Renormalizing the Kardar-Parisi-Zhang equation in $d\geq 3$ in weak disorder

We study Kardar-Parisi-Zhang equation in spatial dimension 3 or larger driven by a Gaussian space-time white noise with a small convolution in space. When the noise intensity is small, it is known that the solutions converge to a random limit as the smoothing parameter is turned off. We identify this limit, in the case of general initial conditions ranging from flat to droplet. We provide strong approximations of the solution which obey exactly the limit law. We prove that this limit has sub-Gaussian lower tails, implying existence of all negative (and positive) moments.

1 Introduction and main results.

KPZ equation and its regularization.
We consider the Kardar-Parisi-Zhang (KPZ) equation written informally as and driven by a totally uncorrelated Gaussian space-time white noise ξ. More precisely, ξ on R + × R d is a family {ξ(ϕ)} ϕ∈S(R + ×R d ) of Gaussian random variables with mean 0 and covariance The equation (1.1) describes the evolution of a growing interface in d+1 dimension [KPZ86,S16] and also appears as the scaling limit for d = 1 of front propagation of the certain exclusion processes ( [BG97,C12]) as well as that of the free energy of the discrete directed polymer ( [AKQ14]). It should be noted that, on a rigorous level, only distribution-valued solutions are expected for (1.1), and thus it is already ill-posed in d = 1 stemming from the inherent non-linearity of the equation and the fundamental problem of squaring or multiplying random distributions. For d = 1, studies related to the above equation have enjoyed a huge resurgence of interest in the last decade starting with the important work [H13] which gave an intrinsic precise notion of a solution to (1.1). We now fix a spatial dimension d ≥ 3. As remarked earlier, since (1.1) is a-priori ill-posed, we will study its regularized version which is driven by the spatially mollified noise with φ ε (·) = ε −d φ(·/ε) being a suitable approximation of the Dirac measure δ 0 and C ε being a suitable divergent (renormalization) constant. We will work with a fixed mollifier φ : R d → R + which is smooth and spherically symmetric, with supp(φ) ⊂ B(0, 1 2 ) and R d φ(x) dx = 1. Then, {ξ ε (t, x)} is a centered Gaussian field with covariance E[ξ ε (t, x)ξ ε (s, y)] = δ(t − s) ε −d V (x − y)/ε , where V = φ ⋆ φ is a smooth function supported in B(0, 1). We also remark that in (1.2), the multiplicative parameter β can be taken to be positive without loss of generality, while by rescaling, no multiplicative parameter is needed in (1.1), see [Q12]. Also in spatial dimensions d ≥ 3, the factor ε d−2 2 is the correct scaling -a small enough β > 0 guarantees a non-trivial random limit of h ε as ε → 0, see the discussion in Section 1.3.
The goal of the present article is to consider general solutions of (1.2), namely the solutions of (1.2) with various initial conditions and prove that as the mollification parameter ε is turned off, the renormalized solution of (1.2) converges to a meaningful random limit as long as β remains small enough. We use Feynman-Kac representation of the solution of stochastic heat equation and results from directed polymers. Not only do we identify the distributional limit of h ε , but we also provide a sequence (indexed by ε) of functions of the noise such that • it is a strong approximation of h ε , i.e. the difference tends to 0 in norm, • all terms in the sequence obey the limit law.
The above functions for the flat initial condition are defined from the martingale limit of a random polymer model taken at some rescaled, shifted and time-reversed version of the noise. The similar functions for other intial conditions can be derived from the martingale limit taken at various version of the noise and the heat kernel. We finally show that it has sub-Gaussian lower tails in this regime, which implies existence of all negative and positive moments of this object. Besides new contributions, we gather and reformulate results which are atomized in the literature, often stated in a primitive form and hidden by necessary technicalities. We end up the introduction with a rather complete account on the state-of-the-art. We now turn to a more precise description of our main results.

Main results.
