Comparing the stochastic nonlinear wave and heat equations: a case study

We study the two-dimensional stochastic nonlinear wave equation (SNLW) and stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional derivative (of order $\alpha>0$) of a space-time white noise. In particular, we show that the well-posedness theory breaks at $\alpha = \frac 12$ for SNLW and at $\alpha = 1$ for SNLH. This provides a first example showing that SNLW behaves less favorably than SNLH. (i) As for SNLW, Deya (2017) essentially proved its local well-posedness for $0<\alpha<\frac 12$. We first revisit this argument and establish multilinear smoothing of order $\frac 14$ on the second order stochastic term in the spirit of a recent work by Gubinelli, Koch, and Oh (2018). This allows us to simplify the local well-posedness argument for some range of $\alpha$. On the other hand, when $\alpha \geq \frac 12$, we show that SNLW is ill-posed in the sense that the second order stochastic term is not a continuous function of time with values in spatial distributions. This shows that a standard method such as the Da Prato-Debussche trick or its variant, based on a higher order expansion, breaks down for $\alpha \ge \frac 12$. (ii) As for SNLH, we establish analogous results with a threshold given by $\alpha = 1$. We point out that the scaling analysis provides a critical value of $\alpha = 2$ for this problem but the existing well-posedness theory breaks down at $\alpha = 1$.

and the stochastic nonlinear heat equation (SNLH) on T 2 : where ∇ = √ 1 − ∆ and α > 0. Namely, both equations are endowed with a quadratic nonlinearity and forced by an α-derivative of a (Gaussian) space-time white noise on T 2 ×R + .
Over the last decade, we have seen a tremendous development in the study of singular stochastic PDEs, in particular in the parabolic setting [22,23,18,5,24,26,7,6,3,4]. Over the last few years, we have also witnessed a rapid progress in the theoretical understanding of nonlinear wave equations with singular stochastic forcing and/or rough random initial data [35,19,20,21,30,33,28,29,31,34,32]. While the regularity theory in the parabolic setting is well understood, the understanding of the solution theory in the hyperbolic/dispersive setting has been rather poor. This is due to the intricate nature of hyperbolic/dispersive problems, where case-by-case analysis is often necessary (for example, to show multilinear smoothing as in Proposition 1.3 below). Let us compare the hyperbolic and parabolic Φ 3 3 -models on the three-dimensional torus T 3 as an example. In the parabolic setting [15], the standard Da Prato-Debussche trick suffices for local well-posedness, while in the wave setting, the situation is much more complicated. In [20], Gubinelli, Koch, and the first author studied the hyperbolic Φ 3 3 -model by adapting the paracontrolled calculus [18] to the hyperbolic/dispersive setting. In particular, it was essential to exploit multilinear smoothing in the construction of stochastic objects and also to introduce paracontrolled operators. While this comparison on the hyperbolic and parabolic Φ 3 3 -model shows that it may require more effort to study SNLW than SNLH, the resulting outcomes (local well-posedness on T 3 with a quadratic nonlinearity forced by a space-time white noise) are essentially the same.
The main purpose of this paper is to investigate further the behavior of solutions to SNLW and SNLH and study the following question: Does the solution theory for SNLW match up with that for SNLH? For this purpose, we study these equations in a simpler setting of a quadratic nonlinearity on the two-dimensional torus T 2 but with noises more singular than a space-time white noise (i.e. α > 0). In this setting, we indeed provide a negative answer to the question above.
When α = 0, the equations (1.1) and (1.2) correspond to the so-called hyperbolic Φ 3 2model and parabolic Φ 3 2 -model, respectively, 1 whose local well-posedness can be obtained by the standard Da Prato-Debussche trick; see [10,19]. In this paper, we compare the behavior of solutions to these equations for more singular noises, i.e. α > 0. We now state a "meta"-theorem.
"Theorem" 1.1. (i) Let 0 < α < 1 2 . Then, the quadratic SNLW (1.1) is locally well-posed. When α ≥ 1 2 , the quadratic SNLW (1.1) is ill-posed in the sense the standard solution theory such as the Da Prato-Debussche trick or its variant based on a higher order expansion does not work.
