Pi(z,\cdot)$ as a tempered distribution supported on
$\Lambda_{T}$. This will be important for Theorem 3.8 Item (3) below.

It turns out that the above symbols $\Pi^{\xi}\tau$ defined above are
supported in the spaces $E_{w}^{|\tau|}$ for the appropriate choice of $w$
(see Theorem 3.8 just below). Assuming this fact, let us now define the
appropriate semigroups on the spaces $E_{\tau}$. The semigroup acting on the
noise is $R_{\epsilon}f(t,x)=\epsilon^{3/2}f(\epsilon^{2}t,\epsilon x).$ For
$\tau\in\mathcal{W}$, define the $E^{|\tau|}_{w}$-valued homogeneous chaos
$\psi_{\tau}(\xi):=\Pi^{\xi}\tau$ as given above. Then by using the fact that
each of the functions
$K(t,x),t\cdot\partial_{t}K(t,x),x\cdot\partial_{x}K(t,x)$ are invariant under
the scaling $f(t,x)\to cf(c^{2}t,cx)$ for $c>0$, one checks that the following
five semigroups $Q^{\tau}$ on $E^{\tau}_{w}$ will satisfy the commutation
relation $Q^{\tau}_{\epsilon}\circ\psi_{\tau}=\psi_{\tau}\circ R_{\epsilon}$
which is needed to apply Theorem 1.3:

$Q^{\Xi}_{\epsilon}f(t,x,s,y)=\epsilon^{3/2}f(\epsilon^{2}t,\epsilon
x,\epsilon^{2}s,\epsilon y),$
$Q_{\epsilon}^{\Xi\mathcal{I}[\Xi]}f(t,x,s,y)=\epsilon
f(\epsilon^{2}t,\epsilon x,\epsilon^{2}s,\epsilon y),$
$Q^{\Xi\mathcal{I}[\Xi\mathcal{I}[\Xi]]}_{\epsilon}f(t,x,s,y)=\epsilon^{1/2}f(\epsilon^{2}t,\epsilon
x,\epsilon^{2}s,\epsilon y),$
$Q_{\epsilon}^{\Xi\mathcal{I}[\Xi\mathcal{I}[\Xi\mathcal{I}[\Xi]]]}f(t,x,s,y)=f(\epsilon^{2}t,\epsilon
x,\epsilon^{2}s,\epsilon y),$
$Q_{\epsilon}^{\Xi\mathcal{I}[X_{1}\Xi]}f(t,x,s,y)=f(\epsilon^{2}t,\epsilon
x,\epsilon^{2}s,\epsilon y)$

where the $(t,x)$ stands for the $z$ variable and the $(s,y)$ stands for the
$w$ variable in the integrals above (note that the $f$ appearing here will be
distributional in the latter variable and thus, as usual, the rescaling of
coordinates needs to be interpreted by integrating against test functions). In
other words one simply has
$Q^{\tau}_{\epsilon}f(t,x,s,y)=\epsilon^{-|\tau|_{0}}f(\epsilon^{2}t,\epsilon
x,\epsilon^{2}s,\epsilon y)$ where $|\tau|_{0}$ denotes the order of $\tau$
without the $\kappa$ term.

We henceforth fix $s,y>0$ and $\alpha\in(0,1/2-4\kappa)$ where $\kappa$ is the
same as specified after (3.8). We will let $\mathcal{C}^{\alpha}_{s,y}$ denote
the closure of smooth functions in the Banach space of Hölder continuous
functions on $B:=[0,s]\times[-y,y]$, whose norm is given by

$\|h\|_{\mathcal{C}^{\alpha}_{s,y}}:=\sup_{(t,x)\in
B}|h(t,x)|+\sup_{(t,x)\neq(t^{\prime},x)\in
B}\frac{|h(t,x)-h(t^{\prime},x^{\prime})|}{|t-t^{\prime}|^{\alpha/2}+|x-x^{\prime}|^{\alpha}}.$
(3.9)

We will now prove Theorem 1.4. The main idea is to first note that in the
notation of that theorem, we see that the function
$\tilde{h}(t,x):=C_{\epsilon}^{-1}\epsilon^{-1/2}h^{\epsilon}(\epsilon^{2}t,\epsilon
x)$ is the Hopf-Cole solution of

$\partial_{t}\tilde{h}(t,x)=\partial_{x}^{2}\tilde{h}(t,x)+(\partial_{x}\tilde{h}(t,x))^{2}+(\log\log(1/\epsilon))^{-1/2}(R_{\epsilon}\xi)(t,x),$

where $R_{\epsilon}$ is the measure-preserving operation defined above.
Therefore when combined with Theorem 1.3, the following result clearly implies
Theorem 1.4. Its proof is based entirely on the extensive framework and
estimates developed in [HP15, HL18]. Moreover, taking the logarithm poses no
issue since we are working on a compact domain and the solution is strictly
positive.

######  Theorem 3.8 ([HL18, HP15]).

  1. (1)

The above symbols $\Pi^{\xi}\tau$ defined above are supported in the spaces
$E_{w}^{|\tau|}$ for the appropriate choice of weight $w$. Furthermore the
semigroups $Q^{\tau}$ as defined above are strongly continuous on the spaces
$E^{|\tau|}_{w}$.

  2. (2)

For all $\delta>0$ replace $\xi$ by $\delta\xi$ in the above stochastic
integrals defining $\Pi^{\xi}\tau$, and denote the resulting 5-tuple of
objects as $(\Pi^{\delta\xi}\tau)_{\tau\in\mathcal{W}}$. There exists a
(nonlinear) closed subset $\mathcal{X}$ of
$\bigoplus_{\tau\in\mathcal{W}}E^{|\tau|}_{w}$ with the property that for all
$\delta>0$ the 5-tuple $(\Pi^{\delta\xi}\tau)_{\tau\in\mathcal{W}}$ is
supported on $\mathcal{X}$, and moreover there exists a deterministic
continuous map $\Phi:\mathcal{X}\to\mathcal{C}_{s,y}^{\alpha}$ such that
$\Phi$ sends $(\Pi^{\delta\xi}\tau)_{\tau\in\mathcal{W}}$ to the Itô solution
of the SPDE given by $\partial_{t}z=\partial_{x}^{2}z+\delta z\xi$, with
initial data $z(0,x)=1$.

  3. (3)

For $\tau\in\mathcal{W}$, define the $E^{|\tau|}_{w}$-valued homogeneous chaos
$\psi_{\tau}(\xi):=\Pi^{\xi}\tau$ as above. Let $(\psi_{\tau})_{hom}(z;f)$ be
defined as $(\psi_{\tau})_{hom}(f)$ evaluated at $z\in\Lambda_{T}$ in the
first coordinate. Then for all space-time Schwartz functions $f$, the function
$(\psi_{\tau})_{hom}(z;f)$ is given by the expression for $\Pi_{z}^{\xi}\tau$
but with $\xi(dw),\xi(da),\xi(db),\xi(dc)$ respectively replaced by
$f(w)dw,f(a)da,f(b)db,f(c)dc$ (so these are now classical integrals, not
stochastic integrals as above).

