# KPZ fluctuations in the planar stochastic heat equation

Jeremy Quastel  Alejandro Ramírez  Bálint Virág

###### Abstract

We use a version of the Skorokhod integral to give a simple and rigorous
formulation of the Wick-ordered (stochastic) heat equation with planar white
noise, representing the free energy of an undirected random polymer. The
solution for all times is expressed as the $L^{1}$ limit of a martingale given
by the Feyman-Kac formula and defines a randomized shift, or Gaussian
multiplicative chaos. The fluctuations far from the centre are shown to be
given by the one-dimensional KPZ equation.

###### Contents

  1. 1 Introduction and main results
    1. 1.1 History
  2. 2 Skorokhod integral
    1. 2.1 White noise
    2. 2.2 Definition of the integral
    3. 2.3 Projections
    4. 2.4 Examples of Skorokhod integrals
    5. 2.5 Classical Skorokhod integral
  3. 3 Solving Wick-ordered heat equations
    1. 3.1 Projections of the equation and their solution
    2. 3.2 Solving the planar Wick-ordered heat equation
    3. 3.3 The SHE in one dimension with space-time noise
  4. 4 Randomized shifts
    1. 4.1 Convergence in $L^{1}$
    2. 4.2 Properties of randomized shifts
    3. 4.3 Existence of the randomized shift
    4. 4.4 Properties of $\sum m_{j}m_{j}^{\prime}$.
    5. 4.5 Convergence of randomized shifts partition functions
    6. 4.6 The polymer measure
  5. 5 Intersection local time
    1. 5.1 Intersection local time for random measures
    2. 5.2 Moment bounds
    3. 5.3 Brownian intersection local time
  6. 6 Exponential moments off a small set for planar Brownian motion
    1. 6.1 Exponential moments for a short time
    2. 6.2 Fixed drift
    3. 6.3 Uniform exponential moments for large drift
  7. 7 Proofs of the main theorems
    1. 7.1 Proof of Theorems 2 and 3
    2. 7.2 Proof of Theorem 4
    3. 7.3 Proof of Theorem 6, tightness and polymer limits

##  1 Introduction and main results

Although the KPZ universality class should describe random planar geometry,
until recently models shown to be universal have all been directed, missing
crucial symmetries. In this article we study the planar Wick-ordered heat
equation

Figure 1: Solution of the planar Wick-ordered heat equation at fixed time
started from $\delta_{0}$

$\partial_{t}u=\tfrac{1}{2}\Delta u+u\,\xi,\qquad u(\cdot,0)=\varsigma$ (1)

where $\xi$ is a planar white noise that does not depend on time (Figure 1).
The precise definition uses the mild form (3) of the heat equation, in which
the white noise term is given by a Skorokhod integral, a close relative of the
Itô integral that does not need a specific time direction. The term ”Wick-
ordered ” refers to the fact that the solutions of such equations are given
in terms of Gaussian exponentials with quadratic correction.

The solution $u(x,t)$ was known to exist and be expressed in terms of a chaos
series, but only up to a critical time, see Nualart and Zakai (1989) and Hu
(2002), after which the $L^{2}$ norm blows up, and the chaos expansion fails
to converge. The blow-up time, $t_{c}$, which coincides with the largest time
for which the mutual intersection local time of a pair of independent standard
planar Brownian motions has a rate one exponential moment, is known exactly.
It is half the optimal constant in the Gagliardo-Nirenberg-Sobolev inequality,
see Chen (2004) and Bass and Chen (2004).

In this article we extend these results in several ways. First, we use an
elementary version of the Skorokhod integral to define the solution (1) for
all times, including $t>t_{c}$. We give a construction of $u$ as a randomized
shift, or as the free energy of an undirected polymer in a random environment.
This holds for all times as an $L^{1}$ functional of the white noise, and
coincides for $t<t_{c}$ with the chaos series. Let $p$ denote the planar heat
kernel,

$p(x,t)=\tfrac{1}{2\pi t}\exp\\{-|x|^{2}/2t\\}$ (2)

and $(\Xi,\mathcal{F},P)$ be a probability space containing a Gaussian
sequence that we will use to define our planar white noise and Skorokhod
integral, _planar_ in this article always referring to $\mathbb{R}^{2}$. We
have the following definition.

######  Definition 1.

Let $\varsigma$ be a finite measure on $\mathbb{R}^{2}$. A measurable function
$u:(0,T]\times\mathbb{R}\times\Xi\to[0,\infty)$ is a solution of the planar
Wick-ordered heat equation (1) with initial condition $\varsigma$ if for
almost all $(x,t)\in(0,T]\times\mathbb{R}^{2}$ the random function of $y$,

$\mathcal{K}_{x,t}(u)(y)=\int_{0}^{t}p(x-y,t-s)\,u(y,s)ds,$

is Skorokhod-integrable (see Definition 7) and satisfies

$u(x,t)=\int p(x-y,t)d\varsigma(y)+\int{\raisebox{-8.50006pt}{\scriptsize\bf
o}}\hskip 6.99997pt\mathcal{K}_{x,t}(u)~{}\xi.$ (3)

By linearity, it suffices to understand Dirac $\delta$ initial conditions. The
explicit solution is given as follows.

######  Theorem 2.

With $\varsigma=\delta_{0}$, the planar Wick-ordered heat equation has the
following unique solution. Let ${\rm e}_{j}$ be bounded functions forming an
orthonormal basis of $L^{2}(\mathbb{R}^{2})$, let $\xi_{j}=\langle{\rm
e}_{j},\xi\rangle$, let $B$ be a planar Brownian bridge from $0$ to $x$ in
time $t$ defined on a probability space $(\Omega,\mathcal{G},Q)$ independent
of $\xi$. Then

$u(x,t)=p(x,t)\lim_{n\to\infty}Z_{n},\quad\\!Z_{n}=E_{Q}\exp\bigg{\\{}\sum_{j=1}^{n}m_{j}\xi_{j}-\tfrac{1}{2}\sum_{j=1}^{n}m_{j}^{2}\bigg{\\}},\quad\\!m_{j}=\int_{0}^{t}{\rm
e}_{j}(B(s))\,ds.$ (4)

$Z_{n}$ is a uniformly integrable martingale, so it converges almost surely
and in $L^{1}(P)$. For $t<t_{c}$, it converges in $L^{2}(P)$. The solution
does not depend on the choice of basis ${\rm e}_{j}$.

The choice of the Skorokhod integral is motivated by the interpretation of (1)
as a polymer free energy. By the Feynman-Kac formula one expects the solution
of the stochastic heat equation (1) to have a representation

$u(x,t)=\lim_{n\to\infty}E_{Q}\exp\bigg{\\{}\int_{0}^{t}\xi_{[n]}(B(s))ds-\operatorname{norm}(n,\omega)\bigg{\\}}p(x,t).$
(5)

where $\xi_{[n]}$ is a mollified version of the noise, and
$\operatorname{norm}$ is some normalization. Our solution (4) is exactly in
this form, with $\xi_{[n]}=\sum_{j=1}^{n}\xi_{j}{\rm e}_{j}$, a Gaussian noise
with $n$ degrees of freedom. We will use this mollification throughout the
paper.

We expect the random polymer measure $M_{\xi}$ to be the a measure on
continuous paths $[0,t]\to\mathbb{R}^{2}$ written as

$M_{\xi}=\lim_{n\to\infty}\exp\bigg{\\{}\int_{0}^{t}\xi_{[n]}(B(s))ds-\operatorname{norm}(n,\omega)ds\bigg{\\}}Q,\qquad
Q:\mbox{ Brownian bridge law.}$ (6)

We might write integral as $\langle\xi_{[n]},X\rangle$ where
$X(A)=\int_{0}^{t}\mathbf{1}(B(s)\in A)ds$ is the occupation measure of the
Brownian bridge up to time $t$. Since $X$ is singular with respect to Lebesgue
measure, $\langle\xi_{[n]},X\rangle$ will not have a limit.

However, we expect that the $L^{2}$ norm $\langle X_{n},X_{n}\rangle$ of the
mollified version $X_{n}=\sum_{j=1}^{n}{\rm e}_{j}\langle X,{\rm
e}_{j}\rangle$ converges, after subtracting a normalizing constant, to twice
the two-dimensional self-intersection local time $\langle X_{n},X_{n}\rangle-
E_{Q_{BM}}\langle X_{n},X_{n}\rangle\to 2\gamma_{t}$. Here $Q_{BM}$ signifies
that the normalization uses Brownian motion from $0$ rather than Brownian
bridge from $0$ to $x$. Such results are shown in Varadhan (1969), Le Gall
(1994) for certain mollifications.

If we take expectation of $u(x,t)$ over the randomness of the white noise, we
get

$E_{P}E_{Q_{BM}}e^{\langle\xi_{[n]},X\rangle}=E_{Q_{BM}}E_{P}e^{\langle\xi,X_{n}\rangle}=E_{Q_{BM}}e^{\tfrac{1}{2}\langle
X_{n},X_{n}\rangle}$ (7)

which suggests that we could use
$\operatorname{norm}(n,\omega)=E_{Q_{BM}}\langle X_{n},X_{n}\rangle/2$. This
breaks the semigroup property slightly, but the problem can be fixed by adding
a $t$-dependent term to get $\operatorname{norm}(n,\omega)=E_{Q_{BM}}\langle
X_{n},X_{n}\rangle/2-\frac{1}{2\pi}t\log t$, see Example 53.

This choice corresponds to renormalizing the solution of the equation
$\partial_{t}u_{n}=\tfrac{1}{2}\Delta u_{n}+u_{n}\xi_{[n]}$ as
$e^{-E_{Q_{BM}}\langle X_{n},X_{n}\rangle/2+\frac{1}{2\pi}t\log t}u_{n}$ to
obtain a limit. The result is called PAM in the literature, for parabolic
Anderson model, see, for example Hairer and Labbé (2015). PAM satisfies the
usual semigroup property. Note however, that from the above construction the
expectation of the polymer measure over the noise $\xi$ is _not_ the original
Brownian bridge $Q$, but instead the re-weighted bridge measure
$e^{\gamma_{t}+\frac{1}{2\pi}t\log t}Q$. So $M_{\xi}$ is a randomized version
of the $e^{\gamma_{t}+\frac{1}{2\pi}t\log t}Q$, instead of $Q$.

Starting from the physical assumption that the polymer measure should be a
randomized version of the original Brownian bridge, one is led to renormalize
by the self-intersections themselves instead of just their expectations. This
corresponds to renormalizing Gaussian exponentials as

$\exp\\{W-\tfrac{1}{2}E_{P}W^{2}\\}.$ (8)

Our normalization is essentially that, up to the $t$-dependent factor: we use
$\operatorname{norm}(n,\omega)=\langle X_{n},X_{n}\rangle/2$ instead of
$E_{Q_{BM}}\langle X_{n},X_{n}\rangle/2-\frac{1}{2\pi}t\log t$. This leads to
(4), and, remarkably, to the Skorokhod interpretation of the planar SHE. The
price is that the resulting solution of (1) is no longer a semigroup, since
the Brownian motions weighted by their renormalized self-intersection local
time no longer have the Markov property.

In summary, one has two choices: keep the expected path measure as Brownian
bridge, or keep the semigroup property. As in the Itô-Stratonovich dichotomy,
the choice depends on which symmetries are more intrinsic to the scientific
problem at hand. One advantage of the Skorokhod approach is that it leads to a
simple construction of PAM as a re-weighting of the paths in this measure by
the exponential of self-intersection local time, see Example 53.

The Skorokhod integral also has some pleasant computational properties. To
solve the planar Wick-ordered heat equation, and prove Theorem 2, we use the
fact that projections of this equation to a finite-dimensional Gaussian space
satisfy a version of (3), a key property of the Skorokhod integral. These
projections are in fact deterministic finite-dimensional linear PDEs, that can
be solved explicitly via the Feynman-Kac formula. The solutions are given by
$p(x,t)Z_{n}(x,t)$ as in Theorem 2, and they are real-analytic in the noise
coordinates.

To understand the convergence of the projections, we think of the sequence
$m_{j}$ as a randomized shift. A general theory around this notion was
developed by Shamov (2016) in studying Gaussian multiplicative chaos. We
develop a simple version of this theory, reviewing and establishing existence
and convergence theorems along the way.

The Gaussian multiplicative chaos in this setting is the random polymer
measure, so we obtain a rigorous version of the polymer measure implied in (5)
for free.

######  Theorem 3 (GMC representation).

With the notation of Theorem 2, the law of the marginal law of $\xi+m$ is
absolutely continuous with respect to the law of $\xi$ and the Radon-Nikodym
derivative is given by the partition function $Z$. The (unnormalized) random
polymer is the unique $P$-random measure $M_{\xi}$ on $\Omega$ satisfying

$E_{P}E_{M_{\xi}}F(\xi,\omega)=E_{P\times Q}F(\xi+m(\omega),\omega).$ (9)

for all bounded measurable functions $F:\Xi\times\Omega\to\mathbb{R}$.

In fact, we can write

$M_{\xi}=\lim_{n\to\infty}e^{\sum_{j=1}^{n}m_{j}(\omega)\xi_{j}-\tfrac{1}{2}\sum_{j=1}^{n}m_{j}(\omega)^{2}}Q$

in $P$-probability with respect to the weak topology of measures on path
space.

The key technical input to Theorems 2 and 3 is to show uniform integrability
of $Z_{n}$. For this we use that, even for $t>t_{c}$, off sets of arbitrarily
small probability, the mutual intersection local time of planar Brownian
motion has exponential moments.

The solution of the one-dimensional stochastic heat equation with space-time
white noise is given by the same simple construction. Consequences of the
construction are a new criterion (Proposition 46) for convergence in the class
of such models and the apparently new observation (13) that the shift
representation holds for the KPZ equation and the continuum random directed
polymer (CDRP). Indeed, while the full polymer measures were studied before in
a mollified setting (see, for example, Mukherjee et al. (2016), Bröker and
Mukherjee (2019)), only the partition function or the endpoint law were
considered without mollification. We will use the randomized shift
representation of the full polymer measure in the proof of Theorems 4 and 6.

Recall the Kardar-Parisi-Zhang equation,

$\partial_{t}h=\tfrac{1}{2}(\partial_{x}h)^{2}+\tfrac{1}{2}\partial_{x}^{2}h+\xi,$
(10)

where $x\in\mathbb{R}$, $t\geq 0$ and $\xi$ is white noise in one space and
one time dimensions. The solution of this equation is defined through the
Hopf-Cole transformation $h(x,t)=\log z(x,t)$ that turns it into the one-
dimensional multiplicative stochastic heat equation

$\partial_{t}z=\tfrac{1}{2}\partial_{x}^{2}z+\xi z,\qquad
z(0,\cdot)=\delta_{0}.$ (11)

Let

$q(x,t)=\tfrac{1}{\sqrt{2\pi t}}e^{-x^{2}/2t}$ (12)

be the one-dimensional heat kernel.

######  Theorem 4.

The Wick-ordered version, Definition 29, of the one-dimensional stochastic
heat equation (11) with space-time noise has the following unique solution:

$z(x,t)=q(x,t)\lim_{n\to\infty}E_{Q}\exp\bigg{\\{}\sum_{j=1}^{n}m_{j}\xi_{j}-\tfrac{1}{2}\sum_{j=1}^{n}m_{j}^{2}\bigg{\\}},\quad
m_{j}=\int_{0}^{t}{\rm e}_{j}(b(s),s)s.$

Here $b$ is a one-dimensional Brownian bridge from $0$ to $x$ in time $t$. The
limit exists in $L^{2}(P)$ and coincides with the Itô solution of (11). The
corresponding polymer measure $M_{\xi}$ coincides with the continuum directed
polymer constructed in Alberts et al. (2014). It has the shift representation

$E_{P}E_{M_{\xi}}F(\xi,\omega)=E_{P\times Q}F(\xi+m(\omega),\omega).$ (13)

Next, we show that on appropriate scales, the solution $u$ of the planar Wick-
ordered heat equation (1) has KPZ fluctuations. In particular, we prove the
following.

######  Theorem 5.

For any $t>0$ and $a\in\mathbb{R}$ as $N\to\infty$, with
$\varsigma=\delta_{0}$ we have

$P(u((0,N^{3/2}t),Nt)\times Ne^{N^{2}t/2}\sqrt{2\pi t}\leq a)\to
F_{\operatorname{\it KPZ}}(t,a),$ (14)

where $F_{\operatorname{\it KPZ}}(t,a)=P(z(0,t)\leq a)$ are the KPZ crossover
distributions, which have a determinantal representation (see, for example,
Amir et al. (2011)).

We also have a process-level version of Theorem 5. To understand the almost
sure dependence on instances of white noise in the next theorem, couple them
as

$\xi_{N}(\cdot,\cdot)=N\xi(N^{-1/2}\cdot,N^{-3/2}\cdot).$ (15)

The law of $\xi_{N}$ does not depend on $N$, but now these white noises are
defined on the same probability space $(\Xi,\mathcal{F},P)$. We introduce
extra parameters to both equations,

$\partial_{t}u=\tfrac{1}{2}(\nu\partial_{x}^{2}+\partial_{y}^{2})u+\beta
u\,\xi_{N},\qquad\partial_{t}z=\tfrac{1}{2}\nu\partial_{x}^{2}z+\beta z\,\xi,$
(16)

and write $u_{\nu,\beta,N}(x_{0},y_{0};x,y;t)$ to be the solution with initial
condition $\delta_{(x_{0},y_{0})}$ Similarly, write
$z_{\nu,\beta}(x_{0},t_{0};x,t)$ for the solution started from
$\delta_{x_{0}}$ at time $t_{0}$. Note that for $u$, the noise $\xi_{N}$ is
used as $\xi_{N}(x,y)$ and for $z$ the noise $\xi$ means $\xi(x,t)$.

######  Theorem 6.

Let $\rho,\nu,\beta,s>0$ and $x,y,t\in\mathbb{R}$. Let ${\rm p}=(x,t)$ and
${\rm q}=(y,t+s)$. Let ${\rm p}_{N}=(N^{1/2}x,N^{3/2}t)$ and let ${\rm
q}_{N}=(N^{1/2}y,N^{3/2}(t+s))$. As $N\to\infty$, we have

$u_{\nu,\beta,N}({\rm p}_{N};{\rm q}_{N};N\rho s)\times\sqrt{2\pi\rho
s}Ne^{N^{2}s/(2\rho)}\to z_{\nu\rho,\beta}({\rm p};{\rm q})\qquad\mbox{ in
}L^{1}(\Xi,\mathcal{F},P).$ (17)

Moreover, the polymer measure $M_{N}$ on paths from ${\rm p}_{N}$ to ${\rm
q}_{N}$ in time $N\rho s$ corresponding to $Z_{N}$ converges to the
1+1-dimensional continuum directed random polymer measure (CDRP, the polymer
measure corresponding to $z$) $M$ on paths $b$ between space-time points ${\rm
p}$ and ${\rm q}$

$\displaystyle\Big{(}(N^{-1/2}$ $\displaystyle B_{N}(N\rho
r)_{1},N^{-3/2}B_{N}(N\rho r)_{2}),\;r\in[0,s]\Big{)}\mbox{ under $M_{N}$}$
$\displaystyle\;\to\;\Big{(}(b(t+r),t+r),\;r\in[0,s]\Big{)}\mbox{ under
$M$,}\qquad\mbox{in $P$-probability.}$

Since (16) is convergence in $L^{1}$, and not just in law, it holds jointly in
all seven parameters $\beta,\nu,\rho,x,y,s,t$. For example, with all
parameters except $y$ fixed, Theorem 6 implies convergence of finite
dimensional distributions to those of the top line of the KPZ line ensemble.

As a result, in the double limit when $t\to\infty$ after the first limit, the
planar Wick-ordered heat equation converges to the Airy process, see Quastel
and Sarkar (2020), Virág (2020). By linearity, Theorem 6 implies convergence
for general initial conditions often considered in KPZ. The KPZ fixed point
appears in the double limit.

The Feynman-Kac representation makes the proof of Theorem 6 transparent and
explains the scaling as well.

###### Proof of Theorem 6.

Fix an orthonormal basis of $L^{2}(\mathbb{R}^{2})$ consisting of bounded
continuous functions ${\rm e}_{j}$, and let

${\rm e}_{N,j}(x,y):=\frac{1}{N}{\rm e}_{j}(N^{-1/2}x,N^{-3/2}y).$

This, together with our choice of the noise coupling (15) is compatible with
the space-time scaling. It is chosen so that the coordinate representation
$\xi_{j}=\langle{\rm e}_{N,j},\xi_{N}(\cdot)\rangle=\langle{\rm
e}_{j},\xi(\cdot)\rangle$ does not depend on $N$.

