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Weak universality of the KPZ equation
M. Hairer, joint with J. Quastel
University of Warwick
Buenos Aires, July 29, 2014
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KPZ equation
Introduced by Kardar, Parisi and Zhang in 1986.
Stochastic partial differential equation:
∂th = ∂2xh+ (∂xh)2 + ξ (d = 1)
Here ξ is space-time white noise: Gaussian generalised randomfield with Eξ(s, x)ξ(t, y) = δ(t− s)δ(y − x).
Model for propagation of nearly flat interfaces.
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KPZ equation
Introduced by Kardar, Parisi and Zhang in 1986.
Stochastic partial differential equation:
∂th = ∂2xh+ (∂xh)2 + ξ (d = 1)
Here ξ is space-time white noise: Gaussian generalised randomfield with Eξ(s, x)ξ(t, y) = δ(t− s)δ(y − x).
Model for propagation of nearly flat interfaces.
![Page 4: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/4.jpg)
KPZ equation
Introduced by Kardar, Parisi and Zhang in 1986.
Stochastic partial differential equation:
∂th = ∂2xh+ (∂xh)2 + ξ (d = 1)
Here ξ is space-time white noise: Gaussian generalised randomfield with Eξ(s, x)ξ(t, y) = δ(t− s)δ(y − x).
Model for propagation of nearly flat interfaces.
![Page 5: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/5.jpg)
KPZ equation
Introduced by Kardar, Parisi and Zhang in 1986.
Stochastic partial differential equation:
∂th = ∂2xh+ (∂xh)2 + ξ (d = 1)
Here ξ is space-time white noise: Gaussian generalised randomfield with Eξ(s, x)ξ(t, y) = δ(t− s)δ(y − x).
Model for propagation of nearly flat interfaces.
Flat interface propagation
Slope ∂xh� 1.
1
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KPZ equation
Introduced by Kardar, Parisi and Zhang in 1986.
Stochastic partial differential equation:
∂th = ∂2xh+ (∂xh)2 + ξ (d = 1)
Here ξ is space-time white noise: Gaussian generalised randomfield with Eξ(s, x)ξ(t, y) = δ(t− s)δ(y − x).
Model for propagation of nearly flat interfaces.
Flat interface propagation
Slope ∂xh� 1.
1
√1 + (∂xh)2
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Strong Universality conjecture
At large scales, the fluctuations of every 1 + 1-dimensional model hof interface propagation exhibits the same fluctuations as the KPZequation. These fluctuations are self-similar with exponents1− 2− 3:
limλ→∞
λ−1h(λ2x, λ3t)− Cλt = c1 limλ→∞
λ−1h(c2λ2x, λ3t)− Cλt .
Spectacular recent progress: Amir, Borodin, Corwin, Quastel,Sasamoto, Spohn, etc. Relies on considering models that are“exactly solvable”. Partial characterisation of limiting “KPZ fixedpoint”: experimental evidence.
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Strong Universality conjecture
At large scales, the fluctuations of every 1 + 1-dimensional model hof interface propagation exhibits the same fluctuations as the KPZequation. These fluctuations are self-similar with exponents1− 2− 3:
limλ→∞
λ−1h(λ2x, λ3t)− Cλt = c1 limλ→∞
λ−1h(c2λ2x, λ3t)− Cλt .
Spectacular recent progress: Amir, Borodin, Corwin, Quastel,Sasamoto, Spohn, etc. Relies on considering models that are“exactly solvable”. Partial characterisation of limiting “KPZ fixedpoint”: experimental evidence.
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Strong Universality conjecture
At large scales, the fluctuations of every 1 + 1-dimensional model hof interface propagation exhibits the same fluctuations as the KPZequation. These fluctuations are self-similar with exponents1− 2− 3:
limλ→∞
λ−1h(λ2x, λ3t)− Cλt = c1 limλ→∞
λ−1h(c2λ2x, λ3t)− Cλt .
Spectacular recent progress: Amir, Borodin, Corwin, Quastel,Sasamoto, Spohn, etc. Relies on considering models that are“exactly solvable”. Partial characterisation of limiting “KPZ fixedpoint”: experimental evidence.
