KPZ equation with fractional derivatives of white noise

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1 KPZ equation with fractional derivatives of white noise Masato Hoshino The University of Tokyo October 21, 2015 Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

2 Background We consider ω-wise solutions of non-liner stochastic PDEs Examples: t h(t, x) = h(t, x) + h(t, x) 2 + ξ(t, x) : KPZ t Φ(t, x) = Φ(t, x) Φ(t, x) 3 + ξ(t, x) : Φ 4 -model ξ : space-time white noise on [0, ) R d The solution of stochastic heat equation: satisfies X(t, ) C 2 d 2 x R d, t 0 t X(t, x) = X(t, x) + ξ(t, x) We cannot define above non-liner terms, because they are related to the product ξη of ξ C α and η C β with α + β 0 Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

3 Subcriticality h(t, x) h δ (t, x) = δ 2 d 2 h(δ 2 t, δx), ξ(t, x) ξ δ (t, x) = δ d+2 2 ξ(δ 2 t, δx) (δ > 0) Changing variables t h(t, x) = h(t, x) + h(t, x) 2 + ξ(t, x) t h δ (t, x) = h δ (t, x) + δ 2 d 2 hδ (t, x) 2 + ξ δ (t, x) Formally, the non-liner term vanish as δ 0, iff d = 1 : subcritical Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

4 Hairer s theory M Haier introduced regularity structure(rs) (2013) We can construct RS from given stochastic PDE iff it is subcritical We can define the renormalization map on RS, and obtain a result as below: Theorem (Hairer, 2013) Let ρ : R 2 R be a smooth, nonnegative, symmetric, and compactly supported function st ρ = 1 Set ρ ϵ (t, x) = ϵ 3 ρ(ϵ 2 t, ϵ 1 x), and ξ ϵ = ξ ρ ϵ Then, there exists a sequence of constants {C ϵ 1 } st the sequence of ϵ solutions h ϵ (local in time) to t h ϵ (t, x) = 2 x h ϵ(t, x) + ( x h ϵ (t, x)) 2 C ϵ + ξ ϵ (t, x), converges to a stochastic process h x T, t 0 Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

5 More rough noise Define γ x ξ = ( 2 x ) γ 2 ξ (γ > 0) as a random distribution st ( 2 x ) γ 2 ξ(ϕ) = ξ(( 2 x ) γ 2 ϕ) ( ϕ C 0 (R 2 )), where ( 2 x ) γ 2 ϕ = F 1 ( ξ γ Fϕ) Does it hold similar renormalization to the following equation? t h(t, x) = x 2 h(t, x) + ( xh(t, x)) 2 + x γ ξ(t, x) Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

6 Main result This eq is subcritical iff γ < 1 2 But similar renormalization holds iff γ < 1 4 Theorem (Hoshino, 2015) Let 0 γ < 1 4 Let ρ : R2 R be a smooth, nonnegative, symmetric, and compactly supported function such that ρ = 1 Then, there exists a sequence of constants {C ϵ ϵ 1 2γ } st the sequence of solutions h ϵ (local in time) to t h ϵ (t, x) = 2 x h ϵ(t, x) + ( x h ϵ (t, x)) 2 C ϵ + γ x ξ ϵ(t, x), converges to a stochastic process h x T, t 0 Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

7 Notations k s := 2k 0 + k 1, k := k 0 t k 1 x (k = (k 0, k 1 ) Z 2 + ) z s := t + x (z = (t, x) R 2 ) For a function ρ on R 2, S δ s,z ρ(z ) := δ 3 ρ(δ 2 (t t), δ 1 (x x)) B r = {ρ C 0 (R2 ); ρ C r 1, suppρ B s (0, 1)} (r N) C α s (R2 ) (α > 0) : locally α-hölder space wrt s C α s (R2 ) (α < 0) : All of distributions ξ st ξ(s δ s,z ρ) δα, uniformly over ρ B r (r = α ) and locally uniformly over z R 2 Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

8 Abstraction of stochastic PDE Mild form t h = 2 x h + ( xh) 2 + η (η : smooth noise), h(0, ) = h 0 h = G (1 t>0 (( x h) 2 + η)) + Gh 0 Formally, h is represented by a sum of distributions with negative regularities, and a remainder We reformulate KPZ eq into an equation of H = α h ατ α (τ α : basis vector, h α : a function on R 2 ) which values in an abstract liner space H = G(1 t>0 (( H) 2 + Ξ)) + Gh 0 Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

