KPZ FLUCTUATIONS IN THE PLANAR STOCHASTIC HEAT EQUATION JEREMY QUASTEL ALEJANDRO RAM IREZ AND B ALINT VIR AG

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KPZ FLUCTUATIONS IN THE PLANAR STOCHASTIC HEAT

EQUATION

JEREMY QUASTEL, ALEJANDRO RAM´

IREZ, AND B ´

ALINT VIR ´

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 L1limit

of a martingale given by the Feyman-Kac formula and deﬁnes a random-

ized shift, or Gaussian multiplicative chaos. The ﬂuctuations far from the

centre are shown to be given by the one-dimensional KPZ equation.

CONTENTS

1. Introduction and main results 2

1.1. History 10

1.2. Structure of the article 11

2. Skorokhod integral 11

2.1. White noise 11

2.2. Deﬁnition of the integral 12

2.3. Projections 13

2.4. Examples of Skorokhod integrals 15

2.5. Classical Skorokhod integral 18

3. Solving Wick-ordered heat equations 23

3.1. Projections of the equation and their solution 23

3.2. Solving the planar Wick-ordered heat equation 27

3.3. The SHE in one dimension with space-time noise 28

4. Randomized shifts 29

4.1. Convergence in L131

4.2. Properties of randomized shifts 32

4.3. Existence of the randomized shift 33

4.4. Properties of Pmjm′

j. 36

4.5. Convergence of randomized shifts partition functions 38

4.6. The polymer measure 40

4.7. Historical comment 44

5. Intersection local time 45

5.1. Intersection local time for random measures 45

5.2. Moment bounds 48

5.3. Brownian intersection local time 50

6. Exponential moments off a small set for planar Brownian motion 54

arXiv:2210.13607v3 [math.PR] 10 Apr 2024

2 JEREMY QUASTEL, ALEJANDRO RAM´

IREZ, AND B ´

ALINT VIR ´

FIGURE 1. Solution of the planar Wick-ordered heat equa-

tion at ﬁxed time started from δ0

6.1. Exponential moments for a short time 55

6.2. Fixed drift 56

6.3. Uniform exponential moments for large drift 59

7. Proofs of the main theorems 63

7.1. Proof of Theorems 2 and 3 63

7.2. Proof of Theorem 4 64

7.3. Proof of Theorem 6, tightness and polymer limits 66

References 67

1. INTRODUCTION AND MAIN RESULTS

Although the KPZ universality class should describe random planar ge-

ometry, until recently models shown to be universal have all been directed,

missing some crucial symmetries. In this article we study the planar Wick-

ordered heat equation

(1) ∂tu=1

2∆u+u ξ, u(·,0) = ς

where ξis a planar white noise that does not depend on time (Figure 1).

The precise deﬁnition uses the mild form (4) of the heat equation, in which

the white noise term is given by a Skorokhod integral, a close relative of the

Itˆ

o 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

KPZ FLUCTUATIONS IN THE PLANAR STOCHASTIC HEAT EQUATION 3

Hu (2002), after which the L2norm blows up, and the chaos expansion fails

to converge. The blow-up time, tc, 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 el-

ementary version of the Skorokhod integral to deﬁne the solution (1) for all

times, including t>tc. We give a construction of uas a randomized shift,

or as the free energy of an undirected polymer in a random environment.

This holds for all times as an L1functional of the white noise, and coincides

for t<tcwith the chaos series. Let pdenote the planar heat kernel,

(2) p(x, t) = 1

2πt exp{−|x|2/2t}

and (Ξ,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 R2. We have the following deﬁnition.

Deﬁnition 1. Let ςbe a ﬁnite measure on R2. Then u∈L1(R2×(0, T ]×

Ξ) is a solution of the planar Wick-ordered heat equation (1) with initial

condition ςif for almost all (x, t, ξ)∈R2×(0, T ]×Ξthe random function

of y,

(3) Kx,t(u)(y) = Zt

p(x−y, t −s)u(y, s)ds,

is Skorokhod-integrable (Deﬁnition 10) and satisﬁes

(4) u(x, t) = Zp(x−y, t)dς(y) + Z

oKx,t(u)ξ.

By linearity, it sufﬁces to understand Dirac δinitial conditions. The ex-

plicit solution is given as follows.

Theorem 2. With ς=δ0, the planar Wick-ordered heat equation has the following

unique solution. Let ejbe bounded functions forming an orthonormal basis of

L2(R2), let ξj=⟨ej, ξ⟩, let Bbe a planar Brownian bridge from 0to xin time t

deﬁned on a probability space (Ω,G, Q)independent of ξ. Then

u(x, t) = p(x, t) lim

n→∞ Zn, Zn=EQexp n

j=1

mjξj−1

j=1

j,(5)

mj=Zt

ej(B(s)) ds.

Znis a uniformly integrable martingale, so it converges almost surely and in

L1(P)to a limit Z. For t < tc, it converges in L2(P). The solution does not

depend on the choice of basis ej.