  4. (4)

For all space-time Schwartz functions $f$ on $\Lambda_{T}$, the map $\Phi$
sends $\big{(}(\psi_{\tau})_{hom}(f)\big{)}_{\tau\in\mathcal{W}}$ to the
classical solution of the deterministic equation
$\partial_{t}z=\partial_{x}^{2}z+zf,$ with initial data $z(0,x)=1$.

###### Proof.

(1) Let $w(t,x)=e^{a(1+|x|)}$ where $a>0$ is arbitrary. The fact that the
symbols $\Pi^{\xi}\tau$ are supported on $E^{|\tau|}_{w}$ follows from [HL18,
Theorem 5.3] by taking $p=\infty$, see Remark 2.7 there (note that since our
initial data is zero which is a smooth function, we do not even need to
consider finite $p$ in their results, which is done only for the sake of
considering singular initial data). The fact that the operator semigroups
$Q^{\tau}$ are strongly continuous on these spaces is follows from Lemma B.9.

(2) One defines the regularity structure $(T,A,G)$ where $T$ is the graded
vector space generated by the list of twelve symbols in [HL18, Figure 1], and
$A$ is the respective list of homogeneities. Note that the list of symbols is
precisely those symbols $\tau$ appearing in $\mathcal{W}$ as defined in (3.8)
plus symbols of the form $\mathcal{I}\tau$ with $\tau\in\mathcal{W}$ as well
as $\Xi X_{i}$ and $\mathcal{I}(\Xi X_{i}).$ The group $G$ will be described
later.

We are going to let $\mathcal{X}$ denote the space of those functions
$\Pi\in\bigoplus_{\tau\in\mathcal{W}}E^{|\tau|}_{w}$ which extend to
admissible models in the sense of [HL18, Definition 2.2] (such $\Pi$ are also
called “minimal models” in [HW15]). For any map $\Pi$ satisfying the
requirements [HL18, (2.2) and (2.5)] of an admissible model, there is a way to
define a corresponding family of invertible transformations $\Gamma_{xy}$ of
$T$ such that one has the relations $\Pi_{x}\Gamma_{xy}=\Pi_{y}$ and
$\Gamma_{xy}\Gamma_{yz}=\Gamma_{xz}$ for all $x,y,z\in\Lambda_{T}$, as well as
[HL18, (2.3)]. The first condition and the fact that
$\Pi_{x}X^{k}(z)=(z-x)^{k}$ forces $\Gamma_{xy}$ to act on elements
$\mathcal{I}[\tau]$ by the formula

$\Gamma_{xy}\mathcal{I}[\tau]=\mathcal{I}[\tau]+\sum_{|k|_{s}\leq|\tau|+2}\frac{X^{k}}{k!}\int_{\mathbb{R}^{d+1}}\big{(}\partial^{k}K(y-a)-\partial^{k}K(x-a)\big{)}(\Pi_{x}\tau)(da)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\sum_{|k|_{s}\leq|\tau|+2}\frac{(X+y-x)^{k}-X^{k}}{k!}\int_{\mathbb{R}^{d+1}}\partial^{k}K(y-a)(\Pi_{x}\tau)(da),$

where the sum is over multi-indices $k=(k_{1},k_{2})$ and
$|k|_{\mathfrak{s}}=2k_{1}+k_{2}.$ Furthermore it is clear that
$\Gamma_{xy}\Xi=\Xi$. Finally one has $\Gamma_{xy}(\Xi
X^{k}\mathcal{I}[\tau])=\Xi X^{k}\Gamma_{xy}\mathcal{I}[\tau]$ and
$\Gamma_{xy}X^{k}=(X+y-x)^{k}.$ The structure group $G$ may then be described
as the group of all $\Gamma_{x,z}$ as $x,z$ range throughout $\Lambda_{T}.$

Since the five symbols $\Pi^{\delta\xi}\tau$ ($\tau\in\mathcal{W}$) may be
checked to verify [HL18, (2.2) and (2.5)] for all $\tau\in\mathcal{W}$, and
since $\Pi\tau$ for $\tau\notin\mathcal{W}$ may be obtained uniquely and
continuously as a function of these five symbols by the same rules [HL18,
(2.2) and (2.5)], we then conclude that those five symbols indeed constitute
an element of $\mathcal{X}$. Now the existence and local Lipchitz continuity
of the solution map $\Phi$ is done in [HL18, Theorem 5.2], and the fact that
$\Phi$ sends $(\Pi^{\delta\xi}\tau)_{\tau\in\mathcal{W}}$ to the Itô solution
of the SPDE given by $\partial_{t}z=\partial_{x}^{2}z+\delta z\xi$, with
initial data $z(0,x)=1$, is stated in [HL18, Theorem 1.1] (the proof is not
given there but rather it is stated that the proof is nearly identical to
[HP15, Theorem 6.2] which is done in a compact setting).

(3) Suppose $(X,H,\mu)$ is any abstract Wiener space, $Y$ is another Banach
space. Suppose that $\psi:X\to Y$ is any “simple” Gaussian chaos of order $k$
given by $x\mapsto(H_{k}\circ g)(x)y$ where $y\in Y$ is fixed, $g\in X^{*}$ is
fixed, and $H_{k}$ denotes the $k^{th}$ Hermite polynomial. Then one may check
using (2.6) that $\psi_{hom}(h)=g(h)^{k}y$. Since “hom” is an additive
operation and since any chaos of order $k$ can be approximated by finite
linear combinations of simple chaoses of order $k$ (Lemma 2.11), the claim
then follows from Lemma 2.12 since the space-time white noise $\xi$ (when
viewed as an element of the space $E^{|\Xi|}_{w})$ has Cameron-Martin space
given by $L^{2}([0,T]\times\mathbb{R})$, see e.g. Lemma B.7.

(4) We note that the 5-tuple
$\big{(}(\psi_{\tau})_{hom}(\cdot;f)\big{)}_{\tau\in\mathcal{W}}$ extends to a
canonical model in the sense of [HL18, Remark 2.3]. This precisely means that
[HL18, (5.3)] is satisfied except that $c_{\epsilon}$ and $c^{(1)}_{\epsilon}$
are both replaced by $0$. Then following through that proof immediately gives
the claim. ∎

##  4\. Further generalizations

Although Theorem 1.3 is fairly general and widely applicable, there are
interesting examples where it does not apply. Here we study such examples,
giving further gerenalizations such as Theorems 4.5 and 4.6 below.

###  4.1. Noise-smoothing and non-measure-preserving systems

In this section we will further generalize the results of the previous
sections. This will culminate in Theorems 4.2 and 4.5, with a specific SPDE
example given in Theorem 4.4.