We will solve for $u$ in the basis ${\rm e}_{N,j}$. Theorem 2 shows that the
solution is basis independent, and, with parameters added in a straightforward
way, we can write

$\displaystyle u({\rm p}_{N},$ $\displaystyle{\rm q}_{N},N\rho s)=p({\rm
q}_{N}-{\rm p}_{N},N\rho s)\lim_{n\to\infty}Z_{N,n},$ $\displaystyle\quad
Z_{N,n}=E_{Q}\exp\bigg{\\{}\sum_{j=1}^{n}m_{N,j}\xi_{j}-\tfrac{1}{2}\sum_{j=1}^{n}m_{N,j}^{2}\bigg{\\}},\quad
m_{N,j}=\beta\int_{0}^{N\rho s}{\rm e}_{N,j}(B_{N}(r))\,dr,$

where $B_{N}$ is a two-dimensional Brownian bridge ${\rm p}_{N}\to{\rm q}_{N}$
in time $N\rho s$ with covariance matrix $\operatorname{diag}(\nu,1)$. We
couple the bridges on a single probability space $Q$ by writing the limiting
spatial and temporal directions separately:

$B_{N}(N\rho r)=(N^{1/2}b_{\rm sp}(t+r),N^{1/2}b_{\rm
ti}(t+r)+(t+r)N^{3/2}),\qquad r\in[0,s],$

where $b_{\rm sp},b_{\rm ti}$ are independent 1-dimensional Brownian bridges
run from time $t$ to $t+s$ from $x$ to $y$ and from $0$ to $0$, and having
variances $\nu\rho$ and $\rho$ respectively. It is convenient to rewrite

$m_{N,j}=\beta\int_{0}^{s}{\rm e}_{j}(b_{\rm sp}(t+r),b_{\rm
ti}(t+r)/N+t+r)\,dr$

as the change-of-variable and scaling factors $N$ cancel. Then for each $j$,

$\lim_{N\to\infty}m_{N,j}=m_{j}=\beta\int_{0}^{s}{\rm e}_{j}(b_{\rm
sp}(t+r),t+r)\,dr,\qquad Q-\mbox{a.s.}$

and by Theorem 4, with parameters added in a straightforward way, we have

$z_{\nu\rho,\beta}({\rm p},{\rm q})=q({\rm q}-{\rm
p})\lim_{n\to\infty}E_{Q}\exp\left(\sum_{j=1}^{n}m_{j}\xi_{j}-\frac{1}{2}m_{j}^{2}\right).$

This establishes what the limit should be, including matching the parameters.
We justify the exchange of limits in $n$ and $N$ and complete the proof in
Section 7.3. ∎

The key technical input for Theorem 6 is again a mutual intersection local
time estimate for a Brownian bridge. This time, we need a _uniform_ bound on
exponential moments off a small set for large $t$ and distant endpoint.
Fortunately, the large drift in bridges to far away points makes them
intersect less and allows us to establish such a bound.

###  1.1 History

Essentially all earlier work on exact KPZ fluctuations has been on directed
models. An exception is the breakthrough work on the boundary fluctuations of
lozenge tilings of polygons Aggarwal and Huang (2021). These models, however,
are still non-intersecting line ensembles, and the challenge there is to
localize previous methods.

In terms of underlying polymer paths, the technical difficulty is to show that
self-intersections do not affect the large-scale behaviour. Previously the
only tool was to identify a regeneration structure (for example, Ioffe and
Velenik (2012)). Besides leading to non-explicit coefficients defined in terms
of the regeneration times, such methods failed to identify universal
fluctuations except in some trivial (Gaussian) situations.

The present paper treats the undirected two dimensional continuous case,
represented by the stochastic heat equation and its underlying polymer. There
is a choice of interpretation of the stochastic integral in the Duhamel form
of the equation, and here we study the model resulting from the Skorokhod
interpretation. The Skorokhod integral was introduced by Skorohod (1975).
Presentations in Nualart (2006), Janson (1997a) and Hairer (2021) are highly
recommended. Gaveau and Trauber (1982) identified it as the adjoint of the
Malliavin derivative, and it is often defined that way. This makes sense for
$L^{p}$, $p>1$, as the Malliavin derivative is closable on the duals $L^{q}$,
$1<q<\infty$. The technical challenge in our work is the necessity to extend
to appropriate $L^{1}$ integrands so that the integration in the Duhamel form
of the stochastic heat equation (1) makes sense. This is achieved by the use
of martingale theory and an appropriately checkable condition for uniform
integrability.

The martingale approximations take the same form as approximations to the
Gaussian multiplicative chaos studied by Shamov (2016). In particular, this
gives a representation of the solution as the Radon-Nikodym derivative of a
shift of the underlying white noise, in our case by the occupation measure of
the underlying paths. Such representations were used in the mollified setting
by, for example, Mukherjee et al. (2016) and Bröker and Mukherjee (2019).

The technical input for the uniform integrability of the approximating
martingales are exponential moments of the mutual intersection local times off
a small set. Geman et al. (1984), Le Gall (1994) Chen (2004) and Bass and Chen
(2004) are recommended references for the mutual intersection local time and
finiteness of rate one exponential moments without the cutoff.

##  2 Skorokhod integral

###  2.1 White noise

Let $(R,\mathcal{B},\mu)$ be a measure space. We will always assume
$L^{2}(R,\mathcal{B},\mu)$ is separable, with orthonormal basis $\\{{\rm
e}_{j}\\}_{j=1,2,\ldots}$. White noise defined on a probability space
$(\Xi,\mathcal{F},P)$, is a linear isometry $\xi:f\mapsto\langle f,\xi\rangle$
from $L^{2}(R,\mathcal{B},\mu)$ to $L^{2}(\Xi,\mathcal{F},P)$ such that
$\langle f,\xi\rangle$, $f\in L^{2}(R,\mathcal{B},\mu)$ form a mean zero
Gaussian family. The isometry property says that

$E_{P}[\langle f,\xi\rangle\langle g,\xi\rangle]=\langle
f,g\rangle_{L^{2}(R,\mathcal{B},\mu)}$ (18)

for all $f,g\in L^{2}(R,\mathcal{B},\mu)$. The bracket notation $\langle
f,\xi\rangle$ suggests but does not rigorously mean an
$L^{2}(R,\mathcal{R},\mu)$ inner product, since $\xi$ cannot be realized in
$L^{2}(R,\mathcal{B},\mu)$.

The white noise can be constructed in the following elementary way, which we
use repeatedly. Define

$\xi_{j}=\langle{\rm e}_{j},\xi\rangle.$ (19)

From the above definition, these are independent Gaussians with mean $0$ and
variance $1$ and for $f\in L^{2}(R,\mathcal{B},\mu)$ with
$f=\sum_{j=1}^{\infty}f_{j}{\rm e}_{j}$ with $f_{j}=\langle f,{\rm
e}_{j}\rangle$, we can realize

$\langle f,\xi\rangle=\sum_{j=1}^{\infty}f_{j}\xi_{j}$ (20)

which converges in $L^{2}(\Xi,\mathcal{F},P)$.

We will often think of functions, measures and noises in terms of their
representations $f_{j}=\langle f,{\rm e}_{j}\rangle$, $\mu_{j}=\langle\mu,{\rm
e}_{j}\rangle$ and $\xi_{j}=\langle\xi,{\rm e}_{j}\rangle$ in the orthonormal
basis, and sometimes refer to these as _coordinate representation_.

Therefore in this paper we will take $(\Xi,\mathcal{F},P)$ be a probability
space given by an independent standard Gaussian sequence:
$\Xi=\mathbb{R}^{\mathbb{N}}$, $\mathcal{F}$ is the Borel $\sigma$-algebra on
product space, and $P$ is independent standard Gaussian product measure. It
comes equipped with a natural filtration

$\mathcal{F}_{n}=\sigma(\xi_{1},\ldots,\xi_{n}).$ (21)

We will mostly be considering _planar white noise_ , which is the case
$R=\mathbb{R}^{2}$, $\mathcal{B}=$ Borel, $\mu=$ Lebesgue measure.

###  2.2 Definition of the integral

The Skorokhod integral is a notion of integral against white noise that does
not need a time ordering. So it is particularly suited to problems involving
higher dimensional time-independent white noise. The definition is elegant,
and the concept is powerful, but it is hard to quickly see the motivation
behind it. We first give a definition, and then provide motivation.

######  Definition 7.

Let $G$ be a $\mathbb{R}^{D}$-valued random variable, where
$D=\\{1,2,\ldots\\}$ is finite or infinite. Let $\xi_{n},n\in D$ be
independent standard Gaussian random variables. Assume that $E|G_{i}|<\infty$
for each $i\in\mathbb{N}$. We say that $G$ is Skorokhod integrable if there
exists

$S=\int{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptG\;\xi\qquad\mbox{ in }L^{1}$

called the Skorokhod integral of $G$ such that for every $n\in D$ and every
bounded differentiable $F:\mathbb{R}^{n}\to\mathbb{R}$ with bounded gradient
$\nabla F$ we have

$ESF(\xi_{1},\ldots\xi_{n})=EG\cdot\nabla F(\xi_{1},\ldots,\xi_{n}).$ (22)

######  Remark 8.

Let $R$ be a measure space, ${\rm e}_{i}$ an orthonormal basis of $L^{2}(R)$
consisting of bounded functions, and $G$ be a random variable on
$(\Xi,\mathcal{F},P)$ taking values in finite total absolute mass, signed
measure on $R$. Define $G_{i}=\langle G,{\rm e}_{i}\rangle$. Then
$G_{1},G_{2},\ldots$ is an $\mathbb{R}^{\mathbb{N}}$-valued random variable,
and if $\int{\small\raisebox{-3.15005pt}{\scriptsize\bf o}}\hskip 5.0ptG\,\xi$
exists in the sense of Definition 7 we will say it is the Skorokhod integral.
This is somewhat closer to the standard usage. What we are doing is thinking
of the Skorokhod integral as a renormalized extension of an $L^{2}$ inner
product and working directly in the coordinate representation.

###  2.3 Projections

For infinite dimensional examples, it helps to define versions of the
Skorokhod integral where only finitely many variables are integrated over. A
natural projection property then determines the integral, see Proposition 11.

######  Definition 9 (Projections).

Let $\mathcal{P}_{n}G$ denote the vector $G$ with all coordinates beyond $n$
set to zero and

$\int_{[n]}{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptG\,\xi:=\int{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997pt\mathcal{P}_{n}G\,\xi.$ (23)

The following proposition can be thought of as a constructive definition of
the Skorokhod integral in nice cases.

######  Proposition 10.

  1. 1.

If $G$ is Skorokhod integrable, then its integral is unique a.s.

  2. 2.

Let $G_{j}$, $\xi_{j}$ be the $j$th coordinate of $G,\xi$, respectively.
Assume that for all $j$, and almost all
$\xi_{1},\ldots,\xi_{j-1},\xi_{j+1},\ldots$, the function $G_{j}$ is
continuous and a.e. differentiable in $\xi_{j}$. If $E|G_{i}|<\infty$ and with

$S_{n}=\sum_{j=1}^{n}G_{j}\xi_{j}-\partial_{\xi_{j}}G_{j},\qquad
E|S_{n}|<\infty$ (24)

then

$\int_{[n]}{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptG\,\xi=S_{n}.$

If $S_{n}$ converges in $L^{1}$ to a limit $S$ as $n\to\infty$, then $G$ is
Skorokhod integrable and

$\int{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip 6.99997ptG\,\xi=S.$

###### Proof.

If $S_{1},S_{2}$ are two Skorokhod integrals, then by the definition
$E[(S_{1}-S_{2})F]=0$ for all bounded differentiable $F$ with bounded
derivatives. Such functions $F:{\mathbb{R}}^{n}\to\mathbb{R}$ are dense in
$L^{\infty}(\mathbb{R}^{n})$. Hence
$E[S_{1}-S_{2}~{}\mid~{}\mathcal{F}_{n}]=0$ for each $n\in\mathbb{N}$, and so
$S_{1}-S_{2}=0$ a.s. For the second claim, let $F$ be as in the definition. We
condition on the $\sigma$-field $\mathcal{J}_{\hat{j}}$ generated by
$\xi_{1},\ldots,\xi_{j-1},\xi_{j+1},\ldots$. Conditionally on this
$\sigma$-field, $G_{j}$ is a continuous, almost everywhere differentiable
function of $\xi_{j}$. By Gaussian integration by parts, We have

$E[FG_{j}\xi_{j}|\mathcal{J}_{\hat{j}}]=E[\partial_{\xi_{j}}(FG_{j})|\mathcal{J}_{\hat{j}}]$

taking expectations, we get

$E[FG_{j}\xi_{j}-F\partial_{\xi_{j}}G_{j}]=E[\partial_{\xi_{j}}(FG_{j})-F\partial_{\xi_{j}}G_{j}]=E[(\partial_{\xi_{j}}F)G_{j}]$

Note that $\partial_{\xi_{j}}F=0$ for $j>n$. Summing over $j\leq m$ this gives
that for $m\geq n$

$E[S_{m}F]=E[G\cdot\nabla F].$

Since $S$ converges in $L^{1}$ and $F$ is bounded, we get

$E[SF]=\lim_{m\to\infty}E[S_{m}F]=E[G\cdot\nabla F]$

as required. ∎

The Skorokhod integral is linear, and has $ES=0$, as we can see by taking
$F=1$ in the definition. It also behaves nicely under conditional
expectations.

######  Proposition 11.

If $G$ is Skorokhod integrable, then $E[\mathcal{P}_{n}G|\mathcal{F}_{n}]$ is
Skorokhod integrable, and

$E\left[\int{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptG\,\xi\,\Big{|}\,\mathcal{F}_{n}\right]=\int{\raisebox{-8.50006pt}{\scriptsize\bf
o}}\hskip
6.99997ptE[\mathcal{P}_{n}\,G\,|\,\mathcal{F}_{n}]\,\xi=\int_{[n]}{\raisebox{-8.50006pt}{\scriptsize\bf
o}}\hskip 6.99997ptE[G|\mathcal{F}_{n}]\,\xi$

For $n=0$ we get $E\int{\small\raisebox{-3.15005pt}{\scriptsize\bf o}}\hskip
5.0ptG\,\xi=0$.

###### Proof.

We need to check that for all test functions $F$ as in the definition, we have

$E\Big{[}F\,E[S|\mathcal{F}_{n}]\Big{]}=E\Big{[}\nabla F\cdot
E[\mathcal{P}_{n}\,G\,|\,\mathcal{F}_{n}]\Big{]}$

By properties of the conditional expectation, the definition of $S$, we have

$E\Big{[}F\,E[S|\mathcal{F}_{n}]\Big{]}=E\Big{[}E[F|\mathcal{F}_{n}]\,S\Big{]}=E\Big{[}\nabla
E[F|\mathcal{F}_{n}]\cdot G\Big{]}.$

Now

$\partial_{j}E[F|\mathcal{F}_{n}]=\begin{cases}E[\partial_{j}F|\mathcal{F}_{n}],\qquad&j\leq
n\\\ 0&j>n\end{cases}$

so we have

$E\Big{[}\nabla E[F|\mathcal{F}_{n}]\cdot
G\Big{]}=\sum_{j=1}^{n}E\Big{[}E[\partial_{j}F|\mathcal{F}_{n}]G_{j}\Big{]}=\sum_{j=1}^{n}E\Big{[}\partial_{j}FE[G_{j}|\mathcal{F}_{n}]\Big{]}=E\Big{[}\nabla
F\cdot E[\mathcal{P}_{n}\,G\,|\,\mathcal{F}_{n}]\Big{]}$

as required, showing the first equality. The last inequality follows by
definition of the finite-dimensional version of the Skorokhod integral. ∎

######  Corollary 12.

Let $G$ be a $\mathbb{R}^{\mathbb{N}}$-valued random variable such that
$E|G_{i}|<\infty$ for each $i\in\mathbb{N}$. Let
$H_{n}=E[\mathcal{P}_{n}G|\mathcal{F}_{n}]$.

The following are equivalent.

  1. 1.

For every $n$, $S_{n}=\int{\small\raisebox{-3.15005pt}{\scriptsize\bf
o}}\hskip 5.0ptH_{n}\,\xi$ exists and the $S_{n}$ are uniformly integrable.

  2. 2.

$G$ is Skorokhod integrable.

If these conditions hold, then $S_{n}$ is a martingale with limit
$\int{\small\raisebox{-3.15005pt}{\scriptsize\bf o}}\hskip 5.0ptG\,\xi$ almost
surely and in $L^{1}$.

###### Proof.

If 2 holds, by Proposition 11, $H_{n}$ is Skorokhod integrable and

$S_{n}=E[\int{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptG\xi|\mathcal{F}_{n}].$

Since these are conditional expectations of a fixed integrable random
variable, we see that $(S_{n})$ are uniformly integrable, so 1 and the final
conclusion hold.

If 1 holds, then the martingale property of $S_{n}$ follows from Proposition
11. Then $S_{n}$ is a Doob martingale and converges to some limit $S$ almost
surely and in $L_{1}$.

$E[FS]=\lim E[FS_{n}]=\lim E[\nabla F\cdot H_{n}]=E[\nabla F\cdot G].$

In the last two equalities, we used that by definition, the coordinate
$(H_{n})_{j}$ for $n\geq j$ is a uniformly integrable martingale with limit
$G_{j}$. Thus $G$ is Skorokhod integrable with integral $S$. ∎

We call a subspace $S\subset\mathbb{R}^{\mathbb{N}}$ finitary if there is some
$n$ so that all coordinates beyond $n$ of all vectors in $S$ are zero. The
projection to a finitary subspace is well-defined: simply restrict vector to
the first $n$ coordinates and then apply Euclidean projection there.
Orthogonal invariance implies the following corollary to Proposition 11.

######  Corollary 13.

Let $\mathcal{P}$ be projection to a finitary subspace of
$\mathbb{R}^{\mathbb{N}}$. Let $\mathcal{S}$ be $\sigma$-field generated by
the random variable $\mathcal{P}\xi$. If $G$ is Skorokhod integrable, then
$E[\mathcal{P}G|\mathcal{S}]$ is Skorokhod integrable, and

$E\left[\int{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptG\,\xi\,\Big{|}\,\mathcal{S}\right]=\int{\raisebox{-8.50006pt}{\scriptsize\bf
o}}\hskip 6.99997ptE[\mathcal{P}\,G\,|\,\mathcal{S}]\,\xi.$

###  2.4 Examples of Skorokhod integrals

We first consider some one-dimensional problems, as these are as simple as
possible.

######  Example 14.

Consider the case when $\Xi=\mathbb{R}$ houses a one-dimensional Gaussian
$\xi$. Let’s solve the recursion

$X_{0}=1,\qquad X_{k}=\int{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptX_{k-1}\,\xi,\qquad k\geq 1.$

Then the solution is $X_{n}=H_{n}(\xi)$, where $H_{n}$ are the Hermite
polynomials. This can be seen by the recursion
$H_{k+1}(x)=xH_{k}(x)-H_{k}^{\prime}(x)$ and the formula (24). See Example 20
for further connection between the Skorokhod integral and Hermite polynomials.

The following example is good to keep in mind when trying to construct
discrete-time Wick-ordered polymers.

######  Example 15.

Consider a one-dimensional Gaussian $\xi$, and the slightly more complex
linear recursion

$X_{0}=1,\qquad X_{k}=X_{k-1}+\beta\int{\raisebox{-8.50006pt}{\scriptsize\bf
o}}\hskip 6.99997ptX_{k-1}\,\xi,\qquad k\geq 1.$

Write the solution of the form $X_{k}=\beta^{k}p_{k}(\xi)$. Then $p_{k}$
satisfies

$p_{k}=\beta^{-1}p_{k-1}+p_{k-1}\xi+p^{\prime}_{k-1},\qquad p_{0}=1$

This is just the Hermite recursion with $x=\beta^{-1}+\xi$, so the solution is
$X_{k}=\beta^{k}H_{k}(\beta^{-1}+\xi)$, where $H_{k}$ are the Hermite
polynomials.

Can basic functions be non-integrable?

######  Example 16.

Still in one-dimension, let $G=\mathbf{1}(\xi\geq 0)$. The Skorokhod integral
tries to be $\xi^{+}+\delta_{0}(\xi)$, but of course this is not a random
variable, as $\delta_{0}$ is not an honest function. It’s a simple exercise to
show that $\int{\small\raisebox{-3.15005pt}{\scriptsize\bf o}}\hskip
5.0ptG\,\xi$ does not exist. Perhaps it could be defined by extending the
probability space, but we do not pursue this direction.

Next, we study the continuous version of Example 14.