9 Regularity structures Definition (Regularity structure) A regularity structure (A, T, G) consists of the following elements: A R, locally finite, bounded from below, 0 A T = α A T α Each T α is a liner space with a norm α G is a group of linear operators T T, such that Γτ τ β<α,β A T β for all Γ G, α A, τ T α α 0 := inf A Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

10 Model Definition (Model) A model (Π, Γ) on R d for a regularity structure (A, T, G) consists of the following elements: Π z (z R d ) : T D (R d ) is linear map such that (Π z τ )(S δ s,z ρ) δα (loc in z), for all α A, τ T α, and ρ B r (r = α 0 ) Γ z,z (z, z R d ) G is an element such that Γ z,z τ β z z α β s (loc in z, z ), for all α, β A (β < α) and τ T α Π z Γ z,z = Π z, Γ z,z Γ z,z = Γ z,z ( z, z, z R d ) Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

11 Reconstruction Definition (Modelled distribution) Let (Π, Γ) be a model f : R d T is in D γ (γ R) iff f(z) Γ z,z f(z ) α z z γ α s (loc in z, z ), for all α A with α < γ Theorem (Reconstruction theorem) Let (Π, Γ) be a model and γ > 0 Then there exists a unique liner map R : D γ C α 0 s (Rd ) st for all f D γ (Rf Π z f(z))(s δ s,z ρ) δγ (loc in z), Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

12 RS for KPZ U:All of basis vectors that describe h V:All of basis vectors that describe ( x h) 2 + η 1, X 0, X 1, Ξ V, Regularities of basis vectors τ V Iτ U (I(X k ) = 0), (X k 0 0 Xk 1 1 ) = k 1X k 0 0 Xk 1 1 1, τ 1, τ 2 U τ 1 τ 2 V 1 = 0, X 0 = 2, X 1 = 1, Ξ = α 0 (to be determined) τ τ = τ + τ Iτ = τ + 2, Iτ = τ + 1 F = U V, T = spanf Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

13 RS for KPZ ξ ϵ ξ in prob in C 3 2 κ s (κ > 0) We use shorthand notations to describe basis vectors We should choose α 0 = 3 κ (κ > 0 : sufficiently small) 2 Ξ,,,,,,,, 1, F Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

14 RS for fractional case x γξ ϵ x γξ in prob in 3 C 2 γ κ s (κ > 0) We should choose α 0 = 3 γ κ (κ > 0 : sufficiently small) 2 0 γ < 1 10 Ξ,,,,,,,, 1, 1 10 γ < 1 6 Ξ,,,,,,,,,,, 1, Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

15 RS for fractional case 1 6 γ < 3 14 Ξ,,,,,,,,,,,,,,,,,,,,, 1, 3 14 γ < 1 4 Ξ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 1, and so on Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

16 Canonical model There is a canonical lift η into (Π η, Γ η ): Π z 1(z ) = 1, Π z Ξ(z ) = η(z ) (1) Π z X 0 (z ) = t t, Π z X 1 (z ) = x x (2) Π z τ τ = (Π z τ )(Π z τ ) (3) Π z Iτ = G Π z τ ( z) k k G Π z τ k! (4) k < Iτ Π z Iτ = x G Π z τ k < Iτ ( z) k k x G Π z τ (5) k! Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

17 Admissible model Assume (1)(2)(4)(5) Admissible model For each admissible model Z, a linear map G : D θ D θ+2 is defined and satisfies RGf = G Rf, G = I + ({X k }-valued part) Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

18 Solution map Theorem (Hairer, 2013) α 0 ( 2, 3 2 ), θ > α 0, ζ (0, α 0 + 2) For each initial condition h 0 and admissible model Z, there exists T > 0 st the equation H = G(1 t>0 (( H) 2 + Ξ)) + Gh 0 has a unique solution H D θ,ζ (permit singularity as t 0+) on t [0, T ] S : (h 0, Z) H is continuous Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

19 Solution map When Z = (Π η, Γ η ) for some smooth noise η, we have RH = G (1 t>0 R(( H) 2 + Ξ)) = G (1 t>0 (( x RH) 2 + η)) So h = RH solves t h = x 2h + ( xh) 2 + η Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

20 Renormalized KPZ eq α 0 = 3 κ (κ > 0 : sufficiently small) 2 For any constants C, C, C, the set of functions Π η z τ = Πη zτ (τ = Ξ,,,,, 1), Π η z = Π η z C, Π η z = Π η z C, Π η z = Π η z C is uniquely extended to an admissible model ( Π η, Γ η ) h = S(h 0, Ẑη ) solves the equation t h = 2 x h + ( xh) 2 (C + C + 4C ) + η Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

