Solving the KPZ equation M. Hairer University of Warwick SSP 2012, - PowerPoint PPT Presentation

Solving the KPZ equation M. Hairer University of Warwick SSP 2012, University of Kansas

Introduction Object of study: KPZ equation of surface growth: x h + λ ( ∂ x h ) 2 + ξ − ∞ ∂ t h = ∂ 2 with either x ∈ R or x ∈ S 1 , and ξ is space-time white noise. 1. Universal model for interface fluctuations. (Shown rigorously only for SOS model, see Bertini-Giacomin 1997.) 2. Free energy for polymer models. 3. Scaling limit of time-dependent parabolic Anderson model. 4. Universal object describing crossover from Edwards-Wilkinson to KPZ. Problem: Right-hand side is badly ill-posed! (Solutions only C α for α < 1 2 !!)

Introduction Object of study: KPZ equation of surface growth: x h + λ ( ∂ x h ) 2 + ξ − ∞ ∂ t h = ∂ 2 with either x ∈ R or x ∈ S 1 , and ξ is space-time white noise. 1. Universal model for interface fluctuations. (Shown rigorously only for SOS model, see Bertini-Giacomin 1997.) 2. Free energy for polymer models. 3. Scaling limit of time-dependent parabolic Anderson model. 4. Universal object describing crossover from Edwards-Wilkinson to KPZ. Problem: Right-hand side is badly ill-posed! (Solutions only C α for α < 1 2 !!)

Introduction Object of study: KPZ equation of surface growth: x h + λ ( ∂ x h ) 2 + ξ − ∞ ∂ t h = ∂ 2 with either x ∈ R or x ∈ S 1 , and ξ is space-time white noise. 1. Universal model for interface fluctuations. (Shown rigorously only for SOS model, see Bertini-Giacomin 1997.) 2. Free energy for polymer models. 3. Scaling limit of time-dependent parabolic Anderson model. 4. Universal object describing crossover from Edwards-Wilkinson to KPZ. Problem: Right-hand side is badly ill-posed! (Solutions only C α for α < 1 2 !!)

Cole-Hopf solution Trick introduced by Cole and Hopf in the 50’s. Write h = λ − 1 log Z , then Z solves ∂ t Z = ∂ 2 x Z + λZ ξ . ( ⋆ ) Idea: Take this as definition of solution, where ( ⋆ ) is interpreted in the Itˆ o sense. Work by Bertini-Giacomin shows that this is the physically relevant solution. Write h = S CH ( h 0 , ω ) , taking values in C ( R + , C ) .

Cole-Hopf solution Trick introduced by Cole and Hopf in the 50’s. Write h = λ − 1 log Z , then Z solves dZ = ∂ 2 x Z dt + λZ dW . ( ⋆ ) Idea: Take this as definition of solution, where ( ⋆ ) is interpreted in the Itˆ o sense. Work by Bertini-Giacomin shows that this is the physically relevant solution. Write h = S CH ( h 0 , ω ) , taking values in C ( R + , C ) .

Cole-Hopf solution Trick introduced by Cole and Hopf in the 50’s. Write h = λ − 1 log Z , then Z solves dZ = ∂ 2 x Z dt + λZ dW . ( ⋆ ) Idea: Take this as definition of solution, where ( ⋆ ) is interpreted in the Itˆ o sense. Work by Bertini-Giacomin shows that this is the physically relevant solution. Write h = S CH ( h 0 , ω ) , taking values in C ( R + , C ) .

Properties of Cole-Hopf Mollify W , so W ε,k = ϕ ( εk ) W k for cutoff ϕ , and set h ε = λ − 1 log Z ε . dZ ε = ∂ 2 x Z ε dt + λZ ε dW ε , Then h ε solves C ε ≈ 1 � ( ∂ x h ε ) 2 − C ε ϕ 2 . ∂ t h ε = ∂ 2 � � x h ε + λ + ξ ε , ε Problems with this notion of solution: 1. Not satisfactory at the formal level. 2. Lack of robustness: no good approximation theory for other modifications (hyperviscosity, time-smoothing, etc). 3. Properties of solutions do not always transform well (regularity of difference for example).

Properties of Cole-Hopf Mollify W , so W ε,k = ϕ ( εk ) W k for cutoff ϕ , and set h ε = λ − 1 log Z ε . dZ ε = ∂ 2 x Z ε dt + λZ ε dW ε , Then h ε solves C ε ≈ 1 � ( ∂ x h ε ) 2 − C ε ϕ 2 . ∂ t h ε = ∂ 2 � � x h ε + λ + ξ ε , ε Problems with this notion of solution: 1. Not satisfactory at the formal level. 2. Lack of robustness: no good approximation theory for other modifications (hyperviscosity, time-smoothing, etc). 3. Properties of solutions do not always transform well (regularity of difference for example).

Some attempts 1. Consider product as Wick product ∂ x h ⋄ ∂ x h (Øksendal & Al 1995): wrong notion of solution ( � = S CH ). Also wrong scaling properties (Chan 2000). 2. Formulate as martingale problem (Assing 2002): no well-posedness, “generator” not shown to be closable. 3. Apply “standard” renormalisation techniques inspired by QFT (Da Prato, Debussche, Tubaro 2007): only works for a regularised equation. 4. Define nonlinearity on some distributional space (Gon¸ calves, Jara 2010, Assing 2011): no uniqueness. No characterisation of class of distributions for which formulation even makes sense.