######  Example 17 (A simple Skorokhod stochastic differential equation).

We still work on the one-dimensional Gaussian space, and consider the
“stochastic” differential equation
$du=\int{\small\raisebox{-3.15005pt}{\scriptsize\bf o}}\hskip 5.0ptu\xi$
written in an integral form

$u(t)=\int_{0}^{t}\left(\int{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptu(s)\,\xi\right)ds,\qquad u(0)=1.$ (25)

This is not a usual SDE, as the entire randomness is based on a single
Gaussian variable $\xi$! To solve this, we introduce the test function
$F=e^{i\lambda\xi}$ satisfying $\nabla F=i\lambda F$, and let
$\hat{u}(t)=E[u(t)F]$. We multiply (25) by $F$, take expectations and use the
definition of the Skorokhod integral to get

$\hat{u}=i\lambda\int_{0}^{t}\hat{u},\qquad\hat{u}(0)=e^{-\lambda^{2}/2}.$

this is just the mild version of the equation
$\hat{u}^{\prime}=i\lambda\hat{u}$ solved uniquely by $\hat{u}=e^{i\lambda
t-\lambda^{2}/2}$. This holds for all $\lambda$, so inverting the Fourier
transform, we get $u(t)=e^{t\xi-t^{2}/2}$.

Remarkably, this method will also solve the Wick-ordered planar stochastic
heat equation (1).

The planar Wick-ordered heat equation will be a continuum version of the
following simple example.

######  Example 18.

(Continuous time, finite state-space Wick-ordered polymer) Next, consider the
evolution of a finite state-space $S$ continuous time Markov chain, think
continuous time symmetric random walk on the $d$-cycle. The generator $K$ is a
$d\times d$ matrix, and the transition probability vector $p$ from a given
site $x$ satisfies the forward equation

$\partial_{t}p(y,t)=(p(\cdot,t)K)(y,t)\qquad p(\cdot,0)=\mathbf{1}_{x}.$

We add a fixed potential $V:S\to\mathbb{R}$, and then the partition function
for a non-random “polymer” should look like

$\partial_{t}u(y,t)=(u(\cdot,t)K)(y)+u(y,t)V(y)\qquad
u(\cdot,0)=\mathbf{1}_{x}.$

The Skorokhod version of this equation is written as

$\partial_{t}u(y,t)=(u(\cdot,t)K)(y)+\int{\raisebox{-8.50006pt}{\scriptsize\bf
o}}\hskip 6.99997ptu(y,t)\mathbf{1}_{y}\mathbb{\,}\xi\qquad
u(\cdot,0)=\mathbf{1}_{x}.$

here $\xi$ is a standard $d$-dimensional Gaussian vector defined on
$\Xi=\mathbb{R}^{d}$. The vector $\mathbf{1}_{y}$ is there so that the noise
corresponds to a diagonal potential.

Multiplying by $F=e^{i\sum_{y\in S}\xi(y)\lambda(y)}$. setting $\hat{u}=E[uF]$
and taking expectations we get

$\partial_{t}\hat{p}(x,t)=\hat{p}(\cdot,t)K(y)+\hat{p}(y,t)i\lambda(y),\qquad
p(\cdot,0)=\mathbf{1}_{x}e^{-\sum_{i}\lambda_{i}^{2}/2}.$

At this point, we leave it to the reader to use the discrete version of the
Feynman-Kac formula and then Fourier inversion to conclude that

$p(y,t)=E_{Q}\bigg{[}\exp\Big{\\{}\sum_{z}X(z)\xi_{z}-X(z)^{2}/2\Big{\\}}\mathbf{1}(X_{t}=y)\bigg{]}.$

The expectation is under a measure $Q$ independent of the Gaussian space
$\Xi$. Under the measure $Q$, the variable $X_{t}$ is the position of the
original Markov chain at time $t$ when started from $x$, and
$X(z)=\int_{0}^{t}\mathbf{1}(X_{s}=z)ds$ is the time spent at site $z$.

The reader will recognise the formula above, without the quadratic terms in
the exponential, as the definition of the random polymer with Gaussian
weights. The quadratic correction makes $Eu(y,t)=p(y,t)$.

###  2.5 Classical Skorokhod integral

To provide some motivation for the Skorokhod integral (our definition, as well
as the classical one), let’s look at the finite dimensional case. A natural
requirement is that for deterministic functions (in this case, vectors $G$ in
$\mathbb{R}^{n}$) the Skorokhod integral should be the inner product,

$\int{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptG\,\xi=G_{1}\xi_{1}+\ldots+G_{n}\xi_{n}.$ (26)

Now let’s say we are in $n=1$, and ask what the integral of a polynomial $G$
in $\xi$ should be. One option might be $G\xi$, but it does not have mean
zero, for example when $G=\xi$. If we want it to have zero expectation, we
need to introduce a correction,
$\int{\small\raisebox{-3.15005pt}{\scriptsize\bf o}}\hskip 5.0ptG\xi=G\xi~{}+$
correction. It is reasonable that the correction would be a polynomial in
$\xi$ of lower degree. Moreover, to be reminiscent of the Ito integral, we may
want iterated integrals of 1 to be orthogonal. For polynomials $G$, these two
requirements define a unique integral

$\int{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptG\xi=G\xi+\partial_{\xi}G,\qquad n=1,$ (27)

and there is a unique continuous extension for general integrands.

To treat the general finite dimensional case, it helps to first recall a very
general construction.

Let $H=L^{2}(R)$. The $n$-dimensional case corresponds to $R=\\{1,\ldots,n\\}$
with counting measure. For each $k\geq 1$, define recursively
$\mathcal{H}_{k}$ as the set of symmetric polynomials in
$\xi_{1},\xi_{2},\ldots$ of degree $k$, orthogonal to $\mathcal{H}_{k-1}$,
under the inner product $E[FG]$. Then (see, for example, Janson (1997b))

$\oplus_{k=0}^{\infty}\mathcal{H}_{k}=L^{2}(\Xi,\mathcal{F},P).$ (28)

Let $\pi_{k}$ denote the orthogonal projection to $\mathcal{H}_{k}$.

$\mathcal{H}_{0}$ consists of non-random functions on $R$, and can be
identified with $H$. For $k\geq 1$, one needs some symmetrization (after all,
the polynomials $\xi_{1}\xi_{2}$ and $\xi_{2}\xi_{1}$ mean the same thing.)
Let
$H^{\otimes_{s}k}=L^{2}\left(R^{k},\mathcal{R}^{k}_{s},\frac{1}{k!}\mu^{k}\right)$,
where $\mathcal{R}^{k}_{s}$ is the $\sigma$-algebra of symmetric
$\mathcal{R}^{k}$-measurable functions on $R^{k}$. We now explain how to
represent $\mathcal{H}_{k}$ by $H^{\otimes_{s}k}$. The symmetric Fock space
$\oplus_{k=0}^{\infty}H^{k\otimes_{s}k}$ is thus identified with
$\oplus_{k=0}^{\infty}\mathcal{H}_{k}$, and therefore by (28), with the
Hilbert space $L^{2}(\Xi,\mathcal{F},P)$ of all functions of
$\xi_{1},\xi_{2},\ldots$ with finite second moment. Given $G\in H^{\otimes
k}$, call

${\rm Sym}(G)=\sum_{\sigma\in S_{k}}G\circ\sigma$

where $S_{k}$ is the symmetric group. In particular, for
$g_{1},\ldots,g_{k}\in H$, $g_{1}\otimes\cdots\otimes g_{k}$ is the element of
$L^{2}(R^{k})$ which takes $(x_{1},\ldots,x_{k})$ to $\prod_{i=1}^{k}g(x_{i})$
and ${\rm Sym}(g_{1}\otimes\cdots\otimes g_{k})$ is the function that takes
$(x_{1},\ldots,x_{k})$ to $\sum_{\sigma\in
S_{k}}\prod_{i=1}^{k}g_{i}(x_{\sigma_{i}})$. Linear combinations of such are
dense in $H^{\otimes_{s}k}$.

For $g\in H$ define $I_{1}(g)=\langle g,\xi\rangle$, as in the definition of
the white noise. For $k\geq 2$, and $g_{1},\ldots,g_{k}\in H$ let

$I_{k}({\rm Sym}(g_{1}\otimes\cdots\otimes g_{k}))=\pi_{k}(I_{1}(g_{1})\cdots
I_{1}(g_{k})).$ (29)

It can then be extended to $H^{\otimes_{s}k}$ by linearity and density. For
$G\in H^{\otimes_{s}k}$, $\int G\xi=I_{k}(G)$ is its multiple stochastic
integral.

######  Definition 19.

Let $G:R\to L^{2}(\Xi,\mathcal{F},P)$. It follows from (28) that
$G(x)=\sum_{k=1}^{\infty}\pi_{k}(G(x))$ in $L^{2}$. Since for each $x$,
$\pi_{k}(G(x))$ is a function of $k$ variables, we can think of $\pi_{k}(G)$
as function of $k+1$ variables and set

$\int\pi_{k}(G)\xi=I_{k+1}({\rm Sym}(\pi_{k}(G))).$ (30)

If $\int G\xi:=\sum_{k=0}^{\infty}\int\pi_{k}(G)\xi$ converges in
$L^{2}(\Xi,\mathcal{F},P)$, we call it the classical Skorokhod integral of $G$
and call $G$ classically Skorokhod integrable. Note that we use the notation
$\int G\xi$ to distinguish it from our previous definition of the Skorokhod
integral $\int{\small\raisebox{-3.15005pt}{\scriptsize\bf o}}\hskip
5.0ptG\xi$.

######  Example 20.

Let $R=\\{1,\ldots,n\\}$ with counting measure. Then
$H=L^{2}(M,\mathcal{M},\mu)$ is $\mathbb{R}^{n}$ with the standard inner
product. $H^{\otimes k}$ is the set of real functions defined on
$\\{1,\ldots,n\\}^{k}$, or, equivalently the positions in $\\{1,\ldots,n\\}$
of $k$ _distinguishable_ particles. $H^{\otimes_{s}k}$ then corresponds to the
set of real functions defined on the occupation measures of $k$
_indistinguishable_ particles on $\\{1,\ldots,n\\}$ represented by
$\eta=(\eta_{1},\ldots,\eta_{n})\in\mathbb{N}^{n}$ with
$|\eta|:=\sum_{i}\eta_{i}=k$, and inner product
$\sum_{|\eta|=k}G(\eta)F(\eta).$ for $G,F\in H^{\otimes_{s}k}$. Let $H_{k}$,
$k=0,1,\ldots$ be the Hermite polynomials. They can be defined through their
generating function

$e^{\lambda
x-\tfrac{1}{2}\lambda^{2}}=\sum_{k=0}^{\infty}H_{k}(x)\frac{\lambda^{k}}{k!}.$
(31)

One checks directly that $I_{k}:H^{\otimes_{s}k}\to\mathcal{H}_{k}$ is given
in this representation by

$I_{k}(G)=\sum_{|\eta|=k}G(\eta)H_{\eta_{1}}(\xi_{1})\cdots
H_{\eta_{n}}(\xi_{n}),$ (32)

so the classical Skorokhod integral can just be written, by linearity, as a
sum over $|\eta|=k$ of

$\int G(x,\eta)H_{\eta_{1}}(\xi_{1})\cdots
H_{\eta_{n}}(\xi_{n})\xi(x)=\sum_{i=1}^{n}{\rm
Sym}(G)(\eta+\mathbf{1}_{i})H_{\eta_{1}}(\xi_{1})\cdots
H_{\eta_{i}+1}(\xi_{i})\cdots H_{\eta_{n}}(\xi_{n}).$ (33)

Using the Hermite polynomial identity

$H_{k+1}(x)=xH_{k}(x)-\partial_{x}H_{k}(x),$

we obtain that for every $G\in H^{\otimes_{s}k}$,

$\int
G\xi=\sum_{i=1}^{n}G_{i}\xi_{i}-\partial_{\xi_{i}}G_{i}=\int{\raisebox{-8.50006pt}{\scriptsize\bf
o}}\hskip 6.99997ptG\xi$

where in the last equality we have used (24).

We can now deduce the following lemma.

######  Lemma 21.

Let $R=\\{1,\ldots,n\\}$ and let $G$ be a function of the symmetric Fock space
$\sum_{k=0}^{\infty}H^{\otimes_{s}k}$. Here $H^{\otimes k}$ is the set of real
functions defined on $\\{1,\ldots,n\\}^{k}$, such that
$\sum_{k=0}^{\infty}\sum_{|\eta|=k}E[\pi_{k}(G(\eta)]<\infty$. Then, if $G$
has a classical Skorokhod integral, it is Skorokhod integrable. Furthermore,
in this case, these integrals are equal.

###### Proof.

From the previous example, we know that the classical Skorokhod integral and
the Skorokhod integral coincide for each $k$ in the space $H^{\otimes k}$. It
follows that for each $K$, if $G_{K}=\sum_{k=0}^{K}\pi_{k}(G)$, then
$S_{K}:=\int G_{K}\xi=\int{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptG_{K}\xi$. Therefore, for all $F:\mathbb{R}^{n}\to\mathbb{R}$ we have
that $E[S_{K}F]=E[G_{K}\cdot\nabla F]$. Taking the limit $K\to\infty$ in the
above inequality, we conclude that $S=\sum_{k=0}^{\infty}\int\pi_{k}(G)\xi$
satisfies

$E[SF]=E[G\cdot\nabla F],$

so that $G$ is Skorokhod integrable and its Skorokhod integral is equal to its
classical Skorokhod integral. ∎

######  Lemma 22.

Let $G:R\to L^{2}(\Xi,\mathcal{F},P)$ be a random function such that
$\int_{\mathbb{R}}E[G(x)^{2}]d\mu<\infty$. Then, if $G$ has a classical
Skorokhod integral, it is Skorokhod integrable. Furthermore, in this case,
these integrals are equal.

###### Proof.

Lemma 21 says that the statement holds in the finite dimensional case. For the
general case, we use the fact that integrands $G$ with

$\sum_{k=0}^{\infty}(k+1)\int_{R}E[\pi_{k}(G)^{2}]d\mu<\infty$ (34)

form the domain of the classical Skorokhod integral (see, for example, Theorem
7.39 of Janson (1997b)). Take such a $G$ and let $G_{n}$ be the truncation of
its coordinate representation.

Let $S_{n}$ be its classical Skorokhod integral. We have shown in Example 20
that this coincides with our Skorokhod integral, so that for all
$F:\mathbb{R}^{n}\to\mathbb{R}$ which are bounded with bounded gradient,

$E[S_{n}F]=E[G_{n}\cdot\nabla F].$ (35)

If $S$ is the classical Skorokhod integral of $G$ then

$E\left|S_{n}-S\right|^{2}=\sum_{k=0}^{\infty}E\left[\left|\int\pi_{k}(G_{n})-\pi_{k}(G))\xi\right|^{2}\right].$
(36)

From the definition (30) of the classical Skorohod integral as a simple
raising operator on the $\mathcal{H}_{k}$-s, we have for any $G$,

$\sum_{k=0}^{\infty}E\left[\left|\int\pi_{k}(G)\xi\right|^{2}\right]=\sum_{k=0}^{\infty}(k+1)\int_{R}E[\pi_{k}(G)^{2}]d\mu.$

Since $G_{n}\to G$ in this space from (34), $S_{n}$ converges to $S$ in
$L^{2}(\Xi,\mathcal{F},P)$. Now we can take limits on both sides of (35) to
conclude that $E[SF]=E[G\cdot\nabla F]$. ∎

Once one has identified the Skorokhod integral on the homogenous chaoses, it
is not a big step to identify the Skorokhod integral with the Ito integral
when the integrands are appropriately non-anticipating. The proof can be found
in Nualart (1998), p. 138.

######  Remark 23 (Itô integral in 1 dimension).

Let $\xi$ be a $d$-dimensional white noise on $\mathbb{R}$ and
$B(t)=\xi(\mathbf{1}_{[0,t]})$, $d$-dimensional Brownian motion on
$(\Xi,\mathcal{F},\mathbf{P})$. Let $\mathcal{F}_{t}=\sigma(B(s),s\in[0,t])$.

The space of $d$-dimensional, square integrable adapted processes $G(t)$ is
the closure in $L^{2}(\Xi,\mathcal{F},\mathbf{P};\mathbb{R}^{d})$ of those
which can be written as
$G(t)=\sum_{i=0}^{n-1}g_{i}\mathbf{1}_{(t_{i},t_{i+1}]}(t)$ where $g_{i}$ are
bounded and measurable with respect to $\mathcal{F}_{[0,t_{i}]}$.

For any such $G(t)$, $\int{\small\raisebox{-3.15005pt}{\scriptsize\bf
o}}\hskip 5.0ptG\xi$ coincides with the Itô integral $\int G(t)dB(t)$.

######  Remark 24 (Itô integral in 1+1 dimensions).

Let $\xi$ be white noise on $\mathbb{R}_{+}\times\mathbb{R}$ and
$B(x,t)=\xi(\mathbf{1}_{[0,t]\times[0,x]})$, $x\geq 0$ and
$B(x,t)=-\xi(\mathbf{1}_{[0,t]\times[x,0]})$, $x<0$ be the Brownian sheet.

Let $\mathcal{F}_{[0,t]}=\sigma(B(s,x),s\in[0,t],x\in\mathbb{R})$. The space
of square integrable, time-adapted processes $G(x,t)$ is the closure in
$L^{2}(\Xi,\mathcal{F},\mathbf{P})$ of those which can be written as

$G(x,t)=\sum_{i,j=0}^{n-1}g_{i,j}\mathbf{1}_{(t_{i},t_{i+1}],(x_{i},x_{i+1}]}(x,t),$

where $g_{i,j}$ are bounded and measurable with respect to
$\mathcal{F}_{[0,t_{i}]}$.

For such $G(x,t)$, $\int{\small\raisebox{-3.15005pt}{\scriptsize\bf o}}\hskip
5.0ptG\xi$ coincides with the Itô integral $\int G(x,t)B(dt,dx)$.

Crucially for us, the second part of the theorem tells us that the standard
$1+1$ dimensional multiplicative stochastic heat equation coincides with its
Skorokhod version. This can also be seen through the chaos representation.

######  Example 25.

We try to solve the integral equation with initial data $\mu=\delta_{0}$ in
the Skorokhod sense

$Z(x,t)=p(x,t)+\int\int_{0}^{t}p(x-y,t-s)Z(y,s)ds\xi(y)dy.$ (37)

We look for a solution $Z(x,t)\in L^{2}(P)$. As above we have
$Z(x,t)=\sum_{k=0}^{\infty}I_{k}(g_{k})$ for some $g_{k}(\cdot;t,x)$ in
symmetric $L^{2}((\mathbb{R}^{2})^{k})$. Using (30) this means that
$I_{0}(g_{0})=p(x,t)$ and

$I_{k+1}(g_{k+1}(x,t))=\int\int_{0}^{t}p(x-y,t-s)I_{k}(g_{k}(y,s))ds\xi(y)dy$
(38)

Here one should think of $g_{k}\in H^{\otimes_{s}k}$ as depending on the extra
parameters $t,x$. It is not hard to see that the solution is

$g_{k}(x_{1},\ldots,x_{k};t,x)=E_{Q}[m(x_{1})\cdots m(x_{k})]$ (39)

where $m$ is the occupation measure of the Brownian bridge from $0$ at time
$0$ to $x$ at time $t$. Of course, this can also be written explicitly as

$\displaystyle\int_{0<s_{1}<\cdots<s_{k}<t}\prod_{j=0}^{k}p(s_{j+1}-s_{j},x_{j+1}-x_{j})$
(40)

where $p(x,t)$ is the planar heat kernel (2) and $y_{k+1}=x$, $y_{0}=0$.

By comparison, in PAM, the $p$’s in (40) are weighted by the exponential of
the renormalized self-intersection local time (see Gu and Huang (2018)).

In $1+1$ dimensions, the analogue of (39) has the occupation measure of
$(b(s),s)$ where $b$ is the one-dimensional bridge. The analogue of (40) can
be found in Amir et al. (2011). One checks directly that the $L^{2}$ norm of
$Z$, which is the sum over $k$ of the $L^{2}$ norms of the $g$’s, is finite in
$1+1$ dimensions, for all $t$, and only finite up to $t_{c}<\infty$ in the
planar case.

##  3 Solving Wick-ordered heat equations

###  3.1 Projections of the equation and their solution

Perhaps the most useful property on the Skorokhod integral is that the
conditional expectation of solutions of (1) with respect to $\mathcal{F}_{n}$
(defined in (21)) are themselves solutions of a stochastic heat equation, but
now with respect to finite dimensional noise. Moreover, we can solve such
equations in an explicit way.

