The Hairer Quastel universality result at stationarity

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1 The Hairer Quatel univerality reult at tationarity Maimiliano Gubinelli Haudorff Center for Mathematic & Intitute for Applied Mathematic Univerität Bonn Nicola Perkowki Intitut für Mathematik Humboldt Univerität zu Berlin May 26, 216 Abtract We ue the notion of energy olution of the tochatic Burger equation to give a hort proof of the Hairer-Quatel univerality reult for a cla of tationary weakly aymmetric tochatic PD. 1 Introduction Conider the tochatic PD t v = v + ε 1/2 x F (v) + x χ ε (1) on [, ) T ε with T ε = /(2πε 1 Z), where χ ε i a Gauian noie that i white in time and patially mooth. The Hairer Quatel univerality reult [HQ15] tate that there exit contant c 1, c 2 uch that the recaled proce ε 1/2 v tε 2((x c 1 ε 1/2 t)ε 1 ) converge to the olution u of the tochatic Burger equation t u = u + c 2 x u 2 + x ξ, where ξ i a pace-time white noie. Here we give an alternative proof of thi reult, baed on the concept of energy olution [GJ13a, GJ13b, GP15a, GP15b]. nergy olution formulate the equilibrium Burger equation a a martingale problem and allow u to give a impler proof than the one of [HQ15]. On the other hand our method only applie at tationarity and moreover we need an explicit control of the invariant meaure. Let u tate the reult more preciely. We modify (1) uch that after recaling ũ ε t (x) = ε 1/2 v tε 2(xε 1 ) we have t ũ ε = ũ ε + ε 1 x Π N F (ε 1/2 ũ ε ) + x Π N ξ, ũ ε = Π N η, (2) where ξ i a pace-time white noie on [, ) T (where T = T 1 ) with variance 2, η i a pace white noie which i independent of ξ, Π N denote the projection onto the Fourier mode < k N, and we alway link N and ε via N = π/ε. Theorem 1. Let F be almot everywhere differentiable and aume that for all ε > there i a unique olution ũ ε to (2) which doe not blow up before T >. Aume alo that F, F Financial upport by the DFG via eearch Unit FO 242 i gratefully acknowledged. 1

2 L 2 (ν) where ν i the tandard normal ditribution. Then u ε t (x) := ũ ε t (x ε 1/2 c 1 (F )t), (t, x) [, T ] T, converge in ditribution to the unique tationary energy olution u of t u = u + c 2 (F ) x u 2 + x ξ, where ξ i a pace-time white noie with variance 2 and for U ν and k and H k the k-th Hermite polynomial c k (F ) = 1 k! [F (U)H k(u)]. emark 2. If F i even, then c 1 (F ) = while c 2 (F ) = if F i odd. emark 3. Note that we introduced a econd regularization in (2) compared to (1) which act on F (ε 1/2 u ε ). The reaon i that we need to keep track of the invariant meaure and thi econd regularization allow u to write it down explicitly. For the moment we are unable to deal with the original equation (1). For implicity here we only conider the mollification operator Π N, but it i poible to extend everything to more general operator ρ(ε x )u = F 1 (ρ(ε )Fu), where F denote the Fourier tranform and ρ i an even, compactly upported, bounded function which i continuou in a neighborhood of and atifie ρ() = 1. We hould then modify the equation a t ũ ε = ũ ε + ε 1 x ρ(εd)ρ(εd)f (ε 1/2 ũ ε ) + x ρ(εd) ξ, ũ ε = ρ(εd)η, to keep control of