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Heuristic picture
Schematic evolution in “space of models” under rescaling (moduloheight shifts):
TASEP
Eden
KPZ
BD
KPZ equation just one model among many...
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All interface fluctuation models
Universality for symmetric interface fluctuation models: exponents1− 2− 4, Gaussian limit. Picture for all interface models:
Gauss KPZ
KPZ equation: red line.
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All interface fluctuation models
Universality for symmetric interface fluctuation models: exponents1− 2− 4, Gaussian limit. Picture for all interface models:
Gauss KPZKPZ
KPZ equation: red line.
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Weak Universality conjecture
Conjecture: the KPZ equation is the only model on the “red line”.
Conjecture: Let hε be any “natural” one-parameter family ofasymmetric interface models with ε denoting the strength of theasymmetry such that propagation speed ≈
√ε.
As ε→ 0, there is a choice of Cε ∼ ε−1 such that√εhε(ε
−1x, ε−2t)− Cεt converges to solutions h to the KPZequation.
Bertini-Giacomin (1995): proof for height function of WASEP.Jara-Goncalves (2010): accumulation points satisfy weak version ofKPZ for generalisations of WASEP.
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Weak Universality conjecture
Conjecture: the KPZ equation is the only model on the “red line”.
Conjecture: Let hε be any “natural” one-parameter family ofasymmetric interface models with ε denoting the strength of theasymmetry such that propagation speed ≈
√ε.
As ε→ 0, there is a choice of Cε ∼ ε−1 such that√εhε(ε
−1x, ε−2t)− Cεt converges to solutions h to the KPZequation.
Bertini-Giacomin (1995): proof for height function of WASEP.Jara-Goncalves (2010): accumulation points satisfy weak version ofKPZ for generalisations of WASEP.
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Weak Universality conjecture
Conjecture: the KPZ equation is the only model on the “red line”.
Conjecture: Let hε be any “natural” one-parameter family ofasymmetric interface models with ε denoting the strength of theasymmetry such that propagation speed ≈
√ε.
As ε→ 0, there is a choice of Cε ∼ ε−1 such that√εhε(ε
−1x, ε−2t)− Cεt converges to solutions h to the KPZequation.
Bertini-Giacomin (1995): proof for height function of WASEP.Jara-Goncalves (2010): accumulation points satisfy weak version ofKPZ for generalisations of WASEP.
![Page 16: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/16.jpg)
Weak Universality conjecture
Conjecture: the KPZ equation is the only model on the “red line”.
Conjecture: Let hε be any “natural” one-parameter family ofasymmetric interface models with ε denoting the strength of theasymmetry such that propagation speed ≈
√ε.
As ε→ 0, there is a choice of Cε ∼ ε−1 such that√εhε(ε
−1x, ε−2t)− Cεt converges to solutions h to the KPZequation.
Bertini-Giacomin (1995): proof for height function of WASEP.Jara-Goncalves (2010): accumulation points satisfy weak version ofKPZ for generalisations of WASEP.
![Page 17: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/17.jpg)
Difficulty
Problem: KPZ equation is ill-posed:
∂th = ∂2xh+ (∂xh)2 + ξ ,
Solution behaves like Brownian motion for fixed times: nowheredifferentiable!
Trick: Write Z = eh (Hopf-Cole) and formally derive
∂tZ = ∂2xZ + Zξ ,
then interpret as Ito equation. WASEP behaves “nicely” under thistransformation. Many other models do not...
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Difficulty
Problem: KPZ equation is ill-posed:
∂th = ∂2xh+ (∂xh)2 + ξ ,
Solution behaves like Brownian motion for fixed times: nowheredifferentiable!
Trick: Write Z = eh (Hopf-Cole) and formally derive
∂tZ = ∂2xZ + Zξ ,
then interpret as Ito equation. WASEP behaves “nicely” under thistransformation. Many other models do not...
![Page 19: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/19.jpg)
Difficulty
Problem: KPZ equation is ill-posed:
∂th = ∂2xh+ (∂xh)2 + ξ ,
Solution behaves like Brownian motion for fixed times: nowheredifferentiable!
Trick: Write Z = eh (Hopf-Cole) and formally derive
∂tZ = ∂2xZ + Zξ ,
then interpret as Ito equation. WASEP behaves “nicely” under thistransformation. Many other models do not...