######  Proposition 26.

Let $u$ be a solution of the Wick-ordered heat equation, Definition 1. Then
$u_{n}(x,t):=E[u(x,t)|\mathcal{F}_{n}]$ solves the following problem: For
almost all $x,t$ the random variable $\mathcal{K}_{x,t}(u_{n})$ is
$\mathcal{F}_{n}$-measurable, Skorokhod integrable, and

$u_{n}(x,t)=\int
p(x-y,t)d\varsigma(y)+\int_{[n]}{\raisebox{-8.50006pt}{\scriptsize\bf
o}}\hskip 6.99997pt\mathcal{K}_{x,t}(u_{n})~{}\xi.$ (41)

###### Proof.

This follows by taking conditional expectation of (3). Only the last term on
the right needs explanation. By Proposition 11,

$E[\int{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997pt\mathcal{K}_{x,t}(u)\,\xi|\mathcal{F}_{n}]=\int_{[n]}{\raisebox{-8.50006pt}{\scriptsize\bf
o}}\hskip 6.99997ptE[\mathcal{K}_{x,t}(u)|\mathcal{F}_{n}]\,\xi.$

Since $\mathcal{K}_{x,t}u$ is Skorokhod-integrable, it must have finite
expectation. Since $u\geq 0$, by Fubini,
$E[\mathcal{K}_{x,t}u|\mathcal{F}_{n}]=\mathcal{K}_{x,t}u_{n}$ as required. ∎

Note that $u_{n}$ is defined on an $n$-dimensional Gaussian space. Thus the
equation (41) is much simpler than a true stochastic PDE. In fact, it simply a
weak version of the deterministic linear PDE for $u(x,t,\xi_{1}\ldots\xi_{n})$
given by

$\partial_{t}u=\left(\tfrac{1}{2}(\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2})+\sum_{j=1}^{n}{\rm
e}_{j}(\xi_{j}-\partial_{\xi_{j}})\right)u$

and the solution is a real analytic function of $\xi_{1},\ldots,\xi_{n}$. The
next proposition connects the Wick-ordered heat equation to polymers.

######  Proposition 27 (Existence, uniqueness and explicit solution for finite
$n$).

The solution $u_{n}$ to (41) exists and has the following representation. For
almost all $t,x$,

$u_{n}(x,t)=\int Z_{n}(y,0;x,t)p(x-y,t)\,d\varsigma(y)$

where, for $Q=Q_{x,t,y}$ the law of a planar Brownian bridge $B(\cdot)$ from
$y$ at time $0$ to $x$ at time $t$

$Z_{n}(0,y;t,x)=E_{Q}\exp\left(\sum_{j=1}^{n}m_{j}\xi_{j}-\tfrac{1}{2}m_{j}^{2}\right),\qquad
m_{j}=\int_{0}^{t}{\rm e}_{j}(B(s))ds.$ (42)

###### Proof.

We solve the equation by introducing the test functions

$F(\xi)=\exp\left(i\sum_{j=1}^{n}\lambda_{j}\xi_{j}\right),\qquad\lambda_{i}\in\mathbb{R}.$

as these have nice partial derivatives. If $G$ is an
$\mathcal{F}_{n}$-measurable random variable taking values in
$L^{1}(\mathbb{R}^{2})$, then by definition of the finite-dimensional
Skorokhod integral,

$E_{P}\left[F\int_{[n]}{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptG\,\xi\right]=E_{P}\left[\nabla
F\cdot\mathcal{P}_{n}\hat{G}\right]=i\sum_{j=1}^{n}\lambda_{j}E[G_{j}F].$

Let us apply this to the Skorokhod integral in our equation, that is, for the
case $G=\mathcal{P}_{x,t}u_{n}$. The summands on the right become

$\displaystyle E[G_{j}F]=E_{P}[(\mathcal{K}_{x,t}u_{n}~{}\cdot{\rm
e}_{j})F]=E_{P}[F\int\mathcal{K}_{x,t}u_{n}(y){\rm e}_{j}(y)dy]$ (43)

We expand, and apply Fubini to get

$\displaystyle\int_{0}^{t}\int p(x-y,t-s){\rm e}_{j}(y)E[u_{n}(y,s)F]dyds$
(44)

After summing, we get

$E[F\int_{[n]}{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptK_{x,t}u_{n}\,\xi]=\int_{0}^{t}\int p(x-y,t-s)V(y)E[u_{n}(y,s)F]dyds$

where

$V(y)=i\sum_{j=1}^{n}\lambda_{j}{{\rm e}_{j}}(y).$

Thus if we multiply (41) by $F$ and take expectations, we see that

$\hat{u}(y,s):=E_{P}[u_{n}(y,s)F]$

solves the Duhamel form of the ordinary heat equation

$\displaystyle\hat{u}(x,t)=(EF)\int
p(x-y,t)d\varsigma(y)+\int\mathcal{K}_{x,t}(V\hat{u})(y)dy$ (45)

where $EF=e^{-\tfrac{1}{2}\sum_{j=1}^{n}\lambda_{j}^{2}}$ is a constant. The
standard form of this equation is

$\partial_{t}\hat{u}=\tfrac{1}{2}(\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2})\hat{u}+V\hat{u}.$

The solution is represented by the Feynman-Kac formula,

${\hat{u}}(x,t)=\int
E_{Q_{x,t,y}}\exp\left(i\sum_{j=1}^{n}\lambda_{j}\int_{0}^{t}{{\rm
e}}_{j}(B(s))ds-\tfrac{1}{2}\lambda_{j}^{2}\right)p(x-y,t)d\varsigma(y)$ (46)

where $B$ is a Brownian bridge in $\mathbb{R}^{2}$ starting at $y$ and ending
at $x$ at time $t$, and ${Q}_{x,t,y}$ is its law. The representation (46) can
be obtained directly from (45) by repeated iteration.

Since the solution is linear in the initial data $\varsigma$, it suffices to
consider $\varsigma=\delta_{y}$, and write any other data as a superposition.
In this case it is convenient to rewrite

$u_{n}(x,t)=Z_{n}(y,0;x,t)p(x-y,t).$ (47)

Now fix $(x,t)$, and note that $\hat{u}$ as a function of the $\lambda_{i}$ is
the Fourier transform of the function $(\xi_{1},\ldots,\xi_{n})\mapsto
u_{n}(x,t,\xi_{1},\ldots,\xi_{n})\varphi(\xi_{1})\ldots\varphi(\xi_{n})$ where
$\varphi$ is the standard Gaussian density.

Inverting the Fourier transform, we conclude that for almost all $t\geq 0$,
$x\in\mathbb{R}^{2}$, with $m_{j}$ as in (42) we have

$Z_{n}(0,y;x,t)=E_{Q_{x,t,y}}\exp\left(\sum_{j=1}^{n}m_{j}\xi_{j}-\tfrac{1}{2}\sum_{j=1}^{n}m_{j}^{2}\right).$
(48)

∎

Remarks.

  1. 1.

$Z_{n}$ is the Radon-Nikodym derivative of the distribution of
$(\xi_{1}+m_{1},\ldots,\xi_{n}+m_{n})$ under the product measure of the laws
of the white noise $P$ and the Brownian bridge $Q$, with respect to that of
the white noise $P$.

  2. 2.

$Z_{n}$ is a non-negative martingale with respect to the filtration
$\mathcal{F}_{n}=\sigma(\xi_{1},\ldots,\xi_{n})$ and hence converges almost
surely to a random variable $Z$.

  3. 3.

If $E_{P}[Z]=1$ we have a candidate for the solution of the Wick-ordered heat
equation (1).

  4. 4.

If $E_{P}[Z]=1$, $Z$ has the interpretation as the Radon-Nikodym derivative of
the distribution $\hat{P}$ of the shifted white noise $\xi+X$ with respect to
the distribution $P$ of $\xi$, where $X$ is the occupation measure of the
Brownian bridge up to time $t$. The expected free energy is equal to the
relative entropy of $\hat{P}$ with respect to $P$,

$E_{\hat{P}}\log Z=H(\hat{P}\mid P).$

  5. 5.

For a.e. realization of the Brownian bridge, $X$ is not in the Cameron-Martin
space; if it were, we should have
$\int\left(\frac{d\mu}{dx}\right)^{2}dx=\sum_{j=1}^{\infty}m_{i}^{2}=2\int_{0\leq
s_{1}<s_{2}\leq t}\delta_{0}(B(s_{2})-B(s_{1}))ds_{1}ds_{2}$, the
$2$-dimensional self-intersection local time. This sum needs a diverging shift
renormalization to exist. $Z\in L^{1}(P)$ because of the expectation over $Q$.
We call $Z$ a _randomized shift_ , see, for example, Shamov (2016). We will
formally introduce randomized shifts in Section 4.

  6. 6.

The solution of (11) does _not_ satisfy the semigroup property,

$u(x,t+s)\neq\int u(y,s)u(x-y,t)dy.$

On the other hand, $\hat{u}(x,t;\vec{\lambda})$ does satisfy the semigroup
property.

###  3.2 Solving the planar Wick-ordered heat equation

Proposition 26 shows that for almost all $x,t$ the solution $u_{n}(x,t)$ is a
martingale in $n$.

######  Proposition 28 (Solving the Wick-ordered heat equation).

Suppose that for almost all $(x,t)$ the martingale $u_{n}(x,t)$ converges in
$L^{1}$. Then the limit $u(x,t)$ is a solution of the Wick-ordered heat
equation (1). The solution is unique $x,t,\xi$-almost everywhere.

Almost everywhere uniqueness here means that any two solutions agree $x,t,\xi$
a.e. Note that there is considerable work involved in showing $L^{1}$
convergence (the assumption of Proposition 28). This will be addressed in the
later sections of the article.

###### Proof.

First we have to show that for almost all $x,t$ if we write

$u(x,t)=\lim_{n\to\infty}u_{n}(x,t)$

then we also have

$\int{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997pt{\mathcal{K}}_{x,t}u~{}\xi=\lim_{n\to\infty}\int_{[n]}{\raisebox{-8.50006pt}{\scriptsize\bf
o}}\hskip 6.99997pt{\mathcal{K}}_{x,t}u_{n}~{}\xi.$ (49)

The finite-dimensional version of the Wick-ordered heat equation (41) says

$\int_{[n]}{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997pt{\mathcal{K}}_{x,t}u_{n}\xi=u_{n}(x,t)-\int p(x-y,t)d\varsigma(y)$
(50)

since $u_{n}(x,t)$ is uniformly integrable, so is
$\small\int_{[n]}{\raisebox{-3.15005pt}{\scriptsize\bf o}}\hskip
9.90007pt{\mathcal{K}}_{x,t}u_{n}$. Thus by Corollary 12, (49) holds almost
surely and in $L^{1}$, and so $u$ satisfies the Wick-ordered heat equation
(3).

The solution is unique, since it is determined by its conditional
expectations, which are unique by Propositions 26 and 27. ∎

###  3.3 The SHE in one dimension with space-time noise

In this section, we outline how the $1+1$ dimensional stochastic heat
equation, usually defined through a chaos expansion, can be rigorously defined
and solved using the Skorokhod integral.

The method we describe works for a general stochastic PDE of the form
$\partial_{t}v=\mathcal{L}v+\xi v$, where $\mathcal{L}$ is a generator for a
stochastic process. _Algebraically_ , such equations are equivalent to Example
18, although in many cases there are technical challenges or even analytic
obstacles to a solution.

There is no significant _algebraic_ difference between the pure space and
space-time version of the processes: the time coordinate can just be added as
an extra spatial coordinate.

######  Definition 29.

Let $\varsigma$ be a finite measaure on $\mathbb{R}$. A measurable function
$z:(0,T]\times\mathbb{R}\times\Xi\to[0,\infty)$ is a solution of the one-
dimensional stochastic heat equation with space-time white noise and initial
condition $\varsigma$ if for almost all $(x,t)\in(0,T]\times\mathbb{R}$ the
random function of $(y,s)$

$q(x-y,t-s)\,z(y,s)$ (51)

where $q(x,t)$ is the one-dimensional heat kernel (12) is Skorokhod-
integrable, and satisfies

$z(x,t)=\int q(x-y,t)d\varsigma(y)+\int{\raisebox{-8.50006pt}{\scriptsize\bf
o}}\hskip 6.99997ptq(t-\cdot,x-\cdot)z~{}\xi.$ (52)

Although we are working in $[0,\infty)\times\mathbb{R}$ we can just think of
it as a subset of $\mathbb{R}^{2}$ and use the same orthonormal basis ${\rm
e}_{j}$ of $L^{2}(\mathbb{R}^{2})$ of bounded functions.

######  Proposition 30.

Let $z$ be a solution of the one-dimensional stochastic heat equation,
Definition 29. Then $z_{n}(x,t):=E[z(x,t)|\mathcal{F}_{n}]$ solves the
following problem: For almost all $x\in\mathbb{R}$ and $t>0$,

$z_{n}(x,t)=\int
q(x-y,t)d\varsigma(y)+\int_{[n]}{\raisebox{-8.50006pt}{\scriptsize\bf
o}}\hskip 6.99997ptz_{n}(y,s)q(x-y,t-s)\xi.$ (53)

###### Proof.

As in the proof of Proposition 26, it is enough to show that

$E[\int{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptz(y,s)q(x-y,t-s)\xi|\mathcal{F}_{n}]=\int_{[n]}{\raisebox{-8.50006pt}{\scriptsize\bf
o}}\hskip 6.99997ptz_{n}(y,s)q(x-y,t-s)\xi.$

This follows as before by Fubini and the fact that $z$ is Skorokhod-
integrable. ∎

In this case $z_{n}$ satisfies the equation

$\partial_{t}z_{n}=\tfrac{1}{2}\partial_{x}^{2}z_{n}+\sum_{j=1}^{n}{\rm
e}_{j}\,(\xi_{j}-\partial_{\xi_{j}})z_{n}$

with ${\rm e}_{j}={\rm e}_{j}(x,t)$.

######  Proposition 31.

(Existence, uniqueness and explicit solution for finite $n$) The solution
$z_{n}$ of equation (53) exists and is unique in the following sense. For
almost all $x,t$ we have

$z_{n}(x,t)=\int Z_{n}(y,0;x,t)q(x-y,t)d\varsigma(y)$

where $Q$ is the law of one-dimensional Brownian bridge $B$ from $y$ at time
$0$ to $x$ at time $t$ and

$Z_{n}(0,y;t,x)=E_{Q}\exp\left(\sum_{j=1}^{n}m_{j}\xi_{j}-\frac{1}{2}m_{j}^{2}\right),\qquad
m_{j}=\int_{0}^{t}e_{j}(B(s),s)ds.$ (54)

###### Proof.

The proof is really the same as Proposition 27. We introduce test functions

$F(\xi)=\exp\left(i\sum_{j=1}\lambda_{j}\xi_{j}\right),\qquad\lambda_{j}\in\mathbb{R}.$

For $G$ a $\mathcal{F}_{n}$-measurable random variable with values in
$L^{1}([0,\infty)\times\mathbb{R})$, we have

$E\left[F\int_{[n]}{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptG\xi\right]=E[\nabla
F\cdot\mathcal{P}_{n}G]=i\sum_{j=1}^{n}\lambda_{j}E[G_{j}F].$

We apply this to the case $G(y,s)=z_{n}(y,s)q(x-y,t-s)$ so that

$\displaystyle E[G_{j}F]$ $\displaystyle=E[F\int
z_{n}(y,s)q(x-y,t-s)e_{j}(x,s)dxds]$ $\displaystyle=\int
q(x-y,t-s)e_{j}(x,s)E[z_{n}(y,s)F]dyds.$

After summing we get

$E[F\int{\raisebox{-8.50006pt}{\scriptsize\bf o}}\hskip
6.99997ptz_{n}(y,s)q(x-y,t-s)\xi]=\int q(x-y,t-s)V(y,s)E[z_{n}(y,s)F]dyds,$

where $V(y,s)=i\sum_{j=1}^{n}\lambda_{j}e_{j}(y,s)$, and it follows that
$\hat{z}(y,s)=E[z_{n}(y,s)F]$ solves

$\partial_{t}\hat{z}(x,t)=(EF)\int q(x-y,t-s)d\varsigma(y)+\int
q(x-y,t-s)z(y,s)\hat{z}(y,s)dyds.$

As before, we invert the Fourier transform to get (54). ∎

To take the limit as $n\to\infty$, we will need the moment bounds established
in the next section. The last step of the solution is given in Section 7.2.

##  4 Randomized shifts

In this section, we consider the solution of the Wick-ordered heat equation
from another angle, that of randomized shifts, or, using different
terminology, Gaussian multiplicative chaos. The definition we use is closely
related to Shamov (2016). We will first define the notion and give motivation
later.

As before, let $(\Xi,\mathcal{F},P)$ be a probability space given by an
independent standard Gaussian sequence: $\Xi=\mathbb{R}^{\mathbb{N}}$, and
$\mathcal{F}$ is the Borel $\sigma$-algebra on product space, and $P$ is
independent standard Gaussian measure.

Let $m=(m_{1},m_{2},\ldots)$ be a sequence of real valued random variables on
another probability space $(\Omega,\mathcal{G},Q)$.

######  Definition 32.

The random variable $Z$ defined on $\Xi$ is called the partition
function111Shamov calls this the total mass of the GMC of the randomized shift
$m$ if for all bounded measurable functions $F:\Xi\to\mathbb{R}$, we have

$E_{P}[ZF]=E_{P\times Q}F(\xi+m).$ (55)

Note that $Z$ depends only on the law of $m$, not $m$ as a random variable;
nonetheless for simplicity we will sometimes indicate the dependence by
writing $Z=Z(m)$.

The definition is much more concrete when we consider $\mathcal{P}_{n}m$, that
is, $m$ with all coordinates beyond the $n$th set to zero. In this case, one
checks directly that

$Z_{n}(m):=Z(\mathcal{P}_{n}m)=E_{Q}\exp\sum_{j=1}^{n}m_{j}\xi_{j}-m_{j}^{2}/2$
(56)

satisfies the definition. Note that the right hand side is well-defined and
finite for arbitrary $m$, moreover, it is a non-negative
$\mathcal{F}_{n}$-martingale and therefore converges almost surely. We will
need to know when the convergence is in $L^{1}$. This requires uniform
integrability.

We have the following representation.

######  Proposition 33.

For any bounded $F\in\mathcal{F}$, and more generally: for any measurable $F$
for which the left hand side is well-defined, for example any $F\geq 0$, with
$\mathcal{P}_{n}$ the projection to the first $n$ coordinates, defined in the
preamble to (23),

$E_{P}\left[FZ_{n}\right]=E_{Q\times P}\,F(\xi+\mathcal{P}_{n}m).$ (57)

In particular,

  1. (a)

$E_{P}Z_{n}=1$;

  2. (b)

$E_{P}\left[Z_{n}^{2}\right]=E_{Q^{\otimes
2}}\exp\sum_{i=1}^{n}m_{i}m^{\prime}_{i}$. The second expectation is over the
product of $Q$ with itself and $m$, $m^{\prime}$ are two independent copies.
Note that this may be infinite, even for finite $n$.

  3. (c)

$E_{P}[Z^{k}_{n}]=E_{Q^{\otimes
k}}\exp\left\\{\sum_{j,\ell=1,j\neq\ell}^{k}\sum_{i=1}^{n}m_{i}^{(j)}m^{(\ell)}_{i}\right\\}$

where $m^{(1)},\ldots,m^{(k)}$ are independent copies with the same caveat as
in (b).

  4. (d)

For any $A\in\mathcal{G}$, $E|Z_{n}(m)-Z_{n}(1_{A}m)|\leq 2Q(A^{c})$.

###### Proof.

1.

$E_{P}[Z_{n}\mid\mathcal{F}_{n-1}]=E_{Q}\Big{[}\exp\\{\sum_{i=1}^{n-1}m_{i}\xi_{i}-\tfrac{1}{2}\sum_{i=1}^{n}m_{i}^{2}\\}E_{P}[\exp\\{m_{n}\xi_{n}-\tfrac{1}{2}m_{n}^{2}]\\}\big{]}=Z_{n-1}.$

(57) is just the definition of the Gaussian measure together with Fubini’s
theorem, which applies because the integrand is non-negative. To prove (a-b-c)
by induction take $F=Z_{n}^{k-1}$ in (57) and expand the exponential. For $d$,
note that if $m_{c}$ is the conditional law of $m$ given $A^{c}$, then

$|Z_{n}(m)-Z_{n}(1_{A}m)|=Q(A^{c})|Z_{n}(m_{c})-1|.\qed$

We already have examples from the previous section.

######  Example 34.

In Example 18 of a continuous time, finite state-space Wick-ordered polymer we
found the following. For the polymer ending at time $t$ and position $y$ the
partition function can be written as

$p(y,t)=E_{Q}\bigg{[}\exp\Big{\\{}\sum_{z}X(z)\xi_{z}-X(z)^{2}/2\Big{\\}}\mathbf{1}(X_{t}=y)\bigg{]}.$

The expectation is under a measure $Q$ independent of the Gaussian space
$\Xi$. Under the measure $Q$, the variable $X_{t}$ is the position of the
original Markov chain at time $t$ when started from $x$, and
$X(z)=\int_{0}^{t}\mathbf{1}(X_{s}=z)ds$ is the time spent at site $z$.

This is exactly in the form of Definition 32 with $m_{j}=X(s_{j})$ for
$j\leq|S|$ and $m_{j}=0$ otherwise.

######  Example 35 (Bayesian statistics).

Consider trying to guess the mean of Gaussian random variable with variance
$1$. The mean, unknown to us, is zero. Our prior distribution on
$\Omega=\mathbb{R}$ is denoted $Q$. Let $n$ be fixed and let

$m_{1}(\omega)=\ldots=m_{n}(\omega)=\omega.$

If we repeatedly observe $\xi_{1},\ldots\xi_{n}$, the likelihood is given by
$Z_{n}$. A good exercise is to check that the partition function $Z=\lim
Z_{n}$ exists if and only if $Q\\{0\\}>0$.