With this notation, the stochastic convolution and the second order stochastic term can be expressed as = I( ∇ α ξ) and = I( ), (1.4) where denotes a renormalized version of 2 . See (3.2) and (5.1) for precise definitions of the stochastic convolutions. In particular, we impose (0) = 0 in the wave case and (−∞) = 0 in the heat case. We then solve the fixed point problem for the residual term v = u − + . See (1.11) and (1.17).
On the other hand, for α ≥ α * we show that the second order term does not belong to C([0, T ]; D (T 2 )) for any T > 0, almost surely (see Propositions 1.5 and 1.7 below). This implies 4 that a solution u would not belong to C([0, T ]; D (T 2 )) if we were to solve the equation via the second (or higher) order expansion (1.3) or the first order expansion (= the Da Prato-Debussche trick): since the second order term appears in case-by-case analysis of the nonlinear contribution for the residual term v = u − . In Subsection 1.2, we go over details for SNLW (1.1). In Subsection 1.3, we discuss the case of SNLH (1.2).
1.2. Stochastic nonlinear wave equation. Stochastic nonlinear wave equations have been studied extensively in various settings; see [11,Chapter 13] for the references therein. In [19], Gubinelli, Koch, and the first author considered SNLW on T 2 with an additive space-time white noise: where k ≥ 2 is an integer. The main difficulty of this problem comes from the roughness of the space-time white noise. In particular, the stochastic convolution , solving the linear stochastic wave equation: is not a classical function but is merely a distribution for the spatial dimension d ≥ 2. This raises an issue in making sense of powers k and a fortiori of the full nonlinearity u k in (1.6). In [19], by introducing an appropriate time-dependent renormalization, the authors proved local well-posedness of (a renormalized version of) (1.6) on T 2 . See [20,21,33,28,31,34,32] for further work on SNLW with singular stochastic forcing. We also mention the work [12,13] by Deya on SNLW with more singular (both in space and time) noises on bounded domains in R d and the work [37] on global well-posedness of the cubic SNLW on R 2 .
We first state a local well-posedness result of the quadratic SNLW (1.1) on T 2 . Given N ∈ N, we define the (spatial) frequency projector π N by where u(n) denotes the Fourier coefficient of u and e n (x) = 1 2π e in·x as in (2.1). We also set Theorem 1.2. Let 0 < α < 1 2 and s > α. Then, the quadratic SNLW (1.1) on T 2 is locally well-posed in H s (T 2 ). More precisely, there exists a sequence of time-dependent constants {σ N (t)} N ∈N tending to ∞ (see (3.5) below ) such that, given any (u 0 , u 1 ) ∈ H s (T 2 ), there exists an almost surely positive stopping time T = T (ω) such that the solution u N to the following renormalized SNLW with a regularized noise: converges almost surely to some limiting process u ∈ C([0, T ]; H −α−ε (T 2 )) for any ε > 0.
In [13], Deya proved Theorem 1.2 on bounded domains on R 2 but the same proof essentially applies on T 2 . 5 For 0 < α < 1 3 , the standard Da Prato-Debussche argument suffices to prove Theorem 1.2. Indeed, with the first order expansion (1.5), the residual term v = u − satisfies At the second equality, we performed the Wick renormalization: 2 . It is easy to see that and have regularities 6 −α− and −2α−, respectively (see Lemma 3.1 below). Then, thanks to one degree of smoothing from the wave Duhamel integral operator, we expect that v has regularity 1 − 2α−. The restriction α < 1 3 appears from (1 − 2α−) + (−α−) > 0 in making sense of the product v in (1.10). 7 Then, by viewing as a given enhanced data set, 8 one can easily prove local well-posedness of (1.10).
