A LOWER-TAIL LIMIT IN THE WEAK NOISE THEORY YIER LIN AND LI-CHENG TSAI Ab.pcs.pct.pcr.pca.pcc.pct.pc. We consider the variational problem associated with the FreidlinWentzell Large Deviation Principle of the

2025-04-24 0 0 861.5KB 12 页 10玖币

A LOWER-TAIL LIMIT IN THE WEAK NOISE THEORY

YIER LIN AND LI-CHENG TSAI

Abstract. We consider the variational problem associated with the Freidlin–Wentzell Large Deviation Principle of the

Stochastic Heat Equation (SHE). The logarithm of the minimizer of the variational problem gives the most probable shape of

the solution of the Kardar–Parisi–Zhang equation conditioned on achieving certain unlikely values. Taking the SHE with the

delta initial condition and conditioning the value of its solution at the origin at a later time, under suitable scaling, we prove

that the logarithm of the minimizer converges to an explicit function as we tune the value of the conditioning to 0. Our result

conﬁrms the physics prediction [KK09,MKV16,KMS16].

1. Introduction

In this paper we study the following variational problem. For a given ρ=ρ(t, x)∈L2([0,2] ×R), consider the

heat equation driven by the potential ρ, with the delta initial condition:

∂tZ=1

2∂xxZ+ρZ,(t, x)∈(0,2) ×R,Z(0,·) = δ(·).(1.1)

We write Z=Z[ρ] = Z[ρ](t, x)for the solution of this equation. The variational problem of interest is

inf 1

2kρk2

2:Z[ρ](2,0) = e−λ,(1.2)

where λ > 0is a parameter, and k·k2denotes the L2norm. It was proven in [Tsa22b, Corollary 2.5(b)] that, for all

large enough λ, the variational problem (1.2) has a unique minimizer ρm=ρm(λ;t, x).

This variational problem describes the Freidlin–Wentzell Large Deviation Principle (LDP) of the Stochastic Heat

Equation (SHE) and the Kardar–Parisi–Zhang (KPZ) equation. Consider the SHE ∂tZε=1

2∂xxZε+√εξZεwith

Zε(0,·) = δ(·), where ξdenotes the spacetime white noise. It was proven in [LT21] that Zεenjoys the LDP with

speed ε−1and the rate function ISHE(f) := inf{1

2kρk2

2:Z[ρ] = f}.Given the LDP, we see that the variational

problem (1.2) corresponds to conditioning the value of Zε(2,0) around e−λ. Accordingly, the function Z[ρm](t, x)

is the most probable path of Zεunder the conditioning. The solution of the SHE produces the solution of the KPZ

equation through taking logarithm. Namely, log Zε=hε, and hεsolves the KPZ equation. Hence h:= log Z[ρm]

gives the most probable path of the solution of the KPZ equation, which we refer to as the most probable shape.

Studying the Freidlin–Wentzell LDPs for the SHE and the KPZ equation through the variational problem goes under

the name of the weak noise theory. There has been much development around the weak noise theory in the physics and

mathematics literature [KK07,KK08,KK09,JKM16,KMS16,MS17,HLDM+18,MV18,SM18,SMS18,ALM19,

HMS19,SMV19,HKLD20,HMS21,GLLT21,LT21], and more recently around the connection of the weak noise

theory to integrable PDEs [Kra20,KLD21,KLD22b,BSM22,KLD22a,MMS22,Tsa22b]. Among the many questions

of interest in the weak noise theory are the behaviors of the most probable shape under certain limits. Particular limits

of interest are sending the conditioned value log Z[ρ](2,0) to +∞or −∞. We refer to them as the (one-point) upper-

and lower-tail limits, respectively. For the delta initial condition considered here, the upper- and lower-tail limits of the

most probability shape (under suitable scaling) have been predicted in the physics works [KK09,MKV16,KMS16].

Later, the work [GLLT21] gave a rigorous proof of the upper-tail limit. The behaviors of the upper- and lower-tail

limits are very diﬀerent, and a rigorous proof of the lower-tail limit remained open.

