# Ergodicity results for the open KPZ equation

Shalin Parekh

(Date: November 2023)

###### Abstract.

We give a new proof of existence as well as two proofs of uniqueness of the
invariant measure of the open-boundary KPZ equation on $[0,1]$, for all
possible choices of inhomogeneous Neumann boundary data. Both proofs yield an
exponential convergence result in total variation when combined with the
strong Feller property which was recently established in [KM22]. An important
ingredient in both proofs is the construction of a compact state space for the
Markov operator to act on, which measurably contains all of the usual Hölder
spaces modulo constants. Along the way, the strong Feller property is extended
to this larger class of initial conditions. The arguments do not rely on exact
descriptions of the invariant measures, and some of the results generalize to
the case of a spatially colored noise and other boundary conditions.

##  1\. Introduction

The open KPZ equation with inhomogeneous Neumann boundary condition is a
stochastic PDE on the domain $(t,x)\in\mathbb{R}_{+}\times[0,1]$ formally
written as

$\partial_{t}h(t,x)=\partial_{x}^{2}h(t,x)+(\partial_{x}h(t,x))^{2}+\xi(t,x),\;\;\;\;\;\;\partial_{x}h(t,0)=A+\frac{1}{2},\;\;\;\;\;\partial_{x}h(t,1)=B-\frac{1}{2},$
(1)

where $A,B\in\mathbb{R}.$ Here $\xi$ is a space-time white noise, the Gaussian
field on $\mathbb{R}_{+}\times[0,1]$ with formal covariance
$\mathbb{E}[\xi(t,x)\xi(s,y)]=\delta(t-s)\delta(x-y)$. This equation appears
as a scaling limit of various probabilistic systems with very simple
microscopic dynamics, for instance ASEP with open boundaries [CS18, Par19].
The equation (1) is ill-posed due to the singularity of the noise, and so one
usually works with the KPZ equation using the Hopf-Cole transform: if $h$
solves (1) then $z:=e^{h}$ formally solves the Robin-boundary stochastic heat
equation with multiplicative noise:

$\partial_{t}z(t,x)=\partial_{x}^{2}z(t,x)+z(t,x)\xi(t,x),\;\;\;\;\partial_{x}z(t,0)=Az(t,0),\;\;\;\;\partial_{x}z(t,1)=Bz(t,1),$
(2)

which is a well-posed Itô SPDE when written in mild form. The extra additive
constant 1/2 in both parameters of (1) is a convention that has been taken in
order to take into account certain “boundary-renormalization” effects that
become apparent when applying the Hopf-Cole transform, see [GH17, Remark 1.11]
and [GPS]. Recently other methods such as regularity structures [Hai14, GH17]
have been successful in making sense of (1) directly, which is relevant to
this work although we will mainly work with the Hopf-Cole transform here.

In a recent work [CK20] proves existence of an invariant measure for (1) for
all choices of boundary parameters $A,B\in\mathbb{R}.$ They prove this by
taking limits of known invariant measures for ASEP and showing that they are
tight under the KPZ scaling considered in [CS18, Par19]. The invariant
measures are nontrivial and have explicit descriptions in terms of certain
functionals of Brownian motions or Brownian bridges for certain specific
values of $A,B$. When $A-B>-1$ the authors characterize the law in terms of a
Laplace transform using the dual Hahn process. In the present work we give an
independent proof of existence of an invariant measure for each
$A,B\in\mathbb{R}$, which also works for other type of Gaussian noises that
are colored in space and white in time (see Corollary 2.5).

In a follow-up paper [KM22] prove the strong Feller property for (1),
following a method developed by [HM17] using the theory of regularity
structures to make sense of KPZ. This made it possible to prove uniqueness of
the invariant measure for certain specific choices of $A,B$, assuming one
knows certain exact information about the invariant measure in advance, in
particular that it has full support. The difficulty with proving uniqueness
for general parameters is that we do not have explicit descriptions of the
laws of the invariant measures for general values of the boundary parameters.
There is a conjecture from [BLD21] but it remains to be shown that the
resultant object is in fact invariant when $A-B\leq-1$. We also mention [BW10,
BW18] for more about the formulaic viewpoint on the invariant measures.

The present work extends the uniqueness result to general values of $A,B$
without relying on any exact descriptions of the invariant measures, thus
completing a final step in the proof of the uniqueness conjecture from [Cor22,
CK20], actually we will go much further and prove that there is a spectral gap
and uniform mixing over all initial data. We give two separate proofs, one via
a support theorem and another via a polymer coupling argument. In the support
proof, the main tools that we use are the Feynman-Kac formula, the Cameron-
Martin theorem, and the strong Feller property on $C[0,1]$. In the polymer
argument, the main tool used is the convolution property for the propagators
of the stochastic heat equation (2), which also gives a one-force one solution
principle for the open KPZ equation.

To describe our results we need to specify the choice of topology. Let
$\mathcal{X}_{0}$ denote the quotient of the space of continuous functions
$[0,1]\to\mathbb{R}$ by the equivalence relation $f\sim g$ if $f-g$ is
constant. Denote by $[f]$ the equivalence class of $f$. For
$[f],[g]\in\mathcal{X}_{0}$ we define their distance

$d([f],[g]):=d_{\text{Proh}}\bigg{(}\frac{e^{f}}{\int_{[0,1]}e^{f}},\frac{e^{g}}{\int_{[0,1]}e^{g}}\bigg{)},$

where $d_{\text{Proh}}$ is a Prohorov metric on the space of probability
measures on $[0,1]$ (i.e., any metric which metrizes the weak topology),
satisfying $d_{\text{Proh}}(\mu,\nu)\leq\|\mu-\nu\|_{TV}$. Finally we denote
$\mathcal{X}$ to be the completion of $\mathcal{X}_{0}$ under the metric $d$,
which is isometric to the space of probability measures on $[0,1]$ (with
Prohorov distance) and is therefore compact. Intuitively the elements of
$\mathcal{X}$ can be viewed as formal logarithms of the measures that are
initial conditions for the Hopf-Cole transform (2), modulo addition of
constants. Thus we still use “$[\mu]$” to denote elements of the completion.
Sometimes we also use a more generic “$\phi$” to denote elements of
$\mathcal{X}$ in situations when it is not particularly important to emphasize
that elements are equivalence classes associated to a Borel measure on
$[0,1]$.

We will generally take the convention of formulating results in terms of
discrete-time Markov semigroups, in order to simplify the exposition and also
to be consistent with [HM17, KM22]. This does not lose any generality. For a
measurable function $F:\mathcal{X}\to\mathbb{R}$ and an initial condition
$\phi\in\mathcal{X}$ the Markov operator $\mathfrak{P}$ for the open KPZ
equation acts by the formula
$\mathfrak{P}F(\phi):=\mathbb{E}_{\phi}[F([h(1,\cdot)])].$ Here
$\mathbb{E}_{\phi}$ denotes the expectation with respect to the measure
$\mathbb{P}_{\phi}$ on the canonical space $\mathcal{X}^{\mathbb{N}}$ that
corresponds to starting the process at initial state $\phi$, and $h(1,\cdot)$
denotes the spatial process $x\mapsto h(1,x)$ with $h$ solving the open KPZ
equation (1). We extend the results of [KM22] by proving the following.

######  Theorem 1.1 (Uniqueness via a support theorem).

Fix boundary parameters $A,B\in\mathbb{R}$. The Markov operator $\mathfrak{P}$
for the open KPZ equation is globally defined and strong Feller on the compact
state space $\mathcal{X}.$ Furthermore there exists a unique invariant measure
$\rho_{A,B}$ for the semigroup. The invariant measure has full support on
$\mathcal{X}$. Moreover, for every deterministic initial condition
$\phi\in\mathcal{X}$ we have the ergodic theorem:

$\mathbb{P}_{\phi}\bigg{(}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}F([h(n,\cdot)])=\int_{\mathcal{X}}F\;d\rho_{A,B}\bigg{)}=1,$

for all bounded measurable $F:\mathcal{X}\to\mathbb{R}.$

This result will still be further strengthened in Theorems 1.2 and 1.3 below.
The existence statement is proved in Corollary 2.5 below, independently of all
previous works. The uniqueness and ergodicity result is proved as Corollary
3.7 below. The proof of the uniqueness result uses the results of [KM22] as an
input, namely that the strong Feller property already holds in some (stronger)
topology. Note in the above theorem that the ergodic theorem holds for every
initial condition (as opposed to almost every initial condition), which rules
out the existence of another invariant measure even if one restricts the
action of the semigroup to some finer metric space $\mathcal{Y}$ which embeds
continuously into $\mathcal{X}$ (e.g. the space $\mathcal{Y}$ of Hölder
continuous functions given in (7) below).

The proof of our theorem is done by showing full support in $\mathcal{X}$ of
the open KPZ equation at any fixed time. To show full support, we use the
Cameron-Martin theorem and the Feynman-Kac formula. In particular we use the
Hopf-Cole transform and rely on the linear structure of the equation. In fact,
throughout the whole paper, we work with only with the Hopf-Cole transform,
using the results of [HM17, KM22] as given and then building off of those
results using only classical techniques without any further use of regularity
structures.

When there exists a natural coupling of the Markov process started from
different initial data, it is nice to have a “one-force-one-solution”
principle where one can show that the effect of the initial data vanishes
uniformly. Therefore we will give a second approach to proving the uniqueness
result without using the strong Feller property, and this approach also gives
an exponential “one-force one-solution principle” for open KPZ. To formulate
the result, we let $\mathcal{X}$ be as above. The following result will be
proved independently of Theorem 1.1.

######  Theorem 1.2 (Uniqueness via polymer coupling).

The following “one-force one-solution” principle holds for open KPZ: fix a
space-time white noise on $(-\infty,0]\times[0,1]$ and let
$h_{N}^{\mu},h_{N}^{\nu}:[-N,0]\times[0,1]\to\mathbb{R}$ denote the solutions
to open KPZ driven by $\xi$ such that their Hopf-Cole transforms are started
from deterministic nonnegative Borel measures $\mu,\nu$ at time $t=-N$. Then
there exist $C,c>0$ such that for all $N\geq 1$ one has

$\mathbb{E}\bigg{[}\sup_{[\mu],[\nu]\in\mathcal{X}}d_{\mathcal{X}}\big{(}[h_{N}^{\mu}(0,\cdot)],[h_{N}^{\nu}(0,\cdot)]\big{)}\bigg{]}\leq
Ce^{-cN}.$

This will be proved as Theorem 4.3 in the main body. Our approach to proving
this result follows [GK20, Section 4] which is done in the periodic case, and
uses the convolution property of (2) in a manner inspired by the approach of
[Sin90, EKMS]. In particular, the proof of Theorem 1.2 still works to show
uniqueness in the case of a more general Gaussian noise that is colored in
space and white in time, and does not rely on the strong Feller property.

By Markov’s inequality and Borel-Cantelli, the expectation bound also implies
almost sure convergence as $N\to\infty$. We actually prove something stronger,
namely that a uniform synchronization result holds in the (much stronger)
Hölder metric $d_{\mathcal{Y}}$ defined in (7) below, more precisely for
$\kappa\in(0,1/2)$,

$\lim_{N\to\infty}\sup_{[\mu],[\nu]\in\mathcal{X}}\big{\|}(h_{N}^{\mu}(0,\cdot)-h_{N}^{\mu}(0,0))-(h_{N}^{\nu}(0,\cdot)-h_{N}^{\nu}(0,0))\big{\|}_{C^{\kappa}[0,1]}=0,\;\;\;\;\;a.s..$
(3)

See Corollary 4.5 for the proof, which is based on combining Theorem 1.2 with
a compactness trick. We remark that the proof of that corollary goes through
in the case of a spatially smooth noise or a periodic boundary condition as
well, thus recovering some results of [Ros19] with a different argument based
on compactness and the convolution property. In case one is only interested in
the synchronization result, its proof is fairly brief and self-contained
within Definition 4.1, Lemma 4.2, Theorem 4.3, and Corollary 4.5.

We now explain the differences of obtaining Theorem 1.2 compared to method of
[Ros19]. That work contains an abstract result which guarantees the one-force-
one-solution principle for SPDEs whose solutions are linear as a function of
the initial data like the one studied here, as long as Assumption 4.2 therein
is satisfied. In [Ros19] Assumption 4.2 is verified for periodic boundary
conditions and space-time white noise using paracontrolled distributions, but
that is not the only possible way of doing so and Assumption 4.2 for (2)
likely follows easily from the results of Section 2 of the current paper.
While that result is not completely independent of the Hopf-Cole transform,
its reliance on it is less crucial and it can be applied to more general
equations. Furthermore, that paper derives slightly stronger synchronization
results than the one we prove here, for instance in that paper (3) is shown to
occur exponentially fast.

We now turn to the question of total variation convergence. We actually give
two proofs that there is a spectral gap, one using the support theorem and the
other using the one-force-one-solution principle. The following result will be
proved as Theorem 3.11 and then independently reproved as Corollary 4.8 below.
Both proofs use the strong Feller property and compactness as an input.

######  Theorem 1.3 (Spectral gap with a uniform multiplying constant).

Let $\mathfrak{P}_{*}$ denote the adjoint Markov operator for open KPZ, which
acts on the space of measures on $\mathcal{X}$. Then there exist $C,c>0$ such
that for all $N$ one has

$\sup_{\phi\in\mathcal{X}}\|\mathfrak{P}_{*}^{N}\delta_{\phi}-\rho_{A,B}\|_{TV}\leq
Ce^{-cN},$

where $\delta_{\phi}$ denotes a Dirac mass at the initial condition
$\phi\in\mathcal{X}$.

For a Markov process, the bound in Theorem 1.3 is an extremely strong form of
geometric ergodicity, where the exponential decay rate and the multiplying
constant are both uniform over all possible choices of initial data (in the
total variation norm). Since our state space $\mathcal{X}$ measurably contains
many other important spaces of possible initial conditions, e.g. Hölder spaces
such as $\mathcal{Y}$ defined in (7) below, the result also implies the
corresponding result with the sup over $\mathcal{X}$ replaced by a sup over
$\mathcal{Y}$ and the TV norm on $\mathcal{M}(\mathcal{X})$ replaced by the TV
norm on $\mathcal{M}(\mathcal{Y})$.

Theorem 1.3 then leads to a number of other interesting questions. In
particular one can then sensibly ask about the optimal values of the constants
$c$ and $C$, which is equivalent to computing (respectively) the spectral gap
and the mixing time of the process. The most interesting formulation of this
question would be to study the dependence of both the spectral gap and the
mixing time on the size of the interval and the values of the boundary
parameters $A,B$. We do not pursue these optimality questions in the present
work. In the case of TASEP and ASEP, the analogous question has recently been
studied in great depth [SS, Sch, ES, GNS]. In those papers the authors observe
some very interesting cutoff phenomena, where for certain boundary parameters
there is a very explicit cutoff for the mixing time as a function of the
boundary parameters. Recent work of [HS] has even calculated the exact profile
of the cutoff, showing that it is given by the cdf of the Tracy-Widom
distribution. We expect that there may be similar results for the open KPZ
equation as well as the periodic KPZ equation, though this is a difficult
question far beyond the scope of this work.

In Theorem 1.3, we remark that our use of the strong Feller property and
compactness tricks to prove the spectral gap seems similar in spirit to works
of [TW] which used such ideas to prove a spectral gap for the dynamic
$\Phi_{2}^{4}$ model. We further remark that the analogous result for the
additive stochastic heat equation is false. That is, if we consider (1)
without the nonlinear term on the right side, then although the constant $c$
would still be finite, the other constant $C$ would become infinite. In other
words, there is still a spectral gap but there is no mixing time uniform over
all initial conditions. And in fact the state space cannot be compactified for
the linearized equation. The finiteness of the supremum in Theorem 1.3 and the
existence of a compact state space seems related to the so-called “coming down
from infinity” property that was proved in [MW] for the dynamic $\Phi_{3}^{4}$
model. See Remark 4.6 for further details.

Another interesting problem for future work is to see what can be proved about
the invariant measures on the half space as opposed to a bounded interval. A
conjectural characterization of the extremal invariant measures was given in
[BC22], but it remains to be proved. A compact state space or a spectral gap
is unlikely in that case, but it may be possible to show uniqueness using some
variant of the arguments given here. For KPZ on the whole line, a
synchronization result was recently considered and proved in [JRS] using a
different approach with Busemann functions, which seems to be the only
synchronization result to date for an SPDE on an unbounded interval driven by
Gaussian space-time white noise. Results such as [BCK, Ba13] were some of the
first works to prove synchronization in a noncompact setting, though these had
a Poissonian kick-forcing.

Let us now say a word about the proofs of the above three theorems, and the
ordering of the logic. As stated above, Theorems 1.1 and 1.2 will first be
derived completely independently of one another. The proofs of both theorems
will ultimately leverage particular properties of (4) that interpret $z$ as
the partition function of a polymer. Once these two theorems are proved, then
we will show that Theorem 1.3 can be deduced from either of these first two
theorems. To go from either of Theorem 1.1 or 1.2 to Theorem 1.3, we will
derive some abstract results about Markov processes on compact state spaces.
Specifically, we have the following two general results.

######  Theorem 1.4 (Automatic criteria for uniform spectral gap on compact
spaces).

  1. (1)

Suppose $(\mathfrak{P}_{t})_{t\geq 0}$ is a strongly continuous semigroup of
Markov operators on a compact metric space $\mathcal{X}$. Assume that
$\mathfrak{P}_{t}$ is strong Feller for each $t>0$ and admits a unique
invariant probability measure $\rho$. Then there exist $C,c>0$ such that for
all $t\geq 0$ one has
$\sup_{\phi\in\mathcal{X}}\big{\|}\rho-\mathfrak{P}_{t}^{*}\delta_{\phi}\big{\|}_{TV}\leq
Ce^{-ct},$ where $\mathfrak{P}_{t}^{*}$ is the adjoint Markov semigroup, and
$\delta_{\phi}$ denotes a Dirac mass at the initial condition
$\phi\in\mathcal{X}$.

  2. (2)

Let $\mathfrak{P}$ be a strong Feller Markov operator on some compact state
space $\mathcal{X}$, and let $\mathfrak{P}_{*}$ denote the adjoint operator on
measures. Assume that
$\mathfrak{P}^{N}_{*}\delta_{\phi}-\mathfrak{P}^{N}_{*}\delta_{\psi}$
converges weakly to $0$ for all $\phi,\psi\in\mathcal{X}$, i.e.,
$\lim_{N\to\infty}|\mathfrak{P}^{N}F(\phi)-\mathfrak{P}^{N}F(\psi)|=0$ for all
$\phi,\psi\in\mathcal{X}$ and all Lipschitz continuous
$F:\mathcal{X}\to\mathbb{R}$. Then there exists $C,c>0$ such that for all
$N\in\mathbb{N}$ one has
$\sup_{\phi,\psi\in\mathcal{X}}\|\mathfrak{P}^{N}_{*}\delta_{\phi}-\mathfrak{P}^{N}_{*}\delta_{\psi}\|_{TV}\leq
Ce^{-cN}.$

The first result states that in continuous time, compactness plus the strong
Feller property automatically allows us to upgrade unique ergodicity to the
seemingly much stronger property of a spectral gap with uniform multiplying
constant. It is clear that Theorem 1.3 can be deduced immediately from Theorem
1.1 using this first result. In discrete time the analogous statement is
simply not true even on finite state spaces. However the second result states
that even in discrete time, compactness plus the the strong Feller property
allows us to upgrade pointwise convergence of differences in the weak topology
to a spectral gap with uniform multiplying constant. It is clear that Theorem
1.3 is immediate from Theorem 1.2 using this second result.

The monograph of Meyn-Tweedie-Glynn [MT, Chapter 16] contains many abstract
and powerful results for Markov processes on general state spaces, however
despite the extensive literature we were not able to find the above two
results in efficient and readily-applicable forms. Since these results are
evidently quite useful in problems concerning KPZ and related areas, we have
included short and self-contained proofs of these abstract results,
independent of one another and different from the arguments of [MT]. See
Theorems 3.11 and 4.7 below.

Outline: Section 2 recalls some standard facts about the Robin-boundary
stochastic heat equation that will be useful going forward. Section 3 will
prove Theorem 1.1. Section 4 will prove Theorem 1.2. The methods and results
of Sections 3 and 4 should be viewed as disjoint from each other, and both
sections contain an independent proof of Theorem 1.3.

Acknowledgements: I thank Yu Gu, Ivan Corwin, and Konstantin Matetski for very
helpful discussions and for reading portions of the draft. I thank Ran Tao for
finding some mistakes and typos.

##  2\. Properties of the Robin SHE and well-posedness of its Markov semigroup
on the quotient space $\mathcal{X}$

In this section we derive some bounds on the Robin-boundary stochastic heat
equation that are standard but will be very important in later sections. We
also prove existence of an invariant measure. The results described here are
still valid for noises that white in time but colored in space, with some
minor modification of the proofs. Some other boundary conditions such as
Dirichlet (zero) conditions on one endpoint or both can also be considered
(these correspond to $A=\infty$ or $B=\infty)$, with the only exception that
the strict positivity result (Proposition 2.4) will not hold in case of the
latter.