4 JEREMY QUASTEL, ALEJANDRO RAM´

IREZ, AND B ´

ALINT VIR ´

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

(6) u(x, t) = lim

n→∞ EQexp Zt

ξ[n](B(s))ds −norm(n, ω)p(x, t).

where ξ[n]is a molliﬁed version of the noise, and norm is some normal-

ization. Our solution (5) is exactly in this form, with ξ[n]=Pn

j=1 ξjej, a

Gaussian noise with ndegrees of freedom. We will use this molliﬁcation

throughout the paper.

We expect the random polymer measure Mξto be the a measure on con-

tinuous paths [0, t]→R2written as

(7) Mξ= lim

n→∞ exp Zt

ξ[n](B(s))ds −norm(n, ω)dsQ

where Qis the Brownian bridge law. We might write the integral as ⟨ξ[n], X⟩

where

X(A) = Zt

1(B(s)∈A)ds

is the occupation measure of the Brownian bridge up to time t. Since Xis

singular with respect to Lebesgue measure, ⟨ξ[n], X⟩will not have a limit,

as the measure Xwill not smooth out the limiting white noise.

However, we expect that the L2norm ⟨Xn, Xn⟩of the molliﬁed ver-

sion Xn=Pn

j=1 ej⟨X, ej⟩converges, after subtracting a normalizing con-

stant, to twice the two-dimensional self-intersection local time ⟨Xn, Xn⟩ −

EQBM ⟨Xn, Xn⟩ → 2γt. Here QBM signiﬁes that the normalization uses

Brownian motion from 0rather than Brownian bridge from 0to x. Such

results are shown in Varadhan (1969), Le Gall (1994) for certain molliﬁca-

tions.

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

we get

(8) EPEQBM e⟨ξ[n],X⟩=EQBM EPe⟨ξ,Xn⟩=EQBM e1

2⟨Xn,Xn⟩

which suggests that we could use norm(n, ω) = EQBM ⟨Xn, Xn⟩/2. This

breaks the semigroup property slightly, but the problem can be ﬁxed by

adding a t-dependent term to get norm(n, ω) = EQBM ⟨Xn, Xn⟩/2−1

2πtlog t,

see Example 59.

This choice corresponds to renormalizing the solution of the equation

∂tun=1

2∆un+unξ[n]as e−EQBM ⟨Xn,Xn⟩/2+ 1

2πtlog tunto obtain a limit. The

result is called PAM in the literature, for parabolic Anderson model, see,

for example Hairer and Labb´

e (2015). PAM satisﬁes the usual semigroup

property. Note however, that from the above construction the expectation

of the polymer measure over the noise ξis not the original Brownian bridge

KPZ FLUCTUATIONS IN THE PLANAR STOCHASTIC HEAT EQUATION 5

Q, but instead the re-weighted bridge measure eγt+1

2πtlog tQ. So Mξis a

randomized version of the eγt+1

2πtlog tQ, 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 expec-

tations. This corresponds to renormalizing Gaussian exponentials as

(9) exp{W−1

2EPW2}.

Our renormalization is essentially that, up to the t-dependent factor: we

use norm(n, ω) = ⟨Xn, Xn⟩/2instead of EQBM ⟨Xn, Xn⟩/2−1

2πtlog t. This

leads to (5), and, remarkably, to the Skorokhod interpretation of the pla-

nar SHE. The price is that the resulting solution of (1) is no longer a semi-

group, since the Brownian motions weighted by their renormalized self-

intersection local time no longer have the Markov property.

In summary, one has to choose whether to keep the expected path mea-

sure as Brownian bridge, or to keep the semigroup property. As in the Itˆ

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, recovering the semigroup property, Example 59. This will be used in

a future article to extend the present results to PAM itself.

The cutoff (5) is not directly related to standard discrete polymer models.

Instead, consider a random environment of i.i.d. random variables ζi,j,

i, j ∈Zand a random walk Ron Z2. The standard energy is

(10)

n=1

ζRn=X

i,j

ζi,jmi,j ,

where mi,j is the number of visits of the walk to site i, j up to time N. To ob-

tain a polymer model which is a discrete analogue of the 2d Wick ordered

polymer, this energy should be balanced by subtracting Pi,j log EPemi,j ξi,j

so that the expectation over the random environment of the exponential

of the energy is unity. Such balanced polymer models can be expected

to converge to our Zin the weak noise (intermediate disorder) limit un-

der appropriate moment conditions, although chaos expansions can only

work for small times, and the methods developed in this paper only apply

directly in the Gaussian case. We leave this for future work.

The Skorokhod integral also has some pleasant computational proper-

ties. 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 (4), a key property of the Skorokhod in-

tegral. These projections are in fact deterministic ﬁnite-dimensional linear

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KPZFLUCTUATIONSINTHEPLANARSTOCHASTICHEATEQUATIONJEREMYQUASTEL,ALEJANDRORAM´IREZ,ANDB´ALINTVIR´AGABSTRACT.WeuseaversionoftheSkorokhodintegraltogiveasimpleandrigorousformulationoftheWick-orderedstochasticheatequationwithplanarwhitenoise,representingthefreeenergyofanundirectedrandompolymer.Thesolutionf...

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KPZ FLUCTUATIONS IN THE PLANAR STOCHASTIC HEAT EQUATION JEREMY QUASTEL ALEJANDRO RAM IREZ AND B ALINT VIR AG

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