After $n$ samples, the posterior distribution $M_{\xi}$ will have the
interpretation as the polymer measure or Gaussian multiplicative chaos of
$Z_{n}$ in this toy setting, but we will leave this to Section 4.6, Example
52.

###  4.1 Convergence in $L^{1}$

Since $Z_{n}$ given in (56) is a non-negative martingale, we have $Z_{n}\to Z$
$P$-almost surely. In order to extend (57) to $Z$, it is important that mass
not be lost in the limit.

######  Lemma 36.

Suppose that $Z_{n}\to Z$ in $L^{1}(\Xi,\mathcal{F},P)$ (or, equivalently,
that $Z_{n}$ is uniformly integrable). Then the marginal law

$\hat{P}(A):=(P\times Q)(\xi+m\in A)$ (58)

of $\xi+m$ on $(\Xi,\mathcal{F})$ is absolutely continuous with respect to
$P$. Also, $Z:=\lim_{n\to\infty}Z_{n}$ in $L^{1}(\Xi,\mathcal{F},P)$ and
$P$-almost surely is the Radon-Nikodym derivative

$Z=\frac{d\hat{P}}{dP}.$ (59)

###### Proof.

For $A\in\mathcal{F}$ define a probability measure
$\tilde{P}(A)=E_{P}[Z\mathbf{1}_{A}]$. We want to show that
$\tilde{P}=\hat{P}$. For $A\in\mathcal{F}_{n}$,
$E_{P}[Z\mathbf{1}_{A}]=E_{P}[Z_{n}\mathbf{1}_{A}]$ by the martingale
property, and

$E_{P}[Z_{n}\mathbf{1}_{A}]=(P\times Q)(\xi+m\in A)$

by the previous proposition. So $\tilde{P}=\hat{P}$ on
$\mathcal{S}=\bigcup_{n=1}^{\infty}\mathcal{F}_{n}$. Since $\mathcal{S}$ forms
a $\pi$-system, by the $\pi-\lambda$ theorem $\tilde{P}=\hat{P}$ on
$\sigma(\mathcal{S})=\mathcal{F}$. ∎

###  4.2 Properties of randomized shifts

######  Lemma 37 (0-1 law).

If each $|m_{i}|\leq b_{i}$ for some deterministic $b_{i}<\infty$, then
$P(Z=0)\in\\{0,1\\}$.

A slightly different 0-1 law for GMC is shown in Mukherjee et al. (2016).

###### Proof.

Fix $k\geq 1$. We write $Z=\lim_{n\to\infty}E_{Q}[XY_{n}]$ where

$X=\exp\sum_{i=1}^{k}\xi_{i}m_{i}-m_{i}^{2}/2,\qquad
Y_{n}=\exp\sum_{i={k+1}}^{n}\xi_{i}m_{i}-m_{i}^{2}/2$

Note that for each fixed $\xi$, $X$ is bounded from above, and the assumption
implies that $X$ is also bounded from below by a positive number. So

$\lim_{n\to\infty}E_{Q}[XY_{n}]=0\;\Leftrightarrow\;\lim_{n\to\infty}E_{Q}[Y_{n}]=0.$

Since the second event is in $\sigma(\xi_{k+1},\xi_{k+2},\ldots)$, so is the
first, and since this holds for all $k$, $\\{Z=0\\}$ is a tail event for the
$\xi$. Kolmogorov’s 0-1 law implies the claim. ∎

Let $W_{n}=\exp\\{-\sum_{i=1}^{n}m_{i}^{2}/2\\}$, and let $w_{n}=E_{Q}W_{n}$.
Let $\tilde{Q}_{n}$ be the distribution of $m_{1},\ldots,m_{n}$, biased by
$W_{n}$, that is, for any test function $F$

$w_{n}E_{\tilde{Q}_{n}}[F]=E_{Q}[W_{n}F].$ (60)

Clearly, the distributions $\tilde{Q}_{n}$ determine the law of $m$. The
following proposition tells us that $Z$ determines $m$.

######  Proposition 38.

Under $\tilde{Q}_{n}$ the $m_{i}$ have all exponential moments. If $E_{P}Z=1$,
then the Laplace transforms satisfy

$w_{n}E_{\tilde{Q}_{n}}[\exp\sum_{i=1}^{n}\xi_{i}m_{i}]=E_{P}[Z\,|\,\xi_{1},\ldots,\xi_{n}].$

Note that the left hand side is analytic in $\xi_{1},\ldots\xi_{n}$, so even
though the right is only defined almost everywhere, it has a continuous
version (which is, of course, unique).

###### Proof.

For the first claim, let $1\leq k\leq n$. We have for all $t\in\mathbb{R}$,

$w_{n}E_{\tilde{Q}_{n}}\exp\\{tm_{k}\\}=E_{Q}\exp\Big{\\{}tm_{k}-\sum_{i=1}^{n}m_{i}^{2}/2\Big{\\}}<\infty.$

since the expression in the expectation is bounded.

For the second, note that by definition (60) the left hand side equals

$Z_{n}=E_{Q}[\exp\sum_{i=1}^{n}\xi_{i}m_{i}-m_{i}^{2}/2]$

our usual martingale approximation of $Z$. So we need to prove
$Z_{n}=E_{P}[Z\,|\,\xi_{1},\ldots,\xi_{n}].$ This is equivalent to uniform
integrability, or $E_{P}Z=1$. ∎

###  4.3 Existence of the randomized shift

As usual, the trouble is that none of the equivalent conditions of Lemma 36
are practical to check. Not surprisingly, there is a simple $L^{2}$ condition,
and a restricted variant.

Define the intersection exponential as

$\mathfrak{e}(m)=\lim_{n\to\infty}E_{P}[Z_{n}^{2}]=\lim_{n\to\infty}E_{Q^{\otimes
2}}\exp\sum_{i=1}^{n}m_{i}m^{\prime}_{i}.$ (61)

Here $m$ and $m^{\prime}$ are independent copies, and $Q^{\otimes 2}$ denotes
the product measure. Since $E_{P}\left[Z_{n}^{2}\right]$ is a martingale
variance, (61) is a limit of an increasing sequence. Hence $\mathfrak{e}(m)$
is always defined, but it may be infinite.

######  Proposition 39.

  1. 1.

If $\mathfrak{e}(m)$ is finite, then the randomized shift exists,

$Z=\lim_{n\to\infty}Z_{n}\qquad P\mbox{-a.s. \;\; and \;\;in
}L^{2}(\Xi,\mathcal{F},P),$

and $E_{P}[Z^{2}]=\mathfrak{e}(m)$.

  2. 2.

If for every $p<1$ there is a $\mathcal{G}$-measurable set $A$ with $QA>p$ so
that $\mathfrak{e}(\mathbf{1}_{A}m)<\infty$, then $Z$ exists and

$Z=\lim_{n\to\infty}Z_{n}\qquad P\mbox{-a.s. \;\; and \;\;in
}L^{1}(\Xi,\mathcal{F},P).$

###### Proof of Proposition 39.

For 1, $Z_{n}$ is an $L^{2}(P)$-bounded martingale, so it converges in
$L^{2}(P)$ and almost surely. Lemma 36 identifies the limit as $Z$.

For 2, write briefly $Z(m)$ for the partition function corresponding to $m$.
Then by Proposition 56 (d),

$E_{P}|Z_{n}(1_{A}m)-Z_{n}(m)|\leq 2Q(A^{c})<2(1-p).$

By the first part, $Z_{n}(m1_{A})$ converges in $L^{2}(P)$, and so it is a
Cauchy sequence in $L^{1}(P)$. By the above bound $Z_{n}(m)$ is also a Cauchy
sequence in $L^{1}$. The rest follows from Lemma 36. ∎

######  Remark 40 (Is there a rabbit?).

A rabbit possibly ran through a field and left its tracks. Can the fox tell if
the rabbit has been there?

In our setting, the field is planar white noise, and the tracks of the rabbit
are the occupation measure of Brownian motion.

The information of the rabbitless field is $\xi$, and with the rabbit’s
tracks, $\xi+m$. The fact that $\xi+m$ is absolutely continuous with respect
to $\xi$ means that the rabbit is lucky: if it passed through the field, the
fox still cannot tell this with probability one.

The fox’s observation $Y$ is a sample either from the law of $\xi+m$ or from
the law of $\xi$. The fox may use the likelihood ratio test to decide which
one. In this test, if the two laws are absolutely continuous with respect to a
common measure, the choice depends on whether the density ratio passes a
certain threshold $a$. In fact, the law of $\xi$ can be the common measure,
and then the likelihood ratio is $Z(Y)$ (here we use the definition of the
random variable $Z$ as a function $\Xi\to\mathbb{R}$).

So the fox decides to keep pursuing when $Z(Y)>a$ for some $a$ depending on
how hungry it is. The Neyman-Pearson Lemma says that this is the most powerful
among tests of the same level.

Now consider a discrete model of a simple random walk as in Example 18. One
may add a parameter $\beta$ in front of white noise, representing the density
of the brush. The regime will be decided based on the rabbit problem. If the
fox has asymptotically no chance, then $Z$ converges to 1 and we expect an
additive heat equation (and a Gaussian regime). On the boundary, we expect a
multiplicative heat equation (crossover regime), and the asymptotic KPZ
fluctuations – the Tracy-Widom fluctuations – probe the regime of the lucky
fox.

See Berger and Peres (2013) for a related problem.

######  Example 41 (The Gaussian multiplicative chaos on the circle).

Heuristically, for $\gamma\in[0,2)$ the Gaussian multiplicative chaos on the
circle is a random finite measure given by

$M(A)=\int_{A}e^{\gamma\zeta(\omega)-\tfrac{\gamma^{2}}{2}E[\zeta^{2}(\omega)]}d\omega,\qquad
A\subset[0,2\pi),\quad\mbox{(heuristic)}$ (62)

where $\zeta$ is Gaussian free field, a generalized Gaussian process with
covariance

$K(\omega,\omega^{\prime})=2\log\frac{1}{|e^{i\omega}-e^{i\omega^{\prime}}|}.$

There are many ways to make this rigorous. It fits in the present setting as
follows. Let $\Omega=[0,2\pi]$ and let $Q$ be uniform measure. Let $\gamma\geq
0$, and define the random variables

$m_{2k-1}(\omega)=\frac{\cos(k\omega)}{\sqrt{k/2}},\qquad
m_{2k}(z)=\frac{\sin(k\omega)}{\sqrt{k/2}},\qquad k=1,2,\ldots$

The random variable $Z(\gamma m)$ is the total mass of the Gaussian
multiplicative chaos (GMC) on the circle with coupling constant $\gamma$. The
random measure $M_{\xi}$ on $[0,2\pi]$ is the GMC itself. A quick computation
gives

$\sum_{m=1}^{\infty}m_{i}(\omega)m_{i}^{\prime}(\omega^{\prime})=K(\omega,\omega^{\prime}),$

which blows up at the diagonal, $\sum m_{i}^{2}=\infty$, so the partition
function has to be defined as a limit. The intersection exponential is given
by

$\mathfrak{e}(\gamma
m)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{|e^{is}-1|^{2\gamma^{2}}}\,ds=\begin{cases}\frac{2\Gamma(-2\gamma^{2})}{\Gamma(1-\gamma^{2})\Gamma(-\gamma^{2})},\qquad&\gamma<1/\sqrt{2},\\\
\infty&\gamma\geq 1/\sqrt{2}.\end{cases}$

The Gaussian sequence $\xi_{0},\xi_{1},\xi_{2},\ldots$ can be identified with
white noise on the circle $\sum{\rm e}_{i}\xi_{i}$ using the basis elements
${\rm e}_{0}=1$, ${\rm e}_{k}=\sqrt{k}m_{k}$, $k\geq 1$. The analogue of the
path measure in this case has state space given by the circle and is the
random function

$X=\sum_{k=1}^{\infty}m_{k}(\omega){\rm
e}_{k}(x)=2\Re\operatorname{Li}_{1/2}(e^{i(x-\omega)})$

which has a square-root singularity, and is not in $L^{2}[0,2\pi]$ for any
$\omega$. The rabbit problem, Remark 40 in this setting says that the marginal
law of $\xi+\gamma X$ is absolutely continuous with respect to white noise
$\xi$ exactly when $\gamma<2$. An alternative formulation is in terms of the
absolute continuity of $\zeta+\gamma K(\omega,\cdot)$ with respect to the GFF
$\zeta$.

###  4.4 Properties of $\sum m_{j}m_{j}^{\prime}$.

In preparation for our main convergence theorem, we need to establish
properties of the sequence
$\alpha_{n}(m,m^{\prime})=\sum_{j=1}^{n}m_{j}m_{j}^{\prime}$, or, more
generally a sequence

$\alpha_{n}(m,w^{\prime})=\sum_{j=1}^{n}m_{j}w_{j}^{\prime}$ (63)

The prime notation here means the following. Consider the product of two
copies of $(\Omega,\mathcal{G},Q)$. For a random variable $U$ defined on
$\Omega$ and $(\omega,\omega^{\prime})\in\Omega^{2}$ we write $U=U(\omega)$ to
define $U$ on $\Omega^{2}$, and write $U^{\prime}=U(\omega^{\prime})$ to
define an independent copy.

It helps to think of the sequence $m_{j}$ as a random signed measure on
$\mathbb{N}$, defined by $m(\\{j\\})=m_{j}$. Let $m^{\otimes k}$ denote the
random product measure on $\mathbb{N}^{k}$, and let $\rho_{k,m}=Em^{\otimes
k}$. We first establish some facts about $\alpha_{n}$ and its moments.

######  Lemma 42.

Assume $E|m_{j}|^{k},E|w_{j}|^{k}<\infty$ for all $j$. Then

  1. 1.

$E\alpha_{n}(m,w^{\prime})^{k}=\langle\mathbf{1}_{\\{1,\ldots,n\\}^{k}}\rho_{k,m},\rho_{k,w}\rangle_{\mathbb{N}^{\otimes
k}}$

  2. 2.

For $k\in 2\mathbb{N}$, $n\geq\ell$ and
$\|\cdot\|_{k}=\|\cdot\|_{L^{k}({Q^{\otimes 2}})}$ we have

$\|\alpha_{n}-\alpha_{\ell}\|^{2k}_{k}\leq\Big{(}\|\alpha_{n}(m,m^{\prime})\|^{k}_{k}-\|\alpha_{\ell}(m,m^{\prime})\|^{k}_{k}\Big{)}\Big{(}\|\alpha_{n}(w,w^{\prime})\|^{k}_{k}-\|\alpha_{\ell}(w,w^{\prime})\|^{k}_{k}\Big{)}.$

###### Proof.

1 follows by expanding the two sums. To prove 2, consider the signed measure
$\nu=\nu_{m,w}$ on $\mathbb{N}^{k}$ with $\nu(x)=\rho_{k,m}(x)\rho_{k,m}(x)$.
When $m=w$, we have $\nu\geq 0$ and

$E\alpha_{n}^{k}=\nu(\\{1,\ldots,n\\}^{k})\geq\nu(\\{1,\ldots,\ell\\}^{k})+\nu(\\{\ell+1,\ldots,n\\}^{k})=E\alpha_{\ell}^{k}+E(\alpha_{n}-\alpha_{\ell})^{k}$
(64)

since the last two sets are disjoint subsets of the first one. The inequality
holds for all $k\in\mathbb{N}$, but without absolute values (64) describes
$L^{k}$ norms only for even $k$.

When $X$ and $Y$ may be different, by Cauchy-Schwarz, with
$A=\\{\ell+1,\ldots,n\\}^{k}$ we have

$(E(\alpha_{n}-\alpha_{\ell})^{k})^{2}=(\nu_{X,Y}(A))^{2}\leq\nu_{X,X}(A)\nu_{Y,Y}(A)$

and the claim follows as before. ∎

######  Proposition 43.

Let $k_{0}\in 2\mathbb{N}$, and assume that
$\rho_{k,m},\rho_{k,w}\in\ell^{2}(\mathbb{N}^{k})$. Then for integers $1\leq
k\leq k_{0}$,

  1. 1.

$\alpha_{n}(m,w^{\prime})$ has a limit $\alpha(m,w^{\prime})$ in
$L^{k}({Q^{\otimes 2}})$.

  2. 2.

$E\alpha(m,w^{\prime})^{k}=\langle\rho_{k,m},\rho_{k,w}\rangle$.

  3. 3.

$E[\alpha(m,w^{\prime})^{k}]^{2}\leq
E[\alpha(m,m^{\prime})^{k}]E[\alpha(w,w^{\prime})^{k}]$

###### Proof.

By Lemma 42 and the assumption, $\alpha_{n}$ is Cauchy in $L^{k_{0}}$, so also
in $L^{k}$. Taking limits of 1 of the previous lemma gives 2. 3 follows from
the $\ell^{2}$ representation in 2. ∎

The following lemma will be useful to understand the intersection exponential.

######  Lemma 44.

If $\rho_{k,m}\in\ell^{2}(\mathbb{N}^{k})$ for all $k$, then
$\alpha_{n}=\alpha_{n}(m,m^{\prime})$ has a limit $\alpha$ in $L^{k}$ for
every $k$. If $E\cosh(\alpha)<\infty$, then

$\mathfrak{e}(m):=\lim_{n\to\infty}Ee^{\alpha_{n}}=Ee^{\alpha}<\infty.$

###### Proof.

The first claim is a consequence of Proposition 43. Using Lemma 42 as well, we
get

$E\alpha_{n}^{2k}=\|\mathbf{1}_{\\{1,\ldots,n\\}^{k}}\rho_{k,m}\|^{2}_{\ell^{2}(\mathbb{N}^{k})}\leq\|\rho_{k,m}\|^{2}_{\ell^{2}(\mathbb{N}^{k})}=E\alpha^{2k}.$

Summing with coefficients, using positivity (the reason for using $\cosh$
instead of $\exp$) Fubini twice, and dominated convergence, we get

$E\cosh\alpha_{n}=E\sum_{k=0}^{\infty}\frac{\alpha_{n}^{2k}}{(2k)!}=\sum_{k=0}^{\infty}\frac{E\alpha_{n}^{2k}}{(2k)!}\to\sum_{k=0}^{\infty}\frac{E\alpha^{2k}}{(2k)!}=E\sum_{k=0}^{\infty}\frac{\alpha_{n}^{2k}}{(2k)!}=E\cosh\alpha<\infty,$

so $\cosh(\alpha_{n})\to\cosh(\alpha)$ in $L^{1}$. But $f(x)=e^{x}/\cosh(x)$
is a bounded continuous function, so $f(\alpha_{n})\cosh(\alpha_{n})\to
f(\alpha)\cosh(\alpha)$ in $L^{1}$. Thus $Ee^{\alpha_{n}}\to Ee^{\alpha}$. ∎

Next, we consider a setting where sequences ${\mathbf{m}}_{\ell}$ converge to
a limiting sequence $\mathbf{m}$.

######  Lemma 45 (Convergence of $\alpha$).

Assume that

  1. (i)

$\lim_{\ell\to\infty}m_{\ell,j}=m_{j}$ in probability for each $j$,

  2. (ii)

for each $\ell$, as $n\to\infty$,

$S_{\ell,n}=\sum_{j=1}^{n}m_{\ell,j}m^{\prime}_{\ell,j}\to\alpha_{\ell},\qquad
S_{n}=\sum_{j=1}^{n}m_{j}m^{\prime}_{j}\to\alpha\quad\mbox{ in
}L^{2}({Q^{\otimes 2}}),$

  3. (iii)

$\lim_{\ell\to\infty}E\alpha_{\ell}^{2}=E\alpha^{2}$.

Then as $\ell\to\infty$, $\alpha_{\ell}\to\alpha$ in probability.

###### Proof.

Let $\varphi(x)=|x|\wedge 1$, so that $E\varphi(X)$ metrizes convergence in
probability. Note that $E\varphi(X)\leq E|X|\leq(EX^{2})^{1/2}$. Let
$q_{\ell,n}=ES_{\ell,n}^{2}$ and $q_{n}=ES_{n}^{2}$. By the $L^{2}$ moment
formulas of Lemma 42 and Proposition 43, we have

$E(\alpha-S_{n})^{2}\leq E\alpha^{2}-q_{n},\qquad
E(\alpha_{\ell}-S_{\ell,n})^{2}\leq E\alpha_{\ell}^{2}-q_{\ell,n}.$

Given $\varepsilon>0$, let $n$ be so that

$E\alpha^{2}-q_{n}<\varepsilon.$

Let $\ell_{0}$ be so that for all $\ell\geq\ell_{0}$

$q_{n}<q_{n,\ell}+\varepsilon,\qquad
E\alpha_{\ell}^{2}<E\alpha^{2}+\varepsilon,\qquad
E\varphi(S_{n,\ell}-S_{n})<\varepsilon.$

In the first inequality we used Fatou’s Lemma. Then

$\displaystyle E\varphi(\alpha_{\ell}-\alpha)$
$\displaystyle\leq(E(\alpha_{\ell}-S_{\ell,n})^{2})^{1/2}+E\varphi(S_{\ell,n}-S_{n})+(E(\alpha-
S_{n})^{2})^{1/2}$
$\displaystyle\leq(E\alpha_{\ell}^{2}-q_{\ell,n})^{1/2}+\varepsilon+(E\alpha^{2}-q_{n})^{1/2}<\sqrt{3\varepsilon}+\varepsilon+\sqrt{\varepsilon}.\qed$

###  4.5 Convergence of randomized shifts partition functions

One of the advantages of the randomized shifts approach is that it is very
effective for proving convergence of models. We will have a sequence of random
sequences $\mathbf{m}_{\ell}$ converging to a random sequence $\mathbf{m}$ in
probability. Since $Z(\mathbf{m}_{\ell})$ only depends on the law of
$\mathbf{m}_{\ell}$, we can couple the $\mathbf{m}_{\ell}$ in any way we like
to get to the present setting.