One may want to consider proving Strassen’s law for a family of processes that
is being diffusively scaled while being simultaneously approximated by
mollified noise, similar to what was done in [HW15]. Specifically consider
$\delta=\delta(\epsilon)$ and look at the family of processes $h^{\epsilon}$
defined with smooth noises at scale $\delta(\epsilon)$, i.e., letting
$L_{2}:=\log\log$ we want to consider the “smooth KPZ equation”

$\partial_{t}h^{\epsilon}=\partial_{x}^{2}h^{\epsilon}+((\partial_{x}h^{\epsilon})^{2}-C_{\delta(\epsilon)/\epsilon})+(L_{2}(1/\epsilon))^{-1/2}\epsilon^{3/2}\xi_{\delta(\epsilon)}(\epsilon^{2}t,\epsilon
x),$

where $\xi_{\delta}=\xi*\varphi_{\delta}$ where $*$ is space-time convolution
and $\varphi_{\delta}(t,x)=\delta^{-3}\varphi(\delta^{-2}t,\delta^{-1}x)$ for
a smooth nonnegative compactly supported function
$\varphi:\mathbb{R}^{2}\to\mathbb{R}$ which integrates to 1. For an
appropriate choice of divergent constants $C_{\epsilon}$, the goal is to try
to find the set of limit points of the sequence of functions given by
$h^{\epsilon}(\epsilon^{2}t,\epsilon x),$ and in particular show that the
compact limit set is the same as in Theorem 1.4. The main result of this
section will be rather intuitive: the limit points in
$\mathcal{C}^{\alpha}_{s,y}$ coincide with those found previously if
$\delta(\epsilon)=\epsilon^{1+u}$ for some $u>0$, they depend on the mollifier
if $\delta(\epsilon)=\epsilon$, and it is a trivial set consisting of a one-
dimensional family of functions if $\delta(\epsilon)=\epsilon^{1-u}$. To prove
this, we need some preliminaries.

######  Definition 4.1.

Consider a strongly continuous (multiplicative) semigroup
$(R_{\epsilon})_{\epsilon\in(0,1]}$ on a Banach space $X$. Writing
$S_{t}:=R_{e^{-t}}$ note that $\|S_{t+s}\|\leq\|S_{t}\|\|S_{s}\|$, by Fekete’s
subadditive lemma we know that the following quantity exists as a real number

$\beta(R):=\lim_{t\to\infty}t^{-1}\log\|S_{t}\|=\inf_{t>0}t^{-1}\log\|S_{t}\|.$

In other words, $\|R_{\epsilon}\|_{X\to
X}\leq\epsilon^{-\beta(R)-o_{\epsilon}(1)},$ and $\beta(R)$ is the optimal
such exponent (here $o_{\epsilon}(1)$ is some non-negative function of
$\epsilon\in(0,1]$ which tends to zero as $\epsilon\to 0)$. Now to illustrate
our results, we will consider a simpler setting first. Suppose we have a
Gaussian measure $\mu$ on $X$ such that $R_{\epsilon}$ is measure-preserving
and satisfies the strong mixing condition on the Cameron-Martin space $H$.
Suppose we are given a family of bounded linear operators $A_{\epsilon}$ on
$X$ (where $(x,\epsilon)\mapsto A_{\epsilon}(x)$ is Borel-measurable, but
$A_{\epsilon}$ is not necessarily a semigroup). Suppose that there exists a
Banach space $Y$ and constants $C,\gamma>0$ such that

  1. (1)

$Y$ embeds continuously into $X$.

  2. (2)

$\mu(Y)=1.$

  3. (3)

$\|A_{\epsilon}y-y\|_{X}\leq C\epsilon^{\beta(R)+\gamma}$ for all $y$ with
$\|y\|_{Y}\leq 1$.

Then we have the bounds
$\|R_{\epsilon}A_{\epsilon}x-R_{\epsilon}x\|_{X}\leq\|R_{\epsilon}\|\|A_{\epsilon}x-x\|\leq\epsilon^{-\beta(R)-o_{\epsilon}(1)}\|A_{\epsilon}x-x\|$.
Now for all $x$ in a set of full measure (namely $x\in Y$) we can bound
$\|A_{\epsilon}x-x\|\leq C\|x\|_{Y}\epsilon^{\beta(R)+\gamma}.$ Consequently,

$\|R_{\epsilon}A_{\epsilon}x-R_{\epsilon}x\|_{X}\leq C\epsilon^{\gamma-
o_{\epsilon}(1)}\|x\|_{Y},$

and thus as long as $x\in Y$ (which is almost every $x$) then the set of limit
points of $(\log\log(1/\epsilon))^{-1/2}R_{\epsilon}A_{\epsilon}x$ coincides
with those of $(\log\log(1/\epsilon))^{-1/2}R_{\epsilon}x$, namely $B(H)$.

This gives the seemingly obvious fact that the Strassen’s law still holds as
long as we mollify fast enough relative to the rescaling operation
$(R_{\epsilon})$. Consider the simple example of two-sided Brownian motion for
instance, say Wiener measure on $X=C[-1,1]$. Let
$A_{\epsilon}f(x)=f*\phi_{\epsilon}$ where $\phi$ is a smooth even mollifier
and $\phi_{\epsilon}(x)=\epsilon^{-1}\phi(\epsilon^{-1}x)$ (by convention we
let $A_{\epsilon}f$ be constant on each of $[-1,-1+\epsilon]$ and
$[1-\epsilon,1]$). Let $R_{\epsilon}f(x)=\epsilon^{-u/2}f(\epsilon^{u}x)$
where $u>0$, so that $\beta(R)=u/2$. If $u<1$ then we may define $Y$ to be the
closure of smooth functions with respect to the Hölder norm of exponent
$(u+1)/4$ and we know that this space supports Wiener measure almost surely.
Moreover an easy computation shows that $\|y*\phi^{\epsilon}-y\|_{C[-1,1]}\leq
C\epsilon^{(u+1)/4}$ as long as $\|y\|_{Y}\leq 1$ (where
$C=\int_{\mathbb{R}}\phi(v)|v|^{(u+1)/4}dv$). Consequently by the discussion
in the previous paragraph, one obtains the same Strassen law as Brownian
motion for the family $R_{\epsilon}A_{\epsilon}$ on $C[-1,1]$. On the other
hand if $u=1$ then the limit points will be the mollifier-dependent compact
set $\\{\phi*f:f(0)=0,\|f^{\prime}\|_{L^{2}[-1,1]}\leq 1\\},$ since one has
that $R_{\epsilon}A_{\epsilon}f=\phi*R_{\epsilon}f$ in this case. Thus we find
that $u<1$ is necessary to obtain a nontrivial limit set that coincides with
the non-mollified case. For $u>1$ the smoothing dominates the rescaling, so
one obtains a trivial limit set consisting only of constant functions of
absolute value bounded above by 1, as may be checked by hand.

