Some Topics in Stochastic Partial Differential Equations Tadahisa - PowerPoint PPT Presentation

Some Topics in Stochastic Partial Differential Equations Tadahisa Funaki University of Tokyo November 26, 2015 L’ H´ eritage de Kiyosi Itˆ o en perspective Franco-Japonaise, Ambassade de France au Japon Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

Plan of talk 1 Itˆ o’s SPDE 2 TDGL equation (Dynamic P ( φ )-model, Stochastic Allen-Cahn equation) 3 Kardar-Parisi-Zhang equation Centennial Anniversary of the Birth of Kiyosi Itˆ o by the Math. Soc. Japan Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

1. Itˆ o’s SPDE Itˆ o was interested in the following problem [2] (Math. Z. ’83): Let { B k ( t ) } ∞ k =1 be independent 1D Brownian motions with common initial distribution µ . Set n n 1 ( [ ]) ∑ ∑ √ n u n ( t , dx ) := δ B k ( t ) ( dx ) − E δ B k ( t ) ( dx ) . k =1 k =1 Then, u n ( t , · ) ⇒ u ( t , · ) dx and u ( t , · ) satisfies the SPDE: ∂ t u = 1 µ ( t , x ) ˙ 2 ∂ 2 (√ ) x u + ∂ x W ( t , x ) , where ˙ W ( t , x ) = ˙ W ( t , x , ω ) is a space-time Gaussian white noise with covariance structure formally given by E [ ˙ W ( t , x ) ˙ W ( s , y )] = δ ( t − s ) δ ( x − y ) , (1) 2 π t e − ( x − y )2 1 ∫ and µ ( t , x ) = µ ( dy ) . √ 2 t R Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

Proof is given as follows: For every test function ϕ ∈ C ∞ 0 ( R ), n n 1 ( [ ]) ∑ ∑ √ n u n ( t , ϕ ) = ϕ ( B k ( t )) − E ϕ ( B k ( t )) . k =1 k =1 Applying Itˆ o’s formula, we have n n 1 ( ∂ x ϕ ( B k ( t )) dB k ( t )+1 x ϕ ( B k ( t )) dt − 1 ) ∑ ∑ ∂ 2 [ ] du n ( t , ϕ ) = √ n 2 E · · · dt . 2 k =1 k =1 drift term = 1 2 u n ( t , ∂ 2 x ϕ ) dt ∫ t ∑ n 1 diffusion term 0 ∂ x ϕ ( B k ( s )) dB k ( s ) has a quadratic √ n k =1 ∫ t ∑ n 1 0 ∂ x ϕ ( B k ( s )) 2 ds which converges as n → ∞ to variation: k =1 n ∫ t ∫ R ∂ x ϕ ( x ) 2 µ ( s , x ) dx by LLN. 0 ds ∫ t µ ( s , x ) ˙ ∫ √ The limit R ∂ x ϕ ( x ) W ( s , x ) dsdx has the same quad.var. 0 This result was extended by H. Spohn (CMP ’86) to the interacting case dX k ( t ) = − 1 under equilibrium: ∑ i ̸ = k ∇ V ( X k ( t ) − X i ( t )) dt + dB k ( t ). 2 Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

2. TDGL equation Time-dependent Ginzburg-Landau (TDGL) equation (cf. Hohenberg-Halperin, Kawasaki-Ohta, Langevin equation) ∂ t u = − 1 δ H δ u ( x )( u ) + ˙ W ( t , x ) , x ∈ R d , 2 ˙ W ( t , x ) : space-time Gaussian white noise ∫ { 1 } 2 |∇ u ( x ) | 2 + V ( u ( x )) H ( u ) = dx . R d Heuristically, Gibbs measure 1 Z e − H du is invariant under these dynamics, where du = ∏ x ∈ R d du ( x ). Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

• Since the functional derivative is given by δ H δ u ( x ) = − ∆ u + V ′ ( u ( x )) , TDGL eq has the form: ∂ t u = 1 2∆ u − 1 2 V ′ ( u ) + ˙ W ( t , x ) . (2) • The noise ˙ W ( t , x ) can be constructed as follows: k =1 : CONS of L 2 ( R d , dx ) and { B k ( t ) } ∞ Take { ψ k } ∞ k =1 : independent 1D BMs, and consider a (formal) Fourier series: ∞ ∑ W ( t , x ) = B k ( t ) ψ k ( x ) . (3) k =1 Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

Stochastic PDEs used in physics are sometimes ill-posed. For TDGL eq (2), 2 − := ∩ δ> 0 C − d +1 2 − δ a.s. W ∈ C − d +1 ˙ Noise is very irregular: 2 − a.s. 2 − d 4 − , 2 − d Linear case (without V ′ ( u )): u ( t , x ) ∈ C Well-posed only when d = 1. Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

Martin Hairer: Theory of regularity structures, systematic renormalization TDGL equation with V ( u ) = 1 4 u 4 : =Stochastic quantization (Dynamic P ( φ ) d -model): ∂ t φ = ∆ φ − φ 3 + ˙ x ∈ R d W ( t , x ) , W ε and For d = 2 or 3, replace ˙ W by a smeared noise ˙ introduce a renormalization factor − C ε φ . Then, the limit of φ = φ ε as ε ↓ 0 exists (locally in time). W ε and their (finitely The solution is continuous in ˙ many) polynomials. Another approaches Gubinelli and others: Paracontrolled distributions (harmonic analytic method) Kupiainen Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