the invariant meaure, ee [FQ15]. emark 4. While our reult only applie in the tationary tate, we have more freedom in chooing the nonlinearity F than [HQ15] who require it to be an even polynomial. Alo, the method of thi paper will extend without great difficulty to the (modified) equation on [, T ]. Notation For k Z we write e k (x) = e ikx / 2π for the k-th Fourier monomial, and for u S, the ditribution on T, we define û(k) = Fu(k) = u, e k. We ue, to denote both the duality pairing in S C (T, C) and the inner product in L 2 (T), o ince we want the notation to be conitent we will alway conider the L 2 (T, ) inner product and not that of L 2 (T, C). That i, even for complex valued f, g we et f, g = f(x)g(x)dx and do T not take a complex conjugate. The Fourier projection operator Π N i given by Π N v = e kˆv(k). 2 Preliminarie < k N Let u tart by making ome baic obervation concerning the olution to (2). Galilean tranformation ecall that ũ ε olve t ũ ε = ũ ε + ε 1 x Π N F (ε 1/2 ũ ε ) + x Π N ξ, and that u ε t (x) = ũ ε t (x ε 1/2 c 1 (F )t). We define the modified tet function ϕ t (x) = ϕ(x + ε 1/2 c 1 (F )t) and then u ε t, ϕ = ũ ε t, ϕ t. The Itô Wentzell formula give d u ε t, ϕ = dũ ε t, ϕ t + ũ ε t, t ϕ t dt = ũ ε t, ϕ t dt + ε 1 x Π N F (ε 1/2 ũ ε ), ϕ t dt + d x M ε t, ϕ t + ε 1/2 c 1 (F )ũ ε t, x ϕ t dt, where M ε t (x) = ΠN ξ(, x)d. Integrating the lat term on the right hand ide by part, we get d u ε t, ϕ = u ε t, ϕ dt + ε 1 x Π N F (ε 1/2 u ε ), ϕ dt ε 1/2 c 1 (F ) x u ε t, ϕ dt + d x M ε t, ϕ t. 2

3 The martingale term ha quadratic variation d[ x M ε, ϕ t ] t = d[ M ε, x ϕ t ] t = 2 Π N x ϕ t 2 L 2dt = 2 ΠN x ϕ 2 L 2dt, which mean that the proce M ε t, ϕ := M ε t, ϕ t i of the form M ε t = ΠN ξ(, x)d for a new pace-time white noie ξ with variance 2. In concluion, u ε olve t u ε = u ε + ε 1 x Π N (F (ε 1/2 u ε ) c 1 (F )ε 1/2 u ε ) + x Π N ξ, u ε = Π N η, (3) o in other word by performing the change of variable u ε t (x) = ũ ε t (x ε 1/2 c 1 (F )t) we replaced the function F by F (x) = F (x) c 1 (F )x, and now it uffice to tudy equation (3). Invariant meaure Note that (3) actually i an SD in the finite dimenional pace Y N = Π N L 2 (T, ) 2N, o that we can apply cheverria criterion to how the tationarity of a given ditribution. The natural candidate i µ ε = law(π N η), where η i a pace white noie, ince we know that the dynamic of the regularized Orntein-Uhlenbeck proce t X ε = X ε + x Π N ξ are invariant and even reverible under µ ε and that for model in the KPZ univerality cla the aymmetric verion often ha the ame invariant meaure a the ymmetric one. Let u write BF ε (u) = ε 1 x Π N (F (ε 1/2 u) c 1 (F )ε 1/2 u) =: ε 1 x Π N F (ε 1/2 u), where F = F c 1 (F )x. Lemma 5. The vector field BF ε : Y N Y N leave the Gauian meaure µ ε invariant. More preciely, if D denote the gradient with repect to the Fourier monomial (e k ) < k N on Y N, then (BF ε (u) DΦ(u))Ψ(u)µ ε (du) = Φ(u)BF ε (u) DΨ(u)µ ε (du) Y N Y N for all Φ, Ψ L 2 (µ ε ) with B ε F DΦ, Bε F DΨ L2 (µ ε ). Proof. In thi proof it i more convenient to work with the orthonormal bai { } 1 1 π in(k ), co(k ), < k N π of Y N, rather