We thus formulate the following abstract result:

######  Theorem 4.2.

Let $(X,H,\mu)$ be an abstract Wiener space, let
$(R_{\epsilon})_{\epsilon\in(0,1]}$ be a family of Borel-measurable a.e.
linear maps from $X\to X$ which are measure-preserving and strongly mixing,
and let $T^{i}:X\to Y_{i}$ be homogeneous of degree $k_{i}$ for $1\leq i\leq
m$. Suppose that there exist strongly continuous semigroups
$(Q^{i}_{\epsilon})_{t\geq 0}$ operators from $Y_{i}\to Y_{i}$ for $1\leq
i\leq m$ with the property that $T^{i}\circ R_{\epsilon}=Q^{i}_{\epsilon}\circ
T^{i}$, $\mu$-a.e. for all $\epsilon\in(0,1]$. Suppose that
$J_{\epsilon}^{i}:X\to Y_{i}$ is a family of measurable maps such that at
least one of the following two conditions holds

  1. (1)

There exist $\gamma>0$ and measurable functions $C^{i}:X\to\mathbb{R}_{+}$
such that $\|J_{\epsilon}^{i}(x)-T^{i}(x)\|_{Y_{i}}\leq
C^{i}(x)\epsilon^{\beta(Q^{i})+\gamma}$ for all $\epsilon\in(0,1].$

  2. (2)

Letting $J^{i}_{0}:=T^{i}$, assume that $J^{i}_{\epsilon}$ is a homogeneous
chaos of order $k_{i}\in\mathbb{N}$ and there exist $C,\gamma>0$ such that
$\int_{X}\|J_{\epsilon}^{i}(x)-J^{i}_{\epsilon^{\prime}}(x)\|_{Y_{i}}\;\mu(dx)\leq
C|\epsilon-\epsilon^{\prime}|^{\beta(Q^{i})+\gamma},$ for all
$\epsilon,\epsilon^{\prime}\in[0,1].$

Let $\mathcal{M}\subset Y_{1}\times\cdots Y_{m}$ be a closed subset, such that
the semigroup $Q_{\epsilon}^{1}\oplus\cdots\oplus Q_{\epsilon}^{m}$ sends
$\mathcal{M}$ to itself, and moreover for all $\delta,\epsilon>0$ assume that
$\mu(\\{x\in
X:(\delta^{k_{1}}J^{1}_{\epsilon}(x),...,\delta^{k_{m}}J^{m}_{\epsilon}(x))\in\mathcal{M}\\})=1.$

Then for all $\delta>0$ we have that $\mu(\\{x\in
X:(\delta^{k_{1}}T^{1}(x),...,\delta^{k_{m}}T^{m}(x))\in\mathcal{M}\\})=1,$
and furthermore the compact set
$K:=\\{(T^{1}_{\mathrm{hom}}(h),...,T^{m}_{\mathrm{hom}}(h)):h\in B(H)\\}$ is
necessarily contained in $\mathcal{M}$. Moreover, for any Banach space $Z$ and
any $\Phi:\mathcal{M}\to Z$ that is uniformly continuous on bounded sets, the
set of cluster points at zero of

$\big{\\{}\Phi\big{(}(2\log\log(1/\epsilon))^{-k_{1}/2}Q^{1}_{\epsilon}J^{1}_{\epsilon}(x),\;.\;.\;.\;,(2\log\log(1/\epsilon))^{-k_{m}/2}Q^{m}_{\epsilon}J^{m}_{\epsilon}(x)\big{)}:\epsilon\in(0,1]\\}$

is almost surely equal to $\Phi(K).$

We recover the result of the discussion above (for first order chaos) when we
set $m=1$ with $X=Y_{1}=Z=\mathcal{M}$ and $J^{1}_{\epsilon}=A_{\epsilon}$ and
$Q^{i}_{\epsilon}=R_{\epsilon}$ and $\Phi=I$ and $C^{1}(x)=C\|x\|_{Y}.$ Note
also that we do not make the requirement that
$Q^{i}_{\epsilon}J^{i}_{\epsilon}=J^{i}_{\epsilon}R_{\epsilon}$ and in general
this will be false. Finally we remark that when $J_{\epsilon}^{i}=T^{i}$ the
above theorem recovers Theorem 2.18, at least in the case that $\Phi$ is
uniformly continuous and $\mathcal{M}$ is invariant under the semigroup (which
is usually true in practice).

###### Proof.

First let us show that $\mu(\\{x\in
X:(\delta^{k_{1}}T^{1}(x),...,\delta^{k_{m}}T^{m}(x))\in\mathcal{M}\\})=1.$
Note that, regardless of whether (1) or (2) holds, we have that
$\|J_{2^{-N}}^{i}(x)-T^{i}(x)\|_{Y_{i}}\to 0$ as $N\to\infty$ for almost every
$x\in X$ (indeed $\beta(Q^{i})$ must be strictly positive under the
assumptions of the theorem, else $\|Q_{\epsilon}^{i}\|_{Y_{i}\to Y_{i}}\leq 1$
so that
$(\log\log(1/\epsilon))^{-k_{i}/2}\|Q^{i}_{\epsilon}T^{i}(x)\|_{Y_{i}}\to 0$,
which contradicts the result of Theorem 2.16). Consequently,
$(\delta^{k_{1}}T^{1}(x),...,\delta^{k_{m}}T^{m}(x))$ is the almost sure limit
of $(\delta^{k_{1}}J^{1}_{2^{-N}}(x),...,\delta^{k_{m}}J^{m}_{2^{-N}}(x))$.
Since the latter is assumed to be an element of $\mathcal{M}$ and since
$\mathcal{M}$ is a closed subset, we conclude that the former is also an
almost sure element of $\mathcal{M}$.

The proof that $K$ is contained in $\mathcal{M}$ assuming that $\mu(\\{x\in
X:(\delta^{k_{1}}T^{1}(x),...,\delta^{k_{m}}T^{m}(x))\in\mathcal{M}\\})=1$ for
all $\delta>0$ is given in the first paragraph of the proof of Corollary 2.17.