######  Proposition 46.

Let $\mathbf{m}_{\ell}=(m_{\ell,1},m_{\ell,2},\ldots)$, $\ell=1,2,\ldots$ be a
random sequence. Suppose that $m_{\ell,j}\to m_{j}$ in probability for each
$j$. Let ${\mathbf{m}}=(m_{1},m_{2},\ldots)$.

  1. 1.

Assume that
$\mathfrak{e}(\mathbf{m}_{\ell})\to\mathfrak{e}(\mathbf{m})<\infty.$ Then
$Z(\mathbf{m}_{\ell})\to Z(\mathbf{m})$ in $L^{2}(\Xi,\mathcal{F},P)$.

  2. 2.

Assume that for all $k\geq 1$ and each $\ell$, as $n\to\infty$,

$S_{\ell,n}=\sum_{j=1}^{n}m_{\ell,j}m^{\prime}_{\ell,j}\to\alpha_{\ell},\qquad
S_{n}=\sum_{j=1}^{n}m_{j}m^{\prime}_{j}\to\alpha\quad\mbox{ in
}L^{k}({Q^{\otimes 2}}).$

Suppose that $E\alpha_{\ell}^{2}\to E\alpha^{2}$ and that for some $\gamma>1$
the following holds. For each $p<1$, there exists $\mathcal{G}$-measurable
sets $A_{\ell}$ with $QA_{\ell}>p$ and

$\limsup_{\ell\to\infty}E\cosh(\gamma\mathbf{1}_{A_{\ell}}\mathbf{\alpha}_{\ell})<\infty.$
(65)

Then $Z(\mathbf{m}_{\ell})\to Z(\mathbf{m})$ in $L^{1}(\Xi,\mathcal{F},P)$.

Here, for clarity, we use $Z(\bf m)$ to denote the dependence on $\bf m$ even
though it really only depends on the law of $\bf m$.

###### Proof.

For 1, we can bound
$\tfrac{1}{3}E_{P}[(Z(\mathbf{m}_{\ell})-Z(\mathbf{m})^{2}]$ by

$\displaystyle
E_{P}[(Z({\mathbf{m}_{\ell}})-Z_{n}({\mathbf{m}}_{\ell}))^{2}]+E_{P}[(Z_{n}({\mathbf{m}_{\ell}})-Z_{n}({\mathbf{m}}))^{2}]+E_{P}[(Z_{n}({\mathbf{m}})-Z({\mathbf{m}}))^{2}]$
(66)

Let

$q_{n,\ell}=E_{P}[(Z_{n+1}({\mathbf{m}_{\ell}})-Z_{n}({\mathbf{m}}_{\ell}))^{2}],\qquad
q_{n}=E_{P}[(Z_{n+1}({\mathbf{m}})-Z_{n}(\mathbf{m}))^{2}].$

The weak convergence of $\mathbf{m}_{\ell}$ to $\mathbf{m}$ and the fact that
$e^{\lambda x-\tfrac{1}{2}x^{2}}$ is bounded and continuous in $x$, means that
for each $n$,

$Z_{n}(\mathbf{m}_{\ell})\to Z_{n}(\mathbf{m})$ (67)

$P$-almost surely. Let $\|\cdot\|_{1}$ denote the sequence $\ell^{1}$\- norm.
By the martingale property,

$\mathfrak{e}(\mathbf{m}_{\ell})=\|q_{\ell,\cdot}\|_{1},\qquad\mathfrak{e}(\mathbf{m})=\|q\|_{1}$

Let $\hat{q}_{n}=\liminf_{\ell\to\infty}q_{n,\ell}$. By Fatou’s lemma applied
to sequences, $\lim_{\ell\to\infty}\|q_{\ell,\cdot}\|_{1}\geq\|\hat{q}\|_{1}$.
By (67) and Fatou’s lemma applied in the probability space $\Xi$ to
$|Z_{n+1}(\mathbf{m}_{\ell})-Z_{n}(\mathbf{m}_{\ell})|^{2}$, we have
$q_{n}\leq\hat{q}_{n}$. After summing over $n$ we get

$\mathfrak{e}(\mathbf{m})=\|q\|_{1}\leq\|\hat{q}\|_{1}\leq\lim_{\ell\to\infty}\|q_{\ell,\cdot}\|_{1}=\lim_{\ell\to\infty}\mathfrak{e}(\mathbf{m}_{\ell})=\mathfrak{e}(\mathbf{m}),\qquad
0\leq q_{n}\leq\hat{q}_{n},$

where the last equality is our assumption. This implies $\hat{q}=q$. It
follows that $q_{n,\ell}\to q_{n}$ coordinate-wise and also in $\ell^{1}$.

From the martingale property, the first and third terms of (66) are given by
$\sum_{i=n}^{\infty}q_{i,\ell}$ and $\sum_{i=n}^{\infty}q_{i}$. The
convergence $\|q_{\ell,\cdot}-q\|_{1}\to 0$ implies that for large enough
$n_{0}$ the first term (and the easier third term) in (66) can be made
uniformly small for all $\ell\geq 1,n\geq n_{0}$. Since the $L^{2}(P)$-norm of
the left hand side converges to that of the right in (67), the claimed
convergence also holds in $L^{2}(P)$. Thus the middle term in (66) vanishes as
$\ell\to\infty$. This proves Case 1.

For 2, by diagonalization, we can find sets $A_{\ell}$ so that $QA_{\ell}\to
1$, and (65) still holds. Let
$\mathbf{\bar{m}}_{\ell}=\mathbf{1}_{A_{\ell}}\mathbf{m}_{\ell}$ and use bar
to denote all truncated quantities with $\mathbf{\bar{m}_{\ell}}$ replacing
$\mathbf{m}_{\ell}$.

By Lemma 45, we have $\alpha_{\ell}\to\alpha$ in probability, which implies
$\bar{\alpha}_{\ell}\to\alpha$ and $\cosh\bar{\alpha}_{\ell}\to\cosh\alpha$ in
probability. By our assumption, $E\cosh(\gamma\bar{\alpha}_{\ell})$ is bounded
for some $\gamma>1$, and therefore $\cosh\bar{\alpha}_{\ell}$ is uniformly
integrable, so $E\cosh\alpha_{\ell}\to E\cosh\alpha<\infty$. Similarly,
$Ee^{\bar{\alpha}_{\ell}}\to Ee^{\alpha}<\infty$.

By dominated convergence, we have
$\bar{\alpha}_{\ell}=\lim_{n\to\infty}\bar{S}_{\ell,n}$ in every
$L^{k}({Q^{\otimes 2}})$, so by Lemma 44, we have
$\mathfrak{e}(\mathbf{\bar{m}}_{\ell})=Ee^{\bar{\alpha}_{\ell}}$, and
similarly, $\mathfrak{e}(\mathbf{m})=Ee^{\alpha}$. Thus

$\mathfrak{e}(\mathbf{\bar{m}}_{\ell})\to\mathfrak{e}(\mathbf{m})<\infty.$

Now Case 1 applies, and we get $L^{2}(P)$ convergence, and hence also
$L^{1}(P)$ convergence

$E_{P}|Z(\mathbf{\mathbf{1}}_{A_{\ell}}\mathbf{m}_{\ell})-Z(\mathbf{m})|\to
0.$

Finally, by taking limits of Proposition 56 (d),

$E_{P}|Z({\mathbf{m}_{\ell}})-Z({\mathbf{1}_{A_{\ell}}\mathbf{m}_{\ell}})|\leq
2Q(A_{\ell}^{c})\to 0$

and the claim follows by the triangle inequality. ∎

######  Remark 47.

$Z(\bf m)$ does not depend on our choice of basis ${\rm e}_{j}$. This is
because it is arbitrarily close in $L^{1}(\Xi,\mathcal{F},P)$ to
$Z(\mathbf{1}_{A}\mathbf{m})$. The latter is basis-independent since it has
$L^{2}(\Xi,\mathcal{F},P)$ as its chaos series is as in Example 25 with (39)
replaced by

$g_{k,A}(x_{1},\ldots,x_{k};t,x)=E_{Q\mathbf{1}_{A}}[m(x_{1})\cdots
m(x_{k})].$

The results in this section, as well as the next, parallel those of Theorem 25
of Shamov (2016). In our setting, it reads as follows:

######  Proposition 48 (Shamov).

Let $\mathbf{m}=(m_{1},m_{2},\ldots)$,
$\mathbf{m}_{\ell}=(m_{\ell,1},m_{\ell,2},\ldots)$, $\ell=1,2,\ldots$ be
random sequences defined on the same probability space. Suppose that

  1. 1.

$Z(\mathbf{m}_{\ell})$ is uniformly integrable

  2. 2.

For all $v\in\ell^{2}$, we have
$\langle\mathbf{m}_{\ell},v\rangle\to\langle\mathbf{m},v\rangle$ in
$Q$-probability.

  3. 3.

$\alpha(\mathbf{m}_{\ell},\mathbf{m}_{\ell}^{\prime})\to\alpha(\mathbf{m},\mathbf{m}^{\prime})$
in $Q^{\otimes 2}$-probability for some
$\alpha(\mathbf{m},\mathbf{m}^{\prime})$.

Then $Z(\mathbf{m}_{\ell})\to Z(\mathbf{m})$ in $L^{1}(\Xi,\mathcal{F},P)$ and
the random measures satisfy $M_{\ell}\to M$ in $L^{1}$ in the following sense.
For any test function $F\in L^{1}(\Omega,\mathcal{G},Q)$, we have

$E_{M_{\ell}}F\to E_{M}F\qquad\mbox{ in }L^{1}(\Xi,\mathcal{F},P).$

Let us comment on the main differences. In Proposition 48, uniform
integrability is _assumed_ , while we are instead giving explicit conditions
which imply the uniform integrability, conditions we verify in later sections.
Also, Assumption 2 of Proposition 48 seems harder to verify than convergence
in probability of the entries of $\mathbf{m}$.

###  4.6 The polymer measure

Up to this point, our main focus was random heat flow in one and two
dimensions. Random paths came up as a tool for solving heat equations through
the Feynman-Kac formula. In fact, one motivation for heat equations, random or
deterministic, is that they describe the partition functions of polymers.

As an example, assign an i.i.d. mean one positive random variables to lattice
site in spacetime $\mathbb{N}\times\mathbb{Z}$, and let the weight of a simple
random walk path be given by the product of the weight of sites its graph
visits. When the law of each weight is $e^{\beta G-\beta^{2}/2}$ for a
standard Gaussian $G$ and $\beta\in\mathbb{R}$, this construction can be
described through a randomized shift.

The remarkable advantage of the random shift formalism is that it works
effortlessly in technically difficult settings like Brownian paths weighted by
white noise. The price we have to pay is a bit of abstraction.

In short, the polymer measure is given by the Gaussian multiplicative chaos on
path space. Recall our setting: a probability space $(\Omega,\mathcal{G},Q)$
(the path space in the example above) on which there is a random sequence $m$.
A Gaussian i.i.d. sequence $\xi$ defined on $(\Xi,\mathcal{F},P)$. In the
special case when the $m_{i}$ are bounded random variables and
$\sum_{i=1}^{\infty}m_{i}^{2}<\infty$, then

$\sum_{i=1}^{\infty}m_{i}(\omega)-m_{i}(\omega)^{2}/2$ (68)

exists as a limit of partial sums $Q$-almost surely, we can define the new
measure

$M_{\xi}=e^{\sum_{i=1}^{\infty}m_{i}(\omega)\xi_{i}-m_{i}(\omega)^{2}/2}Q$
(69)

on $\Omega$. The strength of the GMC theory is in the fact that such measures
behave much better under limiting operations than the weight functions (68).

The direct description (69) fails since in many cases the sum is not defined
and $M_{\xi}$ is in fact singular with respect to $Q$. But it is an exercise
to check that the formula (70) holds in the special case above. It turns out
that it captures enough information in it to be used as a definition.

######  Definition 49.

The polymer measure or Gaussian multiplicative chaos of the randomized shift
$m$ is a random measure $M$ on $\Omega$ parametrized by $\xi$. It is defined
by the property that for all bounded measurable functions $F$ on
$\Xi\times\Omega$

$E_{P}E_{M_{\xi}}F(\xi,\omega)=E_{P\times Q}F(\xi+m(\omega),\omega).$ (70)

Here $\xi,m(\omega)$ are real sequences and $\xi+m(\omega)$ is just their
termwise sum. This formula, simple as it looks, contains the definition of the
Wick-ordered polymers. In that case, $\Omega$ is the space of paths, and
$m_{i}(\omega)$ is the integral of the basis element ${\rm e}_{i}$ with
respect to the occupation measure of $\omega$.

Note that $Z(\xi)=M_{\xi}(\Omega)$ is the partition function, or total mass,
of the random measure $M$, and that the $P$-expectation of $M$ is exactly $Q$.
The normalized version

$M_{\xi}(\cdot)/M_{\xi}(\Omega),$

is a random probability measure.

For the definition to make sense, we need the following, one of the main
results of Shamov (2016).

######  Theorem 50 (Shamov (2016), Theorem 17 and Corollary 18).

If the marginal law of $\xi+m(\omega)$ is absolutely continuous with respect
to $\xi$, then the polymer measure/Gaussian multiplicative chaos exists and is
unique.

######  Example 51 (Discrete lattice polymer in 1+1 dimension).

Let $n(i)$ be an enumeration of the even lattice points in
$\mathbb{N}\times\mathbb{Z}$, and consider discrete time simple random walk
$X$ with $X_{0}=0$ run until time $t_{0}$.

$\mbox{ For }\quad n(i)=(x,t)\qquad\mbox{ let }\quad
m_{i}=\begin{cases}\mathbf{1}(X_{t}=x)&\mbox{ if }t\leq t_{0}\\\
0&\mbox{otherwise,}\end{cases}$

in other words, $m$ encodes the occupation measure of the graph of $X$. In the
polymer measure, each path $\omega$ has weight weight
$\exp\sum_{i}\xi_{i}m_{i}-m_{i}^{2}/2$. So for any test function $F$, we have

$E_{M_{\xi}}F(\xi,\omega)=2^{-n}\sum_{\omega}\Big{(}\exp\sum_{i}\xi_{i}m_{i}(\omega)-\frac{m_{i}(\omega)^{2}}{2}\big{)}F(\xi,\omega).$

using the Radon-Nikodym derivative for the shifted mean Gaussian,
$E_{P}E_{M_{\xi}}F(\xi,\omega)$ can be written as

$2^{-n}\sum_{\omega}E_{P}(\exp\sum_{i}\xi_{i}m_{i}(\omega)-\frac{m_{i}(\omega)^{2}}{2})F(\xi,\omega)=2^{-n}\sum_{\omega}E_{P}(\xi+m,\omega)=E_{P\times
Q}F(\xi+m,\omega)$

so it satisfies (70).

######  Example 52 (Bayesian statistics, continued).

Continuing Example 35, recall that $m_{i}(\omega)=\omega$. After $n$
observations, the Gaussian multiplicative chaos is given by the unnormalized
posterior measure
$\exp\left\\{-n\omega^{2}/2+\omega\sum_{i=1}^{n}\xi_{i}\right\\}Q$. The
normalized posterior distribution is
$\exp\left\\{-n\omega^{2}/2+\omega\sum_{i=1}^{n}\xi_{i}\right\\}Q/Z_{n}$.

This example holds in full generality: one can think of any Gaussian
multiplicative chaos as the posterior distribution of $Q$ after observing the
(usually different) statistics $m_{1},m_{2}\ldots$.

######  Example 53 (Definition of PAM).

The planar parabolic Anderson model, PAM, is closely related to the planar
Wick-ordered polymer, whose partition function is

$Z=\lim_{n\to\infty}Z_{n},\qquad
Z_{n}=E_{Q}\Big{\\{}\exp\sum_{j=1}^{n}m_{j}\xi_{j}-m_{j}^{2}/2\Big{\\}}$ (71)

where $m_{j}=\int_{0}^{t}{\rm e}_{j}(B(s))\,ds$ and ${\rm e}_{j}$ is an
orthonormal basis of $L^{2}(\mathbb{R}^{2})$ consisting of bounded functions.
For this example, we take $Q$ to be standard Brownian motion measure on some
interval $[0,t]$.

Our definition corresponds to smoothing out the noise by replacing it the
finite dimensional Gaussian process with $\sum_{j=1}^{n}\xi_{j}{\rm e}_{j}$,
using it to define a Wick-ordered polymer, and then taking a limit.

PAM should correspond to the partition function

$Z_{PAM}=\lim_{n\to\infty}E_{Q}\Big{\\{}\exp\sum_{j=1}^{n}m_{j}\xi_{j}-c_{n,t}\Big{\\}}\qquad\mbox{(heuristic)}.$

for some constants $c_{n,j}$. This means we are solving the deterministic heat
equation with respect to the smoothed noise $\sum_{j=1}^{n}\xi_{j}{\rm
e}_{j}$, and renormalizing by $e^{-c_{n,t}}$. Proving that this limit exists
can be cumbersome, as, unlike in the Wick-ordered case, there is no natural
martingale to consider. A precise definition can be made in a different way.
While we don’t do it here, it can be shown that

$\gamma(B)=\lim_{n\to\infty}\gamma_{n}(B)-E_{Q}\gamma_{n}(B),\qquad\gamma_{n}(B)=\frac{1}{2}\sum_{j=1}^{n}m_{j}^{2},\qquad\mbox{(heuristic)}$
(72)

exists. The random variable $\gamma(B)$ is called the self-intersection local
time. Heuristically, up to centering, $2\gamma$ wants to be the squared
$L^{2}(\mathbb{R}^{2})$ norm of the occupation measure of $B$. Indeed,
$2\gamma_{n}$ is the squared $L^{2}(\mathbb{R}^{2})$-norm of the occupation
measure projected onto the span of ${\rm e}_{1},\ldots,{\rm e}_{n}$. Instead
of proving (72), we use the definition given in Le Gall (1994) through mutual
intersection local times:

$\gamma(B)=\sum_{n=1}^{\infty}\sum_{k=1}^{2^{n-1}}A_{n,k}-E_{Q}A_{n,k},\qquad
A_{n,k}=\alpha(B_{[\frac{2k-2}{2^{n}}t,\frac{2k-1}{2^{n}}t]},B_{[\frac{2k-1}{2^{n}}t,\frac{2k}{2^{n}}t]}).$

The arguments of $\alpha$ in $A_{n,k}$ are conditionally independent given
$B(\tfrac{2k-1}{2^{n}})$, and so each term is defined as in Section 5.

Considering the normalization (71) one could first try to take $c_{n,t}$ to
equal $E_{Q}\gamma_{n}$. But there is a problem: For the PAM to inherit the
semigroup property from the polymer given by Brownian paths weighted by
$\exp\sum_{j=1}^{n}m_{j}\xi_{j}$, for $0<s<t$ the normalization should be
asymptotically additive:

$\lim_{n\to\infty}c_{n,t}-c_{n,s}-c_{n,t-s}=0.$ (73)

For $0<s<t$ let $B_{1}$ and $B_{2}$ denote the restrictions of $B$ to $[0,s]$
and $[s,t]$, respectively. Then by bilinearity, we have

$E_{Q}\gamma_{n}(B)-E_{Q}\gamma_{n}(B_{1})-E_{Q}\gamma_{n}(B_{2})=E_{Q}\alpha_{n}(B_{1},B_{2})$

which converges to

$\displaystyle E\alpha(B_{1},B_{2})$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{0}^{s}\int_{0}^{t-s}p(x,t_{1})p(x,t_{2})\,dt_{2}\,dt_{1}\,dx$
$\displaystyle=\frac{t\log t-s\log s-(t-s)\log(t-s)}{2\pi}.$

This suggest the time-dependent correction

$c_{n,t}=E\gamma_{n}(B)-\frac{1}{2\pi}t\log t$

and this is indeed asymptotically additive (73).

