Aurélien Deya

A non-linear wave equation with fractional perturbation

We study a d-dimensional wave equation model (d ∈ {2, 3}) with quadratic non-linearity and stochastic forcing given by a space-time fractional noise. Two different regimes are exhibited, depending on the Hurst parameter H = (H0, . . . , H d ) ∈ (0, 1) d+1 of the noise: if

2 , then the equation can be treated directly, while in the case d -3 4 < d i=0 Hi ≤ d -1 2 , the model must be interpreted in the Wick sense, through a renormalization procedure.

 for the two-dimensional white-noise situation, and more generally follow a series of investigations related to stochastic wave models with polynomial perturbation.

Introduction and main results

In this paper, we propose to study the following non-linear stochastic wave equation:

∂ 2 t u -∆u + ρ 2 u 2 = Ḃ , t ∈ [0, T ] , x ∈ R d , u(0, .) = φ0 , ∂tu(0, .) = φ1 , ( 1
)
where φ0, φ1 are (deterministic) initial conditions in an appropriate Sobolev space, ρ : R d → R is a smooth (deterministic) function with support included in a bounded domain D ⊂ R d , d ∈ {2, 3}, and Ḃ ∂t∂x 1 • • • ∂x d B for some space-time fractional Brownian motion B = B H of Hurst index H = (H0, H1, . . . , H d ) ∈ (0, 1) d+1 . For the sake of clarity, let us here recall the specific definition of this process:

Definition 1.1. Fix a dimension parameter d ≥ 1, as well as a complete filtered probability space (Ω, F, P).

For any H = (H0, H1, . . . , H d ) ∈ (0, 1) d+1 , a centered Gaussian process

B : Ω × ([0, T ] × R d ) → R
2H i + |y| 2H i -|x -y| 2H i ) .
In particular, a space-time fractional Brownian motion of Hurst index

H = ( 1 2 , . . . , 1 2 )
is a Wiener process (and in this case the derivative Ḃ is a space-time white noise).

Since the pioneering works of Mandelbrot and Van Ness, fractional noises have been considered as very natural stochastic perturbation models, that offer more flexibility than classical white-noise-driven equations. The involvement of fractional inputs first occured in the setting of standard differential equations and, even in this simple context, the procedure is known to raise numerous difficulties due to the non-martingale nature of the process. Sophisticated alternatives to Itô theory must then come into the picture, whether fractional calculus, Malliavin calculus or rough paths theory, to mention just the most standard methods.

More recently, fractional (multiparameter) noises have also appeared within SPDE models. A first widelyused example is given by white-in-time colored-in-space Gaussian noises, that can be treated in the classical framework of Walsh's martingale-measure theory [START_REF] Walsh | An introduction to stochastic partial differential equations[END_REF], or with Da Prato-Zabczyk's infinite-dimensional approach to stochastic calculus [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF]. Such noise models have thus been applied to a large class of PDE dynamics, and the properties of the solutions to the resulting SPDEs are often well understood (see [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF] and the numerous references therein).

SPDEs involving a fractional-in-time noise are much more delicate to handle (Walsh and Da Prato-Zabczyk theories no longer apply in this case), and the related literature is in fact very scarce:

• In the parabolic setting, one can first mention [START_REF] Tindel | Stochastic evolution equations with fractional Brownian motion[END_REF] for the study of a homogeneous equation with additive fractional Brownian motion, and the series of papers [START_REF] Hu | Heat equation with fractional white noise potentials[END_REF][START_REF] Hu | Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2[END_REF][START_REF] Hu | Feynman-Kac formula for heat equation driven by fractional white noise[END_REF] for the analysis of a linear multiplicative perturbation of the heat equation. Pathwise approaches to the parabolic fractional problem have also been considered in [START_REF] Deya | Non-linear rough heat equations[END_REF][START_REF] Gubinelli | Rough evolution equations[END_REF] using rough-paths ideas, and in [START_REF] Deya | On a modelled rough heat equation[END_REF][START_REF] Deya | Construction and Skorohod representation of a fractional K-rough path[END_REF] with the formalism of Hairer's theory of regularity structures.

• For the wave equation, and to the best of our knowledge, the results are so far limited to the analysis of the specific one-dimensional (d = 1) situation [START_REF] Balan | SPDEs with affine multiplicative fractional noise in space with index 1/4 < H < 1/2[END_REF][START_REF] Caithamer | The stochastic wave equation driven by fractional Brownian noise and temporally correlated smooth noise[END_REF][START_REF] Erraoui | Hyperbolic Stochastic Partial Differential Equations with Additive Fractional Brownian Sheet[END_REF][START_REF] Quer-Sardanyons | The 1-d stochastic wave equation driven by a fractional Brownian motion[END_REF], and to the study of affine models when d ≥ 2: the homogeneous equation with additive fractional noise in [START_REF] Balan | The stochastic wave equation with fractional noise: a random field approach[END_REF] and multiplicative linear noise in [START_REF] Balan | The Stochastic Wave Equation with Multiplicative Fractional Noise: A Malliavin Calculus Approach[END_REF] (when the time-fractional order satisfies H0 > 1/2 and the space covariance structure is given by a Riesz kernel of order α > d -2).

In brief, SPDEs, and especially stochastic hyperbolic equations, driven by a space-time fractional noise remain a widely-open field at this point. Note in particular that the wave-equation case cannot be treated within the recently-introduced framework of regularity structures ( [START_REF] Hairer | A theory of regularity structures[END_REF]), due to the lack of regularization properties for the wave kernel with respect to space-time Sobolev topologies.

With this general background in mind, let us now go back to the consideration of equation [START_REF] Balan | The Stochastic Wave Equation with Multiplicative Fractional Noise: A Malliavin Calculus Approach[END_REF]. Our approach to the model will directly follow a series of investigations [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF][START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local existence theory[END_REF][START_REF] Gubinelli | Renormalization of the two-dimensional stochastic non linear wave equations[END_REF][START_REF] Oh | Invariant Gibbs measure for the 2-d defocusing nonlinear wave equations[END_REF][START_REF] Thomann | Random data Cauchy problem for supercritical Schrödinger equations[END_REF] devoted to the study of stochastic wave (or Schrödinger) equations involving a polynomial drift term. Our study can more specifically be seen as a fractional extension of the results of [START_REF] Gubinelli | Renormalization of the two-dimensional stochastic non linear wave equations[END_REF] for the white-noise situation. In the last five references, and in our study as well, the strategy to handle the equation relies on a central ingredient that is often referred to as the Da Prato-Debussche's trick. Roughly speaking, it consists in regarding the solution u of (1) as some "perturbation" of the solution Ψ to the associated "free" equation

∂ 2 t Ψ -∆Ψ = Ḃ , t ∈ [0, T ] , x ∈ R d , Ψ(0, .) = 0 , ∂tΨ(0, .) = 0 . ( 2
)
In fact, staying at a heuristic level, observe that the difference process v u -Ψ satisfies (morally) the equation

∂ 2 t v -∆v + ρ 2 (v 2 + 2v • Ψ + Ψ 2 ) = 0 , t ∈ [0, T ] , x ∈ R d , v(0, .) = φ0 , ∂tv(0, .) = φ1 . ( 3
)
The key of the method then lies in the fact that, once endowed with a good understanding of the pair (Ψ, Ψ 2 ), equation (3) turns out to be much more tractable than the original equation ( 2), and can be solved with pathwise arguments. The procedure thus emphasizes the following idea: to some extent, the difficulties behind the analysis of equation ( 2) reduce to the difficulties in the study of the two processes Ψ and Ψ 2 . Note in particular that this general approach offers a clear splitting between the stochastic part of the analysis (i.e., the study of (Ψ, Ψ 2 )), and the deterministic part of the problem (i.e., the pathwise study of ( 3)). This decomposition is very reminiscent of the spirit of rough paths (or regularity structures) theory, where the solution of the problem is also built in a deterministic way around a stochastically-constructed object.

The solution Ψ of ( 2) is therefore expected to play a fundamental role in the analysis, and a first step consists of course in providing a clear definition of this process (we recall that the space-time fractional setting is not exactly standard). To this end, we will appeal to a natural approximation procedure and construct Ψ as the limit of a sequence of (classical) solutions driven by a smooth approximation Ḃn of Ḃ (or equivalently a smooth approximation Bn of B). Just as in [START_REF] Deya | On a modelled rough heat equation[END_REF][START_REF] Deya | Construction and Skorohod representation of a fractional K-rough path[END_REF], the approximation that we will consider here is derived from the so-called harmonizable representation of the space-time fractional Brownian motion (see e.g. [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF]), that is the formula (valid for every H = (H0, . . . , H d ) ∈ (0, 1) d+1 )

B(t, x1, . . . , x d ) = cH ξ∈R η∈R d W (dξ, dη) e ıtξ -1 |ξ| H 0 + 1 2 d i=1 e ıx i η i -1 |ηi| H i + 1 2 ,
where cH > 0 is a suitable constant and W stands for the Fourier transform of a space-time white noise in R d+1 , defined on some complete filtered probability space (Ω, F, P). The approximation (Bn) n≥1 of B is then defined as

Bn(t, x1, . . . , x d ) cH |ξ|≤2 n |η|≤2 n W (dξ, dη) e ıtξ -1 |ξ| H 0 + 1 2 d i=1 e ıx i η i -1 |ηi| H i + 1 2 . ( 4
)
It is readily checked that for all fixed H = (H0, H1, . . . , H d ) ∈ (0, 1) d+1 and n ≥ 1, the so-defined process Bn indeed corresponds to a smooth function (almost surely). Accordingly, the associated equation

∂ 2 t Ψn -∆Ψn = Ḃn , t ∈ [0, T ] , x ∈ R d , Ψn(0, .) = 0 , ∂tΨn(0, .) = 0 , ( 5
)
(Ω; L ∞ ([0, T ]; W -α,p (R d ))), for all p ≥ 2 and α > d - 1 2 - d i=0 Hi . ( 6
)
In particular, (ρΨn) n≥1 converges to a limit in

L p (Ω; L ∞ ([0, T ]; W -α,p (R d ))
), that we denote by ρΨ.

