On the 2d KPZ Equation

On the 2d KPZ Equation

Francesco Caravenna

Universit`a degli Studi di Milano-Bicocca

Half day in Stochastic Analysis and applications Padova ∼ 30 October 2019

Collaborators

Nikos Zygouras (Warwick) and Rongfeng Sun (NUS)

Overview

This talk is about a stochastic PDE on R^d: (mainly d = 2)

I theKardar-Parisi-Zhang Equation (KPZ) Very interesting, yet ill-defined object

Plan:

1. Consider a regularized version of the equation

2. Study the limit of the solution, when regularisation is removed

Stochastic Analysis ! Statistical Mechanics

White noise

Space-time white noiseξ = ξ(t, x )on R^1+d

Randomdistribution of negative order (Schwartz) [not a function!]

Gaussian: hφ,ξi = Z

R^1+d

φ(t, x )ξ(t, x ) dt dx ∼ N (0, kφk²_L2)

Cov[ξ(t, x ),ξ(t⁰, x⁰) ] = δ(t − t⁰) δ(x − x⁰)

Case d = 0: ξ(t) = _dt^dB(t) where (Bt) is Brownian motion

The KPZ equation

KPZ

∂_th = ¹₂∆_xh + ₂¹|∇_xh|² +βξ (KPZ)

Model forrandom interface growth

h = h(t, x ) = interface height at time t ≥ 0, space x ∈ R^d

ξ = ξ(t, x ) = space-time white noise β> 0 noise strength

|∇_xh|² ill-defined

Forsmoothξ

u(t, x ) := e^{h(t,x )} (Cole-Hopf)

The multiplicative Stochastic Heat Equation (SHE)

SHE

∂_tu = ¹₂∆_xu + βu ξ (SHE)

Product u ξ ill-defined

(d = 1) SHE is well-posed by Ito integration [Walsh 80’s]

u(t, x ) is a function “KPZ solution” h(t, x ) := log u(t, x ) (d = 1) SHE and KPZ well-understood in arobust sense (“pathwise”) Regularity Structures(Hairer)

Paracontrolled Distributions(Gubinelli, Imkeller, Perkowski) Energy Solutions(Goncalves, Jara), Renormalization(Kupiainen)

Higher dimensions d ≥ 2

In dimensions d ≥ 2 there is no general theory

We mollify the white noise ξ(t, x ) in space on scaleε> 0 ξ^ε(t, ·) := ξ(t, ·) ∗%ε

Solutions h^ε(t, x ), u^ε(t, x ) are well-defined. Convergence as ε ↓ 0 ?

We need to tune disorder strength β = β

βε=









 βˆ 1

p| log ε | (d = 2) βˆε^{d −22} (d ≥ 3)

β ∈ (0, ∞)ˆ

Back to KPZ

We now plug ξ ξ^ε and β βε into KPZ

We also subtract adiverging constant (“Renormalization”)

Renormalized and Mollified KPZ







∂th^ε= ¹₂∆h^ε + ¹₂|∇h^ε|² +βεξ^ε − cβ_ε²ε^−d h^ε(0, ·) ≡ 0

(ε-KPZ)

We present someconvergence results for h^ε(t, x ) as ε ↓ 0 Without renormalization, the solution h^ε(t, x ) does not converge!

Main results

Space dimensiond = 2 βε= βˆ

p| log ε | βˆ∈ (0, ∞)

I. Phase transition

KPZ solution h^ε(t, x ) undergoes aphase transitionatβˆc =√ 2π

II. Sub-critical regime

For all ˆβ <βˆc: convergence of h^ε(t, x ) as ε ↓ 0 (LLN + CLT)

Analogous results for the SHE solution u^ε(t, x )

Critical regime ˆβ =βˆc ? Recent progress for SHE (nothing for KPZ)

Main result I. Phase transition

Space dimension d = 2 β_ε= βˆ

p| log ε | βˆ∈ (0, ∞)

Theorem

I For ˆβ <√

2π h^ε(t, x ) −−→^d

ε↓0 σZ −¹₂σ² Z ∼ N(0, 1) σ²:= log 2π

2π −βˆ² h^ε(t, xi) −−→^d

ε↓0 asympt. independent (for distinct points xi’s)

I For βˆ≥√

2π h^ε(t, x ) −−→^d

ε↓0 − ∞

Law of large numbers

Consider the sub-critical regimeβˆ<√ 2π

I E[h^ε(t, x )] = −¹₂σ²+ o(1)

I h^ε(t, x ) asymptotically independent for distinct x ’s

Corollary: LLN

2π) h^ε(t, ·) −−−→^d

ε↓0 −¹₂σ² as a distribution on R² Z

R²

h^ε(t, x ) φ(x ) dx −−−→^d

ε↓0 −¹₂σ² Z

R²

φ(x ) dx

A picture

x ∈ R² h^(t, x )

0

−¹₂σ² σ

σ²= log 2π 2π −βˆ²

Main result II. Fluctuations

Rescale H^ε(t, x ) := h^ε(t, x ) −E[h^ε]
/βε

Theorem

for ˆβ <√

2π H^ε(t, ·) −−−→^d

ε↓0 v (t, ·) as a distrib.

