Nicolas Schaeffer

Study of a fractional stochastic heat equation

In this article, we study a d-dimensional stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional space-time white noise:

Two types of regimes are exhibited, depending on the ranges of the Hurst index H = (H 0 , ..., H d ) ∈ (0, 1) d+1 . In particular, we show that the local well-posedness of (SNLH) resulting from the Da Prato-Debussche trick, is easily obtained when 2H 0 + d i=1 H i > d. On the contrary, (SNLH) is much more difficult to handle when 2H 0 + d i=1 H i ≤ d. In this case, the model has to be interpreted in the Wick sense, thanks to a time-dependent renormalization. Helped with the regularising effect of the heat semigroup, we establish local well-posedness results for (SNLH) for all dimension d ≥ 1.

Introduction and main results

1.1. General introduction. This paper is devoted to the study of a heat equation with a quadratic nonlinearity, driven by an additive fractional space-time white noise forcing. More precisely, we consider the following nonlinear stochastic heat model:

∂ t u -∆u = ρ 2 u 2 + Ḃ , t ∈ [0, T ] , x ∈ R d , u(0, .) = φ , (1.1)
where φ is a (deterministic) initial condition living in some appropriate Sobolev space and ρ : R d → R is a smooth compactly-supported function fixed once and for all. Before going any further, let us specify the roughness of the stochastic term, namely, Ḃ := ∂ t ∂

d i=1 R H i (x i , y i ) , s, t ∈ [0, T ] , x, y ∈ R d ,
where

R H i (x, y) := 1 2 (|x| 2H i + |y| 2H i -|x -y| 2H i ) .
Stochastic partial differential equations (SPDEs) perturbed by fractional noise terms have become increasingly popular in the recent years. Indeed, these equations can be interpreted as a more flexible alternative to classical SPDEs driven by a space-time white noise, and consequently they can be used to model more complex physical phenomena which are subjects to random perturbations. For instance, fractional noises appear in lots of practical situations, whether in biophysics (see [START_REF] Kou | Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule[END_REF]), in the study of financial time series (see [START_REF] Bayraktar | Estimating the fractal dimension of the SP 500 index using wavelet analysis[END_REF]) or simply in electrical engineering (see [START_REF] Denk | Modeling, simulation and optimization of integrated circuits[END_REF]).

The major motivation of this work is that equation (1.1) is both a physical and mathematically challenging model. In fact, it can be seen as a stochastic reaction diffusion model with p = 2. Recall that the equation:

∂ t u = ∆u + u p , p > 1 (1.2)
is known under the name of Fujita model and has been the subject of many questions. It has already been established (see [START_REF] Brezis | A nonlinear heat equation with singular initial data[END_REF][START_REF] Weissler | Local existence and nonexistence for semilinear parabolic equations in L p[END_REF]) that when φ ∈ L q (R d ) with q ≥ 1 and q > d(p-1)

2

, there exists a constant T = T (φ) > 0 and a unique function u ∈ C([0, T ], L q (R d )) that is a classical solution to (1.1) on [0, T ] × R d whereas when q < d(p-1)

2

, there is no general theory of existence. In this work, we introduce a cut-off function ρ to bring back computations to compact domains, a fractional random perturbation Ḃ and we are interested in the well-posedness of the associated equation (1.1).

Two classical obstacles have to be taken over. First of all, as often in the theory of SPDEs, a standard problem is the roughness of the fractional noise which prevents us from solving directly the associated linear part of the equation, namely:

∂ t -∆ = Ḃ, t ∈ [0, T ], x ∈ R d , (0, .) = 0. (1.3)
Then, the quadratic nonlinearity will require us to give a precise meaning to the notion of solution. Before going into details, let us recall that, over the last decade, we have seen numerous developments in the study of singular stochastic PDEs, in particular in the parabolic setting. Remember that there are three main components to consider: the space dimension d, the nature of the nonlinearity and the irregularity of the noise. For example, polynomial nonlinearities [8, Section 3] have been progressively replaced by sinusoidal and exponential ones. As an application of the theory of regularity structures, Hairer-Shen [START_REF] Hairer | The dynamical sine-Gordon model[END_REF] and Chandra-Hairer-Shen [START_REF] Chandra | The dynamical sine-Gordon model in the full subcritical regime[END_REF] studied the following parabolic sine-Gordon model on T 2 :

∂ t u = 1 2
∆u + sin(βu) + ξ, (1.4) where ξ is a space-time white noise and proved that the local well-posedness depends essentially on the value of the key parameter β 2 > 0. Oh, Robert and Wand [START_REF] Oh | On the parabolic and hyperbolic Liouville equations[END_REF] focused on the parabolic equation in dimension 2:

∂ t u + 1 2 (1 -∆)u + 1 2 λβe βu = ξ, ( 1.5)
where β, λ ∈ R\{0} and they proved that the local well-posedness depends again sensitively on the value of β 2 > 0 as well as the sign of λ. To end with, resorting to a paracontrolled approach, Oh and Okamoto [START_REF] Oh | Comparing the stochastic nonlinear wave and heat equations: a case study[END_REF] worked on a stochastic heat equation with a quadratic nonlinearity but forced by a more irregular noise:

∂ t u + (1 -∆)u + u 2 = ∇ α ξ, (1.6)
where ∇ α ξ denotes a α-derivative (in space) of a (Gaussian) space-time white noise on T 2 ×R + . Our objective is to go one step further that is to study a stochastic heat equation on R d (instead of T d ) driven by the derivative (in space and in time) of a space-time fractional Brownian motion for d ≥ 1. Helped with the regularising effect of the heat semigroup which allows a gain of two derivatives in the fractional Sobolev setting, we will be able to derive existence results about (1.1).

Our results can be summed up in the following way:

(i) Case α H > 0. The equation (1.1) is almost surely locally well-posed in W β,p (R d ) for some β > 0 and p ≥ 2. (ii) Case α H ≤ 0. If α H > -1
4 , then the equation (1.1) can be interpreted in the Wick (renormalized) meaning and it is almost surely locally well-posed in W -β,p (R d ) for some β > 0 and p ≥ 2.

Let us now come back to the analysis of equation (1.1). We first rewrite it under the mild formulation, that is the (formal) equation

u t = e t∆ φ + t 0 e (t-τ )∆ (ρ 2 u 2 τ )dτ + t (1.7)
where e t∆ stands for the heat semigroup, and (morally) t := t 0 e (t-τ )∆ ( Ḃτ )dτ. Here, we adopt Hairer's convention (see [START_REF] Hairer | A theory of regularity structures[END_REF]) to denote the stochastic terms by trees; the vertex in represents the random noise Ḃ and the edge corresponds to the Duhamel integral operator:

I = (∂ t -∆) -1 .
Now, in order to isolate the more irregular term , we resort to the so-called Da Prato-Debussche trick to rewrite the equation under the form:

v t = e t∆ φ + t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ + 2 t 0 e (t-τ )∆ ((ρv τ ) • (ρ τ )) dτ + t 0 e (t-τ )∆ (ρ 2 2 τ ) dτ , t ∈ [0, T ] , (1.8)
where v := u -. Our first and main objective is to define properly the solution related to the linear equation:

∂ t -∆ = Ḃ, t ∈ [0, T ], x ∈ R d , (0, .) = 0. (1.9)
Our strategy is to consider ( Ḃn ) n≥0 a smooth approximation of Ḃ and to show that the associated solutions ( n ) n≥0 verifying

∂ t n -∆ n = Ḃn , t ∈ [0, T ] , x ∈ R d , n (0, .) = 0 , (1.10)
form a Cauchy sequence in a convenient subspace (see Section 4 for more details). This leads us to the construction of the first order stochastic process .

(χ n ) n≥0 converges in the space C([0, T ]; W -α,p (R d )) as soon as 2 ≤ p ≤ ∞ and α > d -2H 0 + d i=1 H i = -α H . (1.11)
The limit of this sequence will be denoted by χ .

It is now clear that the sign of α H will play a major role in the resolution of (1.1). Indeed, when α H > 0, χ (t) will be seen as a function defined on R d whereas, when α H < 0, χ (t) will be seen as a spatial distribution. It is important to remark that the precise nature of is known up to multiplication by χ ∈ C ∞ c (R d ). To put it differently, the stochastic process (t) is (almost surely) only a distribution on R d but we have the more refined result : χ ∈ C([0, T ]; W -α,p (R d )) (see section 4.6 for more details). In the study of our equation, we will thus take χ = ρ.

The regular case.

In the following, we will call regular case the situation where

2H 0 + d i=1 H i > d . (H1)
When this hypothesis is realized, α < 0 will be picked such that condition (1.11) is satisfied, and therefore, resorting to Proposition 1.3, every element χ (χ ∈ C ∞ c (R d )) will be considered as a function of both time and space variables (with probability one) and consequently ρ 2 2 (t) will perfectly make sense as a classical function defined on R d . We thus propose the following natural interpretation of the model: Definition 1.4. Let d ≥ 1 and suppose that condition (H1) is verified. Remember that for all test function χ : R d → R, χ is the process provided by Proposition 1.3.

A stochastic process (u(t, x)) t∈[0,T ],x∈R d is said to be a solution (on [0, T ]) of equation (1.1) if, almost surely, the process v := uis a solution of the mild equation

v t = e t∆ φ + t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ + 2 t 0 e (t-τ )∆ ((ρv τ ) • (ρ τ )) dτ + t 0 e (t-τ )∆ (ρ 2 2 τ ) dτ , t ∈ [0, T ] . (1.12)
We are now in a position to state our first existence result which will be a consequence of a standard fixed-point theorem: Theorem 1.5 (Local well-posedness under (H1)). Suppose that d ≥ 1 and that condition (H1) is satisfied. Let p ≥ 2 and β be such that 0

< β < 2H 0 + d i=1 H i -d and d 2p < 1 + β 2 . Finally, assume that φ ∈ W β,p (R d ).
In this case, the assertions below hold true: (i) Almost surely, there exists a time T 0 (ω) > 0 such that equation (1.1) admits a unique solution u (in the meaning of Definition 1.4) in the subset

S T 0 := + X β,p (T 0 ), where X β,p (T 0 ) := C([0, T 0 ]; W β,p (R d )).
(ii) For any n ≥ 1, let us note u n the smooth solution of (1.1), that is u n is the solution (in the meaning of Definition 1.4) related to ρ n . Then, for all

0 < β < 2H 0 + d i=1 H i -d and for any test function χ : R d → R, the sequence (χu n ) n≥1 converges almost surely in C([0, T 0 ]; W β,p (R d ))
to χu, where u is the solution from item (i).

The rough case.

Let us now focus on the second situation, where

2H 0 + d i=1 H i ≤ d . (1.13)
In this situation, ρ (t) is only a distribution and not a function anymore. So the classical difficulty to properly define ρ 2 2 (t) appears. A Wick-renormalization procedure permits to overcome this issue. To begin with, let us introduce the Wick-renormalized product

n (t, x) := n (t, x) 2 -σ n (t, x) where σ n (t, x) := E n (t, x) 2 . (1.14)
Before looking for a convenient subspace in which (ρ 2 n ) n≥0 would be a Cauchy sequence, let us have a look at the renormalization constant. 2 , the asymptotic equivalence property below is obtained: for every

Proposition 1.6. Let d ≥ 1 and suppose that 2H 0 + d i=1 H i ≤ d. Then, the value of E n (t, x) 2 does not depend on x. Setting σ n (t) := E n (t, x)
(t, x) ∈ (0, T ] × R d , σ n (t, x) := E n (t, x) 2 ∼ n→∞ c 1 H n if 2H 0 + d i=1 H i = d, c 2 H 2 2n(d-[2H 0 + d i=1 H i ]) if 2H 0 + d i=1 H i < d , (1.15)
where c 1 H , c 2 H > 0 are two constants. Remark 1.7. It is interesting to note that the nature of the equivalent (that is linear when

2H 0 + d i=1 H i = d and geometric when 2H 0 + d i=1 H i < d)
is the same as in the wave setting (see [START_REF] Deya | A nonlinear wave equation with fractional perturbation[END_REF]) and in the Schrödinger setting (see [START_REF] Deya | A nonlinear Schrödinger equation with fractional noise[END_REF]). Let us add that, in this case, the term in the right hand-side of (1.15) has no dependence on t. In fact, this results from a dissipative phenomenon of the heat equation (see the proof of Proposition 4.4) contrary to an oscillating one in the case of the Schrödinger equation (see [START_REF] Deya | A nonlinear Schrödinger equation with fractional noise[END_REF]). This kind of phenomenon is well-known in the parabolic setting and has already been observed in [START_REF] Hoshino | KPZ equation with fractional derivatives of white noise[END_REF] for instance in its study of the KPZ equation with fractional derivatives of white noise.

Our definition of , the second order stochastic term, is the following:

Proposition 1.8. Fix d ≥ 1 and (H 0 , H 1 , ..., H d ) ∈ (0, 1) d+1 such that d - 1 4 < 2H 0 + d i=1 H i ≤ d . (H2)
Then, for all test function χ : R d → R, almost surely the sequence (χ 2 n ) n≥1 (defined by (1.14)) converges in the space C([0, T ]; W -2α,p (R d )) as soon as 2 ≤ p ≤ ∞ and α satisfying (1.11).

The limit of this sequence will be denoted by χ 2 .

With the above constructed stochastic processes, the following Wick interpretation of the model naturally comes to mind:

v t = e t∆ φ + t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ + 2 t 0 e (t-τ )∆ ((ρv τ ) • (ρ τ )) dτ + t 0 e (t-τ )∆ (ρ 2 τ ) dτ , t ∈ [0, T ] . (1.16)
In this case, a major difficulty is the treatment of the term (ρv) • (ρ ) that will be understood as the product of a distribution ρ with a regular enough function ρv. Indeed, should ρ be of Sobolev regularity -α and ρv be a function of Sobolev regularity β with β > α, (ρv) • (ρ ) can be defined as a distribution of order -α (see Section 3 for more details). Again, the regularising effect of the heat semigroup will help us to find a stable subspace. The main result of this paper can be stated in the following way: Theorem 1.10 (Local well-posedness under (H2)). Suppose that d ≥ 1 and p ≥ 2 verifies that p ≥ 2d 3 . Assume that d -

1 4 < 2H 0 + d i=1 H i ≤ d. Fix α > 0 such that d -2H 0 + d i=1 H i < α < 1 4 .
(1.17)

Then the assertions below hold true: (i) One can find β > 0 such that almost surely, for every φ ∈ W -α,p (R d ), there exists a time T 0 (ω) > 0 for which equation (1.1) admits a unique Wick solution u (in the meaning of Definition 1.9) in the set

S T 0 := + X α,β (T 0 ) , where X α,β (T 0 ) := C([0, T 0 ]; W -α,p (R d )) ∩ C((0, T 0 ]; W β,p (R d )).
(ii) For any n ≥ 1, let us note u n the smooth Wick solution of (1.1), that is u n is the solution (in the meaning of Definition 1.9) related to the pair (ρ n , ρ 2 n ). Then, for all α satisfying (1.17) and for any test function χ : R d → R, the sequence (χ u n ) n≥1 converges almost surely in C([0, T 0 ]; W -α,p (R d )) to χu, where u is the Wick solution from item (i).