For 1 3 ≤ α < 1 2 , the argument in [13] is based on the second order expansion (1.3). In this case, the residual term v = u − + satisfies If we proceed with a "parabolic thinking", 9 then we expect that has regularity where we gain one derivative from the wave Duhamel integral operator; see (3.6). With this parabolic thinking, we see that the last product in (1.11) makes sense (in a deterministic manner) only for α < 1 3 so that (1 − 2α−) + (−α−) > 0. Nonetheless, for 1 3 < α < 1 2 , one can use stochastic analysis to give a meaning to := · as a random distribution of regularity −α− (inheriting the bad regularity of ). Using the equation (1.11), we expect that v has regularity 1 − α− and, with this regularity of v, all the terms on the right-hand side of (1.11) make sense. Then, by viewing as a given enhanced data set, a standard contraction argument with the energy estimate (Lemma 2.4) yields local well-posedness of (1.11). 5 One may invoke the finite speed of propagation and directly apply the result in [13] to T 2 . We also point out that the paper [13] handles noises with rougher temporal regularity than the space-time white noise and Theorem 1.2 is a subcase of the main result in [13]. 6 In the following, we restrict our attention to spatial regularities. Moreover, we use a− (and a+) to denote a − ε (and a + ε, respectively) for arbitrarily small ε > 0. If this notation appears in an estimate, then an implicit constant is allowed to depend on ε > 0 (and it usually diverges as ε → 0). 7 Recall that a product of two functions is defined in general if the sum of the regularities is positive. 8 Namely, once we have the pathwise regularity property of the stochastic terms and , we can build a continuous solution map: (u0, u1, , ) → v in the deterministic manner. 9 Namely, if we only count the regularity of each of in and put them together with one degree of smoothing from the wave Duhamel integral operator without taking into account the product structure and the oscillatory nature of the linear wave propagator.
In view of "Theorem" 1.1, the restriction α < 1 2 in Theorem 1.2 is sharp. See Proposition 1.5 below. There is, however, one point that we would like to investigate in this well-posedness part. In the discussion above, we simply used a "parabolic thinking" to conclude that has regularity (at least) 1 − 2α−. In fact, by exploiting the explicit product structure and multilinear dispersion, we show that there is an extra smoothing for .
Given N ∈ N, let N to denote the second order term, emanating from the truncated noise π N ∇ α ξ. See (3.7) for a precise definition. We then have the following proposition. (1.13) Then, for any T > 0, N converges to in C([0, T ]; W s,∞ (T 2 )) almost surely. In particular, for any ε > 0, almost surely.
See also Proposition 3.2 below for another instance of multilinear smoothing. In [20], such an extra smoothing property on stochastic terms via multilinear dispersion effect played an essential role in the study of the quadratic SNLW on the three-dimensional torus T 3 . We believe that the multilinear smoothing in Proposition 1.3 is itself of interest since such a multilinear smoothing in the stochastic context for the wave equation is not well understood. See also Remark 1.4 below.
In our current setting, this extra smoothing does not improve the range of α in Theorem 1.2 since, as we will show below, the range α < 1 2 is sharp. Proposition 1.3, however, allows us to simplify the local well-posedness argument for the range 1 3 ≤ α < 5 12 . While the discussion above showed the Da Prato-Debussche argument to study (1.10) breaks down at α = 1 3 , the extra smoothing in Proposition 1.3 allows us to study (1.10) at the level of the Duhamel formulation: where S(t) denotes the linear wave propagator defined in (2.6). Thanks to Proposition 1.3, we expect that v has regularity 5 4 − 2α−, thus allowing us to make sense of the product v as long as 1 3 ≤ α < 5 12 , i.e. ( 5 4 − 2α−) + (−α−) > 0. In this refined Da Prato-Debussche argument, the relevant enhanced data set is given by (1.15) See Theorem 3.3 for a precise statement. Alternatively, we may work with the second order expansion (1.3) and study the equation (1.11). In this case, Proposition 1.3 allows us to make sense of the product in the deterministic manner for α < 5 12 . This in particular shows that for the range 1 3 ≤ α < 5 12 , we can solve (1.11) for v = u − + with a smaller enhanced data set in (1.15). Namely, when α < 5 12 , there is no need to a priori prescribe the last term in (1.12). See Theorem 3.4 (i) for a precise statement.