When ˙ W = 0 (noise is not added) and V = double-well type, TDGL eq (2) is known as Allen-Cahn equation or reaction-diffusion equation of bistable type. Dynamic phase transition, Sharp interface limit as ε ↓ 0 for TDGL equation (=stochastic Allen-Cahn equation): ∂ t u = ∆ u + 1 ε f ( u ) + ˙ x ∈ R d W ( t , x ) , (4) f = − V ′ , Potential V is of double-well type: e.g., f = u − u 3 if V = 1 4 u 4 − 1 2 u 2 − 1 +1 Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

The limit is expected to satisfy: { +1 +1 Γ t u ( t , x ) − → − 1 ε ↓ 0 − 1 A random phase separating hyperplane Γ t appears and its time evolution is studied under proper time scaling. Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

3. Kardar-Parisi-Zhang equation The KPZ (Kardar-Parisi-Zhang, 1986) equation describes the motion of growing interface with random fluctuation. It has the form for height function h ( t , x ): 2 ( ∂ x h ) 2 + ˙ 2 ∂ 2 ∂ t h = 1 x h + 1 W ( t , x ) , x ∈ R . (5) Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

Ill-posedness of KPZ eq (5): The nonlinearity and roughness of the noise do not match. The linear SPDE: x h + ˙ ∂ t h = 1 2 ∂ 2 W ( t , x ) , obtained by dropping the nonlinear term has a solution 1 4 − , 1 2 − ([0 , ∞ ) × R ) a.s. Therefore, no way to define h ∈ C the nonlinear term ( ∂ x h ) 2 in (5) in a usual sense. Actually, the following Renormalized KPZ eq with compensator δ x ( x ) (= + ∞ ) has the meaning: 2 { ( ∂ x h ) 2 − δ x ( x ) } + ˙ ∂ t h = 1 2 ∂ 2 x h + 1 W ( t , x ) , as we will see later. Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

1 3 -power law : Under stationary situation, 2 Var( h ( t , 0)) = O ( t 3 ) 1 as t → ∞ , i. e. the fluctuations of h ( t , 0) are of order t 3 . Subdiffusive behavior different from CLT (=diffusive behavior). (Sasamoto-Spohn) The limit distribution of h ( t , 0) under scaling is given by the so-called Tracy-Widom distribution (different depending on initial distributions). Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

Cole-Hopf solution to the KPZ equation Consider the linear stochastic heat equation (SHE) for Z = Z ( t , x ): x Z + Z ˙ ∂ t Z = 1 2 ∂ 2 W ( t , x ) , (6) with a multiplicative noise. This eq is well-posed (if we understand the multiplicative term in Itˆ o’s sense but ill-posed in Stratonovich’s sense). If Z (0 , · ) > 0 ⇒ Z ( t , · ) > 0. Therefore, we can define the Cole-Hopf transformation: h ( t , x ) := log Z ( t , x ) . (7) This is called Cole-Hopf solution of KPZ equation. Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

Heuristic derivation of the KPZ eq (with renormalization factor δ x ( x )) from SHE (6) under the Cole-Hopf transformation (7): Apply Itˆ o’s formula for h = log z : ∂ t h = Z − 1 ∂ t Z − 1 2 Z − 2 ( ∂ t Z ) 2 = Z − 1 ( ) x Z + Z ˙ 2 ∂ 2 1 − 1 2 δ x ( x ) W by SHE (6) and ( dZ ( t , x )) 2 = ( ZdW ( t , x )) 2 dW ( t , x ) dW ( t , y ) = δ ( x − y ) dt x h + ( ∂ x h ) 2 } + ˙ = 1 2 { ∂ 2 W − 1 2 δ x ( x ) Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

This leads to the Renormalized KPZ eq: 2 { ( ∂ x h ) 2 − δ x ( x ) } + ˙ ∂ t h = 1 2 ∂ 2 x h + 1 W ( t , x ) . (8) The Cole-Hopf solution h ( t , x ) defined by (7) is meaningful, although the equation (5) does not make sense. Goal is to introduce approximations for (8). Hairer (2013, 2014) gave a meaning to (8) without bypassing SHE. Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations

KPZ approximating equation-1: Simple Symmetric convolution kernel Let η ∈ C ∞ 0 ( R ) s.t. ∫ η ( x ) ≥ 0, η ( x ) = η ( − x ) and R η ( x ) dx = 1 be given, and set η ε ( x ) := 1 ε η ( x ε ) for ε > 0. Smeared noise The smeared noise is defined by W ε ( t , x ) = ⟨ W ( t ) , η ε ( x − · ) ⟩ ( = W ( t ) ∗ η ε ( x ) ) . Approximating Eq-1: ( ∂ x h ) 2 − ξ ε ) + ˙ ∂ t h = 1 2 ∂ 2 x h + 1 ( W ε ( t , x ) 2 x Z + Z ˙ ∂ t Z = 1 2 ∂ 2 W ε ( t , x ) , where ξ ε = η ε 2 (0) (:= η ε ∗ η ε (0)). It is easy to show that Z = Z ε converges to the sol Z of (SHE), and therefore h = h ε converges to the Cole-Hopf solution of the KPZ eq. Tadahisa Funaki University of Tokyo Some Topics in Stochastic Partial Differential Equations