This approach of a fractional equation via a regularization procedure is of course a standard strategy, that is also used for instance in rough paths or regularity structure theory (observe that the interpretation of the equation in [START_REF] Hu | Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2[END_REF] leans on an approximation method as well).

Remark 1.3. In [START_REF] Balan | The stochastic wave equation with fractional noise: a random field approach[END_REF], the authors tackle the fractional model (2) using a Malliavin-calculus approach, which provides an interpretation and a solution of the equation that may be considered as more intrinsic (since it does not depend on any approximation of the noise). In fact, we think that this Malliavin-calculus solution to (2) could be identified with the limit process Ψ exhibited in Proposition 1.2, but we will not dwell on this identification procedure, since we find it relatively removed from the purpose of our analysis and also because it would require the introduction of the whole Malliavin-calculus framework. Observe however that the results of [START_REF] Balan | The stochastic wave equation with fractional noise: a random field approach[END_REF] also highlight the threshold

d i=0 Hi = d -1
2 (with the additional assumption H0 > 1 2 ) for Ψ to be either a function or a distribution.

Based on Proposition 1.2, the limit process Ψ will therefore be considered (almost surely) as a function when

d i=0 Hi > d -1
2 and as a distribution otherwise. In the latter situation, and when turning to the study of the auxiliary equation (3), one must then face the problem of interpreting the product Ψ 2 . Just as in [START_REF] Gubinelli | Renormalization of the two-dimensional stochastic non linear wave equations[END_REF][START_REF] Oh | Invariant Gibbs measure for the 2-d defocusing nonlinear wave equations[END_REF], we will actually understand this product in the Wick sense, which, again, can be made rigorous through an approximation method, combined with a renormalization procedure: Proposition 1.4. Let d ≥ 1 and ρ : R d → R be a smooth compactly-supported function. Also, let (H0, H1, . . . , H d ) ∈ (0, 1) d+1 such that

d - 3 4 < d i=0 Hi ≤ d - 1 2 , ( 7
)
and consider the Wick-renormalized product Ψ 2 n (t, y) Ψn(t, y) 2 -σn(t, y), with σn(t, y)

E Ψn(t, y) 2 . Then (ρ 2 Ψ 2 n ) n≥1 is a Cauchy sequence in the space L p (Ω; L ∞ ([0, T ]; W -2α,p (R d ))
), for all p ≥ 2 and

α > d - 1 2 - d i=0 Hi . ( 8
)
In particular,

(ρ 2 Ψ 2 n ) n≥1 converges to a limit in L p (Ω; L ∞ ([0, T ]; W -2α,p (R d ))
), that we denote by ρ 2 Ψ 2 .

Two distinct treatments of the problem (corresponding to the two regimes d -1 2 -

d i=0 Hi < α < 0 and d -1 2 - d i=0 Hi ≥ α ≥ 0 in Proposition 1.
2) are thus to occur in our analysis, with a clear transition phenomenon regarding the interpretation of the product Ψ 2 and the need for renormalization. In order to encompass these two regimes into a single framework, let us slightly extend the formulation of (3) and consider the more general (deterministic) equation

∂ 2 t v -∆v + ρ 2 v 2 + (ρv) • Π 1 + Π 2 = 0 , t ∈ [0, T ] , x ∈ R d , v(0, .) = φ0 , ∂tv(0, .) = φ1 , ( 9
)
where the two "parameters" Π 1 and Π 2 will be either functions or distributions in suitable Sobolev spaces.

Our interpretation of the model (1) can now be expressed as follows:

Definition 1.5. Let Ψ and Ψ 2 be the processes defined in Proposition 1.2 and Proposition 1.4. (i) A stochastic process (u(t, x)) t∈[0,T ],x∈R d is said to be a solution (on [0, T ]) of the equation

∂ 2 t u -∆u + ρ 2 u 2 = Ḃ , t ∈ [0, T ] , x ∈ R d , u(0, .) = φ0 , ∂tu(0, .) = φ1 , ( 10
)
if, almost surely, Ψ is a function and the auxiliary process v := u -Ψ is a mild solution (on [0, T ]) of Equation ( 9) with Π 1 2ρΨ and

Π 2 ρ 2 Ψ 2 . (ii) A stochastic process (u(t, x)) t∈[0,T ],x∈R d is said to be a solution (on [0, T ]) of the Wick-renormalized equation ∂ 2 t u -∆u + ρ 2 : u 2 : = Ḃ , t ∈ [0, T ] , x ∈ R d , u(0, .) = φ0 , ∂tu(0, .) = φ1 , ( 11
)
if, almost surely, the auxiliary process v := u-Ψ is a mild solution (on [0, T ]) of Equation ( 9) with Π 1 2ρΨ and Π 2 ρ 2 Ψ 2 .

The results of Section 3 will in fact allow us to give a clear sense to the notion of a mild solution to (9) (with values in a specific space), thus completing the above definition. With this setting in mind, we can finally state the main results of our study.

Theorem 1.6. Let d ∈ {2, 3} and (φ0, φ1) ∈ H 1 (R d ) × L 2 (R d ).
Then the following picture holds true:

(i) If d i=0 Hi > d -1
2 , then, almost surely, there exists a time T0 > 0 such that the equation ( 10) admits a unique solution u in the set

ST 0 Ψ + X(T0) , where X(T0) L ∞ ([0, T0]; H 1 (R d )) . ( 12
) (ii) If d -3 4 < d i=0 Hi ≤ d -1 2
, then, almost surely, there exists a time T0 > 0 such that the Wickrenormalized equation [START_REF] Deya | Non-linear rough heat equations[END_REF] admits a unique solution u in the set

S s T 0 Ψ + X 1 2 (T0) , where X 1 2 (T0) L ∞ ([0, T0]; H 1 2 (R d )) , ( 13
)
Using the continuity properties of the solution v of ( 9) with respect to (Π 1 , Π 2 ), we will also be able to "lift" the convergence statements for Ψ and Ψ 2 (i.e., the results of Propositions 1.2 and 1.4) at the level of the equation, which will offer the following alternative interpretation of the model:

Theorem 1.7. Let d ∈ {2, 3} and (φ0, φ1) ∈ H 1 (R d ) × L 2 (R d ).
Then the following picture holds true:

(i) If d i=0 Hi > d -1
2 , consider the sequence (un) n≥1 of (classical) solutions to the equation

∂ 2 t un -∆un + ρ 2 u 2 n = Ḃn , t ∈ [0, T0] , x ∈ R d , u(0, .) = φ0 , ∂tu(0, .) = φ1 . ( 14
)
Then, almost surely, there exists a time T0 > 0 and a subsequence of (un) that converges in the space

L ∞ ([0, T0]; L 2 (D)) to the solution u exhibited in Theorem 1.6 (item (i)). (ii) If d -3 4 < d i=0 Hi ≤ d -1 2 , set σn(t) E[Ψn(t, x) 2
] and consider the sequence (un) n≥1 of (classical) solutions to the renormalized equation

∂ 2 t un(t, x) -∆un(t, x) + ρ 2 (x)(un(t, x) 2 -σn(t)) = Ḃn(t, x) , un(0, .) = φ0 , ∂tun(0, .) = φ1 . ( 15
)
for t ∈ [0, T0], x ∈ R d . Then σn(t) n→∞ ∼ c 1 H t 2 2n(d-1 2 - d i=0 H i ) if d i=0 Hi < d -1 2 , c 2 H t n if d i=0 Hi = d -1 2 , ( 16
)
for some constants c 1 H , c 2 H , and, almost surely, there exists a time T0 > 0 and a subsequence of (un) that converges in the space L ∞ ([0, T0]; H -α (D)) to the solution u exhibited in Theorem 1.6 (item (ii)), for every

α > d -1 2 - d i=0
Hi. As far as we know, Theorems 1.6 and 1.7 are the first wellposedness results for a non-linear wave model involving a space-time fractional noise (at least beyond the very specific one-dimensional situation). Observe that the above change-of-regime phenomenon is especially relevant in the fractional setting, where the roughness parameter H can be "continuously" modified in (0, 1) d+1 (contrary to the space parameter d ∈ {2, 3, . . .}).

The rest of the paper is devoted to the proof of these successive statements. Let us just conclude this introduction with a few additional remarks about Theorems 1.6 and 1.7. ). These combinations naturally echo the hyperbolic and parabolic settings, with scaling coefficient s = (1, 1, . . . , 1) and s = (2, 1, . . . , 1), respectively. Remark 1.9. Of course, the two situations (i) and (ii) in Theorem 1.6 and Theorem 1.7 do not cover the whole range of possibilities for the Hurst index H = (H0, . . . , H d ) ∈ (0, 1) d+1 of the noise.

The restricting condition

d i=0 Hi > d -3
4 is first inherited from our computations towards Proposition 1.4 (as reported in ( 7)), where we lean on the possibility to pick α < 1 4 (see (32)), and, due to [START_REF] Deya | On a modelled rough heat equation[END_REF], this can indeed be done only if

d i=0 Hi > d -3
4 . We suspect that, at the price of a sophisticated refinement of the estimations of Section 2.2 (starting from a refinement of the transition from (29) to (30)), the renormalization result of Proposition 1.4 should in fact remain true up to the critical value

d i=0 Hi = d -1.
This conjecture is essentially based on the results of [START_REF] Deya | On a non-linear 2D fractional wave equation[END_REF] for the particular dimension d = 2, where the Wick-renormalization of Ψ 2 is shown to be possible up to the critical situation H0 + H1 + H2 = 1 (see [START_REF] Deya | On a non-linear 2D fractional wave equation[END_REF]Propositions 1.3 and 1.4]).