For the range of α under consideration, i.e. α ≥ 1 3 , the extra gain of regularity in Proposition 1.3 is 1 4 , regardless of the value of α. When 5 12 ≤ α < 1 2 , this extra smoothing is unfortunately not sufficient to make sense of the product in the deterministic manner. Recalling the paraproduct decomposition (see (2.3) below), we see that the resonant product = := = is the only issue here. Thus, for 5 12 ≤ α < 1 2 , we solve (1.11) with an enhanced data set: where we use stochastic analysis to give a meaning to the problematic resonant product = ; see Proposition 3.2. . This 1 4 -difference in two-and three-dimensions seems to come from the effect of Lorentz transformations along null directions. The same situation appears in bilinear estimates for solutions to the linear wave equation; see, for example, Subsection 3.6 in [9]. See also Remark 4.1 for a further discussion, where (i) we show that our computation on T 2 is essentially sharp and (ii) we compute the maximum possible gain of regularity on T d , d ≥ 3. Lastly, we point out that Proposition 1.3 states that the extra smoothing vanishes as α → 0.
Next, let us consider the situation for α ≥ 1 2 . In [13, Proposition 1.4], Deya showed that E (t) 2 H s diverges for any s ∈ R, when α ≥ 1 2 . This can be used to show that the Wick power is not a distribution-valued function of time when α ≥ 1 2 . The following proposition shows that the same result holds for . We point that Proposition 1.5 is by no means to be expected from the bad behavior of for α ≥ 1 2 . For example, in the parabolic Φ 4 3 -model, it is well known that the cubic Wick power does not make sense as a distribution-valued function of time but that = (∂ t − ∆) −1 belongs to C(R; C 1 2 − (T 3 )); see [15,27]. Furthermore, in Proposition 1.7 below, we prove that, for the quadratic SNLH (1.2), (i) the Wick power is not a distribution-valued function for α ≥ 1 2 but (ii) in the heat case makes sense as a distribution-valued function for α < 1. Therefore, we find it rather intriguing that for the wave equation, both and have the same threshold α = 1 2 . In the proof of Proposition 1.5, we show that each Fourier coefficient (n, t) diverges almost surely for α ≥ 1 2 . This divergence comes from the high-to-low energy transfer. Such high-to-low energy transfer was exploited in proving ill-posedness of the deterministic nonlinear wave equations in negative Sobolev spaces; see [8,29,17]. (ii) It is interesting to note that we can prove local well-posedness of SNLW (1.1) for the entire range 0 < α < 1 2 without using the paracontrolled approach as in the three-dimensional case [20].

Stochastic nonlinear heat equation.
In this subsection, we go over the corresponding results for the quadratic SNLH (1.2) on T 2 . With I = ∂ t + (1 − ∆) −1 , let and be as in (1.4) and be the Wick renormalization of 2 . We first state the crucial regularity result for the stochastic terms.
In short, Proposition 1.7 states that is a distribution-valued function if and only if α < 1 2 , while is a distribution-valued function if and only if α < 1. Hence, for the range 1 2 ≤ α < 1, while (t) does not make sense as a spatial distribution, = (∂ t + (1 − ∆)) makes sense as a space-time distribution. As mentioned above, such a phenomenon is already known for the parabolic Φ 4 3 -model; see [15,27]. Proposition 1.7 exhibits sharp contrast with the situation for SNLW discussed earlier (Proposition 1.5 above), where the threshold α = 1 2 applies to both and .
We now state a sharp local well-posedness result for the quadratic SNLH (1.2).
such that, given any u 0 ∈ C s (T 2 ), there exists an almost surely positive stopping time T = T (ω) such that the solution u N to the following renormalized SNLH: converges almost surely to some limiting process u ∈ C([0, T ]; C −α−ε (T 2 )) for any ε > 0.
In [10], Da Prato and Debussche proved Theorem 1.8 for α = 0. The same proof based on the Da Prato-Debussche trick also applies for 0 < α < 2 3 . In this case, with the first order expansion (1.5), the residual term v = u − satisfies where and have regularities −α− and −2α−, respectively. Then, by repeating the analysis in the previous subsection with two degrees of smoothing coming from the heat Duhamel integral operator, v has expected regularity 2 − 2α− and thus the restriction α < 2 3 appears from (2 − 2α−) + (−α−) > 0 in making sense of the product v in (1.16). Then, local well-posedness of (1.16) easily follows with an enhanced data set (u 0 , , ). For 2 3 ≤ α < 1, the proof of Theorem 1.8 is based on the second order expansion (1.3) and proceeds exactly as in the wave case (but without any multilinear smoothing). 10 In this case, the residual term v = u − + satisfies (1.17) 10 Since there is no multilinear smoothing for the heat equation, "parabolic thinking" provides a correct insight.
