This is the accepted version of the journal article: Giordano, Luca M.; Jolis Giménez, Maria; Quer i Sardanyons, Lluís. «SPDEs with linear multiplicative fractional noise : Continuity in law with respect to the Hurst index». Stochastic Processes and their Applications, Vol. 130, Issue 12 (December 2020), p. 7396-7430. DOI 10.1016/j.spa.2020.08.001 This version is available at https://ddd.uab.cat/record/274592 under the terms of the license SPDEs with linear multiplicative fractional noise: continuity in law with respect to the Hurst index ∗3 Luca M. Giordano1 , Maria Jolis2 , and Lluı́s Quer-Sardanyons 1 Department of Mathematics, University of Milano, Via C. Saldini 50, 20133 Milano, Italy, and Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia, Spain. E-mail address: luca.giordano@unimi.it 2,3 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia, Spain. E-mail addresses: mjolis@mat.uab.cat, quer@mat.uab.cat. July 6, 2020 Abstract In this article, we consider the one-dimensional stochastic wave and heat equations driven by a linear multiplicative Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index H ∈ ( 41 , 1). We prove that the solution of each of the above equations is continuous in terms of the index H, with respect to the convergence in law in the space of continuous functions. The proof is based on a tightness criterion on the plane and Malliavin calculus techniques in order to identify the limit law. MSC 2010: 60B10; 60H07; 60H15 Keywords: fractional noise; stochastic heat equation; stochastic wave equation; weak conver- gence; Wiener chaos expansion 1 Introduction In this article, we consider the Hyperbolic Anderson Model  2 H ∂ u ∂ 2 uH (t, x) = (t, x) + uH (t, x)Ẇ H (t, x), (t, x) ∈ R+ × R,   2 2     ∂t ∂x uH (0, x) = η, x ∈ R, (SWE)  H   ∂u (0, x) = 0, x ∈ R,    ∂t and the Parabolic Anderson Model  H 2 H  ∂u (t, x) = 1 ∂ u (t, x) + uH (t, x)Ẇ H (t, x),  (t, x) ∈ R+ × R, ∂t 2 ∂x2 (SHE)  uH (0, x) = η, x ∈ R.  ∗ Corresponding author 1 The initial condition η ∈ R is assumed to be constant. The random perturbation Ẇ H is a Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index H ∈ ( 14 , 1). More precisely, it is given by a family of centered Gaussian random variables W H = {W H (ϕ), ϕ ∈ C0∞ (R+ ×R)}, indexed in the space of C ∞ functions with compact support on R+ × R, with the following covariance structure: Z ∞Z  H H  E W (ϕ)W (ψ) = Fϕ(t, ·)(ξ)Fψ(t, ·)(ξ)µH (dξ)dt, 0 R for any ϕ, ψ ∈ C0∞ (R+ × R), where the measure µH is given by µH (dξ) = cH |ξ|1−2H dξ, with Γ(2H + 1) sin(πH) cH = . (1) 2π We denote by F the Fourier transform in the space variable, which is defined by Z Ff (ξ) = e−iξx f (x)dx, f ∈ L1 (R). R The solutions of (SWE) and (SHE) are understood in the mild Itô sense, as follows. We fix a time horizon T > 0 and we denote by {FtH , t ≥ 0} the filtration generated by the noise W H (conveniently completed). Then, we say that an adapted and jointly measurable random field uH = {uH (t, x), (t, x) ∈ [0, T ] × R} solves (SWE) (resp. (SHE)) if it holds, for all (t, x) ∈ [0, T ] × R: Z tZ uH (t, x) = η + Gt−s (x − y)uH (s, y)W H (ds, dy), P-a.s. (2) 0 R where G is the fundamental solution of the wave (resp. heat) equation in R. We recall that  1    2 1{|x| 0. Let H0 ∈ ( 14 , 1) and {Hn , n ≥ 1} ⊂ ( 14 , 1) be any sequence converging to H0 . Then, uHn converges to uH0 , as n → ∞, in law in the space C([0, T ] × R)) of continuous functions, endowed with the metric of uniform convergence on compact sets. This result, apart from its intrinsic mathematical interest, provides also a good starting point towards the use of such equations in the field of applications. Indeed, if one wishes to model some physical phenomenon either with (SWE) or (SHE)), one of the problems that has to be faced is the estimation of the parameter H. This estimation cannot be exact, due to finiteness of available data in practice. In order to have a robust model, one wishes then that the solution of (SWE) and (SHE) are somewhat stable with respect to small changes in the value of H. The above Theorem 1.1, in this range of ideas, is a first qualitative result about this kind of stability. 2 We point out that we restrict to Hurst indices greater than 14 . This is due to the fact that, as proved in [4, Prop. 3.7], H > 41 is also a necessary condition in order to have a solution to (SWE) and (SHE). The above theorem can be considered a continuation of the results obtained by the authors in [15] (see Theorem 4.2 therein), where the same kind of problem has been addressed for one- dimensional quasi-linear stochastic wave and heat equations with an additive fractional noise as the one described above. The proof of the latter result, which is indeed valid for any H0 ∈ (0, 1), is based on the fact that the solution of the underlying SPDE can be represented as the image of the stochastic convolution through a continuous functional on the space C([0, T ] × R)). In the present paper, this technique cannot be applied anymore because of the structure of the linear multiplicative noise. Instead, we consider the following strategy. First, we prove that the sequence of probability measures induced by {uHn , n ≥ 1} is tight in the space C([0, T ] × R) (see Section 3). Here, we split the proof taking into account that the sequence of Hurst indices is contained in ( 41 , 12 ] or [ 21 , 1), for the definition and properties of the stochastic integral in (2) differ significantly between those two cases. Indeed, the main diffi- culty here is concentrated in the rough case, where we carefully extend some moment estimates appearing in [4] in order to make them uniform with respect to H ∈ ( 14 , 12 ). Secondly, in order to identify the limit law, we prove the convergence of the corresponding finite dimensional distributions (see Section 4). The main problem here comes from the fact that the solution uH is not a Gaussian process, and so identifying its covariance structure is not enough to characterize its law. However, thanks to a spectral representation in law of our noise W H in terms of a complex-valued Gaussian measure (extending of some classical results in [25]), we are able to define the whole family of noises {W H , H ∈ (0, 1)} in a single probability space and then check that, for any fixed (t, x) ∈ [0, T ] × R, uHn (t, x) converges to uH0 (t, x) in L2 (Ω). For this, we will use techniques of the Malliavin calculus, precisely the Wiener chaos expansion of the mild Skorohod solutions of (SWE) and (SHE). In the process of applying this methodology, we provide three preliminary results which have their own interest and turn out to be crucial in our main result’s proof: (i) For any H ∈ (0, 1), we prove that any multiple Wiener integral with respect to W H admits a representation as a multiple Wiener integral with respect to the above-mentioned complex-valued Gaussian measure (see Theorem 2.8). (ii) For any H ∈ ( 14 , 1), we prove an equivalence result between Itô and Skorohod stochastic integrals with respect to W H (see Theorem 2.13). This result has already been proved in [4, Thm. 4.2] for the case H < 21 , and we extend it to H ≥ 21 . We point out that the latter case, in which the noise is more regular, entails some extra difficulties due to the fact that the underlying Hilbert space associated to the noise’s covariance contains distributions. An