Consider the solution of the multiplicative stochastic heat equation with
Robin boundary condition on $[0,1]$:

$\partial_{t}z=\partial_{x}^{2}z+z\xi,\;\;\;\;\partial_{x}z(t,0)=Az(t,0),\;\;\;\;\partial_{x}z(t,1)=Bz(t,1).$
(4)

Here (and henceforth) $A,B$ are just some fixed real numbers. The formal
definition of such a process is a $\mathcal{F}^{\xi}$-adapted process
$z(t,\cdot)$ such that a.s. for all $t\geq 0$ and $x\in[0,1]$ one has

$z(t,x)=\int_{[0,1]}p^{A,B}_{t}(x,y)z_{0}(dy)+\int_{0}^{t}\int_{[0,1]}p^{A,B}_{t-s}(x,y)z(s,y)\xi(ds,dy),$
(5)

where the initial condition $z_{0}$ is a fixed deterministic tempered
distribution supported on $[0,1]$, and the latter is an Itô integral with
$p^{A,B}$ the Robin heat kernel [CS18, Par19], that is, the fundamental
solution of the heat equation whose boundary conditions at $x=0$ and $x=1$
match those of (4).

We are interested in the invariant measures for this process. However, there
is no bona fide invariant measure since $cz$ is a solution for all $c>0$,
whenever $z$ is. Consequently the invariant measures must be infinite measures
and cannot be probability measures.

We fix this problem by constructing state spaces $\mathcal{X}$ and
$\mathcal{Y}$ consisting of equivalence classes of functions, which is where
$z(t,\cdot)$ will live. More precisely, we define $\mathcal{X}$ to be the set
of all finite non-negative non-zero Borel measures on $[0,1]$ modulo the
relation $\mu\sim c\mu$ for all $c>0$. For equivalence classes
$[\mu],[\nu]\in\mathcal{X}$ we define their distance through the metric

$d_{\mathcal{X}}([\mu],[\nu]):=d_{\text{Proh}}\bigg{(}\frac{\mu}{\mu[0,1]},\frac{\nu}{\nu[0,1]}\bigg{)},$
(6)

where $d_{\text{Proh}}$ is a Prohorov metric on the space of probability
measures on $[0,1]$ (i.e., any metric which metrizes the weak topology),
satisfying $d_{\text{Proh}}(\mu,\nu)\leq\|\mu-\nu\|_{TV}$. An explicit example
of such a metric $d_{\text{Proh}}$ is given by

$d_{\text{Proh}}(\mu,\nu)=\sup_{\|g\|_{C^{1}[0,1]}\leq
1}\bigg{|}\int_{[0,1]}g\;d(\mu-\nu)\bigg{|},$

where $\|g\|_{C^{1}[0,1]}:=\|g\|_{C[0,1]}+\|g^{\prime}\|_{C[0,1]}.$ An
important fact about $\mathcal{X}$ is that it is compact (since any collection
of probability measures on $[0,1]$ is automatically tight), which will be a
very useful simplifying tool later.

Now fix some $\kappa\in(0,1/2)$. For separability reasons, we take the
convention that $C^{\kappa}[0,1]$ is defined as the closure of smooth
functions inside the usual Banach space of $\kappa$-Holder continuous
functions $[0,1]\to\mathbb{R}$. Now we define another metric space
$\mathcal{Y}$ to be the set of all strictly positive functions on $[0,1]$
lying in $C^{\kappa}[0,1]$, modulo the equivalence relation $f\sim cf$ for all
$c>0$. We define their distance through the metric

$d_{\mathcal{Y}}([f],[g]):=\bigg{\|}\log(f)-\log(g)-\int_{[0,1]}\big{(}\log(f)-\log(g)\big{)}\bigg{\|}_{C^{\kappa}[0,1]}.$
(7)

For $d_{\mathcal{Y}}$ any equivalent metric could also be used, such as
$\big{\|}\log(f)-\log(g)-\big{(}\log(f(0))-\log(g(0))\big{)}\big{\|}_{C^{\kappa}}.$
One may easily verify that this is a complete metric space and that it embeds
densely into $\mathcal{X}$ by identifying a continuous function $f$ with the
measure $\mu(A)=\int_{A}f(x)dx.$ Certainly closed bounded subsets in
$\mathcal{Y}$ are still closed in $\mathcal{X}$, and are thus compact in
$\mathcal{X}$. This already implies that all closed sets (hence all open sets)
in $\mathcal{Y}$ are Borel measurable in $\mathcal{X}$, which will be
important in the next section.

We will next establish global well-posedness of the Markov semigroup of (4) on
the space of finite Borel measures on $[0,1]$, and show that the process
$z(1,\cdot)$ already takes values in $C^{\kappa}$ almost surely for
$\kappa<1/2$. Furthermore we will derive a bound on the expected $C^{\kappa}$
norm which depends only on the total mass of the initial data, which will in
particular allow us to prove (in Proposition 3.4 and Corollary 3.5) that
$[z(t,\cdot)]$ is strong Feller on $\mathcal{X}$, using the knowledge from
[KM22] that it is strong Feller on $\mathcal{Y}$.

We will need to use some purely deterministic facts about the Robin heat
kernel which we summarize without proof here, see e.g. [Fri, Section 2.5] for
a reference on how to obtain these facts:

  1. (1)

For each $x\in[0,1]$ the function $(t,y)\mapsto p_{t}^{A,B}(x,y)$ =
$p_{t}^{A,B}(y,x)$ solves the heat equation on $\mathbb{R}_{+}\times[0,1]$
with initial data $\delta_{x}$, and furthermore
$\partial_{y}|_{y=0}p_{t}^{A,B}(x,y)=Ap_{t}^{A,B}(x,0)$ and
$\partial_{y}|_{y=1}p_{t}^{A,B}(x,y)=Bp_{t}^{A,B}(x,1)$ for all $x,t$. It is a
symmetric kernel in the sense that $p_{t}^{A,B}(x,y)=p_{t}^{A,B}(y,x).$

  2. (2)

One may write

$p_{t}^{A,B}(x,y)=p_{t}^{neu}(x,y)\mathbf{E}_{Br[0,t]}^{x\to
y}\bigg{[}e^{-AL_{0}^{W}(t)-BL_{1}^{W}(t)}\bigg{]},$ (8)

where $p_{t}^{neu}$ is the Neumann heat kernel $(A=B=0)$ and the expectation
is with respect to a Brownian bridge of diffusive rate 2 on the time interval
$[0,t]$ starting from $x$ and ending at $y$ that is reflected at the
boundaries of $[0,1]$ (so that it always stays inside the interval). Here
$L_{a}^{W}(t)$ equals the local time of $W$ at spatial location $a$ up to time
$t$. This equation reflects a Feynman-Kac decomposition for the Robin boundary
heat equation, where the boundary condition is reinterpreted as a special case
of Neumann boundary heat equation, but with a multiplicative forcing term
consisting of two Dirac masses of weights $-A$ and $-B$ at the respective
boundaries $x=0$ and $x=1$. In particular (8) implies the strict positivity of
the kernel: $p_{t}^{A,B}(x,y)>0$ for all $A,B\in\mathbb{R}$, all $x,y\in[0,1]$
and all $t>0$.

  3. (3)

Similarly to the ordinary heat kernel, for any integers $k_{1},k_{2}\geq 0$
one has the following bound uniformly over all
$(x,y,t)\in[0,1]^{2}\times(0,T]$

$\partial_{t}^{k_{1}}\partial_{x}^{k_{2}}p_{t}^{A,B}(x,y)=\partial_{x}^{2k_{1}+k_{2}}p_{t}^{A,B}(x,y)\leq
C(|x-y|+t^{1/2})^{-(1+2k_{1}+k_{2})},$ (9)

where $C$ may depend on $k_{1},k_{2},T,A,B$ but not on $x,y,t$. Since
$\partial_{y}^{2}p_{t}(x,y)=\partial_{x}^{2}p_{t}(x,y)$, the bound (9) is
actually equivalent to a seemingly more general bound
$\partial_{t}^{k_{1}}\partial_{x}^{k_{2}}\partial_{y}^{k_{3}}p_{t}^{A,B}(x,y)\leq
C(|x-y|+t^{1/2})^{-(1+2k_{1}+k_{2}+k_{3})}$ for $k_{3}$ even. Because of these
bounds, one can hope to sensibly define

$(P^{A,B}\phi)(t,x):=\int_{[0,t)\times[0,1]}p_{t-s}^{A,B}(x,y)\phi(s,y)dyds$

for certain tempered space-time distributions $\phi$ supported on
$[0,\infty)\times[0,1].$ One would intuitively expect $p^{A,B}$ to improve
space-time regularity by two exponents because the singularity of $p^{A,B}$
along the space-time diagonal (with respect to the parabolic metric) is of
order $-1$ by (9), whereas the parabolic dimension of the space-time is $-3$.
Indeed let us define $C_{\mathfrak{s}}^{\alpha}([0,1]\times[0,T])$ to be all
those tempered distributions (supported on that rectangle) that are of
parabolic Hölder regularity of exponent $\alpha\in\mathbb{R}$, more precisely
the completion of smooth functions on $[0,1]\times[0,T]$ with respect to the
norm given by

$\|f\|_{C^{\alpha,\sigma}_{\mathfrak{s}}([0,1]\times[0,T])}:=\sup_{(t,x)\in[0,1]\times[0,T]}\sup_{\lambda\in(0,1]}\sup_{\varphi\in
B_{r}}\lambda^{-\alpha}(f,S^{\lambda}_{(t,x)}\varphi)$

where $(f,\varphi)$ denotes the pairing in $L^{2}([0,1]\times[0,T])$, where
the scaling operators are defined by
$S^{\lambda}_{(t,x)}\phi(s,y)=\lambda^{-3}\phi(\lambda^{-2}(t-s),\lambda^{-1}(x-y)),$
and where if $\alpha<0$ then $r:=-\lfloor\alpha\rfloor$ and $B_{r}$ is the set
of all smooth functions of $C^{r}(\mathbb{R}^{2})$-norm less than 1 with
support contained in the unit ball of $\mathbb{R}^{2}$, and if $\alpha>0$ then
$r=\lceil\alpha\rceil$ and $B_{r}$ is the set of all smooth functions of
$C^{0}$ norm less than 1, supported on the unit ball of $\mathbb{R}^{2}$ that
are orthogonal in $L^{2}(\mathbb{R}^{2})$ to all polynomials of parabolic
degree less than or equal to $r$. Then we have the Schauder estimate which
says that $P^{A,B}$ is a bounded linear operator from
$C_{\mathfrak{s}}^{\alpha}$ to $C_{\mathfrak{s}}^{\alpha+2}$ for all
$\alpha<1$ ($\alpha\notin\mathbb{Z})$, i.e., there exists $C=C(\alpha)>0$ such
that

$\|P^{A,B}\phi\|_{C_{\mathfrak{s}}^{\alpha+2}([0,1]\times[0,T])}\leq
C\|\phi\|_{C_{\mathfrak{s}}^{\alpha}([0,1]\times[0,T])},$ (10)

uniformly over all $\phi\in C_{\mathfrak{s}}^{\alpha}([0,1]\times[0,T]).$ The
proof of the claim (10) follows from a standard and elementary argument using
(9) and following the definition of the Holder spaces, noting e.g. that for
very small $\lambda$ the operation $P^{A,B}$ applied to
$S^{\lambda}_{(t,x)}\varphi$ behaves at worst like
$\lambda^{2}S^{\lambda}_{(t,x)}\psi$ for some other smooth function $\psi$.
Although we will not need to use (10) for $\alpha>1$
($\alpha\notin\mathbb{Z}$), it is still true in that regime except that the
bound only holds for $\phi$ in the closed linear subspace of
$C_{\mathfrak{s}}^{\alpha}([0,1]\times[0,T])$ such that $\phi$ and all of its
space-time derivatives respect the Robin boundary conditions (this allows for
integration by parts without worrying about appearance of boundary terms).
Another corollary of (9) is that for any (purely spatial) Scwhartz
distribution $\varphi$ supported on $[0,1]$ we can also sensibly define the
purely spatial convolution at fixed time $t$ as

$p_{t}^{A,B}\varphi(x):=\int_{[0,1]}p_{t}^{A,B}(x,y)\varphi(dy).$

######  Theorem 2.1 (Existence/Hölder estimates of Mild Solutions).

Fix some deterministic signed Borel measure $\mu$ on $[0,1]$ and let $|\mu|$
denote the sum of its positive and negative parts. Let $\xi$ be a space-time
white noise on $[0,1]\times[0,T]$. There exists a unique $C[0,1]$-valued
process $(z(t,\cdot))_{t\in(0,T]}$ which is adapted to
$\mathcal{F}_{t}:=\sigma(\\{\xi(t,\cdot)\\}_{s\leq t})$, and satisfies (5)
with $z_{0}=\mu$.

For $p\geq 1$ this mild solution satisfies

$\mathbb{E}[|z(t,x)|^{p}]^{2/p}\leq
Ct^{-1/2}\|\mu\|_{TV}(p_{t}^{A,B}|\mu|)(x)\leq Ct^{-1}\|\mu\|_{TV}^{2},$ (11)

for some universal constant $C=C(A,B,p,T)$ not depending on $\mu$ or
$(t,x)\in[0,T]\times[0,1]$. For $\kappa>0$ we furthermore have the existence
of a universal constant $C=C(A,B,\kappa,p,T)>0$ independent of $\mu$ and
$\epsilon\in(0,T)$ such that

$\mathbb{E}\big{[}\|z\|_{C_{\mathfrak{s}}^{1/2-\kappa}([0,1]\times[\epsilon,T])}^{p}\big{]}^{1/p}\leq
C\epsilon^{-\frac{3}{4}}\|\mu\|_{TV}.$ (12)

Finally if $\mu_{n}$ is a sequence of Borel measures on $[0,1]$ converging
weakly to $\mu$ and if $z_{\mu_{n}},z_{\mu}$ denote the associated solutions,
then for all $\epsilon\in(0,T)$ we have the convergence in probability

$\lim_{n\to\infty}\mathbb{P}\big{(}\|z_{\mu_{n}}-z_{\mu}\|_{C_{\mathfrak{s}}^{1/2-\kappa}([0,1]\times[\epsilon,T])}>\delta\big{)}=0,\;\;\;\;\;\text{for
all }\delta>0.$ (13)

###### Proof.

Throughout the proof we abbreviate $p_{t}^{A,B}$ as simply $p_{t}$ and
$p_{t}^{A,B}\mu$ as $p_{t}\mu$. Define a sequence of iterates for $t\leq T$
and $x\in[0,1]$:

$\displaystyle u_{0}(t,x)$ $\displaystyle:=(p_{t}\mu)(x),$ $\displaystyle
u_{n+1}(t,x)$
$\displaystyle:=\int_{0}^{t}\int_{[0,1]}p_{t-s}(x,y)u_{n}(s,y)\xi(dy,ds).$

Note that $|p_{t}\mu(x)|\leq(p_{t}|\mu|)(x)$. To show existence of a solution
it suffices to show that $\sum u_{n}$ converges in the appropriate Banach
space, specifically we choose the space of $\mathcal{F}^{\xi}_{t}$-adapted
processes $z$ such that $\sup_{x\in[0,1],s\leq
T}s^{1/2}\mathbb{E}[|z(s,y)|^{p}]^{1/p}<\infty$. Now we define

$f_{n}(t):=\sup_{\begin{subarray}{c}x\in[0,1]\\\
s\in[0,t]\end{subarray}}s^{1/2}(p_{s}|\mu|(x))^{-1}\mathbb{E}[|u_{n}(s,x)|^{p}]^{2/p}$

where the RHS is defined as $+\infty$ if the stochastic integral defining
$u_{n}$ fails to exist. For the proof it will be crucial that
$(p_{s}|\mu|)(x)>0$ for all $s>0$ and $x\in[0,1]$ which is true as long as
$\mu$ is not the zero measure.

Using the Itô isometry, the definition of $f_{n}$, and the fact that
$p_{t-s}^{A,B}(x,y)\leq C_{T}(t-s)^{-1/2}$ on $[0,1]\times[0,T]$ (see equation
(9) or alternatively [Par19, Proposition 3.31]), we compute by applying
Burkholder-Davis-Gundy and then Minkowski’s inequality (in that order)

$\displaystyle\mathbb{E}[|u_{n+1}(t,x)|^{p}]^{2/p}$ $\displaystyle\leq
C_{p}\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{t}\int_{[0,1]}p_{t-s}(x,y)^{2}u_{n}(s,y)^{2}dyds\bigg{)}^{p/2}\bigg{]}^{2/p}$
$\displaystyle\leq
C_{p}\int_{0}^{t}\int_{[0,1]}p_{t-s}(x,y)^{2}\mathbb{E}[u_{n}(s,y)^{p}]^{2/p}dyds$
$\displaystyle\leq C_{p}\int_{0}^{t}\int_{[0,1]}p_{t-s}(x,y)^{2}\cdot
s^{-1/2}(p_{s}|\mu|)(y)f_{n}(s)dyds$ $\displaystyle\leq
C\int_{0}^{t}(t-s)^{-1/2}s^{-1/2}\bigg{[}\int_{[0,1]}p_{t-s}(x,y)(p_{s}|\mu|)(y)dy\bigg{]}f_{n}(s)ds$
$\displaystyle=C(p_{t}|\mu|)(x)\int_{0}^{t}(t-s)^{-1/2}s^{-1/2}f_{n}(s)ds$

where we used the semigroup property in the final line. Multiplying both sides
by $t^{1/2}(p_{t}|\mu|(x))^{-1}$, we find that

$\displaystyle
t^{1/2}(p_{t}|\mu|(x))^{-1}\mathbb{E}[|u_{n+1}(t,x)|^{p}]^{2/p}$
$\displaystyle\leq Ct^{1/2}\int_{0}^{t}(t-s)^{-1/2}s^{-1/2}f_{n}(s)ds.$

Notice from the definition that $f_{n}$ is an increasing function, therefore
(by making a substitution $s=tu)$ one may see that the RHS of the last
expression is an increasing function of $t$. It follows by taking a supremum
that

$\displaystyle f_{n+1}(t)$ $\displaystyle\leq
Ct^{1/2}\int_{0}^{t}(t-s)^{-1/2}s^{-1/2}f_{n}(s)ds$

which we can iterate twice to obtain

$f_{n+2}(t)\leq C^{\prime}t^{1/2}\int_{0}^{t}s^{-1/2}f_{n}(s)ds.$

Again using the fact that $p_{t}(x,y)\leq C_{T}t^{-1/2}$, we obtain

$(p_{t}|\mu|)(x)\leq C_{T}\|\mu\|_{TV}t^{-1/2}$ (14)

which implies that $\sup_{t\in[0,T]}f_{0}(t)\leq C_{T}\|\mu\|_{TV}$. Then
using this bound we can iterate this recursion to obtain

$f_{n}(t)\lesssim\|\mu\|_{TV}t^{n/2}/(n/2)!$

We have just proved that

$\mathbb{E}[|u_{n}(t,x)|^{p}]^{2/p}\leq
C\|\mu\|_{TV}(p_{t}|\mu|)(x)t^{(n-1)/2}/(n/2)!$

for a constant $C$ not depending on $t\in[0,T]$, $x\in[0,1]$, or
$n\in\mathbb{N}$. Since $z=\sum_{n}u_{n}$, we find that

$\mathbb{E}[|z(t,x)|^{p}]^{1/p}\leq\sum_{n\geq
0}\mathbb{E}[|u_{n}(t,x)|^{p}]^{1/p}\leq
C^{\prime}\|\mu\|_{TV}^{1/2}t^{-1/4}p_{t}|\mu|(x)^{1/2}$

where $C^{\prime}$ is obtained by summing the infinite series (bounding $t$ by
the terminal time $T$), which which will not depend on $\mu,t,x$. This already
proves the first inequality in (11). The second inequality in (11) just comes
by once again using the same bound (14) again.

The proof of uniqueness is very similar. In fact, the difference of two mild
solutions started from the same initial data would itself be a mild solution
started from $\mu=0$, thus by (11) must be zero a.s. for all $(t,x)$. By the
continuity statement which we prove next, the entire profile $(t,x)\mapsto
z(t,x)$ must then be zero almost surely.