When α ≥ 2 3 , we can not make sense of the last product in the deterministic manner. Using stochastic analysis, we can give a meaning to as a distribution of regularity −α− for 2 3 ≤ α < 1. See Lemma 5.2. In this case, v has expected regularity of 2 − α− and thus the restriction α < 1 also appears in making sense of the product v , namely from (2 − α−) + (−α−) > 0. Then, by applying the standard Schauder estimate, we can easily prove local well-posedness of (1.17) with an enhanced data set: Remark 1.9. Let us compare the situations for SNLW (1.1) and SNLH (1.2). In this discussion, we disregard initial data. For the quadratic SNLH (1.2), the required enhanced data set consists of and the Wick power when 0 ≤ α < 2 3 . Namely, it involves only (the powers of) the first order process . When 2 3 ≤ α < 1, it also involves the second order and the third order processes and = . It is interesting to note that for the quadratic SNLW (1.1), thanks to the multilinear smoothing effect (Proposition 1.3), there is now an intermediate regime 1 3 ≤ α < 5 12 , where the required enhanced data set in (1.15) involves only the first and second order processes (but not the third order process). Furthermore, in this range, while the usual Da Prato-Debussche argument with (1.10) fails, the refined Da Prato-Debussche argument (1.14) at the level of the Duhamel formulation works thanks to the multilinear smoothing in Proposition 1.3.
As in [26], we now apply a scaling argument to find a critical value of α. By applying the following parabolic scaling (and the associated white noise scaling for ξ): for λ > 0, we obtain ∂ t u − ∆ u + λ 2−α u 2 = |∇| α ξ. Then, by taking λ → 0, the nonlinearity formally vanishes when α < 2. This provides the critical value of α = 2, (which agrees with the notion of local subcriticality introduced in [23]). It is very intriguing that the well-posedness theory for the quadratic SNLH (1.2) breaks down at α = 1 before reaching the critical value α = 2. For dispersive equations including the quadratic SNLW, the scaling analysis as above does not seem to provide any useful insight, 11 unless appropriate integrability conditions are incorporated. See, for example, [16] for a discussion in the case of the stochastic nonlinear Schrödinger equation. This paper is organized as follows. In Section 2, we introduce some notations and recall useful lemmas. In Section 3, assuming the regularity properties of the stochastic objects, we prove local well-posedness of SNLW (1.1) (Theorem 1.2). We then present details of the construction of the stochastic objects in Section 4. In particular, we prove the multilinear smoothing for (Proposition 1.3) and divergence of (Proposition 1.5). Finally, in Section 5, we present proofs of Proposition 1.7 and Theorem 1.8. 11 For example, applying the hyperbolic scaling (x, t) → (λx, λt), the scaling invariant version of SNLW (1.1) yields a critical value of α = 5 2 , even higher than the heat case but the well-posedness theory for SNLW breaks down at α = 1 2 .

Basic lemmas
In this section, we introduce some notations and go over basic lemmas.
for the orthonormal Fourier basis in L 2 (T 2 ). Given s ∈ R, we define the Sobolev space H s (T 2 ) by the norm: (1.8). Similarly, given s ∈ R and p ≥ 1, we define the L p -based Sobolev space (Bessel potential space) W s,p (T 2 ) by the norm: When p = 2, we have H s (T 2 ) = W s,2 (T 2 ). When we work with space-time function spaces, we use short-hand notations such as We also use a subscript to denote dependence on an external parameter; for example, A α B means A ≤ C(α)B, where the constant C(α) > 0 depends on a parameter α.
Note that H s (T 2 ) = B s 2,2 (T 2 ). We also define the Hölder-Besov space by setting Next, we recall the following paraproduct decomposition due to Bony [2]. See [1,18] for further details. Given two functions f and g on T 2 of regularities s 1 and s 2 , respectively, we write the product f g as 3) The first term f < g (and the third term f > g) is called the paraproduct of g by f (the paraproduct of f by g, respectively) and it is always well defined as a distribution of regularity min(s 2 , s 1 + s 2 ). On the other hand, the resonant product f = g is well defined in general only if s 1 + s 2 > 0.