In order to state our main results, we will introduce the following notation which will be consistently used throughout the sequel. Recall the definition of the space-time white noise ξ ∈ S ′ (R×R d ) which is a random tempered distribution (defined in all times, including negative ones), and for any ϕ ∈ S(R × R d ), ε > 0, t ∈ R and x ∈ R d , Equivalently, so that by invariance under space-time diffusive rescaling, time-reversal and spatially translation, ξ (ε,t,x) is itself a Gaussian white noise and possesses the same law as ξ. This is also the reason why we define the noise above also for negative times. To abbreviate notation, we will also write ξ (ε,t) = ξ (ε,t,0) . (1.4) We also need specify the definition(s) of the critical disorder parameter. Note that (1.2) is inherently non-linear. The Hopf-Cole transformation suggests that solves the linear multiplicative noise stochastic heat equation (SHE) provided that the stochastic integral in (1.6) is interpreted in the classical Itô-Skorohod sense and that we choose equal to the Itô correction below. Then, the generalized Feynman-Kac formula ([K90, Theorem 6.2.5]) provides a solution to (1.6) with E x denoting expectation with respect to the law P x of a d-dimensional Brownian path W = (W s ) s≥0 starting at x ∈ R d , which is independent of the noise ξ. Plugging (1.3) in the previous formula, using Brownian scaling and time-reversal, we get the a.s. equality is the normalized partition function of the Brownian directed polymer is a white noise environment ξ, or equivalently, the total-mass of a Gaussian multiplicative chaos in the Wiener space ([MSZ16, Section 4]).
It follows that there exists β c ∈ (0, ∞) and a strictly positive non-degenerate random variable Z ∞ (x) so that, a.s. as T → ∞, (1.10) See [MSZ16], or [C17] for a general reference. Moreover, (Z T ) T ≥0 is uniformly integrable for β < β c , which we will always assume from now on. Now, let C α (R × R d ) denote the path-space of the white noise (see Appendix for a precise definition) and be any arbitrary representative of the random limit Z ∞ = Z ∞ (0); in particular u(ξ) = Z ∞ . Then, E[u] = 1, and throughout the sequel we will write (recall (1.5) and (1.8)) h = log u . (1.11) Since u is non constant with Eu = 1, we have Eh < 0. Finally, we also define another critical disorder parameter: We are now ready to state our main results.
Theorem 1.1. Assume d ≥ 3 and recall that h is defined in (1.11).
• (Flat initial condition.) Fix β ∈ (0, β c ) and consider the solution • (General initial condition.) Fix β ∈ (0, β L 2 ) and consider the solution h ε to (1.2) with h ε (0, ·) = h 0 (·) for some h 0 : R d → R which is continuous and bounded from above. Then, • (Droplet or narrow-wedge initial condition.) Fix β ∈ (0, β L 2 ) and consider the solution where ρ is the d-dimensional Gaussian kernel, and The deterministic terms in (1.12) and (1.13) are logarithms of solutions to heat equation without noise, and one can see that h → 0 in L 2 as β → 0. This implies that We obtain an immediate corollary to Theorem 1.1.
where h 0 is continuous and bounded from above. Then for any Our next main result is the following which provides a sub-Gaussian upper tail estimate on the limit h defined in (1.11).
In particular, h ∈ L p (P) for any p ∈ R.
From Theorems 1.1 and 1.3, we derive Corollary 1.4. In the hypothesis of Corollary 1.2, we have for any 1.3 Literature remarks and discussion.
In the present set up, by finding a non-trivial limit when letting the regularization parameter vanish we have obtained a non-trivial renormalization of KPZ equation (1.1). Let us stress the main specificity of Theorem 1.1. The approximating sequence (h(ξ (ε,t,x) ); ε > 0) in the case of flat initial condition combines two interesting properties: • it is constant in law for all (ε, t, x), with law given by the one of log Z ∞ ; • it approximates h ε (t, x) in probability.
(Similar properties hold for the other initial conditions). Since it depends on ε, it is not a (strong) limit, but it can be used similarly. In particular, fluctuations can be studied as shown in [CCM18]. This is quite different from using a deterministic centering, e.g.,h ε (t, x) = h ε (t, x)−Eh ε (t, x). As mentioned in [DGRZ18],h ε does not converge to 0 pointwise, but it does as a distribution. Integratingh ε in space against test functions cause oscillations to cancel. On the contrary, in our result h ε (t, x) − h(ξ (ε,t,x) ) → 0 pointwise, and we do not need any averaging in space.
We also emphasize that our results concern studying the asymptotic behavior of the solution to the non-linear equation (1.2), and are not restricted to the linear multiplicative noise stochastic heat equation (see (1.6)). Furthermore, the statements of the results concern the solution itself, without need of integrating spatially against test functions. However, note that the limit obtained in Theorem 1.1 does depend on the smoothing procedure φ as well as on the disorder parameter β and it is not universal (in particular, for β < β L 2 , the variance of exp(h) can be computed from the RHS of (1.14) for x = 0, and it depends on the mollification). Thus the present scenario lies in total contrast with the 1-dimensional spatial case where the limit can be defined by a chaos expansion [AKQ14,AKQ14b] (with the parameter β absorbed by scaling) or via the theory of regularity structures ( [H14]) which also produces a renormalized limit which does not depend on the mollification scheme.
