Weak universality of the KPZ equation

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1 Weak universality of the KPZ equation M. Hairer, joint with J. Quastel University of Warwick Buenos Aires, July 29, 2014

2 KPZ equation Introduced by Kardar, Parisi and Zhang in Stochastic partial differential equation: t h = 2 xh + ( x h) 2 + ξ (d = 1) Here ξ is space-time white noise: Gaussian generalised random field with Eξ(s, x)ξ(t, y) = δ(t s)δ(y x). Model for propagation of nearly flat interfaces.

3 KPZ equation Introduced by Kardar, Parisi and Zhang in Stochastic partial differential equation: t h = 2 xh + ( x h) 2 + ξ (d = 1) Here ξ is space-time white noise: Gaussian generalised random field with Eξ(s, x)ξ(t, y) = δ(t s)δ(y x). Model for propagation of nearly flat interfaces.

4 KPZ equation Introduced by Kardar, Parisi and Zhang in Stochastic partial differential equation: t h = 2 xh + ( x h) 2 + ξ (d = 1) Here ξ is space-time white noise: Gaussian generalised random field with Eξ(s, x)ξ(t, y) = δ(t s)δ(y x). Model for propagation of nearly flat interfaces.

5 KPZ equation Introduced by Kardar, Parisi and Zhang in Stochastic partial differential Flat interface equation: propagation t h = xh 2 + ( x h) 2 + ξ (d = 1) Here ξ is space-time white noise: Gaussian generalised random field with Eξ(s, x)ξ(t, y) = δ(t s)δ(y x). Slope x h 1. 1 Model for propagation of nearly flat interfaces.

6 KPZ equation Introduced by Kardar, Parisi and Zhang in Stochastic partial differential Flat interface equation: propagation t h = xh 2 + ( x h) 2 + ξ (d = 1) 1 + ( x h) 2 Here ξ is space-time white noise: Gaussian generalised random field with Eξ(s, x)ξ(t, y) = δ(t s)δ(y x). Slope x h 1. 1 Model for propagation of nearly flat interfaces.

7 Strong Universality conjecture At large scales, the fluctuations of every dimensional model h of interface propagation exhibits the same fluctuations as the KPZ equation. These fluctuations are self-similar with exponents 1 2 3: lim λ 1 h(λ 2 x, λ 3 t) C λ t = c 1 lim λ λ λ 1 h(c 2 λ 2 x, λ 3 t) C λ t. Spectacular recent progress: Amir, Borodin, Corwin, Quastel, Sasamoto, Spohn, etc. Relies on considering models that are exactly solvable. Partial characterisation of limiting KPZ fixed point : experimental evidence.

8 Strong Universality conjecture At large scales, the fluctuations of every dimensional model h of interface propagation exhibits the same fluctuations as the KPZ equation. These fluctuations are self-similar with exponents 1 2 3: lim λ 1 h(λ 2 x, λ 3 t) C λ t = c 1 lim λ λ λ 1 h(c 2 λ 2 x, λ 3 t) C λ t. Spectacular recent progress: Amir, Borodin, Corwin, Quastel, Sasamoto, Spohn, etc. Relies on considering models that are exactly solvable. Partial characterisation of limiting KPZ fixed point : experimental evidence.

9 Strong Universality conjecture At large scales, the fluctuations of every dimensional model h of interface propagation exhibits the same fluctuations as the KPZ equation. These fluctuations are self-similar with exponents 1 2 3: lim λ 1 h (λ2 x, λ3 t) C λ t = c1 lim λ 1 h(c2 λ2 x, λ3 t) Cλ t. λ λ Spectacular recent progress: Amir, Borodin, Corwin, Quastel, Sasamoto, Spohn, etc. Relies on considering models that are exactly solvable. Partial characterisation of limiting KPZ fixed point : experimental evidence.

10 Heuristic picture Schematic evolution in space of models under rescaling (modulo height shifts): Eden BD TASEP KPZ KPZ equation just one model among many...

11 All interface fluctuation models Universality for symmetric interface fluctuation models: exponents 1 2 4, Gaussian limit. Picture for all interface models: Gauss KPZ KPZ equation: red line.