We have the following product estimates. See [1,25] for details of the proofs in the non-periodic case (which can be easily extended to the current periodic setting).
2.3. Product estimates and discrete convolutions. Next, we recall the following product estimates. See [19] for the proof.
(ii) Suppose that 1 < p, q, r < ∞ satisfy the scaling condition: Note that while Lemma 2.2 (ii) was shown only for 1 p + 1 q = 1 r + α d in [19], the general case 1 p + 1 q ≤ 1 r + α d follows from a straightforward modification of the proof. We also recall the following basic lemma on a discrete convolution.
Then, we have (ii) Let d ≥ 1 and α, β ∈ R satisfy α + β > d. Then, we have By writing (2.4) in the Duhamel formulation, we have where the linear wave propagator S(t) is defined by and the wave Duhamel integral operator I is defined by Then, the following energy estimate follows from (2.5), (2.7), and the unitarity of the linear wave propagator S(t) in H s (T d ).
In [19,31], the authors used the Strichartz estimates to study local well-posedness of the stochastic nonlinear wave equations. Note, however, that the Strichartz estimates are not needed for proving local well-posedness of the quadratic nonlinear wave equation (NLW) in two dimensions. More precisely, the energy estimate (Lemma 2.4), Sobolev's inequality, and a standard contraction argument yield local well-posedness of the quadratic NLW in H s (T 2 ) for s > 0.
Next, we recall the Schauder estimate for the heat equation. Let P (t) = e −t(1−∆) denote the linear heat propagator defined as a Fourier multiplier operator: for t ≥ 0. Then, we have the following Schauder estimate on T d .
for any t > 0.
The bound (2.9) on T d follows from the decay estimate for the heat kernel on R d (see Lemma 2.4 in [1]) and the Poisson summation formula to pass such a decay estimate to T d .

2.5.
Tools from stochastic analysis. Lastly, we recall useful lemmas from stochastic analysis. Let {g n } n∈N be a sequence of independent standard Gaussian random variables defined on a probability space (Ω, F, P ), where F is the σ-algebra generated by this sequence. Given k ∈ N 0 , we define the homogeneous Wiener chaoses H k to be the closure (under L 2 (Ω)) of the span of Fourier-Hermite polynomials ∞ n=1 H kn (g n ), where H j is the Hermite polynomial of degree j and k = ∞ n=1 k n . We also set We say that a stochastic process X : The following lemma will be used in studying regularities of stochastic objects. For the proof, see Proposition 3.6 in [27] and Appendix in [29]. In the following, we state the result in terms of the Sobolev space W s,∞ (T d ) but the same result holds for the Hölder-Besov space C s (T d ).
Lemma 2.6. Let {X N } N ∈N and X be spatially homogeneous stochastic processes : R + → D (T d ). Suppose that there exists k ∈ N such that X N (t) and X(t) belong to H ≤k for each t ∈ R + .
(i) Let t ∈ R + . If there exists s 0 ∈ R such that E | X(n, t)| 2 n −d−2s 0 (2.10) for any n ∈ Z d , then we have X(t) ∈ W s,∞ (T d ), s < s 0 , almost surely. Furthermore, if there exists γ > 0 such that for any n ∈ Z d and N ≥ 1, then X N (t) converges to X(t) in W s,∞ (T d ), s < s 0 , almost surely.

Stochastic nonlinear wave equation with rough noise
In this section, we consider SNLW (1.1). We first state the regularity properties of the relevant stochastic terms and reformulate the problem in terms of the residual term v = u − or v = u − + . We then present a proof of Theorem 1.2. The analysis of the stochastic terms will be presented in Section 4.
3.1. Reformulation of SNLW. Let W denote a cylindrical Wiener process on L 2 (T 2 ): where {β n } n∈Z 2 is a family of mutually independent complex-valued Brownian motions on a fixed probability space (Ω, F, P ) conditioned so that 13 β −n = β n , n ∈ Z 2 . By convention, we normalize β n such that Var(β n (t)) = t. Then, the stochastic convolution = I( ∇ α ξ) in the wave case can be written as We define the truncated stochastic convolution N by where π N denotes the frequency projector defined in (1.7). We then define the Wick power N by 4) where σ N is given by for α > 0. We have the following regularity and convergence properties of N and N whose proofs are presented in Section 4.  13 In particular, we take β0 to be real-valued.