In this paper, we prove the lower-tail limit of the most probable shape. Let us introduce the scaling. Set Zλ[ρ](t, x) :=

Z[λρ(·, λ−1/2·)](t, xλ1/2). Under this scaling, the variational problem becomes

(1.2)=λ5/2inf 1

2kρk2

2:Zλ[ρ](2,0) = e−λ,(1.3)

and its minimizer is ρm

λ(t, x) := λ−1ρm(λ;t, λ1/2x). Let Zλ[ρm

λ](t, x) := Zm

λ(t, x)and hλ(t, x) := 1

λlog Zm

λ(t, x).

The main result is as follows.

Theorem 1.1. Let h∗be given by (2.7)and (2.10)and depicted in Figure 2in Section 2. For any δ > 0,

lim

λ→∞ sup |hλ(t, x)−h∗(t, x)|: (t, x)∈(δ, 2] ×[−δ−1, δ−1]= 0.

arXiv:2210.05629v1 [math.PR] 11 Oct 2022

2 YIER LIN AND LI-CHENG TSAI

Let us provide a context of Theorem 1.1 in terms of Hamilton–Jacobi-Fokker–Planck (HJ-FP) equations. To begin,

one can derive the Euler–Lagrangian equation for the variational problem (1.3) and turn the equation into a system of

Hamilton equations. This derivation is done in the physics literature of the weak noise theory (see [KMS16, Appendix],

[MKV16, Supp. Mat. A], [KLD21, Supp. Mat. A]) and has recently been proven in [Tsa22b, Theorem 2.1] (at the level

of Zm

λ). At the level of hλ, the Hamilton equations are

∂thλ=1

2λ∂xxhλ+1

2(∂xhλ)2+ρm

λ,(1.4a)

−∂tρm

λ=1

2λ∂xxρm

λ−∂x(ρm

λ∂xhλ),(1.4b)

where the second equation is solved backward in time with a suitable terminal condition at t= 2. This system can

be viewed as an instance of the HJ-FP equations studied in mean ﬁeld games [Lio07,Car10,GLL11]. The lower-tail

limit λ→ ∞ here corresponds to the inviscid limit in the language of mean ﬁeld games and PDE. We expect that, for

ρm

λ≤0, the solution of (1.4) converges to the entropy solution of

∂th∗=1

2(∂xh∗)2+ρ∗,(1.5a)

−∂tρ∗=−∂x(ρ∗∂xh∗).(1.5b)

The physic works [KK09,MKV16,KMS16] solved (1.5) for the entropy solution and found an explicit expression for

ρ∗. Accordingly, the function h∗can be expressed in terms of ρ∗; see Section 2.2.

From the PDE perspective, the challenge of proving Theorem 1.1 lies in controlling ρm

λ. We will show in Appen-

dix A.2 that ρm

λ≤0for all λlarge enough, and we will explain in Section 2.2 that the results from [LT21] gives ρm

λ→ρ∗

in L2. These properties alone, however, do not suﬃce for hλ→h∗. For b > 0, set wλ(t, x) := ρ∗(t, x)1{|x|>λ−b},

which is non-positive and converges to ρ∗in L2. It is possible to show that, there exist a small enough band a

neighborhood Ωaround (2,0) such that limλ→∞ infΩ|1

λlog Zλ[wλ]−h∗|>0. In many settings of mean ﬁeld games,

the term ρm

λin (1.4a) is replaced by a better-behaved term, and phenomena like the one just shown does not occur.

Our proof proceeds through the Feynman–Kac formula and bypasses the need to control ρm

λ. The ﬁrst key observation

is that the property ρm

λ→ρ∗in L2alone does suﬃce for proving the lower bound (lim infλ→∞ hλ)≥h∗and in

fact a stronger version of it: Proposition 3.1. Roughly speaking, Proposition 3.1 states that the change of hλalong a

geodesic (deﬁned in Section 2.2) is bounded from below by the change of h∗along the same geodesic. The second

key observation is that the upper bound, which is the more subtle bound, follows by combining Proposition 3.1 and the

property Zm

λ(2,0) = eλhλ(2,0) =e−λ, as well as a Hölder continuity estimate. This observation is manifested in the

proof in Section 5.1.