To get a quick definition of PAM, we use $\gamma$ to re-adjust the weight of
the Wick-ordered polymer paths:

$M_{PAM,\xi}=e^{\gamma(\omega)+\frac{1}{2\pi}t\log t}M_{\xi},\qquad
Z_{PAM}=E_{M_{PAM,\xi}}[\Omega]=E_{M_{\xi}}e^{\gamma+\frac{1}{2\pi}t\log t}.$

By the definition of $M_{\xi}$, we have

$E_{P}Z_{PAM}=E_{P}E_{M_{\xi}}e^{\gamma+\frac{1}{2\pi}t\log
t}=e^{\frac{1}{2\pi}t\log t}E_{Q}e^{\gamma}.$

Le Gall (1994) shows that the right hand side is finite for small $t$, but is
infinite for large $t$. It can be shown that in the latter case, $Z_{PAM}$ is
still finite $P$-almost surely, but $E_{P}Z_{PAM}=\infty.$

The following proposition will be used to show that the planar Wick-ordered
polymer converges to the continuum random directed polymer.

######  Proposition 54.

Assume that condition 1 or 2 of Proposition 46 holds. Then the polymer
measures converge in the following sense. For any bounded test function
$F:\Omega\to\mathbb{R}$

$E_{M_{\ell}}F\to E_{M}F\qquad\mbox{in }L^{1}(P).$ (74)

By (74), $M_{\ell}\to M$ in $P$-probability with respect to weak topology of
measures. The normalized measures $M_{\ell}/Z_{\ell}\to M/Z$ as well, as long
as $P(Z>0)=1$. This holds in our cases of interest by Lemma 37.

###### Proof.

By linearity, we may assume $E_{Q}F=1$, $F\geq 0$. By bounded convergence, the
assumptions of the Proposition 46 hold for the sequences $\mathbf{m}_{\ell}$
and $\mathbf{m}$ under the modified probability measure $FQ$. Let
$Z_{F}(\cdot)$ denote the corresponding partition functions. Then

$E_{M_{\ell}}F=Z_{F}(\mathbf{m}_{\ell})\to
Z_{F}(\mathbf{m})=E_{M}F\qquad\mbox{ in }L^{1}(P).\qed$

##  5 Intersection local time

###  5.1 Intersection local time for random measures

Let $X,Y$ be random variables defined on a probability space
$(\Omega,\mathcal{G},Q)$ taking values on the space of finite measures on a
measure space $(R,\mathcal{B},\mu)$. $R$. In general, we will write $\rho_{X}$
for the density of the deterministic measure $EX$ with respect to $\mu$ if it
exists.

Consider the product of two copies of $(\Omega,\mathcal{G},Q)$. For a random
variable $U$ defined on $\Omega$ and $(\omega,\omega^{\prime})\in\Omega^{2}$
we write $U=U(\omega)$ to define $U$ on $\Omega^{2}$, and write
$U^{\prime}=U(\omega^{\prime})$ to define an independent copy.

We would like to define the mutual intersection local time
$\alpha=\alpha(X,Y^{\prime})$ as the random inner product

$\alpha{=}\int\frac{dX}{d\mu}\frac{dY^{\prime}}{d\mu}d\mu,\qquad\mbox{(heuristic)}$

a random variable on $\Omega^{2}$, even when $X,Y^{\prime}$ do not have
densities. When they do, for any nonnegative bounded random variables $F,G$ on
$\Omega$ we have

$E_{Q^{\otimes 2}}[FG^{\prime}\alpha]{=}\int E_{{Q^{\otimes
2}}}[F\frac{dX}{d\mu}G^{\prime}\frac{dY^{\prime}}{d\mu}]d\mu=\langle\rho_{FX},\rho_{GY}\rangle.$

This gives rise to a definition.

######  Definition 55.

Assume that $\rho_{X}$ and $\rho_{Y}$ exists. If there exists $\alpha\in
L^{1}(\Omega^{2},\sigma(X,Y^{\prime}),{Q^{\otimes 2}})$ so that for all
bounded measurable $F,G\geq 0$ on $\Omega$ we have

$E_{{Q^{\otimes
2}}}[FG^{\prime}\alpha]=\langle\rho_{FX},\rho_{GY}\rangle_{\mu}<\infty,$

then we call $\alpha=\alpha(X,Y^{\prime})$ the mutual intersection local time
of $X,Y^{\prime}$.

######  Proposition 56.

  1. 1.

If the intersection local time exists, then it is nonnegative and unique.

  2. 2.

If $X\leq\bar{X}$, $Y\leq\bar{Y}$ and
$\bar{\alpha}=\alpha(\bar{X},\bar{Y}^{\prime})$ exists, then
$\alpha=\alpha(X,Y^{\prime})$ exists and $\alpha\leq\bar{\alpha}$.

###### Proof.

Let $\alpha,\hat{\alpha}$ be two versions of the intersection local time.
Consider the subset $\mathcal{S}\subset\sigma(\mathcal{G}^{2})$ on which both
random variables have the same expectation, and this integral is nonnegative.
Then $\mathcal{S}$ is a $\pi$-system containing a $\lambda$ system
$\mathcal{G}^{2}$ given by the product sets, since

$E[\mathbf{1}_{A\times B}\,\alpha]=E[\mathbf{1}_{A\times
B}\,\hat{\alpha}]=\int\rho_{\mathbf{1}_{A}X}\rho_{\mathbf{1}_{B}Y}d\mu\geq 0.$

Thus by the $\pi-\lambda$ theorem, $\sigma(\mathcal{G}^{2})=\mathcal{S}$. Then
$\\{\alpha<0\\},\\{\alpha<\hat{\alpha}\\}\in\mathcal{S}$, so

$E[\mathbf{1}_{\alpha<0}\,\alpha]\geq 0,\qquad
E[\mathbf{1}_{\alpha<\hat{\alpha}}\alpha]=E[\mathbf{1}_{\alpha<\hat{\alpha}}\hat{\alpha}]$

so $E\alpha^{-}=0$, and since $\alpha\in L^{1}$, we get
$E(\hat{\alpha}-\alpha)^{+}=0$. Similarly $E(\hat{\alpha}-\alpha)^{-}=0$, and
so $\hat{\alpha}=\alpha\geq 0$ a.s.

For 2, note that by the Caratheodory extension theorem $\mathcal{A}(A\times
B):=\int\rho_{\mathbf{1}_{A}X}\rho_{\mathbf{1}_{B}Y}d\mu$ extends to a measure
$\mathcal{A}$ on $\Omega^{2}$. Then $\bar{\alpha}d\mu^{\otimes 2}$ dominates
$\mathcal{A}$ on product sets. So $\mathcal{A}$ has a density in $L^{1}$,
which fits the definition of $\alpha$. ∎

Definition 55 does not give a construction of intersection local time, nor an
easily verifiable condition for when it should exist.

Recall the notation $X^{\otimes 2}=X\otimes X$ is the $Q$-random product
measure on $R\times R$; it is the product of the same random measure, not
independent copies. As a special case of Proposition 60, we show the
following.

######  Corollary 57.

If the deterministic measure $E_{Q}[X^{\otimes 2}]$ has a square integrable
density with respect to $\mu^{\otimes 2}$, then the mutual intersection local
time $\alpha(X,X^{\prime})$ exists.

Random measures defined on $Q$ form a cone in a real vector space through
$(bX+Y)(A)=bX(A)+Y(A)$ for $b\geq 0$. When they are occupation measures of
random functions, this has nothing to do with adding function values
$bX(t)+Y(t)$ in $\mathbb{R}^{d}$. The following are immediate from the
definition.

######  Lemma 58.

If $d\geq 0$ is a constant and $\alpha(X,Y^{\prime})$, $\alpha(X,Z^{\prime})$
exist, then the following quantities exist and satisfy

$\alpha(cX,Y^{\prime})=c\alpha(X,Y^{\prime}),\qquad\alpha(X,Y^{\prime}+Z^{\prime})=\alpha(X,Y^{\prime})+\alpha(X,Z^{\prime})$

and
$\alpha(X,Y^{\prime})(\omega_{1},\omega_{2})=\alpha(Y,X^{\prime})(\omega_{2},\omega_{1}).$
For $Q$-random bounded scalars $C,D\geq 0$, the following random variable on
$\Omega^{2}$ exists and satisfies

$\alpha(CX,D^{\prime}Y^{\prime})=CD^{\prime}\alpha(X,Y^{\prime}).$ (75)

Corollary 57 is based on a coordinate representation of the random measure
$X$. Consider an orthonormal basis ${\rm e}_{i}$, of
$L^{2}(R,\mathcal{B},\mu)$ consisting of bounded functions. Then with

$m_{j}=m_{X,j}=\int{\rm e}_{j}dX,\qquad X_{n}=\sum_{j=1}^{n}m_{j}{\rm e}_{j},$

we have a signed approximation $X_{n}d\mu$ of the random measure $X$.
Similarly, let $w_{j}$ be the coordinates of $Y$, and let

$\alpha_{n}(X,Y^{\prime})=\int
X_{n}Y^{\prime}_{n}d\mu=\sum_{j=1}^{n}m_{j}w_{j}^{\prime}.$ (76)

We are now in the setting of Section 4.4. Indeed, $\alpha_{n}(X,Y^{\prime})$
as defined here, equals $\alpha_{n}(m,w^{\prime})$ of (63). So in order for
$\alpha_{n}$ to have a limit in $L^{k}({Q^{\otimes 2}})$, $k\in 2\mathbb{N}$,
we just need $\rho_{k,m},\rho_{k,w}\in L^{2}(R^{k})$. But

$\rho_{k,m}(i_{1},\ldots,i_{k})=E_{Q}[m_{i_{1}}\cdots m_{i_{k}}],$

the coefficient of the basis vector ${\rm e}_{i_{1}}\otimes\cdots\otimes{\rm
e}_{i_{k}}$ in the orthonormal basis representation of $\rho_{k,X}$, the
density of $E_{Q}X^{\otimes k}$. Thus we have

$\langle\rho_{k,m},\rho_{k,w}\rangle_{\mathbb{N}^{k}}=\langle\rho_{k,X},\rho_{k,Y}\rangle_{\mu^{\otimes
k}}.$ (77)

We will use the results of Section 4.4 to show that $\alpha_{n}$ converges,
and we will use the following general lemma to identify the limit.

######  Lemma 59.

If $\rho_{X}$, $\rho_{Y}\in L^{2}(R)$ and $\alpha_{n}(X,Y^{\prime})$ is
precompact in $L^{1}({Q^{\otimes 2}})$, then the mutual intersection local
time $\alpha(X,Y^{\prime})$ exists and
$\alpha_{n}(X,Y^{\prime})\to\alpha(X,Y^{\prime})$ in $L^{1}({Q^{\otimes 2}})$.

###### Proof.

Let $\alpha_{\infty}$ be a limit point of $\alpha_{n}$ along a subsequence
$\mathcal{N}$. Let $F,G\geq 0$ be bounded random variables on $\Omega$. By a
direct computation, we have

$E_{{Q^{\otimes
2}}}[FG^{\prime}\alpha_{n}]=\sum_{i=1}^{n}\langle\rho_{FX},e_{i}\rangle\langle\rho_{GY},e_{i}\rangle$

Since $\rho_{FX},\rho_{GY}\in L^{2}(R)$, and $FG^{\prime}\alpha_{n}\to
FG^{\prime}\alpha_{\infty}$ in $L^{1}$ along $\mathcal{N}$, taking limits we
conclude

$E_{{Q^{\otimes
2}}}[FG^{\prime}\alpha_{\infty}]=\langle\rho_{FX},\rho_{GY}\rangle_{\mu}.\qed$

######  Proposition 60.

Let $k_{0}\in 2\mathbb{N}$. If $\rho_{k,X},\rho_{k,Y}\in L^{2}(\mu^{\otimes
k})$, then for all integers $1\leq k\leq k_{0}$,

  1. 1.

The intersection local time $\alpha(X,Y^{\prime})$ exists and
$\alpha_{n}\to\alpha(X,Y^{\prime})$ in $L^{k}({Q^{\otimes 2}})$.

  2. 2.

$\|\alpha\|^{k}_{L^{k}({Q^{\otimes
2}})}=\langle\rho_{k,X},\rho_{k,Y}\rangle_{\mu^{\otimes k}}$.

  3. 3.

$E[\alpha(X,Y^{\prime})^{k}]^{2}\leq
E[\alpha(X,X^{\prime})^{k}]E[\alpha(Y,Y^{\prime})^{k}]$ .

###### Proof.

This is a direct consequence of (77), Proposition 43 and Lemma 59. ∎

In classical settings, with $R=\mathbb{R}^{d}$ and Lebesgue measure $\mu$,
intersection local time is often defined as the value of the continuous
density of the convolution $X*\tilde{Y}^{\prime}$ where $\tilde{Y}=Y(-\cdot)$.
The two notions are essentially the same, as the next lemma shows.

######  Lemma 61.

Let $X,Y$ be random measures on the torus $R=[-\ell,\ell]^{d}$ or on
$R=\mathbb{R}^{d}$. If $X*\tilde{Y}^{\prime}$ has continuous density
$\alpha_{*}(X,Y^{\prime})$ at zero a.s. and $\rho_{2,X},\rho_{2,Y}\in L^{2}$
then $\alpha(X,Y^{\prime})$ exists and ${Q^{\otimes 2}}$-a.s. equals
$\alpha_{*}(X,Y^{\prime})$.

###### Proof.

For the torus, we may assume $\ell=\pi$. Let ${\rm e}_{j}$ be the Fourier
modes ordered by frequency. For $n=n_{j}=(2j+1)^{d}$, the projection to the
span of ${\rm e}_{1},\ldots,{\rm e}_{n}$ commutes with translations, so it is
a convolution by a function $\varphi_{n}$. More explicitly,

$\varphi_{n}=D_{n}^{\otimes d},\qquad\mbox{and}\quad
D_{n}(x)=\frac{1}{2\pi}\sum_{k=-j}^{j}e^{ikx}=\frac{\sin((k+1/2)x)}{2\pi\sin(x/2)}$

is the Dirichlet kernel. Hence $X_{n}\mu=X*\varphi_{n}$, and

$\alpha_{n}(X,Y^{\prime})=\int(X*\varphi_{n})(Y^{\prime}*\varphi_{n})=(X*\varphi_{n}*\tilde{\varphi}_{n}*\tilde{Y}^{\prime})(0)=(\varphi_{n}*X*\tilde{Y}^{\prime})(0)$

since $\varphi_{n}$ is symmetric and a projection, so
$\varphi_{n}*\tilde{\varphi}_{n}=\varphi_{n}$. Since $\varphi_{n}$
approximates $\delta_{0}$, we see that $\alpha_{n_{i}}\to\alpha_{*}$, which
then equals $\alpha$ by Proposition 60.

For the case of $\mathbb{R}^{d}$, apply the torus argument on
$[-2\ell,2\ell]^{d}$ to $X\mathbf{1}_{[-\ell,\ell]^{d}}$,
$Y\mathbf{1}_{[-\ell,\ell]^{d}}$ and use dominated convergence to get the
result. ∎

###  5.2 Moment bounds

The solution and convergence of the planar Wick-ordered heat equation requires
estimates of the exponential moments of the mutual intersection local time of
planar Brownian motion.

Proposition 60 shows that for random measures $X$, the mutual intersection
local time $\alpha(X,X^{\prime})$ exists if for $k=2$ the measure
$E_{Q}[X^{\otimes k}]$ has a density $\rho_{k}$ in $L^{2}.$ When
$\alpha(X,X^{\prime})$ exists, its $k$th moment is given by
$\|\rho_{k}\|^{2}$.

he following bound on the moments holds for any continuous time process with
independent stationary increments (Lévy process) $B$ with joint densities with
respect to some fixed measure. In particular, it holds for Brownian motion
with drift on $R=\mathbb{R}^{d}$. For Brownian motion without drift, related
bounds can be found in Le Gall (1994). But the method there does not readily
generalize to Brownian motion with drift. The statement involves a parameter
$r$, which will be chosen carefully later.

######  Proposition 62.

Let $k$ be a positive integer, $r>0$ and let
$X(A)=\int_{0}^{1}\mathbf{1}_{A}(B(t))\,dt$ be the occupation measure of a
stochastic process $B$ in $R=\mathbb{R}^{d}$ up to time $1$. For a time vector
$\mathbf{s}=(s_{1},\ldots,s_{k})$ let $p(\mathbf{s},\mathbf{x})$ be the
density of the random vector $(B(s_{1}),\ldots,B(s_{k}))$ at
$\mathbf{x}=(x_{1},\ldots,x_{k})$; assume this density exists for almost all
vectors $s$. Then the random $k$-fold product $X^{\otimes k}$ has density

$\rho_{k}(\mathbf{x})=\int_{\mathbf{s}\in[0,1]^{k}}p(\mathbf{x},\mathbf{s})$
(78)

on $R^{k}$. Assume further that $B$ is a Levy process. Let $\tau$ be an
independent exponential rate $r$ random variable, and let $\varphi_{r}(\cdot)$
be the density of $B(\tau)$ on $R$. Then

$\|\rho_{k}\|^{2}\leq k!^{2}e^{2r}\|\varphi_{r}/r\|^{2k}.$ (79)

###### Proof.

For any measurable set $A\subset R^{k}$ we have

$E[X^{\otimes k}](A)=E\int_{\mathbf{s}\in[0,t]^{k}}\mathbf{1}((B(s_{1}),\ldots
B(s_{k}))\in A)=\int_{\mathbf{s}\in[0,t]^{k}}P((B(s_{1}),\ldots,B(s_{k})\in
A))$

Expressing the probability as an integral and using Fubini we get

$E[X^{\otimes k}](A)=\int_{\mathbf{x}\in
A}\int_{\mathbf{s}\in[0,t]^{k}}p(\mathbf{x},\mathbf{s}),$

which implies the density formula (78). Let $[0,1]^{\uparrow k}$ denote the
subset of vectors in $\mathbf{s}\in[0,1]^{k}$ with $s_{1}<\cdots<s_{k}$.

$\|\rho_{k}\|^{2}=\int_{R^{k}}\int_{\mathbf{s},\mathbf{t}\in[0,1]^{k}}p(\mathbf{x},\mathbf{s})p(\mathbf{x},\mathbf{t})=\sum_{\sigma,\eta}\int_{R^{k}}\int_{\mathbf{s}\in[0,1]^{\uparrow
k}}p(\mathbf{x},\sigma\mathbf{s})\int_{\mathbf{t}\in[0,1]^{\uparrow
k}}p(\mathbf{x},\eta\mathbf{t})$

where the sum is over all permutations $\sigma$ and $\eta$. By Cauchy-Schwarz
applied to the $\mathbf{x}$-integral, the summand is at most

$\int_{R^{k}}\left(\int_{\mathbf{s}\in[0,1]^{\uparrow
k}}p(\mathbf{x},\mathbf{\sigma}s)\right)^{2}$ (80)

since $p(\sigma\mathbf{s},\mathbf{x})=p(\mathbf{s},\sigma^{-1}\mathbf{x})$,
changing variables $\mathbf{x}\mapsto\sigma\mathbf{x}$ shows that we can drop
the $\sigma$ from (80). We now bound each factor the same way. Rewrite (80) as

$\int_{R^{k}}\int_{\mathbf{s},\mathbf{t}\in[0,1]^{\uparrow
k}}p(\mathbf{x},\mathbf{s})p(\mathbf{x},\mathbf{t}).$ (81)

Let

$a(s,t)=\int_{R^{k}}\int_{\begin{subarray}{c}(s_{1},\ldots,{s_{k-1}})\in[0,s]^{\uparrow(k-1)}\\\
(t_{1},\ldots,{t_{k-1}})\in[0,t]^{\uparrow(k-1)}\end{subarray}}p(\mathbf{x},\mathbf{s})p(\mathbf{x},\mathbf{t}),$

i.e. the integral on the right hand side of (81) without integrating over the
last two variables $s,t$. Since $1\leq e^{2r}e^{-rs-rt}$ for $s,t\in[0,1]$,

$\int_{\mathbb{R}^{k}}\int_{\mathbf{s},\mathbf{t}\in[0,1]^{\uparrow
k}}p(\mathbf{x},\mathbf{s})p(\mathbf{x},\mathbf{t})=\int_{0}^{1}\int_{0}^{1}a(s,t)dsdt\leq
e^{2r}\int_{0}^{\infty}\int_{0}^{\infty}e^{-rs-rt}a(s,t)dsdt.$

Change variables to the increments $\mathbf{u}$, $\mathbf{v}$ of
$\mathbf{s},\mathbf{t}$ and the increments $\mathbf{y}$ of $\mathbf{x}$ to get

$\displaystyle\int_{0}^{\infty}\int_{0}^{\infty}r^{2k}e^{-rs-rt}a(s,t)dsdt$
$\displaystyle=\int_{R^{k}}\int_{\mathbf{u},\mathbf{v}\in[0,\infty)^{k}}r^{2k}e^{-r\sum_{i=1}^{k}u_{i}+v_{i}}\prod_{i=1}^{k}q(y_{i},u_{i})q(y_{i},v_{i})$
$\displaystyle=\Big{(}\int_{R}\varphi_{r}(x)^{2}\Big{)}^{k}$

where we used the independent increment property of $B$, and $q(x,u)$ is the
density of $B(u)-B(0)$ at $x$ and that $re^{-ru}$ is the density of the
exponential random variable $\tau$. ∎

Next, we apply this to planar Brownian motion, with and without drift. We need
the following fact.