![Page 20: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/20.jpg)
Recent progress
Theory of rough paths / regularity structures / paraproducts givesdirect meaning to nonlinearity in a robust way:
F M
R
×× C(S1) Dγ
·
F
R
× ?
Noise
∈
× C(S1)
I.C.
∈C(S1 ×R+)
RΨ
SA
SC
F : Constant C in ∂th = ∂2xh+ (∂xh)2 + ξε − C.SC : Classical solution to the PDE with smooth input.SA: Abstract fixed point: locally jointly continuous!
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Recent progress
Theory of rough paths / regularity structures / paraproducts givesdirect meaning to nonlinearity in a robust way:
F M
R
×× C(S1) Dγ
·
F
R
× ?
Noise
∈
× C(S1)
I.C.
∈C(S1 ×R+)
RΨ
SA
SC
F : Constant C in ∂th = ∂2xh+ (∂xh)2 + ξε − C.SC : Classical solution to the PDE with smooth input.SA: Abstract fixed point: locally jointly continuous!
![Page 22: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/22.jpg)
Recent progress
Theory of rough paths / regularity structures / paraproducts givesdirect meaning to nonlinearity in a robust way:
F M
R
×× C(S1) Dγ
·
F
R
× ?
Noise
∈
× C(S1)
I.C.
∈C(S1 ×R+)
RΨ
SA
SC
Strategy: find Mε ∈ R such that MεΨ(ξε) converges.
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Universality result for KPZ
Consider the model
∂thε = ∂2xhε +√εP (∂xhε) + η ,
with P an even polynomial, η a Gaussian field with compactlysupported correlations %(t, x) s.t.
∫% = 1.
Theorem (H., Quastel, 2014) As ε→ 0, there is a choice ofCε ∼ ε−1 such that
√εh(ε−1x, ε−2t)− Cεt converges to solutions
to (KPZ)λ with λ depending in a non-trivial way on all coefficientsof P .
Remark: Convergence to KPZ with λ 6= 0 even if P (u) = u4!!
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Universality result for KPZ
Consider the model
∂thε = ∂2xhε +√εP (∂xhε) + η ,
with P an even polynomial, η a Gaussian field with compactlysupported correlations %(t, x) s.t.
∫% = 1.
Theorem (H., Quastel, 2014) As ε→ 0, there is a choice ofCε ∼ ε−1 such that
√εh(ε−1x, ε−2t)− Cεt converges to solutions
to (KPZ)λ with λ depending in a non-trivial way on all coefficientsof P .
Remark: Convergence to KPZ with λ 6= 0 even if P (u) = u4!!
![Page 25: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/25.jpg)
Universality result for KPZ
Consider the model
∂thε = ∂2xhε +√εP (∂xhε) + η ,
with P an even polynomial, η a Gaussian field with compactlysupported correlations %(t, x) s.t.
∫% = 1.
Theorem (H., Quastel, 2014) As ε→ 0, there is a choice ofCε ∼ ε−1 such that
√εh(ε−1x, ε−2t)− Cεt converges to solutions
to (KPZ)λ with λ depending in a non-trivial way on all coefficientsof P .
Nonlinearity λ(∂xh)2
Remark: Convergence to KPZ with λ 6= 0 even if P (u) = u4!!
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Universality result for KPZ
Consider the model
∂thε = ∂2xhε +√εP (∂xhε) + η ,
with P an even polynomial, η a Gaussian field with compactlysupported correlations %(t, x) s.t.
∫% = 1.
Theorem (H., Quastel, 2014) As ε→ 0, there is a choice ofCε ∼ ε−1 such that
√εh(ε−1x, ε−2t)− Cεt converges to solutions
to (KPZ)λ with λ depending in a non-trivial way on all coefficientsof P .
Remark: Convergence to KPZ with λ 6= 0 even if P (u) = u4!!
![Page 27: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/27.jpg)
Universality result for KPZ
Consider the model
∂thε = ∂2xhε +√εP (∂xhε) + η ,
with P an even polynomial, η a Gaussian field with compactlysupported correlations %(t, x) s.t.
∫% = 1.