Remark 1.11. Thanks to the regularising effect of the heat semigroup, we are able to solve equation (1.1) for all dimension d ≥ 1 as soon as α < 1 4 . This situation highly contrasts with the Schrödinger setting (see [START_REF] Deya | A nonlinear Schrödinger equation with fractional noise[END_REF]) where the restriction comes from the deterministic part of the equation. Nonetheless, although the restriction in Theorem 1.10 comes from the stochastic constructions, we will be able to improve this result in dimension d = 2. See subsection 1.1.3 below.

Remark 1.12. Let us go back to the role of the cut-off function ρ in equation (1.1). Contrary to the Schrödinger setting where ρ played an essential role to obtain local regularity, in our case, ρ is only needed to measure precisely the regularity of the stochastic processes ρ (t, .), ρ 2 (t, .) in some W α,p (R d ) spaces.

2H 0 + H 1 + • • • + H d ≤ 3 4 d (1.18)
and let χ : R d → R be a non-zero, smooth and compactly-supported function. Then, for all α > 0 and t > 0, one has

E χ • n (t, .) 2 H -2α (R d ) → ∞ as n → ∞.
Based on the latter explosion phenomenon for n , we can at least conclude that the condition (H2) in Proposition 1.8 and Theorem 1.10 is optimal when d = 1.

When d ≥ 2, one can observe that the situation where

3 4 d < 2H 0 + d i=1 H i ≤ d -1 4
is not covered by the above results. In fact, we must admit that we are not able, for the moment, to offer a complete picture of the situation, that is a general well-posedness result for every d ≥ 2 and for all indexes such that 2H 0 + d i=1 H i > 3 4 d. However, we propose to make a first step toward this ambitious challenge by going further with the analysis in the two-dimensional setting d = 2. In this case, it turns out that several of our arguments and estimates behind the construction of can be slightly refined, leading us to the following statement: Proposition 1.14. Let (H 0 , H 1 , H 2 ) ∈ (0, 1) 3 be such that

0 < H 1 < 3 4 , 0 < H 2 < 3 4 , 3 2 < 2H 0 + H 1 + H 2 ≤ 7 4 . (1.19)
Then, for all smooth compactly-supported function χ :

R 2 → R, T > 0, 2 ≤ p ≤ ∞ and α > 2 -(2H 0 + H 1 + H 2 ) , (1.20)
the sequence (χ 2 n ) n≥1 (defined by (1.14)) converges almost surely in the space C([0, T ]; W -2α,p (R 2 )).

We still denote the limit of this sequence by χ 2 .

According to Proposition 1.13 with d = 2, this construction is optimal. We are now in a position to state our more general result when d = 2:

Theorem 1.15 (Roughest case). Suppose that d = 2 and p ≥ 2. Let (H 0 , H 1 , H 2 ) ∈ (0, 1) 3 be such that 0 < H 1 < 3 4 , 0 < H 2 < 3 4 , 3 2 < 2H 0 + H 1 + H 2 ≤ 7 4 . (1.21) Fix α > 0 such that 2 -(2H 0 + H 1 + H 2 ) < α < 1 2 . (1.22)
Then the assertions below hold true: (i) One can find β > 0 such that almost surely, for every φ ∈ W -α,p (R 2 ), there exists a time T 0 (ω) > 0 for which equation (1.1) admits a unique Wick solution u (in the meaning of Definition 1.9) in the set

S T 0 := + X α,β (T 0 ) , where X α,β (T 0 ) := C([0, T 0 ]; W -α,p (R 2 )) ∩ C((0, T 0 ]; W β,p (R 2 )).
(ii) For any n ≥ 1, let us note u n the smooth Wick solution of (1.1), that is u n is the solution (in the meaning of Definition 1.9) related to the pair (ρ n , ρ 2 n ). Then, for all α satisfying (1.22) and for any test function χ : R 2 → R, the sequence

(χ u n ) n≥1 converges almost surely in C([0, T 0 ]; W -α,p (R 2
)) to χu, where u is the Wick solution from item (i).

Remark 1.16. Again, the restriction comes from the stochastic part of the study (see Proposition 1.14) that tackles with the existence of the stochastic element . This situation is analogous to those in the wave setting (see for example [START_REF] Deya | A nonlinear wave equation with fractional perturbation[END_REF][START_REF] Gubinelli | Renormalization of the two-dimensional stochastic nonlinear wave equation[END_REF][START_REF] Gubinelli | Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity[END_REF][START_REF] Oh | Comparing the stochastic nonlinear wave and heat equations: a case study[END_REF]), where the strategy is limited by the domain of validity of the stochastic constructions.

The above well-posedness theorem can be viewed as the main novelty of our work. We can sum up our results in the following way ; equation

(1.1) is well-posed in some W α,p (R 2 ) space when 3 2 < 2H 0 + H 1 + H 2
and is ill-posed as soon as

2H 0 + H 1 + H 2 ≤ 3 2
in the sense that the second order stochastic term is not a continuous function of time with values in Sobolev spaces. These results can be viewed as a generalization of those obtained by Oh and Okamoto in [START_REF] Oh | Comparing the stochastic nonlinear wave and heat equations: a case study[END_REF] where the parameter α (related to the derivatives of their Gaussian noise) has to satisfy analogous constraints to those of the combination 2H 0 + H 1 + H 2 .

Remark 1.17. It is interesting to compare the above regularity restriction (1.11) for in Proposition 1.3 with the analogous results of [START_REF] Deya | A nonlinear wave equation with fractional perturbation[END_REF] in the wave setting and of [START_REF] Deya | A nonlinear Schrödinger equation with fractional noise[END_REF] in the Schrödinger setting, that is when replacing ∂ t -∆ with ∂ 2 t -∆ (respectively ı∂ t -∆ ) . In these situations, the conditions on α are

α wave > d - 1 2 - d i=0 H i and α schröd > d + 1 -2H 0 + d i=1 H i .
In fact, the method presented in this paper would allow us to define the linear solution of several other fractional problems. Let us briefly describe some results that could be obtained along the same procedure (see Section 4). First of all, for fractional Schrödinger-type equations of the form: i∂ t (t, x) + |∇| σ (t, x) = 0, where |∇| σ is the Fourier multiplier by |ξ| σ and σ ∈ (0, +∞), that generalize the classical one (take σ = 2), one could prove that the associated condition on α becomes

α frac schröd > d + σ 2 -σH 0 + d i=1 H i .
Likewise, we could construct the linear solution for fractional wave-type equations of the form:

∂ 2 t (t, x) + |∇| 2σ (t,
x) = 0, that generalize the standard one (take σ = 1). In this case, the associated condition on α would be

α frac wave > d - σ 2 -σH 0 + d i=1 H i .
To end with, it is possible to construct the first order stochastic process for fractional heat-type equations of the form:

∂ t (t, x) + |∇| σ (t, x) = 0,
as soon as

α frac heat > d -σH 0 + d i=1 H i .
These considerations allow us to understand that the key parameter in the construction of the linear solution is the exponent of the Fourier multiplier that appears in front of H 0 (that described the irregularity of B with respect to the time t) whereas the combination d i=1 H i stays unchanged. The main functional spaces of this article are the Sobolev spaces defined for all s ∈ R and 1 ≤ p ≤ ∞ as

W s,p (R d ) := f ∈ S ′ (R d ) : f W s,p = F -1 ({1 + |.| 2 } s 2 Ff )|L p (R d ) < ∞ ,
where the Fourier transform F, resp. the inverse Fourier transform F -1 , is defined with the following convention: for every f ∈ S(R d ) and x ∈ R d ,

F(f )(x) = f (x) := R d f (y)e -ı x,y dy , resp. F -1 (f )(x) := 1 (2π) d R d
f (y)e ı x,y dy .

We also set

H s (R d ) := W s,2 (R d ).
Now, as soon as space-time functions (or distributions) are concerned, we may use the following shortcut notation: for every T ≥ 0, 1 ≤ p, q ≤ ∞ and s ∈ R,

L p T W s,q := L p ([0, T ]; W s,q (R d )) , . L p T W s,q := . L p ([0,T ];W s,q (R d )) . (1.23)
The notation C([0, T ]; W s,q (R d )) will refer to the set of continuous functions on [0, T ] taking values in W s,q (R d ).

Outline of the study.

The rest of the article is organized as follows. In Section 2, we prove the local well-posedness result in the regular case. Section 3 is devoted to the analysis in the irregular case. Then, in Section 4 and Section 5, we develop the stochastic constructions at the core of the equation. To end with, in Section 6, we combined the results from the previous sections to prove Theorem 1.5, Theorem 1.10 and Theorem 1.15.

Throughout the paper, the notation A B will be used to signify that there exists an irrelevant constant c such that A ≤ cB.

Analysis of the deterministic equation under condition (H1)

The aim of this section is to deal with the model in the regular case, that is when assumption (H1) on the Hurst index is verified, and the linear solution ρ (defined by Proposition 1.3) is a function of both time and space. Recall that in this situation, the equation is interpreted through Definition 1.4. Thus, what we would like to solve in this section is the equation

v t = e t∆ φ + t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ + 2 t 0 e (t-τ )∆ (ρv τ • ρ τ ) dτ + t 0 e (t-τ )∆ (ρ 2 2 τ ) dτ , t ∈ [0, T ] . (2.1)
Contrary to the next sections where several probabilistic arguments will be used, our strategy towards a (local) solution v for (2.1) will be based on deterministic estimates only. To put it differently, we henceforth consider ρ as a fixed (i.e., non-random) element in the space C([0, T ]; W β,p (R d )) for some appropriate 0 < β < 1 (where β = -α is given by Proposition 1.3) and p ≥ 2, and try to solve the deterministic equation below: for

Ψ ∈ C([0, T ]; W β,p (R d )), v t = e t∆ φ + t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ + 2 t 0 e (t-τ )∆ (ρv τ • Ψ τ ) dτ + t 0 e (t-τ )∆ (Ψ 2 τ ) dτ , t ∈ [0, T ] . (2.2)
Let us present the tools related to the two main operations in (2.2).

2.1. W α,p -W α,q estimate. We recall the well-known integrability property of the heat semigroup resulting from the Riesz-Thorin theorem.

Proposition 2.1. Let 1 ≤ q ≤ p ≤ ∞, α ∈ R and u 0 ∈ W α,q (R d ).
Then for all t > 0,

e t∆ u 0 W α,p ≤ (4πt) -d 2 ( 1 q -1 p ) u 0 W α,q .

Pointwise multiplication.

The second tool to deal with equation (2.2) is a refined fractional Leibniz rule that can be found in [START_REF] Grafakos | The Kato-Ponce inequality[END_REF].

Lemma 2.2 (Kato-Ponce inequality). Let 1 ≤ r < ∞ and 1 < p 1 , p 2 , q 1 , q 2 < ∞ verifying 1 r = 1 p 1 + 1 p 2 = 1 q 1 + 1 q 2 .
Let s ≥ 0. Then one has

u • v W s,r (R d ) u W s,p 1 (R d ) v L p 2 (R d ) + u L q 1 (R d ) v W s,q 2 (R d ) .

About the resolution of the deterministic equation.

Our main result regarding equation (2.2) can be formulated in the following way:

Theorem 2.3. Fix β ∈ (0, 1). Assume that d ≥ 1 and p ≥ 2 is such that d 2p < 1 + β 2 .
For every T > 0, define the space X β,p (T ) as

X β,p (T ) := C([0, T ]; W β,p (R d )) , (2.3) equipped with the norm v X(T ) := v X β,p (T ) = v L ∞ T W β,p .
Then for every φ ∈ W β,p (R d ), one can find a time T 0 > 0 such that, for every Ψ ∈ X β,p (T 0 ), equation (2.2) admits a unique solution in X β,p (T 0 ). This local well-posedness result will be the consequence of a classical fixed-point argument. In order to implement this procedure, let us introduce the map Γ defined by the right-hand side of (2.2), that is: for all

φ ∈ W β,p (R d ), v, Ψ ∈ X β,p (T ), T ≥ 0, set Γ T,Ψ (v) t := e t∆ φ + t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ + 2 t 0 e (t-τ )∆ (ρv τ • Ψ τ ) dτ + t 0 e (t-τ )∆ (Ψ 2 τ ) dτ , t ∈ [0, T ] .
Proposition 2.4. In the setting of Theorem 2.3, the bounds below hold true: one can find ε > 0 such that for all

0 ≤ T ≤ 1, φ ∈ W β,p (R d ), Ψ 1 , Ψ 2 , v, v 1 , v 2 ∈ X β,p (T ) := X(T ), Γ T,Ψ 1 (v) X(T ) φ W β,p (R d ) + T ε v 2 X(T ) + Ψ 1 X(T ) v X(T ) + Ψ 1 2 X(T ) , (2.4)
and

Γ T,Ψ 1 (v 1 ) -Γ T,Ψ 2 (v 2 ) X(T ) T ε v 1 -v 2 X(T ) v 1 X(T ) + v 2 X(T ) + Ψ 1 -Ψ 2 X(T ) v 1 X(T ) + Ψ 2 X(T ) v 1 -v 2 X(T ) + Ψ 1 -Ψ 2 X(T ) Ψ 1 X(T ) + Ψ 2 X(T ) , (2.5)
where the proportional constants only depend on β, p and ρ.

Before we focus on the proof of this proposition, let us briefly remember that, once endowed (2.4)-(2.5), the result of Theorem 2.3 follows from a classical application of the Picard fixed-point theorem. Precisely, using (2.4), we can first show that for all T = T (φ, Ψ) > 0 small enough, there exists a ball in X(T ) that is stable through the action of Γ T,Ψ . Then, helped with (2.5) (used with Ψ 1 = Ψ 2 = Ψ), the fact that Γ T,Ψ is actually a contraction on this ball is easily deduced (for T > 0 possibly even smaller), which completes the proof of the theorem.

Let us remark that the continuity of Γ T,Ψ with respect to Ψ (a direct consequence of (2.5)) will be a major ingredient toward item (ii) of Theorem 1.5.

We begin by establishing some estimates.