Let us mention that Zm

λpermits an explicit formula. The expression was derived by [KLD21] based on [Kra20] at a

physics level of rigor, and later proven in [Tsa22b, Theorem 2.3, Corollary 2.5]. Extracting the limit of hλ:= 1

λlog Zm

from this explicit formula is an interesting open problem and will provide another proof of Theorem 1.1.

We conclude this introduction by discussing some related works. There exists another approach to study the Freidlin–

Wentzell LDPs for the SHE and the KPZ equation, based on explicit formulas of the one-point distribution. The analysis

has been carried out in the physics literature for various initial and boundary conditions [LDMRS16,KLD17,KLD18a].

While the explicit formulas does not give direct access to the most probable shape, they allow for studying LDPs for

the KPZ equation in the long time regime [LDMS16,SMP17,CGK+18,KLD18b,KLDP18,Kra19,KLD19,LD20,

CG20b,CG20a,DT21,Kim21,Lin21,CC22,GL22,Tsa22a]. See also [GH22,LW22] for related work.

Outline. In Section 2we prepare some notation and recall some basic tools. In Section 3, we prove a lower bound,

which in particular gives the lower half of Theorem 1.1. In Section 4, we prove a spatial Hölder continuity. In Section 5,

we prove the upper half of Theorem 1.1.

Acknowledgment. We thank Daniel Lacker and Panagiotis Souganidis for useful discussions. The research of Tsai

was partially supported by the NSF through DMS-2243112 and the Sloan Foundation Fellowship.

2. Notation and tools

2.1. The Feynman–Kac formula. Let us introduce the Feynman–Kac formula for Zλ[ρ]. For s≤t∈[0,2], let

BBλ((t, x)→(s, y)) denote the law of the following Brownian bridge, which goes backward in time:

λ−1/2B(t−u)−u−s

t−sB(t−s)+(u−s)x+(t−u)y

t−s, u ∈[s, t],(2.1)

A LOWER-TAIL LIMIT IN THE WEAK NOISE THEORY 3

where Bdenotes a standard Brownian motion. For any ρ∈L2([0,2] ×R), deﬁne the Feynman–Kac expectation

FKλ[ρ](s, y;t, x) := Ehexp Zt

λρ(u, Wλ(u)) dui, Wλ∼BBλ((t, x)→(s, y)).(2.2)

Let pλ(t, x) := exp(−λx2

2t)pλ/(2πt)denote the scaled heat kernel. We have the Feynman–Kac formula

Zλ[ρ](t, x) = pλ(t, x)FKλ[ρ](0,0; t, x),(2.3)

Zλ[ρ](t, x) = ZR

pλ(t−s, x −y)FKλ[ρ](s, y;t, x)Zλ[ρ](s, y) dy, s < t ∈(0,2].(2.4)

For the sake of completeness, we provide a proof of (2.3) in Appendix A.3. The formula (2.4) follows from (2.3) via

conditioning Wλat time s.

2.2. The limits ρ∗and h∗.We begin by introducing ρ∗, which is the limit of ρm

λas λ→ ∞. Let rbe the unique

C1[1,2)-valued solution with r|(1,2) >π

2of the diﬀerential equation

˙r= 21/2π−1/2r2(r−π

2)1/2, r(1) = π

2,(2.5)

and symmetrically extend rto C1(0,2) by setting r(t) := r(2 −t)for t∈(0,1). Set `(t) := 1/r(t). Integrating the

diﬀerential equation by separation of variables and analyzing the result near t= 2 give

r(t) = r(2 −t)=(2

3)2/3(π

2)1/3t−2/3+o(t−2/3)when t→0.(2.6)

In particular, r→+∞as t→0or 2. We hence set `(0) = `(2) := 0 to make `∈C[0,2]. The limit ρ∗is given by

ρ∗(t, x) := −1

2πr(t)1−x2

`(t)2+.(2.7)

This expression was derived in the physics works [KK09,MKV16,KMS16] by solving (1.5) for the entropy solution.