In [CCM18] we have also investigated the rate of the convergence of h ε → h for small enough β and showed that ε law −→ N (0, σ 2 (β)) for each fixed x ∈ R d and t > 0. For larger β, the so-called KPZ regime is expected to take place with different limits, different scaling exponents and non-Gaussian limiting distributions. In particular, the variance in the above Gaussian distribution is given by which already diverges for β > β L 2 indicating that the amplitude of the fluctuations, or at least their distributional nature, changes at this point. However the KPZ regime is not expected before the critical value β c . Hence this region β ∈ (β L 2 , β c ) remains mysterious. Finally, we remark on the correlation structure of the limit u which were computed in [CCM18]. It was shown that, for β small enough, (1.14) with The above correlation structure also underlines that solution u ε (t, x) and u ε (t, y) becomes asymptotically independent so that the so that the spatial averages R d f (x) u ε (t, x) dx → f (x)u(t, x) dx become deterministic and u solves the unperturbed heat equation ∂ t u = 1 2 ∆u. As remarked earlier, the spatially averaged fluctuations We now consider the regularized KPZ equation (1.6) as before, but with different initial data and identify the limit of the solution up to leading order. For notational brevity, we will write We also remind the reader that u ε solves (1.6) and h ε = exp[u ε ] solves (1.2) with C ε = β 2 ε −2 2 V (0). Finally, recall that u ε (t, x) = Z t ε 2 ξ (ε,t) ; x ε with ξ (ε,t) given by (1.3) and (1.4).
Fix continuous functions u 0 : R d → (0, +∞) and h 0 : R d → R which are bounded from above, consider the solution of SHE or, equivalently by the relations u ε = exp h ε and u 0 = exp h 0 , the solution of KPZ As before, we have the Feynman-Kac representation with ξ (ε,t) as above.

Proof. Note that
where the right hand side is a triplet of three independent random variables, with Z (1) t and Z (2) t distributed as W t . Hence, expectation (2.5) vanishes as ε → 0, provided that u 0 is bounded and continuous, and because of uniform integrability which is implied by for β < β L 2 and δ > 0 small enough. The proof is concluded by the observation that We now end the Proof of (1.12): For β < β L 2 , for all t, x, as ε → 0, we first show (2.7) Note that (2.7) follows directly from Lemma 2.1 and (2.4). Then, since u > 0, taking logarithm we deduce the convergence in probability (1.12).
Remark 2.2. Recall that in [MSZ16] it was shown that, for any smooth function f with compact support, Note that unlike the latter statement, no smoothing in space is needed in the present context. In fact, we can recover the spatially averaged statement from above. Indeed, fast decorrelation in space of ξ (ε,t,x) as ε → 0, ergodicity and smoothness justify the equivalence below: 2.2 Narrow-wedge initial condition: Proof of (1.13).
Fix x 0 ∈ R d , and consider the solution of SHE or, equivalently by the relation u ε = exp h ε , the solution of KPZ (2.9) By Feynman-Kac formula, the solution of SHE now admits a Brownian bridge representation: where E t,y 0,x denotes expectation with respect to a Brownian bridge starting at x and conditioned to be found at y at time t, ρ is the d-dimensional Gaussian kernel and so that we again have ξ (ε,x 0 ) law = ξ. The following Lemma follows the approach for proving the local limit theorem as in [S95, V06].
Proof. We will write X = ε −1 x, T = ε −2 t and let m = m ε be a time parameter, such that m ε → ∞ and m ε = o(T ), as ε → 0. We use the notation: Step 1: We first want to approximate E T,X in L 2 -norm, so we compute the difference: To show that the right hand side which goes to 0 as ε → 0, it suffices to observe that, for all a > 0, To prove this, we use Hölder's inequality similarly to (2.6), and apply [CCM18, Lemma 3.5] (alternatively [V06, Corollary 3.8]) and transience of Brownian motion for d ≥ 3, which implies, since m → ∞, that Step 2: We wish to use Markov property and symmetry of the Brownian bridge to show that as ε → 0 for β < β L 2 , by Cauchy-Schwarz inequality and invariance in law of the white noise with a shift by X. Hence, we compute (2.14) After change of variable by setting Y = √ T y in the above integral and since ρ(T s, √ T z) = T −d/2 ρ(s, z), observe that by Jensen's inequality and dominated convergence, we can prove that sup if we can show that for all fixed y ∈ R d , To prove this, we use the density of the Brownian bridge, at truncated time horizon, with respect to the Brownian motion ([CCM18, Lemma 3.4]), to get that: After rescaling, the difference inside the parenthesis goes almost surely to 0, for y fixed and uniformly in |x| ≤ A; we conclude the proof of the lemma using Hölder's inequality.
