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Paracontrolled KPZ equation Nicolas Perkowski Humboldt Universität zu Berlin November 6th, 2015 Eighth Workshop on RDS Bielefeld Joint work with Massimiliano Gubinelli Nicolas Perkowski Paracontrolled KPZ equation 1 / 23

Motivation: modelling of interface growth Figure: Takeuchi, Sano, Sasamoto, Spohn (2011, Sci. Rep.) Nicolas Perkowski Paracontrolled KPZ equation 2 / 23

KPZ equation KPZ equation is a model for random interface growth: h : R + R R, t h(t, x) = κ h(t, x) + λ }{{} x h(t, x) 2 + ξ(t, x) }{{}}{{} diffusion slope-dependence space-time white noise Kardar-Parisi-Zhang (1986): slope-dependent growth F ( x h); F ( x h) = F ( h) + F ( h)( x h h) + 1 2 F ( h)( x h h) 2 +... Highly non-rigorous since x h is a distribution. But: Hairer-Quastel (2015, unpublished) justify it via scaling. Nicolas Perkowski Paracontrolled KPZ equation 3 / 23

KPZ equation KPZ equation is a model for random interface growth: h : R + R R, t h(t, x) = κ h(t, x) + λ }{{} x h(t, x) 2 + ξ(t, x) }{{}}{{} diffusion slope-dependence space-time white noise Kardar-Parisi-Zhang (1986): slope-dependent growth F ( x h); F ( x h) = F ( h) + F ( h)( x h h) + 1 2 F ( h)( x h h) 2 +... Highly non-rigorous since x h is a distribution. But: Hairer-Quastel (2015, unpublished) justify it via scaling. Fluctuations of ε 1/3 h(tε 1, xε 2/3 ) should converge to KPZ fixed point. Only known for one-point distribution, special h 0 (Amir et al. (2011), Sasamoto-Spohn (2010), Borodin et al. (2014)). Nicolas Perkowski Paracontrolled KPZ equation 3 / 23

Weak KPZ universality conjecture t h = h + x h 2 + ξ. KPZ equation for t in KPZ universality class. For t 0 Gaussian (Edwards-Wilkinson class of symmetric growth models). Nicolas Perkowski Paracontrolled KPZ equation 4 / 23

Weak KPZ universality conjecture t h = h + x h 2 + ξ. KPZ equation for t in KPZ universality class. For t 0 Gaussian (Edwards-Wilkinson class of symmetric growth models). Weak KPZ universality conjecture: KPZ equation is only growth model interpolating EW and KPZ. Mathematically: fluctuations of weakly asymmetrical models converge to KPZ. Example: Ginzburg-Landau ϕ model dx j = ( pv (r j+1 ) qv (r j ) ) dt + dw j ; r j = x j x j 1 ; For p = q convergence to t ψ = α ψ + βξ. For p q = ε convergence to KPZ Diehl-Gubinelli-P. (2015, in preparation). Nicolas Perkowski Paracontrolled KPZ equation 4 / 23

How to interpret KPZ? Lh(t, x) = ( t )h(t, x) = x h(t, x) 2 + ξ(t, x). Difficulty: h(t, ) has Brownian regularity, so x h(t, x) 2 =? Nicolas Perkowski Paracontrolled KPZ equation 5 / 23

How to interpret KPZ? Lh(t, x) = ( t )h(t, x) = x h(t, x) 2 + ξ(t, x). Difficulty: h(t, ) has Brownian regularity, so x h(t, x) 2 =? Cole-Hopf transformation: Bertini-Giacomin (1997) set h(t, x) := log w(t, x), where Lw(t, x) = w(t, x)ξ(t, x) (linear Itô SPDE). Correct object but no equation for h. Hairer (2013): series expansion and rough paths/regularity structures, defines x h(t, x) 2. So far on circle (h : R + T R), but certainly soon extended to h : R + R R. Nicolas Perkowski Paracontrolled KPZ equation 5 / 23