Now let us move onto the proof of (12). Consider any smooth nonnegative test
function $\phi(t,x)$ supported in $[\epsilon,T]\times[0,1]$. By applying
Burkholder and then Minkowski (in that order) we find that

$\displaystyle\mathbb{E}\bigg{[}\bigg{|}\int_{[0,1]\times[\epsilon,T]}\phi(s,y)z(s,y)\xi(ds,dy)\bigg{|}^{p}\bigg{]}^{2/p}$
$\displaystyle\leq
C_{p}\mathbb{E}\bigg{[}\bigg{(}\int_{[0,1]\times[\epsilon,T]}\phi(s,y)^{2}z(s,y)^{2}dsdy\bigg{)}^{p/2}\bigg{]}^{2/p}$
$\displaystyle\leq
C_{p}\int_{[0,1]\times[\epsilon,T]}\phi(s,y)^{2}\mathbb{E}[|z(s,y)|^{p}]^{2/p}dsdy$
$\displaystyle\stackrel{{\scriptstyle\eqref{bound1}}}{{\leq}}C\|\mu\|_{TV}^{2}\int_{[0,1]\times[\epsilon,T]}\phi(s,y)^{2}s^{-1}dsdy$
$\displaystyle\leq
C\|\mu\|_{TV}^{2}\epsilon^{-1}\|\phi\|_{L^{2}([0,1]\times[\epsilon,T])}^{2},$

where $C$ is independent of $\mu$ and $\epsilon.$

For $\lambda>0$ and $(t,x)\in\mathbb{R}^{2}$ define the scaling operator
$S^{\lambda}_{(t,x)}:\mathcal{S}(\mathbb{R}^{2})\to\mathcal{S}(\mathbb{R}^{2})$
by

$\big{(}S^{\lambda}_{(t,x)}\phi\big{)}(s,y):=\lambda^{-3}\phi(\lambda^{-2}(s-t),\lambda^{-1}(y-x)).$

The previous bound can now be interpreted as

$\mathbb{E}\big{[}|(z\xi,S^{\lambda}_{(t,x)}\phi)|^{p}\big{]}^{1/p}\leq
C\epsilon^{-1/2}\|\mu\|_{TV}\|S^{\lambda}_{(t,x)}\phi\|_{L^{2}}.$ (15)

We now notice that $\|S^{\lambda}_{(t,x)}\phi\|_{L^{2}}\leq C\lambda^{-3/2}$
independently of $(t,x)\in[\epsilon,T]\times[0,1]$ and independently of
nonnegative smooth $\phi$ supported on $[\epsilon,T]\times[0,1]$ integrating
to one, and independently of $\epsilon$. Using [CW17, Theorem 2.7] this
implies that $z\xi$ can be realized as a random element of
$C^{-3/2-\kappa}_{\mathfrak{s}}([\epsilon,T]\times[0,1])$ and moreover

$\mathbb{E}[\|z\xi\|_{C^{-3/2-\kappa}_{\mathfrak{s}}([\epsilon,T]\times[0,1])}^{p}]^{1/p}\leq
C^{\prime}\epsilon^{-1/2}\|\mu\|_{TV}.$

Here $C^{\prime}$ only depends on the choice of $\kappa>3/p$ and the constant
$C$ appearing in (15), and is therefore independent of $\epsilon,\mu$. Letting
$z_{\epsilon/2}(x):=z(\epsilon/2,x)$, we also have by Minkowski that for all
space-time test functions $\phi$

$\displaystyle\mathbb{E}\bigg{[}\bigg{|}\int_{[0,1]}\phi(y)z(\epsilon/2,y)dy\bigg{|}^{p}\bigg{]}^{1/p}$
$\displaystyle\leq\int_{[0,1]}\phi(y)\mathbb{E}[|z(\epsilon/2,y)|^{p}]^{1/p}dy$
$\displaystyle\stackrel{{\scriptstyle\eqref{bound1}}}{{\leq}}C\int_{0}^{1}\phi(y)\epsilon^{-1/2}\|\mu\|_{TV}dy$
$\displaystyle\leq
C^{\prime}\|\mu\|_{TV}\epsilon^{-1/2}\|\phi\|_{L^{1}[0,1]},$

so that by a similar reasoning we have the bound that
$\mathbb{E}[\|z_{\epsilon/2}\|_{C^{-\kappa}[0,1]}^{p}]^{1/p}\leq
C^{\prime}\epsilon^{-1/2}\|\mu\|_{TV},$ where $C$ may depend on the choice of
$\kappa<-1/p$ but not $\epsilon$ or $\mu.$

For $\kappa>0$ one defines the space $C^{-\kappa}[0,1]$ to be the closure of
smooth functions on $[0,1]$ with respect to a norm given by
$\sup_{x\in[0,1]}\sup_{\lambda\in(0,1]}\sup_{\phi\in
B_{r}}\lambda^{-\alpha}(f,A^{\lambda}_{x}\phi)_{L^{2}[0,1]}$ where
$A^{\lambda}_{x}\phi(y)=\lambda^{-1}\phi(\lambda^{-1}(y-x))$ and $B_{r}$ is
the set of smooth functions supported on $[-1,1]$ of $C^{r}(\mathbb{R})$-norm
less than 1 with $r:=-\lceil\kappa\rceil$. Now by the smoothing effect of the
Robin heat kernel, we have a deterministic bound
$\|p_{t}z_{\epsilon/2}\|_{C^{1/2-\kappa}[0,1]}\leq
Ct^{-1/4}\|z_{\epsilon/2}\|_{C^{-\kappa}[0,1]}$, which is a consequence of (9)
(see also [Hai09, Section 4.3] for another proof via analytic semigroups). In
particular by setting $t=\epsilon/2$ and taking expectation we have

$\mathbb{E}[\|p_{\epsilon/2}z_{\epsilon/2}\|_{C^{1/2-\kappa}[0,1]}^{p}]^{1/p}\leq
C\epsilon^{-3/4}\|\mu\|_{TV}.$

where $C$ is independent of $\epsilon$ or $\mu$. Now let us write for
$t>\epsilon$

$z(t,x)=p_{t-\epsilon}(p_{\epsilon/2}z_{\epsilon/2})(x)+\int_{\epsilon/2}^{t}\int_{[0,1]}p_{t-s}(x,y)z(s,y)\xi(dy,ds)=:z_{1}^{\epsilon}(t,x)+z_{2}^{\epsilon}(t,x).$

Now for $f\in C^{1/2-\kappa}[0,1]$ if we define $P_{f}(t,x)=p_{t}f(x)$ then
one directly shows the bound (independently of $f$) that
$\|P_{f}\|_{C_{\mathfrak{s}}^{1/2-\kappa}([0,T]\times[0,1])}\leq
C_{T}\|f\|_{C^{1/2-\kappa}[0,1]},$ consequently we have that

$\mathbb{E}[\|z_{1}^{\epsilon}\|^{p}_{C_{\mathfrak{s}}^{1/2-\kappa}([\epsilon,T]\times[0,1])}]^{1/p}\leq
C\mathbb{E}[\|p_{\epsilon/2}z_{\epsilon/2}\|_{C^{1/2-\kappa}[0,1]}^{p}]^{1/p}\leq
C\epsilon^{-3/4}\|\mu\|_{TV}.$

Now to bound $z_{2}$ we notice that $z_{2}$ is just the space-time convolution
of the field $z\xi|_{[\epsilon/2,T]\times[0,1]}$ with the Robin heat kernel,
therefore by the Schauder estimate (10) one has

$\mathbb{E}[\|z_{2}\|_{C_{\mathfrak{s}}^{1/2-\kappa}([\epsilon,T]\times[0,1])}^{p}]^{1/p}\leq
C\mathbb{E}[\|z\xi\|_{C^{-3/2-\kappa}_{\mathfrak{s}}([\epsilon/2,T]\times[0,1])}^{p}]^{1/p}\leq
C^{\prime}\epsilon^{-1/2}\|\mu\|_{TV},$

completing the proof of the claim (12).

Finally, to prove (13) we will need to use the notion of propagators. For each
fixed realization of $\xi$, let $z_{t}(a,x)$ denote the solution evaluated at
$(t,x)$ when started from initial condition $\delta_{a}$. We now claim the
existence of a modification of this process such that one has joint continuity
in all three variables $(t,a,x),$ so long as $t$ is bounded away from $0$.
This is easily justified by first noting that for fixed $x$, one has
continuity of the process $(t,a)\mapsto z_{t}(x,a)$ e.g. by (12) with
$\mu=\delta_{x}$, and secondly noting that one has an equality in the sense of
finite-dimensional distributions
$(z_{t}(a,x))_{a\in[0,1]}\stackrel{{\scriptstyle
d}}{{=}}(z_{t}(x,a))_{a\in[0,1]}$ for each fixed $t>0$ and $x\in[0,1]$, which
is clear by a time reversal of the noise on the interval $[0,t]$. Now suppose
$\mu_{n}\to\mu$ weakly as Borel measures on $[0,1]$. By the uniform
boundedness principle this implies that
$\sup_{n\in\mathbb{N}}\|\mu_{n}\|_{TV}<\infty.$ Therefore, by (12) we have
tightness of $(z_{\mu_{n}})_{n\geq 1}$ in the space
$C_{\mathfrak{s}}^{1/2-\kappa}([\epsilon,T]\times[0,1]),$ for every
$\epsilon,\kappa>0$. We just need to show therefore that
$z_{\mu_{n}}(t,x)-z_{\mu}(t,x)$ converges to 0 in probability, for each
individual value of $(t,x)$. We claim that

$z_{\mu_{n}}(t,x)=\int_{[0,1]}z_{t}(a,x)\mu_{n}(da)\to\int_{[0,1]}z_{t}(a,x)\mu(da)=z_{\mu}(t,x).$

The middle convergence statement is clear by continuity of the propagators in
the $a$ variable, thus we just need to justify the two equalities. In other
words we need to show that $\int_{[0,1]}z_{t}(a,x)\mu(da)=z_{\mu}(t,x)$ for
all finite and nonnegative Borel measures on $[0,1].$ The left side is easily
shown to satisfy the mild equation (5), thus by uniqueness of solutions must
agree with the right side. ∎

In the following proposition, $f$ and $\varphi$ will be functions defined on
the unit interval $[0,1]$ and whenever we write $f*\varphi$ it should be
understood as the periodic mollification

$\varphi*f(x)=\int_{[0,1]}f(y)\varphi\big{(}(x-y)\text{mod 1}\big{)}dy.$

######  Proposition 2.2 (Feynman-Kac formula for smooth noise).

Let $\varphi:[0,1]\to\mathbb{R}$ be a smooth function such that
$\int_{[0,1]}\varphi(u)du=1$. Assume that $\varphi$ is such that
$f\mapsto\varphi*f$ is a contraction on $L^{2}[0,1]$ (which is always the case
if $\varphi$ is non-negative). Define $\xi^{\varphi}(t,x)$ to be the spatial
periodic convolution of $\xi$ by $\varphi$. Define the mild solution of the
stochastic heat equation $z^{\varphi}$ driven by $\xi^{\varphi}$ exactly as in
the Duhamel formula (5) but with $\xi$ replaced by $\xi^{\varphi}$. Then the
solution exists and satisfies both (11) and (12) as well as (13) in Theorem
2.1. Moreover the constants appearing in those bounds are independent of
$\varphi$. In particular the Markov semigroup for this equation is also
globally defined on the space $\mathcal{X}$ from (6) and has at least one
invariant measure.

Furthermore we may express the solution started from initial data $\mu$ as

$z^{\varphi}(t,y)=\int_{[0,1]}p_{t}^{neu}(x,y)\mathbf{E}_{Br[0,t]}^{x\to
y}\bigg{[}e^{\int_{0}^{t}\xi^{\varphi}(t-s,W_{s})ds-\frac{1}{2}(\varphi*\varphi)(0)t-AL_{0}^{W}(t)-BL_{1}^{W}(t)}\bigg{]}\mu(dx).$

Here $p_{t}^{neu}$ is the Neumann heat kernel $(A=B=0)$ and
$\mathbf{E}_{Br[0,t]}^{x\to y}$ is the expectation is with respect to a
Brownian bridge of diffusive rate 2 on the time interval $[0,t]$ starting from
$x$ and ending at $y$ that is reflected at the boundaries of $[0,1]$ (so that
it always stays inside the interval), independent of $\xi$. Here
$L_{a}^{W}(t)$ equals the local time of $W$ at spatial location $a$ up to time
$t$. The quantity $\int_{0}^{t}\xi^{\varphi}(t-s,W_{s})ds$ appearing in the
exponential needs to be understood as a stochastic integral.

Finally, if we take a sequence $\varphi_{n}$ of such functions such that
$\|\varphi_{n}*f-f\|_{L^{2}}\to 0$ as $n\to\infty$ for all $f\in L^{2}[0,1]$,
then for any fixed initial data $\mu$, one has convergence of the associated
solutions $z^{\varphi_{n}}$ to the solution of (4) driven by space-time white
noise. Convergence here is in probability with respect to the topology of
$C^{\kappa}([0,1]\times[\epsilon,T])$ for all $\kappa\in(0,1/2)$ and all
$\epsilon<T$.

Here is a rough sketch of the proof: first of all, the fact that $\varphi$ is
a contraction on $L^{2}$ ensures that all of the bounds derived in the proof
of Theorem 2.1 go through unchanged if we replace $\xi$ by $\xi^{\varphi}$ (in
particular the bounds on the expected $p^{th}$ moment of $u_{n}$). Then to
prove the Feynman-Kac representation, one expands the exponential
$e^{\int_{0}^{t}\xi^{\varphi}(t-s,W_{s})ds-\frac{1}{2}(\varphi*\varphi)(0)t}$
as an infinite sum of Hermite powers of the noise. Then by using the
representation (8) of the Robin heat kernel, the $k^{th}$ term in the
expansion may be expressed as a $k$-fold iterated integral of products of the
$p_{t}^{A,B}$ against the mollified noise, which is precisely the mild
solution as defined in (5). The convergence result stated at the end of the
proposition can be proved by showing term-by-term convergence of the resulting
chaos expansions as $n\to\infty$ (using e.g. the master theorem for polynomial
chaos [CSZ, Theorem 2.3]). This would give pointwise convergence in
probability, but it can automatically be upgraded to convergence in
$C^{\kappa}$ since we have the bound (12) where the constant is independent of
$\varphi$ as described in the above proposition statement. We also refer the
reader to the work [BC95] for a similar theorem.

######  Proposition 2.3 (Convolution property).

Consider the propagators $(z_{s,t}(x,y))_{x,y\in[0,1],t\geq s\geq 0}$ where
$(t,y)\mapsto z_{s,t}(x,y)$ solves the stochastic heat equation (4) with
initial data $\delta_{x}$ at time $s$, all coupled to the same noise $\xi$.
Then

$\int_{[0,1]}z_{s,t}(x,r)z_{t,u}(r,y)dr=z_{s,u}(x,y)$

for all $x,y\in[0,1]$ and all $s<t<u$ almost surely. Furthermore, this
property remains true if we replace $\xi$ by its spatial mollification
$\xi^{\varphi}$ as in Proposition 2.2.

The proof is the same as in the full line case, see [AKQ13, Theorem 3.1(vii)].

######  Proposition 2.4 (Existence of negative moments).

Fix a space-time white noise $\xi$ on $[0,1]\times[0,T]$ and fix
$A,B\in\mathbb{R}$. Consider the process
$\mathbf{z}:=(z_{t}(x,y))_{x,y\in[0,1],t\geq 0}$ where $(t,y)\mapsto
z_{t}(x,y)$ solves the stochastic heat equation (4) with initial data
$\delta_{x}$. Then for all $p\geq 1$, all $\kappa<1/2$, and all $T>\epsilon>0$
we have that

$\mathbb{E}\bigg{[}\big{\|}\mathbf{z}\big{\|}_{C^{\kappa}_{\mathfrak{s}}([\epsilon,T]\times[0,1]^{2})}^{p}+\big{\|}\mathbf{z}^{-1}\big{\|}_{C^{\kappa}_{\mathfrak{s}}([\epsilon,T]\times[0,1]^{2})}^{p}\bigg{]}<\infty.$

Here $\mathbf{z}^{-1}=1/\mathbf{z}$ is the reciprocal of the process, to be
interpreted as $+\infty$ if $z$ is not strictly positive.

In particular we have strict positivity of the solution which is expected by
classical results on the whole line such as such as [Mue91]. Although such a
result is standard, it seems that a proof has not been written down elsewhere
for the Robin-boundary case (or even the Neumann case). Therefore we give a
self-contained proof here for the sake of completeness, and also because some
of the ideas in the proof (e.g. Feynman-Kac and approximation theory for the
SHE) are instructive in understanding some of the proofs of the later sections
as well. The proof will give an explicit tail decay on the above Holder norm
proportional to $e^{-c(\log r)^{2}}$ for some $c>0$. Of course, $c$ will
depend on the choice of $\epsilon,\kappa,A,B.$

###### Proof.

We break the proof into three steps.

Step 1. In this step, we realize the solution $z$ of (4) as the limit of a
two-parameter family of Banach-valued martingales. Consider the Cameron-Martin
space $H:=L^{2}([0,T]\times[0,1])$ of the noise $\xi$. It will be convenient
to write $L^{2}([0,T]\times[0,1])=L^{2}[0,T]\otimes
L^{2}[0,1]=:L^{2}_{T}\otimes L^{2}_{X}.$ We consider the orthonormal basis for
$L^{2}_{X}$ given by $e_{2i}(x)=\sqrt{2}\sin(2\pi i(x-1/2))$ and
$e_{2i+1}(x)=\sqrt{2}\cos(2\pi i(x-1/2))$ (here $i\geq 0$ but we exclude
$e_{0}=0$), and we choose some orthonormal basis of smooth functions
$\\{f_{j}(t)\\}_{j\geq 1}$ for $L^{2}_{T}$ whose exact form does not matter.
With this in place, we now sample a space-time white noise on
$[0,1]\times[0,T]$ by

$\xi=\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\xi_{ij}\;f_{j}\otimes e_{i}$

where the $\xi_{ij}$ are IID $N(0,1)$. The infinite sum converges in some
appropriately chosen Banach space whose norm is measurable on $H$ in the sense
of L. Gross [Gro]. We define the following hierarchy of approximations to
$\xi$ for $k,n\in\mathbb{N}$:

$\xi^{k}=\sum_{j=1}^{\infty}\sum_{i=1}^{2k}\xi_{ij}\;f_{j}\otimes e_{i},$
$\xi^{n,k}=\sum_{j=1}^{n}\sum_{i=1}^{2k}\xi_{ij}\;f_{j}\otimes e_{i}.$

We then define $\mathcal{F}_{n,k}$ and $\mathcal{F}_{k}$ to be the
$\sigma$-algebras generated by $\xi^{n,k}$ and $\xi^{k}$, respectively. Note
that $\xi^{k}$ can be written as the spatial mollification of $\xi$ where the
mollifier (which is not necessarily positive or of unit $L^{1}$ norm but still
satisfies the conditions of Proposition 2.2) equals a quotient of sine
functions (the Dirichlet kernel).