v =Gaussian = solution ofEdwards-Wilkinson equation

∂_tv = ¹₂∆_xv + γξ where γ = q

2π 2π−βˆ² > 1

∂tH^ε = ¹₂∆xH^ε + ξ^ε + 

βε|∇xH^ε|² − cβεε⁻²

Last term {. . .} produces “extra” white noise! (Independent of ξ)

Other works

Alternative proof by [Gu 18] via Malliavin calculus (only for smallβ)ˆ [Chatterjee and Dunlap 18] first considered fluctuations for 2d KPZ They provedtightness of H^ε (only for smallβ)ˆ Weidentify the limit(EW) in theentire sub-critical regime ˆβ <√

2π Results in dimensions d ≥ 3 by many authors

References

With Rongfeng Sun and Nikos Zygouras:

I [CSZ 17]Universality in marginally relevant disordered systems Ann. Appl. Probab. 2017

I [CSZ 18a]On the moments of the (2+1)-dimensional directed polymer and Stochastic Heat Equation in the critical window Commun. Math. Phys. (to appear)

I [CSZ 18b] The two-dimensional KPZ equation in the entire subcritical regime

Ann. Probab. (to appear) (d = 2) [Bertini Cancrini 98]

[Chatterjee Dunlap 18] [Gu 18] [Gu Quastel Tsai 19]

(d ≥ 3) [Magnen Unterberger 18] [Gu Ryzhik Zeitouni 18]

[Dunlap Gu Ryzhik Zeitouni 19] [Comets Cosco Mukherjee 18 19a 19b]

Renormalized KPZ vs. SHE

Renormalized and Mollified KPZ







∂th^ε= ¹₂∆h^ε + ¹₂|∇h^ε|² +βεξ^ε − cβ_ε²ε^−d h^ε(0, ·) ≡ 0

(ε-KPZ)

We can write h^ε(t, x ) =: log u^ε(t, x ) and apply Ito’s formula

Mollified SHE







∂tu^ε=¹₂∆u^ε +βεu^εξ^ε u^ε(0, ·) ≡ 1

(ε-SHE)

Facts: u^ε(t, x ) > 0 and E[u^ε(t, x )] ≡ 1 ∃ subseq. limits

Directed Polymers

We can study the SHE solution u^ε(t, x ) viaDirected Polymers

sn

N z

I s = (sn)n≥0simple random walk path

I Indep. standard Gaussian RVsω(n, x ) (Disorder)

I H_N(ω, s) :=

N

X

n=1

ω(n, s_n)

Directed Polymer Partition Functions

(2d )^N

X

s=(s0,...,sN) s.r.w. path with s₀=z

e^{βHN(ω,s)−12β2N}

Directed Polymers and KPZ

Partition functionsZβ(N, z) are discrete analoguesof u^ε(t, x ) (SHE)

I Similar toFeynman-Kac formula for SHE

I They solve alattice version of the SHE

Theorem

We can approximate (in L²)

u^ε(t, x ) ≈ Zβ(N, z) and h^ε(t, x ) ≈ logZβ(N, z) where we set N = ε⁻²t , z = ε⁻¹x , βε= ε^{d −22} β

Our results are first proved for partition functionsZβ(N, z)

In conclusion

We study KPZ using partition functions of Directed Polymers We use tools from “discrete stochastic analysis”

I Polynomial chaos (analogous toWiener chaos)

I 4th Moment Theoremsto prove Gaussianity

I Hypercontractivity+Concentration of Measure

together with “classical” probabilistic techniques, esp.Renewal Theory Robustness + Universality

Next challenges

I Critical regime ˆβ =√ 2π

I Robust (pathwise) analysis of sub-critical regime ˆβ <√ 2π

Thanks.

Renormalization of KPZ

We have considered theRenormalizedMollified KPZ

∂_th^ε= ¹₂∆h^ε + ¹₂|∇h^ε|² + β_εξ^ε − cβ_ε²ε⁻² As ε ↓ 0 we formally obtain (!)

∂th = ¹₂∆h + ¹₂|∇h|² +0ξ − ∞

Are we entitled to “change the equation”? We started from (ill-posed)

∂_th = ¹₂∆h + ¹₂|∇h|² + ξ

For smooth ξ we can look at afamily of equations (A,B∈ R)

∂_th =¹₂∆h + ¹₂|∇h|² + Aξ + B

Renormalization = appropriate “reference frame”A_ε,B_εforξ^ε

Feynman-Kac for SHE

Recall the mollified SHE







∂tu^ε= ¹₂∆u^ε +βεu^εξ^ε u^ε(0, ·) ≡ 1

A stochasticFeynman-Kac formula holds

u^ε(t, x ) = E^d _ε−1x

"

exp



βεε^{−d −22} Z ε⁻²t

0

Z

R²

%(Bs− y ) ξ(ds, dy ) − q.v.

#

where%∈ C_c^∞(R^d) is the mollifier andB = (Bs)s≥0 is Brownian motion

We can identify u^ε(t, x ) ≈ Zβ(N, z) with N = ε⁻²t z = ε⁻¹x βε=ε^{d −22} β