Next, we define the second order stochastic term by Then, Proposition 1.3 shows that is a well-defined distribution and is a limit of the truncated version: provided that 0 < α < 1 2 . As mentioned in Section 1, we need to use stochastic analysis to give a meaning to the resonant product = when 5 12 ≤ α < 1 2 . Proposition 3.2. Let 0 < α < 1 2 and s < s α − α, where s α is as in (1.13). Then, given for any ε > 0, almost surely.
(i) For 0 < α < 5 12 , the solution v depends continuously on the enhanced data set: almost surely belonging to the class X s,ε T defined (3.8). (ii) For 5 12 ≤ α < 1 2 , the solution v depends continuously on the enhanced data set: almost surely belonging to the class: In Subsection 3.3, we present a proof of Theorem 3.4. In view of the pathwise regularities of the relevant stochastic terms, we simply build a continuous map, sending the enhanced data set Ξ to a solution v in the deterministic manner. We point out, however, that, in Theorem 3.4 (i), the extra smoothing on plays an essential role in making sense of the product in the deterministic manner in the range 0 < α < 5 12 . We conclude this subsection by presenting a proof of Theorem 1.2.
for some almost surely finite constant C ω > 0, provided that α < σ < 1 − α. From Proposition 1.3, we also have Similarly, we have x . The rest follows as in the previous subsection. This completes the proof of Theorem 3.4.

On the construction of the relevant stochastic objects
In this section, we go over the construction of the stochastic terms for SNLW (1.1). As in [20], our strategy is to estimate the second moment of the Fourier coefficient and apply Lemma 2.6. In Subsection 4.1, we briefly discuss the regularity and convergence properties of and (Lemma 3.1). By exploiting multilinear dispersive smoothing for , we then present a proof of Proposition 1.3 in Subsection 4.2. In Subsection 4.3, we establish analogous multilinear smoothing for = (Proposition 3.2). Lastly, in Subsection 4.4, we show that, when α ≥ 1 2 , the second order stochastic term (t) is not a spatial distribution almost surely for any t > 0 (Proposition 1.5). Let = I( ∇ α ξ) be the stochastic convolution defined in (3.2). Given n ∈ Z 2 and 0 ≤ t 2 ≤ t 1 , we define σ n (t 1 , t 2 ) by  Moreover, from Wick's theorem (Lemma 2.7), we have E | (n 1 , t 1 )| 2 − σ n 1 (t 1 , t 1 ) | (n 2 , t 2 )| 2 − σ n 2 (t 2 , t 2 ) = 1 n 1 =±n 2 · σ 2 n 1 (t 1 , t 2 ). (4.3) In the following, we fix T > 0.
for any n ∈ Z 2 and 0 ≤ t ≤ T . Also, by the mean value theorem and an interpolation argument as in [20], we have for any θ ∈ [0, 1], n ∈ Z 2 , and 0 ≤ t 2 ≤ t 1 ≤ T with t 1 − t 2 ≤ 1. Hence, from Lemma 2.6, we conclude that ∈ C([0, T ]; W −α−ε,∞ (T 2 )) for any ε > 0 almost surely. Moreover, a slight modification of the argument yields convergence of N to . Since the required modification is exactly the same as in [20], we omit the details here.
In the remaining part of this section, we only establish the estimate (2.10) in Lemma 2.6 for each of , , and = . The time difference estimate (2.11) and the estimates (2.12) and (2.13) follow from a straightforward modification as in [20].

4.2.
Proof of Proposition 1.3. Let 0 < α < 1 2 and let s α be as in (1.13). In view of Lemma 2.6, it suffices to show for any n ∈ Z 2 and 0 ≤ t ≤ T . Our argument follows closely to that in the proof of Proposition 1.6 in [20] up to Case 2 below, where our argument diverges. We, however, present details for readers' convenience. See also Remark 4.1 below. By the definition (3.6), we have Let us first consider the case n = 0. It follows from (4.11) and (4.6) that By symmetry, (4.3), and (4.1), we obtain provided that α < 1 2 . This proves (4.10) when n = 0.