Some attempts 1. Consider product as Wick product ∂ x h ⋄ ∂ x h (Øksendal & Al 1995): wrong notion of solution ( � = S CH ). Also wrong scaling properties (Chan 2000). 2. Formulate as martingale problem (Assing 2002): no well-posedness, “generator” not shown to be closable. 3. Apply “standard” renormalisation techniques inspired by QFT (Da Prato, Debussche, Tubaro 2007): only works for a regularised equation. 4. Define nonlinearity on some distributional space (Gon¸ calves, Jara 2010, Assing 2011): no uniqueness. No characterisation of class of distributions for which formulation even makes sense.

Some attempts 1. Consider product as Wick product ∂ x h ⋄ ∂ x h (Øksendal & Al 1995): wrong notion of solution ( � = S CH ). Also wrong scaling properties (Chan 2000). 2. Formulate as martingale problem (Assing 2002): no well-posedness, “generator” not shown to be closable. 3. Apply “standard” renormalisation techniques inspired by QFT (Da Prato, Debussche, Tubaro 2007): only works for a regularised equation. 4. Define nonlinearity on some distributional space (Gon¸ calves, Jara 2010, Assing 2011): no uniqueness. No characterisation of class of distributions for which formulation even makes sense.

Some attempts 1. Consider product as Wick product ∂ x h ⋄ ∂ x h (Øksendal & Al 1995): wrong notion of solution ( � = S CH ). Also wrong scaling properties (Chan 2000). 2. Formulate as martingale problem (Assing 2002): no well-posedness, “generator” not shown to be closable. 3. Apply “standard” renormalisation techniques inspired by QFT (Da Prato, Debussche, Tubaro 2007): only works for a regularised equation. 4. Define nonlinearity on some distributional space (Gon¸ calves, Jara 2010, Assing 2011): no uniqueness. No characterisation of class of distributions for which formulation even makes sense.

A robustness result Theorem (H. 2011): For α > 0 arbitrary one can build the following objects: • A metric space X and a measurable map Ψ: Ω → X ; • A “blow-up time” T ⋆ : C α × X → (0 , ∞ ] (lower semi-continuous); • A jointly locally Lipschitz continuous solution map S R : C α × X → C ( R + , ¯ C α ) . These are such that one has T ⋆ ( h 0 , Ψ( ω )) = + ∞ almost surely and the factorisation S CH ( h 0 , ω ) = S R ( h 0 , Ψ( ω )) holds almost surely.

Some corollaries of the proof 1. One can construct h ⋆ explicitly from multilinear expressions of 2 − δ for every δ > 0 . 3 Gaussians such that h t − h ⋆ t ∈ C 2. One can construct a Gaussian process Φ such that e − 2 λ Φ t ∂ x h t − h ⋆ ∈ C 1 − δ � � t for every δ > 0 . 3. Solutions form a perfect cocycle.

Some corollaries of the proof 1. One can construct h ⋆ explicitly from multilinear expressions of 2 − δ for every δ > 0 . 3 Gaussians such that h t − h ⋆ t ∈ C 2. One can construct a Gaussian process Φ such that e − 2 λ Φ t ∂ x h t − h ⋆ ∈ C 1 − δ � � t for every δ > 0 . 3. Solutions form a perfect cocycle.

Some corollaries of the proof 1. One can construct h ⋆ explicitly from multilinear expressions of 2 − δ for every δ > 0 . 3 Gaussians such that h t − h ⋆ t ∈ C 2. One can construct a Gaussian process Φ such that e − 2 λ Φ t ∂ x h t − h ⋆ ∈ C 1 − δ � � t for every δ > 0 . 3. Solutions form a perfect cocycle.

A deterministic result Consider solutions h ε to x h ε + ( ∂ x h ε ) 2 + ε − 3 / 2 g ( ε − 1 x − ε − 2 t ) − K ε , ∂ t h ε = ∂ 2 for centred periodic g and suitable constants K ε . Then, one can compute K such that h ε → h solving x h + ( ∂ x h ) 2 + K∂ x h . ∂ t h = ∂ 2 Proof: Just show that Ψ( g ε ) converges to a limit in X ...

A deterministic result Consider solutions h ε to x h ε + ( ∂ x h ε ) 2 + ε − 3 / 2 g ( ε − 1 x − ε − 2 t ) − K ε , ∂ t h ε = ∂ 2 for centred periodic g and suitable constants K ε . Then, one can compute K such that h ε → h solving x h + ( ∂ x h ) 2 + K∂ x h . ∂ t h = ∂ 2 Proof: Just show that Ψ( g ε ) converges to a limit in X ...

Ideas of technique Idea: Perform Wild expansion of solution: define ∂ t Y ε = ∂ 2 x Y ε + ξ ε . For any binary tree τ = [ τ 1 , τ 2 ] , define Y τ ε recursively by ∂ t Y τ x Y τ ε + ∂ x Y τ 1 ∂ x Y τ 2 − C τ ε = ∂ 2 ε . ε ε Formal calculation shows that � λ | τ |− 1 Y τ h ε ( t ) = ε ( t ) , τ τ C τ provided that � ε = C ε .

Ideas of technique Idea: Perform Wild expansion of solution: define ∂ t Y ε = ∂ 2 x Y ε + ξ ε . For any binary tree τ = [ τ 1 , τ 2 ] , define Y τ ε recursively by ∂ t Y τ x Y τ ε + ∂ x Y τ 1 ∂ x Y τ 2 − C τ ε = ∂ 2 ε . ε ε Formal calculation shows that � λ | τ |− 1 Y τ h ε ( t ) = ε ( t ) , τ τ C τ provided that � ε = C ε .