Let $z_{t}(x,y)$ be as in the theorem statement. Because of the identity
$\sum_{i=1}^{2k}e_{i}(x)^{2}=2k$ for all $x\in[0,1],$ one may show that
$\mathbb{E}[z_{t}(x,y)|\mathcal{F}_{k}]$ is given by the solution
$(t,y)\mapsto z_{t}^{k}(x,y)$ of the Ito equation
$\partial_{t}z=\partial_{x}^{2}z+z\xi^{k}$ started from $\delta_{x}$ (with the
same boundary conditions). By Proposition 2.2, this can also be written as

$\mathbb{E}[z_{t}(x,y)|\mathcal{F}_{k}]=p_{t}^{neu}(x,y)\mathbf{E}_{Br[0,t]}^{x\to
y}\bigg{[}e^{\int_{0}^{t}\xi^{k}(t-s,W_{s})ds-kt-
AL_{0}^{W}(t)-BL_{1}^{W}(t)}\bigg{]}$

where the expectation on the right side is with respect to a Brownian bridge
on the time interval $[0,t]$ starting from $x$ and ending at $y$, reflected at
the boundaries of $[0,1]$. Here $p_{t}^{neu}$ is the Neumann heat kernel on
$[0,1]$. The integral $\xi^{k}(t-s,W_{s})$ is well-defined as an Ito-integral
for any continuous path $W$. Applying Fubini’s theorem, we can also write

$\displaystyle z_{t}^{n,k}(x,y)$
$\displaystyle:=\mathbb{E}[z_{t}(x,y)|\mathcal{F}_{n,k}]$
$\displaystyle=\mathbb{E}[z_{t}^{k}(x,y)|\mathcal{F}_{n,k}]$
$\displaystyle=p_{t}^{neu}(x,y)\mathbf{E}_{Br[0,t]}^{x\to
y}\bigg{[}\mathbb{E}\bigg{[}e^{\int_{0}^{t}\xi^{k}(t-s,W_{s})ds}\bigg{|}\mathcal{F}_{n,k}\bigg{]}e^{-kt-
AL_{0}^{W}(t)-BL_{1}^{W}(t)}\bigg{]}$
$\displaystyle=p_{t}^{neu}(x,y)\mathbf{E}_{Br[0,t]}^{x\to
y}\bigg{[}e^{\int_{0}^{t}\xi^{n,k}(t-s,W_{s})ds+\frac{1}{2}\sum_{i=1}^{2k}\sum_{j=n+1}^{\infty}\big{(}\int_{0}^{t}f_{j}(t-s)e_{i}(W_{s})ds\big{)}^{2}-kt-
AL_{0}^{W}(t)-BL_{1}^{W}(t)}\bigg{]}$
$\displaystyle=p_{t}^{neu}(x,y)\mathbf{E}_{Br[0,t]}^{x\to
y}\bigg{[}e^{\int_{0}^{t}\xi^{n,k}(t-s,W_{s})ds-\frac{1}{2}\sum_{i=1}^{2k}\sum_{j=1}^{n}\big{(}\int_{0}^{t}f_{j}(t-s)e_{i}(W_{s})ds\big{)}^{2}-AL_{0}^{W}(t)-BL_{1}^{W}(t)}\bigg{]},$

where we used Gaussianity of the stochastic integrals (conditional on $W$) in
the third equality and then orthonormality of $f_{j}$ in the final equality.
The above expression is a martingale in the $n$ variable (for fixed $k$). By
the Banach-valued martingale convergence theorem [Ch, Theorem 1], we conclude
that $z^{k}$ converges as $k\to\infty$ to $z$ in
$C^{\kappa}([\epsilon,T]\times[0,1]^{2})$ for all $\kappa<1/2$ and
$\epsilon>0$, and we also conclude that (for each fixed $k\in\mathbb{N})$
$z^{n,k}$ converges to $z^{k}$ as $n\to\infty$ in
$C^{\kappa}([\epsilon,T]\times[0,1]^{2})$. All convergences are almost sure
and in $L^{p}$. Furthermore by Jensen’s inequality, any conditional
expectation necessarily contracts the $L^{p}$ norm, so we also have the
following bounds for all $p\geq 1$:

$\displaystyle\sup_{n\geq 1}\mathbb{E}[z_{t}^{n,k}(x,y)^{p}]$
$\displaystyle\leq\mathbb{E}[z_{t}^{k}(x,y)^{p}],\;\;\;\forall k\in\mathbb{N}$
(16) $\displaystyle\sup_{k\geq 1}\mathbb{E}[z_{t}^{k}(x,y)^{p}]$
$\displaystyle\leq\mathbb{E}[z_{t}(x,y)^{p}].$ (17)

Step 2. Now we establish the intermediate claim that

$\sup_{(t,x,y)\in[\epsilon,T]\times[0,1]^{2}}\mathbb{E}[z_{t}(x,y)^{-p}]<\infty.$
(18)

To do this, we will roughly follow the approach of [HL] which in turn was
inspired by [MF]. Let $e_{i},f_{j}$ be as in the previous step and define

$Q_{n,k}(s,t,x,y):=\sum_{j=1}^{n}\sum_{i=1}^{2k}f_{j}(s)f_{j}(t)e_{i}(x)e_{i}(y)=\mathbb{E}[\xi^{n,k}(t,x)\xi^{n,k}(s,y)].$

Then for continuous paths $W^{1},W^{2}$ from $[0,t]\to[0,1]$, define

$\mathscr{Q}_{n,k}(t,W^{1},W^{2}):=\int_{0}^{t}\int_{0}^{t}Q_{n,k}(t-s,t-r,W^{1}_{r},W^{2}_{s})drds.$

Also define for continuous paths $W$ the quantity

$\Gamma_{t}^{n,k}(W,\xi^{n,k}):=e^{\int_{0}^{t}\xi^{n,k}(t-s,W_{s})ds-\frac{1}{2}\mathscr{Q}_{n,k}(t,W,W)-AL_{0}^{W}(t)-BL_{1}^{W}(t)}.$

By the results of Step 1, we can write

$z^{n,k}_{t}(x,y)=p_{t}^{neu}(x,y)\mathbf{E}_{Br[0,t]}^{x\to
y}[\Gamma^{n,k}_{t}(W,\xi^{n,k})].$

We now define a measure on paths by a Radon-Nikodym derivative with respect to
the Brownian bridge measure from $x$ to $y$ reflected at the boundaries of
$[0,1]$:

$\frac{d\mathbf{P}^{n,k,t}_{\xi,x,y}}{d\mathbf{P}_{Br[0,t]}^{x\to
y}}(W)=\frac{\Gamma^{n,k}_{t}(W,\xi^{n,k})}{p_{t}^{neu}(x,y)^{-1}z^{n,k}_{t}(x,y)}.$

This measure is well-defined because $\xi^{n,k}$ is a smooth function. Next,
for all $\lambda>0$ and $(t,x,y)\in[0,\infty)\times[0,1]^{2}$ define the event

$A^{\lambda,n,k}_{t,x,y}:=\big{\\{}\xi:z_{t}^{n,k}(x,y)\geq\frac{1}{2}p_{t}^{A,B}(x,y),\;\;\;\;\big{(}\mathbf{E}_{\xi,x,y}^{n,k,t}\big{)}^{\otimes
2}[\mathscr{Q}_{n,k}(t,W^{1},W^{2})]\leq\lambda\big{\\}},$

where the expectation is with respect to two independent paths, each sampled
from the path measure $\mathbf{P}^{n,k,t}_{\xi,x,y}$ defined above. We claim
that there exists $\lambda>0$ such that

$\inf_{n,k\geq
1}\inf_{x,y\in[0,1]}\inf_{t\in[\epsilon,T]}\mathbb{P}(A^{\lambda,n,k}_{t,x,y})>0.$
(19)

To prove this, we write

$\mathbb{P}(A^{\lambda,n,k}_{t,x,y})\geq\mathbb{P}\big{(}z_{t}^{n,k}(x,y)\geq\frac{1}{2}p_{t}^{A,B}(x,y)\big{)}-\mathbb{P}(B^{\lambda,n,k}_{t,x,y}),$
(20)

where

$B^{\lambda,n,k}_{t,x,y}:=\big{\\{}\xi:z_{t}^{n,k}(x,y)\geq\frac{1}{2}p_{t}^{A,B}(x,y),\;\;\;\;\big{(}\mathbf{E}_{\xi,x,y}^{n,k,t}\big{)}^{\otimes
2}[\mathscr{Q}_{n,k}(t,W^{1},W^{2})]>\lambda\big{\\}}.$

The Paley-Zygmund inequality says that
$\mathbb{P}(F>(1-\theta)\mathbb{E}[F])\geq(1-\theta)^{2}\mathbb{E}[F]^{2}/\mathbb{E}[F^{2}]$,
so we have that

$\mathbb{P}\big{(}z_{t}^{n,k}(x,y)\geq\frac{1}{2}p_{t}^{A,B}(x,y)\big{)}\geq\frac{1}{4}\frac{p_{t}^{A,B}(x,y)^{2}}{\mathbb{E}[z^{n,k}_{t}(x,y)^{2}]}\geq\frac{1}{4}\frac{p_{t}^{A,B}(x,y)^{2}}{\mathbb{E}[z_{t}(x,y)^{2}]},$
(21)

where we use (16) and (17) in the second inequality. Now to get an upper bound
for $\mathbb{P}(B^{\lambda,n,k}_{t,x,y}),$ notice that if
$z_{t}^{n,k}(x,y)\geq\frac{1}{2}p_{t}^{A,B}(x,y)$, then
$\big{(}\mathbf{E}_{\xi,x,y}^{n,k,t}\big{)}^{\otimes
2}[\mathscr{Q}_{n,k}(t,W^{1},W^{2})]$ is equal to

$\displaystyle
p_{t}^{neu}(x,y)^{2}z_{t}^{n,k}(x,y)^{-2}\big{(}\mathbf{E}_{Br[0,t]}^{x\to
y}\big{)}^{\otimes
2}\big{[}\mathscr{Q}_{n,k}(t,W^{1},W^{2})\Gamma^{n,k}_{t}(W^{1},\xi^{n,k})\Gamma^{n,k}_{t}(W^{2},\xi^{n,k})]$
$\displaystyle\leq
4p_{t}^{neu}(x,y)^{2}p_{t}^{A,B}(x,y)^{-2}\big{(}\mathbf{E}_{Br[0,t]}^{x\to
y}\big{)}^{\otimes
2}\big{[}\mathscr{Q}_{n,k}(t,W^{1},W^{2})\Gamma^{n,k}_{t}(W^{1},\xi^{n,k})\Gamma^{n,k}_{t}(W^{2},\xi^{n,k})].$

Taking expectation $\mathbb{E}$ over the noise $\xi$ and applying Fubini gives

$\displaystyle\mathbb{E}$
$\displaystyle\bigg{[}\big{(}\mathbf{E}_{Br[0,t]}^{x\to y}\big{)}^{\otimes
2}\big{[}\mathscr{Q}_{n,k}(t,W^{1},W^{2})\Gamma^{n,k}_{t}(W^{1},\xi^{n,k})\Gamma^{n,k}_{t}(W^{2},\xi^{n,k})]\bigg{]}$
$\displaystyle\leq\big{(}\mathbf{E}_{Br[0,t]}^{x\to y}\big{)}^{\otimes
2}\big{[}\mathscr{Q}_{n,k}(t,W^{1},W^{2})\mathbb{E}[\Gamma^{n,k}_{t}(W^{1},\xi^{n,k})\Gamma^{n,k}_{t}(W^{2},\xi^{n,k})]\big{]}$
$\displaystyle=\big{(}\mathbf{E}_{Br[0,t]}^{x\to y}\big{)}^{\otimes
2}\big{[}\mathscr{Q}_{n,k}(t,W^{1},W^{2})e^{\mathscr{Q}_{n,k}(t,W^{1},W^{2})-A(L_{0}^{W^{1}}(t)+L_{0}^{W^{2}}(t))-B(L_{1}^{W^{1}}(t)+L_{1}^{W^{2}}(t))}\big{]},$

where in the last equality we use the fact that
$\sum_{i=1,2}\int_{0}^{t}\xi^{n,k}(t-s,W_{s}^{i})ds$ is a Gaussian random
variable conditional on $W^{1},W^{2},$ whose variance can be explicitly
computed in terms of $\mathscr{Q}_{n,k}$. Now use the bound $a\leq e^{a}$ and
we obtain that the last expression is bounded above by

$\big{(}\mathbf{E}_{Br[0,t]}^{x\to y}\big{)}^{\otimes
2}\big{[}e^{2\mathscr{Q}_{n,k}(t,W^{1},W^{2})-A(L_{0}^{W^{1}}(t)+L_{0}^{W^{2}}(t))-B(L_{1}^{W^{1}}(t)+L_{1}^{W^{2}}(t))}\big{]}=p_{t}^{neu}(x,y)^{-2}\mathbb{E}[z^{n,k}_{t,(\sqrt{2})}(x,y)^{2}],$

where $z_{t,(\sqrt{2})}$ denotes the solution of the stochastic heat equation
where the noise has an extra factor of $\sqrt{2}$ in the front, and
$z^{n,k}_{t,(\sqrt{2})}$ is the respective approximation given in Step 1. In
particular (16) and (17) still hold and we have
$\mathbb{E}[z^{n,k}_{t,(\sqrt{2})}(x,y)^{2}]\leq\mathbb{E}[z_{t,(\sqrt{2})}(x,y)^{2}]$
for all $n,k,x,y,t$. Summarizing these computations and applying Markov’s
inequality, we have

$\displaystyle\mathbb{P}(B^{\lambda,n,k}_{t,x,y})$
$\displaystyle\leq\mathbb{P}\bigg{(}4p_{t}^{neu}(x,y)^{2}p_{t}^{A,B}(x,y)^{-2}\big{(}\mathbf{E}_{Br[0,t]}^{x\to
y}\big{)}^{\otimes
2}\big{[}\mathscr{Q}_{n,k}(t,W^{1},W^{2})\Gamma^{n,k}_{t}(W^{1},\xi^{n,k})\Gamma^{n,k}_{t}(W^{2},\xi^{n,k})]>\lambda\bigg{)}$
$\displaystyle\leq
4p_{t}^{neu}(x,y)^{2}p_{t}^{A,B}(x,y)^{-2}\lambda^{-1}\mathbb{E}\bigg{[}\big{(}\mathbf{E}_{Br[0,t]}^{x\to
y}\big{)}^{\otimes
2}\big{[}\mathscr{Q}_{n,k}(t,W^{1},W^{2})\Gamma^{n,k}_{t}(W^{1},\xi^{n,k})\Gamma^{n,k}_{t}(W^{2},\xi^{n,k})]\bigg{]}$
$\displaystyle\leq
4p_{t}^{A,B}(x,y)^{-2}\lambda^{-1}\mathbb{E}[z_{t,(\sqrt{2})}(x,y)^{2}],$

where we remark that all instances of $p_{t}^{neu}(x,y)$ have cancelled out.
Note that $\lambda$ so far is independent of $n,k,x,y,t$, so now we exercise
our freedom to set it equal to

$\lambda:=32\bigg{(}\inf_{x,y\in[0,1]}\inf_{t\in[\epsilon,T]}p_{t}^{A,B}(x,y)^{4}\mathbb{E}[z_{t,(\sqrt{2})}(x,y)^{2}]^{-1}\mathbb{E}[z_{t}(x,y)^{2}]^{-1}\bigg{)}^{-1}.$
(22)

The infimum is strictly positive because $\mathbb{E}[z_{t}(x,y)^{2}]\leq
Ct^{-1}$ by the first bound in Theorem 2.1. With this value of $\lambda$ we
then see that

$\mathbb{P}(B^{\lambda,n,k}_{t,x,y})\leq\frac{1}{8}\frac{p_{t}^{A,B}(x,y)^{2}}{\mathbb{E}[z_{t}(x,y)^{2}]}$

uniformly over all $n,k\geq 1$, $x,y\in[0,1]$, and $t\in[\epsilon,T].$ By
combining this with (20) and (21), this shows that

$\mathbb{P}(A^{\lambda,n,k}_{t,x,y})\geq\frac{1}{8}\frac{p_{t}^{A,B}(x,y)^{2}}{\mathbb{E}[z_{t}(x,y)^{2}]}$

uniformly over all $n,k\geq 1$, $x,y\in[0,1]$, and $t\in[\epsilon,T].$ This
proves the claim (19), since by the first bound in Theorem 2.1 we have
$\mathbb{E}[z_{t}(x,y)^{2}]\leq Ct^{-1}$ with $C$ independent of $t,x,y$. With
this lower bound established on the probability of $A^{\lambda,n,k}_{t,x,y}$,
we next claim that

$z_{t}^{n,k}(x,y)\geq\frac{1}{2}p_{t}^{A,B}(x,y)e^{-\lambda^{1/2}dist_{L^{2}}(\xi,A^{\lambda,n,k}_{t,x,y})},$
(23)

for all $\lambda,t,x,y,n,k.$ In this expression we are viewing $\xi$ as lying
in some abstract Wiener space and the quantity $dist_{L^{2}}(\xi,A)$ is
defined as $\inf\\{\|\xi-\eta\|_{L^{2}([0,T]\times[0,1])}:\eta\in A\\},$ to be
understood as $+\infty$ if $\xi-\eta\notin L^{2}([0,T]\times[0,1])$ for all
$\eta$ in the abstract Wiener space. To prove this, write
$z_{t}^{n,k}(x,y;\xi)$ to emphasize the dependence on the noise. Then note
that for any $\eta\in A^{\lambda,n,k}_{t,x,y},$ we have that

$\displaystyle z_{t}^{n,k}(x,y;\xi)$
$\displaystyle=z_{t}^{n,k}(x,y;\eta)\mathbf{E}_{\eta,x,y}^{n,k,t}\bigg{[}e^{\int_{0}^{t}\big{[}\xi^{n,k}-\eta^{n,k}\big{]}(t-s,W_{s})ds}\bigg{]}$
$\displaystyle\geq\frac{1}{2}p_{t}^{A,B}(x,y)e^{\mathbf{E}_{\eta,x,y}^{n,k,t}\big{[}\int_{0}^{t}\big{[}\xi^{n,k}-\eta^{n,k}\big{]}(t-s,W_{s})ds\big{]}}.$

where we use Jensen’s inequality and the definition of
$A^{\lambda,n,k}_{t,x,y}$ in the second bound. Next we note by Cauchy-Schwarz
that if $\eta\in A^{\lambda,n,k}_{t,x,y},$ then

$\displaystyle\bigg{|}\mathbf{E}_{\eta,x,y}^{n,k,t}\big{[}$
$\displaystyle\int_{0}^{t}\big{[}\xi^{n,k}-\eta^{n,k}\big{]}(t-s,W_{s})ds\big{]}\bigg{|}=\bigg{|}\sum_{i=1}^{n}\sum_{j=1}^{2k}(\xi_{ij}-\eta_{ij})\mathbf{E}_{\eta,x,y}^{n,k,t}\bigg{[}\int_{0}^{t}f_{j}(t-s)e_{i}(W_{s})ds\bigg{]}\bigg{|}$
$\displaystyle\leq\bigg{[}\sum_{i=1}^{n}\sum_{j=1}^{2k}(\xi_{ij}-\eta_{ij})^{2}\bigg{]}^{1/2}\bigg{[}\sum_{i=1}^{n}\sum_{j=1}^{2k}\bigg{(}\mathbf{E}_{\eta,x,y}^{n,k,t}\bigg{[}\int_{0}^{t}f_{j}(t-s)e_{i}(W_{s})ds\bigg{]}\bigg{)}^{2}\bigg{]}^{1/2}$
$\displaystyle=\|\xi^{n,k}-\eta^{n,k}\|_{L^{2}([0,T]\times[0,1])}\big{(}\mathbf{E}_{\eta,x,y}^{n,k,t}\big{)}^{\otimes
2}[\mathscr{Q}_{n,k}(t,W^{1},W^{2})]^{1/2}$
$\displaystyle\leq\|\xi-\eta\|_{L^{2}([0,T]\times[0,1])}\cdot\lambda^{1/2}.$

This already gives the claim (23) by combining with the previous expression.

Now we are finally in a position to establish the claim (18) stated at the
beginning of this step. This will be done by combining (23) and (19) with the
Gaussian isoperimetric inequality. The isoperimetric inequality states that if
$A$ is a Borel measurable subset of any abstract Wiener space $(X,H,\mu)$ then
for all $t>0$ one has that $\mu(\\{\xi:dist_{H}(\xi,A)>t\\})\leq
1-\Phi\big{(}\Phi^{-1}(\mu(A))+t\big{)},$ where $\Phi$ is the cdf of a
standard Gaussian [Hai09, Theorem 3.48]. Note that this bound is uniform over
all Borel sets $A$.

Take $\lambda=\lambda(\epsilon,T)$ as in (22), and then based on this value of
$\lambda$, choose some $a=a(\epsilon,T)\in\mathbb{R}$ so that the infimum in
(19) is precisely equal to $\Phi(a)$ where $\Phi$ is the cdf of a standard
Gaussian. Finally, choose $\delta=\delta(\epsilon,T)>0$ so that
$\delta<\inf_{x,y\in[0,1]}\inf_{t\in[\epsilon,T]}\frac{1}{2}p_{t}^{A,B}(x,y)$.
Then for all $r>0$, we see by (23) that

$\displaystyle\sup_{n,k}\sup_{x,y\in[0,1]}\sup_{t\in[\epsilon,T]}\mathbb{P}(z_{t}^{n,k}(x,y)<r)$
$\displaystyle\leq\sup_{n,k}\sup_{x,y\in[0,1]}\sup_{t\in[\epsilon,T]}\mathbb{P}\big{(}dist_{L^{2}}(\xi,A^{\lambda,n,k}_{t,x,y})>\lambda^{-1/2}\log(r/\delta)\big{)}$
$\displaystyle\leq 1-\Phi\big{(}a+\lambda^{-1/2}\log(r/\delta)\big{)},$

where we use (19) and the isoperimetric inequality in the second bound. Now
since $\Phi(x)\sim 1-e^{-x^{2}/2}$ for large $x$, this establishes a tail
bound of order $e^{-c(\log r)^{2}}$. This is a stronger decay than any power
and uniform over all $n,k$, and therefore by taking a limit (first letting
$n\to\infty$ for fixed $k$, then letting $k\to\infty$ as described in Step 1)
immediately gives (18) as desired.

Step 3. In this last step we need to justify how to turn the pointwise bound
(18) into a Holder bound as in the theorem statement. To do this, note that

$\displaystyle|z_{t}(x,a)^{-1}-z_{t}(x,b)^{-1}|$
$\displaystyle=z_{t}(x,a)^{-1}z_{t}(x,b)^{-1}|z_{t}(x,a)-z_{t}(x,b)|$ (24)
$\displaystyle|z_{t}(a,y)^{-1}-z_{t}(b,y)^{-1}|$
$\displaystyle=z_{t}(a,y)^{-1}z_{t}(b,y)^{-1}|z_{t}(a,y)-z_{t}(b,y)|$ (25)
$\displaystyle|z_{t}(x,y)^{-1}-z_{s}(x,y)^{-1}|$
$\displaystyle=z_{t}(x,y)^{-1}z_{s}(x,y)^{-1}|z_{t}(x,y)-z_{s}(x,y)|$ (26)

for all $a,b,x,y,s,t.$ Consequently we have by Holder’s inequality that

$\displaystyle\mathbb{E}\big{[}|z_{t}(x,a)^{-1}-z_{t}(x,b)^{-1}|^{p}\big{]}$
$\displaystyle=\mathbb{E}\big{[}z_{t}(x,a)^{-p}z_{t}(x,b)^{-p}|z_{t}(x,a)-z_{t}(x,b)|^{p}\big{]}$
$\displaystyle\leq\mathbb{E}[z_{t}(x,a)^{-3p}]^{1/3}\mathbb{E}[z_{t}(x,b)^{-3p}]^{1/3}\mathbb{E}[|z_{t}(x,a)-z_{t}(x,b)|^{3p}]^{1/3}.$

Now for any $x,y,a,b\in[0,1]$ and any $t\in[\epsilon,T]$ we know by (18) that
$\mathbb{E}[z_{t}(x,a)^{-3p}]^{1/3}\mathbb{E}[z_{t}(x,b)^{-3p}]^{1/3}$ is
bounded by some universal constant $C=C(\epsilon,T)$. by applying (12) in
Theorem 2.1, the term $\mathbb{E}[|z_{t}(x,a)-z_{t}(x,b)|^{3p}]^{1/3}$ can be
bounded by $C|a-b|^{1/2-\delta}$ where $C$ is a universal constant. This gives
us a bound on (24) that will be conducive to applying Kolmogorov-Chentsov. To
bound (25), note that for each fixed $t$ we have
$(z_{t}(x,y))_{x,y\in[0,1]}\stackrel{{\scriptstyle
d}}{{=}}(z_{t}(y,x))_{x,y\in[0,1]}$ (by a time reversal of the noise) and so
the same bound holds for (25). Finally, to bound (26), we can use a similar
argument applying Holder’s inequality in conjuction with (18) and (12) to get
a bound of the form $C|t-s|^{1/4-\delta}.$ Consequently by Kolmogorov-Chentsov
(multiparameter version), we conclude the desired Holder continuity of
$\mathbf{z}^{-1}$ appearing in the proposition statement.