12 All interface fluctuation models Universality for symmetric interface fluctuation models: exponents 1 2 4, Gaussian limit. Picture for all interface models: Gauss KPZ KPZ KPZ equation: red line.

13 Weak Universality conjecture Conjecture: the KPZ equation is the only model on the red line. Conjecture: Let h ε be any natural one-parameter family of asymmetric interface models with ε denoting the strength of the asymmetry such that propagation speed ε. As ε 0, there is a choice of C ε ε 1 such that ε hε (ε 1 x, ε 2 t) C ε t converges to solutions h to the KPZ equation. Bertini-Giacomin (1995): proof for height function of WASEP. Jara-Gonçalves (2010): accumulation points satisfy weak version of KPZ for generalisations of WASEP.

14 Weak Universality conjecture Conjecture: the KPZ equation is the only model on the red line. Conjecture: Let h ε be any natural one-parameter family of asymmetric interface models with ε denoting the strength of the asymmetry such that propagation speed ε. As ε 0, there is a choice of C ε ε 1 such that ε hε (ε 1 x, ε 2 t) C ε t converges to solutions h to the KPZ equation. Bertini-Giacomin (1995): proof for height function of WASEP. Jara-Gonçalves (2010): accumulation points satisfy weak version of KPZ for generalisations of WASEP.

15 Weak Universality conjecture Conjecture: the KPZ equation is the only model on the red line. Conjecture: Let h ε be any natural one-parameter family of asymmetric interface models with ε denoting the strength of the asymmetry such that propagation speed ε. As ε 0, there is a choice of C ε ε 1 such that ε hε (ε 1 x, ε 2 t) C ε t converges to solutions h to the KPZ equation. Bertini-Giacomin (1995): proof for height function of WASEP. Jara-Gonçalves (2010): accumulation points satisfy weak version of KPZ for generalisations of WASEP.

16 Weak Universality conjecture Conjecture: the KPZ equation is the only model on the red line. Conjecture: Let h ε be any natural one-parameter family of asymmetric interface models with ε denoting the strength of the asymmetry such that propagation speed ε. As ε 0, there is a choice of C ε ε 1 such that ε hε (ε 1 x, ε 2 t) C ε t converges to solutions h to the KPZ equation. Bertini-Giacomin (1995): proof for height function of WASEP. Jara-Gonçalves (2010): accumulation points satisfy weak version of KPZ for generalisations of WASEP.

17 Difficulty Problem: KPZ equation is ill-posed: t h = 2 xh + ( x h) 2 + ξ, Solution behaves like Brownian motion for fixed times: nowhere differentiable! Trick: Write Z = e h (Hopf-Cole) and formally derive t Z = 2 xz + Zξ, then interpret as Itô equation. WASEP behaves nicely under this transformation. Many other models do not...

18 Difficulty Problem: KPZ equation is ill-posed: t h = 2 xh + ( x h) 2 + ξ, Solution behaves like Brownian motion for fixed times: nowhere differentiable! Trick: Write Z = e h (Hopf-Cole) and formally derive t Z = 2 xz + Zξ, then interpret as Itô equation. WASEP behaves nicely under this transformation. Many other models do not...

19 Difficulty Problem: KPZ equation is ill-posed: t h = 2 xh + ( x h) 2 + ξ, Solution behaves like Brownian motion for fixed times: nowhere differentiable! Trick: Write Z = e h (Hopf-Cole) and formally derive t Z = 2 xz + Zξ, then interpret as Itô equation. WASEP behaves nicely under this transformation. Many other models do not...

20 Recent progress Theory of rough paths / regularity structures / paraproducts gives direct meaning to nonlinearity in a robust way: F R S A M C(S 1 ) D γ Ψ R F? C(S 1 ) S C C(S 1 R + ) R Noise I.C. F : Constant C in t h = 2 xh + ( x h) 2 + ξ ε C. S C : Classical solution to the PDE with smooth input. S A : Abstract fixed point: locally jointly continuous!

21 Recent progress Theory of rough paths / regularity structures / paraproducts gives direct meaning to nonlinearity in a robust way: F R S A M C(S 1 ) D γ Ψ R F? C(S 1 ) S C C(S 1 R + ) R Noise I.C. F : Constant C in t h = 2 xh + ( x h) 2 + ξ ε C. S C : Classical solution to the PDE with smooth input. S A : Abstract fixed point: locally jointly continuous!

