Invariant measures in coupled KPZ equations Tadahisa Funaki Waseda University/University of Tokyo June 14, 2017 Stochastic dynamics out of equilibrium, IHP, Paris Tadahisa Funaki Waseda University/University of Tokyo Invariant measures in coupled KPZ equations
Plan of the talk Coupled KPZ (Kardar-Parisi-Zhang) equations – Motivation: nonlinear fluctuating hydrodynamics Quick overview of results with Hoshino (JFA 273 , 2017) – Two approximating equations – Trilinear condition (T) for coupling constants Γ – Invariant measure – Global-in-time existence Role of (T) – Invariant measure, renormalizations (for 4th order terms) Extensions of Erta¸ s-Kardar’s example, not satisfying (T) but having Invariant measure Tadahisa Funaki Waseda University/University of Tokyo Invariant measures in coupled KPZ equations
Multi-component coupled KPZ equation R d -valued KPZ eq for h ( t , x ) = ( h α ( t , x )) d α =1 on T = [0 , 1): ∂ t h α = 1 x h α + 1 βγ ∂ x h β ∂ x h γ + σ α 2 ∂ 2 2 Γ α β ξ β ( σ, Γ) KPZ We use Einstein’s convention. ≡ ˙ ξ ( t , x ) = ( ξ α ( t , x )) d ( ) is an R d -valued W ( t , x ) α =1 space-time Gaussian white noise with covariance structure: E [ ξ α ( t , x ) ξ β ( s , y )] = δ αβ δ ( x − y ) δ ( t − s ) . Coupled KPZ is ill-posed, since noise is irregular and doesn’t 1 4 − , 1 2 − match with nonlinear term. ( h ∈ C a.s. when Γ = 0) t , x We need to introduce approximations with smooth noises and renormalization for ( σ, Γ) KPZ . Indeed, one can introduce two types of approximations: one is simple, the other is suitable to study invariant measures ( d = 1: F-Quastel 2015). Tadahisa Funaki Waseda University/University of Tokyo Invariant measures in coupled KPZ equations
The constants Γ α βγ satisfy bilinear condition Γ α βγ = Γ α γβ for all α, β, γ, and (sometimes) trilinear condition γβ = Γ γ Γ α βγ = Γ α βα for all α, β, γ. ( T ) (cf. Ferrari-Sasamoto-Spohn 2013, Kupiainen-Marcozz 2017) σ = ( σ α β ) is an invertible matrix. Tadahisa Funaki Waseda University/University of Tokyo Invariant measures in coupled KPZ equations
Since σ is invertible, ˆ h = σ − 1 h transforms ( σ, Γ) KPZ to ( I , ˆ Γ = σ ◦ Γ) KPZ , where β ′ γ ′ σ β ′ ( σ ◦ Γ) α βγ := ( σ − 1 ) α α ′ Γ α ′ β σ γ ′ γ . Thus, the KPZ equation with σ = I is considered as a canonical form. The operation (coordinate change) Γ �→ σ ◦ Γ keeps the bilinearity, but not the trilinearity. We should say ( σ, Γ) satisfies trilinear condition, iff ˆ Γ := σ ◦ Γ satisfies (T). In the following, we assume σ = I . Tadahisa Funaki Waseda University/University of Tokyo Invariant measures in coupled KPZ equations