Lemma 2.5. Fix β ∈ (0, 1). Assume that d ≥ 1 and p ≥ 2 is such that d 2p < 1 + β 2 . There exists ε > 0 such that for all 0 ≤ T ≤ 1, f i = ρv or f i = ρ , t 0 e (t-τ )∆ (f 1 • f 2 (τ ))dτ X β,p (T ) T ε f 1 X β,p (T ) f 2 X β,p (T ) . Proof. Let 0 ≤ t ≤ T, 1 ≤ q < p and 1 ≤ r < ∞ be such that 1 q = 1 p + 1 r .
Using successively Minkowsky inequality, Proposition 2.1 and Lemma 2.2, we get

t 0 e (t-τ )∆ (f 1 • f 2 (τ ))dτ W β,p ≤ t 0 e (t-τ )∆ (f 1 • f 2 (τ )) W β,p dτ t 0 (t -τ ) -d 2 ( 1 q -1 p ) f 1 • f 2 (τ ) W β,q dτ t 0 (t -τ ) -d 2 ( 1 q -1 p ) [ f 1 (τ ) W β,p f 2 (τ ) L r + f 2 (τ ) W β,p f 1 (τ ) L r ]dτ. Now, as soon as 1 q > max 2 p -β d , 1 p ), observe that β ≥ d( 1 p -1 r
) and consequently we have the Sobolev embedding

W β,p (R d ) ֒→ L r (R d ) that leads to t 0 e (t-τ )∆ (f 1 • f 2 (τ ))dτ W β,p t 0 (t -τ ) -d 2r f 1 (τ ) W β,p f 2 (τ ) W β,p dτ T 1-d 2r f 1 X β,p (T ) f 2 X β,p (T ) T 1-d 2p + β 2 f 1 X β,p (T ) f 2 X β,p (T ) .
Taking the supremum over t ∈ [0, T ], we obtain the desired conclusion:

t 0 e (t-τ )∆ (f 1 • f 2 (τ ))dτ X β,p (T ) T 1-d 2p + β 2 f 1 X β,p (T ) f 2 X β,p (T ) .
Proof of Proposition 2.4. Let us bound each of the four terms in the expression of Γ T,Ψ :

Γ T,Ψ (v) t := e t∆ φ + t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ + 2 t 0 e (t-τ )∆ (ρv τ • Ψ τ ) dτ + t 0 e (t-τ )∆ (Ψ 2 τ ) dτ , t ∈ [0, T ] .
On the one hand, for all 0 ≤ t ≤ T , using the expression of the heat semigroup, it holds that

e t∆ φ W β,p (R d ) φ W β,p (R d ) , yielding to e t∆ φ X β,p (T ) φ W β,p (R d ) . (2.6)
On the other hand, the bound concerning the three other terms is a straight consequence from Lemma 2.5 and provides us with the bound (2.4) we are looking for. The second one (2.5) can be obtained with quite the same arguments.

Analysis of the deterministic equation under condition (H2)

The objective of this section is to cope with the wellposedness issue in the rough situation, that is when condition (H2) on the Hurst indexes is verified. We recall that in this rough case, the model is understood in the meaning of Definition 1.9, that is as

v t = e t∆ φ + t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ + 2 t 0 e (t-τ )∆ (ρv τ • ρ τ ) dτ + t 0 e (t-τ )∆ (ρ 2 τ ) dτ , t ∈ [0, T ] . (3.1)
where the processes ρ and ρ 2 are those defined in Proposition 1.3 and Proposition 1.8.

In order to handle (3.1), we intend to follow the same deterministic strategy as in Section 2. To put it differently, we will consider the pair (ρ , ρ 2 ) as a fixed element in the subspace

R α,p := L ∞ [0, T ]; W -α,p (R d ) × L ∞ [0, T ]; W -2α,p (R d ) , ( 3.2)
where 0 < α < 1 4 (coming from Propositions 1.3 and 1.8), p ≥ 2 and then the aim will be to solve the more general deterministic equation: for (Ψ,

Ψ 2 ) ∈ R α,p , v t = e t∆ φ + t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ + 2 t 0 e (t-τ )∆ (ρv τ • Ψ τ ) dτ + t 0 e (t-τ )∆ (Ψ 2 τ ) dτ , t ∈ [0, T ] . (3.3)
The main originality of the situation (compared to Section 2) is the lack of regularity of Ψ τ and Ψ 2 τ , that can only be seen as negative-order Sobolev terms (remark indeed that α > 0 in (3.2)).

Regularising effect of the heat semigroup.

The use of the fractional Sobolev spaces allows us to quantify the regularising effect of the heat semigroup (see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]).

Proposition 3.1. Let 1 < p < ∞. For any s 1 < s 2 , there exists a constant C(s 1 , s 2 ) < ∞ such that e t∆ φ W s 2 ,p (R d ) ≤ C(s 1 , s 2 )(1 + t -s 2 -s 1 2 ) φ W s 1 ,p (R d )
holds for all φ ∈ W s 1 ,p (R d ) and t > 0.

Pointwise multiplication.

The solution we propose to deal with the product (ρv τ ) • (Ψ τ ) in (3.3) will be to resort to the following general multiplication property in Sobolev spaces (see e.g. [START_REF] Runst | Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations[END_REF]Section 4.4.3] for a proof of this result) which garantees that (ρv τ ) • (Ψ τ ) exists and inherits the worst regularity. More precisely:

Lemma 3.2. Fix d ≥ 1. Let α, β > 0 and 1 ≤ p, p 1 , p 2 < ∞ be such that 1 p = 1 p 1 + 1 p 2 and 0 < α < β . If f ∈ W -α,p 1 (R d ) and g ∈ W β,p 2 (R d ), then f • g ∈ W -α,p (R d ) and f • g W -α,p f W -α,p 1 g W β,p 2 .

About the resolution of the auxiliary deterministic equation.

For all T ≥ 0, α, β > 0 and p ≥ 2, introduce the space

X α,β,p (T ) := C([0, T ]; W -α,p (R d )) ∩ C((0, T ]; W β,p (R d )) , ( 3.4)
equipped with the norm

v X(T ) := v X α,β,p (T ) = v L ∞ T W -α,p + v Y (T )
where the Y (T )-seminorm is given by

v Y (T ) = sup 0<t≤T t β+α 2 v(t) W β,p (R d ) .
Also, remember that the subspace R α,p has been defined in (3.2).

To end with, we are in a position to formulate (and prove) the main result of this section:

Theorem 3.3. Suppose that d ≥ 1 and p ≥ 2 is such that d 2p < 1. In addition, assume that α, β > 0 verify α < β < min 2 -α - d p , 2 -2α . (3.5)
Then for every φ ∈ W -α,p (R d ) and (Ψ, Ψ 2 ) ∈ R α,p , one can find a time T > 0 for which equation (3.3) admits a unique solution in the above-defined set X α,β,p (T ).

In the same way as in Section 2.3, the proof of Theorem 3.3 is of course a direct consequence of the estimates below for the map Γ T,Ψ,Ψ 2 defined for all T ≥ 0 and (Ψ,

Ψ 2 ) ∈ R α,p by Γ T,Ψ,Ψ 2 (v) := e t∆ φ + t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ + 2 t 0 e (t-τ )∆ (ρv τ • Ψ τ ) dτ + t 0 e (t-τ )∆ (Ψ 2 τ ) dτ , t ∈ [0, T ] . (3.6) Proposition 3.4. Suppose that d ≥ 1 and p ≥ 2 is such that d 2p < 1.
In addition, assume that α, β > 0 verify condition (3.5). Then there exists ε > 0 such that, setting X(T ) := X α,β,p (T ), the following bounds hold true: for all 0

≤ T ≤ 1, φ ∈ W -α,p (R d ), (Ψ 1 , Ψ 2 1 ) ∈ R α,p , (Ψ 2 , Ψ 2 2 ) ∈ R α,p and v, v 1 , v 2 ∈ X(T ), Γ T,Ψ 1 ,Ψ 2 1 (v) X(T ) φ W -α,p + T ε v 2 X(T ) + Ψ 1 L ∞ T W -α,p v X(T ) + Ψ 2 1 L ∞ T W -2α,p , (3.7) and Γ T,Ψ 1 ,Ψ 2 1 (v 1 ) -Γ T,Ψ 2 ,Ψ 2 2 (v 2 ) X(T ) T ε v 1 -v 2 X(T ) { v 1 X(T ) + v 2 X(T ) } + Ψ 1 -Ψ 2 L ∞ T W -α,p v 1 X(T ) + Ψ 2 L ∞ T W -α,p v 1 -v 2 X(T ) + Ψ 2 1 -Ψ 2 2 L ∞ T W -2α,p , (3.8)
where the proportional constants depend only on ρ, p, α and β.

Proof of Proposition 3.4. Let us bound each of the four terms in the expression of Γ T,Ψ,Ψ 2 separately.

Bound on e t∆ φ: For all 0 ≤ t ≤ T , using the expression of the heat semigroup, it holds that

e t∆ φ W -α,p (R d ) φ W -α,p (R d ) , yielding to e t∆ φ L ∞ T W -α,p φ W -α,p (R d ) .
Now, thanks to Proposition 3.1, for any 0

< t ≤ T , e t∆ φ W β,p (R d ) t -β+α 2 φ W -α,p (R d ) , which implies t β+α 2 e t∆ φ W β,p (R d ) φ W -α,p (R d ) and e t∆ φ Y (T ) φ W -α,p (R d ) leading to e t∆ φ X(T ) φ W -α,p (R d ) .
(3.9)

Bound on t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ : Let 0 ≤ t ≤ T. Using successively Minkowsky inequality, Proposition 2.1 and Lemma 3.2 (since α < β), we get

t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ W -α,p ≤ t 0 e (t-τ )∆ (ρ 2 v 2 τ ) W -α,p dτ t 0 1 (t -τ ) d 2p ρ 2 v 2 τ W -α, p 2 dτ t 0 1 (t -τ ) d 2p v τ W β,p v τ W -α,p dτ t 0 (t -τ ) -d 2p τ -β+α 2 dτ v Y (T ) v L ∞ T W -α,p T 1-d 2p -β+α 2 v 2 X(T ) .
Taking the supremum over t ∈ [0, T ], we deduce:

t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ L ∞ T W -α,p T 1-d 2p -β+α 2 v 2 X(T ) . Now, as e (t-τ )∆ = e t-τ 2 ∆ e t-τ
2 ∆ , the heat kernel allows a gain of both regularity (Proposition 3.1) and integrability (Proposition 2.1) showing that for all 0 < t ≤ T ,

t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ W β,p ≤ t 0 e (t-τ )∆ (ρ 2 v 2 τ ) W β,p dτ t 0 (t -τ ) -β+α 2 e t-τ 2 ∆ (ρ 2 v 2 τ ) W -α,p dτ t 0 (t -τ ) -β+α 2 -d 2p ρ 2 v 2 τ W -α, p 2 dτ t 0 (t -τ ) -β+α 2 -d 2p v τ W β,p v τ W -α,p dτ t 0 (t -τ ) -β+α 2 -d 2p τ -β+α 2 dτ v Y (T ) v L ∞ T W -α,p t 1-d 2p -β-α v 2 X(T ) , leading to t β+α 2 t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ W β,p t 1-d 2p -β+α 2 v 2 X(T ) T 1-d 2p -β+α 2 v 2 X(T ) . Taking the supremum over t ∈ (0, T ], it holds t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ Y (T ) T 1-d 2p -β+α 2 v 2 X(T ) , which entails t 0 e (t-τ )∆ (ρ 2 v 2 τ )dτ X(T ) T 1-d 2p -β+α 2 v 2 X(T ) . ( 3.10)
Bound on t 0 e (t-τ )∆ (ρv τ • Ψ τ ) dτ : By repeating the same arguments as above (one of two ρv being replaced by Ψ), we obtain

t 0 e (t-τ )∆ (ρv τ • Ψ τ )dτ X(T ) T 1-d 2p -β+α 2 v X(T ) Ψ L ∞ T W -α,p . (3.11)
Bound on t 0 e (t-τ )∆ (Ψ 2 τ )dτ : Fix 0 ≤ t ≤ T . Using successively Minkowsky inequality and Proposition 3.1, we write

t 0 e (t-τ )∆ (Ψ 2 τ )dτ W -α,p ≤ t 0 e (t-τ )∆ (Ψ 2 τ ) W -α,p dτ t 0 (t -τ ) -α 2 Ψ 2 τ W -2α,p dτ T 1-α 2 Ψ 2 L ∞ T W -2α,p , which entails t 0 e (t-τ )∆ (Ψ 2 τ )dτ L ∞ T W -α,p T 1-α 2 Ψ 2 L ∞ T W -2α,p . (3.12)
Then, for all 0 < t ≤ T , thanks to the regularising effect of the heat semigroup and the assumption β

+ 2α < 2, t 0 e (t-τ )∆ (Ψ 2 τ )dτ W β,p ≤ t 0 e (t-τ )∆ (Ψ 2 τ ) W β,p dτ t 0 (t -τ ) -β+2α 2 Ψ 2 τ W -2α,p dτ t 1-β+2α 2 Ψ 2 L ∞ T W -2α,p , which leads to t β+α 2 t 0 e (t-τ )∆ (Ψ 2 τ )dτ W β,p T 1-α 2 Ψ 2 L ∞ T W -2α,p . Thus, t 0 e (t-τ )∆ (Ψ 2 τ )dτ Y (T ) T 1-α 2 Ψ 2 L ∞ T W -2α,p (3.13)
and, combining the identities (3.12) and (3.13),

t 0 e (t-τ )∆ (Ψ 2 τ )dτ X(T ) T 1-α 2 Ψ 2 L ∞ T W -2α,p . ( 3

On the construction of the relevant stochastic objects

The objective of this section is to prove Proposition 1.3, Proposition 1.8 and to give an asymptotic estimate of the quantity E n (t, x) 2 (Proposition 1.6). We recall that the spacetime fractional white noise B has been defined as a Gaussian process whose mean and covariance function are well-known (see Definition 1.1). This definition can seem a bit abstract and, in order to realize computations with this noise, we need a representation formula. We will resort to the harmonizable representation of the fractional Brownian motion presented in [START_REF] Samoradnitsky | Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance[END_REF]. Let us recall the procedure. First of all, we fix, on some complete probability space (Ω, F, P), a space-time white noise W on R d+1 . Then let H = (H 0 , H 1 , . . . , H d ) ∈ (0, 1) d+1 and set, for any t ∈ [0, T ] and

x ∈ R d , B(t, x 1 , . . . , x d ) := c H ξ∈R η∈R d e ıtξ -1 |ξ| H 0 + 1 2 d i=1 e ıx i η i -1 |η i | H i + 1 2 W (dξ, dη) , ( 4.1)
where c H > 0 is a constant, and where W stands for the Fourier transform of W . Then, for an appropriate value of c H , B is a space-time fractional Brownian motion of index H (in the sense of Definition 1.1). Unfortunately, we know that B suffers from a lack of regularity measured by its Hurst index. That is why we are looking for a smooth approximation (B n ) n≥0 . Our strategy is based on a truncation of the integration domain in (4.1): precisely, we set B 0 ≡ 0, and for n ≥ 1,

B n (t, x 1 , . . . , x d )(ω) := c H |ξ|≤2 2n |η|≤2 n e ıtξ -1 |ξ| H 0 + 1 2 d i=1 e ıx i η i -1 |η i | H i + 1 2 W (dξ, dη) . (4.2)
It is then clear than B n defines a smooth process for any n ≥ 0 and that for all (t, x)

∈ [0, T ]×R d , B n (t, x) n→∞ -→ B(t, x) in L 2 (Ω) .

Linear solution associated with the model.

With a view to constructing the first order process , the solution of the linear equation

∂ t -∆ = Ḃ, t ∈ [0, T ], x ∈ R d , (0, .) = 0, (4.3)
we are now interested in giving a rigorous meaning to the sequence ( n ) n≥0 of the regularised version of (4.3), that is the sequence of solutions to

∂ t n -∆ n = Ḃn , t ∈ [0, T ] , x ∈ R d , n (0, .) = 0 , ( 4.4)
where, for all n ≥ 0, Ḃn denotes the classical derivative Ḃn :

= ∂ t ∂ x 1 • • • ∂ x d B n .
Despite the regularity of Ḃn , its a priori lack of integrability over the whole space R d prevents us from using some general theorem which would guarantee the existence of n . We thus propose a stochastic construction of n . Should this latter exist, its expression would by given the formula n (t, x) = t 0 e (t-s)∆ ( Ḃn (s))(x)ds.