than with Fourier monomial. We write (ϕ k ),...,2N for an enumeration of thee trigonometric function. Then B ε F D can alo be expreed in term of the (ϕ k), and we have Φ(u) = f( u, ϕ 1,..., u, ϕ 2N ), Ψ(u) = g( u, ϕ 1,..., u, ϕ 2N ) for ome f, g : 2N. We aume that f and g are continuouly differentiable, with polynomial growth of the firt order derivative. The general cae then follow by an approximation argument (note that Hermite polynomial of linear combination of ( u, ϕ k ) k form an orthogonal bai of L 2 (µ ε )). Identifying Y N with 2N, we can write µ ε (du) = γ 2N (u)du, where γ 2N i the denity of a 2N-dimenional tandard normal variable. Integrating by part we therefore have (BF ε (u) DΦ(u))Ψ(u)µ ε (du) = (BF ε (u) DΨ(u))Φ(u)µ ε (du) Y N Y N 2N ( u,ϕk BF ε (u), ϕ k BF ε (u), ϕ k u, ϕ k )Ψ(u)Φ(u)µ ε (du) (4) Y N 3

4 and it uffice to how that the zero order differential operator term on the right hand ide vanih. For the firt one of them we have 2N u,ϕk B ε F (u), ϕ k = = 2N 2N 2N = u,ϕk ε 1 x Π N F (ε 1/2 u), ϕ k ε 1/2 x (Π N F (ε 1/2 u)ϕ k ), ϕ k = ε 1/2 2 ε 1/2 Π N F (ε 1/2 u)ϕ k, x ϕ k Π N F 2N (ε 1/2 u), x ϕ 2 k, and ince in(mx) 2 + co(mx) 2 = 1 the um of the quare of the ϕ k doe not depend on x o it derivative i. For the remaining term in (4) we get µ ε -almot urely 2N B ε F (u), ϕ k u, ϕ k = B ε F (u), u = ε 1 x Π N F (ε 1/2 u), u = ε 1 x F (ε 1/2 u), Π N u = ε 1 F (ε 1/2 u), x Π N u. Now oberve that there exit G with G = F, and that under µ ε we have u = Π N u almot urely, which yield ε 1 F (ε 1/2 u), x Π N u = ε 1 G (ε 1/2 Π N u), x Π N u = ε 3/2 x G(εΠ N u), 1 =, and therefore the proof i complete. The previou lemma, together with the reveribility of the Orntein-Uhlenbeck dynamic under µ ε, implie that the Itô SD (3) ha µ ε a invariant meaure and that for T > the time revered proce û ε t = û ε T t olve with a time-revered pace-time white noie ˆξ. t û ε = û ε ε 1 x F (ε 1/2 Π N û ε ) + x Π N ˆξ (5) 3 Boltzmann-Gibb principle In the theory of interacting particle ytem the phenomenon that local quantitie of the microcopic field can be replaced in time average by imple functional of the conerved quantitie i called Boltzmann Gibb principle. In thi ection we invetigate a imilar phenomenon in order to control the antiymmetric drift term ε 1 x F (ε 1/2 u ε (x))d (6) a N +. Note that ince ε = π/n and u ε = Π N u ε we have [(ε 1/2 u ε (x)) 2 ] = 1 for all N, and therefore the Gauian random variable (ε 1/2 u ε (x)) N tay bounded in L 2 for fixed (, x), but for large N there will be wild fluctuation in (, x). We how that the quantity in (6) can be replaced by impler expreion that are contant, linear, or quadratic in u ε. 3.1 A firt computation In the following we ue η to denote a generic pace white noie and we write µ for it law, and G C(, ) denote a generic continuou function. A firt intereting computation i 4