On the other hand, and in light of the assumptions in the subsequent Proposition 3.5, it is clear that the deterministic part of our analysis can only be applied if α < 1 3 . Keeping condition [START_REF] Deya | On a modelled rough heat equation[END_REF] in mind, this would here lead to the restriction d i=0 Hi > d - 5 6 , and so, in brief, we think that the "second-order" results of Theorems 1.6 and 1.7 should remain valid if d - 5 6 < d i=0 Hi ≤ d -3 4 . To our opinion, extending such properties to the case d -1 < d i=0 Hi ≤ d - 5 6 can only be done through the consideration of higher-order expansions of the equation, as performed in [START_REF] Deya | On a non-linear 2D fractional wave equation[END_REF] for the particular dimension d = 2.

Remark 1.10. The forthcoming proofs (and accordingly the above results) could certainly be extended to more general covariance structures, such as the ones considered for instance in [START_REF] Balan | The stochastic wave equation with fractional noise: a random field approach[END_REF]. Our arguments are indeed based on a Fourier-type analysis, which suggests that a suitable control on the Fourier transform of the covariance function might be sufficient for the computations to remain valid. Besides, we think that, just as in rough paths or regularity structures results, the above properties are in fact relatively independent of the choice of the approximation Bn. For instance, using an appropriate Fourier transformation, the results should be the same when starting from an approximation of the form Bn ϕn * B, for a given mollifying sequence (ϕn) n≥1 (the only possible difference may be the value of the constants c 1 H , c 2 H in ( 16), as classically observed in regularity structures theory).

Remark 1.11. For d = 2, our results cover the white-noise situation H0 = H1 = H2 = 1 2 , and so we can consider Theorem 1.6 as a fractional extension of [14, Theorem 1.1] in the quadratic case (as far as the non-linearity). Our study thus offers an additional illustration of the flexibility of the general two-step procedure described above (i.e., we first study the free equation ( 2) and then the auxiliary equation ( 9)). Observe that the white-noise situation for d = 2 corresponds here to a "border case", that is a case for which

d i=0 Hi = d -1
2 , with specific rate of divergence in [START_REF] Gubinelli | Rough evolution equations[END_REF].

Remark 1.12. As the reader may have guessed it, the involvement of the smooth function ρ in (1) is only meant to bring the computations back to a compact space-time domain (which will be often esssential in the sequel). Thus, our results should morally be read as local results, both in time and in space, for the real "target" equation, that is the equation with ρ ≡ 1. What refrained us to formulate the problem on a torus (just as in [START_REF] Gubinelli | Renormalization of the two-dimensional stochastic non linear wave equations[END_REF][START_REF] Oh | Invariant Gibbs measure for the 2-d defocusing nonlinear wave equations[END_REF]) is the consideration of the fractional noise, which is more convenient to define and handle on the whole Euclidean space.

As we already pointed it out, our analysis will be clearly divided into a stochastic and a deterministic part. The organization of the paper will follow this splitting. Section 2 is first devoted to the stochastic analysis, that is the study of Ψn and the proof of Propositions 1.2 and 1.4. The estimation ( 16) of the renormalization constant (which is directly related to Ψn) will also be carried out in this section. In Section 3, we will focus on the deterministic study of the auxiliary equation [START_REF] Deya | Construction and Skorohod representation of a fractional K-rough path[END_REF], first in the "regular" case where Π 1 and Π 1 are functions (Proposition 3.3), and then in the distributional situation (Proposition 3.5). We will finally combine these successive results in Section 4 in order to derive the proof of Theorem 1.6 and Theorem 1.7.

Throughout the paper, and for any normed space E, the notation N [v; E] will refer to the norm of v ∈ E.

Acknowledgements. I am deeply grateful to two anonymous reviewers for their careful reading and their enthusiastic comments about this study.

Study of the (stochastic) linear equation

We here propose to tackle the issues related to the solution Ψn of the regularized equation [START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local existence theory[END_REF].

For a fixed dimension d ≥ 1, let us denote by G the Green function associated with the standard ddimensional wave equation and recall that the (space) Fourier transform of G is explicitly given for all t ≥ 0 and x ∈ R d by the formula

R d dy e -ı x,y Gt(y) = sin(t|x|) |x| .
Now, the solution Ψn of ( 5) can be written as

Ψn(t, x) = t 0 ds (Gt-s * Ḃn(s))(x) = c |ξ|≤2 n |η|≤2 n W (dξ, dη) ξ |ξ| H 0 + 1 2 d i=1 ηi |ηi| H i + 1 2 t 0 ds R d dy Gt-s(x -y)e ıξs e ı η,y = c |ξ|≤2 n |η|≤2 n W (dξ, dη) ξ |ξ| H 0 + 1 2 d i=1 ηie ıη i x i |ηi| H i + 1 2 γt(ξ, |η|) , ( 17
)
where for all t ≥ 0, ξ ∈ R and r > 0, we define the quantity γt(ξ, r) as γt(ξ, r) e ıξt t 0 ds e -ıξs sin(sr) r . [START_REF] Hu | Heat equation with fractional white noise potentials[END_REF] Let us also set γs,t(ξ, r) γt(ξ, r) -γs(ξ, r).

With these notations in hand, our computations towards Proposition 1.2 and Proposition 1.4 will extensively rely on the two following elementary estimates.

Lemma 2.1. For all 0 ≤ s ≤ t, ξ ∈ R, r > 0 and κ, λ ∈ [0, 1], it holds that |γs,t(ξ, r)| min |ξ| κ |t -s| κ t 2 + |t -s|t, |t -s| |ξ| + |t -s| κ t |ξ| 1-κ , |t -s| κ {r κ + |ξ| κ }t λ(1-κ) r ||ξ| -r| 1-λ(1-κ) . ( 19
)
Proof. First, one has obviously

e ırt -e ıξt ξ -r - e -ırt -e ıξt ξ + r , ( 20
)
which easily leads to

|γs,t(ξ, r)| |t -s| κ {r κ + |ξ| κ }t λ(1-κ) r ||ξ| -r| 1-λ(1-κ) . Corollary 2.2. For all 0 ≤ s ≤ t ≤ 1, H ∈ (0, 1), r > 0, ε ∈ (0, 1) and κ ∈ [0, min(H, 1-ε 2 )), it holds that R dξ |γs,t(ξ, r)| 2 |ξ| 2H-1 |t -s| 2κ min 1, 1 r 2+2(H-κ) + 1 r 1+2(H-κ)-ε .
Proof. The two bounds follow from [START_REF] Hu | Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2[END_REF]. First,

R dξ |γs,t(ξ, r)| 2 |ξ| 2H-1 |t -s| 2κ |ξ|≤1 dξ |ξ| 2H-1 + |ξ|≥1 dξ |ξ| 2(H-κ)+1
|t -s| 2κ .

Then consider the decomposition

R dξ |γs,t(ξ, r)| 2 |ξ| 2H-1 = ||ξ|-r|≥ |ξ| 2 dξ |γs,t(ξ, r)| 2 |ξ| 2H-1 + ||ξ|-r|≤ |ξ| 2 dξ |γs,t(ξ, r)| 2 |ξ| 2H-1 .
On the one hand, it holds that

||ξ|-r|≥ |ξ| 2 dξ |γs,t(ξ, r)| 2 |ξ| 2H-1 |t -s| 2κ r 2 {|ξ|≤ 2 3 r}∪{|ξ|≥2r} dξ r 2κ + |ξ| 2κ |ξ| 2H-1 ||ξ| -r| 2 |t -s| 2κ r 2+2(H-κ) {|ξ|≤ 2 3 }∪{|ξ|≥2} dξ 1 + |ξ| 2κ |ξ| 2H-1 ||ξ| -1| 2 |t -s| 2κ r 2+2(H-κ) .
On the other hand, for any λ ∈ [0, 1], one has

||ξ|-r|≤ |ξ| 2 dξ |γs,t(ξ, r)| 2 |ξ| 2H-1 |t -s| 2κ r 2 2 3 r≤|ξ|≤2r dξ r 2κ + |ξ| 2κ |ξ| 2H-1 ||ξ| -r| 2-2λ(1-κ) |t -s| 2κ r 2+2(H-κ)-2λ(1-κ) 2 3 ≤|ξ|≤2 dξ |ξ| 2H-1 ||ξ| -1| 2-2λ(1-κ) ,
and we get the conclusion by taking λ = 1+ε 2(1-κ) ∈ [0, 1].

Proof of Proposition 1.2.

For the sake of clarity, we shall assume that T ≤ 1 and set, for all m, n ≥ 1, Ψn,m Ψm -Ψn.

Step 1: Let us show that for all m ≥ n ≥ 1, 0 ≤ s < t ≤ 1 and ε > 0 small enough, one has

R d dx E F -1 {1 + |.| 2 } -α 2 F(ρ[Ψn,m(t, .) -Ψn,m(s, .)]) (x) 2p 2 -2nεp |t -s| 2εp , ( 21
)
where the proportional constant only depends on ρ, α and p.