A formal computation leads us to the following expression:

n (t, x) = t 0 e (t-s)∆ ( Ḃn (s))(x)ds = t 0 F -1 (e -|ξ| 2 (t-s) Ḃn (s, ξ))(x)ds = t 0 F -1 (e -|ξ| 2 (t-s) ) ⋆ Ḃn (s, x)ds = t 0 R d F -1 (e -|ξ| 2 (t-s) )(x -y) Ḃn (s, y)dyds = c H t 0 ds R d dy |ξ|≤2 2n |η|≤2 n F -1 (e -|ξ| 2 (t-s) )(x -y)i d+1 ξ |ξ| H0+ 1 2 d i=1 η i |η i | Hi+ 1 2
e iξs e i η,y W (dξ, dη)

= c H i d+1 |ξ|≤2 2n |η|≤2 n ξ |ξ| H0+ 1 2 d i=1 η i |η i | Hi+ 1 2 e i η,x • t 0 ds e iξs R d dy F -1 (e -|ξ| 2 (t-s) )(x -y)e -i η,x-y W (dξ, dη) = c H i d+1 |ξ|≤2 2n |η|≤2 n ξ |ξ| H0+ 1 2 d i=1 η i |η i | Hi+ 1 2 e i η,x • t 0 ds e iξ(t-s) R d dy F -1 (e -|ξ| 2 s )(y)e -i η,y
W (dξ, dη).

We have finally obtained that

n (t, x) = c H i d+1 |ξ|≤2 2n |η|≤2 n ξ |ξ| H0+ 1 2 d i=1 η i |η i | Hi+ 1 2 e i η,x γ t (ξ, |η|) W (dξ, dη),
where for every t ≥ 0, ξ ∈ R and r > 0, we introduce the quantity γ t (ξ, r) as γ t (ξ, r) := e iξt t 0 e -sr 2 e -iξs ds .

(4.5)

Moreover, let us observe that is real-valued. Indeed, let f t,x the function defined for all ξ ∈ R, η ∈ R d by

f t,x (ξ, η) := c H i d+1 ½ |ξ|≤2 2n ½ |η|≤2 n ξ |ξ| H 0 + 1 2 d i=1 η i |η i | H i + 1 2 e i η,x γ t (ξ, |η|), in such a way that n (t, x) = R d+1 f t,x (ξ, η) W (dξ, dη). As f t,x (ξ, η) = f t,x (-ξ, -η), the Fourier transform of f is real-valued that implies n (t, x) = R d+1 f t,x (ξ, η) W (dξ, dη) = R d+1 f t,x (ξ, η)W (dξ, dη) = R d+1 f t,x (ξ, η)W (dξ, dη) = R d+1 f t,x (ξ, η)W (dξ, dη) = R d+1 f t,x (ξ, η) W (dξ, dη) = n (t, x).
We are now in a position to define properly n .

Definition 4.1. We call a solution of equation (4.4) (or linear solution associated with (1.1)) any centered real Gaussian process

n (s, x), n ≥ 1, s ≥ 0, x ∈ R d
whose covariance function is given by the relation: for all n, m ≥ 1, s, t ≥ 0 and x, y ∈ R d ,

E n (s, x) m (t, y) = c 2 H (ξ,η)∈Dn∩Dm 1 |ξ| 2H 0 -1 d i=1 1 |η i | 2H i -1 γ s (ξ, |η|)γ t (ξ, |η|)e ı η,x-y dξdη , (4.6) where D n := B 1 2n × B d n with B k ℓ := λ ∈ R k : |λ| ≤ 2 ℓ .

Some preliminary estimates on γ.

Before proving that χ n is a Cauchy sequence in a convenient subspace, we need to establish some bounds on the quantity γ t (ξ, r) which contains all the information with respect to the heat semigroup: γ t (ξ, r) := e iξt t 0 e -sr 2 e -iξs ds .

(4.7)

We first state some estimates on the variation

γ s,t (ξ, r) := γ t (ξ, r) -γ s (ξ, r). (4.8) Lemma 4.2. For all 0 ≤ s ≤ t ≤ 1, ξ ∈ R, r > 0 and ε ∈ [0, 1]
, the following bound holds true:

|γ s,t (ξ, r)| min |ξ| ε |t -s| ε + |t -s|, |t -s|r 2 |ξ| + |t -s| ε {1 + r 2 } |ξ| 1-ε , |t -s| ε {r 2ε + |ξ| ε } r 4 + ξ 2 .
Proof. First of all, we write

γ s,t (ξ, r) = {e iξt -e iξs } s 0 e -ur
|γ s,t (ξ, r)| r 2 |t -s| |ξ| + |t -s| ε |ξ| 1-ε + r 2 |t -s| ε |ξ| 1-ε + r 2 |ξ| |t -s| r 2 |t -s| |ξ| + {1 + r 2 } |t -s| ε |ξ| 1-ε .
To end with, it can be verified that

γ t (ξ, r) = e iξt -e -r 2 t r 2 + iξ , which yields |γ s,t (ξ, r)| = 1 r 4 + ξ 2 {e -r 2 t -e -r 2 s } -{e iξt -e iξs } |t -s| ε r 4 + ξ 2 {r 2ε + |ξ| ε } . Corollary 4.3. For any 0 ≤ s ≤ t ≤ 1, H ∈ (0, 1), r > 0 and ε ∈ [0, H), it holds that R |γ s,t (ξ, r)| 2 |ξ| 2H-1 dξ |t -s| 2ε 1 + r 4(H-ε) .
Proof. The desired inequality comes of course from a relevant use of the estimates shown in Lemma 4.2.

For 0 < r < 1, we have

R |γ s,t (ξ, r)| 2 |ξ| 2H-1 dξ |t -s| 2ε |ξ|≤1 dξ |ξ| 2H-1 + |ξ|≥1 dξ |ξ| 2(H-ε)+1 |t -s| 2ε .
Then, for r > 1, it holds that

R |γ s,t (ξ, r)| 2 |ξ| 2H-1 dξ |t -s| 2ε R r 4ε + |ξ| 2ε (r 4 + ξ 2 )|ξ| 2H-1 dξ |t -s| 2ε r 4(H-ε) R 1 + |ξ| 2ε (1 + ξ 2 )|ξ| 2H-1 dξ |t -s| 2ε r 4(H-ε) .

Construction of the first order stochastic process.

Proof of Proposition 1.3. Without loss of generality, we will suppose during the whole proof that T = 1 and we set, for all m, n ≥ 0, n,m := mn . Also, along the statement of the proposition, we fix α verifying (1.11).

Step 1: The first objective is to show that for any p ≥ 1, 1 ≤ n ≤ m, 0 ≤ s ≤ t ≤ 1 and ε > 0 small enough, it holds

R d E F -1 {1 + |.| 2 } -α 2 F χ n,m (t, .) -n,m (s, .) (x) 2p dx 2 -4nεp |t -s| 2εp , (4.9)
where the proportional constant only depends on p, α and χ.

Resorting to the same kind of arguments as those used in [7, Proof of Prop 1.2] , we obtain the estimate

R d dx E F -1 {1 + |.| 2 } -α 2 F χ n,m (t, .) -n,m (s, .) (x) 2p (ξ,η)∈Dn,m 1 |ξ| 2H 0 -1 d i=1 1 |η i | 2H i -1 {1 + |η| 2 } -α |γ s,t (ξ, |η|)| 2 dξdη p .

Now we can split the latter integral into

(ξ,η)∈Dn,m 1 |ξ| 2H 0 -1 d i=1 1 |η i | 2H i -1 {1 + |η| 2 } -α |γ s,t (ξ, |η|)| 2 dξdη p 2 2n ≤|ξ|≤2 2m |η|≤2 m 1 |ξ| 2H 0 -1 d i=1 1 |η i | 2H i -1 {1 + |η| 2 } -α |γ s,t (ξ, |η|)| 2 dξdη p + |ξ|≤2 2m 2 n ≤|η|≤2 m 1 |ξ| 2H 0 -1 d i=1 1 |η i | 2H i -1 {1 + |η| 2 } -α |γ s,t (ξ, |η|)| 2 dξdη p =: (I n,m (s, t)) p + (II n,m (s, t)) p . (4.10)
Let us focus on the estimation of I n,m (s, t). To this end, we fix ε > 0, so that

I n,m (s, t) ≤ 2 -4nε R dξ R d dη 1 |ξ| 2H 0 -2ε-1 d i=1 1 |η i | 2H i -1 {1 + |η| 2 } -α |γ s,t (ξ, |η|)| 2 , (4.11)
which, after a classical hyperspherical change of variables, leads us to

I n,m (s, t) 2 -4nε ∞ 0 dr 1 + r 2 -α r 2(H 1 +•••+H d )-2d+1 R dξ |γ s,t (ξ, r)| 2 |ξ| 2H 0 -2ε-1 . (4.12)
We can now apply Corollary 4.3 with H := H 0ε, which gives, for all 0 < ε < H 0 2 , I n,m (s, t)

2 -4nε |t -s| 2ε ∞ 0 dr 1 + r 2 -α r 2(H 1 +•••+H d )-2d+1 1 1 + r 4H 0 -8ε 2 -4nε |t -s| 2ε 1 0 1 r 2(H 1 +•••+H d )-2d+1 dr + ∞ 1 1 r 2α+2(2H 0 +H 1 +•••+H d )-2d+1-8ε dr . (4.13)
Due to our assumption (1.11), we can in fact pick ε small enough so that

4ε < α -d -2H 0 + d i=1 H i ,
and for such a parameter, the two integrals in (4.13) are finite, yielding

I n,m (s, t) 2 -4nε |t -s| 2ε .
It is not difficult to see that the previous estimates could also be used to control II n,m (s, t), yielding the very same estimate

II n,m (s, t) 2 -4nε |t -s| 2ε .
Coming back to (4.10), we get the desired bound (4.9).

Step 2: The estimate we proved in the previous step can be reformulated in the following way

E χ n,m (t, .) -χ n,m (s, .) 2p W -α,2p 2 -4nεp |t -s| 2εp , (4.14) for all p ≥ 1, 1 ≤ n ≤ m, 0 ≤ s ≤ t ≤ 1 and ε > 0 small enough.
Combining Kolmogorov continuity criterion with the classical Garsia-Rodemich-Rumsey estimate (see [START_REF] Garsia | A Real Variable Lemma and the Continuity of Paths of Some Gaussian Processes[END_REF]), we easily show that for any p ≥ 1 large enough,

χ n,m L 2p (Ω;C T W -α,2p ) 2 -2nε . (4.15) In particular, (χ n ) n≥1 is a Cauchy sequence in L 2p (Ω; C([0, T ]; W -α,2p (R d ))) (for any p ≥ 1 large enough). As L 2p (Ω; C([0, T ]; W -α,2p (R d ))
) is a Banach space, we deduce the convergence of (χ n ) n≥1 in this space to a limit χ . Coming back to (4.15), we also have Proof of Proposition 1.8. We will follow the same general strategy as in the proof of Proposition 1.3. Let us again suppose that T = 1, and set, for every m, n ≥ 0, n,m := mn .

χ -χ n L 2p (Ω;C T W -α,2p ) 2 -
Step 1: Our first objective here is to prove that for every p ≥ 1, 0 ≤ n ≤ m, 0 ≤ s ≤ t ≤ 1, ε > 0 small enough, and for every α verifying

d -2H 0 + d i=1 H i < α < 1 4 , (4.16) it holds R d E F -1 {1 + |.| 2 } -α F χ 2 n,m (t, .) -n,m (s, .) (x) 2p dx 2 -4nεp |t -s| εp , (4.17)
where the proportional constant only depends on p, α, and χ.

In order to reduce the length of the proof, we will only show estimate (4.17) for n = 0, that is we will focus on the bound for the time-variation m (t, .)m (s, .), with m ≥ 1. The extended result to all m ≥ n ≥ 0 could in fact be easily deduced from the estimates below combined with the bounding argument used (for example) in (4.11).

Resorting to the same kind of arguments as those used in [5, Proof of Prop 1.6], we obtain the estimate

R d dx E F -1 {1 + |.| 2 } -α F χ 2 m (t, .) -m (s, .) (x) 2 p 4 ℓ=1 J ℓ m;s,t p (4.18)
where Let us focus on the treatment of J 1 m;s,t .

J ℓ m;s,t := (ξ,η)∈Dm dξdη ( ξ,η)∈Dm d ξdη 1 |ξ| 2H 0 -1 d i=1 1 |η i | 2H i -1 1 | ξ| 2H 0 -1 d i=1 1 | ηi | 2H i -1 Γ ℓ m;s,t (ξ, |η|; ξ, |η|) 1 + |η -η| 2 -2α , ( 4
J 1 m;s,t (ξ,η)∈Dm dξdη ( d ξdη 1 + |η -η| 2 -2α 1 |ξ| 2H 0 -1 d i=1 1 |η i | 2H i -1 |γ t (ξ, |η|)||γ s,t (ξ, |η|)| • 1 | ξ| 2H 0 -1 d i=1 1 | ηi | 2H i -1 |γ t ( ξ, |η|)| 2 (R×R d ) 2 dξdηd ξdη 1 + ||η| -|η|| 2 -2α 1 |ξ| 2H 0 -1 d i=1 1 |η i | 2H i -1 |γ t (ξ, |η|)||γ s,t (ξ, |η|)| • 1 | ξ| 2H 0 -1 d i=1 1 | ηi | 2H i -1 |γ t ( ξ, |η|)| 2 , ( 4.20)
where we have used the trivial bound |η -η| ≥ ||η| -|η||.