5 to conider the random field x G(ε 1/2 Π N η(x)) and to derive it chao expanion in the variable (η k ) k where η k = η, e k are the Fourier coordinate of η. To do o conider the tandard Gauian random variable η N (x) = ε 1/2 Π N η(x) = ε 1/2 e k (x)η k, < k N and oberve that the chao expanion in L 2 (law(η N (x))) yield G(η N (x)) = n c n (G)H n (η N (x)), where H n i the n-th Hermite polynomial and c n (G) = 1 n! [G(ηN (x))h n (η N (x))] = 1 n! G(x)H n (x)γ(x)dx, where γ i the tandard Gauian denity. Since H n (x) = ( 1) n e x2 /2 n x e x2 /2, we get c n (G) = 1 n! G(x)( 1) n x n γ(x)dx = ψ(n) G (), n! where ψ G (λ) = [G(λ + η N (x))]. Our next aim i to relate the Hermite polynomial of η N (x) with the Wick power of the family (η k ) k. To do o we oberve that, on one hand the monomial H n (η N (x)) are the coefficient of the power of λ in exp(λη N (x) λ 2 /2), and on the other hand n λ n n! H n(η N (x)) = exp(λη N (x) λ 2 /2) = exp (λε 1/2 < k N e k (x)η k 1 4π < k N (λε 1/2 ) 2). Writing n for the projection onto the n-th homogeneou chao generated by η, we have ( exp < k N µ k η k 1 2 < k N ) µ k µ k = µ k1 µ kn η k1 η kn n, n! k 1 k n where the um on the right hand ide and all the following um in k 1... k n are over < k 1,..., k n N. Setting µ k = ε 1/2 λe k (x) and identifying the coefficient for different power of λ, we get H n (ε 1/2 Π N η(x)) = ε n/2 ei(k1+ +kn)x η (2π) n/2 k1 η kn n, k 1 k n which can alo be obtained by writing H n (ε 1/2 Π N η(x)) = (ε 1/2 Π N η(x)) n n and expanding the power ( ) n inide the projection. We can thu repreent the function G(η N (x)) a G(η N (x)) = c n (G)H n (ε 1/2 Π N η(x)) = c n (G)ε n/2 ei(k1+ +kn)x η (2π) n/2 k1 η kn n. n n k 1,...,k n If ϕ C (T) i a tet function, we get G(η N ), ϕ = n c n (G)ε n/2 ˆϕ( k 1 k n ) η (2π) (n 1)/2 k1 η kn n. (7) k 1,...,k n So in particular the q-th Littlewood-Paley block of G(η N ) i given by q G(η N )(x) = n c n (G)ε n/2 k 1,...,k n θ q (k k n ) ei(k1+ +kn)x (2π) (n 1)/2 η k 1 η kn n, 5

6 where (θ q ) q 1 i a dyadic partition of unity, and [ q (G(η N ) c (G))(x) 2 ] c n (G) 2 z n ε n (2π) n 1 θ q (k k n ) 2 n 1 k 1,...,k n n 1 c n (G) 2 ε n (2N) n 1 z n (2π) n 1 (2 q N) ε c n (G) 2 z n (2 q N), n 1 where z n = max k1...k n [ η k1 η kn n 2 ] n! i a combinatorial factor. We thu obtain uniformly in N, and then [ [ q (G(η N ) c (G)) 2 L 2 (T) ] min{ε2q, 1}, 2] q (G(ε 1/2 u ε r(x)) c (G))dr t t 2 min{ε2 q, 1}, [ q (G(ε 1/2 u ε r(x)) c (G)) 2 ]dr where in the lat tep we ued that ε 1/2 u ε r ha the ame ditribution a η N, which eaily implie the following reult. Lemma 6. Aume that [ G(U) 2 ] < for a tandard normal variable U, and let c (G) = [G(U)]. Then lim N G(ε 1/2 u ε (x))d = c (G)t, where the convergence i in C([, T ], H ). If c (G) =, then i bounded in C([, T ], H 1/2 ). ε 1/2 G(ε 1/2 u ε (x))d To analye the cae where c (G) = we need a more refined argument which i provided by the technique of regularization by noie for controlled path. 3.2 egularization by noie Let u write L ε for the generator of the mollified Orntein Uhlenbeck proce t X ε = X ε + x Π N ξ. The baic tool which allow u to control time integral uch a G(ε1/2 u ε (x))d i the following Itô trick. Define for Ψ L 2 (µ ε ) ε (Ψ) := k 2 D k Ψ 2, < k N where D k i the derivative in the direction e k. Lemma 7 (Itô trick). For Ψ dom (L ε ) and T >, p 1 we have [ p] up L ε Ψ(u ε )d T p/2 [ ε (Ψ) p/2 ]. t [,T ] The proof i given in [GJ13b, GP15b] and extend without difficulty to our etting, o we do not repeat the argument here. To apply the Itô trick we need to olve the Poion equation. In our etting thi can be done efficiently by uing the chao expanion (7). ecall that we wrote η k = η, e k 6