In the mathematics literature, [LT21] gives the L2convergence

lim

λ→∞ kρm

λ−ρ∗k2= 0.(2.8)

More precisely, this follows by combining kρm

λ−ρ∗k2

2=kρm

λk2

2− kρ∗k2

2+ 2hρ∗, ρ∗−ρm

λiand Equations (4.28) and

(4.32) in [LT21]. Note that the expression within the limsup in [LT21, Equation (4.28)] is 1

2kρm

λk2

We now turn to h∗. The function is deﬁned by

h∗(t, x) := −inf nZt

2˙γ2(s)−ρ∗(s, γ(s)) ds:γ∈H1((0,0) →(t, x))o,(2.9)

with the convention inf ∅:= ∞. Hereafter H1((s, y)→(t, x)) denotes the space of paths γ: [s, t]→Rsuch

that (γ(s), γ(t)) = (y, x)and Rt

s˙γ2du < ∞. To understand the motivation of this deﬁnition, recall that hλ:=

λlog Zλ[ρm

λ]. In view of (2.8), it is natural to expect (but non-trivial to prove) that this function approximates

λlog Z[ρ∗]as λ→ ∞. Since ρ∗is uniformly continuous except near (0,0) and (2,0), through the Feynman–Kac

formula (2.3), the last expression can be analyzed by Varadhan’s lemma, which yields (2.9) as λ→ ∞.

Crucial to our analysis is the notion of geodesics. Proposition 4.3 in [LT21] shows that the inﬁmum is achieved in

H1((0,0) →(t, x)) for all (t, x)∈(0,2] ×R. We call a path realizing the inﬁmum a geodesic. Proposition 4.3 in

[LT21] characterizes all geodesics, and they are depicted in Figure 1; see the caption there. We write (s, y)geod.

−−−→ (t, x)

for “(s, y)and (t, x)are connected by a geodesic with s≤t∈[0,2]”, and write (s, y)geod.θ

−−−→ (t, x)to signify that the

geodesic is θ. We can rewrite (2.9) more explicitly as

h∗(t, x) = Zt

0−1

2˙

θ2(u) + ρ∗(u, θ(u))du, (0,0) geod.θ

−−−→ (t, x).(2.10)

Note that, for any (t, x)6= (2,0) with t > 0, there exists a unique geodesic that connects (0,0) and (t, x); when

(t, x) = (2,0), the expression (2.10) holds for θ=a`, for all a∈[−1,1]; when t= 0, by deﬁnition h∗(0, x) =

01{x=0}− ∞1{x6=0}; a plot of h∗is shown in Figure 2. Further, for any given (s, y)geod.θ

−−−→ (t, x), the geodesic θcan

always be extended backward in time to (0,0). This together with (2.10) gives

h∗(t, x)−h∗(s, y) = Zt

s−1

2˙

θ2(u) + ρ∗(u, θ(u))du, (s, y)geod.θ

−−−→ (t, x).(2.11)

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ALOWER-TAILLIMITINTHEWEAKNOISETHEORYYIERLINANDLI-CHENGTSAIAbstract.WeconsiderthevariationalproblemassociatedwiththeFreidlinWentzellLargeDeviationPrincipleoftheStochasticHeatEquation(SHE).ThelogarithmoftheminimizerofthevariationalproblemgivesthemostprobableshapeofthesolutionoftheKardarParisiZhange...

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A LOWER-TAIL LIMIT IN THE WEAK NOISE THEORY YIER LIN AND LI-CHENG TSAI Ab.pcs.pct.pcr.pca.pcc.pct.pc. We consider the variational problem associated with the FreidlinWentzell Large Deviation Principle of the

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