Using the hypercontractivity property of Gaussian variables, we can first assert that

E F -1 {1 + |.| 2 } -α 2 F(ρ[Ψn,m(t, .) -Ψn,m(s, .)]) (x) 2p ≤ cp E F -1 {1 + |.| 2 } -α 2 F(ρ[Ψn,m(t, .) -Ψn,m(s, .)]) (x) 2 p
, where the constant cp only depends on p. Then write

E F -1 {1 + |.| 2 } -α 2 F(ρ[Ψn,m(t, .) -Ψn,m(s, .)]) (x) 2 = R d dλ R d dy R d d λ R d dỹ e ı x,λ {1 + |λ| 2 } -α 2 e -ı λ,y e -ı x, λ {1 + | λ| 2 } -α 2 e ı λ,ỹ ρ(y)ρ(ỹ)E {Ψn,m(t, y) -Ψn,m(s, y)}{Ψn,m(t, ỹ) -Ψn,m(s, ỹ)} .
At this point, and with expression [START_REF] Hairer | A theory of regularity structures[END_REF] in mind, note that

E {Ψ n,m (t, y) -Ψ n,m (s, y)}{Ψ n,m (t, ỹ) -Ψ n,m (s, ỹ)} = c (ξ,η)∈Dm,n dξdη 1 |ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 |γ s,t (ξ, |η|)| 2 e ı η,y e -ı η,ỹ , ( 22
)
where

D m,n (B 1 m × B d m )\(B 1 n × B d n ), B k {λ ∈ R k , |λ| ≤ 2 },
and accordingly

E F -1 {1 + |.| 2 } -α 2 F(ρ[Ψ n,m (t, .) -Ψ n,m (s, .)]) (x) 2 = c (ξ,η)∈Dm,n dξdη 1 |ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 |γ s,t (ξ, |η|)| 2 R d dλ R d d λ e ı x,λ-λ ρ(λ -η)ρ(η -λ){1 + |λ| 2 } -α 2 {1 + | λ| 2 } -α 2 ,
which gives

E F -1 {1 + |.| 2 } -α 2 F(ρ[Ψ n,m (t, .) -Ψ n,m (s, .)]) (x) 2 p = c p j=1 (ξ j ,η j )∈Dm,n dξ j dη j 1 |ξ j | 2H0-1 d i=1 1 |η j i | 2Hi-1 |γ s,t (ξ j , |η j |)| 2 R d dλ j R d d λj e ı x,λ j -λj ρ(λ j -η j )ρ(η j -λj ){1 + |λ j | 2 } -α 2 {1 + | λj | 2 } -α 2 . ( 23
)
Now,

R d dx p j=1 R d dλ j R d d λj e ı x,λ j -λj ρ(λ j -η j )ρ(η j -λj ){1 + |λ j | 2 } -α 2 {1 + | λj | 2 } -α 2 = R d dx p j=1 R d dλ j R d d λj e ı x,λ j -λj ρ(λ j )ρ(-λj ){1 + |η j + λ j | 2 } -α 2 {1 + |η j + λj | 2 } -α 2 = p-1 j=1 R d dλ j R d d λj ρ(λ j )ρ(-λj ){1 + |η j + λ j | 2 } -α 2 {1 + |η j + λj | 2 } -α 2 R d dλ p ρ(λ p )ρ p-1 k=1 ( λk -λ k ) + λ p {1 + |η p + λ p | 2 } -α 2 1 + η p + p-1 k=1 ( λk -λ k ) + λ p 2 -α 2 . ( 24
)
The absolute value of this product can in fact be bounded by

c p i=1 {1 + |η j | 2 } -α ,
for some constant c > 0, due to

ρ(λ){1 + |η + λ| 2 } -α 2 = ρ(λ){1 + |η + λ| 2 } -α 2 1 {|λ|> 1 2 |η|} + ρ(λ){1 + |η + λ| 2 } -α 2 1 {|λ|< 1 2 |η|} ≤ c α |ρ(λ)|1 {|λ|> 1 2 |η|} + |ρ(λ)|{1 + |η| 2 } -α 2 1 {|λ|< 1 2 |η|} ≤ c α,ρ,κ {1 + |λ| 2 } -κ {1 + |λ| 2 } -α 2 1 {|λ|> 1 2 |η|} + {1 + |λ| 2 } -κ {1 + |η| 2 } -α 2 1 {|λ|< 1 2 |η|} ≤ c α,ρ,κ {1 + |λ| 2 } -κ {1 + |η| 2 } -α 2 , ( 25
)
for all λ, η ∈ R d and κ > 0.

Going back to [START_REF] Runst | Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations[END_REF], we get that

E F -1 {1 + |.| 2 } -α 2 F(ρ[Ψ n,m (t, .) -Ψ n,m (s, .)]) (x) 2 p (ξ,η)∈Dm,n dξdη 1 |ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 {1 + |η| 2 } -α |γ s,t (ξ, |η|)| 2 p 2 n ≤|ξ|≤2 m dξ |η|≤2 m dη 1 |ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 {1 + |η| 2 } -α |γ s,t (ξ, |η|)| 2 p + |ξ|≤2 m dξ 2 n ≤|η|≤2 m dη 1 |ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 {1 + |η| 2 } -α |γ s,t (ξ, |η|)| 2 p I m,n (s, t) p + II m,n (s, t) p . ( 26
)
Let us focus on the estimation of I m,n (s, t) (the treatment of II m,n (s, t) can be done along similar arguments). Using an elementary spherical change-of-variable for the η i -coordinates, we get that for any 0 < ε < H 0 ,

I m,n (s, t) ≤ 2 -2nε R dξ |ξ| 2H0-2ε-1 ∞ 0 dr {1 + r 2 } -α r 2(H1+...+H d )-2d+1 |γ s,t (ξ, r)| 2 [0,2π] d-1 dθ 1 • • • dθ d-1 d-1 i=1 1 | cos(θ i )| 2Hi-1 | sin(θ i )| 2(Hi+1+...+H d )-2d+2i+1 , and since max(2H i -1, 2(H i+1 + . . . + H d ) -2d + 2i + 1) < 1 for every i ∈ {1, . . . , d -1}, this yields I m,n (s, t) 2 -2nε R dξ ∞ 0 dr 1 |ξ| 2H0-2ε-1 {1 + r 2 } -α r 2(H1+...+H d )-2d+1 |γ s,t (ξ, r)| 2 .
Now, by applying Corollary 2.2, we can assert that for all 0 < ε < min(H 0 , 1 2 ) and 0

< κ < min(H 0 -ε, 1 2 -ε), R dξ ∞ 0 dr 1 |ξ| 2H0-2ε-1 {1 + r 2 } -α r 2(H1+...+H d )-2d+1 |γ s,t (ξ, r)| 2 |t -s| 2κ 1 0 dr r 2(H1+...+H d )-2d+1 + ∞ 1 dr 1 r 2α+2(H0+...+H d )-2d+2-2κ-4ε . ( 27
)
The conclusion is straightforward: the two integrals involved in [START_REF] Walsh | An introduction to stochastic partial differential equations[END_REF] are indeed finite as soon as

2ε + κ < α -d - 1 2 - d i=0 H i .
Step 2: We have thus shown that

E ρΨ n,m (t, .) -ρΨ n,m (s, .) 2p W -α,2p (R d ) 2 -2nεp |t -s| 2εp ,
and we can now conclude by applying the classical Garsia-Rodemich-Rumsey estimate: for any ε 0 > 0,

E N ρΨ n,m ; C ε0 ([0, T ]; W -α,2p (R d )) 2p [0,1] 2 dsdt E ρΨ n,m (t, .) -ρΨ n,m (s, .) 2p W -α,2p (R d ) |t -s| 2ε0p+2 2 -2nεp [0,1] 2 dsdt |t -s| -2(ε-ε0)p+2 ,
noting that the latter integral is finite for all 0 < ε 0 < ε and p large enough. [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF], we can (and will) assume in the sequel that α < 1 4 , which will be of importance in our estimates (see (32)). Also, for the sake of clarity, we shall again assume that T ≤ 1. Finally, let us set, for all m, n ≥ 1 and 0

Proof of Proposition 1.4. Due to condition

≤ s, t ≤ 1, Ψ n,m Ψ m -Ψ n , Ψ 2 n,m Ψ 2 m -Ψ 2 n and f (s, t; .) f (t, .) -f (s, .) for f ∈ {Ψ n , Ψ n,m , Ψ 2 n , Ψ 2
n,m }. Just as in [START_REF] Gubinelli | Renormalization of the two-dimensional stochastic non linear wave equations[END_REF], the success of the renormalization procedure essentially lies in the following elementary property, which can be readily derived from the classical Wick formula:

Lemma 2.3. For all m, n ≥ 1, s, t ≥ 0 and y, ỹ ∈ R, it holds that E Ψ 2 m (t, y) Ψ 2 n (s, ỹ) = 2E Ψ m (t, y)Ψ n (s, ỹ) 2 .
We can now turn to the proof of Proposition 1.4, that we present as a two-step procedure (just as the proof of Proposition 1.2).

Step 1: Let us show that for all m ≥ n ≥ 1, 0 ≤ s ≤ t ≤ 1 and ε > 0 small enough, one has

R d dx E F -1 {1 + |.| 2 } -α F(ρ 2 Ψ 2 n,m (s, t; .)) (x) 2p 2 -2nεp |t -s| 2εp , ( 28
)
where the proportional constant only depends on ρ, α and p.