7 for the Fourier coefficient of a truncated patial white noie Π N η (which therefore ha law µ ε ), and that n denote the projection onto the n-th chao. We need to compute L ε η k1... η kn n, a thee are the random variable appearing in a general chao expanion. Let u tart by conidering ϕ Y N = Π N L 2 (T, ) with ϕ L 2 = 1 for which we have η, ϕ n n = H n ( η, ϕ ), where H n i the n-th Hermite polynomial. Itô formula give dh n ( X ε t, ϕ ) = H n( X ε t, ϕ ) X ε t, ϕ dt + H n( X ε t, ϕ ) Π N x ϕ, Π N x ϕ dt + dm t, with a quare integrable martingale M. The Hermite polynomial atify H n = nh n 1, o we get H n( X ε t, ϕ ) X ε t, ϕ + H n( X ε t, ϕ ) Π N x ϕ, Π N x ϕ = nh n 1 ( X ε t, ϕ )H 1 ( X ε t, ϕ ) n(n 1)H n 2 ( X ε t, ϕ ) Π N ϕ, Π N ϕ. The projection onto the n-th chao of the firt term i explicitly given by H n 1 ( X ε t, ϕ )H 1 ( X ε t, ϕ ) n = X ε t, ϕ n 1 n 1 X ε t, ϕ 1 n = X ε t, ϕ n 1 n 1 X ε t, ϕ 1 (n 1) X ε t, ϕ n 2 n 2 Π N ϕ, Π N ϕ, which i obtained by contracting X ε t, ϕ with each of the n 1 variable X ε t, ϕ inide the projector n 1. Therefore, we have which how that dh n ( X ε t, ϕ ) = n H n 1 ( X ε t, ϕ )H 1 ( X ε t, ϕ ) n dt + dm t = n X ε t, ϕ n 1 X ε t, ϕ n dt + dm t, L ε η, ϕ n n = n η, ϕ n 1 η, ϕ n. So far we aumed ϕ L 2 = 1, but actually thi lat formula i invariant under caling o it extend to all ϕ Π N L 2 (T, ), and then to ϕ Π N L 2 (T, C), and for general product we obtain by polarization L ε η, ϕ 1... η, ϕ n n = So finally we deduce that n η, ϕ 1... η, ϕ k... η, ϕ n η, ϕ k n. L ε η k1 η kn = (k k 2 n) η k1 η kn (8) for all < k 1,..., k n N. Combining that formula with (7), we obtain the following lemma. Lemma 8. Conider a function of the form Φ(η) = G(ε 1/2 Π N η), ϕ and aume that [G(U)] =, where U i a tandard normal variable, or that ˆϕ() =. Then the olution Ψ to the Poion equation L ε Ψ = Φ i explicitly given by Ψ(η) = c n (G)ε n/2 ˆϕ( k 1 k n ) η k1 η kn n (2π) (n 1)/2 (k n 1 k k2 1 k n n), where the um i over all < k 1,..., k n N. emark 9. Incidentally note that the olution can be repreented a Ψ(η) = = dt c n (G)ε n/2 n 1 k 1 k n e (k dtg(ε 1/2 (e t Π N η)(x)) k2 n )t ei(k1+ +kn)x (2π) n/2 η k1 η kn n

8 To apply the Itô trick we need to compute (Ψ) = k k2 D k ΨD k Ψ for the olution Ψ of the Poion equation. For that purpoe conider again ϕ Y N with ϕ L 2 = 1 and H n ( η, ϕ ) = η, ϕ n n, for which we have D k H n ( η, ϕ ) = H n( η, ϕ ) e k, ϕ = nh n 1 ( η, ϕ ) e k, ϕ = n η, ϕ n 1 n 1 e k, ϕ, o by polarization D k η k1 η kn n = j 1 kj=k η k1 η kj η kn n 1. (9) To prove the Boltzmann Gibb principle we need one more auxiliary reult. Lemma 1. For all M N, l Z and < t < we have the etimate Proof. We imply bound [ [ 2] x (Π M u ε r) 2, e l dr l 2 t 2 M. 2] x (Π M u ε r) 2, e l dr t [ x (Π M u ε r) 2, e l 2 ]dr, and ince we can replace (Π M u ε r) 2 by (Π M u ε r) 2 [(Π M u ε r) 2 ], the integrand i given by [ x (Π M u ε