Now assume Condition (1) holds true. One makes the bound

$\|Q^{i}_{\epsilon}J^{i}_{\epsilon}(x)-Q^{i}_{\epsilon}T^{i}(x)\|\leq\|Q^{i}_{\epsilon}\|_{Y^{i}\to
Y^{i}}\|J^{i}_{\epsilon}(x)-T^{i}(x)\|\leq\|Q^{i}_{\epsilon}\|_{Y^{i}\to
Y^{i}}C^{i}(x)\epsilon^{\beta(Q^{i})+\gamma}\leq C^{i}(x)\epsilon^{\gamma-
o_{\epsilon}(1)}$

where we used by definition of $\beta(Q^{i})$ that
$\|Q^{i}_{\epsilon}\|\epsilon^{\beta(Q^{i})}\leq\epsilon^{-o_{\epsilon}(1)}$
(here $o_{\epsilon}(1)$ is some non-negative function of $\epsilon\in(0,1]$
which tends to zero as $\epsilon\to 0)$. Multiplying by
$(\log\log(1/\epsilon))^{-k_{i}/2}$ on both sides, we see that the set of
limit points of the desired sequence must coincide with that of Theorem 1.3
since the difference tends to zero as an element of $Z$. Note that one needs
the additional assumption of uniform continuity of $\Phi$ on bounded subsets
of $\mathcal{M}$ to argue this last part.

Now assume Condition (2) holds true instead. Since $J^{i}_{\epsilon}$ is a
homogeneous chaos, (2.2) gives

$\bigg{(}\int_{X}\|J_{\epsilon}^{i}(x)-J^{i}_{\epsilon^{\prime}}(x)\|_{Y_{i}}^{p}\mu(dx)\bigg{)}^{1/p}\leq
C_{p}|\epsilon-\epsilon^{\prime}|^{\beta(Q^{i})+\gamma}$

for all $p\geq 1$. By Kolmogorov-Chentsov, for a.e. $x\in X$ it is clear that
$\epsilon\mapsto J^{i}_{\epsilon}(x)$ is Hölder continuous on $[0,1]$ of any
desired exponent strictly less than $\beta(Q^{i})+\gamma$. In particular, we
may conclude that Condition (1) holds with $\gamma$ replaced by $\gamma/2$. ∎

Now let us argue how this result allows us to obtain the Strassen law for a
version of KPZ driven by a smooth noise mollified in both time and space. Let
$\xi$ be a standard space-time white noise on $\mathbb{R}_{+}\times\mathbb{R}$
and define the following distribution-valued chaoses for
$z\in\mathbb{R}_{+}\times\mathbb{R}$ and $\delta>0$:

$\displaystyle\Pi_{z}^{\xi,\epsilon}\Xi(\psi)$
$\displaystyle:=\int_{\mathbb{R}_{+}\times\mathbb{R}}\psi(w)\xi_{\epsilon}(dw),$
$\displaystyle\Pi_{z}^{\xi,\epsilon}\big{[}\Xi\mathcal{I}[\Xi]\big{]}(\psi)$
$\displaystyle:=\int_{(\mathbb{R}_{+}\times\mathbb{R})^{2}}\psi(w)(K(w-a)-K(z-a))\xi_{\epsilon}(da)\diamond\xi_{\epsilon}(dw),$
$\displaystyle\Pi_{z}^{\xi,\epsilon}\big{[}\Xi\mathcal{I}[\Xi\mathcal{I}[\Xi]]\big{]}(\psi)$
$\displaystyle:=\int_{(\mathbb{R}_{+}\times\mathbb{R})^{3}}\psi(w)(K(w-a)-K(z-a))(K(a-b)-K(z-b))\xi_{\epsilon}(db)\diamond\xi_{\epsilon}(da)\diamond\xi_{\epsilon}(dw),$
$\displaystyle\Pi_{z}^{\xi,\epsilon}$
$\displaystyle\big{[}\Xi\mathcal{I}[\Xi\mathcal{I}[\Xi\mathcal{I}[\Xi]]]\big{]}(\psi):=\int_{(\mathbb{R}_{+}\times\mathbb{R})^{4}}\psi(w)(K(w-a)-K(z-a))(K(a-b)-K(z-b))$
$\displaystyle\cdot\big{(}K(b-c)-K(z-c)-(t_{z}-t_{c})\partial_{t}K(z-c)-(x_{z}-x_{c})\partial_{x}K(z-c)\big{)}\xi_{\epsilon}(dc)\diamond\xi_{\epsilon}(db)\diamond\xi_{\epsilon}(da)\diamond\xi_{\epsilon}(dw),$
$\Pi_{z}^{\xi,\epsilon}\big{[}\Xi\mathcal{I}[X_{1}\Xi]\big{]}(\psi):=\int_{(\mathbb{R}_{+}\times\mathbb{R})^{2}}\psi(w)K(w-a)(x_{z}-x_{a})\xi_{\epsilon}(da)\diamond\xi_{\epsilon}(dw)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$

where $z=(t_{z},x_{z}),a=(t_{a},x_{a}),b=(t_{b},x_{b}),$ and
$c=(t_{c},x_{c})\in\mathbb{R}_{+}\times\mathbb{R}.$ The “$\diamond$” products
above should be interpreted as follows: first one interprets the integrals as
classical integrals against the smooth noise $\xi_{\delta}$, then one projects
the resultant object onto the homogeneous chaos of order given by the number
of times that $\xi_{\delta}$ appears in the integral. For $\tau\in\mathcal{W}$
from (3.8) we let $k_{\tau}$ be the number of instances of $\Xi$ in the symbol
$\tau$. We then have the following result.

######  Proposition 4.3.

Let $\mathcal{X}$ and $\Phi$ be as in Theorem 3.8, and let $\mathcal{W}$ be as
in (3.8). Then for all $\delta,\epsilon>0$ the 5-tuple
$(\delta^{k_{\tau}}\Pi^{\xi,\epsilon}\tau)_{\tau\in\mathcal{W}}$ is almost
surely an element of $\mathcal{X}$. Furthermore
$\Phi\big{(}(\delta^{k_{\tau}}\Pi^{\xi,\epsilon}\tau)_{\tau\in\mathcal{W}}\big{)}$
is given by the solution to the classical equation

$\partial_{t}z=\partial_{x}^{2}z+z(\delta\xi_{\epsilon}-C_{\epsilon,\delta}),$

with $z(0,x)=1$ and where

$C_{\epsilon,\delta}=c\delta^{2}\epsilon^{-1}+\delta^{4}c_{\epsilon}^{\prime}$

where $c$ is a constant depending only on the mollifier (not
$\epsilon,\delta)$ and where $c_{\epsilon}^{\prime}$ may depend on both
$\epsilon$ and the mollifier but is convergent to a finite constant as
$\epsilon\to 0$.

###### Proof.

This is proved in [HL18, Theorem 5.3]. The constants are explicitly written
out in [HL18, (1.1)], and their $\epsilon\to 0$ asymptotics are explained in
the subsequent paragraphs. ∎

With this established, we can prove the following theorem, the main result of
this subsection.