Using the hypercontractivity property of Wiener chaoses, we can first assert that

E F -1 {1 + |.| 2 } -α F(ρ 2 Ψ 2 n,m (s, t; .)) (x) 2p ≤ c p E F -1 {1 + |.| 2 } -α F(ρ 2 Ψ 2 n,m (s, t; .)) (x) 2 p
, where the constant c p only depends on p. Then write

E F -1 {1 + |.| 2 } -α F(ρ 2 Ψ 2 n,m (s, t; .)) (x) 2 = R d dλ R d dy R d d λ R d dỹ e ı x,λ {1 + |λ| 2 } -α e -ı λ,y e -ı x, λ {1 + | λ| 2 } -α e ı λ,ỹ ρ 2 (y)ρ 2 (ỹ)E Ψ 2 n,m (s, t; y) Ψ 2 n,m (s, t; ỹ) ,
and, using Lemma 2.3, we can check that

1 2 E Ψ 2 n,m (s, t; y) Ψ 2 n,m (s, t; ỹ) = E Ψ n,m (t, y)Ψ n,m (s, t; ỹ) E Ψ m (t, y){Ψ m + Ψ n }(t, ỹ) +E Ψ n,m (t, y)Ψ n,m (s, ỹ) E Ψ m (t, y){Ψ m + Ψ n }(s, t; ỹ) +E Ψ n (t, y)Ψ n,m (s, t; ỹ) E Ψ n,m (t, y){Ψ m + Ψ n }(t, ỹ) +E Ψ m (t, y)Ψ n,m (s, ỹ) E Ψ n,m (t, y){Ψ m + Ψ n }(s, t; ỹ) +E Ψ n,m (s, y)Ψ n,m (t, s; ỹ) E Ψ m (s, y){Ψ m + Ψ n }(s, ỹ) +E Ψ n,m (s, y)Ψ n,m (t, ỹ) E Ψ m (s, y){Ψ m + Ψ n }(t, s; ỹ) +E Ψ n (s, y)Ψ n,m (t, s; ỹ) E Ψ n,m (s, y){Ψ m + Ψ n }(s, ỹ) +E Ψ m (s, y)Ψ n,m (t, ỹ) E Ψ n,m (s, y){Ψ m + Ψ n }(t, s; ỹ) i=1,...,8
A i n,m (s, t; y, ỹ) .

It turns out that the eight terms derived from A i m,n (s, t; y, ỹ) (i ∈ {1, . . . , 8}) can be handled with the same arguments, and therefore we will only focus on the treatment of A 1 m,n (s, t; y, ỹ). In fact, just as in [START_REF] Quer-Sardanyons | The 1-d stochastic wave equation driven by a fractional Brownian motion[END_REF], one has

E Ψ n,m (t, y)Ψ n,m (s, t; ỹ) = c (ξ,η)∈Dm,n dξdη 1 |ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 e ı η,y e -ı η,ỹ γ t (ξ, |η|)γ s,t (ξ, |η|) and E Ψ m (t, y){Ψ m + Ψ n }(t, ỹ) = c R×R d d ξdη 1 | ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 |γ t ( ξ, |η|)| 2 e ı η,y e -ı η,ỹ 1 ( ξ,η)∈B 1 m ×B d m + 1 ( ξ,η)∈B 1 n ×B d n , where D m,n (B 1 m × B d m )\(B 1 n × B d n ) and B k {λ ∈ R k , |λ| ≤ 2 }.
Besides, one has obviously

R d dλ R d dy R d d λ R d dỹ e ı x,λ {1 + |λ| 2 } -α e -ı λ,y e -ı x, λ {1 + | λ| 2 } -α e ı λ,ỹ ρ 2 (y)ρ 2 (ỹ)e ı η,y e -ı η,ỹ e ı η,y e -ı η,ỹ = R d dλ R d d λ e ı x,λ-λ {1 + |λ| 2 } -α {1 + | λ| 2 } -α F ρ 2 (λ -η -η)F ρ 2 (η + η -λ) .
Using the same arguments as in the proof of Proposition 1.2 (see ( 23)-( 25)), we end up with

R d dx R d dλ R d dy R d d λ R d dỹ e ı x,λ {1 + |λ| 2 } -α e -ı λ,y e -ı x, λ {1 + | λ| 2 } -α e ı λ,ỹ A 1 m,n (s, t; y, ỹ) p (ξ,η)∈Dm,n dξdη 1 |ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 γ t (ξ, |η|) γ s,t (ξ, |η|) R×R d d ξdη 1 | ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 |γ t ( ξ, |η|)| 2 1 ( ξ,η)∈B 1 m ×B d m + 1 ( ξ,η)∈B 1 n ×B d n {1 + |η -η| 2 } -2α p |ξ|≥2 n dξ R d dη 1 |ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 |γ t (ξ, |η|)||γ s,t (ξ, |η|)| R×R d d ξdη 1 | ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 |γ t ( ξ, |η|)| 2 {1 + |η -η| 2 } -2α p + R dξ |η|≥2 n dη 1 |ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 |γ t (ξ, |η|)||γ s,t (ξ, |η|)| R×R d d ξdη 1 | ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 |γ t ( ξ, |η|)| 2 {1 + |η -η| 2 } -2α p I n (s, t) p + II n (s, t) p .
As in the proof of Proposition 1.2, we will restrict our attention to I n (s, t). For 0 < ε < H 0 , one has

In(s, t) 2 -2nε R×R d dξdη R×R d d ξdη {1 + |η -η| 2 } -2α 1 |ξ| 2(H 0 -ε)-1 d i=1 1 |ηi| 2H i -1 |γt(ξ, |η|)||γs,t(ξ, |η|)| 1 | ξ| 2H 0 -1 d i=1 1 |ηi| 2H i -1 |γt( ξ, |η|)| 2 (29) 2 -2nε R×R d dξdη R×R d d ξdη {1 + ||η| -|η|| 2 } -2α 1 |ξ| 2(H 0 -ε)-1 d i=1 1 |ηi| 2H i -1 |γt(ξ, |η|)||γs,t(ξ, |η|)| 1 | ξ| 2H 0 -1 d i=1 1 |ηi| 2H i -1 |γt( ξ, |η|)| 2 . ( 30
)
Now let us split the integration domain as

(R × R d ) 2 = D 1 ∪ D 2 , with D 1 {(ξ, η, ξ, η) : |η| 2 < |η| < 3|η| 2 }
and

D 2 {(ξ, η, ξ, η) : 0 < |η| < |η| 2 or |η| > 3|η| 2 } .
For (ξ, η, ξ, η) ∈ D 2 , one has ||η| -|η|| > max |η| 2 , |η| 3 , and so

D2 dξdηd ξdη {1 + ||η| -|η|| 2 } 2α 1 |ξ| 2(H0-ε)-1 d i=1 1 |η i | 2Hi-1 |γ t (ξ, |η|)||γ s,t (ξ, |η|)| 1 | ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 |γ t ( ξ, |η|)| 2 R×R d dξdη {1 + |η| 2 } α 1 |ξ| 2(H0-ε)-1 d i=1 1 |η i | 2Hi-1 |γ t (ξ, |η|)||γ s,t (ξ, |η|)| R×R d d ξdη {1 + |η| 2 } α 1 | ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 |γ t ( ξ, |η|)| 2 R×R d dξdη {1 + |η| 2 } α 1 |ξ| 2(H0-ε)-1 d i=1 1 |η i | 2Hi-1 |γ t (ξ, |η|)| 2 1/2 R×R d dξdη {1 + |η| 2 } α 1 |ξ| 2(H0-ε)-1 d i=1 1 |η i | 2Hi-1 |γ s,t (ξ, |η|)| 2 1/2 R×R d d ξdη {1 + |η| 2 } α 1 | ξ| 2H0-1 d i=1 1 |η i | 2Hi-1 |γ t ( ξ, |η|)| 2 . ( 31
)
At this point, observe that we are exactly in the same position as in the proof of Proposition 1.2 (see [START_REF] Tindel | Stochastic evolution equations with fractional Brownian motion[END_REF]), and so we can rely on the same arguments to assert that for ε > 0 small enough, the above integral (over D 2 ) is indeed bounded by c|t -s| ε , for some finite constant c.

In order to deal with the integral over the domain D 1 , observe first that

|η| 2 <|η|< 3|η| 2 dη {1 + ||η| -|η|| 2 } 2α |γ t ( ξ, |η|)| 2 d i=1 1 |η i | 2Hi-1 = |η| -2(H1+...+H d )+2d 1 2 <|η|< 3 2 dη {1 + |η| 2 (1 -|η|) 2 } 2α |γ t ( ξ, |η||η|)| 2 d i=1 1 |η i | 2Hi-1 |η| -2(H1+...+H d )+2d 3 2 1 2 dr {1 + |η| 2 (1 -r) 2 } 2α |γ t ( ξ, |η|r)| 2 ,
and so

D 1 dξdηd ξdη {1 + ||η| -|η|| 2 } 2α 1 |ξ| 2(H 0 -ε)-1 d i=1 1 |ηi| 2H i -1 |γt(ξ, |η|)||γs,t(ξ, |η|)| 1 | ξ| 2H 0 -1 d i=1 1 |ηi| 2H i -1 |γt( ξ, |η|)| 2 R d dη |η| 2(H 1 +...+H d )-2d d i=1 1 |ηi| 2H i -1 3 2 1 2 dr {1 + |η| 2 (1 -r) 2 } 2α R dξ |γt(ξ, |η|)||γs,t(ξ, |η|)| |ξ| 2(H 0 -ε)-1 R d ξ |γt( ξ, |η|r)| 2 | ξ| 2H 0 -1 R d dη |η| 2(H 1 +...+H d )-2d d i=1 1 |ηi| 2H i -1 3 2 1 2 dr {1 + |η| 2 (1 -r) 2 } 2α R dξ |γt(ξ, |η|)| 2 |ξ| 2(H 0 -ε)-1 1/2 R dξ |γs,t(ξ, |η|)| 2 |ξ| 2(H 0 -ε)-1 1/2 R d ξ |γt( ξ, |η|r)| 2 | ξ| 2H 0 -1 ∞ 0 dρ ρ 4(H 1 +...+H d )-4d+1 3 2 1 2 dr {1 + ρ 2 (1 -r) 2 } 2α R dξ |γt(ξ, ρ)| 2 |ξ| 2(H 0 -ε)-1 1/2 R dξ |γs,t(ξ, ρ)| 2 |ξ| 2(H 0 -ε)-1 1/2 R d ξ |γt( ξ, ρr)| 2 | ξ| 2H 0 -1 |t -s| κ 1 0 dρ ρ 4(H 1 +...+H d )-4d+1 + ∞ 1 dρ ρ 4(H 0 +...+H d )-4d+3-8ε-κ 3 2 1 2 dr {1 + ρ 2 (1 -r) 2 } 2α
for all 0 < ε < min(H 0 , 1 2 ) and 0 < κ < min(H 0 -ε, 1 2 -ε), where we have used Corollary 2.2 to derive the last inequality. Finally, since α < 1 4 , it is readily checked that for all 0 < ε < min(H 0 , 1 2 ) and 0