We also point out that the calculation above can easily be extended to the higher dimensional case. More precisely, the right-hand side of (4.27) is unchanged on T d since we did not perform any summation. By setting and repeating the same computation on T d , the power on the right-hand side of (4.29) becomes − 11 2 + d 2 + 4α. By writing this computation indicates that the regularity of on T d is at best When d = 3 and α = 0, this agrees with the 1 2 -smoothing shown in [20].
Note that for α < 1 2 , the sums over n 2 and n 2 in (4.34) are absolutely convergent. This makes our analysis simpler than the proof of Proposition 1.8 in [20], where the corresponding sums in n 2 and n 2 were not absolutely convergent and hence, it was crucially to exploit the oscillatory nature of the problem and also apply some symmetrization argument.

4.4.
Divergence of the stochastic terms. In this subsection, we present the proof of Proposition 1.5. By (3.6) and (4.2), for n ∈ Z 2 and t > 0, we can write (n, t) = 1 2π where denotes the lexicographic ordering of Z 2 and X k (n, t) : Note that X k (n, t)'s are independent. We show that the sum in (4.41) diverges almost surely. We only consider the case |k| ∼ |n − k| |n|. Otherwise, we have either |k| ∼ |n| |n − k| or |n − k| ∼ |n| |k|. In either case, for fixed n ∈ Z 2 , the sum in k is a finite sum and hence is almost surely convergent. This allows us to focus on the case |k| ∼ |n − k| |n|. In particular, we assume k = n 2 . As in (4.12), we have From (4.1) and |k| ∼ |n − k|, we have This shows that Proposition 1.5 is a consequence of E | (n, t)| 2 = ∞.
In particular, the lower bound (4.47) is also valid on T d . From this observation, we conclude that / ∈ C([0, T ]; D (T d )) almost surely if α ≥ 1 − d 4 .

Stochastic nonlinear heat equation with rough noise
In this section, we consider SNLH (1.2). In Subsection 5.1, we first state the regularity properties of the relevant stochastic terms and present a proof of Theorem 1.8 by reformulating the problem in terms of the residual term v = u − + . We then proceed with the construction of the stochastic terms in the remaining part of this section. This includes the divergence of (and , respectively) for α ≥ 1 2 (and α ≥ 1, respectively) stated in Proposition 1.7.
5.1. Reformulation of SNLH. Let α > 0. We define the stochastic convolution = I( ∇ α ξ) by for t ≥ 0, where P (t), e n , and W (t) are as in (2.8), (2.1), and (3.1), respectively. We then define the truncated stochastic convolution N and the Wick power N by where π N is as in (1.7) and κ N is defined by Then, by proceeding as in the proof of Lemma 3.1 (i), we have the following regularity and convergence property of N . Since the argument is standard, we omit details.
We now define the second order stochastic term: Then, a slight modification of the proof of Proposition 1.7 (ii) presented below shows that N converges to in C([0, T ]; C 2−2α− (T 2 )) almost surely, provided that 0 < α < 1. From the regularities 2 − 2α− and −α− of and , there is an issue in making sense of the resonant product = in the deterministic manner when α ≥ 2 3 . For the range 2 3 ≤ α < 1, we instead use stochastic analysis to give a meaning to the resonant product = = = .
Theorem 5.3. Let 0 < α < 1 and s > −α − ε for sufficiently small ε > 0. Then, the Cauchy problem (5.4) is locally well-posed in C s (T 2 ). More precisely, given any u 0 ∈ C s (T 2 ), there exist an almost surely positive stopping time T = T (ω) and a unique solution v to (5.4) in the class: where −s < σ < s + 2. Furthermore, the solution v depends continuously on the enhanced data set: almost surely belonging to the class: Once we prove Theorem 5.3, Theorem 1.8 follows from the same lines as in the proof of Theorem 1.2 and thus we omit details.
Taking a supremum of the left-hand side of (5.9) over 0 < t ≤ T , it follows from (5.7) and (5.9) that By a similar computation, we also obtain a difference estimate: Therefore, we conclude from (5.10) and (5.11) that a standard contraction argument yields local well-posedness of (5.4). Moreover, an analogous computation shows that the solution v ∈ X(T ) depends continuously on the enhanced data set Σ = u 0 , , , = .