Two coupled KPZ approximating equations ( d = 1: FQ ’15) We replace the noise by smooth one: η ε = 1 ε η ( x ε ) → δ 0 as usual. Approx. eq-1 (usual): h α = h ε,α ∂ t h α = 1 x h α + 1 βγ ( ∂ x h β ∂ x h γ − c ε δ βγ − B ε,βγ ) + ξ α ∗ η ε , 2 ∂ 2 2 Γ α (1) where c ε = 1 ε )) and B ε,βγ (= O (log 1 L 2 ( R ) (= O ( 1 ε ∥ η ∥ 2 ε ) in general) is another renormalization factor. h α = ˜ Approx. eq-2 (suitable to study inv meas): ˜ h ε,α h α = 1 h α + 1 h γ − c ε δ βγ − ˜ 2 + ξ α ∗ η ε , ∂ t ˜ 2 ∂ 2 x ˜ βγ ( ∂ x ˜ h β ∂ x ˜ 2 Γ α B ε,βγ ) ∗ η ε (2) 2 = η ε ∗ η ε . with a renormalization factor ˜ B ε,βγ , where η ε The idea behind (2) is the fluctuation-dissipation relation. Renorm-factor c ε ≡ = O ( 1 ε ) is from 2nd order terms in the expansion, while R-factors B ε,βγ and ˜ B ε,βγ = O ( log 1 ε ) are from 4th order terms involving C ε = , D ε = . Tadahisa Funaki Waseda University/University of Tokyo Invariant measures in coupled KPZ equations
Quick overview of results on coupled KPZ eq (F-Hoshino, JFA 2017) Convergence of h ε and ˜ h ε and Local well-posedness of coupled KPZ eq ( σ, Γ) KPZ by applying paracontrolled calculus due to Gubinelli-Imkeller-Perkowski 2015 (Cole-Hopf doesn’t work for coupled eq. in general. In 1D, we used it and showed Boltzmann-Gibbs principle, FQ 2015) 2nd approx. fits to identify invariant measure under (T) Global solvability for a.s.-initial data under an invariant measure under (T) (similar to Da Prato-Debussche) Strong Feller property (due to Hairer-Mattingly 2016) Global well-posedness (existence, uniqueness) under (T) ergodicity and uniqueness of invariant measure A priori estimates for 1st approximation (1) under (T) Tadahisa Funaki Waseda University/University of Tokyo Invariant measures in coupled KPZ equations
Convergence of h ε and ˜ h ε and Local well-posedness of coupled C κ = ( B κ ∞ , ∞ ( T )) d , κ ∈ R KPZ eq ( σ, Γ) KPZ (we take σ = I ): denotes R d -valued Besov space on T . Theorem 1 (1) Assume h 0 ∈ ∪ δ> 0 C δ , then a unique solution h ε of (1) exists up to some T ε ∈ (0 , ∞ ] and ¯ T = lim inf ε ↓ 0 T ε > 0 holds. With a proper choice of B ε,βγ , h ε converges in prob. to 1 2 − δ ) for every δ > 0 and 0 < T ≤ ¯ some h in C ([0 , T ] , C T. h ε of (2) with some (2) Similar result holds for the solution ˜ h. Under proper choices of B ε,βγ and ˜ limit ˜ B ε,βγ , we can actually make h = ˜ h. ∂ t h α = 1 x h α + 1 βγ ( ∂ x h β ∂ x h γ − c ε δ βγ − B ε,βγ ) + ξ α ∗ η ε 2 ∂ 2 2 Γ α (1) h α = 1 h α + 1 h γ − c ε δ βγ − ˜ 2 + ξ α ∗ η ε ∂ t ˜ x ˜ βγ ( ∂ x ˜ h β ∂ x ˜ 2 ∂ 2 2 Γ α B ε,βγ ) ∗ η ε (2) Tadahisa Funaki Waseda University/University of Tokyo Invariant measures in coupled KPZ equations
Results under (T): Cancellation in Log-Renormalizations, Invariant measure = Wiener measure, difference of two limits. Theorem 2 Assume the trilinear condition ( T ) . B ε,βγ = O (1) so that the solutions of (1) with (1) Then, B ε,βγ , ˜ B = 0 and (2) with ˜ B = 0 converge. In the limit, we have ˜ h α ( t , x ) = h α ( t , x ) + c α t , 1 ≤ α ≤ d , where c α = 1 ∑ α ′ α ′′ Γ α ′ γγ ′ Γ α ′′ Γ α γγ ′ . 24 γ,γ ′ (2) Moreover, the distribution of { ∂ x B } x ∈ T (B = periodic BM) is invariant under the tilt process u = ∂ x h (or periodic Wiener 1 2 − δ / ∼ where h ∼ h + c). measure on the quotient space C Tadahisa Funaki Waseda University/University of Tokyo Invariant measures in coupled KPZ equations