Now, let us decompose the latter integration domain into

(R × R d ) 2 := D 1 ∪ D 2 ,
where

D 1 := (ξ, η, ξ, η) : 0 ≤ |η| ≤ |η| 2 or |η| ≥ 3|η| 2
and

D 2 := (ξ, η, ξ, η) : |η| 2 < |η| < 3|η| 2 .
Concerning the integral on D 1 , the inequality ||η| -|η|| ≥ max( |η| 2 , |η| 3 ) (valid for all (ξ, η, ξ, η) ∈ D 1 ) allows us to write

A 1 := D 1 dξdηd ξdη {1 + ||η| -|η|| 2 } 2α 1 |ξ| 2H 0 -1 d i=1 1 |η i | 2H i -1 |γ t (ξ, |η|)||γ s,t (ξ, |η|)| • 1 | ξ| 2H 0 -1 d i=1 1 | ηi | 2H i -1 |γ t ( ξ, |η|)| 2 R×R d dξdη {1 + |η| 2 } α 1 |ξ| 2H 0 -1 d i=1 1 |η i | 2H i -1 |γ t (ξ, |η|)||γ s,t (ξ, |η|)| • R×R d d ξdη {1 + |η| 2 } α 1 | ξ| 2H 0 -1 d i=1 1 | ηi | 2H i -1 |γ t ( ξ, |η|)| 2 R×R d dξdη {1 + |η| 2 } α 1 |ξ| 2H 0 -1 d i=1 1 |η i | 2H i -1 |γ t (ξ, |η|)| 2 1 2 • R×R d dξdη {1 + |η| 2 } α 1 |ξ| 2H 0 -1 d i=1 1 |η i | 2H i -1 |γ s,t (ξ, |η|)| 2 1 2 • R×R d d ξdη {1 + |η| 2 } α 1 | ξ| 2H 0 -1 d i=1 1 | ηi | 2H i -1 |γ t ( ξ, |η|)| 2 ,
where Cauchy-Schwarz inequality has been used to get the last estimate. Now remark that we are here coping with the same integral as in the proof of Proposition 1.3 (see in particular (4.11)) and therefore we can reproduce the arguments in (4.12)-(4.13) to obtain the estimate we are looking for, namely: for all ε > 0 small enough,

A 1 |t -s| ε .
Concerning the integral over D 2 , a hyperspherical change of variable entails

|η| 2 <|η|< 3|η| 2 dη {1 + ||η| -|η|| 2 } 2α |γ t ( ξ, |η|)| 2 d i=1 1 | ηi | 2H i -1 = |η| -2(H 1 +...+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 | ηi | 2H i -1 |η| -2(H 1 +...+H d )+2d 1 2 <r< 3 2 dr {1 + |η| 2 (1 -r) 2 } 2α |γ t ( ξ, |η|r)| 2 .
As a consequence,

A 2 := D 2 dξdηd ξdη {1 + ||η| -|η|| 2 } 2α 1 |ξ| 2H 0 -1 d i=1 1 |η i | 2H i -1 |γ t (ξ, |η|)||γ s,t (ξ, |η|)| • 1 | ξ| 2H 0 -1 d i=1 1 | ηi | 2H i -1 |γ t ( ξ, |η|)| 2 R d dη |η| 2(H 1 +...+H d )-2d d i=1 1 |η i | 2H i -1 1 2 <r< 3 2 dr {1 + |η| 2 (1 -r) 2 } 2α • R dξ |γ t (ξ, |η|)||γ s,t (ξ, |η|)| |ξ| 2H 0 -1 • R d ξ |γ t ( ξ, |η|r)| 2 | ξ| 2H 0 -1 R d dη |η| 2(H 1 +...+H d )-2d d i=1 1 |η i | 2H i -1 1 2 <r< 3 2 dr {1 + |η| 2 (1 -r) 2 } 2α • R dξ |γ t (ξ, |η|)| 2 |ξ| 2H 0 -1 1 2 • R dξ |γ s,t (ξ, |η|)| 2 |ξ| 2H 0 -1 1 2 • R d ξ |γ t ( ξ, |η|r)| 2 | ξ| 2H 0 -1 .
Using again a hyperspherical change of variable (with respect to η), we obtain

A 2 ∞ 0 dρ ρ 4(H 1 +...+H d )-4d+1 1 2 <r< 3 2 dr {1 + ρ 2 (1 -r) 2 } 2α R dξ |γ t (ξ, ρ)| 2 |ξ| 2H 0 -1 1 2 • R dξ |γ s,t (ξ, ρ)| 2 |ξ| 2H 0 -1 1 2 • R d ξ |γ t ( ξ, ρr)| 2 | ξ| 2H 0 -1 ,
and we can now use Corollary 4.3 to assert that

A 2 |t -s| ε 1 0 dρ ρ 4(H 1 +...+H d )-4d+1 + ∞ 1 dρ ρ 4(2H 0 +H 1 +...+H d )-4d+1-8ε 1 2 <r< 3 2 dr {1 + ρ 2 (1 -r) 2 } 2α for all 0 < ε < H 0 . Now, remark that ∞ 1 dρ ρ 4(2H 0 +H 1 +...+H d )-4d+1-8ε 1 2 <r< 3 2 dr {1 + ρ 2 (1 -r) 2 } 2α ≤ ∞ 1 dρ ρ 4α+4(2H 0 +H 1 +...+H d )-4d+1-8ε 1 2 <r< 3 2 dr (1 -r) 4α . ( 4.21)
Thanks to our hypothesis on α (see (4.16)), we know that 4α < 1 and we can choose ε > 0 such that 4α + 4(2H

0 + H 1 + ... + H d ) -4d + 1 -8ε > 1 .
For such a choice of parameter, the two integrals in the right-hand side of (4.21) are clearly finite, and finally we have proved that

A 2 |t -s| ε .
Going back to (4.20), we have thus shown that, uniformly over m,

J 1 m;s,t |t -s| ε .
It is now easy to realize that the other three integrals {J i m;s,t , i = 2, 3, 4} (as defined in (4.19)) could be controlled with the very same arguments (with the very same resulting bound).

Injecting the above estimates into (4.18) provides us with the desired conclusion (4.17).

Step 2: Conclusion. Let α satisfying (1.11) 1 4 , then condition (4.16) is satisfied, and so the moment estimate (4.17) holds true. Thanks to this bound, we can reproduce the arguments used in Step 2 of Section 4.3 to get that (χ 2 n ) n≥1 converges almost surely in the space C([0, T ]; W -2α,p (R d )), for all 2 ≤ p < ∞. If α ≥ 1 4 , remark that, due to assumption (H2), we can choose α ′ verifying α ′ < α and

, that is α > d -2H 0 + d i=1 H i . If α <
d-2H 0 + d i=1 H i < α ′ < 1 4 (that is, α ′ satisfies (4. 16
)). By executing the above scheme again, we deduce that the sequence (χ

2 n ) n≥1 converges almost surely in C([0, T ]; W -2α ′ ,p (R d ))
, and therefore it converges almost surely in C([0, T ]; W -2α,p (R d )) as well, for all 2 ≤ p < ∞.

Finally, the (a.s.

) convergence in C([0, T ]; W -2α,∞ (R d )) is the result of the Sobolev embedding W -2α+ d p +η,p (R d ) ⊂ W -2α,∞ (R d ),
for any η > 0, which ends the proof of Proposition 1.8.

Asymptotic estimate of the renormalization constant.

Fix d ≥ 1 and (H 0 , ..., H d ) ∈ (0, 1) d+1 such that

2H 0 + d i=1 H i ≤ d .
The objective of this subsection is to give an equivalent of the quantity

σ n (t, x) = E[ n (t, x) 2 ].
Let us rewrite it under the integral form:

σ n (t, x) = E[ n (t, x) 2 ] = c 2 |ξ|≤4 n dξ |ξ| 2H 0 -1 |η|≤2 n d i=1 dη i |η i | 2H i -1 |γ t (ξ, |η|)| 2 = C 2 n 0 dr r 2(H 1 +...+H d )-2d+1 |ξ|≤4 n dξ |ξ| 2H 0 -1 |γ t (ξ, r)| 2 ,
where the previous identity is obtained after a hyperspherical change of variables. As remarked in Proposition 1.6, the above formula shows the surprising fact that σ n does not depend on x.

The desired estimate (1.15) is now a consequence of the following technical result (applied with α := 2H 0 ∈ (0, 2) and κ :

= d -[2H 0 + d i=1 H i ] ≥ 0): Proposition 4.4. Fix t > 0.
For all α ∈ (0, 2) and κ ≥ 0, there exists two constants c 1 and c 2 depending only on α and κ such that

2 n 0 dr r -2α-2κ+1 |ξ|≤4 n dξ |ξ| α-1 |γ t (ξ, r)| 2 ∼ n→∞ c 1 4 nκ if κ > 0 c 2 n if κ = 0 .
Proof. It is quite easy to verify from the expression

γ t (ξ, r) := e iξt t 0 e -sr 2 e -iξs ds (4.22) that |γ t (ξ, r)| 2 = 1 -2 cos(ξt)e -r 2 t + e -2r 2 t r 4 + ξ 2 .
With the change of variables u = r 2 and using the parity in ξ of the function |γt(ξ,r)| 2 |ξ| α-1 , we rewrite the integral we are dealing with as

2 n 0 dr r -2α-2κ+1 |ξ|≤4 n dξ |ξ| α-1 |γ t (ξ, r)| 2 = 4 n 0 dr r -α-κ+1 4 n 0 dξ ξ α-1 1 -2 cos(ξt)e -rt + e -2rt r 2 + ξ 2 = 4 nκ 1 0 dr r -α-κ+1 1 0 dξ ξ α-1 1 -2 cos(4 n ξt)e -4 n rt + e -2.4 n rt r 2 + ξ 2 .
We perform the change of variables (r, ξ) → T (r, ξ) defined by

T : (r, ξ) → T (r, ξ) = r ξ , ξ .
It is readily checked that T is a one-to-one map of (0, 1) 2 on

{(x, y) ∈ R 2 , x > 0, 0 < y < min 1, 1 x }.
Moreover, its reverse is explicitly given by T -1 : (x, y) → (xy, y) whose absolute value of the Jacobian equals y.

2 n 0 dr r -2α-2κ+1 |ξ|≤4 n dξ |ξ| α-1 |γ t (ξ, r)| 2 = 4 nκ 1 0 x α+κ-1 1 + x 2 dx 1 0 dy y 1-κ 1 -2 cos(4 n yt)e -4 n xyt + e -2.4 n xyt + 4 nκ +∞ 1 x α+κ-1 1 + x 2 dx 1 x 0 dy y 1-κ 1 -2 cos(4 n yt)e -4 n xyt + e -2.4 n xyt .
First case: κ > 0. By Lebesgue's dominated convergence theorem, we observe that

1 0 x α+κ-1 1 + x 2 dx 1 0 dy y 1-κ -2 cos(4 n yt)e -4 n xyt + e -2.4 n xyt + +∞ 1 x α+κ-1 1 + x 2 dx 1 x 0 dy y 1-κ -2 cos(4 n yt)e -4 n xyt + e -2.4 n xyt → n→∞ 0, leading to 2 n 0 dr r -2α-2κ+1 |ξ|≤4 n dξ |ξ| α-1 |γ t (ξ, r)| 2 ∼ n→∞ c 1 4 nκ , where c 1 = 1 0 x α+κ-1 1 + x 2 dx 1 0 dy y 1-κ + +∞ 1 x α+κ-1 1 + x 2 dx 1 x 0 dy y 1-κ = 1 κ 1 0 x α+κ-1 1 + x 2 dx + +∞ 1 x α-1 1 + x 2 dx .
A more visual expression of c 1 can be found in the appendix.

Second case: κ = 0. Let us recall that

2 n 0 dr r -2α+1 |ξ|≤4 n dξ |ξ| α-1 |γ t (ξ, r)| 2 = 1 0 x α-1 1 + x 2 dx 1 0 dy y 1 -2 cos(4 n yt)e -4 n xyt + e -2.4 n xyt + +∞ 1 x α-1 1 + x 2 dx 1 x 0 dy y 1 -2 cos(4 n yt)e -4 n xyt + e -2.4 n xyt .
Let us introduce the function f defined for all T > 0 by To deal with this derivative with ease, we will resort to the following technical lemma: Lemma 4.5 with a = 1 and a = 1 x readily entails:

f (T ) = 1 0 x α-1 1 + x 2 dx
f ′ (T ) = 1 T +∞ 0 x α-1 1 + x 2 dx + 1 0 x α-1 1 + x 2 dx e -2T x -2 cos(T )e -T x + +∞ 1 x α-1 1 + x 2 dx e -2T -2 cos T x e -T .
By Lebesgue's dominated convergence theorem, we deduce since α ∈ (0, 2) that x α-1 1 + x 2 dx. For the sake of beauty, let us compute the value of the constant c 2 . We recall the classical result, coming from complex analysis, stating that for all β ∈]0, 1[,

+∞ 0 dt t β (1 + t) = π sin πβ .
It yields:

f ′ (T ) ∼ T →∞ π 2 sin( απ 2 )T .
We can finally use a standard comparison argument (see Lemma 4.6 below) to assert that

f (T ) ∼ T →∞ π 2 sin( απ 2 )
ln(T ).

We obtain the equivalent we are looking for, namely As in [START_REF] Deya | A nonlinear Schrödinger equation with fractional noise[END_REF], the statement of Proposition 1.3 provides us with a local definition of the process , that is, up to multiplication by χ ∈ C ∞ c (R d ). To put it differently, what is actually given by the proposition is the set of the limit elements {χ , χ ∈ C ∞ c (R d )}. Let us briefly recall how those elements can be gathered into a single process .

Fix p ≥ 2 and α verifying (1.11). We denote by P the set of sequences σ = (σ k ) k≥1 such that for every k ≥ 1, σ k : R d → R is a smooth function verifying

σ k (x) = 1 if x ≤ k , 0 if x ≥ k + 1 .
Let us fix such a sequence σ, and for all k ≥ 1, we call (σ k ) the limit of the sequence (σ k n ) n≥1 in the space C([0, T ]; W -α,p (R d )), as given by Proposition 1.3. As (σ k ) is defined on a probability space Ω (σ k ) of full measure 1, Ω (σ) := ∩ k≥1 Ω (σ k ) is still of measure 1.

For each time t ∈ [0, T ], we are now able to define the random distribution

(σ) (t) : Ω (σ) → D ′ (R d )
as follows: for all smooth compactly-supported function ϕ : R d → R such that supp(ϕ) ⊂ B(0, k) (for some k ≥ 1), (σ) (t), ϕ := (σ k ) (t), ϕ .

Proposition 4.7.

(1) The above distribution (σ) is well defined, i.e. for every 1 ≤ k ≤ ℓ and for all smooth compactly-supported function ϕ : R d → R with supp(ϕ) ⊂ B(0, k) ⊂ B(0, ℓ), one has (σ k ) (t), ϕ = (σ ℓ ) (t), ϕ on Ω (σ) .

(2) For any smooth compactly-supported function χ : R d → R, one has, on Ω (σ) ,

χ • n → n→∞ χ • (σ) in C([0, T ]; W -α,p (R d )) .
(3) If σ, γ ∈ P, it holds that

(σ) = (γ) on Ω (σ) ∩ Ω (γ) .
Thanks to the latter identification property, we can set := (σ) , as soon as σ ∈ P is a fixed element.

Remark 4.8. The latter procedure can of course be used to give a rigorous meaning to the second-order process χ 2 as a distribution-valued function .

Rougher stochastic constructions when the space dimension equals 2

The aim of this section is to establish the proofs of Proposition 1.14 and Proposition 1.13, that is to construct the second order stochastic process in the roughest case, namely when

3 2 < 2H 0 + H 1 + H 2 ≤ 7 4
, and to show that this construction is in some sense optimal insofar as

E χ • n (t, .) 2 H -2α -→ n→+∞ +∞,
for all compactly-supported function χ and t > 0 when 2H 0 + H 1 + H 2 ≤ 3 2 .

Additional notations.

To begin with, let us introduce some notations that will be frequently used in the proofs of Proposition 1.14 and Proposition 1.13. For all τ ∈ , , 0 ≤ n ≤ m and 0 ≤ s, t ≤ 1, let us set τ n,m := τ mτ n , and then, for f ∈ {τ n , τ m , τ n,m }, f s,t := f tf s .