important consequence of Theorem 2.13 is that mild Itô and Skorohod solutions to (SWE) (resp. (SHE)) coincide, and the corresponding Picard iteration scheme admits a (finite) Wiener chaos decomposition. (iii) In the setting H ∈ ( 14 , 21 ), we prove a Sobolev embedding-type result for the norms of the Banach space on which we define our solutions (see Lemma 2.19). This result is similar to classical embedding results, e.g. the ones appearing in [14], but takes into account the different nature of the Sobolev norm in our setting. The above strategy will be made clearer in Section 4 below, but let us remark at this point that the main strategy in this part of the paper does not require a separate analysis for the cases 3 H < 21 and H ≥ 12 . Furthermore, the methodology used in both results on tightness and the limit identification cover equations (SWE) and (SHE) at the same time. In the case of the stochastic heat equation (SHE) with H > 12 , the result in Theorem 1.1 is a particular case of [6, Thm. 1], where the author considers a general non-linear coefficient σ(uH (t, x)) in front of the noise. We believe that such diffusion coefficient could be also con- sidered in the case of the wave equation with H > 21 , but we have chosen to stick to the linear multiplicative noise in order to find a unified result that covers also the case H < 12 , which is more mathematically demanding. Concerning other related results, we point out the recent article [22], in which the authors prove strong regularity properties in H ∈ (0, 1) of the Mandelbrot-van Ness representation of the fractional Brownian motion. As a consequence, it is proved that the solution of a scalar stochastic differential equation driven by the fractional Brownian motion is differentiable with respect to the Hurst parameter. We also mention that continuity in law with respect to the Hurst index has been focused in other type of contexts beyond stochastic equations. For instance, in the series of papers [19, 20, 21], the authors study weak continuity with respect to H for different types of integrals with respect to fractional Brownian motion. In [18, 30], the same kind of continuity property has been tackled for the local time of the fractional Brownian motion and other Gaussian fields. Eventually, in the recent paper [1], the continuity property has been shown for additive functionals of the sub-fractional Brownian motion. Finally, let us make some comments on a possible extension of our result. Namely, one could consider an SPDE with a Caputo derivative in time, which corresponds to an interpolation between the wave and heat equations (see, e.g., [23, 8, 9, 10]). Nevertheless, for this type of equations, existence and uniqueness of solution, as well as pathwise continuity, should be first established in the case where the Hurst index is less that 12 . The paper is organized as follows. In Section 2, we give some preliminary tools that will be needed throughout the paper. Namely, we introduce the basic elements of the Malliavin calculus, we provide a new integral representation for the multiple Wiener integral with respect to W H , we recall the construction of the stochastic integral with respect to W H and, finally, we report about the existing well-posedness results for equations (SWE) and (SHE). Section 3 is devoted to prove the tightness property of the family of laws induced by the solution uH , H ∈ ( 14 , 1). In Section 4, we deal with the limit identification, which allows us to conclude the proof of Theorem 1.1. In the Appendix, we collect some technical results and a tightness criterion that are used in the paper. 2 Preliminaries 2.1 Malliavin calculus In this section, we recall some elements of Malliavin calculus and a useful result of [28]. We refer the reader to [24] for more details. We will work in the Gaussian space determined by the noise W H , which is defined as follows. Let hϕ, ψiH := E W H (ϕ)W H (ψ) and define HH as the completion of C0∞ (R+ × R) with  respect to the inner product h·, ·iH . Then HH defines a Hilbert space and it is well-known that, if H ≤ 12 , it is a space of functions, while for H > 12 it contains distributions (see [5, Thm. 4.3] and [17, Prop. 4.2]). Then, {W H (ϕ), ϕ ∈ C0∞ (R+ × R)} can be extended to a family of Gaussian random variables indexed on the space HH , which we denote again by W H = {W H (ϕ), ϕ ∈ HH }. This family defines an isonormal Gaussian process on the Hilbert space HH : for any ϕ ∈ HH , 4 W H (ϕ) is a centered Gaussian variable and E W H (ϕ), W H (ψ) = hϕ, ψiH ,   ϕ, ψ ∈ HH . Let G H be the σ-algebra generated by {W H (ϕ), ϕ ∈ HH }. Then, any G H -measurable random variable F ∈ L2 (Ω) admits the representation X F = JnH F, (4) n≥0 where JnH F is the projection of F on the n-th Wiener chaos space HH,n , for n ≥ 1, and J0H F = E[F ]. We denote by InH the multiple Wiener integral of order n with respect to W H , which defines ⊗n a linear and continuous operator from HH onto HH,n . We briefly recall the construction of InH , since we will use some of its steps in the sequel. Let {ek , k ≥ 1} be an orthonormal basis of HH ⊗n and consider an elementary element of HH of the form ˆ ···⊗ ϕ = e i1 ⊗ ˆ ein , (5) where ⊗ ˆ denotes the symmetrized tensor product, for some i1 , . . . , in ≥ 1. Recall that the set ⊗n of finite linear combinations of elementary elements is dense in HH . An elementary element of the form (5) can be more conveniently written as ϕ = e⊗k 1 ˆ ˆ ⊗km j1 ⊗ . . . ⊗ ejm , (6) where all j1 , . . . jm ≥ 1 are different and k1 + · · · + km = n. The n-th order multiple Wiener integral of ϕ is defined as follows: InH (ϕ) = Pk1 W H (ej1 ) · · · Pkm W H (ejm ) ,   (7) where we denote by Pk the normalized k-th Hermite polynomial. The multiple Wiener integral is then extended by linearity to all finite linear combinations of elementary elements, and finally extended to the whole space HH by density. We also remind that any element in the n-th chaos HH,n can be represented as InH (f ), for ⊗n some f ∈ HH . Hence, representation (4) can be written as follows: X F = E[F ] + InH (fn ), n≥1 ⊗n ⊗n where fn ∈ HH , for all n ≥ 1. We recall that, for any f ∈ HH , h i E |InH (f )|2 = E |InH (f˜)|2 = n! kf˜k2H⊗n ,   H where f˜ stands for the symmetrization of f . We also remind that, for a general element f of ⊗n HH , the norm kf kH⊗n is given by H Z Z kf k2H⊗n = |Ff (t1 , ·, t2 , ·, . . . , tn , ·)(ξ1 , . . . , ξn )|2 µ(dξ1 ) · · · µ(dξn )dt1 · · · dtn . H Rn + Rn Here, we still denoted by F the Fourier transform on the space of tempered distributions in Rn . 