Ergodicity for Ginzburg-Landau equation with complex-valued space-time white noise on two-dimensional torus

We are interested in the stochastic complex Ginzburg-Landau equation on the two-dimensional torus defined as

$$\begin{cases}\partial_{t}u=(\mathrm{i}+\mu)\Delta u-\nu|u|^{2m}u+\tau u+\xi,&t>0,x\in\mathbb{T}^{2},\\ u(0,\cdot)=u_{0},\end{cases}$$

(1)

where $\mu>0$, $\nu,\tau\in\mathbb{C}$, $\mathrm{Re}\,\nu>0$, $m\geq 1$ is an integer and $\xi$ is the complex-valued space-time white noise as introduced in Section 2. The definition of $\xi$ can be found in (3), and a construction for the complex-valued space-time white noise is provided in (5). Here, we call $\mathrm{i}\Delta u$ the dispersion term and $\mu\Delta u$ the dissipation term. We aim to get its global well-posedness and ergodicity.

The complex Ginzburg-Landau equation is one of the most crucial nonlinear partial differential equations (PDEs) which describes various physical phenomena such as nonlinear waves and superconducting phase transition and superfluidity, see [1]. In addition to wide applications in physics, it has rich research background in mathematics as a parabolic equation. In [29], Parisi and Wu proposed to construct a quantum field theory based on the invariant measure of a random process, which is called stochastic quantisation. We establish the global well-posedness and ergodicity of (1), which is important to study the complex-valued $\Phi_{2}^{2(m+1)}$ measure in quantum field theory.

There has been a substantial amount of research on complex Ginzburg-Landau equations with random forcing. For non-white or multiplicative noise, see [3, 4, 21, 28] for example. In [15], Hairer studied the complex cubic Ginzburg-Landau equation driven by a real-valued space-time white noise in one spatial dimension. When the spatial dimension is larger than one, due to the roughness of complex-valued space-time white noise, complex Ginzburg-Landau equation driven by complex-valued space-time white noise is a singular stochastic partial differential equation and its solution is expected to be a distribution-valued stochastic process. For this reason, the non-linear term $|u|^{2m}u$ is ill-defined and needs to be renormalized. The local and global well-posedness of the complex Ginzburg-Landau equation on the three-dimensional torus driven by complex-valued space-time white noise are obtained in [18, 19] using the theory of regularity structure (see [16]) and paracontrolled distribution (see [14]).

For two-dimensional case, in [34], Trenberth worked with the generalized Laguerre polynomials for the renormalization of (1) and obtained its global well-posedness. Different from the way of renormalization using generalized Laguerre polynomials in [34], we utilize complex multiple Wiener-Itô integral to renormalize (1) more clearly in Section 3. We show the regularity of stochastic heat equation with dispersion (see Proposition 3.1) combining the fact that a complex multiple Wiener-Itô integral can be viewed as a two-dimensional real multiple Wiener-Itô integral (see [9, Theorem 3.3]), with [27, Proposition 5], which concerning the regularity of real-valued random processes. This idea that utilizing the theory of two-dimensional real Wiener-Itô integral to solve the theoretical and practical problems involving complex Wiener-Itô integral, is what we have always wanted to express to the reader in [7] and [8]. Compared to [25] where the regularity is obtained using [26, Lemma 9], our proof is shorter and more concise. Motivated by [35], Matsuda in [25] proved the global well-posedness and strong Feller property of (1) for $m=1$. Combining with the support theorem proved in [24], Matsuda got its exponential ergodicity in [25]. In our setting, we consider a general nonlinearity ($m\geq 1$) and prove the ergodicity of (1) using an alternative approach, namely an asymptotic coupling argument (see [17]), under the assumption that dissipation coefficient $\mu$ is greater than some constant (see (68)).

This paper is organized as follows. Section 2 introduces some notions of Besov space and complex-valued space-time white noise. In Section 3, we show the regularity of Wick product associated with stochastic heat equation with dispersion (6). In Section 4, inspired by the Da Prato-Debussche method (see [10]) and combining with the priori estimates in Section 3, we obtain the global well-posedness of the stochastic complex Ginzburg-Landau equation (1) in the sense of renormalization. In Section 5, by the asymptotic (or generalized) coupling approach developed in [17, 22], we show the existence and uniqueness of the invariant measure of the renormalization solution to (1).

To characterize the regularity of the stationary solution of the stochastic heat equation with dispersion (6), we introduce the Besov space. We refer readers to [2, Chapter 2] and [13, Chapter 3] for more details about the Besov space and Fourier analysis on the $d$-dimensional torus $\mathbb{T}^{d}$, where $d\geq 1$ is an integer. We denote by $C^{\infty}\left(\mathbb{T}^{d},\mathbb{C}\right)$ the space of all complex-valued smooth functions defined over $\mathbb{T}^{d}$. For $p\geq 1$, let $L^{p}:=L^{p}\left(\mathbb{T}^{d},\mathbb{C}\right)$ and $L^{p}\left(\mathbb{R}\times\mathbb{T}^{d},\mathbb{C}\right)$ be the space of all complex-valued functions that are $p$-integrable over $\mathbb{T}^{d}$ and $\mathbb{R}\times\mathbb{T}^{d}$, respectively. For $f,g\in L^{2}(G,\mathbb{C})$, where $G=\mathbb{T}^{d}\mbox{ or }\mathbb{R}\times\mathbb{T}^{d}$, we set

$$\left\langle f,g\right\rangle=\int_{G}f(x)g(x)\mathrm{d}x$$

in this chapter. Note that we do not take complex conjugate for $g$.