######  Lemma 63.

Let $B$ be planar Brownian motion with covariance $I$ and drift $(0,N)$, and
let $\varphi_{\mu,r}$ be the density of $B$ at independent rate $r$
exponential random time. Then we have

$\|\varphi_{N,r}\|^{2}\;\begin{cases}=\frac{r}{2\pi},&\;N=0;\\\
\leq\frac{r}{2\pi}\wedge\frac{r^{3/2}}{\sqrt{8}N},&\;N\geq 0\end{cases}$ (82)

###### Proof.

Let $\tau$ be the exponential variable, and let prime denote independent
copies. Then $\|\varphi_{N,r}\|^{2}$ is the density of
$B_{\tau}-B^{\prime}_{\tau^{\prime}}$ at zero. Conditionally on $\tau$ and
$\tau^{\prime}$, $B_{\tau}-B^{\prime}_{\tau^{\prime}}$ is a Gaussian with mean
$(\tau-\tau^{\prime})N$ and variance $\tau+\tau^{\prime}$. Therefore

$\|\varphi_{N,r}\|^{2}=\int_{0}^{\infty}\int_{0}^{\infty}\frac{r^{2}e^{-r(t+t^{\prime})}}{2\pi(t+t^{\prime})}\exp\Big{\\{}-\frac{(t-t^{\prime})^{2}N^{2}}{2(t+t^{\prime})}\Big{\\}}\,dt\,dt^{\prime}.$

Evaluate at $0$ to get $\frac{1}{2\pi}$ and note that it is decreasing in $N$
to obtain the first part of (82). To get the second statement, change
variables $u=t+t^{\prime}$, $v=N(t-t^{\prime})$. The Jacobian is $2N$, and we
get

$\frac{r^{2}}{4\pi
N}\int_{0}^{\infty}\frac{e^{-ru}}{u}\int_{-Nu}^{Nu}e^{-\frac{v^{2}}{2u}}\,dv\,du\leq\frac{r^{2}}{4\pi
N}\int_{0}^{\infty}\frac{e^{-ru}}{u}\int_{-\infty}^{\infty}e^{-\frac{v^{2}}{2u}}\,dv\,du=\frac{r^{2}}{\sqrt{8r}N}.\qed$

###  5.3 Brownian intersection local time

Next we consider the special case of $R=\mathbb{R}^{2}$, $\mathcal{B}$ the
Borel sets, $\mu=$Lebesgue measure.

######  Proposition 64.

Let $B$ be planar Brownian motion with drift $\mu$ run until time $t$. Then
for every $k\geq 1$, $\alpha(B,B^{\prime})$ exists as a limit in $L^{k}$, and
we have

$E[\alpha(B,B^{\prime})^{k}]\leq ek!\sqrt{k}(te/(2\pi))^{k}.$ (83)

Note that in the driftless case Le Gall (1994) shows $a_{1}^{k}k!\leq
E\alpha(B,B^{\prime})^{k}\leq a_{2}^{k}k!$, so (83) is not far from the truth.

###### Proof.

By scaling, it suffices to show this for $t=1$. By Proposition 62 and Lemma
63,

$\|\rho_{k}\|^{2}\leq
k!^{2}e^{2r}\|\varphi_{r}/r\|^{2k}\leq\frac{k!^{2}e^{2r}}{(2\pi r)^{k}}.$ (84)

Since this is finite for all $k$, Proposition 60 implies that the intersection
local time $\alpha$ exists and $\|\rho_{k}\|^{2}=E\alpha^{k}$. Setting $r=k$
and using the inequality version of Stirling’s formula $k!\leq
ek^{1/2+k}e^{-k}$ for $k\geq 1$ gives (83). ∎

######  Proposition 65 (From BM to BB).

Let $R$ be a Brownian bridge with covariance $\Sigma$ from $p$ to $p+\mu s$ in
$\mathbb{R}^{d}$ in time $s$. Let $a\in(0,1)$ and let $F(R_{[0,as]})$ be a
nonnegative test function depending on the initial part of $R$. Let $B$ be a
Brownian motion with drift $\mu$ started at $p$. Then

$EF(R_{[0,\alpha s]})\leq(1-a)^{-d/2}EF(B_{[0,\alpha s]}).$

By time reversal, a similar claim holds for the last $a$ portion of $R$.

###### Proof.

The conditional distributions $R_{[0,as]}$ given $R(as)$, and $B_{[0,as]}$
given $B(as)$ are the same: both are Brownian bridges with the given
endpoints. So it suffices to bound the Radon-Nikodym derivative of $R(as)$
with respect to $B(as)$. They are Gaussians with the same mean and variance
$as\Sigma$ and $as(1-a)\Sigma$, respectively. The maximal derivative does not
depend on a common scale or shift, and it is $(1-a)^{-d/2}$ when $s=1$,
$\Sigma=I$. ∎

We are now ready to bound intersection local time for Brownian bridges.

######  Proposition 66.

Consider planar Brownian bridge $R$ from $0$ at time $0$ to $\mu t$ at time
$t$. Then for all $k\in\mathbb{N}$, the intersection local time
$\alpha(R,R^{\prime})$ exists as a limit in $L^{k}$ and

$E\alpha^{k}\leq 4ek!\sqrt{k}(te/\pi)^{k}.$

###### Proof.

Let $R_{1}$, $R_{2}$ denote the bridge restricted to $[0,t/2]$ and $[t/2,t]$,
respectively. Let $Q$ denote the law of $R_{1}$. Then $Q^{\otimes 2}$ is
absolutely continuous with respect to $Q_{BM}^{\otimes 2}$ of Brownian motion
with drift $\mu$ run until time $t$ with derivative bounded by $4$, see
Proposition 65. In particular, since $\alpha_{n}$ converges in
$L^{k}(Q_{BM}^{\otimes 2})$, it also converges in $L^{k}(Q^{\otimes 2})$, and
with $\alpha=\alpha(R_{1},R_{1}^{\prime})$

$E_{Q^{\otimes 2}}\alpha^{k}\leq 4E{Q_{BM}^{\otimes 2}}\alpha^{k}\leq
4ek!\sqrt{k}(te/(4\pi))^{k}.$

The same bound applies to $R_{2}$, and Lemma 58 implies that
$\alpha(R,R^{\prime})=\sum_{i,j=1}^{2}\alpha(R_{i},R^{\prime}_{j})$ exists.
Expanding the $k$-th power, and using Proposition 60.3, Hölder’s inequality
and symmetry, we see that $E\alpha(R,R^{\prime})^{k}\leq
4^{k}E\alpha(R_{1},R_{1}^{\prime})^{k}$, as claimed. ∎

######  Remark 67.

The mutual intersection local time is classically defined as the value at $0$
of the continuous density of the random measure

$A\mapsto\int_{0}^{t}\int_{0}^{t}\mathbf{1}_{A}(B(s)-B^{\prime}(s^{\prime}))dsds^{\prime},\qquad
A\in\mathscr{B}(\mathbb{R}^{2}),$ (85)

see Geman, Horowitz and Rosen (1984). By Lemma 61 this coincides with our
definition of $\alpha$ as a random variable on ${Q^{\otimes 2}}$.

Finally, we consider the case where $X$ is the occupation measure up to time
$t$ of $(b(s),s)$ where $b(s)$ is a one-dimensional Brownian bridge. The
computations are the same and we can identify the limiting $\alpha$ again by
its moments. Let $q(x,t)$ be the density at $x$ of the centered Gaussian with
variance $t$.

######  Proposition 68.

Let $X$ be the occupation measure of $(b(s),s)$ where $b(s)$ is a one-
dimensional Brownian bridge from $y_{0}=(0,0)$ to $y_{*}=(x,t)$. Then for
every $k\in\mathbb{N}$,

  1. 1.

The measure $E_{Q}[X^{\otimes k}]$ is supported on
$(\mathbb{R}\times[0,t])^{k}$ and has density

$\rho_{k,X}(\mathbf{y})=\frac{1}{q(y_{*})}\prod_{i=1}^{k+1}q(y_{(i)}-y_{(i-1)}),$

where $y_{k+1}=y_{*}$ and $y_{(i)}$ are the order statistics of $y$ ordered
increasingly by the second (time) coordinate.

  2. 2.

With $s_{0}=0,s_{k+1}=t$ we have

$\displaystyle\|\rho_{k,X}\|_{L^{2}((\mathbb{R}^{2})^{k})}^{2}$
$\displaystyle=\frac{2^{-k/2}k!}{q(0,t)}\int_{0<s_{1}<\cdots<s_{k}<t}\prod_{i=1}^{k+1}q(0,s_{i}-s_{i-1})$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!=\frac{\sqrt{t}k!}{(4\pi)^{k/2}}\int_{0<s_{1}<\cdots<s_{k}<t}\prod_{i=1}^{k+1}\frac{1}{\sqrt{s_{i}-s_{i-1}}}=\frac{\sqrt{\pi}t^{k/2}k!}{2^{k}\Gamma(\tfrac{1}{2}(k+1))}\leq
c\,\frac{(tk/(2e))^{k/2}}{\sqrt{k}}.$

  3. 3.

The intersection local time $\alpha$ exists, $\alpha=\lim\alpha_{n}$ in
$L^{k}(Q^{\otimes 2})$ and $E_{Q^{\otimes
2}}[\alpha^{k}]=\|\rho_{k}\|_{L^{2}((\mathbb{R}^{2})^{k})}^{2}$.

  4. 4.

For all $\gamma>0$ we have $E\cosh(\gamma\alpha)<\infty$.

###### Proof.

Fubini gives 1. To get 2, note that for $s$ fixed, the $x$ integral gives the
squared $L^{2}$-norm of the density of $(b(s_{1}),\ldots,b(s_{k}))$. The same
computation also gives the density of the self-convolution at zero, which is
exactly $2^{-k/2}$ times the original density at $0$ for any Gaussian vector.
Up to scaling, the integral computes the normalizing constant of a
Dirichlet$(1/2,\ldots,1/2)$ distribution. Proposition 60 implies 3. Writing
the Taylor series of $\cosh$, using Fubini and the bounds 2 gives 4. ∎

######  Remark 69.

As in Remark 67, by Lemma 61, $\alpha$ coincides with the classical
definition, the value at $0$ of the continuous density of the random measure

$A\mapsto\int_{0}^{t}\mathbf{1}_{A}(b(s)-b^{\prime}(s))ds,\qquad
A\in\mathscr{B}(\mathbb{R}),$ (86)

Let $L$ be a $d\times d$ matrix with $\det L\not=0$. Then $X\circ L(A)$ and
$X\circ L^{-1}(A)$ define the pullback and pushforward of the measure $X$ by
the linear transformation $L$.

The definition implies that $\alpha(X,X^{\prime})$ scales like
$\|\rho_{X}\|_{2}^{2}$. More precisely, we have the following.

######  Lemma 70 (Scaling $\alpha$).

Let $X,Y$ be a random measures on $\mathbb{R}^{d}$, so that
$\alpha(X,Y^{\prime})$ The mean densities satisfy $\langle\rho_{X\circ
L},\rho_{Y\circ L}\rangle_{\mu}=|\\!\det
L|\,\langle\rho_{X},\rho_{Y}\rangle_{\mu}$ and

$\alpha(X\circ L,Y^{\prime}\circ L)=|\\!\det(L)|\,\alpha(X,Y^{\prime}).$

Now let $t>0$, $(B(s),s\in[0,1])$ be a process on $\mathbb{R}^{d}$ and let
$\hat{B}(s)=LB(s/t)$ on $[0,t]$. Then the occupation measures satisfy

$\hat{X}(A)=|\hat{B}^{-1}(A)|=t|B^{-1}L^{-1}(A)|=tX\circ
L^{-1}(A),\qquad\alpha(\hat{X},\hat{X}^{\prime})=\frac{t}{|\\!\det
L|}\,\alpha(X,X^{\prime}).$

######  Proposition 71.

Let $B^{\nu}$, be a planar Brownian bridge on the time interval $[0,1]$ from
$(0,0)$ to $(y,1)$ with covariance $\operatorname{diag}(1,\nu^{2})$. Then for
every $y$,

$\phi(\nu,y):=E\alpha(B^{\nu,y},(B^{\nu,y})^{\prime}$ (87)

is continuous at $\nu=0$. Furthermore,

$\phi(0,y)=E[\alpha(b^{y},(b^{y})^{\prime})^{2}]$ (88)

where $b^{y}$ is a Brownian bridge from $0$ to $y$ in time $1$. Moreover,
$\phi(0,y)=\phi(0,0)$.

###### Proof.

By Propositions 60, 62, and 66, for $\nu>0$ we have the moment formula
$\phi(\nu,y)=\|\rho_{2,B^{\nu,y}}\|_{2}^{2}$ with $\rho_{2}$ as in (78). For
$\nu>0$, it can be written using Fourier transform as

$\phi(\nu,y)=\|\rho_{2,B^{\nu,y}}\|_{2}^{2}=\int_{\mathbb{R}^{4}}\prod_{j,k=1}^{2}\frac{dz^{j}_{k}}{2\pi}\int_{[0,1]^{4}}\prod_{j,k=1}^{2}dt^{j}_{k}\exp\\{\Psi(z,t)\\},$
(89)

where

$\displaystyle\Psi=-\frac{\nu^{2}}{2}{\rm
Var}(\sum_{j=1}^{2}z_{2}^{j}(b(t^{1}_{j})-{b^{\prime}}(t^{2}_{j}))-\frac{1}{2}{\rm
Var}(\sum_{j=1}^{2}z_{1}^{j}(b(t^{1}_{j})-{b^{\prime}}(t^{2}_{j}))+i\sum_{j=1}^{2}(z_{2}^{j}+yz_{1}^{j})(t^{1}_{j}-t^{2}_{j}).$

(see Geman and Horowitz (1980) p. 43). Dropping the imaginary part of $\Psi$
just corresponds to the same problem with the bridges going from $(0,0)$ to
$(0,0)$. Since for such bridges $E\alpha^{2}<\infty$ by Proposition 66, we see
that (89) is absolutely integrable for $\nu>0$. So we can use Fubini and
perform the $z_{2}^{j}$ integrations. We have

${\rm
Var}(\sum_{j=1}^{2}z_{2}^{j}(b(t^{1}_{j})-{b^{\prime}}(t^{2}_{j}))={z}_{2}^{T}C{z}_{2},\qquad
C=\sum_{j=1}^{2}\scriptsize{\begin{pmatrix}t_{1}^{j}(1-t_{1}^{j})&(t_{1}^{j}\wedge
t_{2}^{j})(1-t_{1}^{j}\vee t_{2}^{j})\\\ (t_{1}^{j}\wedge
t_{2}^{j})(1-t_{1}^{j}\vee t_{2}^{j})&t_{2}^{j}(1-t_{2}^{j})\end{pmatrix}},$

so the $z_{2}^{j}$ integrations give

$\int_{\mathbb{R}^{2}}\prod_{j=1}^{2}\frac{dz^{j}_{1}}{2\pi}\int_{[0,1]^{4}}\prod_{j=1}^{2}dt_{j}^{1}dt_{j}^{2}\frac{e^{-\frac{\nu^{-2}}{2}(t^{1}-t^{2})^{T}C^{-1}(t^{1}-t^{2})}}{2\pi\nu^{-2}\sqrt{\det
C}}e^{-\frac{1}{2}{\rm
Var}(\sum_{j=1}^{2}z_{1}^{j}(b(t^{1}_{j})-{b^{\prime}}(t^{2}_{j}))+iy\sum_{j=1}^{2}z_{1}^{j}(t^{1}_{j}-t^{2}_{j})}.$

Let $\nu\to 0$ to get

$\int_{\mathbb{R}^{2}}\prod_{j=1}^{2}\frac{dz^{j}_{1}}{2\pi}\int_{[0,1]^{2}}\prod_{j=1}^{2}dt_{j}\exp\\{-\frac{1}{2}{\rm
Var}(\sum_{j=1}^{2}z_{1}^{j}(b(t_{j})-{b^{\prime}}(t_{j}))\\}=\phi(0,y),$

which we identify as the Fourier transform expression for
$\|\rho_{2,b}\|_{2}^{2}$ given in Proposition 68. The last claim follows from
Lemma 70. ∎

##  6 Exponential moments off a small set for planar Brownian motion

In this section we use the moment bounds of Section 5.2 to bound the
exponential moments of intersection local times off some small exceptional
sets. We need this in two different settings. The first is for fixed drift as
the size of the exceptional set tends to zero, and the second is a uniform
bound as the drift increases. The first is needed to define the solution of
the Wick-ordered heat equation for large times, the second is needed for the
convergence to the KPZ equation.

###  6.1 Exponential moments for a short time

We start with establishing exponential moment bounds for short times.

######  Lemma 72.

Let $B_{1},B_{2}^{\prime}$ be independent planar Brownian motions on time
interval $[0,t]$ with variance $1$ with possibly different drifts and started
at possibly different locations. Then

$Ee^{\gamma\alpha(B_{1},B_{2}^{\prime})}\leq 1+\frac{2\pi\gamma
t}{(2\pi/e-\gamma t)^{2}},\qquad\gamma t\in[0,2\pi/e).$

For independent Brownian bridges $R_{1},R_{2}^{\prime}$ with variance 1, run
until time $t$ but an arbitrary start and endpoint,

$Ee^{\gamma\alpha(R_{1},R_{2}^{\prime})}\leq 1+\frac{4\pi\gamma
t}{(\pi/e-\gamma t)^{2}},\quad\gamma t\in[0,\pi/e),\qquad
Ee^{\gamma\alpha(R_{1},R_{2})}\leq e^{10\gamma},\quad\gamma t\in[0,1].$

Moreover, there is a constant $c$ such that if the Brownian motions have a
shared drift $(0,N),N\geq 0$ then

$E[\alpha(B_{1},B_{2}^{\prime})^{2}]\leq\frac{ct}{1+N^{2}t},\qquad
Ee^{\gamma\alpha(B_{1},B_{2}^{\prime})}\leq 1+\frac{c\gamma
t}{1+N\sqrt{t}},\qquad\gamma t\in[0,\pi/4].$

###### Proof.

By scaling, Lemma 70, we may assume $t=1$. With prime denoting independent
copies, by Proposition 60.3 we have

$E\alpha(B_{1},B_{2}^{\prime})^{k}\leq(E\alpha(B_{1},B_{1}^{\prime})^{k}E\alpha(B_{2},B_{2}^{\prime})^{k})^{1/2}\leq
ek!\sqrt{k}(e/(2\pi))^{k}$

Using Proposition 64 and $\sqrt{k}\leq k$, we have

$\displaystyle Ee^{\gamma\alpha(B_{1},B_{2}^{\prime})}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{\gamma^{k}}{k!}E\alpha(B_{1},B_{2}^{\prime})^{k}\leq
1+e\sum_{k=0}^{\infty}{(e\gamma/(2\pi))^{k}}k=1+\frac{2\pi\gamma}{(2\pi/e-\gamma)^{2}}.$

The bridge bound works the same way, just with the moment bounds of
Proposition 66; the exponential bounds the first bound. For the second bound,
By Proposition 62 with $k=2,r=1$ and Lemma 63,

$E\alpha(B_{1},B_{1}^{\prime})^{2}\leq\frac{c}{1+N^{2}}.$ (90)

For the third bound, for any $x\geq 0$ we have $e^{x}-1\leq xe^{x}$. By
Cauchy-Schwarz, for any nonnegative random variable $X$, we have
$(Ee^{X}-1)^{2}\leq EX^{2}Ee^{2X}$. So by what we have shown already
$Ee^{\gamma\alpha}-1\leq(E[(\gamma\alpha)^{2}]Ee^{2\gamma\alpha})^{1/2}\leq
c\gamma/(1+N).$ ∎

###  6.2 Fixed drift

The goal of this section is to show Corollary 75: on sets of probability
arbitrarily close to one, large exponential moments of the intersection local
time exist. This is used in the solution of the planar Wick-ordered heat
equation for large times.

We will control paths through two parameters. Fix $\kappa\in(0,1/2)$, and let
$\operatorname{box}(\varepsilon)$ be the set of $\varepsilon\times\varepsilon$
closed boxes with corners in $\varepsilon\mathbb{Z}^{2}$.

$\|B\|_{\kappa}=\sup_{0\leq s<t\leq
1}\frac{|B(t)-B(s)|}{(t-s)^{\kappa}},\qquad
M(B,\varepsilon,\delta)=\max_{K\in\operatorname{box}(\varepsilon)}\sum_{i=0}^{\lfloor