We can now conclude the Proof of (1.13): Let h ε be the narrow-wedge height function solution of of (2.9). We need to show that for β < β L 2 , for all t, x, as ε → 0, with ξ (ε,x 0 ) in (2.12). We use the representation (2.11) and the property ξ (ε,x 0 ) law = ξ, so that we can exchange ξ with ξ (ε,x 0 ) , in convergence (2.13) taken with endpoint ε −1 (x − x 0 ). This leads to the above convergence in probability for the logarithm, proving (1.13).
We focus on showing that for β < β L 2 , log Z ∞ admits a sub-Gaussian lower tails estimate, that is, for some C ∈ (0, ∞) and any θ > 0, We invoke a second moment method combined with the Talagrand's concentration inequality as in [CH02] (see also [M14, Section 2.2]). As the Cauchy-Schwarz inequality, which is a central tool in this proof is not directly available in the continuous setting, we choose to introduce a discretization of the white noise to recover it.
Consider R n a tiling of [0, 2 n ]×[−2 n , 2 n ] d , composed of cubes of length 2 −n , such that every cube of R n can be divided in 2 d+1 cubes of R n+1 . We define a discrete version of Z T through: where, is in a cube R of R n , and 0 otherwise. We stress that φ (n) W is non-decreasing with n and converges almost surely to φ W . Using the Gaussian covariance structure, we have that For β < β L 2 and since φ (n) W ≤ φ W , we immediatly obtain from the monotone convergence theorem that the right-hand side goes to zero in the limit T → ∞ followed by n → ∞. By Doob's L 2 inequality applied to the martingale Z − Z (n) , this implies in particular that Since φ (n) W is set to be 0 outside of [0, 2 n ] × [−2 n , 2 n ] d , the set C n containing the centers of the cubes of R n is finite. Hence, as each cube has volume 2 −(d+1)n , we can write that where the ξ n (i, x) are independent centered Gaussian random variables of variance 1. Then, we define the polymer measure P β,ξn of renormalized partition function Z (n) ∞ : where we have set, using (3.3), Finally, we let E β,ξn denote expectation corresponding to P β,ξn . Now, we can compare the free energies of two realizations of the noise ξ n (i, x) and ξ ′ n (i, x): where d(·, ·) denotes the euclidean distance on R Card(Cn) , and where we used Jensen's and Cauchy-Schwarz inequalities for respectively the first and second lower bounds.
Remark 3.1. After finishing the writing of the present article, we learned that another proof of the negative moments of Z ∞ has been recently proposed in [HL18,DGRZ18] using a continuous approximation of the white noise. A proof of the corresponding result for the KPZ equation in dimension 2, which relies on the convexity of the free energy and the Malliavin derivative, can be found in [CSZ18].
A Appendix.
We will include some elementary facts regarding the regularity properties of space-time white noise ξ. For any z, z ′ ∈ R × R d , we will denote by · the parabolic distance given by where z = (t, x) and z ′ = (t ′ , x ′ ). Recall that the Hölder space of positive exponent α ∈ (0, 1) consists of all functions u : R × R d → R such that for any compact set K ⊂ R × R d , The corresponding Hölder (Besov) space of negative regularity is defined as follows. First for any k ∈ N, let B k denote the space of all smooth functions ϕ : R × R d → R which are supported on the unit ball in (R × R d , · ) such that Then for any fixed α < 0, we define the space C α to be the space of all tempered distributions η ∈ S ′ (R × R d ) such that for any compact set K ⊂ R × R d , where k = ⌈−α⌉ and (Θ λ z u)(s, y) = λ −(d+2) u λ −2 (t − s), λ −1 (y − x) z = (t, x).
A crucial estimate on the Besov norm · C α (K) is given by where K is also a compact set slightly larger than K and u is a single, well-chosen test function (which can be constructed by wavelets, see [H14]). Recall that if ξ is space-time white noise (i.e, E[ ξ, ϕ 1 ξ, ϕ 2 ] = ϕ 1 (t, x)ϕ 2 (t, x) dt dx), then with