5 ⊗n Let A ∈ B([0, ∞)). We define, for every f ∈ HH , the element f 1⊗n ⊗n A ∈ HH in the following ⊗n way: if f is a function, we define it obviously as the function f 1A . If f is a general element ⊗n ⊗n ⊗n of HH , we take any sequence {fk , k ≥ 1} of functions in HH such that fk → f in HH , as k → ∞, and we set f 1⊗n ⊗n A := lim fk 1A . k→∞ ⊗n This limit exists; indeed, we have that {fk , k ≥ 1} is Cauchy in HH and kfk 1⊗n ⊗n A − f` 1A kH⊗n ≤ kfk − f` kH⊗n , H H which implies that {fk 1⊗n ⊗n A , k ≥ 1} is also a Cauchy sequence in HH . The limit clearly does not depend on the chosen approximating sequence. On the other hand, we define the σ-field FAH = σ{W H (1D ϕ), D ∈ B0 (R+ ), D ⊂ A, ϕ ∈ C0∞ (R)} ∨ N , where N are the null sets of F and B0 (R+ ) are the bounded Borel sets of R+ . We have the following result: Lemma 2.1. Let F ∈ L2 (Ω) with Wiener chaos expansion given by F = E[F ] + H P n≥1 In (fn ), ⊗n where fn ∈ HH are symmetric, and let A ∈ B([0, ∞)). Then, it holds  X H E F |FAH = In fn 1⊗n   A . n≥0 Proof. The proof follows exactly as that of [4, Lem. A.1]. We only need to observe that, if ⊗n h ∈ HH is symmetric, it can be written as the limit of a sequence of symmetric functions, which in turn can be written as the limit of finite linear combinations of functions of the type f ⊗n , where f ∈ HH and kf kHH = 1. Let us now introduce the Malliavin derivative operator and the Skorohod integral. Let S be the class of random variables F of the form F = f (W H (ϕ1 ), . . . , W H (ϕn )), where f ∈ Cb∞ (Rn ) and ϕj ∈ HH , for every j = 1, . . . , n. For any F ∈ S, we define the Malliavin derivative of F as the HH -valued random variable DF given by n X ∂f DF = (W H (ϕ1 ), . . . , W H (ϕn ))ϕj . ∂xj j=1 1 h i1 2 If we endow S with the norm kF kD1,2 := E |F |2 2 +E kDF k2HH , it turns out that the operator  D can be extended to the completion of S with respect to k · kD1,2 , which we will denote by D1,2 . We define now the divergence operator δ, which is the adjoint of D. The divergence operator is defined on its domain Dom(δ), which is the space of HH -valued random variables such that u ∈ L2 (Ω; HH ) and 1 |E [hDF, uiH ]| ≤ cE |F |2 2 , for all F ∈ D1,2 ,  where the constant c depends on u. Being the adjoint of D, the divergence operator δ(u) is defined for any u ∈ Dom(δ) by the duality relation, holding for every F ∈ D1,2 : h i E hDF, uiH = E[F δ(u)]. 6 From the duality relation one can deduce that E[δ(u)] = 0, for every u ∈ Dom(δ). For any u ∈ Dom(δ), δ(u) is called the Skorohod integral of u and is denoted by Z ∞Z u(t, x)W H (δt, δx) := δ(u). 0 R We will need the following two results involving the Skorohod integral (cf. Propositions 1.3.3 and 1.3.6 in [24]). Lemma 2.2. Let F ∈ D1,2 and u ∈ Dom(δ) such that F u ∈ L2 (Ω; HH ). Then, F u ∈ Dom(δ) and it holds δ(F u) = F δ(u) − hDF, uiH . Lemma 2.3. Let u ∈ L2 (Ω; HH ) and {un , n ≥ 1} ⊂ Dom(δ) such that lim E ||un − u||2HH = 0.   n→∞ Suppose that there exists a random variable G ∈ L2 (Ω) such that, for all F ∈ S, E [δ(un )F ] → E[GF ]. Then u ∈ Dom(δ) and δ(u) = G. ⊗n We now define the contraction ⊗1 . For h ∈ HH and any element e1 ⊗· · ·⊗en of the canonical ⊗n basis of HH , we define (e1 ⊗ · · · ⊗ en ) ⊗1 h := (e1 ⊗ · · · ⊗ en−1 )hen , hiH , ⊗n and we extend it to a generic f ∈ HH by linearity and density. The following lemma can be found in [28, Thm. 4.3.8]: Lemma 2.4. Let F ∈ L2 (Ω) with Wiener chaos expansion F = E[F ] + n≥1 InH (fn ), where P ⊗n fn ∈ HH is symmetric, for all n ≥ 1. Then F ∈ D1,2 if and only if X n n!kfn k2H⊗n < ∞. H n≥1 In this case, for every h ∈ HH , we have X H hDF, hiH = nIn−1 (fn ⊗1 h). n≥1 2.2 Spectral representation of W H This section is devoted to prove that any multiple Wiener integral with respect to the noise W H admits a representation as a multiple Wiener integral with respect to a complex-valued Gaussian measure. For this, we will provide a suitable spectral representation of W H in terms of such a complex-valued Gaussian measure. We point out that the results in the present section will only be used in Section 4 in order to identify the underlying limit law. Recall that {W H (ϕ), ϕ ∈ HH } denotes the underlying isonormal Gaussian process associated to our noise W H . Using an approximation argument, one proves that, for any t > 0 and x ∈ R, 1[0,t]×[0,x] ∈ HH . Then, we can define the random field (making an abuse of notation) W H (t, x) := W H 1[0,t]×[0,x] , (t, x) ∈ R+ × R,  (8) 7 which is Gaussian, centered and satisfies, for all s, t > 0 and x, y ∈ R:  1 E W H (t, x)W H (s, y) = (s ∧ t) |x|2H + |y|2H − |x − y|2H .   2 The latter equality is a consequence of the representation in law of the fractional Brownian motion as a Wiener type integral with respect to a complex Brownian motion (see, for instance, [25, p. 257]). As in [25, Sec. 3.1], let W̃ : B0 (R+ × R) → C be a complex-valued Gaussian measure which can be written as W̃ = W̃1 + iW̃2 , where W̃1 and W̃2 are independent real-valued centered Gaussian measures such that W̃1 and W̃2 are symmetric and antisymmetric measures in the space variable, respectively, and, for any A, B ∈ B(R+ × R),   |A ∩ B| E W̃j (A)W̃j (B) = , j = 1, 2, 2 where |A ∩ B| is the Lebesgue measure of A ∩ B. In particular, E |W̃ (A)|2 = |A|, for all   A ∈ B0 (R+ × R). Note that W̃1 and W̃2 are essentially white noises in the sense of [12, Page 6, Example 3.13]. One can define the integral of any deterministic function f ∈ L2 (R+ × R; C) with respect to W̃ , as follows: Z Z Z Z Z Z f (t, x)W̃ (dt, dx) := f (t, x)W̃1 (dt, dx) + i f (t, x)W̃2 (dt, dx), R+ R R+ R R+ R and, for j = 1, 2, Z Z Z Z Z Z f (t, x)W̃j (dt, dx) := Re[f ](t, x)W̃j (dt, dx) + i Im[f ](t, x)W̃j (dt, dx). R+ R R+ R R+ R The latter integrals can be interpreted, e.g., as integrals with respect to a martingale measure (see [29]). It holds that, for any f, g ∈ L2 (R+ × R; C), Z Z Z Z  Z Z E f (t, x)W̃ (dt, dx) g(t, x)W̃ (dt, dx) = f (t, x)g(t, x) dxdt. R+ R R+ R R+ R This yields, for all f ∈ L2 (R+ × R; C), the isometry property "Z Z # Z Z 2 E f (t, x)W̃ (dt, dx) = |f (t, x)|2 dxdt. R+ R R+ R We have the following result, whose proof follows immediately. Proposition 2.5. Set, for any (t, x) ∈ R+ × R, Z tZ √ 1 H W̃ (t, x) := cH F[1[0,x] ](ξ)|ξ| 2 −H W̃ (ds, dξ). (9) 0 R Then, W̃ H is a Gaussian process which has the same distribution as the random field W H defined in (8). Remark 2.6. Though the right hand-side of (9) may seem to be complex-valued, it is indeed a real-valued quantity. In order to verify this, one uses the properties of W̃ and the fact that sin(xξ) cos(xξ) − 1 F[1[0,x] ](ξ) = +i . ξ ξ 8 At this point, we aim to extend the random field W̃ H defined in (9) to an isonormal Gaussian process in HH . We need the following corollary of [5, Thm. 4.3]: Proposition 2.7. The space of finite linear combinations of functions of the form f (r, z) = 1(s,t]×(x,y] (r, z), with 0 ≤ s < t and x < y, is dense in the Hilbert space HH . Proof. The result is a direct consequence of [5, Thm. 4.3]. Indeed, in the latter paper it is proved that any predictable process {X(t, x), (t, x) ∈ R+ × R} belonging to L2 (Ω; HH ) can be approx- imated by finite linear combinations of processes of the form (r, z, ω) 7→ 1G (ω)1(s,t] (r)1(x,y] (z), for some G ∈ F. To prove our result, it suffices to observe that, if we choose a deterministic element ϕ in their proof, also its approximating sequence ϕn is deterministic, and the norm in the space L2 (Ω; HH ) coincides with the norm in HH for deterministic elements. Let us now define, for any (t, x) ∈ R+ × R, W̃ H 1[0,t]×[0,x] := W̃ H (t, x)  (again making an abuse of notation). This definition can be extended by linearity to any simple function on R+ × R. Then, thanks to Proposition 2.7 and using an approximation argument, one constructs an isonormal Gaussian process {W̃ H (ϕ), ϕ ∈ HH } which has exactly the same law as {W H (ϕ), ϕ ∈ HH }. For the remainder of the paper, we will assume, without any loss of generality, that our Gaussian setting is the one determined by the isonormal Gaussian process W̃ H = {W̃ H (ϕ), ϕ ∈ HH }. For the sake of simplicity, we will use again the