Remark (F-Quastel 2015, stationary case): When d = 1 (i.e., scalar-valued eq), (T) is automatic and solutions of two approx. eqs without log-renormalizations satisfy ε ↓ 0 h ε + t = h CH + t h ε = lim ( ) ˜ lim . 24 24 ε ↓ 0 Tadahisa Funaki Waseda University/University of Tokyo Invariant measures in coupled KPZ equations
Global existence for a.s.-initial values under stationary measure We assume (T) and initial value h (0) is given by h (0 , 0) = 0 and u (0) := ∂ x h (0) = law ( ∂ x B ) x ∈ T . Then, similarly to Da Prato-Debussche, u = ∂ x h satisfies Theorem 3 For every T > 0 , p ≥ 1 , κ > 0 , we have [ ] ∥ u ( t ; u 0 ) ∥ p E sup < ∞ − 1 2 − κ t ∈ [0 , T ] In particular, T survival ( u (0)) = ∞ for a.a.-u (0) . Global existence for all given u (0): In the scalar-valued case, this is immediate, since the limit is Cole-Hopf solution. Hairer-Mattingly 2016 proved this for coupled eq. by showing the strong Feller property on C α − 1 , α ∈ (0 , 1 2 ). Tadahisa Funaki Waseda University/University of Tokyo Invariant measures in coupled KPZ equations
Cancellation of Log-Renorm’s, ∃ Invariant measure without (T) Example (Erta¸ s and Kardar 1992: d = 2) ∂ t h 1 = 1 x h 1 + 1 2 { λ 1 ( ∂ x h 1 ) 2 + λ 2 ( ∂ x h 2 ) 2 } + ξ 1 , 2 ∂ 2 (EK) ∂ t h 2 = 1 x h 2 + λ 1 ∂ x h 1 ∂ x h 2 + ξ 2 2 ∂ 2 Γ satisfies (T) only when λ 1 = λ 2 . However, under the transform ˆ h = sh with ( λ 1 ( λ 1 λ 2 ) 1 / 2 ) s = , (EK) is transformed into λ 1 − ( λ 1 λ 2 ) 1 / 2 h α = 1 h α + 1 h α ) 2 + s α ∂ t ˆ x ˆ 2 ( ∂ x ˆ 2 ∂ 2 β ξ β . (EK T ) ˆ Γ = s ◦ Γ in (EK T ) is given by ˆ Γ α αα = 1 , = 0 otherwise, so that ˆ Γ satisfies (T). But, (EK) is the canonical form (with σ = I ) and not (EK T ). Tadahisa Funaki Waseda University/University of Tokyo Invariant measures in coupled KPZ equations
(EK) doesn’t satisfy (T). However, since nonlinear term is decoupled in (EK T ), the Cole-Hopf transform Z α = exp ˆ h α works for each component so that global well-posedness follows. Log-renormalization factors are unnecessary. Invariant measure exists whose marginals are Wiener measures, but the joint distribution of such invariant measure is unclear (presumably non-Gaussian). Indeed, with the help of Rellich type theorem, one can easily check the tightness on the space C δ − 1 / ∼ of the 0 ∫ T a ro mean µ T = 1 Ces` 0 µ ( t ) dt of the distributions µ ( t ) T of ∂ x ˆ h ( t ) having an initial distribution ⊗ α µ α , so that the limit of µ T as T → ∞ is an invariant measure. Invariance of marginals means that of E [Φ( h ( t ))] in t only for a subclass of Φ s.t. Φ = Φ( h α ) for α = 1 or 2. Tadahisa Funaki Waseda University/University of Tokyo Invariant measures in coupled KPZ equations
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