r) 2, e l 2 ] = l 2 dx dx [ (Π M u ε r(x)) 2 2 (Π M u ε r(x )) 2 2 ] T T l 2 dx dx [Π M u ε r(x)π M u ε r(x )] 2. The expectation on the right hand ide can be explicitly computed a [Π M u ε r(x)π M u ε r(x )] = < k M T e ik(x x ) min{2m, C x x 1 }, T = co(m(x x )) co((m + 1)(x x )) 1 co(x x 1 ) for ome contant C < +, for which dx dx min{2m, C x x 1 } 2 dx 2M, and therefore the claim follow. T T Propoition 11 (Boltzmann Gibb principle). Let G, G L 2 (ν), where ν denote the law of a tandard normal variable. Then for all l Z and < t + 1 and all κ > [ ] 2 ε 1 x Π N G(ε 1/2 u ε r) ε 1/2 c 1 (G) x Π N u ε r, e l dr t 3/2 κ l 2 uniformly in N N, and for all M N/2 [ ε 1 x Π N G(ε 1/2 u ε r) ε 1/2 c 1 (G) x Π N u ε r c 2 (G) x (Π M u ε r) 2, e l dr t l 2 (M 1 + ε log 2 N) G (x) 2 ν(dx). G (x) 2 ν(dx) 2 ] 8

9 Proof. We firt how the econd bound. Toward thi end note that x (Π M η) 2 = Π N x (Π M η) 2 for M N/2 and that by Lemma 8 the olution Ψ to i given by L ε Ψ(η) = ε 1 G(ε 1/2 Π N η) c 1 (G)ε 1/2 Π N η c 2 (G)(ε 1/2 Π M η) 2, x Π N e l Ψ(η) = c 2 (G) k 1,k 2 1 k1 k 2 >M 1 < l N (il) 1 k 1+k 2=l (2π) 1/2 η k1 η k2 2 (k k2 2 ) + c n (G)ε n/2 1 n 3 k 1 k n 1 < l N (il) 1 k1+ +kn=l (2π) (n 1)/2 η k1 η kn n (k k2 n), where it i undertood that all um um in k i are over < k i N. Therefore (9) yield for < l N D k Ψ(η) = c 2 (G)2 k 1 1 k k1 >M il 1 k+k 1=l (2π) 1/2 η k1 1 (k 2 + k 2 1 ) Applying the Itô trick we get [ + c n+1 (G)ε (n 1)/2 (n + 1) n 2 il 1 k+k1+ +kn=l (2π) n/2 k 1 k n η k1 η kn n (k 2 + k k2 n). ε 1 x Π N G(ε 1/2 u ε r) ε 1/2 c 1 (G) x Π N u ε r c 2 (G) x (Π M u ε r) 2, e l dr t k 2 [ D k Ψ 2 ] = t < k N < k N + t = t n 1 A n, k 2 c 2 (G) l 2 k 1 < k N k 2 n 2 1 k+k1=l [ η k1 1 2 ] 1 k k1 >M 2π (k 2 + k1 2)2 c n+1 (G) 2 ε (n+1) 2 (n + 1) 2 l 2 k 1 k n 1 k+k1+ +kn=l 2 ] [ η k1 η kn n 2 ] (2π) n (k 2 + k k2 n) 2 where the (A n ) are implicitly defined by the equation. Now [ η k1 η kn n 2 ] n! for all k 1,..., k n, o that A 1 < k, k 1 N c 2 (G) 2 l 2 k 2 c 2 (G) 2 l 2 1 k k1 >M 1 k+k1=l (k 2 + k1 2 )2 < k < c 2 (G) 2 l 2 M 1, < k, k 1 N 1 l k 1 k l k >M k 2 + (l k) 2 c 2 (G) 2 l 2 < k < ( c 2 (G) 2 l 2 1 k k1 >M 1 k+k1=l k 2 + k1 2 1 l k M 2 + (l k) l k k 2 + M 2 ) 9

10 while for n > 1 A n = k 2 c n+1 (G) 2 ε n 1 (n + 1) 2 l 2 < k N εn 1 (2π) n l2 (n + 1) 2 c n+1 (G) 2 n! εn 1 (2π) n l2 (n + 1) 2 c n+1 (G) 2 n! εn 1 (2π) n l2 (n + 1) 2 c n+1 (G) 2 n! 1 k+k1+ +kn=l [ η k1 η kn n 2 ] (2π) n (k 2 + k k k2 1 k n n) 2 < k, k 1,..., k n N < k 1,..., k n N < k 1,..., k n N = εn 1 (2π) n l2 (n + 1) 2 c n+1 (G) 2 n!(2n) n 2 n=2 k 2 1 k+k1+ +k n=l (k 2 + k k2 n) 2 1 k k2 n 1 k k2 2 < k 1, k 2 N 1 k k2 2 The um over n i bounded by c n+1 (G) 2 n!