notation W H instead of W̃ H . So, the main implications of this setting are that, first, we have the representation Z tZ  √ 1 W H 1[0,t]×[0,x] = cH F[1[0,x] ](ξ)|ξ| 2 −H W̃ (ds, dξ) (10) 0 R and, secondly, the whole family of processes {W H , H ∈ (0, 1)} are defined in a single probability space, which is the one where the Gaussian measure W̃ is defined. This last fact will be crucial in Section 4. The main result of the section is the following: ⊗n Theorem 2.8. Let n ≥ 1, f ∈ HH and InH (f ) be the multiple Wiener integral of f with respect to W . Let fˆ be the function defined by H n 1 1 fˆ(t1 , x1 , t2 , x2 , . . . , tn , xn ) = (cH ) 2 F[f (t1 , ·, t2 , ·, . . . , tn , ·)](x1 , . . . , xn ) |x1 | 2 −H · · · |xn | 2 −H , where we recall that the constant cH is given in (1) Then, it holds that InH (f ) = I˜n (fˆ), P-a.s., (11) where I˜n is the n-th order Wiener integral with respect to the complex Gaussian measure W̃ . Proof. We first check that the result is true for the first-order Wiener integral I1H . We aim to prove that, for any ϕ ∈ HH , 1  1  I1H (ϕ) = (cH ) 2 I˜1 Fϕ(t, ·)(x) |x| 2 −H , (12) 9 which means that Z Z Z Z 1 1 ϕ(t, x)W H (dt, dx) = (cH ) 2 Fϕ(t, ·)(x) |x| 2 −H W̃ (dt, dx). R+ R R+ R By (10) and the linearity of the Wiener integral, the latter equality clearly holds in the case where ϕ(t, x) = 1(r,s]×(y,z] (t, x), for 0 ≤ r < s and y < z. Moreover, owing to Proposition 2.7, it can also be extended to the whole space HH , hence proving the theorem’s statement for first-order Wiener integrals. ⊗n Let us now prove (11) for n > 1. We first consider the case where f ∈ HH is an elementary element of the form (6). In this case, we use the definition of the multiple Wiener integral (see (7)) and the validity of the case n = 1 (see (12)), as follows: InH (f ) = Pk1 W H (ej1 ) · · · Pkm W H (ejm )       = Pk1 I˜1 (êj1 ) · · · Pkm I˜1 (êjm )  ⊗  ⊗ = I˜n êj1 1 ⊗ · · · ⊗ êjmkm k = I˜n (fˆ). ⊗n The extension to any element of HH can be proved by recalling that the set of finite linear ⊗n combinations of elementary elements of the form (6) is dense in HH . Remark 2.9. We have used a representation of our noise in terms of a complex Gaussian measure. This may seem artificial, and even rather complicated, and one could think of finding instead a representation using a real Gaussian measure. For instance, let us consider the random field defined by Z tZ h H− 1 H− 1 i H W (t, x) := C1 (H) (x − y)+ 2 − (−y)+ 2 W (ds, dy), (t, x) ∈ [0, T ] × R, (13) 0 R where C1 (H) is some positive constant and W is a Brownian sheet. Then, the Gaussian process H W has the same law as W H defined in (8) (see [25]). Furthermore, the authors of the latter reference established that H− 1 H− 1  1  H− 1  (x − y)+ 2 − (−y)+ 2 = Γ H + J 21 [0,x) (y). 2 1 H− 12 When H > 21 , J H− 2 = I− is the fractional integral of order H − 21 , which is defined by Z  H− 12  1 H− 32 I− φ (y) = φ(u)(u − y) − du, y ∈ R. Γ(H − 21 ) R 1 1 −H In the case H = 21 , J 0 is the identity operator. Finally, if H < 21 , J H− 2 = D−2 corresponds to the Marchaud fractional derivative operator, defined by 1 Z ∞  12 −H  −H φ(y) − φ(y + u) D− φ (y) = lim 2 du. ε→0+ Γ(H + 1 ) ε u3/2−H 2 The above results and representation (13) imply that Z TZ Z TZ H  H− 1  f (t, x) W (dt, dx) = C1 (H) J 2 f (t, ·) (x)W (dt, dx), (14) 0 R 0 R 10 for all T > 0 and any elementary function f . Compared to the representation (11) with respect to the complex Gaussian measure, in the right hand-side of (14) the function f appears to be affected by an operator that depends on H, while in (11) the dependence of H is shown only 1 through terms of the form |ξ| 2 −H . This has been very suitable to study continuity properties with respect to H. 2.3 Itô and Skorohod stochastic integrals This section is devoted to recall the definition of stochastic integrals with respect to W H , both in the case H < 12 and H ≥ 12 , and to prove that the Skorohod integral with respect to W H of an adapted process coincides with the corresponding Itô integral (see Theorem 2.13 below). This result will allow us to express any Picard iteration associated to our underlying SDPEs as a finite sum of multiple Wiener integrals, and this fact will be used in the proof of Theorem 4.1 in Section 4. Recall that we have a complete probability space (Ω, F, P) in which we have our complex- valued Gaussian measure W̃ (see Section 2.2). Recall that our isonormal Gaussian process W H = {W H (ϕ), ϕ ∈ HH } has been defined in such a way that we may assume that W H is defined in (Ω, F, P), for all H ∈ (0, 1). Regarding adaptability, we consider the natural filtration generated by W̃ , which we denote by {Ft , t ≥ 0} and can be defined as Ft = σ(W̃ (s, x), (s, x) ∈ [0, t] × R), where Z Z W̃ (s, x) := 1[0,s]×[0,x] (r, z)W̃ (dr, dz). R+ R Fix a time horizon T > 0. We denote by E the space of simple processes on [0, T ] × R, that is the space of finite linear combinations of processes of the form g(t, x, ω) := Y (ω)1(r,s]×(y,z] (t, x), (15) for some 0 ≤ r ≤ s ≤ T and y ≤ z, and for some Fr -measurable random variable Y . The (Itô) stochastic integral of g with respect to W H is defined as follows: for any t ∈ [0, T ], set Z tZ g(τ, x)W H (dτ, dx) := Y W H (t ∧ s, z) − W H (t ∧ s, y) − W H (t ∧ r, z) + W H (t ∧ r, y) .  0 R This definition can be extended to all elements of E by linearity. Following [11] and [2], we endow E with the norm   Z T Z  12 2 1−2H kgk0 := E cH |Fg(t, ·)(ξ)| |ξ| dξdt , 0 R and we define P0T as the completion of E with respect to the norm k · k0 . It turns out that P0T is the space of predictable processes g for which kgk0 < ∞. The stochastic integral can be extended to the whole space P0T . The following result is a particular case of [13, Prop. 2.9]: Theorem 2.10. Suppose that H ∈ [ 21 , 1). Let Γ : [0, T ] × R → R be such that, for all t ∈ (0, T ], the function Γ(t, ·) defines non-negative distribution with rapid decrease and Z T Z |FΓ(t, ·)(ξ)|2 |ξ|1−2H dξdt < ∞. 0 R 11 Moreover, we assume that, for all t ∈ [0, T ], Γ(t, dx) := Γ(t, x)dx defines a non-negative measure on R such that sup Γ(t, R) < ∞. t∈[0,T ] Let Z = {Z(t, x), (t, x) ∈ [0, T ] × R} be a predictable stochastic process satisfying E |Z(t, x)|2 < ∞.   sup (t,x)∈[0,T ]×R Then, the process {S(t, x) := Z(t, x)Γ(t, x), (t, x) ∈ [0, T ] × R} belongs to P0T . Furthermore, if Z satisfies, for some p ≥ 2, that sup E [|Z(t, x)|p ] < ∞, (t,x)∈[0,T ]×R then we have the following Burkholder-Davis-Gundy’s inequality: p "Z Z # T E S(s, x)W H (ds, dx) (16) 0 R p Z T Z −1 p ≤ zp (νT,H ) 2 sup E [|Z(s, x)| ] cH |FΓ(s, ·)(ξ)|2 |ξ|1−2H dξds, 0 x∈R R where the constant zp is the one in the classical Burkholder-Davis-Gundy inequality for contin- uous martingales, and νT,H is given by Z T Z νT,H := cH |FΓ(s, ·)(ξ)|2 |ξ|1−2H dξds. 0 R 1 As far as the case H < 2 is concerned, we have the following result (see [2, Thm. 2.9]). Theorem 2.11. Suppose that H ∈ (0, 12 ). Let {S(t, x), (t, x) ∈ [0, T ] × R} be a predictable process such that, for every (ω, t), S(ω, t, ·) defines a tempered function whose Fourier transform FS(ω, t, ·) is a locally integrable function satisfying Z T Z  2 1−2H E |FS(t, ·)(ξ)| |ξ| dξdt < ∞. 0 R Then, S ∈ P0T and we have the isometry 2 "Z Z # T Z T Z  H 2 1−2H E S(t, x)W (dt, dx) =E |FS(t, ·)(ξ)| cH |ξ| dξdt . 