How to interpret KPZ? Lh(t, x) = ( t )h(t, x) = x h(t, x) 2 + ξ(t, x). Difficulty: h(t, ) has Brownian regularity, so x h(t, x) 2 =? Cole-Hopf transformation: Bertini-Giacomin (1997) set h(t, x) := log w(t, x), where Lw(t, x) = w(t, x)ξ(t, x) (linear Itô SPDE). Correct object but no equation for h. Hairer (2013): series expansion and rough paths/regularity structures, defines x h(t, x) 2. So far on circle (h : R + T R), but certainly soon extended to h : R + R R. Martingale problem: Assing (2002), Gonçalves-Jara (2014), Gubinelli-Jara (2013) define energy solutions of equilibrium KPZ. Uniqueness long open, solved in Gubinelli-P. (2015). Nicolas Perkowski Paracontrolled KPZ equation 5 / 23

Solution concepts and weak KPZ universality Cole-Hopf: equation for e h ; most systems behave badly under exponential transformation. Only very specific models: Bertini-Giacomin (1997), Dembo-Tsai (2013), Corwin-Tsai (2015). Pathwise approach: needs precise control of regularity, so far only semilinear S(P)DEs: Hairer-Quastel (2015), Hairer-Shen (2015), Gubinelli-P. (2015). Martingale problem: powerful for universality of equilibrium fluctuations Gonçalves-Jara (2014), Gonçalves-Jara-Sethuraman (2015), Diehl-Gubinelli-P. (2015, in preparation). Before only tightness and martingale characterization of limits. Now: uniqueness proves convergence. Nicolas Perkowski Paracontrolled KPZ equation 6 / 23

Aims for the rest of the talk Equivalent derivation of Hairer s solution, replacing rough paths by paracontrolled distributions. Nicolas Perkowski Paracontrolled KPZ equation 7 / 23

Aims for the rest of the talk Equivalent derivation of Hairer s solution, replacing rough paths by paracontrolled distributions. New stochastic optimal control formulation of the KPZ equation. Nicolas Perkowski Paracontrolled KPZ equation 7 / 23

Aims for the rest of the talk Equivalent derivation of Hairer s solution, replacing rough paths by paracontrolled distributions. New stochastic optimal control formulation of the KPZ equation. Uniqueness of equilibrium KPZ martingale problem. Nicolas Perkowski Paracontrolled KPZ equation 7 / 23

1 Paracontrolled formulation of the equation 2 KPZ as HJB equation 3 Uniqueness of the martingale solution Nicolas Perkowski Paracontrolled KPZ equation 8 / 23

Formal expansion of the KPZ equation Lh(t, x) = ( t )h(t, x) = x h(t, x) 2 + ξ(t, x), Perturbative expansion around linear solution: h = Y + h 1 with Y C 1/2, LY = ξ, thus Lh 1 = x Y 2 + 2 }{{} x Y x h 1 + }{{} x h 1 2. }{{} C 1 =B, 1 C 1/2 C 0 Continue expansion: set LY = x Y 2 and then LY = x Y x Y and in general LY (τ 1τ 2 ) = x Y τ 1 x Y τ 2. Formally: h = τ c(τ)y τ. Seems very difficult to make this rigorous. Nicolas Perkowski Paracontrolled KPZ equation 9 / 23

Truncated expansion Following Hairer (2013), truncate expansion and set h = Y + Y + 2Y + h P, where h P is paracontrolled by P with LP = x Y, write h P = h } {{ P } + }{{} h, C 3/2 C 2 Nicolas Perkowski Paracontrolled KPZ equation 10 / 23