For any $f\in L^{1}$ and $k\in\mathbb{Z}^{d}$, we write

$$\mathscr{F}f(k)=\hat{f}(k):=\int_{\mathbb{T}^{d}}f(x)e^{-2\pi\mathrm{i}k\cdot x}\mathrm{d}x,$$

for the Fourier coefficient of $f$ with frequency $k$.

We now introduce distributions on the torus. The set of test functions on the torus is defined as

$$\mathscr{S}\left(\mathbb{T}^{d},\mathbb{C}\right):=\left\{f\in{C}^{\infty}\left(\mathbb{T}^{d},\mathbb{C}\right):\exists M_{\alpha}<\infty,\text{ s.t., }\left\|\partial^{\alpha}f\right\|_{L^{\infty}}\leq M_{\alpha}<\infty,\alpha\in\mathbb{N}^{d}\right\},$$

where $M_{\alpha}$ is a constant depending on the multi-index $\alpha$. We define the semi-norms as

$$\left\|f\right\|_{m}:=\sup_{|\alpha|\leq m}\left\|\partial^{\alpha}f\right\|_{L^{\infty}},\quad m\in\mathbb{N},$$

where for a multi-index $\alpha=\left(\alpha_{1},\ldots,\alpha_{d}\right)\in\mathbb{N}^{d}$, $|\alpha|=\alpha_{1}+\cdots+\alpha_{d}$. The space of all complex-valued linear and continuous functionals on $\mathscr{S}\left(\mathbb{T}^{d},\mathbb{C}\right)$ is denoted by $\mathscr{S}^{\prime}\left(\mathbb{T}^{d},\mathbb{C}\right)$ and is called the space of distributions. For $u\in\mathscr{S}^{\prime}\left(\mathbb{T}^{d},\mathbb{C}\right)$, we define the Fourier coefficient with frequency $k\in\mathbb{Z}^{d}$ as

$$\hat{u}(k):=\left\langle u,e^{-2\pi\mathrm{i}k\cdot(\cdot)}\right\rangle,$$

where $\left\langle u,\cdot\right\rangle$ stands for the action of $u$ on $\mathscr{S}\left(\mathbb{T}^{d},\mathbb{C}\right)$. This is well-defined since $e^{-2\pi\mathrm{i}k\cdot(\cdot)}\in\mathscr{S}\left(\mathbb{T}^{d},\mathbb{C}\right)$.

We denote by $\left\{\chi_{j}\right\}_{j=-1}^{\infty}$ a dyadic partition of unity satisfying the following properties

• •

  $\chi_{-1},\chi_{0}:\mathbb{R}^{d}\rightarrow[0,1]$ are radial and infinitely differentiable;

• •

  $\mathrm{supp}(\chi_{-1})\subseteq B(0,\frac{4}{3})$ and $\mathrm{supp}(\chi_{0})\subseteq B(0,\frac{8}{3})\setminus B(0,\frac{3}{4})$, where $B(0,r):=\left\{x\in\mathbb{R}^{d}:|x|<r\right\}$;

• •

  for $j\geq 0$, $\chi_{j}(\cdot)=\chi_{0}(\cdot/2^{j})$;

• •

  for any $x\in\mathbb{R}^{d}$, $\sum_{j=-1}^{\infty}\chi_{j}(x)=1$.

The existence of dyadic partitions of unity can be found in [2, Proposition 2.10].

For any $f\in C^{\infty}(\mathbb{T}^{d},\mathbb{C})$ and $j\geq-1$, we define the $j$-th Littlewood-Paley block as

$$\delta_{j}f:=\sum_{k\in\mathbb{Z}^{d}}\chi_{j}(k)\hat{f}(k)e^{2\pi\mathrm{i}k\cdot x}.$$

The space of $\mathcal{B}_{p,q}^{\alpha}$ for $\alpha\in\mathbb{R}$ and $p,q\in[1,\infty]$ is defined as the completion of $C^{\infty}\left(\mathbb{T}^{d},\mathbb{C}\right)$ with respect to the norm

$$\|f\|_{\mathcal{B}_{p,q}^{\alpha}}:=\left(\sup_{j\geq-1}2^{\alpha jq}\left\|\delta_{j}f\right\|_{L^{p}}^{q}\right)^{\frac{1}{q}}.$$