The desired bound on Holder continuity of $\mathbf{z}$ itself is similar, but
one does not even need to consider the negative moments: the argument follows
immediately from (12) together with
$(z_{t}(x,y))_{x,y\in[0,1]}\stackrel{{\scriptstyle
d}}{{=}}(z_{t}(y,x))_{x,y\in[0,1]}$. ∎

For a compact metric space $X$ we define $C(X)$ to be the Banach space of all
continuous functions $X\to\mathbb{R}.$

######  Corollary 2.5.

Let $\mathcal{X}$ be the state space defined in (6). Define the Markov
operator $\mathfrak{P}:C(\mathcal{X})\to C(\mathcal{X})$ as follows: for $F\in
C(\mathcal{X})$ and $\phi\in\mathcal{X}$ let

$\mathfrak{P}F(\phi):=\mathbb{E}_{\phi}[F([z(1,\cdot)]_{\mathcal{X}})]$ (27)

where $z$ solves (4) and $[\mu]_{\mathcal{X}}$ denotes the equivalence class
of the nonzero measure $\mu$, and the expectation on the right side is taken
with respect to the process $z$ started at any representative of the
equivalence class $\phi$. Then $\mathfrak{P}$ is globally well-defined on the
quotient space $\mathcal{X}$, and has at least one invariant measure.
Furthermore any invariant measure is supported on the space $\mathcal{Y}$
given by (7).

###### Proof.

Proposition 2.4 implies that the solution $z$ of (4) does not collapse to zero
in finite time. Thus $z(1,\cdot)$ is indeed a nonzero and non-negative
function on $[0,1]$, consequently it does indeed yield a nonzero random
measure on $[0,1]$ by identifying it with the measure
$A\mapsto\int_{A}z(1,x)dx$. The fact that $\mathfrak{P}F$ is well-defined on
equivalence classes of $\mathcal{X}$ is immediate from the fact that $cz$
solves (4) for all $c>0$, whenever $z$ solves (4), e.g. by uniqueness of mild
solutions stated in Theorem 2.1.

Note that $\mathcal{X}$ is compact, and therefore any collection of
probability measures on $\mathcal{X}$ is tight. Thus existence follows
immediately from the Krylov-Bogoliubov criterion, since we have already shown
global well-posedness of the Markov operator of $[z(t,\cdot)]$ on
$\mathcal{X}$.

The fact that any invariant measure is supported on $\mathcal{Y}$ follows from
the positivity of the solution (see Proposition 2.4) together with the
estimate (12) in Theorem 2.1. More precisely, positivity and (12) imply that
if we start from any deterministic initial data $[\mu]\in\mathcal{X}$, then
the time-one solution $[z^{\mu}(1,\cdot)]$ will be supported on $\mathcal{Y}$
almost surely. In particular, if we start the process from an invariant
measure, then the time-one solution will be in $\mathcal{Y}$ almost surely. ∎

##  3\. Support theorem and proof of Theorem 1.1

In this section we will prove Theorem 1.1, which will be reformulated as
Corollary 3.7 below. In this section the strong Feller property will be used,
so it is crucially important that the driving noise is space-time white noise
(not colored in space). First let us prove full support of the invariant
measure that was guaranteed to exist in Corollary 2.5.

######  Definition 3.1 (Continuum directed random polymer measure).

Fix $t>0$. For each realization of the space-time white noise $\xi$ on
$[0,1]\times[0,t]$, and each finite nonnegative measure $\mu$ on $[0,t]$ let
$\mathbf{P}^{\xi,\mu}_{X^{t,x}}$ be the unique path measure on $C[0,t]$ such
that for all $0\leq t_{1}<...<t_{k}\leq t$ and $x_{1},...,x_{k}\in[0,1]$ one
has

$\mathbf{P}^{\xi,\mu}_{X^{t,x}}\big{(}X_{t_{1}}\in dx_{1},...,X_{t_{k}}\in
dx_{k}\big{)}=\frac{\int_{[0,1]}z_{0,t-t_{k}}(x_{k+1},x_{k})\cdots
z_{t-t_{2},t-t_{1}}(x_{2},x_{1})z_{t-t_{1},t}(x_{1},x)\mu(dx_{k+1})}{\int_{[0,1]}z_{0,t}(x_{k+1},x)\mu(dx_{k+1})}dx_{1}\cdots
dx_{k}.$

Here $(X_{s})_{s\in[0,t]}$ denotes the canonical process on $C[0,t]$, and
$z_{s,t}(x,\bullet)$ for $t>s$ denotes the solution at time $t$ of (4) started
from $\delta_{x}$ at time $s$, all coupled to the same realization of $\xi$.
We call this measure the continuum directed random polymer measure on
$C[0,t]$, starting from $x$ at $t=0$ and finishing with with endpoint
distribution $z_{0,t}\mu/\int z_{0,t}d\mu$.

In other words, $\mathbf{P}^{\xi,\mu}_{X^{t,x}}$ is the measure induced by
leveraging the convolution property of the propagators $z_{s,t}(x,y)$ to
specify the finite dimensional distributions for each individual realization
of $\xi$, as done in [AKQ13, Definition 4.1]. We have introduced an extra time
reversal because it will be useful for our purposes. The fact that such a
measure is indeed supported on $C[0,t]$ is non-obvious and was shown in
Section 5 of [AKQ13]. On a purely formal level one has the formal Radon-
Nikodym derivative with respect to the law $\mathbf{P}_{W^{t,x}}$ of a
Brownian motion of diffusion rate 2 on $[0,t]$ started at $x$ and reflected at
the boundaries of $[0,1]$:

$\frac{d\mathbf{P}^{\xi,\mu}_{X^{t,x}}}{d\mathbf{P}_{W^{t,x}}}(W)\propto\mu(W(t))e^{-AL_{0}^{W}(t)-BL_{1}^{W}(t)+\int_{0}^{t}\xi(t-s,W(s))ds},$
(28)

where $L_{a}^{W}$ are the local times at the boundaries and $A,B$ are the
Robin boundary parameters.

######  Lemma 3.2.

Fix a deterministic continuous function $h:[0,T]\times[0,1]\to\mathbb{R}$, and
consider the mild solution of the SPDE given by

$\partial_{t}z=\partial_{x}^{2}z+zh+z\xi,\;\;\;\;\partial_{x}z(t,0)=Az(t,0),\;\;\;\;\partial_{x}z(t,1)=Bz(t,1),\;\;\;z(0,dx)=\mu(dx)$
(29)

Then for a.e $\xi\in E$ the solution $z^{h}$ can be decomposed as
$z^{h}(t,x)=z^{0}(t,x)v^{h}(t,x)$, where $z^{0}$ is just the mild solution of
(29) with $h=0$ (i.e., the solution of (4)), and

$v^{h}(t,x)=\mathbf{E}^{\xi,\mu}_{X^{t,x}}\bigg{[}\exp\bigg{(}\int_{0}^{t}h(t-s,X(s))ds\bigg{)}\bigg{]},$
(30)

where the expectation is taken with respect to the continuum directed random
polymer measure $\mathbf{P}^{\xi,\mu}_{X^{t,x}}$ on $C[0,t]$.

Formally at the level of Feynman-Kac, the decomposition claimed in Lemma 3.2
is obvious by (28) and some simple algebraic manipulation, but because of the
roughness of $\xi$ one needs some work to give a rigorous proof.

###### Proof.

First of all, we note that the mild solution $z^{h}$ can be shown to exist
using the Cameron-Martin theorem together with Theorem 2.1 (the $h=0$ case).
To rigorously establish the decomposition (30) one first proves that it is
true when $\xi$ is replaced by a noise $\xi^{\gamma}$ which is smooth in space
and white in time, in which case we have an associated decomposition
$z=z^{0,\gamma}\cdot v^{h}_{\gamma}$ for the solution of (29) where
$z^{0,\gamma}$ solves (29) with $h=0$ and $\xi$ replaced by $\xi^{\gamma}$,
and $v^{h}_{\gamma}$ is the same as in (30) except the path measure
$\mathbf{P}^{\xi,\mu}_{X^{t,x}}$ is replaced by the appropriate measure
$\mathbf{P}^{\xi,\mu}_{X^{t,x}_{\gamma}}$ associated to $z^{0,\gamma}$.

In this case the claimed decomposition is straightforward to show from
applying Feynman-Kac, because by Proposition 2.2 the density formula (28) is
actually rigorous modulo two caveats. Firstly $\mu(W(t))$ should be
interpreted in terms of a disintegration of Brownian bridge measures, so that
the endpoint distribution of the Brownian path is proportional to
$p_{t}^{neu}(x,y)\mu(dx)$ as in Proposition 2.2. Secondly $\xi(t-s,W(s))$
needs to be replaced by $\xi^{\gamma}(t-s,W(s))-C_{\gamma}$ for a large
constant $C_{\gamma}$ as described in Proposition 2.2. Then as $\gamma\to 0$
the solutions $z^{0,\gamma}(t,x)$ converge in law (in probability in the
topology of $C([\epsilon,T]\times[0,1])$ for all $T>\epsilon>0$) to
$z^{0}(t,x)$ as stated in Proposition 2.2.

Since $h$ is bounded and continuous, the path functionals
$G(X):=\exp\big{(}\int_{0}^{t}h(t-s,X(s))ds\big{)}$ are bounded and continuous
on $C[0,1]$ so it just remains to show that for a.e. fixed realization of
$\xi$, the path measures $\mathbf{P}^{\xi,\mu}_{X^{t,x}_{\gamma}}$ also
converge in law as $\gamma\to 0$, as measures on $C[0,1]$ to the path measure
$\mathbf{P}^{\xi,\mu}_{X^{t,x}}$. The finite-dimensional marginals certainly
converge, simply by convergence of the $z^{0,\gamma}$ to $z^{0}$. Thus one
only needs to establish a.s. tightness of these path measures on $C[0,1]$. For
this one can use Fatou’s lemma to note that for $\kappa<1/2$ one has

$\mathbb{E}\big{[}z^{0}(t,x)\liminf_{\gamma\to
0}\mathbf{E}^{\xi,\mu}_{X^{t,x}_{\gamma}}[\|X\|_{C^{\kappa}[0,t]}]\big{]}\leq\liminf_{\gamma\to
0}\mathbb{E}\big{[}z^{0,\gamma}(t,x)\mathbf{E}^{\xi,\mu}_{X^{t,x}_{\gamma}}[\|X\|_{C^{\kappa}[0,t]}]\big{]}=\mathbf{E}_{BM}[\|X\|_{C^{\kappa}[0,t]}]]<\infty.$

Here the outer expectation $\mathbb{E}$ is over the noise $\xi$, and
$\mathbf{E}_{BM}$ is the expectation with respect to a standard Brownian
motion of rate 2. By strict positivity of $z_{0}(t,x)$, the above bound
establishes $\xi$-a.s. finiteness of $\liminf_{\gamma\to
0}\mathbf{E}^{\xi,\mu}_{X^{t,x}_{\gamma}}[\|X\|_{C^{\kappa}[0,t]}].$ In the
last equality we are using that for each $\gamma>0$ one has

$\mathbb{E}\big{[}z^{0,\gamma}(t,x)\mathbf{E}^{\xi,\mu}_{X^{t,x}_{\gamma}}[\|X\|_{C^{\kappa}[0,t]}]\big{]}=\mathbf{E}_{BM}[\|X\|_{C^{\kappa}[0,t]}],$

which holds by [AKQ13, Lemma 4.2]. That result is for $\gamma=0$ and full
space, but the result for the Robin case and $\gamma>0$ is completely
analogous. This proves the desired a.s. tightness (at least along some random
subsequence), thus establishing the validity of the a.s. decomposition claimed
above, with $v^{h}$ given by (30). ∎

With the above preliminaries established, we can now prove our support
theorem.

######  Theorem 3.3 (Support theorem).

Fix any deterministic initial condition $[\mu]\in\mathcal{X}$. Let $z(t,x)$
denote the Itô solution of (4) started from $\mu$. Then the
$\mathcal{X}$-valued random variable $[z(1,\cdot)]$ has full support in
$\mathcal{X}$. In particular, any invariant measure for the
$\mathcal{X}$-valued Markov process $[z(t,\cdot)]$ necessarily has full
support in $\mathcal{X}.$

###### Proof.

We break the proof into three steps. In this proof, we will set the terminal
time $T=1$ and we will consider $\xi$ as living in some Banach space $E$ which
can be explicitly taken as the closure of smooth functions with respect to the
norm of some parabolic Hölder space
$\mathcal{C}^{-3/2-\kappa}_{\mathfrak{s}}.$ The exact choice of space is not
important here but what is important is that there is a continuously embedded
Hilbert space $H=L^{2}([0,1]\times[0,T])$ (the Cameron-Martin space) in $E$
with the property that if $T_{h}(\xi)=\xi+h$ then the pushforward
$T_{h}^{*}\mu$ of the law $\mu$ of $\xi$ on $E$ is absolutely continuous (in
fact equivalent) with respect to $\mu$ and one has
$\frac{d(T_{h}^{*}\mu)}{d\mu}(\xi)=e^{\langle\xi,h\rangle-\frac{1}{2}\|h\|_{H}^{2}},$
where $\langle\cdot,h\rangle$ is the stochastic integral against $\xi$.

Step 1. We claim that the law of any $z$ solving (4) under $T_{h}^{*}\mu$
equals that of the solution of

$\partial_{t}z=\partial_{x}^{2}z+zh+z\xi,\;\;\;\;\partial_{x}z(t,0)=Az(t,0),\;\;\;\;\partial_{x}z(t,1)=Bz(t,1),\;\;\;z(0,dx)=\mu(dx)$
(31)

under the original measure with the same initial condition. Here $zh$ is an
un-renormalized or “classical” product.

To prove this, note by the Cameron-Martin theorem that under $T_{h}^{*}\mu$
the random field $\eta:=\xi-h$ is a standard space-time white noise. Write the
equation in mild form as in (5), and note that the integral against
$z(s,y)\xi(ds,dy)$ can be written as an integral against
$z(s,y)\big{[}\eta(ds,dy)+h(s,y)dsdy\big{]}$, and we immediately obtain the
claim.

Step 2.  Assume for this step that $(E,H,\gamma)$ is any abstract Wiener
space, and $\mathcal{X}$ is another separable metric space. Let $\xi$ be
sampled from $\gamma$, let $F:E\to\mathcal{X}$ be any measurable function, and
let $D\subset H$ be any countable subset. Then we claim that the random set
$F(\xi+D):=\\{F(\xi+h):h\in D\\}$ is almost surely contained in the support of
$F(\xi)$.

To prove this, let $U\subset\mathcal{X}$ be the complement of the support of
$F(\xi)$. Then it is clear from the Cameron-Martin theorem that for every
$h\in H$ one has

$\gamma(\\{\xi:F(\xi+h)\in U\\})=0.$

Thus by countability of $D$ we have that

$\gamma(\\{\xi:F(\xi+h)\in U\text{ for some }h\in D\\})\leq\sum_{h\in
D}\gamma(\\{\xi:F(\xi+h)\in U\\})=0,$

which proves the claim.

This step will be used in the following form below: if there exists a
deterministic countable subset $D\subset H$ such that the random set
$F(\xi+D):=\\{F(\xi+h):h\in D\\}$ is almost surely dense in $\mathcal{X}$,
then $F(\xi)$ necessarily has full support in $\mathcal{X}$.

Step 3. We now let $F:E\to\mathcal{X}$ be the functional
$F(\xi):=[z(1,\cdot)],$ where now $\mathcal{X}$ is the space given by (6) and
$z$ solves (4) driven by $\xi$ starting from some arbitrary initial state
$[\mu]\in\mathcal{X}$ (here the “$\cdot$” is in the spatial variable). In this
case let us fix some smooth function $\varphi_{0}:[0,1]\to\mathbb{R}$, and
then define $h_{n}^{\varphi_{0}}\in H=L^{2}([0,1]\times[0,T])$ by setting

$h_{n}^{\varphi_{0}}(t,x):=n1_{[1-n^{-1},1]}(t)\varphi_{0}(x).$

In this case by (31) and Lemma 3.2, the random variable
$F(\xi+h_{n}^{\varphi_{0}})$ is given by $[z_{0}(1,x)v_{n}(1,x)]$ where
$z_{0}$ solves (4) and

$v_{n}(1,x)=\mathbf{E}^{\xi,\mu}_{X^{1,x}}\bigg{[}\exp\bigg{(}n\int_{0}^{1/n}\varphi_{0}(X(s))ds\bigg{)}\bigg{]}.$

Now, since $X^{1,x}(0)=x$ and $X^{1,x}$ is a continuous path, as $n\to\infty$
the expression inside the expectation converges to $\exp(\varphi_{0}(x)).$
Moreover the expression inside the expectation is deterministically bounded
above by $\exp(\|\varphi_{0}\|_{\infty}),$ and consequently

$\sup_{n\in\mathbb{N}}\sup_{x\in[0,1]}v_{n}(1,x)\leq\exp(\|\varphi_{0}\|_{\infty}).$

By the bounded convergence theorem we find that $v_{n}(1,x)$ converges a.s. to
$e^{\varphi_{0}(x)},$ for each fixed $x\in[0,1]$. Likewise, the same bound may
be used to conclude that
$\int_{[0,1]}v_{n}(1,x)\psi(x)dx\stackrel{{\scriptstyle
a.s.}}{{\to}}\int_{[0,1]}e^{\varphi_{0}(x)}\psi(x)dx$ for all $\psi\in
C[0,1]$.

Now we recall that $\mathcal{X}$ is a compact space, and therefore any
collection of random variables in $\mathcal{X}$ is tight, so from the
convergences stated in the previous paragraph, we can actually also conclude
that $F(\xi+h_{n}^{\varphi_{0}})=[z_{0}(1,x)v_{n}(1,x)]$ converges in
probability (with respect to the topology of $\mathcal{X}$) as $n\to\infty$ to
$[z_{0}(1,x)e^{\varphi_{0}(x)}].$ Note here that $z_{0}$ has no dependence on
$\varphi_{0}$ since only $v$ depends on $h$ in the decomposition from Step 2.

Letting $D_{0}\subset C[0,1]$ be some countable dense subset consisting of
smooth functions, we define $D:=\\{h^{\varphi}_{n}\in
H:n\in\mathbb{N},\varphi\in D_{0}\\}$. We have shown that the closure in
$\mathcal{X}$ of the random set $F(\xi+D)$ almost surely contains all elements
of the form $[z(1,\cdot)e^{\varphi}]$ with $\varphi\in D_{0}$. Since
$z(1,\cdot)$ is just a fixed positive function with no dependence on
$\varphi\in D_{0}$, letting $\varphi$ vary throughout $D_{0}$ certainly gives
a dense subset of $\mathcal{X}$, and therefore we conclude the proof of the
theorem using the result of Step 3. ∎

Although Theorem 3.3 was proved using an argument specific to the KPZ
equation, we remark that more abstract techniques such as those of [HS19]
could be used to prove a general Stroock-Varadhan type support theorem in much
greater generality, in particular characterizing the support for the entire
temporal trajectory (as opposed to just fixed time as we have done above) and
in arbitrarily strong topologies.

The above theorem is the only input to the uniqueness result that is specific
to KPZ which is needed in order to prove the uniqueness theorem. The remainder
of this section will focus on general results about strong Feller processes
with invariant measures of full support, which will be useful in concluding
the proof of the uniqueness theorem.

Recall that a Markov operator $\mathfrak{P}$ on a Polish space $\mathcal{X}$
is called strong Feller if it maps all bounded measurable functions to bounded
continuous functions. Going forward, we will work extensively with this
property. The next proposition allows us to determine when the strong Feller
property can be “upgraded” from a smaller space with a finer topology to a
larger space with a weaker topology.