Truncated expansion Following Hairer (2013), truncate expansion and set h = Y + Y + 2Y + h P, where h P is paracontrolled by P with LP = x Y, write h P = h } {{ P } + }{{} h, C 3/2 C 2 where for k k-th Littlewood-Paley block: h P = i h j P. i<j 1 Intuitively: h P is frequency modulation of P plus smoother remainder; more intuitively: on small scales h P looks like P. Nicolas Perkowski Paracontrolled KPZ equation 10 / 23

Paracontrolled differential equation Paracontrolled ansatz: h D rbe if h = Y + Y + 2Y + h P with h P = h P + h. Nicolas Perkowski Paracontrolled KPZ equation 11 / 23

Paracontrolled differential equation Paracontrolled ansatz: h D rbe if h = Y + Y + 2Y + h P with h P = h P + h. Theorem (Gubinelli, Imkeller, P. (2015)) For paracontrolled h D rbe the square x h 2 is well defined, depends continuously on h and (Y, Y, Y, Y, Y, x P x Y ), and we have x h 2 = lim ε 0 ( x (δ ε h) 2 c ε ). Theorem (Gubinelli, P. (2015)) Local-in-time existence and uniqueness of paracontrolled solutions. Solution depends locally Lipschitz continuously on extended data (Y, Y, Y, Y, Y, x P x Y ). Agrees with Hairer s solution. Nicolas Perkowski Paracontrolled KPZ equation 11 / 23

1 Paracontrolled formulation of the equation 2 KPZ as HJB equation 3 Uniqueness of the martingale solution Nicolas Perkowski Paracontrolled KPZ equation 12 / 23

Formal derivation Cole-Hopf: h = log w, where Lw(t, x) = w(t, x)(ξ(t, x) ), w(0) = e h(0). Nicolas Perkowski Paracontrolled KPZ equation 13 / 23

Formal derivation Cole-Hopf: h = log w, where Lw(t, x) = w(t, x)(ξ(t, x) ), w(0) = e h(0). Feynman-Kac: w(t, x) = E x [ exp ( h 0 (B t ) + t 0 )] (ξ(t s, B s ) )ds. Nicolas Perkowski Paracontrolled KPZ equation 13 / 23

Formal derivation Cole-Hopf: h = log w, where Lw(t, x) = w(t, x)(ξ(t, x) ), w(0) = e h(0). Feynman-Kac: w(t, x) = E x [ exp Boué-Dupis (1998): ( h 0 (B t ) + [ log E[e F (B) ] = sup E F (B + v t 0 )] (ξ(t s, B s ) )ds. 0 v s ds) 1 4 t 0 ] vs 2 ds. Nicolas Perkowski Paracontrolled KPZ equation 13 / 23

Formal derivation Cole-Hopf: h = log w, where Lw(t, x) = w(t, x)(ξ(t, x) ), w(0) = e h(0). Feynman-Kac: w(t, x) = E x [ exp Boué-Dupis (1998): ( h 0 (B t ) + [ log E[e F (B) ] = sup E F (B + v t 0 Thus (see also E-Khanin-Mazel-Sinai (2000)): h(t, x) = sup E x [h 0 (γt v ) + v t 0 )] (ξ(t s, B s ) )ds. 0 v s ds) 1 4 t (ξ(t s, γ v s ) )ds 1 4 0 ] vs 2 ds. t 0 ] vs 2 ds, where γ v s = x + B s + s 0 v r dr. (But of course nothing was rigorous!) Nicolas Perkowski Paracontrolled KPZ equation 13 / 23

Let s make it rigorous Regularize ξ: Lh ε = x h ε 2 c ε + ξ ε. Then h ε (t, x) = sup E x [h 0 (γt v ) + v t 0 (ξ ε (t s, γ v s ) c ε )ds 1 4 t 0 ] vs 2 ds. Nicolas Perkowski Paracontrolled KPZ equation 14 / 23