< κ < min(H 0 -ε, 1 2 -ε) such that 8ε + κ < 4 α -d -1 2 - d i=0 H i , ∞ 1 dρ ρ 4(H0+...+H d )-4d+3-8ε-κ 3 2 1 2 dr {1 + ρ 2 (1 -r) 2 } 2α ≤ ∞ 1 dρ ρ 4α+4(H0+...+H d )-4d+3-8ε-κ 3 2 1 2 dr (1 -r) 4α < ∞ . ( 32
)
Going back to (30), we have thus shown (28).

Step 2: Once endowed with estimate (28), we can of course use the same arguments as in Step 2 of the proof of Proposition 1.2 to obtain that, for 0 < ε 0 < ε and p large enough,

E N Ψ 2 n,m ; C ε0 ([0, T ]; W -2α,2p (D)) 2p 2 -nεp ,
which completes the proof our assertion.

Estimation of the renormalization constant.

Let us conclude this section with the asymptotic analysis of the renormalization constant σ n (t) E Ψ n (t, x) 2 at the core of the above renormalization procedure. In other words, our aim here is to show [START_REF] Gubinelli | Rough evolution equations[END_REF]. To this end, fix d ≥ 2 and (H 0 , . . . , H d ) ∈ (0, 1) d+1 such that

d - 3 4 < d i=0 H i ≤ d - 1 2 ,
and, with expression (17) in mind, write the renormalization constant as

σ n (t) = E Ψ n (t, x) 2 = c |ξ|≤2 n dξ |ξ| 2H0-1 |η|≤2 n d i=1 dη i |η i | 2Hi-1 |γ t (ξ, |η|)| 2 = c 2 n 0 dr r 2(H1+...+H d )-2d+1 |ξ|≤2 n dξ |ξ| 2H0-1 |γ t (ξ, r)| 2 .
The asymptotic estimate ( 16) is now a straightforward consequence of the following technical result (take α 2H 0 ∈ (0, 2) and κ 2(d

-1 2 - d i=0 H i ) ∈ [0, 1
)): Proposition 2.4. There exists a constant c > 0 such that for all α ∈ (0, 2) and κ ∈ [0, 1), one has, as n tends to infinity,

2 n 0 dr r -α-κ |ξ|≤2 n dξ |ξ| α-1 |γ t (ξ, r)| 2 = c t 2 n 1 dr r 1-κ + O(1) . ( 33
)
Proof. First, observe that using [START_REF] Hu | Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2[END_REF], we have

1 0 dr r -α-κ |ξ|≤2 n dξ |ξ| α-1 |γ t (ξ, r)| 2 {1 + t 4 } 1 0 dr r -2H0-κ |ξ|≤1 dξ |ξ| α-1 + t 2 1 0 dr r -α-κ |ξ|≥1 dξ |ξ| 1+α ,
accordingly it suffices to focus on the estimation of the integral

2 n 1 dr r -α-κ |ξ|≤2 n dξ |ξ| α-1 |γ t (ξ, r)| 2 .
To this end, we will rely on the following expansion, which can be readily derived from [START_REF] Hu | Feynman-Kac formula for heat equation driven by fractional white noise[END_REF]: =: Γ t (ξ, r) + Γ t (ξ, r) .

|γ t (ξ, r)| 2 = c r 2 1 -cos(t(ξ -r)) (ξ -r) 2 - cos (
For obvious symmetry reasons, we have in fact

2 n 1 dr r -α-κ |ξ|≤2 n dξ |ξ| α-1 |γ t (ξ, r)| 2 = 2 2 n 1 dr r -α-κ |ξ|≤2 n dξ |ξ| α-1 Γ t (ξ, r) = 2 2 n 1 dr r -α-κ r 0 dξ |ξ| α-1 Γ t (ξ, r) + 2 2 n 1 dr r -α-κ 2 n r dξ |ξ| α-1 Γ t (ξ, r) +2 2 n 1 dr r -α-κ 2 n 0 dξ |ξ| α-1 Γ t (-ξ, r) =: J 1 n,t + J 2 n,t + J 3 n,t . ( 34
)
Study of J 1 n,t . Let us introduce the additional notation

Γ 1 t (ξ, r) := 1 -cos(t(ξ -r)) r 2 (ξ -r) 2 , Γ 2 t (ξ, r) := cos(tr){cos(tξ) -cos(tr)} r 2 (ξ -r)(ξ + r) , so that Γ t (ξ, r) = c Γ 1 t (ξ, r) + Γ 2 t (ξ, r) . ( 35
)
Now on the one hand, for any 0 < ε < 1,

2 n 1 dr r -α-κ r 0 dξ |ξ| α-1 Γ 2 t (ξ, r) = 2 n 1 dr r 2-κ 1 0 dξ |ξ| α-1 cos(tr)(cos(trξ) -cos(tr)) (ξ -1)(ξ + 1) t ε ∞ 1 dr r 2-κ-ε 1 0 dξ |ξ| α-1 |1 -ξ| 1-ε |1 + ξ|
and the latter integrals are finite for any ε > 0 such that κ + ε < 1, which shows that

2 n 1 dr r -α-κ r 0 dξ |ξ| α-1 Γ 2 t (ξ, r) = O(1) .
On the other hand,

2 n 1 dr r -α-κ r 0 dξ |ξ| α-1 Γ 1 t (ξ, r) = 2 n 1 dr r 2-κ 1 0 dξ |ξ| α-1 1 -cos(tr(1 -ξ)) (1 -ξ) 2 = 2 n 1 dr r 2-κ 1 2 0 dξ |1 -ξ| α-1 1 -cos(trξ) ξ 2 + 2 n 1 dr r 2-κ 1 1 2 dξ |1 -ξ| α-1 1 -cos(trξ) ξ 2 , with 2 n 1 dr r 2-κ 1 1 2 dξ |1 -ξ| α-1 1 -cos(trξ) ξ 2 ∞ 1 dr r 2-κ 1 1 2 dξ |1 -ξ| α-1 < ∞ and 2 n 1 dr r 2-κ 1 2 0 dξ |1 -ξ| α-1 1 -cos(trξ) ξ 2 = t 2 n 1 dr r 1-κ rt 2 0 dξ |1 -ξ rt | α-1 1 -cos(ξ) ξ 2 = t ∞ 0 dξ 1 -cos(ξ) ξ 2 2 n 1 dr r 1-κ +t 2 n 1 dr r 1-κ rt 2 0 dξ |1 -ξ rt | α-1 1 -cos(ξ) ξ 2 - ∞ 0 dξ 1 -cos(ξ) ξ 2 .
By applying the below technical Lemma 2.5, we can easily conclude that

J 1 n,t = c t 2 n 1 dr r 1-κ + O(1) . ( 36
)
Study of J 2 n,t . We will here use the (readily-checked) decomposition

Γ t (ξ, r) = c 2Γ 3 t (ξ, r) + Γ 4 t (ξ, r) (37) with Γ 3 t (ξ, r) := 1 -cos(t(ξ -r)) r(ξ -r) 2 (ξ + r) and Γ 4 t (ξ, r) := 1 -cos(t(ξ -r)) -cos(tr)(cos(tr) -cos(tξ)) r 2 (ξ -r)(ξ + r) .
Now observe on the one hand that for every ε ∈ (0, 1),

2 n 1 dr r -α-κ 2 n r dξ |ξ| α-1 Γ 4 t (ξ, r) t ε 2 n 1 dr r 2-α-κ ∞ r dξ |ξ| α-1 |ξ -r| 1-ε |ξ + r| t ε ∞ 1 dr r 2-κ-ε ∞ 1 dξ |ξ| α-1 |1 -ξ| 1-ε |1 + ξ|
and the latter integrals are finite provided 0 < ε < min(1 -κ, α). On the other hand,

2 n 1 dr r -α-κ 2 n r dξ |ξ| α-1 Γ 3 t (ξ, r) = 2 n 1 dr r 1-α-κ 2 n -r 0 dξ |ξ + r| α-1 |ξ + 2r| 1 -cos(tξ) ξ 2 = t 2 2 n 1 dr r 1-κ ∞ 0 dξ 1 -cos(ξ) ξ 2 + 2 n 1 dr r 1-α-κ 2 n -r 0 dξ |ξ + r| α-1 |ξ + 2r| 1 -cos(tξ) ξ 2 - 1 2r α ∞ 0 dξ 1 -cos(tξ) ξ 2 . ( 38
)
Using Lemma 2.6, we can then assert that for ε ∈ (0, min(α, 1 -κ)),