######  Theorem 4.4.

Let $\varphi_{\delta}(t,x)=\delta^{-3}\varphi(\delta^{-2}t,\delta^{-1}x)$ for
a smooth nonnegative compactly supported function
$\varphi:\mathbb{R}^{2}\to\mathbb{R}$ which integrates to 1. Consider the
sequence of solutions to the classical PDE

$\partial_{t}h(t,x)=\partial_{x}^{2}h(t,x)+(\partial_{x}h(t,x))^{2}-C_{\epsilon^{s-1},\delta(\epsilon)}+\delta(\epsilon)\epsilon^{3/2}\xi_{\epsilon^{s}}(\epsilon^{2}t,\epsilon
x)$

where $h(0,x)=0$ and $\xi_{\epsilon}=\xi*\varphi_{\epsilon}$ and
$\delta(\epsilon)=(\log\log(1/\epsilon))^{-1/2}$. If $s>1$ then the set of
cluster points in $\mathcal{C}^{\alpha}_{t,y}$ as $\epsilon\to 0$ is equal to
the same compact limit set $K$ as in Theorem 1.4, for any $\alpha<1/2$.

###### Proof.

Let $R_{\epsilon}$ and $Q^{\tau}_{\epsilon}$ ($\tau\in\mathcal{W})$ be the
same multiplicative semigroups as defined in the previous subsection. We
define $J^{\tau}_{\epsilon}(\xi)$ to be equal to $\Pi^{\xi,\epsilon}\tau$. We
also fix some exponent $s>0$. We claim for each $\epsilon>0$ that
$v:=\Phi\big{(}(\delta^{k_{\tau}}Q^{\tau}_{\epsilon^{s}}J^{\tau}_{\epsilon^{s}}\xi)_{\tau\in\mathcal{W}}\big{)}$
is the solution of the equation

$\partial_{t}v(t,x)=\partial_{x}^{2}v(t,x)+v(\delta\epsilon^{3/2}\xi_{\epsilon^{s}}(\epsilon^{2}t,\epsilon
x)-C_{\epsilon^{s-1},\delta})$

where $v(0,x)=1$ and $\xi_{\epsilon}=\xi*\varphi_{\epsilon}$. Indeed one
verifies that
$Q^{\tau}_{\epsilon^{s}}J^{\tau}_{\epsilon^{s}}\xi=J^{\tau}_{\epsilon^{s-1}}(R_{\epsilon}\xi)$,
so that the claim is immediate from Proposition 4.3 and the fact that
$\big{(}(R_{\epsilon}\xi)*\varphi_{\epsilon^{s-1}}\big{)}(t,x)=\epsilon^{3/2}\xi_{\epsilon^{s}}(\epsilon^{2}t,\epsilon
x)$ (note that this calculation shows why we need $s>1$).

Now to finish our analysis of the mollified equation, one needs as in
assumption (2) of Theorem 4.2 to show a bound of the type
$\mathbb{E}\|J^{\tau}_{\epsilon}(\xi)-J^{\tau}_{\epsilon^{\prime}}(\xi)\|_{E^{|\tau|}_{w}}\leq
C|\epsilon-\epsilon^{\prime}|^{\beta(Q^{\tau})+\gamma}$. We first remark that
for the semigroups $Q^{\tau}$ on the spaces $E^{|\tau|}_{w}$ one has
$\beta(Q^{\tau})=k_{\tau}\kappa$ where $\kappa$ on the right side is the same
as in the previous subsection, which may be verified by hand. By choosing
$\kappa,\gamma$ to be sufficiently small relative to the desired exponent
$\alpha$ from the theorem statement, the desired bound follows (as noted in
[HL18, Theorem 5.3]) by following the logic in [HP15, Theorem 4.5] (see also
[Tem23, (3.7) and (3.8)]). By taking a logarithm and using compactness of the
domain, we obtain the claim. ∎

We remark that mollification at scale $\epsilon^{s}$ is not the only example
of where Theorem 4.2 is useful. One can also use it, for instance, when
looking at a noise $\xi$ that is spatially periodic with some large period
$\epsilon^{-1}$, by defining the chaoses $J^{i}_{\epsilon}$ to be the
spatially periodized version of the chaoses $T^{i}$ (which may be checked to
satisfy a bound of the desired form (2) in Theorem 4.2). This would be useful
in showing a Strassen law for KPZ (and potentially other singular models) on
periodic domains such as a torus.

While the preceding result illustrates the use of Theorem 4.2, we note that it
can be generalized even further by considering some family of maps
$\Phi_{\epsilon}:\mathcal{M}\to Z$ where $\epsilon\in(0,1]$ and then imposing
suitably strong convergence conditions on $\Phi_{\epsilon}$ as $\epsilon\to
0$. Additionally, we remark that in Condition (1) the chaoses
$J^{i}_{\epsilon}$ do not need to be assumed to be deterministic functions of
the Gaussian noise $x$. In particular we have the following result which can
be viewed as a generalization of all of other results thus far, whose interest
will be explained shortly.

######  Theorem 4.5.

Let $(X,H,\mu)$ be an abstract Wiener space, let
$(R_{\epsilon})_{\epsilon\in(0,1]}$ be a family of Borel-measurable a.e.
linear maps from $X\to X$ which are measure-preserving and strongly mixing,
and let $T^{i}:X\to Y_{i}$ be homogeneous of degree $k_{i}$ for $1\leq i\leq
m$. Suppose that there exist strongly continuous semigroups
$(Q^{i}_{\epsilon})_{t\geq 0}$ operators from $Y_{i}\to Y_{i}$ for $1\leq
i\leq m$ with the property that

$\displaystyle T^{i}\circ R_{\epsilon}=Q^{i}_{\epsilon}\circ
T^{i},\;\;\;\;\;\;\;\;\mu\text{-a.e.}\;\;\;\;\;\;\text{ for all
}\epsilon\in(0,1].$

Assume that there exists some probability space $(\Omega,\mathcal{F},P)$ which
admits a coupling of the following form: there is a map $x:\Omega\to X$ such
that the law of $x$ is $\mu$ (i.e., $P_{*}x=\mu$) and there is a collection of
measurable functions
$(\mathfrak{J}^{1}_{\epsilon},...,\mathfrak{J}^{m}_{\epsilon})_{\epsilon\in(0,1]}$
from $\Omega\to\mathcal{M}$ such that

$\|\mathfrak{J}^{i}_{\epsilon}(\omega)-Q^{i}_{\epsilon}T^{i}(x(\omega))\|_{Y_{i}}\leq
C^{i}(\omega)$ (4.1)

for some collection of measurable functions $C^{i}:\Omega\to[0,\infty)$. Let
$\mathcal{M}\subset Y_{1}\times\cdots Y_{m}$ be a closed subset, such that for
all $\delta,\epsilon>0$

$P((\delta^{k_{1}}\mathfrak{J}^{1}_{\epsilon},...,\delta^{k_{m}}\mathfrak{J}^{m}_{\epsilon})\in\mathcal{M})=1,\;\;\text{
and }\;\;\mu(\\{x\in
X:(\delta^{k_{1}}T^{1}(x),...,\delta^{k_{m}}T^{m}(x))\in\mathcal{M}\\})=1.$

Then the compact set
$K:=\\{(T^{1}_{\mathrm{hom}}(h),...,T^{m}_{\mathrm{hom}}(h)):h\in B(H)\\}$ is
necessarily contained in $\mathcal{M}$. Moreover, consider any Banach space
$Z$ and any family of maps $\\{\Phi_{\epsilon}\\}_{\epsilon\in(0,1]}$ from
$\mathcal{M}\to Z$ that converges uniformly on bounded subsets of
$\mathcal{M}$ as $\epsilon\to 0$ to a limit $\Phi$ that is uniformly
continuous on bounded subsets of $\mathcal{M}$. Then the set of cluster points
as $\epsilon\to 0$ of
$\Phi_{\epsilon}\big{(}((\log\log(1/\epsilon))^{-k_{i}/2}\mathfrak{J}^{i}_{\epsilon}(\omega))_{1\leq
i\leq m}\big{)}$ is equal to $\Phi(K)$ for a.e. $\omega\in\Omega$.