Let s make it rigorous Regularize ξ: Lh ε = x h ε 2 c ε + ξ ε. Then h ε (t, x) = sup E x [h 0 (γt v ) + v t Fix singular part of optimal control: Then Itô gives 0 (ξ ε (t s, γ v s ) c ε )ds 1 4 dζ v s = 2 x (Y ε + Y ε )(t s, ζ v s )ds + v s ds + db s, h ε (t, x) = (Y ε + Y ε + Y R ε )(t, x) where Y R ε + sup E x [h 0 (ζt v ) + v t solves a linear paracontrolled equation. 0 t 0 ] vs 2 ds. ( x Yε R (t s, ζs v )v s 1 4 v s 2) ] ds, Nicolas Perkowski Paracontrolled KPZ equation 14 / 23

Singular control problem h ε (t, x) = (Y ε + Y ε + Y R ε )(t, x) + sup E x [h 0 (ζt v ) + v t 0 ( x Yε R (t s, ζs v )v s 1 4 v s 2) ] ds, dζ v s = 2 x (Y ε + Y ε )(t s, ζ v s )ds + v s ds + db s. Robust formulation that allows ε 0; get quantitative pathwise bounds in terms of linear equation, no blowup for all realizations of ξ and all initial conditions; techniques of Delarue-Diel (2014), Cannizzaro-Chouk (2015) allow to formulate control problem in the limit, get variational representation of KPZ. Nicolas Perkowski Paracontrolled KPZ equation 15 / 23

Singular control problem h ε (t, x) = (Y ε + Y ε + Y R ε )(t, x) + sup E x [h 0 (ζt v ) + v t 0 ( x Yε R (t s, ζs v )v s 1 4 v s 2) ] ds, dζ v s = 2 x (Y ε + Y ε )(t s, ζ v s )ds + v s ds + db s. Robust formulation that allows ε 0; get quantitative pathwise bounds in terms of linear equation, no blowup for all realizations of ξ and all initial conditions; techniques of Delarue-Diel (2014), Cannizzaro-Chouk (2015) allow to formulate control problem in the limit, get variational representation of KPZ. Result independent of Cole-Hopf, only used to abbreviate derivation. Nicolas Perkowski Paracontrolled KPZ equation 15 / 23

1 Paracontrolled formulation of the equation 2 KPZ as HJB equation 3 Uniqueness of the martingale solution Nicolas Perkowski Paracontrolled KPZ equation 16 / 23

Burgers generator Burgers equation: t u = u + x u 2 + x ξ. Invariant measure µ = law(white noise). Nicolas Perkowski Paracontrolled KPZ equation 17 / 23

Burgers generator Burgers equation: t u = u + x u 2 + x ξ. Invariant measure µ = law(white noise). Formally: generator L 0 + B, L 0 symmetric in L 2 (µ) and B antisymmetric. L 0 is generator of OU process t ψ = ψ + x ξ. So for u(0) µ, backward process û(t) = u(t t) should solve t û = û x û 2 + x ˆξ for new white noise ˆξ. Difficult to make rigorous. Nicolas Perkowski Paracontrolled KPZ equation 17 / 23

Gubinelli-Jara controlled processes Gubinelli-Jara (2013): u is called controlled by the OU process if 1 u t µ for all t; 2 for all ϕ S u t (ϕ) = u 0 (ϕ) + t 0 u s ( ϕ)ds + A t (ϕ) + M t (ϕ), M(ϕ) martingale with M(ϕ) t = 2t x ϕ L 2 and A(ϕ) 0; 3 û t = u T t of same type with backward martingale ˆM, Â t = (A T A T t ). Define T 0 xus 2 ds via martingale trick: F (u T ) = F (u 0 ) + F (û T ) = F (û 0 ) + T 0 T so 2 T 0 L 0F (u s )ds = M F T ˆM F T. 0 L 0 F (u s )ds + L 0 F (u s )ds + T 0 T 0 DF (u s )da s + M F T, DF (û s )dâ s + ˆM F T, Nicolas Perkowski Paracontrolled KPZ equation 18 / 23