######  Proposition 3.4.

Let $\mathfrak{P}$ be a Markov operator on some complete metric space
$\mathcal{X}$. Let $\mathcal{Y}$ be another complete metric space contained
densely in $\mathcal{X}$ (but with a separate metric) such that the inclusion
map $\mathcal{Y}\hookrightarrow\mathcal{X}$ is continuous and such that open
sets in $\mathcal{Y}$ are Borel measurable in $\mathcal{X}$. Suppose that the
following assumptions hold:

  * •

$\mathfrak{P}(1_{\mathcal{Y}})=1$ identically, and the Markov operator
$\bar{\mathfrak{P}}$ on $\mathcal{Y}$ defined by
$\bar{\mathfrak{P}}f=\mathfrak{P}(f1_{\mathcal{Y}})|_{\mathcal{Y}}$ is strong
Feller.

  * •

The family of probability measures $\\{\nu_{\phi}\\}_{\phi\in\mathcal{X}}$ on
$\mathcal{Y}$ defined by $\nu_{\phi}(A):=\mathfrak{P}1_{A}(\phi)$, satisfies
the following regularity condition: whenever
$\phi_{n}\stackrel{{\scriptstyle\mathcal{X}}}{{\longrightarrow}}\phi$ the we
have the weak convergence $\nu_{\phi_{n}}\to\nu_{\phi}$ as measures on
$\mathcal{Y}.$

Then $\mathfrak{P}^{2}$ is a strong Feller operator on the larger space
$\mathcal{X}$.

We remark that $\nu_{\phi}=\mathfrak{P}_{*}\delta_{\phi}$ simply equals the
law of the associated Markov process at time 1, when started from initial
state $\phi$. The condition $\mathfrak{P}(1_{\mathcal{Y}})=1$ identically is
the same as saying that the $\mathcal{X}$-valued Markov process associated to
$\mathfrak{P}$ takes values in $\mathcal{Y}$ almost surely, regardless of the
initial state. Then $\bar{\mathfrak{P}}$ is the Markov operator of the process
viewed in the state space $\mathcal{Y}$. We will apply the theorem when
$\mathcal{X}$ and $\mathcal{Y}$ are given by (6) and (7) respectively. Using a
result of [KM22], this will then allow us to extend the strong Feller property
to the the entire space $\mathcal{X}$ in Corollary 3.5 knowing that it holds
on $\mathcal{Y}$.

###### Proof.

Suppose $\phi_{n}\stackrel{{\scriptstyle\mathcal{X}}}{{\longrightarrow}}\phi.$
By hypothesis
$\int_{\mathcal{Y}}gd\nu_{\phi_{n}}\to\int_{\mathcal{Y}}gd\nu_{\phi}$ for all
bounded continuous $g:\mathcal{Y}\to\mathbb{R}.$

Now since $\mathfrak{P}(1_{\mathcal{Y}})=1$, for any bounded measurable
$f:\mathcal{X}\to\mathbb{R}$ and any $\phi\in\mathcal{X}$ we have the identity

$\mathfrak{P}^{2}f(\phi)=\int_{\mathcal{Y}}\bar{\mathfrak{P}}(f1_{\mathcal{Y}})d\nu_{\phi}.$

By the strong Feller property of $\bar{\mathfrak{P}}$ the integrand is a
bounded continuous function on $\mathcal{Y}$, and therefore we conclude that
if $\phi_{n}\stackrel{{\scriptstyle\mathcal{X}}}{{\longrightarrow}}\phi$ then

$\lim_{n\to\infty}\mathfrak{P}^{2}f(\phi_{n})=\lim_{n\to\infty}\int_{\mathcal{Y}}\bar{\mathfrak{P}}(f1_{\mathcal{Y}})d\nu_{\phi_{n}}=\int_{\mathcal{Y}}\bar{\mathfrak{P}}(f1_{\mathcal{Y}})d\nu_{\phi}=\mathfrak{P}^{2}f(\phi),$

proving the claim. ∎

######  Corollary 3.5.

The process $[z(t,\cdot)]$ where $z$ solves (4) is strong Feller on the space
$\mathcal{X}$ given by (6).

###### Proof.

In [KM22, Theorem 1.1] the strong Feller property was proved on the finer
space (7). Now thanks to the convergence statement in (13), we can immediately
apply Proposition 3.4 to “upgrade” the strong Feller property to the coarser
space $\mathcal{X}$, since we know it holds on $\mathcal{Y}$. Note here that
we are implicitly using the fact that if $f_{n},f$ are strictly positive and
continuous random functions, and if $\|f_{n}-f\|_{C^{\kappa}[0,1]}\to 0$ in
probability, then $\|\log(f_{n})-\log(f)\|_{C^{\kappa}[0,1]}\to 0$ in
probability, as is needed in (7). ∎

######  Proposition 3.6.

Let $\mathfrak{P}$ be a strong Feller Markov operator on some connected Polish
space $\mathcal{X}$. Let $\rho$ be an invariant probability measure for
$\mathfrak{P}$ with full topological support on $\mathcal{X}$. If
$(X_{n})_{n}$ is the associated Markov process started from any deterministic
initial condition in $\mathcal{X}$, and if $f:\mathcal{X}\to\mathbb{R}$ is
bounded and measurable, then as $N\to\infty$

$N^{-1}\sum_{j=1}^{N}f(X_{j})\to\int_{\mathcal{X}}f\;d\rho,\;\;\;\;a.s..$

Let $\mathcal{Y}$ be another Polish space contained in $\mathcal{X}$ (but with
a separate topology) such that the inclusion map
$\mathcal{Y}\hookrightarrow\mathcal{X}$ is continuous and such that open sets
in $\mathcal{Y}$ are Borel measurable in $\mathcal{X}$. If
$\mathfrak{P}(1_{\mathcal{Y}})=1$ identically, then $\rho(\mathcal{Y})=1$ and
moreover, the Markov semigroup $\bar{\mathfrak{P}}$ on $\mathcal{Y}$ defined
by $\bar{\mathfrak{P}}f=\mathfrak{P}(f1_{\mathcal{Y}})|_{\mathcal{Y}}$ is also
strong Feller and $\rho$ is the unique invariant measure for
$\bar{\mathfrak{P}}.$

Before proving the proposition, we remark that the above convergence holds for
every (not just $\mu$-almost every) initial condition, which is very important
because we do not assume that $\mu$ has full support in the finer space
$\mathcal{Y}$, only in the coarser space $\mathcal{X}$. The relevance of this
proposition is that (under the strong Feller assumption) even if we can prove
that the invariant measure of some Markov process (usually some SPDE) has full
support in some relatively weak topology (that of $\mathcal{X})$, then that
automatically implies uniqueness of the invariant measure for the process in
any stronger topology (that of $\mathcal{Y})$ as long as the Markov process
takes values in the stronger topological space and the open sets of the
stronger topological space are Borel measurable in the weaker topology. This
proposition therefore generalizes [HM17, Corollaries 3.8 and 3.9]. Note
however that our proposition does not necessarily imply full support of the
invariant measure on $\mathcal{Y},$ and in general this can actually be false,
but the uniqueness in $\mathcal{Y}$ still holds nonetheless (regardless of
whether or not the full support holds).

###### Proof.

Let $\rho$ be the invariant measure for $\mathfrak{P}$ with full support. By
invariance we see that

$\rho(\mathcal{Y})=\int_{\mathcal{X}}\mathfrak{P}(1_{\mathcal{Y}})\;d\rho=\int_{\mathcal{X}}1\;d\rho=1.$

The fact that $\bar{\mathfrak{P}}$ is strong Feller is immediate from the fact
that if $g:\mathcal{Y}\to\mathbb{R}$ is bounded and measurable, then
$g1_{\mathcal{Y}}$ is bounded and measurable from $\mathcal{X}\to\mathbb{R}$
(thanks to the fact that Borel subsets of $\mathcal{Y}$ are also Borel subsets
of $\mathcal{X})$. Now $\mathfrak{P}(g1_{\mathcal{Y}})$ is bounded and
continuous on $\mathcal{X}$, and since the topology of $\mathcal{Y}$ is finer
than that of $\mathcal{X}$ by assumption, the restriction
$\mathfrak{P}(g1_{\mathcal{Y}})|_{\mathcal{Y}}$ is automatically continuous
from $\mathcal{Y}\to\mathbb{R}.$

The proof of Corollary 3.9 in [HM17] shows (using connectedness of
$\mathcal{X}$ and the full support property) that $\rho$ is necessarily
ergodic thanks to the support assumption plus the strong Feller property. Thus
if $(X_{n})_{n}$ is the Markov process in $\mathcal{X}$ started from the
measure $\rho$, then $\frac{1}{N}\sum_{n=1}^{N}f(X_{n})\to\int f\;d\rho$
almost surely for all bounded measurable $f:\mathcal{X}\to\mathbb{R}$. Letting
$\mathbb{P}_{\phi}$ denote the law (on the canonical space
$\mathcal{X}^{\mathbb{N}})$ of the process $(X_{n})_{n}$ started from some
deterministic point $\phi\in\mathcal{X}$, this means that for each such $f$
one has

$\int_{\mathcal{X}}\mathbb{P}_{\phi}\bigg{(}\frac{1}{N}\sum_{n=1}^{N}f(X_{n})\to\int
f\;d\rho\bigg{)}\rho(d\phi)=1.$ (32)

Fix some bounded measurable function $f:\mathcal{X}\to\mathbb{R}$, and let
$E_{f}$ be the Borel set defined by

$E_{f}:=\bigg{\\{}\phi\in\mathcal{X}:\mathbb{P}_{\phi}\bigg{(}\frac{1}{N}\sum_{n=1}^{N}f(X_{n})\to\int
f\;d\rho\bigg{)}=1\bigg{\\}},$

so that $\rho(E_{f})=1$ by (32). Note by the strong Feller property that
$\mathfrak{P}1_{E_{f}}$ is a continuous function from
$\mathcal{X}\to\mathbb{R}.$ Moreover by invariance of $\rho$ we have that

$\int_{\mathcal{X}}\mathfrak{P}1_{E_{f}}\;d\rho=\rho(E_{f})=1,$

and therefore $\mathfrak{P}1_{E_{f}}(\phi)=1$ for $\rho$-a.e.
$\phi\in\mathcal{X}$. Since $\rho$ has full support on $\mathcal{X}$, any set
of full measure is necessarily dense in $\mathcal{X}$. Thus the continuous
function $\mathfrak{P}1_{E_{f}}$ equals 1 on a dense set, and therefore is
identically 1 on all of $\mathcal{X}$. This means that

$\mathbb{P}_{\phi}\bigg{(}N^{-1}\sum_{n=1}^{N}f(X_{n+1})\to\int
f\;d\rho\bigg{)}=1$ (33)

for every $\phi\in\mathcal{X}$ (not just on a set of full $\rho$-measure), so
that $E_{f}$ is actually the entire space $\mathcal{X}$. The fact that $\rho$
is the unique invariant measure for $\bar{\mathfrak{P}}$ on $\mathcal{Y}$ is
then immediate from (33) and the fact that $g1_{\mathcal{Y}}$ is a bounded
measurable function on $\mathcal{X}$ whenever $g$ is a bounded measurable
function on $\mathcal{Y}.$ ∎

######  Corollary 3.7.

Let $\mathcal{X}$ and $\mathcal{Y}$ be the state spaces defined in (6) and
(7), respectively. For any $A,B\in\mathbb{R}$, consider the
$\mathcal{X}$-valued Markov process $X_{A,B}(t):=[z(t,0)]$ where $z$ solves
(4). Then, regardless of whether we view the state space as $\mathcal{X}$ or
as $\mathcal{Y}$, the process $X_{A,B}$ has a unique ergodic invariant measure
$\rho_{A,B}$ which is supported on $\mathcal{Y}$. Furthermore starting from
any deterministic initial condition $[\mu]\in\mathcal{X}$, and for any bounded
Borel-measurable function $F:\mathcal{X}\to\mathbb{R}$, one has the a.s.
convergence

$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}F(X_{A,B}(n))=\int_{\mathcal{Y}}F\;d\rho_{A,B}.$

###### Proof.

Existence of an invariant measure for all $A,B\in\mathbb{R}$ follows
immediately by compactness of the state space $\mathcal{X}$ as remarked in
Corollary 2.5. The strong Feller property of the process on $\mathcal{X}$ was
shown in Corollary 3.5. The fact that $\rho_{A,B}$ has full support in
$\mathcal{X}$ follows from Theorem 3.3. Therefore we can conclude the
uniqueness and ergodic theorem on both $\mathcal{X}$ and $\mathcal{Y}$ by
applying Proposition 3.6. ∎

Let $C_{+}[0,1]$ denote all positive continuous functions on $[0,1]$. We note
that if $F:C_{+}[0,1]\to\mathbb{R}$ is any measurable functional, then it
gives rise to a family of measurable functionals from
$\mathcal{X}\to\mathbb{R}$ by setting for instance
$\bar{F}_{1}([\mu]):=F(f/f(1))$, $\bar{F}_{2}([\mu])=F(f/\int_{[0,1]}f)$,
$\bar{F}_{3}([\mu])=F(f/\|f\|_{L^{p}[0,1]})$, etc, where all of these are
understood to be zero unless the measure $\mu$ is absolutely continuous with
continuous density $f$. Another important class of measurable functions on
$\mathcal{X}$ are induced by measurable functions
$G:C^{-1/2-\kappa}[0,1]\to\mathbb{R}$ in which case we may define
$\bar{G}:\mathcal{X}\to\mathbb{R}$ given by $\bar{G}([\mu])=G(\partial_{x}\log
f),$ again understood to be zero unless $\mu$ is absolutely continuous with
strictly positive density $f\in C^{1/2-\kappa}[0,1]$. Such functionals
$\bar{G}$ are related to the so-called stochastic Burgers equation with
inhomogeneous Dirichlet boundary conditions on $[0,1].$ Consequently there is
a fairly rich class of observables for which the above ergodic theorem
applies.

Another corollary is the existence of an almost sure exponential growth rate
independent of the initial data:

######  Corollary 3.8.

Starting from any initial data $\nu$, the limit

$\lim_{N\to\infty}\frac{1}{N}\log z(N,0)$

exists and is constant almost surely. The constant equals
$\mathbb{E}_{\mathbb{P}_{\rho_{A,B}}}[\log z(1,0)-\log z(0,0)]$ and is
therefore independent of the initial data $\nu$. Here
$\mathbb{P}_{\rho_{A,B}}$ denotes the law on $\mathcal{X}^{\mathbb{N}}$ of the
process started from stationarity.

###### Proof.

Define the canonical shift map
$\theta:\mathcal{X}^{\mathbb{N}}\to\mathcal{X}^{\mathbb{N}}$ by sending
$\theta((\phi_{j})_{j})=(\phi_{j+1})_{j}.$ Let $\mathbb{P}_{\phi}$ denote the
measure on $\mathcal{X}^{\mathbb{N}}$ corresponding to the Markov process with
initial state $\phi$. An easy consequence of Corollary 3.7 is that for any
$\phi\in\mathcal{X}$ and any measurable $F\in L^{2}(\mathbb{P}_{\phi})$ that
depends on only a finite number of coordinates, one has the convergence

$\frac{1}{N}\sum_{n=1}^{N}F\theta^{n}\to\int_{\mathcal{X}^{\mathbb{N}}}F\;d\mathbb{P}_{\rho_{A,B}},\;\;\;\;\;\;\;\mathbb{P}_{\phi}\text{-}a.s..$

Letting $F(([z_{j}])_{j}):=\log z_{1}(0)-\log z_{0}(0)$ now immediately gives
the claim (note that $F$ is well defined on equivalence classes, since scaling
$z_{0}$ by some constant also scales $z_{1}$ by the same constant
$\mathbb{P}_{\phi}$-almost surely). ∎

Next let us address the question of total variation convergence.

######  Definition 3.9.

Let $\mathcal{X}$ be a Polish space. If $\mathfrak{P}$ is a Markov operator on
$\mathcal{X}$ and $\phi\in\mathcal{X}$, let
$\nu_{\phi}(A):=\mathfrak{P}1_{A}(\phi),$ in other words
$\nu_{\phi}=\mathfrak{P}_{*}\delta_{\phi}.$ Then $\mathfrak{P}$ is called
ultra Feller if $\|\nu_{\phi_{n}}-\nu_{\phi}\|_{TV}\to 0$ whenever
$\phi_{n}\stackrel{{\scriptstyle\mathcal{X}}}{{\to}}\phi$.

######  Proposition 3.10.

[Hai07, Theorem 1.6.6] Let $\mathcal{X}$ be a Polish space and let
$\mathfrak{P}$ be a strong Feller Markov operator on $\mathcal{X}$. Then
$\mathfrak{P}^{2}$ is ultra Feller.

If $\mathcal{M}_{1}(\mathcal{X})$ denotes the space of probability measures on
the Polish space $\mathcal{X}$, then we claim that the ultra Feller property
is equivalent to the adjoint operator $\mathfrak{P}_{*}$ being continuous from
the topology of weak convergence on $\mathcal{M}_{1}(\mathcal{X})$ to the
topology of total variation convergence on $\mathcal{M}_{1}(\mathcal{X})$. The
backward implication is obvious. To prove the forward implication suppose
$\gamma_{n}\in\mathcal{M}_{1}(\mathcal{X})$ converges weakly to
$\gamma\in\mathcal{M}_{1}(\mathcal{X})$. Then by Skorohod’s representation
theorem we can choose $\mathcal{X}$-valued random variables
$\boldsymbol{\phi}_{n},\boldsymbol{\phi}$ such that the law of each
$\boldsymbol{\phi}_{n}$ is $\gamma_{n}$, the law of $\boldsymbol{\phi}$ is
$\gamma$, and $\boldsymbol{\phi}_{n}\to\boldsymbol{\phi}$ almost surely in the
topology of $\mathcal{X}$. By assumption
$\|\mathfrak{P}_{*}(\delta_{\boldsymbol{\phi}_{n}}-\delta_{\boldsymbol{\phi}})\|_{TV}\to
0$ almost surely. By the bounded convergence theorem this implies that
$\mathbb{E}[\|\mathfrak{P}_{*}(\delta_{\boldsymbol{\phi}_{n}}-\delta_{\boldsymbol{\phi}})\|_{TV}]\to
0$ which is strong enough to prove the claim since for each $n\geq 1$ we have

$\|\mathfrak{P}_{*}(\gamma_{n}-\gamma)\|_{TV}=\big{\|}\mathbb{E}[\mathfrak{P}_{*}(\delta_{\boldsymbol{\phi}_{n}}-\delta_{\boldsymbol{\phi}})]\big{\|}_{TV}\leq\mathbb{E}[\|\mathfrak{P}_{*}(\delta_{\boldsymbol{\phi}_{n}}-\delta_{\boldsymbol{\phi}})\|_{TV}].$

######  Theorem 3.11 (An automatic spectral gap from unique ergodicity).

Suppose that $(\mathfrak{P}_{t})_{t\geq 0}$ is a strongly continuous semigroup
of Markov operators on a compact metric space $\mathcal{X}$. Assume that
$\mathfrak{P}_{t}$ is strong Feller for each $t>0$ and admits a unique
invariant probability measure $\rho$. Then there exist $C,c>0$ such that for
all $t\geq 0$

$\sup_{\phi\in\mathcal{X}}\big{\|}\rho-\mathfrak{P}_{t}^{*}\delta_{\phi}\big{\|}_{TV}\leq
Ce^{-ct},$

where $\mathfrak{P}_{t}^{*}$ is the adjoint Markov semigroup, and
$\delta_{\phi}$ denotes a Dirac mass at the initial condition
$\phi\in\mathcal{X}$. In particular this is true for the Markov operator of
the open KPZ equation on the space $\mathcal{X}$ given in (6).

In other words, in continuous time, compactness plus the strong Feller
property automatically allows us to upgrade unique ergodicity to geometric
ergodicity (in fact a spectral gap). The intuitive idea of the proof is that
compactness of $\mathcal{X}$ plus the strong Feller property makes the adjoint
semigroup exhibit properties that are similar to those of a finite-state
Markov chain, in particular an automatically positive spectral gap.

The reason that we are breaking our usual convention and using a continuous
time parameter here is because we need it for the proof. The discrete time
analogue is simply false: consider the simple random walk on
$\mathbb{Z}/4\mathbb{Z}$ or even $\mathbb{Z}/2\mathbb{Z}$. But also notice how
introducing exponential holding times immediately fixes the problem in either
case. It might be possible to formulate some other reasonable statement in
discrete time but it seems more difficult.