Then, we set for all H = (H 0 , H 1 , H 2 ) ∈ (0, 1)

3 , η ∈ R 2 , K H (η) := |η 1 | 1-2H 1 |η 2 | 1-2H 2 1 + |η| 4H 0 . ( 5.1)
For all a = (a 1 , a 2 ), resp. b = (b 1 , b 2 ), with a i ∈ {n, m, {n, m}}, resp. b i ∈ {s, t, {s, t}}, we write

L H,a b (η) := 1 |η 1 | 2H 1 -1 |η 2 | 2H 2 -1 (ξ,η)∈D a 1 ∩D a 2 dξ γ b 1 (ξ, |η|)γ b 2 (ξ, |η|) |ξ| 2H 0 -1
where

D n := B 1 2n × B 2 n with B k ℓ := λ ∈ R k : |λ| ≤ 2 ℓ and D n,m := D m \D n ,
in such a way that for all y, ỹ ∈ R 2 ,

E a 1 b 1 (y) a 2 b 2 (ỹ) = c 2 H R 2
dη e ı η,y e -ı η,ỹ L H,a b (η) .

(5.2)

In the following, we will resort to several technical lemmas whose statements and proofs can be found in the Appendix. In particular, Lemma 7.2 is of major interest since it describes in some way the role of the cut-off function χ which allows a gain of integrability.

5.2. Proof of Proposition 1.14. The strategy is exactly the same as the one developed in the proof of Proposition 1.3 and results from the combination of Kolmogorov criterion and Garsia-Rodemich-Rumsey lemma. Let us write

E F -1 {1 + |.| 2 } -α F χ 2 • n,m s,t (x) 2 = 1 (2π) 4 (R 2 ) 2 dλd λ {1 + |λ| 2 } α {1 + | λ| 2 } α e ı x,λ-λ (R 2 ) 2 dξd ξ χ 2 (λ -ξ) χ 2 ( λ -ξ)Q n,m;s,t (ξ, ξ) , (5.3) with Q n,m;s,t (ξ, ξ) := E F n,m s,t (ξ)F n,m s,t ( ξ) = (R 2 ) 2
dydỹ e -ı ξ,y e ı ξ,ỹ E n,m s,t (y) n,m s,t (ỹ) .

Resorting to Wick formula, we derive

E n,m s,t (y) n,m s,t (ỹ) = (a,b)∈A c a,b E a 1 b 1 (y) a 2 b 2 (ỹ) E a 3 b 3 (y) a 4 b 4 (ỹ) = (a,b)∈A c a,b (R 2 ) 2 dηdη e ı η,y e -ı η,ỹ e ı η,y e -ı η,ỹ L H,(a 1 ,a 2 ) b 1 ,b 2 (η)L H,(a 3 ,a 4 ) b 3 ,b 4 (
Q n,m;s,t (ξ, ξ) = (a,b)∈A c a,b (R 2 ) 2 dηdη L H,(a 1 ,a 2 ) b 1 ,b 2 (η)L H,(a 3 ,a 4 ) b 3 ,b 4 (η)δ {ξ=η+η} δ { ξ=η+η} .
Consequently,

E F -1 {1 + |.| 2 } -α F χ 2 • n,m s,t (x) 2 = (a,b)∈A φ a,b , with φ a,b = c a,b (R 2 ) 2 dλd λ {1 + |λ| 2 } α {1 + | λ| 2 } α e ı x,λ-λ (R 2 ) 2 dηdηL H,(a 1 ,a 2 ) b 1 ,b 2 (η)L H,(a 3 ,a 4 ) b 3 ,b 4 (η) χ 2 (λ -(η + η)) χ 2 ( λ -(η + η)). (5.4)
With the help of Lemma 7.2 and the hypercontractivity of Gaussian chaoses, we obtain

R 2 dx E F -1 {1 + |.| 2 } -α F χ 2 • n,m s,t (x) 2p   (a,b)∈A ψ a,b   p ,
where

ψ a,b = (R 2 ) 2 dηdη {1 + |η + η| 2 } -2α |L H,(a 1 ,a 2 ) b 1 ,b 2 (η)||L H,(a 3 ,a 4 ) b 3 ,b 4 (η)|. ( 5.5)
Now, for 0 < ε < H 0 , Lemma 7.1 provides us with the bound

ψ a,b 2 -2nε |t -s| ε (R 2 ) 2 dηdη {1 + |η + η| 2 } -2α K H ε,0 (η) + K H ε,0,1 (η) + K H ε,0,2 (η) K H ε,0 (η) + K H ε,0,1 (η) + K H ε,0,2 (η) . ( 5.6)
According to Lemma 7.3, the latter quantity is finite as soon as ε is small enough and we have finally obtained:

R 2 dx E F -1 {1 + |.| 2 } -α F χ 2 • n,m s,t (x) 2p 2 -2nεp |t -s| εp .
We can mimic the arguments at the end of the proof of Proposition 1.3 to get the result. 5.3. Proof of Proposition 1.13. Suppose that d ≥ 1 and that (H 0 , H 1 , ..., H d ) ∈ (0, 1) d+1 verifies the condition 2H

0 + H 1 + • • • + H d ≤ 3 4 d.
As usual, we will use the notation A B whenever one can find a constant c > 0 such that A ≥ cB. Let us also introduce the additional notation

Γ H 0 ,n t (r) := 4 n -4 n dξ |γ t (ξ, r)| 2 |ξ| 2H 0 -1 . ( 5.7)
Fix t > 0. Using (5.2) and then Wick formula, we get that

E χ • n (t, .) 2 H -2α = c |η|≤2 n dη |η|≤2 n dη d i=1 1 |η i | 2H i -1 d i=1 1 |η i | 2H i -1 Γ H 0 ,n t (|η|)Γ H 0 ,n t (|η|) R d dξ {1 + |ξ| 2 } 2α χ(ξ -(η -η)) 2 .
With the change of variables ξ := ξ -(ηη), it holds that

R d dξ {1 + |ξ| 2 } 2α χ(ξ -(η -η)) 2 = R d dξ {1 + |ξ + (η -η)| 2 } 2α χ(ξ) 2 1 {1 + |η -η| 2 } 2α R d dξ {1 + |ξ| 2 } 2α χ(ξ) 2 .
As χ is a non-zero compactly-supported function, the support of χ contains a ball of radius R > 0 that guarantees

R d dξ {1 + |ξ| 2 } 2α χ(ξ) 2 > 0. Performing the changes of variables r 1 = η 1 -η 1 η 1 , • • • , r d = η d -η d η d (second inequality) followed by r1 = η 1 r 1 , • • • , rd = η d r d (third inequality), we write E χ • n (t, .) 2 H -2α 2 n √ d 0 dη 1 η 1 1 2 η 1 dη 1 • • • 2 n √ d 0 dη d η d 1 2 η d dη d Γ H 0 ,n t (|η|)Γ H 0 ,n t (|η|) {1 + |η -η| 2 } 2α d i=1 1 |η i | 2H i -1 d i=1 1 |η i | 2H i -1 2 n √ d 0 • • • 2 n √ d 0 dη 1 • • • dη d d i=1 1 |η i | 4H i -3 1 2 0 • • • 1 2 0 dr 1 • • • dr d {1 + η 2 1 r 2 1 + • • • + η 2 d r 2 d } 2α Γ H 0 ,n t (|η|)Γ H 0 ,n t η 2 1 (1 -r 1 ) 2 + • • • + η 2 d (1 -r d ) 2 2 n √ d 0 • • • 2 n √ d 0 dη 1 • • • dη d d i=1 1 |η i | 4H i -2 1 2 η 1 0 • • • 1 2 η d 0 dr 1 • • • dr d {1 + r 2 1 + • • • + r 2 d } 2α Γ H 0 ,n t (|η|)Γ H 0 ,n t η 2 1 1 - r 1 η 1 2 + • • • + η 2 d 1 - r d η d 2 .
The hyperspherical change of variables below

                         |η| = r η 1 = r cos(θ 1 ) η 2 = r sin(θ 1 ) cos(θ 2 ) . . . η d-1 = r sin(θ 1 ) • • • sin(θ d-2 ) cos(θ d-1 ) η d = r sin(θ 1 ) • • • sin(θ d-2 ) sin(θ d-1 )
whose absolute value of the Jacobian equals r d-1

d i=1 | sin(θ i )| d-1-i entails that E χ • n (t, .) 2 H -2α [ π 8 , π 4 ] d-1 dθ 1 • • • dθ d-1 2 n 2 dr r 4(H 1 +•••+H d )-3d+1 1 2 r cos θ 1 0 • • • 1 2 r sin(θ 1 )••• sin(θ d-1 ) 0 dr 1 • • • dr d {1 + r 2 1 + • • • + r 2 d } 2α Γ H 0 ,n t (r)Γ H 0 ,n t r 2 cos 2 θ 1 1 - r 1 r cos θ 1 2 + • • • + r 2 sin 2 θ 1 • • • sin 2 θ d-1 1 - r d r sin θ 1 • • • sin θ d-1 2 .
A quick view on the integration domain reveals that

r ≥ rθ := r 2 cos 2 θ 1 1 - r 1 r cos θ 1 2 + • • • + r 2 sin 2 θ 1 • • • sin 2 θ d-1 1 - r d r sin θ 1 • • • sin θ d-1 2 ≥ 1 2 r ≥ 1 .
Resorting to Lemma 5.1, the (forthcoming) lower bound (5.8) leads to

E χ • n (t, .) 2 H -2α [ π 8 , π 4 ] d-1 dθ 1 • • • dθ d-1 2 n 2 dr r 4(H 1 +•••+H d )-3d+1 1 2 r cos θ 1 0 • • • 1 2 r sin(θ 1 )••• sin(θ d-1 ) 0 dr 1 • • • dr d {1 + r 2 1 + • • • + r 2 d } 2α 1 r 8H 0 (1 -2e -t + e -2t ) 2 (1 -2e -t + e -2t ) 2 cos π 8 0 • • • sin( π 8 ) d-1 0 dr 1 • • • dr d {1 + r 2 1 + • • • + r 2 d } 2α 2 n 2 dr r 4(2H 0 +H 1 +•••+H d )-3d+1 . As 2H 0 + H 1 + • • • + H d ≤ 3 4 d, 4(2H 0 + H 1 + • • • + H d ) -3d + 1 ≤ 1, and consequently 2 n 2 dr r 4(2H 0 +H 1 +•••+H d )-3d+1 -→ n→+∞ +∞ .
We then get the desired conclusion

E χ • n (t, .) 2 H -2α -→ n→+∞ +∞. Lemma 5.1. For all H 0 ∈ (0, 1), n ≥ 1, t > 0 and r ∈ [1, 2 n ],
Γ H 0 ,n t (r) 1 r 4H 0 (1 -2e -t + e -2t ) > 0.

(5.8)

Proof. Remember that |γ t (ξ, r)| 2 = 1 -2 cos(ξt)e -r 2 t + e -2r 2 t r 4 + ξ 2 .

It is clear that Γ H 0 ,n t (r) = (1 + ξ 2 )ξ 2H 0 -1 dξ ≥ 2 r 4H 0 1 0 dξ (1 + ξ 2 )ξ 2H 0 -1 (1 -2e -r 2 t + e -2r 2 t ) > 0. Indeed, if we set for all t ≥ 0, r ≥ 1, h(t, r) := 1 -2e -r 2 t + e -2r 2 t , it is quite easy to check that, when r ≥ 1, h(., r) is strictly increasing on [0, +∞) and, since h(0, r) = 0, for every t > 0, h(t, r) = 1 -2e -r 2 t + e -2r 2 t > 0. Moreover, we can verify that if t ≥ 0, h(t, .) is increasing on [1, +∞) that provides the conclusion. i) The statement of Proposition 1.3 with α := -β and χ := ρ ensures the existence of a measurable set Ω of measure one on which ρ (ω) ∈ X β,p (T ). Now, the well-posedness result of Theorem 1.5 comes from the application of Theorem 2.3 (in an almost sure way) to Ψ := ρ .

ii) By reproducing the arguments that can be found in the proof of [START_REF] Deya | A nonlinear wave equation with fractional perturbation[END_REF]Theorem 1.7], we see that the convergence property is a consequence from the continuity of Γ T,Ψ with respect to Ψ (along (2.5)) and the almost sure convergence of χ n to χ . As p ≥ 2, observe that α < 1 2 ≤ 1 -2 2p and we can pick α < β < min 2α -d p , 2 -2α . i) Proposition 1.3 and Proposition 1.8 applied with α > 0 guarantees the existence of a measurable set Ω of measure one on which (ρ (ω), ρ 2 (ω)) ∈ R α . Again, Theorem 1.10 results from the application of Theorem 2.3 (in an almost sure way) to (Ψ, Ψ 2 ) := (ρ , ρ 2 ). ii) As before, the convergence property is a consequence from the continuity of Γ T,Ψ,Ψ 2 with respect to (Ψ, Ψ 2 ) and the almost sure convergence of (χ n , χ 2 n ) to (χ , χ 2 ).

| 2H 1 -1 |η 2 | 2H 2 -1 ½ |η|≥2 n 2 -2nε 1 |η 1 | 2(H 1 -ε)-1 |η 2 | 2H 2 -1 + 2 -2nε 1 |η 1 | 2H 1 -1 |η 2 | 2(H 2 -ε)-1
which, thanks to Corollary 4.3, immediately implies II m,n (s, t, u) 2 -nε |t -s| ε K H ε,0,1 (η) + K H ε,0,2 (η) .

The lemma below brings back computations on compact domains and its proof can be found in [START_REF] Deya | A nonlinear Schrödinger equation with fractional noise[END_REF]Lemma 2.6]. Lemma 7.2. Let χ : R d → R be a test function and fix σ ∈ R. Then, for every p ≥ 1 and for all η 1 , . . . , η p ∈ R d , it holds that

R d dx p i=1 (R d ) 2 dλ i d λi {1 + |λ i | 2 } σ 2 {1 + | λi | 2 } σ 2 e ı x,λ i -λi χ(λ i -η i ) χ( λi -η i ) p i=1 1 {1 + |η i | 2 } σ ,
where the proportional constant only depends on χ and σ.

We end this subsection by a highly technical lemma that permits us to construct the second order stochastic process when d = 2 in the roughest case.

(R 2 ) 2 dηdη {1 + |η -η| 2 } 2α K H (η)K H (η) < ∞ .

Proof.

It is a simple adaptation of the proof of [START_REF] Deya | On a non-linear 2D fractional equation[END_REF]Lemma 3.3].

1

 1

1. 2 .

 2 Notations. Let d ≥ 1. Throughout the paper, a test function (on R d ) will be any function ρ : R d → R that is smooth and compactly-supported. Naturally, let us denote by S(R d ) the space of Schwartz functions on R d .

. 14 )

 14 Putting together the four inequalities (3.9), (3.10),(3.11) and(3.14) provides us with the desired bound(3.7). The second one (3.8) can be obtained with similar arguments.

  2nε , and from there, a classical use of the Borell-Cantelli lemma justifies the desired almost sure convergence of (χ n ) n≥1 to χ in C([0, T ]; W -α,p (R d )), for every 2 ≤ p < ∞. To end with, the convergence in C([0, T ]; W -α,∞ (R d )) is the result of the Sobolev embedding W -α+ d p +η,p (R d ) ⊂ W -α,∞ (R d ), for any η > 0. 4.4. Construction of the second order stochastic process.

1 - 1 x α-1 1 + x 2 dx 1 x 0 dy y 1 - 1 0x α-1 1 + x 2 dx 1 0 1 x α-1 1 + x 2 dx 1 x 0 dy 2

 11111111102 2 cos(T y)e -T xy + e -2T xy + +∞ 2 cos(T y)e -T xy + e -2T xy . Then, one has by derivation f ′ (T ) = dy 2 sin(T y)e -T xy + 2x cos(T y)e -T xy -2xe -2T xy + +∞ sin(T y)e -T xy + 2x cos(T y)e -T xy -2xe -2T xy .