Uniqueness of energy solutions I Call controlled u energy solution if A = 0 xu 2 s ds. Gubinelli-Jara (2013): existence. Uniqueness difficult because energy formulation gives little control. Easy: uniqueness of paracontrolled energy solutions. Then we read Funaki-Quastel (2014) who study invariant measure for KPZ via Sasamoto-Spohn discretization: Mollify discrete model to safely pass to continuous limit Cole-Hopf: w ε = e hε t h ε = h ε + δ ε δ ε ( x h ε c ε ) 2 + δ ε ξ; solves t w ε = w ε + w ε( ( x w ε ) 2 ( x w ε ) 2 ) δ ε δ ε w ε w ε + w ε (δ ε ξ). Use Boltzmann-Gibbs principle to show convergence of nonlinearity. Nicolas Perkowski Paracontrolled KPZ equation 19 / 23

Uniqueness of energy solutions II Implement Funaki-Quastel strategy for energy solutions: u ε = δ ε u. Itô: w ε = e 1 x u ε solves dwt ε = w ε dt + wt ε x 1 (dmt ε + da ε t) wt ε (ut ε ) 2 dt + wt ε c ε dt. x 1 t Mt ε ξ. If rest converges to c R, then t w = w + w(ξ + c). Since x log w c 1 = x log w c 2, u is unique. Nicolas Perkowski Paracontrolled KPZ equation 20 / 23

Uniqueness of energy solutions II Implement Funaki-Quastel strategy for energy solutions: u ε = δ ε u. Itô: w ε = e 1 x u ε solves dwt ε = w ε dt + wt ε x 1 (dmt ε + da ε t) wt ε (ut ε ) 2 dt + wt ε c ε dt. x 1 t Mt ε ξ. If rest converges to c R, then t w = w + w(ξ + c). Since x log w c 1 = x log w c 2, u is unique. Remains to study (d x 1 A ε t (ut ε ) 2 dt + c ε dt). Nicolas Perkowski Paracontrolled KPZ equation 20 / 23

Uniqueness of energy solutions III Convergence of (d x 1 A ε t (ut ε ) 2 dt + c ε dt): A ε = δ ε A = 0 δ ε x us 2 ds, so where Π 0 ϕ = ϕ T ϕdx. (d x 1 A ε t (ut ε ) 2 dt + c ε dt) = Π 0 (δ ε (ut 2 ) (δ ε u t ) 2 )dt + (c ε (δ ε u t ) 2 dx)dt Remains to control integrals like T 0 F (u s)ds. Kipnis-Varadhan extends to controlled processes, so t E[ sup t T 0 F (u s )ds 2 ] sup G where L 0 is OU generator, u 0 white noise. T {2E[F (u 0 )G(u 0 )] E[G(u 0 )( L 0 G)(u 0 )]}, Nicolas Perkowski Paracontrolled KPZ equation 21 / 23

Uniqueness of energy solutions IV Control sup G {2E[F (u 0 )G(u 0 )] E[G(u 0 )( L 0 G)(u 0 )]}, where L 0 is OU generator, u 0 white noise. For us: F in second chaos of white noise. Use Gaussian IBP to reduce to deterministic integral over explicit kernel. Nicolas Perkowski Paracontrolled KPZ equation 22 / 23

Uniqueness of energy solutions IV Control sup G {2E[F (u 0 )G(u 0 )] E[G(u 0 )( L 0 G)(u 0 )]}, where L 0 is OU generator, u 0 white noise. For us: F in second chaos of white noise. Use Gaussian IBP to reduce to deterministic integral over explicit kernel. Theorem (Gubinelli, P. (2015)) There exists a unique controlled process u which is an energy solution to Burgers equation. Nicolas Perkowski Paracontrolled KPZ equation 22 / 23

Thank you Nicolas Perkowski Paracontrolled KPZ equation 23 / 23