###### Proof.

Note that the assumptions are equivalent to the following statement: for all
bounded sequences
$\\{\boldsymbol{y}_{\epsilon}\\},\\{\boldsymbol{y}_{\epsilon}^{\prime}\\}\subset\mathcal{M}$,
we have

$\lim_{\epsilon\to
0}\|\Phi_{\epsilon}(\boldsymbol{y}_{\epsilon})-\Phi(\boldsymbol{y}_{\epsilon}^{\prime})\|_{Z}=0\;\;\;\text{
whenever }\;\;\;\lim_{\epsilon\to
0}\|\boldsymbol{y}_{\epsilon}-\boldsymbol{y}_{\epsilon}^{\prime}\|_{Y_{1}\oplus...\oplus
Y_{m}}=0.$ (4.2)

Note that $(\log\log(1/\epsilon))^{-k_{i}/2}Q^{i}_{\epsilon}T^{i}(x(\omega))$
remains almost surely bounded as $\epsilon\to 0$ by Theorem 2.16. Thus we have
by (4.2) that

$\big{\|}\Phi_{\epsilon}\big{(}((\log\log(1/\epsilon))^{-k_{i}/2}\mathfrak{J}^{i}_{\epsilon}(\omega))_{1\leq
i\leq
m}\big{)}-\Phi\big{(}((\log\log(1/\epsilon))^{-k_{i}/2}Q^{i}_{\epsilon}T^{i}(x(\omega)))_{1\leq
i\leq m}\big{)}\big{\|}_{Z}\to 0$

as $\epsilon\to 0$. The term on the right has the desired set of limit points
by Theorem 1.3. ∎

Theorem 4.2 is recovered by taking $\Omega=X$ and
$\mathfrak{J}^{i}_{\epsilon}=Q^{i}_{\epsilon}J^{i}_{\epsilon}$ and
$\Phi_{\epsilon}=\Phi$. Such a general statement would be useful in showing
(for instance) that Strassen’s law holds if we replace $\xi_{\epsilon}$ by
some non-Gaussian approximation of a Gaussian noise, for instance a Poisson
noise or some other discrete approximation of the desired limiting system,
e.g., random walk approximations of Brownian motion, or statistical mechanical
approximations of SPDEs such as KPZ.

When applying the theorem in practice, one can usually obtain a stronger bound
than (4.1), typically an “almost sure invariance principle” (ASIP) which gives
a bound of the form $C^{i}(\omega)\epsilon^{\alpha}$ where $\alpha>0$. For
instance, in a random walk approximation of standard Brownian motion, the KMT
coupling [KMT76] will give such a bound with $\alpha$ slightly less than 1/2
on the space constructed for the large-time regime in Example 3.1. For more
complicated examples involving higher chaos, the discretized analogue
$\mathfrak{J}^{i}_{\epsilon}$ of each of the $m$ chaoses also needs to be
constructed, and then the joint ASIP for these chaoses needs to be proved “by
hand.” This is rather difficult in general, see for instance [GMW23] for some
recent progress on a discrete approximation theory for Gaussian chaoses by
general martingale chaoses. The mollification example above may be viewed a
particular case where the ASIP for the chaoses is particularly simple to
prove, compared to other choices of prelimiting objects such as
discretizations.

###  4.2. Iterated processes

In this subsection we consider a different generalization of the main result
(Theorem 1.3) than those described in the previous subsection, specifically we
study the effect of reparametrizing the $\epsilon$ variable, which destroys
the semigroup property. To motivate why one might be interested in such a
reparametrization of $\epsilon$, we consider iterated processes.

Burdzy in [Bur93] proved that if $B,W$ are independent two-sided Brownian
motions, then

$\limsup_{t\to
0}\frac{B(W(t))}{t^{1/4}\log(\log(1/t))^{3/4}}=2^{5/4}3^{-3/4}.$

In [CCFR95], the authors extend this to prove a functional version of this
theorem, namely that that the set of limit points of the family of random
functions $Z^{\epsilon}:[-1,1]\to\mathbb{R}$ defined by

$Z^{\epsilon}(t):=\epsilon^{-1/4}(\log\log(1/\epsilon))^{-3/4}B(W(\epsilon
t)),$ (4.3)

as $\epsilon\to 0$ equals the compact set of functions in $C[0,1]$ given by

$Q:=\\{f\circ g:f,g\in
C([-1,1]),f(0)=g(0)=0,\int_{-1}^{1}f^{\prime}(t)^{2}+g^{\prime}(t)^{2}\leq
1\\}.$

Note that the composition is well-defined because the integral condition
implies that $g$ maps $[-1,1]$ to itself.

We would like to recover this result using our abstract framework. For
$X,Y\subset\mathbb{R}$ let $C(X,Y)$ denote continuous maps from $X$ to $Y$ and
$CX:=C(X,\mathbb{R})$. Define $R_{\epsilon}f(t)=\epsilon^{-1/2}f(\epsilon t).$
Note that the map $\Phi$ from $C[-2,2]\times C([-1,1],[-2,2])\to C[-1,1]$
defined by $(f,g)\mapsto f\circ g$ is continuous. Furthermore we note that

$Z^{\epsilon}=\Phi\big{(}(\log\log(1/\epsilon))^{-1/2}R_{(\epsilon\log\log(1/\epsilon))^{1/2}}B\;\;,\;\;(\log\log(1/\epsilon))^{-1/2}R_{\epsilon}W\big{)}.$
(4.4)

The latter expression makes sense because Lemmas 2.8 and 2.15 ensure that with
probability $1$, $R_{\epsilon}W$ has image contained in $[-2,2]$ for small
$\epsilon$ almost surely.