Theorem (H., Quastel, 2014) As ε→ 0, there is a choice ofCε ∼ ε−1 such that
√εh(ε−1x, ε−2t)− Cεt converges to solutions
to (KPZ)λ with λ depending in a non-trivial way on all coefficientsof P .
Remark: Convergence to KPZ with λ 6= 0 even if P (u) = u4!!
![Page 28: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/28.jpg)
Case P (u) = u4
Write hε(x, t) =√εh(ε−1x, ε−2t)− Cεt. Satisfies
∂thε = ∂2xhε + ε(∂xhε)4 + ξε − Cε ,
with ξε an ε-approximation to white noise.
Fact: Derivatives of microscopic model do not converge to 0 asε→ 0: no small gradients! Heuristic: gradients have O(1)fluctuations but are small on average over large scales... Generalformula:
λ =1
2
∫P ′′(u)µ(du) , Cε =
1
ε
∫P (u)µ(du) +O(1) ,
with µ a Gaussian measure, explicitly computable variance.
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Case P (u) = u4
Write hε(x, t) =√εh(ε−1x, ε−2t)− Cεt. Satisfies
∂thε = ∂2xhε + ε(∂xhε)4 + ξε − Cε ,
with ξε an ε-approximation to white noise.
Fact: Derivatives of microscopic model do not converge to 0 asε→ 0: no small gradients! Heuristic: gradients have O(1)fluctuations but are small on average over large scales... Generalformula:
λ =1
2
∫P ′′(u)µ(du) , Cε =
1
ε
∫P (u)µ(du) +O(1) ,
with µ a Gaussian measure, explicitly computable variance.
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Case P (u) = u4
Write hε(x, t) =√εh(ε−1x, ε−2t)− Cεt. Satisfies
∂thε = ∂2xhε + ε(∂xhε)4 + ξε − Cε ,
with ξε an ε-approximation to white noise.
Fact: Derivatives of microscopic model do not converge to 0 asε→ 0: no small gradients! Heuristic: gradients have O(1)fluctuations but are small on average over large scales... Generalformula:
λ =1
2
∫P ′′(u)µ(du) , Cε =
1
ε
∫P (u)µ(du) +O(1) ,
with µ a Gaussian measure, explicitly computable variance.
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Main step in proof
Rewrite general equation in integral form as
H = P(E(DH)4 + a(DH)2 + Ξ
),
with E an abstract integration operator of order 1.
Find two-parameter lift of noise η 7→ Ψα,c(η) so that h = RHsolves
∂th = ∂2xh+ αH4(∂xh, c) + aH2(∂xh, c) + η
= ∂2xh+ α(∂xh)4 + (a− 6αc)(∂xh)2 + (3αc2 − ac) + η .
Show that Ψε,1/ε(ξε) converges to same limit as Ψ0,1/ε(ξε)!(Actually Ψβε,1/ε(ξε) for every β ∈ R...)
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Main step in proof
Rewrite general equation in integral form as
H = P(E(DH)4 + a(DH)2 + Ξ
),
with E an abstract integration operator of order 1.
Find two-parameter lift of noise η 7→ Ψα,c(η) so that h = RHsolves
∂th = ∂2xh+ αH4(∂xh, c) + aH2(∂xh, c) + η
= ∂2xh+ α(∂xh)4 + (a− 6αc)(∂xh)2 + (3αc2 − ac) + η .
Show that Ψε,1/ε(ξε) converges to same limit as Ψ0,1/ε(ξε)!(Actually Ψβε,1/ε(ξε) for every β ∈ R...)
![Page 33: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/33.jpg)
Main step in proof
Rewrite general equation in integral form as
H = P(E(DH)4 + a(DH)2 + Ξ
),
with E an abstract integration operator of order 1.
Find two-parameter lift of noise η 7→ Ψα,c(η) so that h = RHsolves
∂th = ∂2xh+ αH4(∂xh, c) + aH2(∂xh, c) + η
= ∂2xh+ α(∂xh)4 + (a− 6αc)(∂xh)2 + (3αc2 − ac) + η .
Show that Ψε,1/ε(ξε) converges to same limit as Ψ0,1/ε(ξε)!(Actually Ψβε,1/ε(ξε) for every β ∈ R...)