(n + 1) 2 = nc n (G) 2 n! n=1 εl 2 (n + 1) 2 c n+1 (G) 2 n! log 2 N. G (x) 2 ν(dx), o that overall we get [ ] 2 ε 1 x Π N G(ε 1/2 u ε r) ε 1/2 c 1 (G) x Π N u ε r c 2 (G) x (Π M u ε r) 2, e l dr t l 2 (M 1 + ε log 2 N) G (x) 2 ν(dx), (1) which i our econd claimed bound. To get the firt bound, we take M t 1/2 in (1) (which require N > t 1/2 ), and combine thi with Lemma 1 to obtain [ ] 2 ε 1 x Π N G(ε 1/2 u ε r) ε 1/2 c 1 (G) x Π N u ε r, e l dr t l 2 (M 1 + ε log 2 N + t M) G (x) 2 ν(dx) t 3/2 κ l 2 G (x) 2 ν(dx). If N t 1/2 we ue another etimate: a in the proof of Lemma 1 we have [ ] 2 ε 1 x Π N G(ε 1/2 u ε r) ε 1/2 c 1 (G) x Π N u ε r, e l dr t 2 [ ε 1 x Π N G(ε 1/2 u ε ) ε 1/2 c 1 (G) x Π N u ε, e l 2 ] t 2 l 2 ε 2 c n (G) 2 dx dx [H n (ε 1/2 u ε (x))h n (ε 1/2 u ε (x ))], n 2 T T t 2 l 2 ε 2 n!c n (G) 2 dx dx [ε 1/2 u ε (x)ε 1/2 u ε (x )] n n 2 T T t 2 l 2 ε n 2 n!c n (G) 2 dx dx min{2n, C x x 1 } n n 2 T T t 2 l 2 ε n 2 n!c n (G) 2 (2N) n 1 t 2 l 2 ε 1 c n (G) 2 n! n 2 n 2 l 2 t 3/2 G (x) 2 ν(dx), where in the lat tep we ued that t 1/2 N

11 4 The invariance principle We now have all the tool to prove the convergence of (u ε ) to an energy olution of the tochatic Burger equation. We proceed in two tep. Firt we etablih the tightne of (u ε ), and in a econd tep we how that every weak limit i an energy olution. Uing the uniquene of energy olution, we therefore obtain the convergence of (u ε ). Tightne Let (u ε ) olve (3) and write F (x) = F (x) c 1 (F )x. To prove the tightne of (u ε ) it uffice to how that for all l Z the complex-valued proce ( u ε, e l ) i tight and atifie a polynomial bound in l, uniformly in ε. We decompoe u ε t, e l a u ε t, e l = u ε, e l + u ε, e l d Π N ξ, x e l d ε 1 Π N F (ε 1/2 u ε ), x e l d =: u ε, e l + S ε t, e l + A ε t, e l + M ε t, e l, (11) where S ε, A ε, M ε tand for ymmetric, antiymmetric and martingale part, repectively, and we how tightne for each term on the right hand ide eparately. The convergence of u ε t, e l at a fixed time (in particular t = ) follow from the fact that the law of u ε t i that of µ ε for all t, and (µ ε ) obviouly converge to the law of the white noie a ε. The linear term i tight becaue [ ] p u ε r, e l dr t p 1 [ u ε r, l 2 e l p ]dr t p 1 [ u ε r, l 2 e l 2 ] p/2 dr = t p l 2p. For all ε > the martingale term i a mollified pace time white noie, o it convergence i immediate. Only the nonlinear contribution to the dynamic i nontrivial to control. Here we ue the Boltzmann Gibb principle tated in Propoition 11 to get [ ε 1 Π N F ] 2 (ε 1/2 u ε r), x e l dr t 3/2 κ l 2 F (x) 2 ν(dx). Thi bound give readily tightne in C([, T ], C) and alo that any limit point ha zero quadratic variation. Similarly we have for the time revered proce û ε t = u ε T t û ε t, e l = û ε, e l + û ε, e l d + Π N ˆξ, x e l d ε 1 Π N F (ε 1/2 û ε ), x e l d =: û ε, e l + Ŝε t, e l + Âε t, e l + ˆM ε t, e l, (12) and the ame argument a before how that each term on the right hand ide i tight in C([, T ], C), atifie a uniform polynomial bound, and that any limit point of Âε, e l ha zero quadratic variation. Since we have uitable moment bound for each term, we actually get the joint tightne: Lemma 12. Conider the decompoition (11), (12). Then the tuple (u ε, û ε, S ε, Ŝε, A ε, Âε, M ε, ˆM ε ) i tight in (S ) 2 C ([, T ], S ) 6. For every weak limit (u, û, S, Ŝ, A, Â, M, ˆM) and any ϕ C (T) the procee A, ϕ and Â, ϕ have zero quadratic variation and atify Ât = (A T A T t ). Moreover, u t = u + S t + A t + M t, t [, T ], i for every fixed time a patial white noie. 11