2 n 1 dr r 1-α-κ 2 n -r 0 dξ |ξ + r| α-1 |ξ + 2r| 1 -cos(tξ) ξ 2 - 1 2r α ∞ 0 dξ 1 -cos(tξ) ξ 2 t ε 2 n 1 dr r 2-κ-ε + t ε 2 n 1 dr r 1-κ |2 n -r| 1-ε t ε ∞ 1 dr r 2-κ-ε + t ε 2 -n(1-κ-ε) 1 0 dr r 1-κ |1 -r| 1-ε .
Going back to (38), we have thus shown that

J 2 n,t = c t 2 n 1 dr r 1-κ + O(1) . ( 39
)
Study of J 3 n,t . Using decomposition (35), it is readily checked that for ε ∈ (0, 1) and for all ξ, r > 0,

Γ t (-ξ, r) 1 r 2 1 |ξ + r| 2 + t ε |ξ + r||ξ -r| 1-ε and so 2 n 1 dr r -α-κ 2 n 0 dξ |ξ| α-1 Γ t (-ξ, r) 2 n 1 dr r 2-α-κ 2 n 0 dξ |ξ| α-1 1 |ξ + r| 2 + t ε |ξ + r||ξ -r| 1-ε ∞ 1 dr r 2-κ-ε ∞ 0 dξ |ξ| α-1 1 |ξ + 1| 2 + t ε |ξ + 1||ξ -1| 1-ε .
The latter integrals being finite as soon as 0 < ε < min(1 -κ, α), this shows that J 3 n,t = O(1). Injecting this result, together with (36) and (39), into (34) yields the expected decomposition (33). Lemma 2.5. Given α ∈ (0, 2) and ε ∈ (0, 1), one has, for all r > 0,

r 2 0 dξ |1 -ξ r | α-1 1 -cos(ξ) ξ 2 - ∞ 0 dξ 1 -cos(ξ) ξ 2 ≤ c α,ε 1 r + 1 r 1-ε . Proof. Write the difference as r 2 0 dξ 1 |1 -ξ r | α-1 -1 1 -cos(ξ) ξ 2 + ∞ r 2 dξ 1 -cos(ξ) ξ 2 .
Then observe that

∞ r 2 dξ 1 -cos(ξ) ξ 2 1 r 1-ε ∞ 0 dξ 1 -cos(ξ) |ξ| 1+ε and r 2 0 dξ 1 |1 -ξ r | α-1 -1 1 -cos(ξ) ξ 2 r 2 0 dξ ||r| α-1 -|r -ξ| α-1 | |r -ξ| α-1 1 -cos(ξ) ξ 2 r 2 0 dξ 1 -cos(ξ) |r -ξ||ξ| 1 + |log r| 1 {r>1} r ,
hence the conclusion.

Lemma 2.6. Given α ∈ (0, 2) and 0 < ε < min(α, 1), one has, for all t > 0, n ≥ 0 and 1 < r < 2 n ,

2 n -r 0 dξ |ξ + r| α-1 |ξ + 2r| 1 -cos(tξ) ξ 2 - 1 2r α ∞ 0 dξ 1 -cos(tξ) ξ 2 ≤ c α,ε t ε r 1+α-ε + t ε r α |2 n -r| 1-ε .
Proof. Let us decompose the difference as

2 n -r 0 dξ 1 |ξ + r| α-1 - 1 r α-1 1 -cos(tξ) |ξ + 2r|ξ 2 + 1 r α-1 2 n -r 0 dξ 1 |ξ + 2r| - 1 2r 1 -cos(tξ) ξ 2 - 1 2r α ∞ 2 n -r dξ 1 -cos(tξ) ξ 2 .
Then observe that

2 n -r 0 dξ 1 |ξ + r| α-1 - 1 r α-1 1 -cos(tξ) |ξ + 2r|ξ 2 1 r α ∞ 0 dξ |ξ + 2r| 1 -cos(tξ) |ξ| 1 r 1+α-ε ∞ 0 dξ 1 -cos(tξ) |ξ| 1+ε , and 1 r α-1 2 n -r 0 dξ 1 |ξ + 2r| - 1 2r 1 -cos(tξ) ξ 2 = 1 2r α 2 n -r 0 dξ ξ |ξ + 2r| 1 -cos(tξ) ξ 2 1 r 1+α-ε ∞ 0 dξ 1 -cos(tξ) |ξ| 1+ε .
Finally, one has of course

∞ 2 n -r dξ 1 -cos(tξ) ξ 2 1 |2 n -r| 1-ε ∞ 0 dξ 1 -cos(tξ) |ξ| 1+ε .

Study of the (deterministic) auxiliary equation

Let us now turn to the analysis of the deterministic equation associated with our quadratic model [START_REF] Balan | The Stochastic Wave Equation with Multiplicative Fractional Noise: A Malliavin Calculus Approach[END_REF], that is the equation

∂ 2 t v -∆v + ρ 2 v 2 + (ρv) • Π 1 + Π 2 = 0 , t ∈ [0, T ] , x ∈ R d , v(0, .) = φ 0 , ∂ t v(0, .) = φ 1 , ( 40
)
where Π 1 , Π 2 are two (fixed) elements living in appropriate Sobolev spaces. We are actually interested in the exhibition of a unique (local) mild solution to (40), which will be achieved by means of a standard fixed-point argument. In other words, for fixed Π (Π 1 , Π 2 ) and T > 0, we will focus on the study of the map Γ T,Π defined as

Γ T,Π (v) t ∂ t (G t * x φ 0 ) + G t * x φ 1 + (G * t,x [ρ 2 v 2 + Π 1 • ρv + Π 2 ]) t , (41)
where G stands for the Green function of the standard d-dimensional wave equation. Putting the fixed components aside, this map is thus essentially built upon two successive operations: multiplication of v with itself or with Π 1 , and convolution with G. Accordingly, before we specify the space in which we will study Γ T,Π , let us recall a few general results on pointwise multiplication and convolution with the wave kernel. If f ∈ W -α,p1 (R d ) and g ∈ W s,p2 (R d ), then f • g ∈ W -α,p (R d ) and f • g W -α,p f W -α,p 1 g W s,p 2 .

Regularization effect of the wave operator.

The wave kernel G is known to satisfy specific regularization properties in the scale of Sobolev spaces (see e.g. [START_REF] Ginibre | Generalized Strichartz inequalities for the wave equation[END_REF]). In the analysis of (40), we will only rely on the following controls.

E Π = (Π 1 , Π 2 ) ∈ L ∞ ([0, T ]; L ∞ (R d )) 2 :
Π 1 and Π 2 are both supported by a compact set D .

Thus, for the moment, Π 1 and Π 2 are merely (bounded and compactly-supported) functions. When going back to the stochastic model ( 1) and with the result of Proposition 1.2 in mind, this situation will later correspond to the "regular" case Then, for all T > 0, (φ 0 , φ

1 ) ∈ H 1 (R d ) × L 2 (R d ), Π 1 = (Π 1 1 , Π 2 1 ) ∈ E, Π 2 = (Π 1 2 , Π 2
2 ) ∈ E and v, v 1 , v 2 ∈ X(T ), the following bounds hold true: where the proportional constants only depend on s and the norm . is naturally defined as

N [Γ T,Π1 (v); X(T )] φ 0 H 1 + φ 1 L 2 + T 1 2 N [v; X(T )] 2 + T 1 2 Π 1 N [v; X(T )] + T
Π = Π E N [Π 1 ; L ∞ ([0, T ]; L ∞ (R d ))] + N [Π 2 ; L ∞ ([0, T ]; L ∞ (R d ))] .
By combining the two bounds (45) and (46), it is now easy to see that for any fixed Π ∈ E and any time T 0 > 0 small enough, the map Γ T0,Π : X(T 0 ) → X(T 0 ) is a contraction on a appropriate stable ball of X(T 0 ), which immediately yields the expected (local) well-posedness result:

φ 0 H 1 + φ 1 L 2 + N [ρ 2 v 2 ; L 2 ([0, T ]; L 2 (R d ))] + N [Π 1 1 • ρv; L 2 ([0, T ]; L 2 (R d ))] + N [Π 2 1 ; L 2 ([0, T ]; L 2 (R d ))] . ( 47
)
In order to bound the last three quantities, we will merely appeal to the elementary continuous embedding: for all compact domain D ⊂ R d , p 0 , p 1 ≥ 1 and s 0 ≥ s 1 , W s0,p0 (D) ⊂ W

Remark 1 . 8 .

 18 The consideration of the linear combination d i=0 Hi in the above splitting must be compared with the role of the linear combination 2H0 + d i=1 Hi in the study of the fractional heat equation (see e.g. [8, Theorem 1.2]

c r 2 1

 2 tr){cos(tr) -cos(tξ)} (ξ -r)(ξ + r) + -cos(t(ξ + r)) (ξ + r) 2 -cos(tr){cos(tr) -cos(tξ)} (ξ -r)(ξ + r)

Proposition 3 . 2 .

 32 Fix d ≥ 2. (i) For all 0 ≤ T ≤ 1, s > 0 and w ∈ L 2 ([0, T ]; H s-1 (R d )), it holds that N G * t,x w; L ∞ ([0, T ]; H s (R d )) N [w; L 2 ([0, T ]; H s-1 (R d ))] .(42)(ii) For all 0 ≤ T ≤ 1, s > 0 and (φ 0 , φ 1 ) ∈ H s (R d ) × H s-1

Proposition 3 . 3 .

 33 d i=0 H i > d -12 (along the splitting of Theorem 1.6 or Theorem 1.7). Fix d ∈ {2, 3} and setX(T ) L ∞ ([0, T ]; H 1 (R d )) .