22 Recent progress Theory of rough paths / regularity structures / paraproducts gives direct meaning to nonlinearity in a robust way: F R S A M C(S 1 ) D γ Ψ R F? C(S 1 ) S C C(S 1 R + ) R Noise I.C. Strategy: find M ε R such that M ε Ψ(ξ ε ) converges.

23 Universality result for KPZ Consider the model t h ε = 2 xh ε + εp ( x h ε ) + η, with P an even polynomial, η a Gaussian field with compactly supported correlations ϱ(t, x) s.t. ϱ = 1. Theorem (H., Quastel, 2014) As ε 0, there is a choice of C ε ε 1 such that εh(ε 1 x, ε 2 t) C ε t converges to solutions to (KPZ) λ with λ depending in a non-trivial way on all coefficients of P. Remark: Convergence to KPZ with λ 0 even if P (u) = u 4!!

24 Universality result for KPZ Consider the model t h ε = 2 xh ε + εp ( x h ε ) + η, with P an even polynomial, η a Gaussian field with compactly supported correlations ϱ(t, x) s.t. ϱ = 1. Theorem (H., Quastel, 2014) As ε 0, there is a choice of C ε ε 1 such that εh(ε 1 x, ε 2 t) C ε t converges to solutions to (KPZ) λ with λ depending in a non-trivial way on all coefficients of P. Remark: Convergence to KPZ with λ 0 even if P (u) = u 4!!

25 Universality result for KPZ Consider the model t h ε = 2 xh ε + εp ( x h ε ) + η, with P an even polynomial, η a Gaussian field with compactly supported correlations ϱ(t, x) s.t. ϱ = 1. Nonlinearity λ( x h) 2 Theorem (H., Quastel, 2014) As ε 0, there is a choice of C ε ε 1 such that εh(ε 1 x, ε 2 t) C ε t converges to solutions to (KPZ) λ with λ depending in a non-trivial way on all coefficients of P. Remark: Convergence to KPZ with λ 0 even if P (u) = u 4!!

26 Universality result for KPZ Consider the model t h ε = 2 xh ε + εp ( x h ε ) + η, with P an even polynomial, η a Gaussian field with compactly supported correlations ϱ(t, x) s.t. ϱ = 1. Theorem (H., Quastel, 2014) As ε 0, there is a choice of C ε ε 1 such that εh(ε 1 x, ε 2 t) C ε t converges to solutions to (KPZ) λ with λ depending in a non-trivial way on all coefficients of P. Remark: Convergence to KPZ with λ 0 even if P (u) = u 4!!

27 Universality result for KPZ Consider the model t h ε = 2 xh ε + εp ( x h ε ) + η, with P an even polynomial, η a Gaussian field with compactly supported correlations ϱ(t, x) s.t. ϱ = 1. Theorem (H., Quastel, 2014) As ε 0, there is a choice of C ε ε 1 such that εh(ε 1 x, ε 2 t) C ε t converges to solutions to (KPZ) λ with λ depending in a non-trivial way on all coefficients of P. Remark: Convergence to KPZ with λ 0 even if P (u) = u 4!!

28 Case P (u) = u 4 Write h ε (x, t) = εh(ε 1 x, ε 2 t) C ε t. Satisfies t hε = 2 xh ε + ε( x hε ) 4 + ξ ε C ε, with ξ ε an ε-approximation to white noise. Fact: Derivatives of microscopic model do not converge to 0 as ε 0: no small gradients! Heuristic: gradients have O(1) fluctuations but are small on average over large scales... General formula: λ = 1 P (u) µ(du), C ε = 1 P (u) µ(du) + O(1), 2 ε with µ a Gaussian measure, explicitly computable variance.

29 Case P (u) = u 4 Write h ε (x, t) = εh(ε 1 x, ε 2 t) C ε t. Satisfies t hε = 2 xh ε + ε( x hε ) 4 + ξ ε C ε, with ξ ε an ε-approximation to white noise. Fact: Derivatives of microscopic model do not converge to 0 as ε 0: no small gradients! Heuristic: gradients have O(1) fluctuations but are small on average over large scales... General formula: λ = 1 P (u) µ(du), C ε = 1 P (u) µ(du) + O(1), 2 ε with µ a Gaussian measure, explicitly computable variance.