F M ×× C(S1) Dγ
·
F × ?
Noise
∈
× C(S1)
I.C.
∈C(S1 ×R+)
RΨα,c
SA
SC
![Page 34: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/34.jpg)
Main step in proof
Rewrite general equation in integral form as
H = P(E(DH)4 + a(DH)2 + Ξ
),
with E an abstract integration operator of order 1.
Find two-parameter lift of noise η 7→ Ψα,c(η) so that h = RHsolves
∂th = ∂2xh+ αH4(∂xh, c) + aH2(∂xh, c) + η
= ∂2xh+ α(∂xh)4 + (a− 6αc)(∂xh)2 + (3αc2 − ac) + η .
Show that Ψε,1/ε(ξε) converges to same limit as Ψ0,1/ε(ξε)!(Actually Ψβε,1/ε(ξε) for every β ∈ R...)
![Page 35: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/35.jpg)
Main step in proof
Rewrite general equation in integral form as
H = P(E(DH)4 + a(DH)2 + Ξ
),
with E an abstract integration operator of order 1.
Find two-parameter lift of noise η 7→ Ψα,c(η) so that h = RHsolves
∂th = ∂2xh+ αH4(∂xh, c) + aH2(∂xh, c) + η
= ∂2xh+ α(∂xh)4 + (a− 6αc)(∂xh)2 + (3αc2 − ac) + η .
Show that Ψε,1/ε(ξε) converges to same limit as Ψ0,1/ε(ξε)!(Actually Ψβε,1/ε(ξε) for every β ∈ R...)
![Page 36: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/36.jpg)
Outlook
Some open questions:
1. Strong Universality without exact solvability???
2. Convergence of particle models. (Are “weak solutions”unique??)
3. Convergence on whole space instead of circle (cf. Labbe).
4. Models with non-Gaussian noise.
5. Fully nonlinear continuum models.
6. Control over larger scales to see convergence to KPZ fixedpoint.
![Page 37: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/37.jpg)
Outlook
Some open questions:
1. Strong Universality without exact solvability???
2. Convergence of particle models. (Are “weak solutions”unique??)
3. Convergence on whole space instead of circle (cf. Labbe).
4. Models with non-Gaussian noise.
5. Fully nonlinear continuum models.
6. Control over larger scales to see convergence to KPZ fixedpoint.
![Page 38: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/38.jpg)
Outlook
Some open questions:
1. Strong Universality without exact solvability???
2. Convergence of particle models. (Are “weak solutions”unique??)
3. Convergence on whole space instead of circle (cf. Labbe).
4. Models with non-Gaussian noise.
5. Fully nonlinear continuum models.
6. Control over larger scales to see convergence to KPZ fixedpoint.
![Page 39: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/39.jpg)
Outlook
Some open questions:
1. Strong Universality without exact solvability???
2. Convergence of particle models. (Are “weak solutions”unique??)
3. Convergence on whole space instead of circle (cf. Labbe).
4. Models with non-Gaussian noise.
5. Fully nonlinear continuum models.
6. Control over larger scales to see convergence to KPZ fixedpoint.
![Page 40: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/40.jpg)
Outlook
Some open questions:
1. Strong Universality without exact solvability???
2. Convergence of particle models. (Are “weak solutions”unique??)
3. Convergence on whole space instead of circle (cf. Labbe).
4. Models with non-Gaussian noise.
5. Fully nonlinear continuum models.
6. Control over larger scales to see convergence to KPZ fixedpoint.
![Page 41: Weak universality of the KPZ equationmate.dm.uba.ar/~probab/spa2014/pdf/hairer.pdfStrong Universality conjecture Atlarge scales, the uctuations of every 1+1-dimensional model ~h of](https://reader033.vdocuments.net/reader033/viewer/2022042811/5fa5d2679c711609de31484e/html5/thumbnails/41.jpg)
Outlook
Some open questions:
1. Strong Universality without exact solvability???
2. Convergence of particle models. (Are “weak solutions”unique??)
3. Convergence on whole space instead of circle (cf. Labbe).
4. Models with non-Gaussian noise.
5. Fully nonlinear continuum models.
6. Control over larger scales to see convergence to KPZ fixedpoint.