###### Proof.

We define $\mathcal{M}_{0}(\mathcal{X})$ to be the Banach space consisting of
all signed Borel measures $\gamma$ on $\mathcal{X}$ with
$\gamma(\mathcal{X})=0$, equipped with total variation norm. For a linear
operator $T:\mathcal{M}_{0}(\mathcal{X})\to\mathcal{M}_{0}(\mathcal{X}),$
define $\|T\|_{\text{op}}:=\sup_{\|\gamma\|_{TV}=1}\|T(\gamma)\|_{TV}$ to be
the operator norm. Note that $\mathfrak{P}_{t}^{*}$ is a bounded linear
operator from $\mathcal{M}_{0}(\mathcal{X})\to\mathcal{M}_{0}(\mathcal{X})$
with $\|\mathfrak{P}_{t}^{*}\|_{\text{op}}\leq 1.$

For all $\phi\in\mathcal{X}$ and $t>0$, we claim that
$\frac{1}{N}\sum_{n=1}^{N}\mathfrak{P}_{nt}^{*}\delta_{\phi}$ converges weakly
to $\rho.$ Indeed by compactness of $\mathcal{X}$, any subsequence is tight,
hence has a further subsequence that converges weakly. Any such weak limit may
be verified to be an invariant probability measure by directly applying
$\mathfrak{P}_{t}^{*}-I$ and noting that the telescoping sums cancel. Hence
any such weak limit must be equal to $\rho$. As every subsequence has a
further subsequence converging to $\rho$, the original sequence must converge
to $\rho$.

From here it is clear that
$\frac{1}{N}\sum_{n=1}^{N}\mathfrak{P}_{nt}^{*}\lambda$ converges weakly to
$\lambda(\mathcal{X})\rho$ for all nonnegative Borel measures $\lambda$ on
$\mathcal{X}.$ Consequently for all $\gamma\in\mathcal{M}_{0}(\mathcal{X})$,
we have weak convergence of $N^{-1}\sum_{n=1}^{N}\mathfrak{P}_{nt}^{*}\gamma$
to the zero measure (use the Hahn-Jordan decomposition theorem to write
$\gamma=\gamma_{+}-\gamma_{-}$ with
$\gamma_{+}(\mathcal{X})=\gamma_{-}(\mathcal{X})$, and then apply
$\frac{1}{N}\sum_{n=1}^{N}\mathfrak{P}_{nt}^{*}$ both components and take the
limit).

We claim that $\mathfrak{P}_{t}^{*}$ is a compact operator on
$\mathcal{M}_{0}(\mathcal{X})$ for each $t>0$. To prove this, first note that
the sphere of radius 2 in $\mathcal{M}_{0}(\mathcal{X})$ is compact with
respect to the topology of weak convergence. This is true by Prohorov’s
theorem since each $\gamma\in\mathcal{M}_{0}(\mathcal{X})$ with
$\|\gamma\|_{TV}=2$ is a difference of probability measures, and because
compactness of $\mathcal{X}$ ensures that any family of probability measures
is automatically tight. The ultra Feller property guarantees that
$\mathfrak{P}_{t}^{*}$ is continuous from the topology of weak convergence on
probability measures to the topology of total variation convergence on
probability measures, which proves compactness of the image under
$\mathfrak{P}_{t}^{*}$ of the sphere of radius 2 in
$\mathcal{M}_{0}(\mathcal{X})$.

For this paragraph and the next, we will work with the complexification of
$\mathfrak{P}_{t}^{*}$, which is an operator on the space
$\mathcal{M}_{0}^{\mathbb{C}}(\mathcal{X})$ of finite complex-valued Borel
measures $\gamma$ on $\mathcal{X}$ with $\gamma(\mathcal{X})=0$ (we will not
distinguish the complexified operator from the real one). The norm on
$\mathcal{M}_{0}^{\mathbb{C}}(\mathcal{X})$ is defined by
$\|\gamma\|_{TV}:=\sup\\{|\gamma(f)|:f\in C(\mathcal{X};\mathbb{C})$ and
$\sup|f|\leq 1\\}$, where the bars now denote the complex modulus and $f$ can
be a complex-valued function. Note that under this norm, the complexified
operator still satisfies $\|\mathfrak{P}_{t}^{*}\|_{\text{op}}\leq 1$ with the
same proof. The complexified operator is still compact since it can be viewed
as the direct sum of two copies of the real operator. By the spectral theory
of compact operators, the complex spectrum of $\mathfrak{P}_{t}^{*}$ consists
of zero and some countable number of eigenvalues whose only accumulation point
can be zero, and furthermore each of the eigenspaces must be finite
dimensional. Since $\|\mathfrak{P}_{t}^{*}\|_{\text{op}}\leq 1$ all
eigenvalues must have complex modulus less than or equal to 1.

We claim that 1 cannot be an eigenvalue of $\mathfrak{P}_{t_{0}}^{*}$ for any
$t_{0}>0$. Indeed if $\mathfrak{P}_{t_{0}}^{*}\gamma=\gamma,$ then
$\gamma=N^{-1}\sum_{n=1}^{N}\mathfrak{P}_{nt_{0}}^{*}\gamma$ for all $N$, and
we already know that the latter converges weakly to the zero measure as
$N\to\infty$ (that was in the case of a real-valued measure, but the complex
case follows by applying the result individually to the real and imaginary
parts of the measure). More generally we claim that $e^{i\theta}$ cannot be an
eigenvalue of $\mathfrak{P}_{t_{0}}^{*}$ for any $\theta\in(0,2\pi]$ and any
$t_{0}>0$ (this is the step where we need a continuous time parameter). To
obtain a contradiction, note that any eigenspace of $\mathfrak{P}_{t_{0}}$ is
invariant under $\mathfrak{P}_{t}$ for all $t>0$. This is because
$\mathfrak{P}_{t_{0}}$ and $\mathfrak{P}_{t}$ commute, so it follows that
$\mathfrak{P}_{t}$ sends any eigenspace of $\mathfrak{P}_{t_{0}}$ to itself.
Identify the $e^{i\theta}$-eigenspace of $\mathfrak{P}_{t_{0}}^{*}$ with
$\mathbb{C}^{d}$, then we see that $\mathfrak{P}_{t}^{*}$ are linear maps from
$\mathbb{C}^{d}\to\mathbb{C}^{d}$ that are continuous in $t\geq 0$ satisfying
$\mathfrak{P}^{*}_{t+s}=\mathfrak{P}^{*}_{t}\mathfrak{P}^{*}_{s}$ for all
$s,t\geq 0$ and $\mathfrak{P}^{*}_{0}=I_{d\times d}$, and
$\mathfrak{P}_{t_{0}}^{*}=e^{i\theta}I_{d\times d}$. A $C_{0}$-semigroup on a
finite-dimensional vector space has a generator, so we can write
$\mathfrak{P}^{*}_{t}=e^{tA}$ for some complex $d\times d$ matrix $A$. Then
$e^{(t_{0}A-i\theta I)}=I$, which implies that $t_{0}A-i\theta I$ must be
conjugate to a diagonal matrix with diagonal entries $2\pi ik_{1},...,2\pi
ik_{d}$ for some $k_{1},...,k_{d}\in\mathbb{Z}$ (by e.g. Jordan canonical
form). In particular $A$ itself is conjugate to a diagonal matrix with entries
$t_{0}^{-1}(2\pi k_{1}+\theta)i,...,t_{0}^{-1}(2\pi k_{d}+\theta)i.$ In
particular $\mathfrak{P}_{t}^{*}=e^{tA}$ has 1 as an eigenvalue whenever
$t=2\pi jt_{0}/(\theta+2\pi k_{1})$ for any $j\in\mathbb{Z}$ such that $2\pi
jt_{0}/(\theta+2\pi k_{1})>0$, which is a contradiction.

Consequently there are no eigenvalues of modulus 1, so the spectral radius of
the compact operator $\mathfrak{P}_{1}^{*}$ is strictly less than 1. By
Gelfand’s spectral radius formula, this means that
$\lim_{N\to\infty}\|\mathfrak{P}_{N}^{*}\|_{\text{op}}^{1/N}<1,$ or in other
words $\|\mathfrak{P}_{N}^{*}\|_{\text{op}}\leq Ce^{-cN}$ for some $C,c>0$
independent of $N$. Since the invariant measure satisfies
$\rho=\mathfrak{P}_{N}^{*}\rho$ for all $N$, and since
$\delta_{\phi}-\rho\in\mathcal{M}_{0}(\mathcal{X})$ for all
$\phi\in\mathcal{X}$, this exponential bound on the operator norm then clearly
gives

$\sup_{\phi\in\mathcal{X}}\big{\|}\mathfrak{P}_{N}^{*}\delta_{\phi}-\rho\big{\|}_{TV}=\sup_{\phi\in\mathcal{X}}\big{\|}\mathfrak{P}_{N}^{*}\big{(}\delta_{\phi}-\rho\big{)}\big{\|}_{TV}\leq\|\mathfrak{P}_{N}^{*}\|_{\text{op}}\cdot\sup_{\phi\in\mathcal{X}}\|\delta_{\phi}-\rho\|_{TV}\leq
Ce^{-cN}\cdot 2,$

proving the claim.

For the special case of KPZ, Corollary 3.5 shows that $\mathfrak{P}$ is strong
Feller on $\mathcal{X}$. Moreover, Corollary 3.7 clearly guarantees that
unique ergodicity holds. Strong continuity of the semigroup follows from the
Duhamel formula (5) for the Hopf-Cole transform and noting that the stochastic
part vanishes in the topology of $\mathcal{X}$ (for any initial condition) so
that only the deterministic heat flow matters when taking the $t\to 0$ limit.
∎

We remark that $e^{i\theta}$ can indeed be an eigenvalue of
$\mathfrak{P}_{1}^{*}$ on $\mathcal{M}_{0}(\mathcal{X})$ in the discrete-time
case: for example $e^{\pm 2\pi i/n}$ are eigenvalues in the case of the
deterministic Markov chain on $\mathbb{Z}/n\mathbb{Z}$ where one always moves
clockwise. We are uncertain if irrational multiples of $\pi$ are possible when
the strong Feller property holds (this was the real obstruction to the
formulation of a discrete time result).

##  4\. A coupling argument and proof of Theorem 1.2

Here we describe a different approach to proving uniqueness and geometric
ergodicity of the stationary measure for open KPZ, which relies more heavily
on the linear structure of the Hopf-Cole transform (2), and the interpretation
through polymers. A version of this method was implemented in [GK20] for the
periodic case. The methods described here still work to show uniqueness of the
invariant measure in the case of driving noises that are colored in space, so
long as they are still white in time (the only properties needed are negative
moments of the solution and the convolution property).

######  Definition 4.1.

A continuous probability kernel (CPK) on $[0,1]$ is any function
$\mathfrak{p}:[0,1]\times[0,1]\to\mathbb{R}$ that is continuous in both
variables, strictly positive, and satisfies
$\int_{[0,1]}\mathfrak{p}(x,y)dy=1$ for all $x\in[0,1]$.

We remark that any two CPKs $\mathfrak{p}_{1},\mathfrak{p}_{2}$ can be
“convolved” or “multiplied” to form another CPK

$(\mathfrak{p}_{1}\mathfrak{p}_{2})(x,y):=\int_{[0,1]}\mathfrak{p}_{1}(x,z)\mathfrak{p}_{2}(z,y)dz.$

This is an associative binary operation on CPKs, thus we may unambiguously
write $\mathfrak{p}_{1}\cdots\mathfrak{p}_{N}.$ We have the following general
result which essentially gives total variation bounds of exponential form for
such products of CPKs. The main idea is that the Doeblin-type conditions
implicit in the definition of the CPK can be bootstrapped to give nice bounds.

######  Lemma 4.2.

Let $\\{\mathfrak{p}_{n}\\}_{n\in\mathbb{N}}$ be any collection of CPKs, and
define $\mathfrak{q}_{N}:=\mathfrak{p}_{1}\cdots\mathfrak{p}_{N}$. Then for
all $\delta>0$ one has

$\sup_{x,y\in[0,1]}\|\mathfrak{q}_{N}(x,\cdot)-\mathfrak{q}_{N}(y,\cdot)\|_{L^{1}[0,1]}\leq
2(1-\delta)^{J_{N}(\delta)},$

where

$J_{N}(\delta):=\\#\\{k\leq
N:\inf_{x,y\in[0,1]}\mathfrak{p}_{k}(x,y)>\delta\\}.$

###### Proof.

Let $A:=\\{k\in\mathbb{N}:\inf_{x,y\in[0,1]}\mathfrak{p}_{k}(x,y)>\delta\\}.$
For $n\in A$ we define

$\tilde{p}_{n}(x,y):=\frac{\mathfrak{p}_{n}(x,y)-\delta}{1-\delta}.$

Next we will construct a Markov chain $(S_{n},T_{n})_{n\geq 0}$ on $[0,1]^{2}$
whose marginals at time $N$ have law $\mathfrak{q}_{N}(x,\cdot)$ and
$\mathfrak{q}_{N}(y,\cdot)$. Define $S_{0}=x$ and $T_{0}=y$. Inductively
assuming $(S_{k},T_{k})_{k\leq n}$ have already been defined, we choose
$(S_{n+1},T_{n+1})$ according to four different cases as follows:

  * •

if $S_{n}=T_{n}$ then sample $S_{n+1}=T_{n+1}$ according to
$\mathfrak{p}_{n}(S_{n},\cdot).$

  * •

if $n\notin A$ and $S_{n}\neq T_{n}$ then independently sample $S_{n+1}$ and
$T_{n+1}$ from $\mathfrak{p}_{n}(S_{n},\cdot)$ and
$\mathfrak{p}_{n}(T_{n},\cdot)$ respectively.

  * •

if $n\in A$ and $S_{n}\neq T_{n}$, flip an independent coin that is heads with
probability $\delta$:

$\bullet$ if heads, then sample $S_{n+1}=T_{n+1}$ uniformly at random from
$[0,1]$.

$\bullet$ if tails, then independently sample $S_{n+1}$ and $T_{n+1}$
according to $\tilde{p}_{n}(S_{n},\cdot)$ and $\tilde{p}_{n}(T_{n},\cdot)$
respectively.

One may check that both $S_{n}$ and $T_{n}$ are Markov chains whose transition
kernel at step $n$ is precisely $\mathfrak{p}_{n}$, and consequently $S_{n}$
has law $\mathfrak{q}_{N}(x,\cdot)$ and $T_{n}$ has law
$\mathfrak{q}_{N}(y,\cdot)$. Furthermore the trajectories of $S,T$ coincide as
soon as the first heads appears. The probability of exceeding some
deterministic time $N$ before reaching a heads is equal to
$(1-\delta)^{J_{N}}.$

From this coupling, we have shown that

$\sup_{x,y}\|\mathfrak{q}_{N}(x,\cdot)-\mathfrak{q}_{N}(y,\cdot)\|_{TV}\leq
2(1-\delta)^{J_{N}}.$

For absolutely continuous measures, the total variation is just the $L^{1}$
norm, i.e.,

$\|\mathfrak{q}_{N}(x,\cdot)-\mathfrak{q}_{N}(y,\cdot)\|_{TV}=\|\mathfrak{q}_{N}(x,\cdot)-\mathfrak{q}_{N}(y,\cdot)\|_{L^{1}[0,1]}.$

∎

As an interesting corollary of the previous lemma, note that if
$\mathfrak{p}_{n}$ are random CPKs which are chosen IID from some probability
distribution on the space of CPKs, then clearly $J_{N}(\delta)$ is distributed
as Binomial$(N,\epsilon)$ for some choice of $\delta,\epsilon>0$ and therefore
the above lemma shows the exponential convergence

$\mathbb{E}[\sup_{x,y}\|\mathfrak{q}_{N}(x,\cdot)-\mathfrak{q}_{N}(y,\cdot)\|_{L^{1}}]\leq
Ce^{-cN}$

for some constants $c,C$ depending on the distribution of $\mathfrak{p}_{1}$
but not $N$. Then using Borel-Cantelli and Markov’s inequality, this
expectation bound also implies almost-sure statements such as

$\sup_{x,y}\|\mathfrak{q}_{N}(x,\cdot)-\mathfrak{q}_{N}(y,\cdot)\|_{L^{1}}\to
0,\;\;\;\;\;a.s..$

This will be the main idea going forward, however the IID case is very simple
and we will actually need to apply the theorem in a more complicated
situation.

######  Theorem 4.3 (One-force one-solution principle).

Fix a space-time white noise on $(-\infty,0]\times[0,1]$ and let
$z_{N}^{\mu},z_{N}^{\nu}:[-N,0]\times[0,1]\to\mathbb{R}$ denote the solutions
to open KPZ driven by $\xi$ started from any deterministic nonnegative Borel
measures $\mu,\nu$ at time $t=-N$. Then there exist $C,c>0$ such that for all
$N\geq 1$ one has

$\mathbb{E}\bigg{[}\sup_{[\mu],[\nu]\in\mathcal{X}}d_{\mathcal{X}}\big{(}[z_{N}^{\mu}(0,\cdot)],[z_{N}^{\nu}(0,\cdot)]\big{)}\bigg{]}\leq
Ce^{-cN}.$

Here $\mathcal{X}$ is the space defined in (6).

###### Proof.

Fix a realization of $\xi$ on $(-\infty,0]\times[0,1]$, and let $z_{s,t}(x,y)$
denote the propagators of the stochastic heat equation for $s<t\leq 0$ and
$x,y\in[0,1]$. Fix some positive continuous deterministic
$f:[0,1]\to\mathbb{R}_{+}.$ Then define for $n\geq 0$

$\mathfrak{p}_{n}^{f}(x,y):=\frac{z_{-n-1,-n}(x,y)\int_{[0,1]}z_{-n,0}(y,a)f(a)da}{\int_{[0,1]}z_{-n-1,0}(x,a)f(a)da},$

where $z_{0,0}$ is understood to be $\delta_{0}$ so the integral in the
numerator for $n=0$ is understood to be $f(y)$. Note by Proposition 2.3 that
$\mathfrak{p}_{n}^{f}$ is indeed a CPK. Then for $N\geq 1$ let
$\mathfrak{q}_{N}^{f}:=\mathfrak{p}_{N-1}^{f}\cdots\mathfrak{p}_{0}^{f},$ so
that

$\mathfrak{q}_{N}^{f}(x,y)=\frac{z_{-N,0}(x,y)f(y)}{\int_{[0,1]}z_{-N,0}(x,a)f(a)da}.$

For $\delta>0$ and $n\geq 0$ we define $\Omega_{n}$ to be the event

$\displaystyle\big{\\{}\inf_{u,x,y\in[0,1]}$ $\displaystyle
z_{-n-1,-n}(x,y)\wedge z_{-n,1-n}(y,u)\wedge
z_{-n-1,1-n}(x,u)>\delta\big{\\}}$
$\displaystyle\cap\big{\\{}\sup_{u,x,y\in[0,1]}z_{-n-1,-n}(x,y)\vee
z_{-n,1-n}(y,u)\vee z_{-n-1,1-n}(x,u)<\delta^{-1}\big{\\}}$

From now onward, fix some $\delta>0$ (small enough) so that the event
$\Omega_{n}$ has strictly positive probability $\epsilon>0$ (note that
$\epsilon$ does not depend on $n$ by temporal stationarity of the noise
$\xi$). Note foremost that $\Omega_{n}$ is measurable with respect to the
noise $\xi$ restricted to $[-n-2,-n]\times[0,1]$. Consequently
$\\{\Omega_{n}\\}_{n\in 2\mathbb{N}}$ are independent events. Furthermore on
$\Omega_{n}$ we have the bound

$\displaystyle\mathfrak{p}_{n}^{f}(x,y)$
$\displaystyle=\frac{z_{-n-1,-n}(x,y)\int_{[0,1]}z_{-n,0}(y,a)f(a)da}{\int_{[0,1]}z_{-n-1,0}(x,a)da}$
$\displaystyle>\frac{\delta\int_{[0,1]}z_{-n,0}(x,a)da}{\int_{[0,1]}z_{-n-1,0}(y,a)f(a)da}$
$\displaystyle=\frac{\delta\int_{[0,1]}\big{(}\int_{[0,1]}z_{-n,1-n}(y,u)z_{1-n,0}(u,a)du\big{)}f(a)da}{\int_{[0,1]}\big{(}\int_{[0,1]}z_{-n-1,1-n}(x,u)z_{1-n,0}(u,a)du\big{)}f(a)da}$
$\displaystyle=\frac{\delta\int_{[0,1]}z_{-n,1-n}(y,u)\big{(}\int_{[0,1]}z_{1-n,0}(u,a)f(a)da\big{)}du}{\int_{[0,1]}z_{-n-1,1-n}(x,u)\big{(}\int_{[0,1]}z_{1-n,0}(u,a)f(a)da\big{)}du}$
$\displaystyle>\frac{\delta\int_{[0,1]}\delta\big{(}\int_{[0,1]}z_{1-n,0}(u,a)f(a)da\big{)}du}{\int_{[0,1]}\delta^{-1}\big{(}\int_{[0,1]}z_{1-n,0}(u,a)f(a)da\big{)}du}$
$\displaystyle=\delta^{3}.$

Note that this bound holds uniformly over all $x,y\in[0,1]$ and $f\in
C_{+}[0,1]$. In the notation of Lemma 4.2 this means that

$J_{N}(\delta^{3})\geq\sum_{n\leq N/2}1_{\Omega_{2n}}.$

Therefore by Lemma 4.2 we find that

$\sup_{\begin{subarray}{c}x,\bar{x}\in[0,1]\\\ f\in
C_{+}[0,1]\end{subarray}}\|\mathfrak{q}_{N}^{f}(x,\cdot)-\mathfrak{q}_{N}^{f}(\bar{x},\cdot)\|_{L^{1}[0,1]}<(1-\delta^{3})^{J_{N}(\delta^{3})}\leq(1-\delta^{3})^{\sum_{n\leq\lfloor
N/2\rfloor}1_{\Omega_{2n}}}.$