0 R 0 R Moreover, we have the Burkholder-Davis-Gundy inequality: for any p ≥ 2, "Z Z T p # p Z T Z  p2 H 2 1−2H E S(t, x)W (dt, dx) ≤ zp cH E 2 |FS(t, ·)(ξ)| |ξ| dξdt , (17) 0 R 0 R where the constant zp is the constant appearing in the classical Burkholder-Davis-Gundy inequal- ity for continuous martingales. 12 Remark 2.12. Owing to [2, Prop. 2.8], the isometry property in the above Theorem 2.11 can be equivalently written as 2 "Z Z # T  Z TZ  H E S(t, x)W (dt, dx) = E c̃H |S(t, x) − S(t, y)|2 |x − y|2H−2 dydxdt , 0 R 0 R2 where c̃H = H(1−2H) 2 . Hence, (17) becomes "Z Z T p # p Z T Z  p2 H E S(t, x)W (dt, dx) ≤ zp c̃H E 2 |S(t, x) − S(t, y)|2 |x − y|2H−2 dydxdt . 0 R 0 R The following result is an extension of [4, Thm. 4.2] to the case H > 12 . Note that, in this latter case, though the noise is more regular in space than a white noise, the corresponding Hilbert space HH may be rather big, and indeed contains genuine distributions. This makes our proof different compared to the one of [4, Thm. 4.2], in which HH is a space of functions (because H < 12 ). Theorem 2.13. Let H ∈ [ 14 , 1) and u = {u(t, x), (t, x) ∈ [0, ∞) × R} be a stochastic process such that, restricted to t ∈ [0, T ], belongs to P0T . Then, for any t > 0, u1[0,t] ∈ Dom(δ) and its Skorohod integral coincides with the Itô integral, that is Z ∞Z Z tZ u(s, x)1[0,t] (s)W H (δs, δx) = u(s, x)W H (ds, dx), P-a.s. 0 R 0 R Proof. The proof is an adaptation of that of [4, Thm. 4.2]. The only difference is that, here, a general element of HH is not necessarily a function. It is enough to prove the statement in the case where u is an elementary process of the form (15). The extension to any arbitrary element of P0T can be done exactly as in Case 2 of the proof of [4, Thm. 4.2]. Let g be an elementary process of the form g(τ, x, ω) = Y (ω)1(r,s] (τ )1(y,z] (x), with 0 ≤ r < s ≤ T and y < z, where we assume that Y is Fr -measurable, bounded and belongs to D1,2 . We have to check that g1[0,t] ∈ Dom(δ) and it holds Z tZ g(τ, x)W H (dτ, dx).  δ g1[0,t] = 0 R First, we note that g1[0,t] = Y 1[r∧t,s∧t]×[y,z] . Since Y ∈ D1,2 and 1[r∧t,s∧t]×[y,z] ∈ Dom(δ), we can apply Lemma 2.2 to conclude that g1[0,t] ∈ Dom(δ) and δ(g1[0,t] ) = Y δ(1[r∧t,s∧t]×[y,z] ) − hDY, 1[r∧t,s∧t]×[y,z] iH , if the right-hand side above belongs to L2 (Ω). We clearly have that Y δ(1[r∧t,s∧t]×[y,z] ) ∈ L2 (Ω), and we will show that hDY, 1[r∧t,s∧t]×[y,z] iH = 0, which will allow us to conclude the proof. Let h := 1[r∧t,s∧t]×[y,z] . Since Y is Fr -measurable, we have, by Lemma 2.1, X InH gn 1⊗n  Y = E[Y |Fr ] = [0,r] , n≥0 ⊗n for some symmetric gn ∈ HH , n ≥ 1. By Lemma 2.4 we have that X H gn 1⊗n  hDY, hiH = nIn−1 [0,r] ⊗1 h . n≥1 13 We claim that g1⊗n ⊗n [0,r] ⊗1 h = 0, for all g ∈ HH . Indeed, if g = e ⊗n for some function e ∈ H , H we have ⊗(n−1) ⊗(n−1) e⊗n 1⊗n [0,r] ⊗1 h = e 1[0,r] he1[0,r] , hiH , and we observe that Z ∞Z he1[0,r] , hiH = Fe(s, ·)(ξ)1[0,r] (s)F1[y,z] (ξ)1[r∧t,s∧t] (s)dξds = 0. 0 R ⊗n This can be extended to a generic element in HH by linearity and density (using Lemma 2.3). 2.4 Existence and uniqueness of solution This section is devoted to recall the well-posedness results for equations (SWE) and (SHE) and prove that the corresponding Picard iterations admit a suitable finite Wiener chaos expansion. First, we recall that the solution to our equations is understood in the mild sense. Namely, an adapted and jointly measurable random field uH = {uH (t, x), (t, x) ∈ [0, T ] × R} solves (SWE) (resp. (SHE)) if it holds, for all (t, x) ∈ [0, T ] × R: Z tZ H u (t, x) = η + Gt−s (x − y)uH (s, y)W H (ds, dy), (18) 0 R where G is the fundamental solution of the wave (resp. heat) equation in R (see (3)). The following result is a particular case of [13, Thm. 4.3], which covers the case H ≥ 21 . Theorem 2.14. Let H ∈ [ 12 , 1). There exists a unique mild solution uH to equation (18). Moreover, the solution uH is L2 (Ω)-continuous and satisfies, for every p ≥ 1, E |uH (t, x)|p < ∞.   sup (t,x)∈[0,T ]×R Remark 2.15. The case H = 21 corresponds to the space-time white noise, while in the case H ∈ ( 12 , 1) the noise’s spatial correlation is given by a Riesz kernel of order 2 − 2H. The case H ∈ ( 14 , 21 ) has been considered in [2, Thm. 1.1]. In the latter reference, the authors proved that condition H > 14 is necessary and sufficient in order to have a solution (see [2, Prop. 3.7]). Theorem 2.16. Let H ∈ ( 41 , 21 ). There exists a unique mild solution uH to (18). Moreover, the solution uH is L2 (Ω)-continuous and satisfies, for every p ≥ 2, E |uH (t, x)|p < ∞   sup (19) (t,x)∈[0,T ]×R and 2 E |uH (s, y) − uH (s, z)|p p Z T Z  sup G2t−s (x − y) dydzds < ∞. (20) (t,x)∈[0,T ]×R 0 R2 |y − z|2−2H Remark 2.17. In the case H ∈ ( 14 , 12 ), the solution uH satisfies, in addition to (19), the further constraint (20). This comes from the fact that, in [2], the solution of (18) was proved to exist in the space of L2 (Ω)-continuous, adapted and jointly measurable processes endowed with a Sobolev’s type norm which included a term of the form (20). 14 In the case H ∈ ( 14 , 21 ), the solution uH of (SWE) (and (SHE)) has been found in [2] as a limit of the Picard iteration scheme, which is defined by uH 0 (t, x) := η Z tZ uH m+1 (t, x) := η + Gt−s (x − y)uH H m (s, y)W (ds, dy), m ≥ 0, 0 R where (t, x) ∈ [0, T ]×R. The limit is found in the Banach space χpH , for p ≥ 2, which is defined as the space of L2 (Ω)-continuous, adapted and jointly measurable processes Y = {Y (t, x), (t, x) ∈ [0, T ] × R} such that kY kχp := kY kχp1 + kY kχp < ∞, H H,2 where, 1 kY kχp1 := sup E [|Y (t, x)|p ] p (t,x)∈[0,T ]×R and Z tZ 2 !1 p ]) p 2 (E [|Y (s, y) − Y (s, z)| kY kχp := sup c̃H G2t−s (x − y) dydzds . H,2 (t,x)∈[0,T ]×R 0 R2 |y − z|2−2H We recall that the constant c̃H has been defined in Remark 2.12. Notice that the Lp -part k · kχp1 of the norm k · kχp does not depend on H, as it is also pointed out by the notation itself, while H the Gagliardo-type part k · kχp does depend on H. H,2 Remark 2.18. In [2], the norm k · kχp is defined without the constant c̃H = H(1−2H) 2 . Since H,2 the two definitions give rise to equivalent norms, the results about existence and uniqueness of solution for equation (18) when H ∈ ( 41 , 12 ) still hold true. On the other hand, we will see how adding this normalizing constant helps us proving some uniform (in H) results that will be needed in the sequel. Before stating the main result of the section, we consider the following Sobolev-type embed- ding for the space χpH , which could be of independent interest. 1 Lemma 2.19. Let p ≥ 2 and 4 < α ≤ β < 12 . Then, it holds: χpα ,→ χpβ . This means that there exists a constant C such that, for every adapted, jointly measurable and L2 (Ω)-continuous process Y , we have kY kχp ≤ CkY kχpα . (21) β Moreover, it holds the following stronger property for the Gagliardo-type seminorm k · kχp : β,2 sup kY kχp ≤ C̃kY kχpα β,2 β∈[α, 12 ) where the constant C̃ only depends on p and T . 15 Proof. We follow the same lines as in the proof of [14, Prop. 2.1]. It suffices to prove (21) for the k · kχp -part of the norm. It holds: H,2 Z tZ 2 ! 12 (E [|Y (s, y) − Y (s, z)|p ]) p c̃β G2t−s (x − y) dydzds 0 R2 |y − z|2−2β Z tZ 2 ! 12 (E [|Y (s, y) − Y (s, y − z)|p ]) p = c̃β G2t−s (x − y) dydzds 0 R2 |z|2−2β ≤ C(I1 + I2 ), (22) where we label I1 the term where we integrate in the variable z in the region |z| ≥ 1, and I2 the term where we integrate in the region |z| < 1. First, we have Z tZ Z 2 ! 12 (E [|Y (s, y) − Y (s, y − z)|p ]) p I1 = c̃β G2t−s (x − y) dzdyds 0 R |z|≥1 |z|2−2β Z tZ Z !1 2 1 1 ≤ Cp sup (E [|Y (t, x)|p ]) p c̃β G2t−s (x − y) dzdyds . (t,x)∈[0,T ]×R 0 R |z|≥1 |z|2−2β 1 2 R Note that |z|≥1 |z|2−2β dz = 1−2β . Hence, Z tZ Z 1 c̃β G2t−s (x − y) dzdyds 0 R |z|≥1 |z|2−2β Z tZ CT ≤β G2t−s (x − y) dyds ≤ βCT ≤ . 0 R 2 Thus, we can conclude that 1 I1 ≤ Cp,T sup (E [|Y (t, x)|p ]) p . (t,x)∈[0,T ]×R Regarding I2 , we observe that Z tZ Z 2 ! 12 (E [|Y (s, y) − Y (s, y − z)| p ]) p I2 = c̃β G2t−s (x − y) dzdyds 0 R |z|<1 |z|2−2β Z tZ Z 2 ! 12 (E [|Y (s, y) − Y (s, y − z)| ]) p p ≤ c̃α G2t−s (x − y) dzdyds 0 R |z|<1 |z|2−2α Z tZ Z 2 ! 12 (E [|Y (s, y) − Y (s, y − z)| p ]) p ≤ c̃α G2t−s (x − y) dzdyds 0 R R |z|2−2α Z tZ Z 2 ! 12 (E [|Y (s, y) − Y (s, y − z)|p ]) p ≤ sup c̃α G2t−s (x − y) dzdyds (t,x)∈[0,T ]×R 0 R R |z|2−2α = kY kχpα,2 . 16 Notice that both the estimate for I1 and I2 are independent of (t, x) ∈ [0, T ] × R and β ∈ [α, 21 ). Therefore, we can take the supremum with respect to (t, x) ∈ [0, T ] × R and β ∈ [α, 12 ) in the left-hand side of (22) and we conclude sup kY kχp ≤ Cp,T kY kχp1 + kY kχpα,2 ≤ C̃kY kχpα , β,2 β∈[α, 12 ) which obviously implies kY kχp ≤ (Cp,T + 1)kY kχp1 + kY kχpα,2 ≤ CkY kχpα , β for some constant C. The path Hölder-continuity of the solution to (18) has been proved in [3] in the case H ∈ ( 14 , 21 ), while the case H ∈ [ 21 , 1) follows from the results in [29, 26, 27]. For the sake of completeness, we state a result which unifies both cases, and whose proof follows, indeed, as an immediate consequence of the stronger results Proposition 3.1 and Proposition 3.8 proven in Section 3. Theorem 2.20. Let H ∈ ( 14 , 1). Then, the solution uH to (18) satisfies the following: for any p ≥ 2, there exists a constant Cp > 0 (which indeed does not depend on H) such that, for all t, t0 ∈ [0, T ] and x, x0 ∈ R, it holds sup E |uH (t0 , x) − uH (t, x)|p ≤ Cp |t0 − t|γp   x∈R and sup E |uH (t, x0 ) − uH (t, x)|p ≤ Cp |x0 − x|Hp ,   t∈[0,T ] where γ = H for the wave equation and γ = H2 for the heat equation. Thus, the process uH has a modification whose trajectories are almost surely γ 0 -Hölder continuous in time, for all γ 0 < γ, and H 0 -Hölder continuous in space for all H 0 < H. Proof. As already mentioned, the result follows from Propositions 3.1 and 3.8 in Section 3, where the same kind of estimates have obtained uniformly with respect to H. The above Theorems 2.16 and 2.14, together with Theorem 2.13 on the equivalence between Itô and Skorohod integrals, allow us to prove that equations (SWE) and (SHE) admit a unique Skorohod mild solution. By definition, it is a square integrable random field {uH (t, x), (t, x) ∈ R+ × R} such that, for all (t, x) ∈ R+ × R, Z tZ H u (t, x) = η + Gt−s (x − y)uH (s, y)W H (δs, δy), P-a.s., (23) 0 R that is, the process v (t,x) := {1[0,t] (s)Gt−s (x − y)uH (s, y), (s, y) ∈ R+ × R} belongs to Dom(δ) and uH (t, x) = η + δ v (t,x) .  Theorem 2.21. Let H ∈ ( 41 , 1) and T > 0. Equation (23) admits a unique adapted solution in [0, T ] × R. Proof. This result has already been proved in [4, Thm. 4.3] for the wave equation in the case H ∈ ( 14 , 12 ). In [16, p. 49], the authors notice that it is also true for the heat equation, still under the constraint H ∈ ( 14 , 21 ). The statement’s validity in the case H ∈ [ 12 , 1) follows combining Theorems 2.13 and 2.14. 17 Finally, the following result will be crucial in order to identify the limit law in Theorem 4.1. Theorem 2.22. Let H ∈ ( 14 , 1) and uH be the solution to (18). Recall that the corresponding Picard iteration scheme is defined as follows: for any m ≥ 0, set uH 0 (t, x) := η, Z tZ uH m+1 (t, x) := η + Gt−s (x − y)uH H m (s, y)W (ds, dy), 0 R where (t, x) ∈ [0, T ] × R. Then, for any m ≥ 0, it holds m X uH m (t, x) = InH (gn (·, t, x)), n=0 where InH is the n-th multiple Wiener integral with respect to W H and the kernel gn (·, t, x) is given by gn (t1 , x1 , t2 , x2 , . . . , tn , xn , t, x) := Gt−tn (x − xn ) · · · Gt2 −t1 (x2 − x1 ) η1{0 0 such that, for every |h| ≤ h0 , it holds: E |uH (t, x + h) − uH (t, x)|p ≤ C|h|η1 p   sup sup H∈K (t,x)∈[0,T ]×R and E |uH (t + h, x) − uH (t, x)|p ≤ C|h|η̃1 p ,   sup sup H∈K (t,x)∈[0,T ∧(T −h)]×R η1 where C is a constant depending only on p, η̃1 = η1 for the wave equation and η̃1 = 2 for the heat equation. Now, we have all needed ingredients to prove our tightness result Proposition 3.1. Proof of Proposition 3.1. We will apply Theorem A.5. First, we notice that condition (i) in this criterion is clearly satisfied, since uH (0, 0) is deterministic and independent of H. In order to check (ii) in Theorem A.5, we apply Proposition 3.7 and we deduce that, for any t, t0 ∈ [0, T ] and x, x0 ∈ R such that |t0 − t| < h0 and |x0 − x| < h0 , it holds: E |uH (t0 , x0 ) − uH (t, x)|p ≤ C(|t0 − t|pη̃1 + |x0 − x|pη1 ).   (28) One can easily deduce that estimate (28) holds for any t, t0 ∈ [0, T ] and any x, x0 in a compact set. 3.2 Tightness in the case [ 12 , 1) We aim to prove an analogous tightness result as Proposition 3.1 for the case H ≥ 12 . We state it in Proposition 3.8 below. Now, we suppose that the limiting exponent H0 ∈ [ 21 , 1), so whenever Hn → H0 we can suppose without loss of generality that Hn ∈ K, where K is of the form [η1 , η2 ], with η1 , η2 ∈ [ 21 , 1) and η1 ≤ η2 . As we already observed at the beginning of Section 3.1, if we prove the tightness of the family of laws of {uH , H ∈ K} also for K of the form considered here, this will include also the case in which H0 = 21 and Hn → H0 either from above or from below. The following tightness result will be proved directly, i.e. without going through the corre- sponding Picard iteration scheme. This is because the Burkholder-Davies-Gundy type inequality (16) is more practical than its rough counterpart (17). 22 Proposition 3.8. Let UK := {uH , H ∈ K} be the family of solutions of (18), where K is of the form [η1 , η2 ], with η1 , η2 ∈ [ 12 , 1) and η1 ≤ η2 . Then, the family UK is tight in C([0, T ] × R), endowed with the metric of uniform convergence on compact sets. Proof. We will apply again Theorem A.5. We split the proof in three steps. Step 1: We show the uniform estimate E |uH (t, x)|p < ∞.   sup sup (29) H∈[η1 ,η2 ] (t,x)∈[0,T ]×R We have   Z tZ p   H p  H H E |u (t, x)| ≤ C 1 + E Gt−s (x − y)u (s, y)W (ds, dy) . 0 R By Theorem 2.10, we obtain that the expectation in the right hand-side above can be bounded, up to some positive constant, by Z t Z p −1  H p |FGt−s (x − ·)(ξ)|2 |ξ|1−2H dξds,  cH (νt,H ) 2 sup sup E |u (s, x)| (30) 0 H∈[η1 ,η2 ] x∈R R where νt,H is defined by Z tZ νt,H = cH |FGs (ξ)|2 |ξ|1−2H dξds. 0 R Γ(2H+1) sin(πH) 1 We recall that cH = 2π , which is bounded by π, for all H. Moreover, by Lemma A.1, it holds that sup sup νt,H < ∞. H∈[η1 ,η2 ] t∈[0,T ] Note that this holds for both wave and heat equations. On the other hand, regarding the integral in dξ in (30), we can argue as follows. In the case of the wave equation, we have Z ∞ sin2 ((t − s)ξ) Z 2 1−2H |FGt−s (x − ·)(ξ)| |ξ| dξ = 2 dξ R 0 ξ 1+2H = 2(t − s)2H 22H−1 C1−2H ≤ T 2H 22H C1−2H , where the constant C1−2H is the same one appearing in Lemma A.1. As showed in the proof of [15, Thm. 2.8], C1−2H defines a continuous function with respect to H ∈ (0, 1), so it can be bounded by a constant when H ∈ [η1 , η2 ]. Thus, for the wave equation we can conclude that Z t ! sup sup E |u (t, x)|p ≤ C 1 +  H sup sup E |uH (s, x)|p ds .    H∈[η1 ,η2 ] x∈R 0 H∈[η1 ,η2 ] x∈R Hence, Grönwall lemma implies (29). In the case of the heat equation, we have Z Z 2 |FGt−s (x − ·)(ξ)|2 |ξ|1−2H dξ = e−(t−s)|ξ| |ξ|1−2H dξ R R Z ∞ 1 = (t − s) H−1 e−y y −H dy 2 0 = Γ(1 − H)(t − s)H−1 . 23 Observe that, for all H ∈ [η1 , η2 ], it holds Γ(1 − H)(t − s)H−1 ≤ g(t − s), where ( rη1 −1 , r < 1 g(r) := Γ(1 − η2 ) 1, r > 1. Therefore, ! Z t H p H p     sup sup E |u (t, x)| ≤C 1+ sup sup E |u (s, x)| g(t − s)ds . H∈[η1 ,η2 ] x∈R 0 H∈[η1 ,η2 ] x∈R The Grönwall type lemma proved in [11, Lem. 15] let us conclude that (29) is also fulfilled in the case of the heat equation. Step 2: In this part of the proof, we deal with the moments of the space increments of the solution uH . Precisely, owing to Theorem 2.10, the estimate (29) and Lemma A.2, we can infer that, for all p ≥ 2 and |h| ≤ 1, E |uH (t, x + h) − uH (t, x)|p   p Z t Z  p2 2 1−2H ≤ C cH2 |F (Gt−s (x − ·) + Gt−s (x + h − ·)) (ξ)| |ξ| dξds 0 R p Z t Z  p2 2 1−2H = C cH 2 (1 − cos(hξ))|FGs (ξ)| |ξ| dξds 0 R p ≤ C C̃H2 |h|Hp . The constant C̃H is the same appearing in [2, Lem. 3.4], and it is given by Z 1 1 C̃H := (1 − cos(θ))|θ|−1−2H dθ < + ≤ C, R H 1 − H provided that H ∈ [η1 , η2 ]. Thus, we have proved that E |uH (t, x + h) − uH (t, x)|p ≤ C|h|η1 p .   sup sup H∈[η1 ,η2 ] (t,x)∈[0,T ]×R Step 3: Here, we aim to prove that, for any p ≥ 2 and |h| < 1, ( C|h|η1 p wave equation, E |uH (t + h, x) − uH (t, x)|p ≤   sup sup η1 H∈[η1 ,η2 ] (t,x)∈[0∨(−h),T ∧(T −h)]×R C|h| 2 p heat equation. (31) Assume that h > 0 (the case h < 0 is completely analogous). Then, h i E |uH (t + h, x) − uH (t, x)|p ≤ C(B1 + B2 ), where  Z tZ p H H B1 := E [Gt+h−s (x − y) − Gt−s (x − y)]u (s, y)W (ds, dy) , 0 R p "Z # t+h Z B2 := E Gt+h−s (x − y)uH (s, y)W H (ds, dy) . t R 24 Theorem 2.10, (29) and Lemma A.3 yield p Z t Z  p2 2 B1 ≤ C c H2 |F (Gt+h−s (x − ·) − Gt−s (x − ·)) (ξ)| |ξ|1−2H dξds 0 R p Z T Z  p2 2 1−2H ≤ C cH2 |FGs+h (ξ) − FGs (ξ)| |ξ| dξds 0 R ( Hp |h| , wave equation, ≤C H (32) |h| 2 p , heat equation. Regarding the term B2 , we can argue as before but we apply Lemma A.1. Indeed, we have that p Z t+h Z  p2 2 1−2H B2 ≤ C cH 2 |FGt+h−s (x − ·)(ξ)| |ξ| dξds t R p Z hZ = C cH 2 |FGs (ξ)|2 |ξ|1−2H dξds ( 01+2HR |h| 2 p , wave equation, ≤C H (33) |h| 2 p , heat equation. Putting together (32) and (33), and taking into account that H ∈ [η1 , η2 ], we end up with (31). Finally, the results in Steps 2 and 3 let us conclude that, for any t, t0 ∈ [0, T ] and x, x0 in a compact of R, we have (  H 0 0 H p  |t0 − t|η1 p + |x0 − x|η1 p , wave equation, E |u (t , x ) − u (t, x)| ≤ C 0 η1 p 0 η p . |t − t| 2 + |x − x| , heat equation 1 Thus, it suffices to take p > η41 for the heat equation and p > 2 η1 for the wave equation to be able to apply the tightness criterion Theorem A.5. The following result extends Corollary 3.5 to the case H ≥ 21 . Its proof is very similar to that of [11, Thm. 13], and the terms that need to be estimated uniformly with respect to H are completely analogous as those appearing in Step 1 of the proof of the above Proposition 3.8. Lemma 3.9. Let H ≥ 21 and {uH m , m ≥ 0} be the sequence of Picard iterations corresponding to the mild formulation (18). Then, for any p ≥ 2, uH p m converges in L (Ω) to the solution u H uniformly with respect to H ∈ K, i.e. E |uH H p   lim sup sup m (t, x) − u (t, x)| m→∞ H∈K (t,x)∈[0,T ]×R = 0. 4 Identification of the limit Let H0 ∈ ( 41 , 1) and {Hn , n ≥ 1} be any sequence such that Hn → H0 , as n → ∞. We may assume that there exists a compact set K ⊂ ( 14 , 1) such that Hn ∈ K, for all n ≥ 1. The tightness results proved in Propositions 3.1 and 3.8 imply that there exists a subsequence {Hnk , k ≥ 1} such that {uHnk , k ≥ 1} converges in law in the space C([0, T ] × R) of contiuous functions. This section is devoted to prove that the limit law is the distribution of uH0 . 25 Our strategy can be summarized as follows. We will verify that the finite dimensional distributions of uHn converge to those of uH0 (see [7, Thm. 2.6]). For this, it suffices to prove that, for any fixed (t, x) ∈ [0, T ] × R, uHn (t, x) converges to uH0 (t, x) in L2 (Ω). This can be done thanks to the fact that the whole family of noises {W H , H ∈ (0, 1)} can be defined on a single probability space (see Section 2.2). In order to prove the above L2 (Ω)-convergence, we will check the same convergence for any of the corresponding Picard iterates, that is, for any m ≥ 1, we show that uH H0 2 m (t, x) → um (t, x) in L (Ω), as n → ∞, and we will take into account n that the Picard iteration scheme converges to the solution uniformly with respect to the Hurst index H. At this point, we recall (invoking Theorem 2.22) that any Picard iterate admits the following Wiener chaos expansion: m X uH n m (t, x) = IjHn (gj (·, t, x)), j=0 where the latter is a finite sum of multiple Wiener integrals of order up to m and the kernels gj are given by (24). Therefore, it will be sufficient to prove the L2 (Ω)-convergence, as n → ∞, of any of the above multiple Wiener integrals, for which we will make use of the representation result given in Theorem 2.8. Here is the main result of the section: Theorem 4.1. Let H0 ∈ ( 41 , 1) and {Hn , n ≥ 1} be any sequence such that Hn → H0 , as n → ∞. Let uH n and u n H0 be the solutions of (18) corresponding the Hurst parameters H and n H0 , respectively. Then, the finite dimensional distributions of uHn converge to those of uH0 , as n → ∞. Proof. We split the proof in three steps. Step 1: To start with, we recall that, owing to Corollary 3.5 and Lemma 3.9 in the particular case p = 2, we have: E |uH H 2   lim sup sup m→∞ H∈K (t,x)∈[0,T ]×R m (t, x) − u (t, x)| = 0, (34) where uH m denotes the associated mth Picard iterate. As we already explained, in order to assure the statement’s validity it is sufficient to show the following pointwise convergence in L2 (Ω): for any fixed (t, x) ∈ [0, T ] × R, it holds lim E |uHn (t, x) − uH0 (t, x)|2 = 0.   n→∞ Note that E |uHn (t, x) − uH0 (t, x)|2   ≤ C E |uHn (t, x) − uH 2    Hn H0 2   H0 H0 2  m (t, x)| + E |um (t, x) − um (t, x)| + E |um (t, x) − u (t, x)| n =: I1 (m, n) + I2 (m, n) + I3 (m). By (34), we can infer that, for any ε > 0, we can choose m0 big enough such that, for every m ≥ m0 , we have sup {I1 (n, m) + I3 (m)} < ε. n≥1 Thus, we are left to show that I2 (m0 , n) tends to zero as n → ∞. This means, in particular, that the m0 -th Picard iterate is continuous in L2 (Ω), with respect to H. 26 Theorem 2.22 implies that, for any H ∈ ( 14 , 1), uH m0 has the Wiener chaos expansion m0 X uH m0 (t, x) = IjH (gj (·, t, x)), j=0 where the functions gj are defined by (24). Hence, in order to check that I2 (m0 , n) tends to zero, it is enough to show that, for any j = 1, . . . , m0 , IjHn (gj (·, t, x)) converges to IjH0 (gj (·, t, x)) in L2 (Ω), as n → ∞. Indeed, by Theorem 2.8 we have that IjHn (gj (·, t, x)) − IjH0 (gj (·, t, x)) Z   1 1 1 1 = cjHn |ξ1 | 2 −Hn · · · |ξj | 2 −Hn − cjH0 |ξ1 | 2 −H0 · · · |ξj | 2 −H0 {[0,T ]×R}j × Fgj (t1 , ·, . . . , tj , ·, t, x)(ξ1 , . . . , ξn )W̃ (dt1 , dξ1 ) · · · W̃ (dtj , dξj ). Hence   2 E IjHn (gj (·, t, x)) − IjH0 (gj (·, t, x)) Z 1 1 1 1 2 = cjHn |ξ1 | 2 −Hn · · · |ξj | 2 −Hn − cjH0 |ξ1 | 2 −H0 · · · |ξj | 2 −H0 {[0,T ]×R}j × |Fgj (t1 , ·, . . . , tj , ·, t, x)(ξ1 , . . . , ξn )|2 dξ1 · · · dξj dt1 · · · dtj . We show that the last integral converges to 0 when n → ∞. To do this, we have to compute explicitly the Fourier transform appearing in the above expression. Precisely, as detailed in [4, p. 10], we have Fgj (t1 , ·, . . . , tj , ·, t, x)(ξ1 , . . . , ξj ) = ηe−i(ξ1 +···+ξj )x FGt2 −t1 (ξ1 ) FGt3 −t2 (ξ1 + ξ2 ) · · · FGt−tj (ξ1 + · · · + ξj ) 1{0 1/4 and define the functions f0 , f1 , f2 : R+ → R+ as follows: f0 (r) = 1 and ( r1−2β , r ≥ 1, f1 (r) = 1, r < 1, ( r2(1−2β) , r ≥ 1, f2 (r) = 1, r < 1. We also set, for every α`,n ,  0,  α`,n = 0, N (α`,n ) := 1, α`,n = 1 − 2Hn ,  2, α`,n = 2(1 − 2Hn ).  Then, we have the following estimate for the term (36):   j j ! 2 X Y Y 1−2Hn  α`,n  |η1 | |η` | FGt` −t`−1 (η` ) α∈Dj `=1 `=1   j j ! 2 X Y Y ≤ f1 (|η1 |)  fN (α`,n ) (|η` |) FGt` −t`−1 (η` ) . α∈Dj `=1 `=1 28 We have to prove that this function is integrable. To check this last fact, it is sufficient to show it for a single integrand of the form j j Y 2 Y FGt` −t`−1 (η` ) |η1 |β |η` |α` `=1 `=1 where, now, αj does not take values in a discrete set, but they satisfy the weaker constraints: α1 ∈ K1 ⊂ [0, 1/2), and α` ∈ K2 ⊂ [0, 1), for ` = 2, . . . , j, where K1 = [0, 1 − 2 minn≥1 Hn ] and K2 = [0, 2(1 − 2 minn≥1 Hn )] (we are assuming implicitly that minn≥1 Hn < 21 ; if this is not the case, then the entire sequence falls in the case Hn ≥ 12 , which will be studied afterwards). It is important to notice that the sets K1 , K2 do not depend on n. The fact that 1 − 2 minn≥1 Hn < 12 and 2(1 − 2 minn≥1 Hn ) < 1 turns out to be crucial for our estimates. Thus, we want to prove that Z Z j Z Y  2 β+α1 2 α` |FGt2 −t1 (η1 )| |η1 | dη1 |FGt`+1 −t` (η` )| |η` | dη` dt1 · · · dtj < ∞. (37) Tj (t) R `=2 R At this point, we have to consider separately the case of the wave equation case from that of the heat equation. It holds that, for any γ ∈ (−1, 1) (see the proof of Proposition 3.8): ( Cγ0 (2 − γ)t1−γ , wave equation, Z |FGt (ξ)|2 |ξ|γ dξ ≤ (γ+1) R Cγ00 1−γ 2 t − 2 , heat equation. We recall that the constants Cγ0 and Cγ00 are continuous with respect to γ ∈ (−1, 1). We will apply the above estimate with γ = 1 − 2H and γ = 2(1 − 2H), and still we can bound them uniformly with respect to H ∈ K ⊂ ( 14 , 21 ], with K compact. Hence, for the heat equation, the integral in (37) can be estimated by Z j −β−α1 Y −α` −1 (t2 − t1 ) 2 (t`+1 − t` ) 2 dt1 · · · dtj , Tj (t) `=2 which is finite because all exponents are strictly greater than −1. For the wave equation, we end up with Z j Y 1−β−α1 (t2 − t1 ) (t`+1 − t` )1−α` dt1 · · · dtj , Tj (t) `=2 which is also finite since all exponents are even greater than 0. This concludes the proof in the case H ∈ ( 41 , 21 ]. Step 3: In the case of Hn ≥ 12 , let us now go back to expression (35), where we resettle the variables ξ` by means of the change of variables ξ` = η` − η`−1 . That is, we aim to bound the following term: j 2 Y |ξ1 |1−2Hn · · · |ξj |1−2Hn FGt`+1 −t` (ξ1 + · · · + ξ` ) . (38) `=1 29 Here, the fact that 1 − 2Hn ≤ 0 helps us. Indeed, we can define the bounding function in a quite straightforward way: ( 1, r ≥ 1, g(r) := 1−2(max H ) r n≥1 n , r < 1. Clearly, the integrand function in (38) is bounded, for any n ≥ 1, by j 2 Y g(|ξ1 |) · · · g(|ξj |) FGt`+1 −t` (ξ1 + · · · + ξ` ) . `=1 We check that this upper bound function is integrable, namely Z Z j−1 2 Y FGt`+1 −t` (ξ1 + · · · + ξ` ) g(|ξ` |) Tj−1 (tj ) Rj−1 `=1 ! Z t Z 2 × FGt−tj (ξ1 + · · · + ξj ) g(|ξj |)dξj dtj dξ1 · · · dξj−1 dt1 · · · dtj−1 < ∞. (39) tj−1 R We have that Z t Z 2 FGt−tj (ξ1 + · · · + ξj ) g(|ξj |)dξj dtj tj−1 R Z t Z 2 = FGt−tj (ξ1 + · · · + ξj ) dξj dtj tj−1 |ξj |>1 Z t Z 2 + FGt−tj (ξ1 + · · · + ξj ) |ξj |1−2 minn≥1 Hn dξj dtj . tj−1 |ξj |≤1 We do the computations separately for the wave and heat equations. To start with, in the case of the wave equation, it clearly holds that sin(t|ξ|) |FGt (ξ)| = ≤ t, |ξ| for all (t, x) ∈ [0, T ] × R. Thus, we have Z t Z 2 FGt−tj (ξ1 + · · · + ξj ) |ξj |1−2 minn≥1 Hn dξj dtj tj−1 |ξj |≤1 Z t Z ≤ |t − tj |2 |ξj |1−2 minn≥1 Hn dξj dtj tj−1 |ξj |≤1 CT 3 ≤ < ∞, 1 − minn≥1 Hn and Z t Z 2 FGt−tj (ξ1 + · · · + ξj ) |ξj |1−2 minn≥1 Hn tj−1 |ξj |>1 Z t sin2 [(t − tj )|ξ1 + · · · + ξj |] Z ≤ dξj dtj tj−1 R |ξ1 + · · · + ξj |2 Z t ≤C (t − tj )dtj < ∞, tj−1 30 2 since R sin|x|(t|x|) R 2 dx = πt. Therefore, we have got rid of the integral with respect to dξj dtj in (39). Iterating this procedure one proves that the whole integral (39) is finite. It remains to prove the analogous result for the heat equation. Here, we have t|ξ|2 |FGt (ξ)| = e− 2 ≤ 1, for all (t, x) ∈ [0, T ] × R. Thus, Z tZ 2 FGt−tj (ξ1 + · · · + ξj ) |ξj |1−2 minn≥1 Hn dξj dtj tj |ξj |≤1 Z tZ ≤ |ξj |1−2 minn≥1 Hn dξj dtj tj |ξj |≤1 T ≤ < ∞, 1 − minn≥1 Hn and Z tZ 2 FGt−tj (ξ1 + · · · + ξj ) |ξj |1−2 minn≥1 Hn dξj dtj tj |ξj |>1 Z tZ exp −(t − tj )|ξ1 + · · · + ξj |2 dξj dtj  ≤ tj R Z tp =C t − tj dtj < ∞, tj which, again by iterating this computation, shows that the integral in (39) is bounded also in the heat equation case. This completes the proof. A Auxiliary results In this section, we state some results that have been applied throughout the paper. We start with four technical lemmas, proved in [2], which provide explicit estimates, depending on H, for the norm in the space L2 (R; µH ) of terms involving the Fourier transforms of the fundamental solutions of the wave and heat equations. Finally, we will also state a tightness criterion which will be applied in Section 3. We recall that, for the wave and heat equations, we have, respectively: sin(t|ξ|)  −tξ 2  FGt (ξ) = and FGt (ξ) = exp , t > 0, ξ ∈ R. |ξ| 2 In the following three lemmas, we will denote either one of these two functions by FGt (ξ). We recall that the spatial spectral measure is given by µH (dξ) = cH |ξ|1−2H dξ . Lemma A.1 ([2], Lemma 3.1). Let T > 0. Then, the integral Z T Z AT (α) := |FGt (ξ)|2 |ξ|α dξ dt 0 R 31 converges if and only if α ∈ (−1, 1). In this case, it holds: 1  21−α Cα  T 2−α for the wave equation,   2−α AT (α) =  2 Γ α + 1 T (1−α)/2 for the heat equation,      1−α 2 where the constant Cα is given by  Γ(α)  1 − α sin(πα/2),    α ∈ (−1, 1) \ {0}, Cα = π ,   α = 0.  2 Lemma A.2 ([2], Lemma 3.4). Let T > 0 and α ∈ (−1, 1). Then, for any h > 0, it holds: Z TZ ( 2 α C|h|1−α for the heat equation, (1 − cos(ξh)) |FGt (ξ)| |ξ| dξ dt ≤ 1−α 0 R CT |h| for the wave equation, − cos η)|η|α−2 dη. R where C = R (1 Lemma A.3 ([2], Lemma 3.5). Let T > 0 and α ∈ (−1, 1). Then, for any h > 0, it holds: Z TZ ( Cα |h|(1−α)/2 for the heat equation, |FGt+h (ξ) − FGt (ξ)|2 |ξ|α dξ dt ≤ 0 R Cα T |h|1−α for the wave equation, where 2 (1 − e−η /2 )2 Z Cα = dη for the heat equation, and R |η|2−α min(1, |η|2 ) Z Cα = 4 dη for the wave equation. R |η|2−α Lemma A.4 ([2], Lemma D.2). For any H ∈ (0, 12 ) and for any ξ ∈ R, we have: |1 − e−iξx |2 Z 2Γ(2H + 1) sin(πH) 2−2H dx = |ξ|1−2H R |x| H(1 − 2H) The following tightness criterion on the plane was proved in [31, Prop. 2.3]. Theorem A.5. Let {Xλ }λ∈Λ be a family of random functions indexed on the set Λ and taking values in the space C([0, T ] × R), in which we consider the metric of uniform convergence over compact sets. 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