30 Case P (u) = u 4 Write h ε (x, t) = εh(ε 1 x, ε 2 t) C ε t. Satisfies t hε = 2 xh ε + ε( x hε ) 4 + ξ ε C ε, with ξ ε an ε-approximation to white noise. Fact: Derivatives of microscopic model do not converge to 0 as ε 0: no small gradients! Heuristic: gradients have O(1) fluctuations but are small on average over large scales... General formula: λ = 1 P (u) µ(du), C ε = 1 P (u) µ(du) + O(1), 2 ε with µ a Gaussian measure, explicitly computable variance.

31 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 Ψ α,c (η) so that h = RH solves t h = 2 xh + αh 4 ( x h, c) + ah 2 ( x h, c) + η = 2 xh + α( x h) 4 + (a 6αc)( x h) 2 + (3αc 2 ac) + η. Show that Ψ ε,1/ε (ξ ε ) converges to same limit as Ψ 0,1/ε (ξ ε )! (Actually Ψ βε,1/ε (ξ ε ) for every β R...)

32 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 Ψ α,c (η) so that h = RH solves t h = 2 xh + αh 4 ( x h, c) + ah 2 ( x h, c) + η = 2 xh + α( x h) 4 + (a 6αc)( x h) 2 + (3αc 2 ac) + η. Show that Ψ ε,1/ε (ξ ε ) converges to same limit as Ψ 0,1/ε (ξ ε )! (Actually Ψ βε,1/ε (ξ ε ) for every β R...)

33 Main step in proof Rewrite general equation in integral form as H = P ( E(DH) 4 + a(dh) 2 + Ξ ), F M C(S 1 S A ) D γ with E an abstract integration operator of order 1. Ψ α,c R Find two-parameter lift of noise η Ψ α,c (η) so that h = RH solves F? C(S 1 S C ) C(S 1 R + ) t h = xh 2 + αh 4 ( x h, c) + ah 2 ( x h, c) + η = xh 2 Noise + α( x h) 4 I.C. + (a 6αc)( x h) 2 + (3αc 2 ac) + η. Show that Ψ ε,1/ε (ξ ε ) converges to same limit as Ψ 0,1/ε (ξ ε )! (Actually Ψ βε,1/ε (ξ ε ) for every β R...)

34 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 Ψ α,c (η) so that h = RH solves t h = 2 xh + αh 4 ( x h, c) + ah 2 ( x h, c) + η = 2 xh + α( x h) 4 + (a 6αc)( x h) 2 + (3αc 2 ac) + η. Show that Ψ ε,1/ε (ξ ε ) converges to same limit as Ψ 0,1/ε (ξ ε )! (Actually Ψ βε,1/ε (ξ ε ) for every β R...)

35 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 Ψ α,c (η) so that h = RH solves t h = 2 xh + αh 4 ( x h, c) + ah 2 ( x h, c) + η = 2 xh + α( x h) 4 + (a 6αc)( x h) 2 + (3αc 2 ac) + η. Show that Ψ ε,1/ε (ξ ε ) converges to same limit as Ψ 0,1/ε (ξ ε )! (Actually Ψ βε,1/ε (ξ ε ) for every β R...)

36 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. Labbé). 4. Models with non-gaussian noise. 5. Fully nonlinear continuum models. 6. Control over larger scales to see convergence to KPZ fixed point.

37 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. Labbé). 4. Models with non-gaussian noise. 5. Fully nonlinear continuum models. 6. Control over larger scales to see convergence to KPZ fixed point.

38 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. Labbé). 4. Models with non-gaussian noise. 5. Fully nonlinear continuum models. 6. Control over larger scales to see convergence to KPZ fixed point.

39 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. Labbé). 4. Models with non-gaussian noise. 5. Fully nonlinear continuum models. 6. Control over larger scales to see convergence to KPZ fixed point.

40 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. Labbé). 4. Models with non-gaussian noise. 5. Fully nonlinear continuum models. 6. Control over larger scales to see convergence to KPZ fixed point.

41 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. Labbé). 4. Models with non-gaussian noise. 5. Fully nonlinear continuum models. 6. Control over larger scales to see convergence to KPZ fixed point.

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