Note that Theorem 1.3 is not directly applicable here due to the expression
$R_{(\epsilon\log\log(1/\epsilon))^{1/2}}B.$ This is of the form
$R_{g(\epsilon)}B$ where one still has that $\log(g(\epsilon))/\log(\epsilon)$
remains bounded away from $0$ and $+\infty$ as $\epsilon\to 0$, but where
$R_{g(\epsilon)}$ does not respect the multiplicative semigroup property since
$g(\epsilon)g(\delta)\neq g(\epsilon\delta)$. The main problem which we now
face is that the variable $\epsilon$ in the expression $R_{\epsilon}W$ has not
been reparametrized in the same way. This motivates the following abstract
result.

######  Theorem 4.6.

For $1\leq j\leq r$ let $(X_{j},H_{j},\mu_{j})$ be abstract Wiener spaces, let
$(R^{j}_{\epsilon})_{\epsilon\in(0,1]}$ ($1\leq j\leq r)$ be a family of
Borel-measurable a.e. linear maps from $X_{j}\to X_{j}$ which are measure-
preserving and strongly mixing, and let $T^{ij}:X_{j}\to Y_{ij}$ be
homogeneous of degree $k_{i,j}$ for $1\leq i\leq m$ and $1\leq j\leq r$, where
$Y_{ij}$ are Banach spaces. Assume that
$\mathcal{M}\subset\bigoplus_{i,j}Y_{ij}$ is a closed subset. Assume that
there exist strongly continuous semigroups $Q^{ij}$ on $Y_{ij}$ such that one
has $Q^{ij}_{\epsilon}T^{ij}=T^{ij}R^{j}_{\epsilon}$ $\mu_{j}$-a.e. for all
$i,j$ and $\epsilon\in(0,1]$. Also assume that for all $\delta>0$,

$(\mu_{1}\otimes\cdots\otimes\mu_{r})\bigg{(}\big{\\{}(x_{1},...,x_{r}):\big{(}\delta^{k_{i,j}}T^{ij}(x_{j})\big{)}_{i,j}\in\mathcal{M}\big{\\}}\bigg{)}=1.$

Then the compact set
$K:=\\{\big{(}(T^{i,j})_{hom}(h_{j})\big{)}_{i,j}:\|h_{1}\|_{H_{1}}^{2}+...+\|h_{r}\|_{H_{r}}^{2}\leq
1\\}$ is contained in $\mathcal{M}$. Moreover consider any collection
$\\{g^{j}\\}_{j=1}^{r}$ of maps from $(0,1]\to(0,1]$ such that
$\log(g^{j}(\epsilon))/\log(\epsilon)$ remains bounded away from $0$ and
$+\infty$ as $\epsilon\to 0$. Sample $(\xi_{1},...,\xi_{r})$ from
$\mu_{1}\otimes\cdots\otimes\mu_{r}$. Then for any Banach space $Z$ and any
continuous function $\Phi:\mathcal{M}\to Z$, the set of cluster points as
$\epsilon\to 0$ of
$\Phi\big{(}\big{(}(\log\log(1/\epsilon))^{-k_{i,j}/2}Q^{ij}_{g^{j}(\epsilon)}T^{ij}(\xi_{j})\big{)}_{i,j}\big{)}$
is a.s. equal to $\Phi(K).$

Before the proof, we remark by (4.4) that this theorem immediately gives the
desired compact limit set for the iterated process $Z^{\epsilon}$ from (4.3),
since $B$ and $W$ are assumed to be independent. Note that we may also obtain
the result in every Hölder norm of exponent less than $1/4$ by noting that for
all $0<\alpha<1$ the composition map is continuous from
$C^{\alpha}[-2,2]\times C^{\alpha}([-1,1],[-2,2])\to C^{\alpha^{2}}[-1,1]$,
where $C^{\alpha}$ is the closure of smooth functions in the Hölder-$\alpha$
norm.

###### Proof.

The proof essentially follows the same line of logic as the proof of Theorem
1.3. We will indicate the places in the argument that need to be modified in
order for the theorem to be proved.

Firstly, in Lemma 2.1 we need weaker assumptions since we no longer have the
structure of a stationary sequence. More precisely, thanks to the assumption
that $\log(g^{j}(\epsilon))/\log(\epsilon)$ remains bounded, it suffices to
show that if $(X_{j})_{j\geq 0}$ is a (possibly non-stationary) sequence of
joint mean-zero Gaussians of unit variance such that
$\lim_{N\to\infty}\max_{|i-j|>N}|\mathbb{E}[X_{i}X_{j}]|=0,$ then one has that
$\lim_{n\to\infty}(2\log n)^{-1/2}X_{n}=1.$ One may verify that this is indeed
true by again using Slepian’s lemma, alternatively see the main result of
[Man76], where this was first observed.

This then leads to the following generalization of Corollary 2.2. If
$\mathbf{X}_{n}:=(X_{n}^{ij})_{1\leq i\leq N,1\leq j\leq
r}\in\mathbb{R}^{N\times r}$ is a jointly Gaussian sequence such that the $r$
distinct collections given by
$(X_{n}^{i1})_{i=1}^{N},...,(X_{n}^{ir})_{i=1}^{N}$ are independent of each
other, and each satisfies
$\mathrm{cov}(X_{0}^{ij},X_{0}^{i^{\prime}j})=\delta_{ii^{\prime}}$ and
$\lim_{A\to\infty}\max_{|m-n|>A}|\mathbb{E}[X_{m}^{ij}X_{n}^{i^{\prime}j}]|=0$,
then the unit sphere of $\mathbb{R}^{N\times r}$ is contained in the set of
limit points of the sequence $((2\log n)^{-1/2}\mathbf{X}_{n})_{n\geq 2}.$

From here, the line of proof will then use the same strategy as in Subsections
2.2 and 2.3, first proving the discrete-time formulation (see Theorems 2.14
and 2.17) along the subsequence $\epsilon=2^{-n}$ by using the finite-rank
approximations $T^{ij}_{N}$ from (2.4) and using the result of the previous
paragraph. The remainder of the proofs are then straightforward to modify, but
taking into account that there are now $r$ independent noises as opposed to
just $1$, thus giving image of the closed unit ball of the Hilbert space
$H_{1}\oplus...\oplus H_{r}$ for the limit sets. ∎

######  Example 4.7.

We now give another example using iterated processes derived from higher
chaoses. This will recover the main result of [Neu98], where the author proves
that if $W,B_{1},B_{2}$ are independent two-sided standard Brownian motions,
and if $A(t):=\frac{1}{2}\int_{0}^{t}B_{2}(s)dB_{1}(s)-B_{1}(s)dB_{2}(s),$
then

$\limsup_{t\to
0}\frac{A(W(t))}{t^{1/2}(\log\log(1/t))^{3/2}}=(2/3)^{-3/2}/\pi.$

We can obtain a functional version of this using the above theorem. Define the
process
$Z^{\epsilon}(t):=\epsilon^{-1/2}(\log\log(1/\epsilon))^{-3/2}A(W(\epsilon
t)),$ and note that