But the $\Omega_{2n}$ are independent, so the random variable
$\sum_{n\leq\lfloor N/2\rfloor}1_{\Omega_{n}}$ is distributed as
Binomial$(\lfloor N/2\rfloor,\epsilon).$ Thus there exist constants $C,c>0$
independent of $N$ such that

$\mathbb{E}[(1-\delta^{3})^{\sum_{n\leq\lfloor
N/2\rfloor}1_{\Omega_{2n}}}]<Ce^{-cN}.$

So far this shows that

$\mathbb{E}\bigg{[}\sup_{\begin{subarray}{c}x,\bar{x}\in[0,1]\\\ f\in
C_{+}[0,1]\end{subarray}}\int_{[0,1]}\bigg{|}\frac{z_{-N,0}(x,y)}{\int_{[0,1]}z_{-N,0}(x,a)f(a)da}-\frac{z_{-N,0}(\bar{x},y)}{\int_{[0,1]}z_{-N,0}(\bar{x},a)f(a)da}\bigg{|}f(y)dy\bigg{]}<Ce^{-cN}.$

Now notice that for any $f,g\in C_{+}[0,1]$ we can write $g=(g/f)\cdot f$ so
that

$\displaystyle\sup_{x,\bar{x}\in[0,1]}\int_{[0,1]}$
$\displaystyle\bigg{|}\frac{z_{-N,0}(x,y)}{\int_{[0,1]}z_{-N,0}(x,a)f(a)da}-\frac{z_{-N,0}(\bar{x},y)}{\int_{[0,1]}z_{-N,0}(\bar{x},a)f(a)da}\bigg{|}g(y)dy$
$\displaystyle\leq\|g/f\|_{C[0,1]}\sup_{x,\bar{x}\in[0,1]}\int_{[0,1]}\bigg{|}\frac{z_{-N,0}(x,y)}{\int_{[0,1]}z_{-N,0}(x,a)f(a)da}-\frac{z_{-N,0}(\bar{x},y)}{\int_{[0,1]}z_{-N,0}(\bar{x},a)f(a)da}\bigg{|}f(y)dy.$

This will be the key estimate going forward, and from now onwards it is mostly
a matter of elementary algebraic manipulations. For $M>0$ (large) define

$\mathscr{C}_{M}:=\\{(f,g)\in C_{+}[0,1]\times
C_{+}[0,1]:\|g/f\|_{C[0,1]}+\|f/g\|_{C[0,1]}<M\\}.$

Then, because of the factor $\|g/f\|_{C[0,1]}$ appearing in the bound above,
we see that

$\mathbb{E}\bigg{[}\sup_{\begin{subarray}{c}x,\bar{x}\in[0,1]\\\
(f,g)\in\mathscr{C}_{M}\end{subarray}}\int_{[0,1]}\bigg{|}\frac{z_{-N,0}(x,y)}{\int_{[0,1]}z_{-N,0}(x,a)f(a)da}-\frac{z_{-N,0}(\bar{x},y)}{\int_{[0,1]}z_{-N,0}(\bar{x},a)f(a)da}\bigg{|}g(y)dy\bigg{]}\leq
CMe^{-cN}.$

By a time reversal of the noise we have for each $N$ the distributional
equality $(z_{-N,0}(x,y))_{x,y\in[0,1]}\stackrel{{\scriptstyle
d}}{{=}}(z_{-N,0}(y,x))_{x,y\in[0,1]}$, so the preceding expectation bound can
be rewritten as

$\mathbb{E}\bigg{[}\sup_{\begin{subarray}{c}x,\bar{x}\in[0,1]\\\
(f,g)\in\mathscr{C}_{M}\end{subarray}}\int_{[0,1]}\bigg{|}\frac{z_{-N,0}(y,x)}{\int_{[0,1]}z_{-N,0}(a,x)f(a)da}-\frac{z_{-N,0}(y,\bar{x})}{\int_{[0,1]}z_{-N,0}(a,\bar{x})f(a)da}\bigg{|}g(y)dy\bigg{]}\leq
CMe^{-cN}.$

Let $z_{-N,0}^{f}(x):=\int_{[0,1]}z_{-N,0}(a,x)f(a)da$ which is the solution
at time 0 of the stochastic heat equation started from initial condition $f$
at time $-N$. Notice that

$\displaystyle\bigg{|}\frac{z_{-N,0}^{g}(x)}{z_{-N,0}^{f}(x)}-\frac{z_{-N,0}^{g}(\bar{x})}{z_{-N,0}^{f}(\bar{x})}\bigg{|}$
$\displaystyle=\bigg{|}\int_{[0,1]}\bigg{(}\frac{z_{-N,0}(y,x)}{\int_{[0,1]}z_{-N,0}(a,x)f(a)da}-\frac{z_{-N,0}(y,\bar{x})}{\int_{[0,1]}z_{-N,0}(a,\bar{x})f(a)da}\bigg{)}g(y)dy\bigg{|}$
$\displaystyle\leq\int_{[0,1]}\bigg{|}\frac{z_{-N,0}(y,x)}{\int_{[0,1]}z_{-N,0}(a,x)f(a)da}-\frac{z_{-N,0}(y,\bar{x})}{\int_{[0,1]}z_{-N,0}(a,\bar{x})f(a)da}\bigg{|}g(y)dy,$

therefore we see from the previous bound that

$\mathbb{E}\bigg{[}\sup_{\begin{subarray}{c}x,\bar{x}\in[0,1]\\\
(f,g)\in\mathscr{C}_{M}\end{subarray}}\bigg{|}\frac{z_{-N,0}^{g}(x)}{z_{-N,0}^{f}(x)}-\frac{z_{-N,0}^{g}(\bar{x})}{z_{-N,0}^{f}(\bar{x})}\bigg{|}\bigg{]}\leq
CMe^{-cN}.$

Setting $\bar{x}=0$ then implies that

$\mathbb{E}\bigg{[}\sup_{(f,g)\in\mathscr{C}_{M}}\bigg{\|}\frac{z^{g}_{-N,0}}{z^{f}_{-N,0}}-\frac{z^{g}_{-N,0}(0)}{z^{f}_{-N,0}(0)}\bigg{\|}_{C[0,1]}\bigg{]}\leq
CMe^{-cN}.$

Next notice that

$\bigg{|}\frac{\int z^{g}_{-N,0}}{\int
z^{f}_{-N,0}}-\frac{z^{g}_{-N,0}(0)}{z^{f}_{-N,0}(0)}\bigg{|}=\frac{\big{|}\int
z^{f}_{-N,0}\big{(}\frac{z^{g}_{-N,0}}{z^{f}_{-N,0}}-\frac{z^{g}_{-N,0}(0)}{z^{f}_{-N,0}(0)}\big{)}\big{|}}{\int
z^{f}_{-N,0}}\leq\bigg{\|}\frac{z^{g}_{-N,0}}{z^{f}_{-N,0}}-\frac{z^{g}_{-N,0}(0)}{z^{f}_{-N,0}(0)}\bigg{\|}_{C[0,1]}.$

The last two convergences imply

$\mathbb{E}\bigg{[}\sup_{(f,g)\in\mathscr{C}_{M}}\bigg{\|}\frac{z^{g}_{-N,0}}{z^{f}_{-N,0}}-\frac{\int
z^{g}_{-N,0}}{\int z^{f}_{-N,0}}\bigg{\|}_{C[0,1]}\bigg{]}\leq CMe^{-cN}.$
(34)

Note that $f$ and $g$ are symmetric in the definition of $\mathscr{C}_{M}$,
and therefore we also have

$\mathbb{E}\bigg{[}\sup_{(f,g)\in\mathscr{C}_{M}}\bigg{\|}\frac{z^{f}_{-N,0}}{z^{g}_{-N,0}}-\frac{\int
z^{f}_{-N,0}}{\int z^{g}_{-N,0}}\bigg{\|}_{C[0,1]}\bigg{]}\leq CMe^{-cN}.$
(35)

For all bounded measurable functions $\varphi:[0,1]\to\mathbb{R}$ one has that

$\displaystyle\bigg{|}\frac{\int z^{g}_{-N,0}\varphi}{\int
z^{g}_{-N,0}}-\frac{\int z^{f}_{-N,0}\varphi}{\int z^{f}_{-N,0}}\bigg{|}$
$\displaystyle=\bigg{|}\frac{\int
z^{f}_{-N,0}\varphi\big{(}\frac{z^{g}_{-N,0}}{z^{f}_{-N,0}}-\frac{\int
z^{g}_{-N,0}}{\int z^{f}_{-N,0}}\big{)}}{\int z^{g}_{-N,0}}\bigg{|}$
$\displaystyle\leq\|\varphi\|_{L^{\infty}[0,1]}\frac{\int z_{-N,0}^{f}}{\int
z^{g}_{-N,0}}\bigg{\|}\frac{z^{g}_{-N,0}}{z^{f}_{-N,0}}-\frac{\int
z^{g}_{-N,0}}{\int z^{f}_{-N,0}}\bigg{\|}_{C[0,1]}.$

The left side is symmetric in $f$ and $g$, thus we also have

$\bigg{|}\frac{\int z^{g}_{-N,0}\varphi}{\int z^{g}_{-N,0}}-\frac{\int
z^{f}_{-N,0}\varphi}{\int
z^{f}_{-N,0}}\bigg{|}\leq\|\varphi\|_{L^{\infty}[0,1]}\frac{\int
z_{-N,0}^{g}}{\int
z^{f}_{-N,0}}\bigg{\|}\frac{z^{f}_{-N,0}}{z^{g}_{-N,0}}-\frac{\int
z^{f}_{-N,0}}{\int z^{g}_{-N,0}}\bigg{\|}_{C[0,1]}.$

Note that $\min\big{\\{}\frac{\int z_{-N,0}^{g}}{\int z^{f}_{-N,0}},\frac{\int
z_{-N,0}^{f}}{\int z^{g}_{-N,0}}\big{\\}}\leq 1$, and therefore the last two
expressions combined give

$\bigg{|}\frac{\int z^{g}_{-N,0}\varphi}{\int z^{g}_{-N,0}}-\frac{\int
z^{f}_{-N,0}\varphi}{\int
z^{f}_{-N,0}}\bigg{|}\leq\|\varphi\|_{L^{\infty}[0,1]}\max\bigg{\\{}\bigg{\|}\frac{z^{g}_{-N,0}}{z^{f}_{-N,0}}-\frac{\int
z^{g}_{-N,0}}{\int
z^{f}_{-N,0}}\bigg{\|}_{C[0,1]},\bigg{\|}\frac{z_{-N,0}^{f}}{z^{g}_{-N,0}}-\frac{\int
z^{f}_{-N,0}}{\int z^{g}_{-N,0}}\bigg{\|}_{C[0,1]}\bigg{\\}}.$

Combining this with (34) and (35), and taking a sup over
$\|\varphi\|_{L^{\infty}}\leq 1$, gives us that

$\mathbb{E}\bigg{[}\sup_{\begin{subarray}{c}(f,g)\in\mathscr{C}_{M}\\\
\|\varphi\|_{L^{\infty}[0,1]}\leq 1\end{subarray}}\bigg{|}\frac{\int
z^{g}_{-N,0}\varphi}{\int z^{g}_{-N,0}}-\frac{\int z^{f}_{-N,0}\varphi}{\int
z^{f}_{-N,0}}\bigg{|}\bigg{]}\leq CMe^{-cN},$

or in other words

$\mathbb{E}\bigg{[}\sup_{(f,g)\in\mathscr{C}_{M}}\bigg{\|}\frac{z^{g}_{-N,0}}{\int
z^{g}_{-N,0}}-\frac{z^{f}_{-N,0}}{\int
z^{f}_{-N,0}}\bigg{\|}_{L^{1}[0,1]}\bigg{]}\leq CMe^{-cN}.$ (36)

Since the $L^{1}$ bounded below by that of $\mathcal{X}$ (recall we assumed in
(6) that $d_{\text{Proh}}(\mu,\nu)\leq\|\mu-\nu\|_{TV}$), this actually
implies

$\mathbb{E}\bigg{[}\sup_{(f,g)\in\mathscr{C}_{M}}d_{\mathcal{X}}\big{(}[z^{g}_{-N,0}],[z^{f}_{-N,0}]\big{)}\bigg{]}\leq
CMe^{-cN}.$ (37)

Now we are in a position to finally prove the theorem. Let
$\mathcal{M}_{1}[0,1]$ denote the set of probability measures on $[0,1]$. Let
$N$ be fixed henceforth (but note that all constants are still independent of
$N$). Let $z_{-N,-n}^{\mu}=\int_{[0,1]}z_{-N,-n}(x,\cdot)\mu(dx)\in
C_{+}[0,1]$ denote the solution at time $-n$ of the stochastic heat equation
started from $\mu$ at initial time $-N$. For $M>0$ let $\Theta_{M}$ be the
event $\\{(z_{-N,1-N}^{\mu},z_{-N,1-N}^{\nu})\in\mathscr{C}_{M}$ for all
$\mu,\nu\in\mathcal{M}_{1}[0,1]\\}$, and we let
$\Gamma_{M}:=\Theta_{M}\backslash\Theta_{M-1}$. Notice by the Markov property
of the stochastic heat equation (looking at the solution from time $N-1$
onwards) that

$1_{\Gamma_{M}}\cdot\sup_{[\mu],[\nu]\in\mathcal{X}}d_{\mathcal{X}}\big{(}[z^{\mu}_{-N,0}],[z^{\nu}_{-N,0}]\big{)}\leq
1_{\Gamma_{M}}\cdot\sup_{(f,g)\in\mathscr{C}_{M}}d_{\mathcal{X}}\big{(}[z^{f}_{1-N,0}],[z^{g}_{1-N,0}]\big{)}.$

Note that for all $M$ the random variable $1_{\Gamma_{M}}$ is measurable with
respect to the noise on $[-N,1-N]$, whereas the random variable
$\sup_{(f,g)\in\mathscr{C}_{M}}d_{\mathcal{X}}\big{(}[z^{f}_{1-N,0}],[z^{g}_{1-N,0}]\big{)}$
is measurable with respect to the noise on $[1-N,0]$, therefore the two are
independent and so

$\displaystyle\mathbb{E}\bigg{[}1_{\Gamma_{M}}\cdot\sup_{[\mu],[\nu]\in\mathcal{X}}d_{\mathcal{X}}\big{(}[z^{\mu}_{-N,0}],[z^{\nu}_{-N,0}]\big{)}\bigg{]}$
$\displaystyle\leq\mathbb{P}(\Gamma_{M})\mathbb{E}\bigg{[}\sup_{(f,g)\in\mathscr{C}_{M}}d_{\mathcal{X}}\big{(}[z^{g}_{1-N,0}],[z^{f}_{1-N,0}]\big{)}\bigg{]}$
$\displaystyle\leq\mathbb{P}(\Gamma_{M})\cdot CMe^{-c(N-1)},$ (38)

where we applied (37). Now we let $\Gamma_{\infty}$ be the complement of
$\bigcup_{M}\Gamma_{M}=\bigcup_{M}\Theta_{M}$ and we obtain that

$\displaystyle\mathbb{E}\bigg{[}\sup_{[\mu],[\nu]\in\mathcal{X}}d_{\mathcal{X}}\big{(}[z^{\mu}_{-N,0}],[z^{\nu}_{-N,0}]\big{)}\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}1_{\bigcup_{M\in\mathbb{N}\cup\\{\infty\\}}\Gamma_{M}}\cdot\sup_{[\mu],[\nu]\in\mathcal{X}}d_{\mathcal{X}}\big{(}[z^{\mu}_{-N,0}],[z^{\nu}_{-N,0}]\big{)}\bigg{]}$
$\displaystyle\leq\sum_{M\in\mathbb{N}\cup\\{\infty\\}}\mathbb{E}\bigg{[}1_{\Gamma_{M}}\cdot\sup_{[\mu],[\nu]\in\mathcal{X}}d_{\mathcal{X}}\big{(}[z^{\mu}_{-N,0}],[z^{\nu}_{-N,0}]\big{)}\bigg{]}$
$\displaystyle\stackrel{{\scriptstyle\eqref{ref1}}}{{\leq}}Ce^{-cN}\sum_{M\in\mathbb{N}\cup\\{\infty\\}}M\mathbb{P}(\Gamma_{M})$
$\displaystyle\leq
Ce^{-cN}\sum_{M\in\mathbb{N}}M\mathbb{P}(\Theta_{M-1}^{c}),$

where $\Theta_{M}^{c}$ denotes the complement. Therefore we just need to
estimate the sum over $M$ (and in particular show that it is finite). To do
that, we see that

$\sup_{\begin{subarray}{c}\mu\in\mathcal{M}_{1}[0,1]\\\
y\in[0,1]\end{subarray}}z^{\mu}_{-N,1-N}(y)=\sup_{\begin{subarray}{c}\mu\in\mathcal{M}_{1}[0,1]\\\
y\in[0,1]\end{subarray}}\int_{[0,1]}z_{-N,1-N}(x,y)\mu(dx)\leq\sup_{x,y\in[0,1]}z_{-N,1-N}(x,y),$
$\inf_{\begin{subarray}{c}\mu\in\mathcal{M}_{1}[0,1]\\\
x\in[0,1]\end{subarray}}z^{\mu}_{-N,1-N}(y)=\inf_{\begin{subarray}{c}\mu\in\mathcal{M}_{1}[0,1]\\\
y\in[0,1]\end{subarray}}\int_{[0,1]}z_{-N,1-N}(x,y)\mu(dx)\geq\inf_{x,y\in[0,1]}z_{-N,1-N}(x,y),$

and therefore

$\Theta_{M}\subset\\{M^{-1/2}\leq\inf_{x,y\in[0,1]}z_{-N,1-N}(x,y)\leq\sup_{x,y\in[0,1]}z_{-N,1-N}(x,y)\leq
M^{1/2}\\}.$

Therefore by taking complements and applying Markov’s inequality with a sixth
moment bound, we find that

$\displaystyle\mathbb{P}(\Theta_{M}^{c})\leq\frac{\mathbb{E}[\sup_{x,y\in[0,1]}z_{0,1}(x,y)^{6}]}{M^{3}}+\frac{\mathbb{E}\big{[}\big{(}\inf_{x,y\in[0,1]}z_{0,1}(x,y)\big{)}^{-6}\big{]}}{M^{3}}.$

where we used the fact that $z_{-N,1-N}\stackrel{{\scriptstyle
d}}{{=}}z_{0,1}.$ That the above moments are finite follows from Proposition
2.4. This is enough to prove convergence of the above sum over $M$, proving
the theorem. ∎

######  Remark 4.4.

Define $\bar{\mathcal{X}}$ to be the space consisting of all finite
nonnegative $f\in L^{1}[0,1]$, modulo the relation $f\sim cf$ for $c>0$, and
equipped with the complete metric

$d_{\bar{\mathcal{X}}}\big{(}[f],[g]\big{)}:=\bigg{\|}\frac{f}{\|f\|_{L^{1}}}-\frac{g}{\|g\|_{L^{1}}}\bigg{\|}_{L^{1}}.$

Note by (36) that in (37) we may replace $d_{\mathcal{X}}$ by
$d_{\bar{\mathcal{X}}}$ and the statement is still true. Therefore the
remainder of the proof still goes through with $\mathcal{X}$ replaced by
$\bar{\mathcal{X}}$, which then gives that

$\mathbb{E}\bigg{[}\sup_{[\mu],[\nu]\in\mathcal{X}}d_{\bar{\mathcal{X}}}\big{(}[z_{N}^{\mu}(0,\cdot)],[z_{N}^{\nu}(0,\cdot)]\big{)}\bigg{]}\leq
Ce^{-cN}.$

This gives a one-force-one-solution principle with respect to a stronger
topology. This was the choice of topology in [GK20]. In fact one has the
following corollary in the much stronger Hölder norm.

######  Corollary 4.5 (Synchronization in Hölder norm).

Fix a space-time white noise on $(-\infty,0]\times[0,1]$ and let
$z_{N}^{\mu},z_{N}^{\nu}:[-N,0]\times[0,1]\to\mathbb{R}$ denote the solutions
to the stochastic heat equation (4) driven by $\xi$ started from any
deterministic $\mu,\nu\in\mathcal{X}$ at time $t=-N$. Then one has that

$\sup_{[\mu],[\nu]\in\mathcal{X}}d_{\mathcal{Y}}\big{(}[z_{N}^{\mu}(0,\cdot)],[z_{N}^{\nu}(0,\cdot)]\big{)}\to
0,\;\;\;\;\;almost\;surely.$

Here $\mathcal{Y}$ is the space defined in (7).

We remark that this proves [KM22, Conjecture 1.1] in a much stronger form,
namely with a supremum over all possible initial data.

###### Proof.

Let $\mathcal{M}_{1}[0,1]$ denote the set of probability measures on $[0,1]$.
Let $d_{\text{Proh}}$ denote the Prohorov metric on $\mathcal{M}_{1}[0,1]$.