Lemma 4 . 5 .

 45 Let a, x and T three positive numbers. It holds that: a 0 sin(T y)e -T xy dy = -xe -T ax sin(T a) (1 + x 2 )T + 1e -T ax cos(T a) (1 + x 2 )T ; a 0 cos(T y)e -T xy dy = e -T ax sin(T a) (1 + x 2 )T + x(1e -T ax cos(T a)) (1 + x 2 )T ; a 0 e -2T xy dy = 1e -2T xa 2T x .

1 0x α-1 1 + 1 x α-1 1 +

 1111 x 2 dx e -2T x -2 cos(T )e -T x + +∞ x 2 dx e -2T -

4 n - 4 n 2 -r 2 2 -r 2 2 0 1 -

 44222221 |γ t (ξ, r)| 2 |ξ| 2H 0 -1 dξ ≥ r |γ t (ξ, r)| 2 |ξ| 2H 0 -1 dξ . Now, by parity, r |γ t (ξ, r)| 2 |ξ| 2H 0 -1 dξ = 2 r |γ t (ξ, r)| 2 |ξ| 2H 0 -1 dξ 2 cos(r 2 ξt)e -r 2 t + e -2r 2 t

6 . 1 . 5 .

 615 Proof of the main results6.Proof of Theorem 1.Suppose that d ≥ 1. Let p ≥ 2 and β be such that 0< β < 2H 0 + d i=1 H id and d 2p < 1 + β 2 . Recall that for every T ≥ 0, X β,p (T ) := C([0, T ]; W β,p (R d )) .(6.1)

  x 1 ...∂ x d B where B = B H is a space-time fractional Brownian motion of Hurst index H = (H 0 , H 1 , ..., H d ) ∈ (0, 1) d+1 . The definition of this process is the following:

Definition 1.1. Fix d ≥ 1 a space dimension, T ≥ 0 a positive time and (Ω, F, P) a complete filtered probability space. On this space, and for any fixed

H = (H 0 , H 1 , ..., H d ) ∈ (0, 1) d+1 , a centered Gaussian process B : Ω × ([0, T ] × R d ) → R

is said to be a space-time fractional Brownian motion of Hurst index H if its covariance function is given by the formula: E B(s, x 1 , ..., x d )B(t, y 1 , ..., y d ) = R H 0 (s, t)

Details about the definition of the linear solution.

  Fix a ∈ R and let g : [a, +∞[→ R, h : [a, +∞[→ (0, ∞), be two continuous functions. If g(t) ∼

	Lemma 4.6. t→∞	h(t) and +∞ a	h(t)dt = ∞, then
			T		T
			a	g(t)dt ∼ T →∞	a	h(t)dt .
	4.6.
	0	2 n	dr r -2α+1 |ξ|≤4 n	dξ |ξ| α-1 |γ t (ξ, r)| 2 ∼ n→∞	π ln(2) sin( απ 2 )	n.

  η) , for some index set A such that a i ∈ {n, m, {n, m}}, b i ∈ {s, t, {s, t}}, and one has both {a 1 , . . . , a 4 } ∩ {{n, m}} = ∅ and {b 1 , . . . , b 4 } ∩ {{s, t}} = ∅. It leads us to

  6.2. Proof of Theorem 1.10. Suppose that d ≥ 1 and p ≥ 2 verifies that d 2p ≤ 3 4 . Assume that 2H 0+ d i=1 H i ≤ d. Fix α > 0 such that d -2H 0 + -d 2p and we can pick α < β < min 2α -d p , 2 -2α . Recall that R α,p := L ∞ [0, T ]; W -α,p (R d ) × L ∞ [0, T ]; W -2α,p (R d ) . (6.2) i) Proposition 1.3 and Proposition 1.8 applied with α > 0 guarantees the existence of a measurable set Ω of measure one on which (ρ (ω), ρ 2 (ω)) ∈ R α,p . Now, the statement of Theorem 1.10 results from the application of Theorem 2.3 (in an almost sure way) to (Ψ, Ψ 2 ) := (ρ , ρ 2 ). ii) Again, we remark that the convergence property is a consequence from the continuity of Γ T,Ψ,Ψ 2 with respect to (Ψ, Ψ 2 ) and the almost sure convergence of (χ n , χ 2 n ) to (χ , χ 2 ). 6.3. Proof of Theorem 1.15. Suppose that d = 2 and p ≥ 2. Let (H 0 , H 1 , H 2 ) ∈ (0, 1) 3 be

		d i=1	H i < α <	1 4	.
	As d 2p ≤ 3 4 , observe that α < 1 4 ≤ 1 such that 0 < H 1 < 3 4 , 0 < H 2 <	3 4	,	3 2	< 2H 0 + H 1 + H 2 ≤	7 4	.	(6.3)
	Fix α > 0 such that
	2 -(2H

0 + H 1 + H 2 ) < α < 1 2

.

  In this subsection, we state the three technical lemmas at the core of the proof of Proposition 1.14.H ε,0 (η) + K H ε,0,1 (η) + K H ε,0,2 (η) , (7.2)whereH ε,0 := (H 0ε, H 1 , H 2 ), H ε,0,1 := (H 0ε, H 1ε, H 2 ), H ε,0,2 := (H 0ε, H 1 , H 2ε),and the proportional constants do no depend on (n, m), (s, t), u and η.2H 1 -1 |η 2 | 2H 2 -1 (ξ,η)∈D m dξ |γ t (ξ, |η|)| 2 |ξ| 2H 0 -1 1 |η 1 | 2H 1 -1 |η 2 | 2H 2 -1 R dξ |γ t (ξ, |η|)| 2 |ξ| 2H 0 -1 K H ε,0 (η).The second one is a bit more technical. Recall that one has2H 1 -1 |η 2 | 2H 2 -1 (ξ,η)∈D n,m ∩D m dξ γ s,t (ξ, |η|)γ u (ξ, |η|) |ξ| 2H 0 -1 2H 1 -1 |η 2 | 2H 2 -1 (ξ,η)∈D n,m dξ |γ s,t (ξ, |η|)||γ u (ξ, |η|)| |ξ| 2H 0 -1 1 |η 1 | 2H 1 -1 |η 2 | 2H 2 -1 |ξ|≥2 2n dξ |γ s,t (ξ, |η|)||γ u (ξ, |η|)| |ξ| 2H 0 -1 + 1 |η 1 | 2H 1 -1 |η 2 | 2H 2 -1 ½ |η|≥2 n R dξ |γ s,t (ξ, |η|)||γ u (ξ, |η|)| |ξ| 2H 0 -1 .Let us call I m,n (s, t, u) (resp. II m,n (s, t, u)) the first (resp. the second) integral. Since 0 < ε < H 0 , Corollary 4.3 combined with Cauchy-Schwarz inequality entails: I m,n (s, t, u) 2 -2nε |t -s| ε 2 K H ε,0 (η), whereas, keeping in mind the identity {|η| ≥ 2 n } ⊂ {|η| ≥ 2 n , |η 1 | ≥

	H,(m,m) t,t 2 -nε |t -s| Proof. The first inequality is a straight consequence of Corollary 4.3 : (η) K H ε,0 (η) and |L H,((n,m),m) (s,t),u (η) L H,(m,m) t,t (η) = 1 H,((n,m),m) (s,t),u (η) := 1 |η 1 | which leads to L H,((n,m),m) (s,t),u (η) 1 |η 1 | 1 √ 2 |η|} ∪ {|η| ≥ 2 n , |η 2 | ≥ it holds 1 |η 1 | L |η 1	1 √ 2	|η|},	(7.1)

7. Appendix 7.1. Technical lemmas. Lemma 7.1.

For all H = (H 0 , H 1 , H 2 ) ∈ (0, 1) 3 , ε ∈ (0, H 0 ), 0 ≤ n ≤ m, 0 ≤ s, t, u ≤ 1 and η ∈ R 2 , it holds that |L ε 2 K

  Lemma 7.3. For all H = (H 0 , H 1 , H 2 ), H = ( H0 , H1 , H2 ) ∈ (0, 1) 3 verifying

	0 < H 1 , H1 <	3 4	, 0 < H 2 , H2 <	3 4	,	3 2	< 2H 0 +H 1 +H 2 ≤	7 4	,	3 2	< 2 H0 + H1 + H2 ≤ (7.3) 7 , 4
	and any it holds that		α ∈ max 2 -(2H 0 + H 1 + H 2 ), 2 -(2 H0 + H1 + H2 ) ,	1 2	,	(7.4)

By developing the integrand in Taylor series, we write

.

To continue the computation, we need to introduce two well-known special functions, namely:

Definition 7.4. The Gamma function Γ and the digamma function Ψ are defined for all x > 0 by the formulas

We are now in a position to recall the following classical result from analytic number theory:

Lemma 7.5. For all a, b > 0, it holds that:

Coming back to (7.5), with the help of Lemma 7.5, we deduce

and, finally, for your viewing pleasure,

.


Anne Bonneau
email: anne.bonneau@loria.fr

Voicing assimilations by French speakers of German in stop-fricative sequences

Keywords: voicing assimilation, L2, obstruents, French/German interferences, speech rate

Voicing assimilations inside groups of obstruents occur in opposite directions in French and German, where they are respectively regressive and progressive. The aim of the study is to investigate (1) whether non native speakers (here French learners of German) are apt to acquire subtle L2 specificities like assimilation direction, although they are not aware of their very existence, or (2) whether their productions depend essentially upon other factors, in particular consonant place of articulation. To that purpose, a corpus made up of groups of obstruents (/t/ followed by /z/, /v/ or /f/) embedded into sentences has been recorded by 16 French learners of German (beginners and advanced speakers). The consonants are separated by a word or a syllable boundary. Results, derived from the analysis of consonant periodicity and duration, do not stand for an acquisition of progressive assimilation, even by advanced speakers, and do not show differences between the productions of advanced speakers and beginners. On the contrary the boundary type and the consonant place of articulation play an important role in the presence or absence of voicing inside obstruent groups. The role of phonetic, universal mechanisms against linguistic specific rules is discussed to interpret the data.

Introduction

The phonological contrast opposing fortis to lenis consonants differs a lot in French and German, both at the phonological and phonetic levels. The presence of fortis aspirated stops [START_REF] Jessen | Phonetics and phonology of tense and lax obstruents in German[END_REF] and the neutralization of the fortis/lenis opposition in final position [START_REF] Wiese | The phonology of German[END_REF] [START_REF] Port | Neutralization of syllable-final voicing in German[END_REF], which characterize German and not French, are well known. It is probably less known that the direction of voicing assimilations occurring between two obstruents C1 and C2, with C1 in final and C2 in initial positions, are different in both languages. Note that, in this paper, the term "voicing" refers to the presence of vocal fold vibration, not to the phonological contrast. In French, in this context, voicing assimilations are in general regressive, and occur from the second consonant to the first one [START_REF] Snoeren | Perceptual processing of partially and fully assimilated words in French[END_REF] [START_REF]From speech physiology to linguistics phonetics[END_REF]. In German, assimilations between obstruents are progressive and occur from the first consonant, in final position and thus fortis, to a following lenis fricative in initial position [START_REF] Kuzla | Prosodic strengthening of German fricatives in duration and assimilatory devoicing[END_REF]. The differences in assimilation directions raise the question as to whether non native speakers modify their L1 direction. We consider here the case of French learners of German uttering C1C2 sequences that elicit progressive assimilation in German (i.e. made up of a fortis consonant followed by a lenis fricative). The realization of assimilation phenomena, their frequency (the number of occurrences) and magnitude (such as the percentage of segment duration affected by the phenomenon), depend upon a number of factors [START_REF] Nespor | Prosodic phonology[END_REF], including notably the prosodic boundaries which separate the potentially assimilated and assimilating segments, and speech rate. The frequency and the extent of assimilations should indeed be more important at weaker boundaries than at higher ones, and should decrease with decreasing speech rates.

Other factors have a direct impact on voicing and affect in turn assimilations, not only their frequency and magnitude, but also the conditions necessary for their emergence. Let us first mention the German phonological system. In word initial position, there is a voicing contrast between /f/ and /v/, but not between /s/ and /z/ (/s/ is not present at word onsets apart from loanwords) [START_REF] Jessen | Phonetics and phonology of tense and lax obstruents in German[END_REF]. This means that, in this context, /z/ can be devoiced with no or minor impact on intelligibility. Thus, we can expect relatively frequent and (sometimes) important devoicing of /z/, especially when the consonant is preceded by a fortis consonant. Kuzla et al. [START_REF] Kuzla | Prosodic strengthening of German fricatives in duration and assimilatory devoicing[END_REF] have shown that German speakers indeed devoiced more /z/ than /v/, which can be partly due to the German phonological system. A major source of pronunciation errors is due to orthographic interferences [START_REF] Koda | Effects of L1 orthographic representation on L2 phonological coding strategies[END_REF]. In word initial position, the letter "s" corresponds to /z/ in German but to /s/ in French, which is a source of errors for beginners. Finally, aerodynamic phenomena also play a role in the devoicing of lenis fricatives. It is indeed difficult to maintain voicing during the production of fricatives [START_REF] Fuchs | Temporal and spatial aspects concerning the voicing contrast in alveolar and postalveolar fricatives[END_REF]. We can thus expect that /z/ (and, to a lesser extend, /v/, which is produced with less frication) will be at least slightly devoiced during their productions due to aerodynamic phenomena.

Assimilations are not taught in foreign language classes and speakers are not aware of their very existence, even in their own language. They might thus be difficult to acquire. The few studies on L2 assimilation acquisition yield conflicting results. On the one hand, the results of Schmidt [START_REF] Schmidt | Contextual Variation in L2 Spanish: Voicing Assimilation in Advanced Learner Speech[END_REF] concerning the acquisition of regressive assimilation of voicing by English learners of Spanish showed that few learners, even the most advanced ones, exhibited L2 assimilation in their productions. On the other hand, in a series of perceptual experiments about assimilation compensation, Darcy et al. [START_REF] Darcy | Bilinguals play by the rules: perceptual compensation for assimilation in late L2learners[END_REF] showed that, whereas beginners applied their native compensation pattern to L2, advanced speakers were able to compensate for L2 assimilation processes. In a preliminary experiment investigating German C1C2 sequences made up, in particular, of a fortis cluster followed by /z/ and pronounced by French learners, we found that about half of the French speakers devoice the alveolar fricative [START_REF] Bonneau | German obstruent sequences by French L2 leaners[END_REF]. In the light of what was previously said, we can wonder whether such results can be attributed to the acquisition of German assimilation or to the characteristics of /z/. The aim of the study, which extends the previous one to all the fricatives present in German at word boundary (/v/, /z/ and /f/) and to different boundary types, is to investigate whether (1) French learners of German are apt to modify such subtle L2 specificities as assimilation direction, or, on the contrary (2) keep their L1 habits, and, finally (3) whether their productions depend upon other factors, in particular prosodic boundary, or consonant place of articulation.