12 ecall the definition of energy olution to the tochatic Burger equa- Convergence tion [GJ13b]: Definition 13. (Controlled proce) Denote with Q the pace of continuou tochatic procee (u, A) on [, T ] with value in S uch that i) the law of u t i the white noie µ for all t [, T ]; ii) For any tet function ϕ S the proce t A t, ϕ i almot urely of zero quadratic variation, A, ϕ = and the pair ( u, ϕ, A, ϕ ) atifie the equation u t, ϕ = u, ϕ + u, ϕ d + A t, ϕ M t, x ϕ (13) where ( M t, x ϕ ) t T i a martingale with repect to the filtration generated by (u, A) with quadratic variation [ M, x ϕ ] t = 2t x ϕ 2 L 2 (T) ; iii) the revered procee û t = u T t, Â t = (ÂT A T t ) atify the ame equation with repect to their own filtration (the backward filtration of (u, A)). The pair (u, A) i called controlled ince for A we imply get the Orntein Uhlenbeck proce, o in general u i a zero quadratic variation perturbation of that proce. Uing the Itô trick, it i not hard to how that for controlled procee the Burger nonlinearity i well defined: Lemma 14 ([GJ13b], Lemma 1). Aume that (u, A) Q and et for M N B M t, ϕ = (Π M u ) 2, x ϕ d. Then (B M t ) converge in probability in C ([, T ], S ) and we denote the limit by x u 2 d, ϕ. A controlled proce (u, A) i a olution to the tochatic Burger equation t u = u + c x u 2 + x ξ if A = c xu 2 d. According to [GP15b, Theorem 2], there i a unique energy olution. The following theorem thu implie our main reult, Theorem 1. Theorem 15. Let (u, A) be a in Lemma 12. Then (u, A) Q and u i the unique energy olution to t u = u + c 2 (F ) x u 2 + x ξ. Proof. The tuple (u ε, û ε, S ε, Ŝε, A ε, Âε, M ε, ˆM ε ) converge along a ubequence ε n, but to implify notation we till denote thi ubequence by the ame ymbol. Since (u ε, S ε, A ε, M ε ) converge jointly and for every fixed ε the proce u ε olve (2), we get for ϕ C (T) u t, ϕ = u, ϕ + S t, ϕ + A t, ϕ + M t, ϕ, and ince St ε, ϕ = uε, ϕ d alo S t, ϕ = u, ϕ d. The ame argument work for the backward proce, o that (u, A) Q. It remain to how that A = c 2 (F ) x u 2, which follow from the Boltzmann Gibb principle, Propoition 11. For all ε > and M N/2 = π/(2ε), [ 2] A ε r c 2 (F ) x (Π M u ε r) 2, e l dr t l 2 (M 1 + ε log 2 N) F (x) 2 ν(dx), 12

13 o by Fatou lemma [ 2] [ A c 2 (F ) x (Π M u r ) 2, e l dr lim inf ε A ε r c 2 (F ) x (Π M u ε r) 2, e l dr t l 2 M 1 F (x) 2 ν(dx). It now uffice to end M. eference [FQ15] Funaki, T. and Quatel, J. KPZ equation, it renormalization and invariant meaure. Stochatic Partial Differential quation: Analyi and Computation, 3.2 (215): [GJ13a] Gonçalve, P., and Jara, M. Nonlinear Fluctuation of Weakly Aymmetric Interacting Particle Sytem. Archive for ational Mechanic and Analyi, 212(2): , dec 213. [GJ13b] Gubinelli, M., and Jara, M. egularization by noie and tochatic Burger equation. Stochatic Partial Differential quation: Analyi and Computation, 1.2 (213): [GP15a] Gubinelli, M., and Perkowki, N. Matematico, 29 (215): ] Lecture on ingular tochatic PD. naio [GP15b] Gubinelli, M., and Perkowki, N. nergy olution of KPZ are unique. arxiv preprint arxiv: (215). [HQ15] Hairer, M., and Quatel, J. A cla of growth model recaling to KPZ. arxiv preprint arxiv: (215). 13