1 2 Π 1 , 1 2N 1 2 Π 1 - 1 2 Π 2 N 1 2 Π 1 -

 111111211 (45) andN [Γ T,Π1 (v 1 ) -Γ T,Π2 (v 2 ); X(T )] T [v 1 -v 2 ; X(T )]{N [v 1 ; X(T )] + N [v 2 ; X(T )]} + T Π 2 N [v 1 ; X(T )] + T [v 1 -v 2 ; X(T )] + T Π 2 , (46)

Corollary 3 . 4 .

 34 Under the assumptions of Proposition 3.3, and for all (fixed) (φ 0 , φ 1 ) ∈ H 1 (R d ) × L 2 (R d ), Π = (Π 1 , Π 2 ) ∈ E,there exists a time T 0 > 0 such that Equation (40) admits a unique solution in X(T 0 ). Proof of Proposition 3.3. Using the estimates in Proposition 3.2, we obtain first N [Γ T,Π1 (v); X(T )]

  falls within the class of standard hyperbolic systems, for which a unique (global) solution Ψn is known to exist. Our first result now reads as follows: Let d ≥ 1 and ρ : R d → R be a smooth compactly-supported function. Then, for every (H0, H1, . . . , H d ) ∈ (0, 1) d+1 , (ρΨn) n≥1 is a Cauchy sequence in the space L p

	Proposition 1.2.

  3.1. Pointwise multiplication. Recall that, with the results of Section 2 in mind, one of our purposes is to handle situations where the elements Π 1 , Π 2 involved in (41) are not functions but only distributions. Thus, even if we expect the solution v itself to be a function, we will need to control the (non-standard) multiplication of a function with a distribution at some point of the procedure. To this end, we can rely on the following general statement, borrowed from [23, Section 4.5.1]. Fix d ≥ 1. Let α, s > 0 and 1 ≤ p, p 1 , p 2 < ∞ be such that

	Proposition 3.1. 1 p	=	1 p 1	+	1 p 2	and 0 < α < s.

  d dξ {1 + |ξ| 2 } s F x G * t,x w (t, .) (ξ) dξ {1 + |ξ| 2 } s-1 |(F x w s )(ξ)| 2 , hence the conclusion. (ii) The bound for N [G t * x φ 1 ; L ∞ ([0, T ]; H s (R d ))] follows from similar arguments as above. Then, along the same idea, observe thatF x ∂ t (G . * x φ 0 )(t, .) (ξ) = ∂ t (F x G) t (ξ)(F x φ 0 )(ξ) = cos(t|ξ|)(F x φ 0 )(ξ) ,and so, for every t ∈ [0, T ],R 2 dξ {1 + |ξ| 2 } s F x ∂ t (G . * x φ 0 )(t, .) (ξ) 2 ≤ R 2 dξ {1 + |ξ| 2 } s |(F x φ 0 )(ξ)| 2 .3.3. First situation. Let us first consider the situation where the pair Π = (Π 1 , Π 2 ) involved in (40) (or in (41)) belongs to the space

								(44)
	Proof. Both (42) and (43) lean on elementary estimates.
	(i) Observe first that
	F 2	≤	0	t	ds	R 2	dξ {1 + |ξ| 2 } s sin 2 ((t -s)|ξ|) |ξ| 2	|(F x w s )(ξ)| 2
				t
					ds
			0			R d

(R d ), it holds that N [∂ t (G t * x φ 0 ); L ∞ ([0, T ]; H s (R d ))] φ 0 H s (R d )

(43)

and

N [G t * x φ 1 ; L ∞ ([0, T ]; H s (R d ))] φ 1 H s-1 (R d ) . x G * t,x w (t, .) (ξ) = t 0 ds F x G t-s (ξ) F x w s (ξ) ,

so that, for every t ∈ [0, T ], R

Institut Élie Cartan, Université de Lorraine, BP 70239, 54506 Vandoeuvre-lès-Nancy, France. Email: aurelien.deya@univ-lorraine.fr

Bound on N [ρ 2 v 2 ; L 2 ([0, T ]; L 2 (R d ))]. As ρ is supported by some fixed compact domain D, one has

Since d ∈ {2, 3}, we can assert, by (48), that H 1 (D) ⊂ L 4 (D), and thus

] , and then

1 is compactly-supported, we immediately have

Injecting the above estimates into (47) provides us with (45). It is then clear that (46) can be derived from similar arguments: for instance,

3.4. Second situation. We now turn to the "irregular" case of our analysis, that will later correspond to item (ii) in Theorem 1.6 or Theorem 1.7. With the result of Proposition 1.4 in mind, we are thus led to consider the situation where the pair Π = (Π 1 , Π 2 ) in (41) belongs to the space

Π 1 and Π 2 are both supported by a compact set D , for some positive coefficient α and some integer p ≥ 2. In particular, Π 1 and Π 2 are now both regarded as distributions. Our main result in this setting reads as follows.

Proposition 3.5. Fix d ∈ {2, 3}, α ∈ (0, 1 4 ) and s ∈ (0, 1) such that

and let p > d be defined by the relation

Then, for all T > 0, (φ 0 , φ

2 ) ∈ E α,p and v, v 1 , v 2 ∈ X s (T ), the following bounds hold true:

and

where the proportional constants depending only on α, s, and where the norm . understood as

Remark 3.6. Let us briefly compare this result with the situation treated in [14, Proposition 3.5].

At the level of the process Ψ (and so at the level of Π in the above formulation), the situation in [14, Proposition 3.5] corresponds to taking α = ε, for ε > 0 as small as one wishes. The latter possibility allows the authors of [START_REF] Gubinelli | Renormalization of the two-dimensional stochastic non linear wave equations[END_REF] to consider a general non-linearity of order k in the model (instead of the quadratic non-linearity in (1)): morally, the condition 2α+s < 1 in (49) turns into kε+s < 1, which, by taking ε small enough, can indeed be satisfied. Our aim here, with the result of Proposition 1.4 in mind, is to handle situations where α may be close to 1 4 , which accounts for our restriction to a non-linearity of low order.

Just as in the previous section, we easily deduce from Proposition 3.5: Corollary 3.7. Under the assumptions of Proposition 3.5, and for all (fixed

there exists a time T 0 > 0 such that Equation (40) admits a unique solution in X s (T 0 ).

Proof of Proposition 3.5. Using the estimates in Proposition 3.2, we obtain first

Let us now bound the last three quantities separately.

]. Thanks to condition (49), and using the general embedding result (48), one can easily check that for r :

which allows us to write

Using (48), one can check that W -α,r (D) ⊂ H s-1 (D), and so

We are now in a position to apply Proposition 3.1, which yields

]. We know by (49) that s -1 < -2α, and so, since Π 2 1 is compactly supported, we get immediately

Proof of the main results

It remains us to combine the (stochastic) results of Section 2 with the (deterministic) results of Section 3 in order to derive the proof of our main theorems. 4.1. Proof of Theorem 1.6. Fix d ∈ {2, 3} and (φ 0 , φ 1 ) ∈ H 1 (R d ) × L 2 (R d ). Now consider the two situations of the statement:

2 and then p large enough so that the continuous embedding W λ,p (D) ⊂ L ∞ (D) holds true. By Proposition 1.2, this puts us in a position to apply Corollary 3.4 (almost surely) with Π 1 2ρΨ, Π 2 ρ 2 Ψ 2 . The result immediately follows. 1 4 , so that, using Proposition 1.2 and Proposition 1.4, one has, for every

The result now follows from the application of Corollary 3.7 with Π 1 2ρΨ, Π 2 ρ 2 Ψ 2 and s = 1 2 .

4.2. Proof of Theorem 1.7. Fix d ∈ {2, 3} and (φ 0 , φ 1 ) ∈ H 1 (R d ) × L 2 (R d ). Now consider the two situations of the statement:

(i) Let (u n ) be the sequence of classical solutions of ( 14) and set v n u n -Ψ n , so that for each fixed n ≥ 1, v n clearly satisfies Equation (40) with

n . We can thus apply (45) and assert that for every T > 0,

) where the proportional constant only depends on ρ. Besides, using a standard Sobolev embedding, we know that for all λ > 0 and p large enough (depending on λ),

) , (54) and so, using Proposition 1.2, we get that for a subsequence of (ρΨ n ) (that we still denote by (ρΨ n )), Π n -Π → 0 almost surely. In particular, sup n Π n < ∞ a.s. Going back to (53) and setting f n (T ) N [v n ; X(T )], we deduce that for all 0 < T 0 ≤ 1 and 0 < T ≤ T 0 , 2 , for some (random) constant C 1 ≥ 1, and where we have set A

, in such a way that for every 0 < T 0 ≤ T 1 , the equation C 1 T 1/2 0 x 2 -x + C 1 A = 0 admits two solutions x 1,T0 , x 2,T0 satisfying 0 < x 1,T0 < x 2,T0 . As a result, for all 0 < T 0 ≤ inf(1, T 1 ) and 0 < T ≤ T 0 , one has either f n (T ) ≤ x 1,T0 or f n (T ) ≥ x 2,T0 . In fact, due to the continuity of T → f n (T ) (a straighforward consequence of the regularity of v n ), one has either sup T ∈[0,T0] f n (T ) ≤ x 1,T0 or inf T ∈[0,T0] f n (T ) ≥ x 2,T0 . Moreover, if we define T 2 > 0 as the largest time such that (sup 0≤T ≤T2 2C 1 T 1/2 φ 0 H1 ) ≤ 1, it can be explicitly checked that for every 0 < T 0 ≤ inf(1, T 1 , T 2 ), f n (0) = φ 0 H 1 ≤ x 1,T0 , and we are therefore in a position to assert that for such a time T 0 (that we fix from now on),

By injecting this uniform bound into (46), we easily derive that, for some time 0

where the proportional constant is also uniform in n.