21 Convergence of renormalized models Theorem Choose proper C τ (ϵ) (τ =,, ) Let Z (ϵ) be a model lifted from ξ ϵ Let Ẑ(ϵ) be the renormalized model Then there exists an admissible random model Ẑ st Ẑ (ϵ) Ẑ in prob (ϵ 0) Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

22 For the fractional case Theorem Let 0 γ < 1 4 Choose proper C τ (ϵ) for all τ F with τ = 2, 4, 6 Let Z (ϵ) be a model lifted from x γξ ϵ Let Ẑ(ϵ) be the renormalized model Then there exists an admissible random model Ẑ st Ẑ (ϵ) Ẑ in prob (ϵ 0) Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

23 Proof of renormalization Π (ϵ) 0 τ (Sλ s,0ϕ) convergences for all τ 0 ( Π (ϵ), Γ (ϵ) ) convergences For each τ F, (ϵ) Π 0 τ has the form Π (ϵ) 0 τ (z) = I k (Ŵ(ϵ,k) τ (z;,, )) I k :kth multiple Wiener-Itô integral Ŵ (ϵ,k) τ (z) (L 2 (R T)) k By Itô isometry, we have (ϵ) E Π 0 τ (Sλ s,0 ϕ) S λ s,0 ϕ(z)sλ s,0 ϕ(z ) Ŵ(ϵ,k) τ (z), Ŵ(ϵ,k) τ (z ) (L 2 (R T)) kdzdz Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

24 Proof of renormalization Assume that there exist Ŵ(k) τ (z) st Ŵ(k) τ (z), Ŵ(k) τ (z ) (L 2 (R T)) k z z 2 τ +δ s δŵ(ϵ,k) τ (z), δŵ(ϵ,k) τ (z ) (L 2 (R T)) k ϵδ z z 2 τ +δ s (δw (ϵ,k) = W (k) W (ϵ,k) ) with 2 τ > 3, for some δ > 0 Note that Ss,0 λ ϕ(z)sλ s,0 ϕ(z ) z z Then Ẑ(ϵ) Ẑ 2 τ +δ s dzdz 2 τ +δ λ Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

25 Proof of renormalization Π (ϵ) 0 (z) = x G γ x ξ(z) = I 1( γ x xg ρ ϵ (z )) Ŵ (ϵ,1) (z; ) = γ x xg ρ ϵ (z ) Ŵ (1) (z; ) = γ x xg(z ) γ x xg(z) z 2 γ s around 0 Ŵ(1) (z), Ŵ(1) (z ) (L 2 (R T)): 1 2γ > 2 ¼ z z 1 2γ s Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

26 Proof of renormalization Π (ϵ) 0 (z) = I 2 ( ) + Π (ϵ) 0 (z) = I 2 ( Π 0 (z) = I 2 ( ) ), C (ϵ) = Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

27 Proof of renormalization Ŵ(2) (z), Ŵ(2) (z ) (L 2 (R T)) 2: z z 2 4γ s 2 4γ > 2 2 4γ > 3 γ < 1 4 ½ ¾ ϵ 1 2γ Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

28 Proof of renormalization Π (ϵ) 0 (z) = I 4 ( ) + I 2 ( ) + Π (ϵ) 0 (z) = I 4 ( ) + I 2 ( ), C (ϵ) = Π (ϵ) 0 (z) = I 4 ( ) + I 2 ( ), Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

29 Proof of renormalization Ŵ(4) (z), Ŵ(4) (z ) (L 2 (R T)) 4: z z 8γ s Ŵ(2) (z), Ŵ(2) (z ) (L 2 (R T)) 2: ¾ + ¾ ¾ ¾ ¾ ¾ ¾ z z 8γ s Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

30 Proof of renormalization 8γ > 2 ¾ ϵ 4γ Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

31 References M Hairer A theory of regularity structures Invent Math (2014) M Hairer Singular stochastic PDEs ArXiv: Masato Hoshino (The University of Tokyo) KPZ equation with fractional derivatives of white noise October 21, / 31

Structures de regularité. de FitzHugh Nagumo

Structures de regularité. de FitzHugh Nagumo Nils Berglund nils.berglund@univ-orleans.fr http://www.univ-orleans.fr/mapmo/membres/berglund/ Strasbourg, Séminaire Calcul Stochastique Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo