Central limit theorems for nonlinear stochastic wave equations in dimension three

Abstract

In this paper, we consider three-dimensional nonlinear stochastic wave equations driven by the Gaussian noise which is white in time and has some spatial correlations. Using the Malliavin–Stein’s method, we prove the Gaussian fluctuation for the spatial average of the solution under the Wasserstein distance in the cases where the spatial correlation is given by an integrable function and by the Riesz kernel. In both cases we also establish functional central limit theorems.

Appendix

In this section we gather some technical estimates needed for our proof. Recall that $\langle x \rangle = \sqrt{1 + |x|^2}$.

We first record the useful inequality known as Peetre’s inequality. See [13, Lemma 34.34, p. 738] for the proof.

Lemma A.1

(Peetre’s inequality) For any $k \in {\mathbb {R}}$ and $x,y \in {\mathbb {R}}^3$, we have

$$\begin{aligned} \langle x + y \rangle ^k \leqslant 2^{\frac{|k|}{2}}\langle x \rangle ^k\langle y \rangle ^{|k|}. \end{aligned}$$

The following two lemmas are easy to check and we state without the proof.

Lemma A.2

Let $\alpha \in {\mathbb {R}}$ and $R \geqslant 2$. Then, we have

$$\begin{aligned} \int _{B_R}\langle x \rangle ^{\alpha } dx&\lesssim {\left\{ \begin{array}{ll} R^{3+\alpha } &{}(\alpha> -3)\\ \log R &{}(\alpha = -3)\\ 1 &{}(\alpha< -3) \end{array}\right. },\\ \int _{{B^2_R}}\langle x-y \rangle ^{\alpha }dxdy&\lesssim {\left\{ \begin{array}{ll} R^{6+\alpha } &{}(\alpha> -3)\\ R^3\log R &{}(\alpha = -3)\\ R^3 &{}(\alpha< -3) \end{array}\right. },\\ \int _{{B^3_R}}\langle x-y \rangle ^{\alpha }\langle y-z \rangle ^{\alpha }dxdydz&\lesssim {\left\{ \begin{array}{ll} R^{9+2\alpha } &{}(\alpha > -3)\\ R^3(\log R)^2 &{}(\alpha = -3)\\ R^3 &{}(\alpha < -3) \end{array}\right. }. \end{aligned}$$

Lemma A.3

For any $\alpha \in {\mathbb {R}}$, we have

$$\begin{aligned} \sup _{t \in [0,T]}\int _{{\mathbb {R}}^3}\langle x \rangle ^{\alpha }G(t,dx)< \infty ,\\ \sup _n\sup _{t \in [0,T]}\int _{{\mathbb {R}}^3}\langle x \rangle ^{\alpha }G_n(t,dx) < \infty . \end{aligned}$$

Using Lemmas A.2 and A.3, we can prove the following estimate which is needed for the proof of Lemma 4.4.

Lemma A.4

Let $f,g \in L^1({\mathbb {R}}^3)$ be nonnegative functions, $\alpha >0$, and $R\geqslant 2$. Suppose that nonnegative functions $\gamma$ and F satisfy

$$\begin{aligned} F(x) \lesssim \theta _1 f(x) + \theta _2 \langle x \rangle ^{-\alpha }, \end{aligned}$$

(A.1)

$$\begin{aligned} \gamma (x) \lesssim \theta _3 g(x) + \theta _4 \langle x \rangle ^{-\alpha }, \end{aligned}$$

(A.2)

where $\theta _i$ is a constant such that $\theta _i \in \{0,1\}$ for all $1 \leqslant i \leqslant 4$. Then, for any $0 \leqslant s \leqslant r \leqslant T$, we have

$$\begin{aligned}&\int _{{\mathbb {R}}^{12}}dydy'dzdz' \varphi _{n,t_1,R}(r,y)\varphi _{n,t_2,R}(s,z)\nonumber \\&\times \gamma (y-z) \varphi _{n,t_1,R}(r,y')\varphi _{n,t_2,R}(s,z')\gamma (y'-z')F(z-z') \nonumber \\&\lesssim {\left\{ \begin{array}{ll} \theta _1\theta _3R^3 + \theta _3(\theta _1\theta _4 + \theta _2)R^{6-\alpha } + \theta _4(\theta _1+\theta _2\theta _3)R^{9-2\alpha } \\ + \theta _2\theta _4R^{12-3\alpha } &{}(0< \alpha < 3)\\ \theta _1\theta _3R^3 + \theta _3(\theta _1\theta _4 + \theta _2)R^3(\log R)+ \theta _4(\theta _1\\ +\theta _2\theta _3)R^3(\log R)^2 + \theta _2\theta _4 R^3(\log R)^3 &{}(\alpha = 3)\\ (\theta _1 + \theta _2)(\theta _3 + \theta _3\theta _4 + \theta _4)R^3 &{}(\alpha > 3) \end{array}\right. }. \end{aligned}$$

(A.3)

Proof

By (A.1), we have

$$\begin{aligned}&\int _{{\mathbb {R}}^{12}}dydy'dzdz' \varphi _{n,t_1,R}(r,y)\varphi _{n,t_2,R}(s,z)\\&\quad \times \gamma (y-z) \varphi _{n,t_1,R}(r,y')\varphi _{n,t_2,R}(s,z')\gamma (y'-z')F(z-z')\\&\lesssim \theta _1\int _{{\mathbb {R}}^{12}}dydy'dzdz'\int _{{B^4_R}}dxdx'dwdw'\\&\quad \times G_n(t_1-r,x-y)G_n(t_2-s,x'-z)\gamma (y-z)\\&\quad \times G_n(t_1-r,w-y')G_n(t_2-s,w'-z')\gamma (y'-z')f(z-z')\\&\quad + \theta _2\int _{{\mathbb {R}}^{12}}dydy'dzdz'\int _{{B^4_R}}dxdx'dwdw'\\&\quad \times G_n(t_1-r,x-y)G_n(t_2-s,x'-z)\gamma (y-z)\\&\quad \times G_n(t_1-r,w-y')G_n(t_2-s,w'-z')\gamma (y'-z')\langle z-z' \rangle ^{-\alpha }\\&{=}{:}\,\theta _1\mathbf {A_1} + \theta _2\mathbf {A_2}. \end{aligned}$$

For the term $\mathbf {A_1}$, we have from (A.2) that

$$\begin{aligned} \mathbf {A_1}&\lesssim \theta _3^2\int _{{\mathbb {R}}^{12}}dydy'dzdz'\int _{{B^4_R}}dxdx'dwdw'G_n(t_1{-}r,x{-}y)G_n(t_2{-}s,x'{-}z)g(y{-}z)\\&\quad \times G_n(t_1-r,w-y')G_n(t_2-s,w'-z')g(y'-z')f(z-z')\\&\quad + \theta _3\theta _4\int _{{\mathbb {R}}^{12}}dydy'dzdz'\int _{{B^4_R}}dxdx'dwdw'G_n(t_1{-}r,x{-}y)G_n(t_2{-}s,x'{-}z)g(y{-}z)\\&\quad \times G_n(t_1-r,w-y')G_n(t_2-s,w'-z')\langle y'-z' \rangle ^{-\alpha }f(z-z')\\&\quad {+}\theta _3\theta _4\int _{{\mathbb {R}}^{12}}dydy'dzdz'\int _{{B^4_R}}dxdx'dwdw'G_n(t_1{-}r,x{-}y)G_n(t_2{-}s,x'{-}z)\langle y-z \rangle ^{-\alpha }\\&\quad \times G_n(t_1-r,w-y')G_n(t_2-s,w'-z')g(y'-z')f(z-z')\\&\quad {+}\theta _4^2\int _{{\mathbb {R}}^{12}}dydy'dzdz'\int _{{B^4_R}}dxdx'dwdw'G_n(t_1{-}r,x{-}y)G_n(t_2{-}s,x'-z)\langle y-z \rangle ^{-\alpha }\\&\quad \times G_n(t_1-r,w-y')G_n(t_2-s,w'-z')\langle y'-z' \rangle ^{-\alpha }f(z-z')\\&{=}{:}\, \theta _3^2\mathbf {A_{11}}+\theta _3\theta _4\mathbf {A_{12}}+\theta _3\theta _4\mathbf {A_{13}}{+}\theta _4^2\mathbf {A_{14}}. \end{aligned}$$

Similarly, for the term $\mathbf {A_2}$, we have from (A.2) that

$$\begin{aligned} \mathbf {A_2}&\lesssim \theta _3^2\int _{{\mathbb {R}}^{12}}dydy'dzdz'\int _{{B^4_R}}dxdx'dwdw'G_n(t_1-r,x-y)G_n(t_2-s,x'-z)g(y-z)\\&\quad \times G_n(t_1-r,w-y')G_n(t_2-s,w'-z')g(y'-z')\langle z-z' \rangle ^{-\alpha }\\&\quad +\theta _3\theta _4\int _{{\mathbb {R}}^{12}}dydy'dzdz'\int _{{B^4_R}}dxdx'dwdw'G_n(t_1-r,x-y)G_n(t_2-s,x'-z)g(y-z)\\&\quad \times G_n(t_1-r,w-y')G_n(t_2-s,w'-z')\langle y'-z' \rangle ^{-\alpha }\langle z-z' \rangle ^{-\alpha }\\&\quad +\theta _3\theta _4\int _{{\mathbb {R}}^{12}}dydy'dzdz'\int _{{B^4_R}}dxdx'dwdw'G_n(t_1-r,x-y)G_n(t_2-s,x'-z)\langle y-z \rangle ^{-\alpha }\\&\quad \times G_n(t_1-r,w-y')G_n(t_2-s,w'-z')g(y'-z')\langle z-z' \rangle ^{-\alpha }\\&\quad +\theta _4^2\int _{{\mathbb {R}}^{12}}dydy'dzdz'\int _{{B^4_R}}dxdx'dwdw'G_n(t_1-r,x-y)G_n(t_2-s,x'-z)\langle y-z \rangle ^{-\alpha }\\&\quad \times G_n(t_1-r,w-y')G_n(t_2-s,w'-z')\langle y'-z' \rangle ^{-\alpha }\langle z-z' \rangle ^{-\alpha }\\&{=}{:}\, \theta _3^2\mathbf {A_{21}}+\theta _3\theta _4\mathbf {A_{22}}+\theta _3\theta _4\mathbf {A_{23}}+\theta _4^2\mathbf {A_{24}}. \end{aligned}$$

First we consider $\mathbf {A_{11}}$. Integrating in the order $dw, dw', dy', dz', dx, dy,dz,dx'$, we get

$$\begin{aligned} \mathbf {A_{11}}&\lesssim _T \int _{{\mathbb {R}}^{12}}dydy'dzdz'\int _{{B^2_R}}dxdx'G_n(t_1-r,x-y)G_n(t_2-s,x'-z)g(y-z)\\&\qquad \times g(y'-z')f(z-z')\\&\lesssim \Vert g\Vert _{L^1({\mathbb {R}}^3)}\Vert f\Vert _{L^1(\mathbb R)^3}\int _{{\mathbb {R}}^{6}}dydz\int _{{B^2_R}}dxdx'\\&\qquad \times G_n(t_1-r,x-y)G_n(t_2-s,x'-z)g(y-z)\lesssim _T R^3. \end{aligned}$$

Next we estimate $\mathbf {A_{12}}$. Integrating in the order $dx,dx',dy,dz$, we obtain

$$\begin{aligned} \mathbf {A_{12}}&\lesssim _T \Vert g\Vert _{L^1({\mathbb {R}}^3)}\Vert f\Vert _{L^1(\mathbb R)^3}\int _{{\mathbb {R}}^{6}}dy'dz'\int _{{B^2_R}}dwdw'\\&\times G_n(t_1-r,w-y')G_n(t_2-s,w'-z')\langle y'-z' \rangle ^{-\alpha }. \end{aligned}$$

Furthermore, Using Peetre’s inequality and integrating with respect to $dy', dz'$, we get

$$\begin{aligned} \mathbf {A_{12}}&\lesssim \int _{{\mathbb {R}}^{6}}dy'dz'\int _{{B^2_R}}dwdw'G_n(t_1-r,w-y')\langle w-y' \rangle ^{\alpha }\\&\qquad \qquad \qquad \quad \times G_n(t_2-s,w'-z')\langle w'-z' \rangle ^{\alpha }\langle w-w' \rangle ^{-\alpha }\\&\lesssim _T \int _{{B^2_R}}dwdw'\langle w-w' \rangle ^{-\alpha }\\&\lesssim {\left\{ \begin{array}{ll} R^{6-\alpha } &{}(0<\alpha < 3)\\ R^3\log R &{}(\alpha = 3)\\ R^3 &{}(\alpha > 3 ) \end{array}\right. }. \end{aligned}$$

$\mathbf {A_{13}}$ can be estimated in the same way as $\mathbf {A_{12}}$. Now we consider the term $\mathbf {A_{14}}$. Applying Peetre’s inequality and integrating with respect to $dy,dy'$ lead to

$$\begin{aligned} \mathbf {A_{14}}&\lesssim \int _{{\mathbb {R}}^{12}}dydy'dzdz'\int _{{B^4_R}}dxdx'dwdw'\\&\quad \qquad \times G_n(t_1-r,x-y)\langle x-y \rangle ^{\alpha }G_n(t_2-s,x'-z)\langle x'-z \rangle ^{\alpha }\\&\quad \qquad \times G_n(t_1-r,w-y')\langle w-y' \rangle ^{\alpha }G_n(t_2-s,w'-z')\langle w'-z' \rangle ^{\alpha }\\&\quad \qquad \times \langle x-x' \rangle ^{-\alpha }\langle w-w' \rangle ^{-\alpha }f(z-z')\\&\lesssim _T \int _{{\mathbb {R}}^{6}}dzdz'\int _{{B^4_R}}dxdx'dwdw'\\&\quad \qquad \times G_n(t_2-s,x'-z)\langle x'-z \rangle ^{\alpha }G_n(t_2-s,w'-z')\langle w'-z' \rangle ^{\alpha }\\&\quad \qquad \times \langle x-x' \rangle ^{-\alpha }\langle w-w' \rangle ^{-\alpha }f(z-z')\\&= \int _{{\mathbb {R}}^{6}}dzdz'\int _{{B^2_R}}dwdw'G_n(t_2-s,w'-z')\langle w'-z' \rangle ^{\alpha }\langle w-w' \rangle ^{-\alpha }f(z-z')\\&\quad \qquad \times \left\{ \int _{B_R}G_n(t_2-s,x'-z)\langle x'-z \rangle ^{\alpha }\left( \int _{B_R(-x')}\langle x \rangle ^{-\alpha }dx \right) dx' \right\} \\&\leqslant \int _{{\mathbb {R}}^{6}}dzdz'\int _{{B^2_R}}dwdw'G_n(t_2-s,w'-z')\langle w'-z' \rangle ^{\alpha }\langle w-w' \rangle ^{-\alpha }f(z-z')\\&\quad \qquad \times \left( \int _{B_R}G_n(t_2-s,x'-z)\langle x'-z \rangle ^{\alpha }dx' \right) \left( \int _{B_{2R}}\langle x \rangle ^{-\alpha }dx \right) . \end{aligned}$$

Therefore, integrating in the order $dx',dz,dz'$, we have

$$\begin{aligned} \mathbf {A_{14}}&\lesssim _T \Vert f\Vert _{L^1({\mathbb {R}})}\int _{B_{2R}}\langle x \rangle ^{-\alpha }dx\int _{{B^2_R}} \langle w-w' \rangle ^{-\alpha }dwdw'\\&\lesssim {\left\{ \begin{array}{ll} R^{9-2\alpha } &{}(0<\alpha < 3)\\ R^3(\log R)^2 &{}(\alpha = 3)\\ R^3 &{}(\alpha > 3 ) \end{array}\right. }. \end{aligned}$$

Next we estimate $\mathbf {A_{21}}$. Integrating in the order $dx,dw,dy,dy'$, we get

$$\begin{aligned} \mathbf {A_{21}}&\lesssim _T \Vert g\Vert _{L^1({\mathbb {R}}^3)}^2 \int _{{\mathbb {R}}^{6}}dzdz'\int _{{B^2_R}}dx'dw'\\&\times G_n(t_2-s,x'-z) G_n(t_2-s,w'-z')\langle z-z' \rangle ^{-\alpha }. \end{aligned}$$

Thus, using Peetre’s inequality and integrating with respect to $dz, dz'$, we obtain

$$\begin{aligned} \mathbf {A_{21}}&\lesssim \int _{{\mathbb {R}}^{6}}dzdz'\int _{{B^2_R}}dx'dw'\langle x'-w' \rangle ^{-\alpha }\\&\qquad \times G_n(t_2-s,x'-z)\langle x'-z \rangle ^{\alpha } G_n(t_2-s,w'-z')\langle w'-z' \rangle ^{\alpha }\\&\lesssim _T \int _{{B^2_R}}dx'dw'\langle x'-w' \rangle ^{-\alpha }\\&\lesssim {\left\{ \begin{array}{ll} R^{6-\alpha } &{}(0<\alpha < 3)\\ R^3\log R &{}(\alpha = 3)\\ R^3 &{}(\alpha > 3 ) \end{array}\right. }. \end{aligned}$$

Next we consider $\mathbf {A_{22}}$. Integrating in the order dx, dy, we have

$$\begin{aligned} \mathbf {A_{22}}&\lesssim _T \Vert g\Vert _{L^1({\mathbb {R}}^3)} \int _{{\mathbb {R}}^{9}}dy'dzdz'\int _{{B^3_R}}dx'dwdw'\langle y'-z' \rangle ^{-\alpha }\langle z-z' \rangle ^{-\alpha }\\&\quad \qquad \qquad \quad \times G_n(t_2-s,x'-z)G_n(t_1-r,w-y')G_n(t_2-s,w'-z'). \end{aligned}$$

Then, applying Peetre’s inequality repeatedly and integrating with respect to $dz,dy',dz'$, we get

$$\begin{aligned} \mathbf {A_{22}}&\lesssim \int _{{\mathbb {R}}^{9}}dy'dzdz'\int _{{B^3_R}}dx'dwdw'\langle w-w' \rangle ^{-\alpha }\langle x'-w' \rangle ^{-\alpha }\\&\qquad \times G_n(t_2-s,x'-z)\langle x'-z \rangle ^{\alpha }G_n(t_1-r,w-y')\langle w-y' \rangle ^{\alpha }\\&\qquad \times G_n(t_2-s,w'-z')\langle w'-z' \rangle ^{2\alpha }\\&\lesssim _T \int _{{B^3_R}}dx'dwdw'\langle w-w' \rangle ^{-\alpha }\langle x'-w' \rangle ^{-\alpha }\\&\lesssim {\left\{ \begin{array}{ll} R^{9-2\alpha } &{}(0<\alpha < 3)\\ R^3(\log R)^2 &{}(\alpha = 3)\\ R^3 &{}(\alpha > 3 ) \end{array}\right. }. \end{aligned}$$

$\mathbf {A_{23}}$ can be estimated as well as $\mathbf {A_{22}}$. Finally, we estimate the term $\mathbf {A_{24}}$. Using Peetre’s inequality iteratively and integrating with respect to $dy,dz,dy',dz'$, we obtain

$$\begin{aligned} \mathbf {A_{24}}&\lesssim \int _{{\mathbb {R}}^{12}}dydy'dzdz'\int _{{B^4_R}}dxdx'dwdw'\langle x-x' \rangle ^{-\alpha }\langle x'-w' \rangle ^{-\alpha }\langle w'-w \rangle ^{-\alpha }\\&\quad \qquad \times G_n(t_1-r,x-y)\langle x-y \rangle ^{\alpha }G_n(t_2-s,x'-z)\langle x'-z \rangle ^{2\alpha }\\&\quad \qquad \times G_n(t_1-r,w-y')\langle w-y' \rangle ^{\alpha }G_n(t_2-s,w'-z')\langle w'-z' \rangle ^{2\alpha }\\&\lesssim _T \int _{{B^4_R}}dxdx'dwdw'\langle x-x' \rangle ^{-\alpha }\langle x'-w' \rangle ^{-\alpha }\langle w'-w \rangle ^{-\alpha }\\&\leqslant \int _{B_{2R}}\langle x \rangle ^{-\alpha }dx\int _{{B^3_R}}dx'dwdw'\langle x'-w' \rangle ^{-\alpha }\langle w'-w \rangle ^{-\alpha }\\&\lesssim {\left\{ \begin{array}{ll} R^{12-3\alpha } &{}(0<\alpha < 3)\\ R^3(\log R)^3 &{}(\alpha = 3)\\ R^3 &{}(\alpha > 3 ) \end{array}\right. }. \end{aligned}$$

As a consequence, combining altogether we have (A.3). $\square$