Experimental protocol 2.1. Corpus

The corpus is made up of groups of C1C2 consonants, where the first consonant, C1 is the fortis stop /t/ and the second consonant is the fricative /z/, /v/ or /f/. Groups of consonants appear at two different linguistic boundaries 1) inside a carrier sentence, at word final and initial positions and 2) inside words, at syllable final and initial positions. In the first context, the carrier sentence has the following structure « X hat Y gemalt » (« X has painted Y »), where X is a first name and Y is either Wälder, Senken, or Felder, which represents three different sentences. We have taken here a structure very similar to that of a series of sentences chosen by Kuzla et al. [START_REF] Kuzla | Prosodic strengthening of German fricatives in duration and assimilatory devoicing[END_REF], so as to allow comparisons between results. In the second context, C1 and C2 are respectively in syllable final and initial position inside a word. The three words chosen were «Tatsache » (/tz/ sequence), «etwas » (/tv/ sequence) and «Bootfarhen » (/tf/ sequence).

Sixteen speakers, seven advanced (C1-C2) and nine beginners (A2-B1) recorded the corpus. There was approximately the same number of male and female speakers in each category. There were five repetitions for each sentence of the corpus in the first context (word boundary) and three repetitions in the second context (syllable boundary), for a total amount of 384 groups (4 repetitions on the average x 2 contexts x 16 speakers x 3 consonant categories).

During the recording session, the subjects were seated in a quiet room and read the sentences from the screen of a Windows laptop, with a headset microphone (AKG C520) connected to an Audiobox (M Audio Fast track). The subjects uttered, in the following order, (1) a series of six sentences to get used to the recording procedure, (2) the sentences with obstruents at word boundary, (3) the words with obstruents at syllable boundary. The sentences and the words were presented in a random order. We used the software Corpus Recorder for calibration and randomization [START_REF] Colotte | Corpus Recorder[END_REF].

Acoustical analysis

Segmentation have been done by hand by the author, using Praat. We have evaluated two acoustic cues: duration and periodicity. Duration, given by segmentation, has been estimated for C1 and C2, their surrounding vowels, and the whole sentence. We analysed the periodicity (glottal pulses) of the signal for each consonant of the consonantal sequence. It has been estimated by Praat "Voicing Report" function which provides the fraction of locally unvoiced frames during a segment, from which we deduced the fraction (the percentages) of voiced frames during the obstruents. We checked the results manually, and corrected them when necessary (there was few corrections, made for obvious mistakes).

Results

At word boundary only, there was sometimes a silent pause between the end of the first word, after C1, and the beginning of the second one, after C2. This concerns about 9% of all the realizations (both boundary types confounded). We separated these realizations from the others, and considered three conditions: C1C2 in word condition, with the consonants separated by pause (first condition), C1C2 in word condition, without pause (second condition), and C1C2 in syllable position (third condition).

Results per consonant.

We build four linear mixed models, one for each numeric factor (C1 and C2 % of voiced frames, C1 and C2 durations), with the numeric factor as the dependent variable; three fixed factors corresponding to the consonant category, the condition and the level, as well as a random factor (the speaker), as the predictors. Note that, for C1 (/t/), the factor "consonant category" corresponds to the category of the following consonant (C2). We then submitted each model to an anova analysis (Satterthwaite's method). For post hoc tests, we used Kenward-Roger's method to evaluate the degree of freedom. All the statistical procedures were made with R [START_REF] Team | R: A Language and Environment for Statistical Computing[END_REF]. We begin by a short presentation of duration's data and discuss important differences between duration values.

C2 consonant duration.

There is a main effect of the category (F(2, 352)= 60, p < 2.2e-16) and a two-way interaction between the category and the condition (F(4, 352) = 6.5, p < 3.8e-05). As expected, the fortis consonants (median at 130 ms) are significantly longer than others [START_REF] Jessen | Phonetics and phonology of tense and lax obstruents in German[END_REF] (medians at 95 for /z/ and 79 ms for /v/), this is true in each condition (p < 0.0001, in ecah case), with only one exception (/f/ and /z/ in the first condition). Concerning the lenis consonants, we find that /z/ is significantly longer than /v/ in the first (p < 0.003) and the third condition (p < 0.001), but not in the second. C1 consonant duration. There is a main effect of the condition (F(2,352) = 128.7, p <2e-16), and a two-way interaction involving the level and the condition (F(2, 352) = 4.2, p < 0.015). Post hoc analyses determining the effect of the condition show that, for each level, the realizations of /t/ are highly significantly longer in the first condition (presence of a pause) than in each other condition (p <0.001, for each comparison). The difference between the second and the third condition is also significant for beginners (p <0.001), these speakers producing slightly longer dentals at word boundary than at syllable boundary. The differences due to the presence or absence of a pause are striking : the median for the three consonants is at 174 ms, in the presence of a pause, and 83 ms in its absence, and there is a very small overlap area between both contexts. This is mainly due to the nature of the boundary, the duration is longer at important boundaries [START_REF] Wightman | Segmental durations in the vicinity of prosodic phrase boundaries[END_REF], and the nature of the sound since for stops, the burst can be fully released and very long in this context. The importance of durations in the first condition, which are far above those observed by Kuzla et al. [START_REF] Kuzla | Prosodic strengthening of German fricatives in duration and assimilatory devoicing[END_REF] when /t/ is preceded by a pause, is probaby also specific to L2 realizations.

Percent of voiced frames for C2.

There is a significant main effect of the category of the consonant (F(2,352) = 53.5, p <2.2e-16), the condition (F(2,352) = 12.2, p <7.31e-06) as well as a significant interaction (F(4,352) = 5.3, p <0.0004) between both factors. The speakers' level has no significant effect on C2 values. Figure 1 represents, for C2, the % of voicing for each consonant in each condition. Post hoc analyses examine the effect of the category (a main effect) as a function of the condition, because of the interaction between both factors, and for the same reasons, the effect of the condition as a function of the category. Concerning the effect of the category on voicing values, the analyses reveal that, for a given condition, the differences between the three consonants, /f/, /v/ and /z/, are significant (p<0.01, for all significant cases) but in two cases among nine: (1) /v/ and /z/ in the first condition, and, (2) /f/ and /z/ in the third condition.

Figure 1. Boxplots for C2 % of voiced frames as a function of the category and the condition (e.g. "v1" represents /v/ in the first condition).

We examine now the effect of the condition for each given consonant, which amounts to examining whether the values for each consonant vary significantly as a function of the nature of the boundary. There is no significant difference between conditions for the consonant /f/, and there is also no significant difference between the first and the second conditions for /v/ and /for /z/. On the contrary, the differences between the third condition and each other condition for /v/ and for /z/ are all significant (p <0.002, in each case). In these cases, the consonant which is at the lower boundary (third condition) has a lower amount of voicing. The rank order of the median values calculated for the three consonants is in agreement with what is expected since, for a given condition: (1) /f/, always unvoiced, has a lower median than /v/ and /z/ (but for the third condition, where /f/ and /z/ do not differ significantly, and the median for /z/ is slightly lower than that of /f/) and (2) /v/ is more voiced than /z/ (the difference is not significant in the first condition). Whereas the medians for /v/ are always above 50% and those for /f/ under 50 %, we remark that the medians for /z/ in the second, and, especially, the third conditions are under 50 % (see Figure 1). Furthermore, there is a very important dispersion of values for lenis consonants with a number of data under 50%, especially for /z/ in the second condition, but also for /v/ in the second and the third condition (as shown in Figure 1). These low values for lenis consonants can be attributed to a progressive assimilation of devoicing. We will come back below to the very low values observed for /z/ in the third condition. The differences observed between the conditions for the lenis consonants, showing a lower amount of voicing at the lower boundary, reinforces the hypothesis of the presence of a devoicing assimilation phenomenon. The result obtained for /z/ in the third condition seems strange (at syllable boundary the amounts of voicing for /z/ and /f/ are not significantly different, and the median for /z/, at 17%, is very low). Although this could be interpreted in terms of a very strong devoicing assimilation from /t/ to /z/ at syllable boundary, we tend to think that this is mostly due to an error due to the spelling of /z/, which is written "s". Most learners probably consider "s" as an unvoiced consonant when "s" appears inside a word and is preceded by a fortis consonant.

Percent of voiced frames for C1.

As it is the case for C2 voicing, there is no effect of the speakers' level, but a main effect of the category of the following consonant (F(2,352) = 7.3, p <0.0007), the condition (F(2,352) = 13.7, p <1.7e-06) as well as a significant interaction between both factors (F(4,352) = 6.9, p <2.1e-05). Figure 2 represents, for C1, the % of voicing for each consonant in each condition.

Figure 2. Boxplots for C1 % of voiced frames as a function of the context and the condition (e.g. "tv1" represents /t/ followed by /v/ in the first condition)

Let us first examine the effect of the following context (C2 category) on /t/ realizations, per each condition. In the first condition, there is no difference due to C2 category. In the second condition, there are highly significant differences between the realizations of /t/ followed by the fortis /f/, and those followed by each lenis consonant, (p <0.001, in each case), and no significant difference between the lenis contexts. In the third condition, there are significant differences between the context /v/ and each other context (p <0.002, in each case), and no significant differences between /f/ and /z/ contexts. What is the impact of the condition on /t/ realizations, for a given consonantal context ? When /t/ is followed by /f/, the differences due to the conditions are either non significant or marginal. When /t/ is followed by /v/, the differences between the first condition and each other condition are very highly significant (all p <0.001), whereas there is no difference between the second and the third condition. When /t/ is followed by /z/, all the differences due to conditions are significant (all p <0.01). These results are consistent with the presence of a regressive assimilation of voicing from the lenis consonants to the fortis /t/, when the context is favourable to the emergence of this phenomenon, i.e. when the consonant /t/ is followed by a lenis consonant, and when the consonants C1 and C2 directly follow each other. As already noted, a pause strongly reduces or eliminates assimilation. In the first condition indeed, where C1 and C2 are separated by a pause, the realizations of /t/ are not affected by the nature (fortis or lenis) of the following consonant. The median values estimated for each context are around 20%, which is characteristic of an unvoiced consonant. In the second condition, whereas /t/ stays unvoiced when followed by /f/, equally unvoiced (median at 32%), the median values observed in lenis contexts (61 and 50%, for /v/ and /z/) clearly indicate that, in most cases, /t/ undergoes a regressive assimilation of voicing. This assimilation is not systematic, as shown by the large dispersion of values in lenis contexts (see Figure 2). In the third condition, we observed the same kind of realizations than in the second condition for /t/ followed by /f/ and /v/, but not for /t/ followed by /z/. In this context, the median (36%) indicated that most realizations of /t/ stays unvoiced. This observation, as well as the highly significant difference between /z/ contexts in word and syllable conditions strengthen our hypothesis of a possible spelling error for /z/ in the third condition. Finally the highly significant and visible differences between the first condition (presence of a pause) and the other ones for /t/ followed by /v/, and between the first and the second condition for /t/ followed by /z/, is probably due to the absence of assimilation when a pause separates the consonants, and also to the differences in consonant length (/t/ being significantly longer before a pause, as underlined above).

Results for C1C2 voicing patterns.

After having analysed the voicing of each consonant separately, this section examine the whole C1C2 sequences, inside which assimilation occurs. Before the analysis of the results, it is important to underline here that 1) the extent of assimilation is variable, and only a part of the assimilated consonant might be affected by the phenomenon; studies on lenis fricatives preceded by fortis consonants and pronounced by German speakers [START_REF] Kuzla | Prosodic strengthening of German fricatives in duration and assimilatory devoicing[END_REF] [START_REF] Shih | Contextual effect on consonant voicing profiles: a cross-linguistic study[END_REF] indeed show that /v/ and /z/ are not entirely assimilated, 2) assimilation is not systematic, and its occurrence depends upon a number of factors, as mentioned in the introduction. Table 1 displays, for the sequences /tv/ and /tz/ in condition 2 and 3, favourable to the appearance of assimilation phenomena, the percentages of C1C2 groups for which 1) each consonant is mainly voiced ("VV" groups), 2) each consonant is mainly unvoiced ("UU" groups), 3) the first is mainly unvoiced and the second mainly voiced ("UV") and 4) the first is mainly voiced and the second mainly unvoiced ("VU"). We have considered that a consonant is mainly voiced when its percent of voicing is above 50%, and mainly unvoiced in the other case. We add two configurations, "FU" (fully unvoiced), for which both consonants percentages of voicing are under 20%, and "FV" (fully voiced), for which both consonants percentages of voicing are above 80%.

Discussion and concluding remarks

Results show that there is no systematic voicing assimilations in either direction, progressive or regressive, and that there are no significant differences between advanced speakers and beginners. This tends to corroborate the results obtained by Schmidt [START_REF] Schmidt | Contextual Variation in L2 Spanish: Voicing Assimilation in Advanced Learner Speech[END_REF], in her study about L2 assimilation production, and does not correspond to the perceptual results obtained by Darcy et al. [START_REF] Darcy | Bilinguals play by the rules: perceptual compensation for assimilation in late L2learners[END_REF], presented in the introduction. The absence of systematic assimilations in one direction is clearly shown by the results obtained for /v/ and /z/ since, in the second condition, favourable to the presence of assimilation and inside which both consonants are considered by learners as lenis, the ratios between VV (both consonants are mainly voiced) and UU (both consonants are mainly unvoiced) sequences are of 50/17 for /v/ against 37/43 for /z/ (see Table 1). In French, assimilation in obstruent groups is mostly regressive in /tv/ and /tz/ sequences. In German, results obtained by Kuzla et al. [START_REF] Kuzla | Prosodic strengthening of German fricatives in duration and assimilatory devoicing[END_REF], in word condition (more precisely in the "WORD" prosodic boundary, relatively similar to condition 2), exhibited a very important degree of devoicing assimilation for /z/ and, to a lesser extent, also for /v/ (median for the % of voicing frames around 20%), which is often devoiced by native speakers. The degree of devoicing assimilation is more important in Kuzla's study than in ours. The absence of clear L1 or L2 patterns could be attributed to a progressive acquisition of L2 patterns. But such an explanation is contradicted by the absence of difference between advanced speakers and beginners. We can also consider that the results are explained by more phonetic, universal mechanisms, and less language-specific rules, notably put into evidence, for L1, by Gow [17]. This explanation would be compatible with the differences observed between /tv/ and /tz/ sequences, the first being more voiced and the second more unvoiced, which corresponds to the differences between /v/ and /z/, due to aerodynamics (see introduction), and with the absence of difference between advanced speakers and beginners.

Table 1 Percentages

 1

			of voicing configurations
	(UU,FU, VV,FV,UV,VU) as a function of C1C2
	sequences and condition. See text for more
		explanations.
		UU (FU)	VU UV	VV (FV)
	tv cond. 2	17 (2)	10	23	50 (37)
	tz cond. 2	43 (17)	8	12	37 (32)
	tv3 cond. 3	27 (0)	16	16	41 (39)
	tz3 cond.3	68 (29)	19	2	11 (11)
	Let us first give results for sequences not mentioned in Table
	1. As expected, all /tf/ sequences are mainly unvoiced, since
	we obtained 100%, 95% and 80% of UU sequences, from

Acknowledgements

I am very grateful to Anne Gégout-Petit, Professor at IECL-UL-Nancy, expert in statistics, for her help with regard to the statistical analysis, and to all the speakers.