When $p=q=\infty$, we denote by $\mathcal{C}^{\alpha}$ for $\alpha\in\mathbb{R}$ the space $\mathcal{B}_{\infty,\infty}^{\alpha}$. That is, the space $\mathcal{C}^{\alpha}$ for $\alpha\in\mathbb{R}$ is defined as the completion of $C^{\infty}\left(\mathbb{T}^{d},\mathbb{C}\right)$ with respect to the norm

$$\|f\|_{\mathcal{C}^{\alpha}}:=\sup_{j\geq-1}2^{\alpha j}\left\|\delta_{j}f\right\|_{L^{\infty}}.$$

This space is a subspace of $\mathscr{S}^{\prime}\left(\mathbb{T}^{d},\mathbb{C}\right)$. We set $\mathcal{C}^{\alpha-}:=\bigcap_{\epsilon>0}\mathcal{C}^{\alpha-\epsilon}$ for $\alpha\in\mathbb{R}$.

We refer reader to [2, 26, 33] for the following properties of the Besov space.

The smoothing effect of the heat semigroup $\left(P_{t}=\mathrm{e}^{At}\right)_{t\geq 0}$ with generator $A:=\left(\mathrm{i}+\mu\right)\Delta-1$ is shown in the following proposition.

The above proposition shows that, if $0<\beta-\alpha\leq 2$ and $f\in\mathcal{B}_{p,q}^{\beta}$, then the mapping $t\mapsto P_{t}f$ is continuous as a mapping from $[0,\infty)$ to $\mathcal{B}_{p,q}^{\alpha}$. When $\beta=\alpha$, this is still valid. That is, if $\alpha\in\mathbb{R}$ and $f\in\mathcal{B}_{p,q}^{\alpha}$, then the mapping $t\mapsto P_{t}f$ is continuous as a mapping from $[0,\infty)$ to $\mathcal{B}_{p,q}^{\alpha}$.

The multiplicative structure and duality property of Besov space are presented in the following two propositions.

For every $\alpha\in(0,1]$, the following gradient estimate shows that the Besov norm $\left\|\cdot\right\|_{\mathcal{B}_{1,1}^{\alpha}}$ can be controlled by the norms $\|\cdot\|_{L^{1}}$ and $\|\nabla(\cdot)\|_{L^{1}}$.

In particular, by [26, Remark 19], there exists $C<\infty$ such that

$$\|f\|_{\mathcal{B}_{1,1}^{\alpha}}\leq C\left(\|\nabla f\|_{L^{1}}+\|f\|_{L^{1}}\right).$$

(2)

Set $\Lambda=(-A)^{\frac{1}{2}}$. For $s\geq 0$ and $1\leq p\leq\infty$, we denote by $H_{p}^{s}$ the subspace of $L^{p}$, consisting of all $f$ that can be written as $f=\Lambda^{-s}g$, where $g\in L^{p}$. The $H_{p}^{s}$ norm of $f$ is defined as the $L^{p}$ norm of $g$, that is, $\left\|f\right\|_{H_{p}^{s}}=\left\|\Lambda^{s}g\right\|_{L^{p}}$.

Recall the following Sobolev embedding results for the Sobolev space $H_{p}^{s}$ (see [5, Theorem 6.4.4] and [33, Theorem 2.13]).

The following interpolation inequality and multiplicative inequality for the Sobolev space $H_{p}^{s}$ can be found in [5, Theorem 6.4.5 ], [30, Lemma A.4] and [31, Lemma 2.1].

Next, we introduce the definition of complex-valued space-time white noise over $\mathbb{R}\times\mathbb{T}^{d}$, where $d\geq 1$ is an integer. A complex-valued space-time white noise defined on the probability space $(\Omega,\mathcal{F},P)$ over $\mathbb{R}\times\mathbb{T}^{d}$ is a family of centered complex Gaussian random variables $\left\{\xi(\varphi):\varphi\in L^{2}\left(\mathbb{R}\times\mathbb{T}^{d},\mathbb{C}\right)\right\}$ such that for any $\varphi,\psi\in L^{2}\left(\mathbb{R}\times\mathbb{T}^{d},\mathbb{C}\right)$,

$$\mathrm{E}\left[\xi(\varphi)\xi(\psi)\right]=0,\quad\mathrm{E}\left[\xi(\varphi)\overline{\xi(\psi)}\right]=\left\langle\varphi,\overline{\psi}\right\rangle_{L^{2}\left(\mathbb{R}\times\mathbb{T}^{d},\mathbb{C}\right)}.$$

(3)

Let $\left(\zeta(\cdot,k)=\left(B_{1}(\cdot,k)+\mathrm{i}B_{2}(\cdot,k)\right)/\sqrt{2}\right)_{k\in\mathbb{Z}^{d}}$ be a family of independent complex-valued Brownian motions over $\mathbb{R}$, where $\left(B_{1}(\cdot,k),B_{2}(\cdot,k)\right)_{t\in\mathbb{R}}$ is a two-dimensional standard Brownian motion for a fixed $k\in\mathbb{Z}^{d}$. We define $\xi$ as the time derivative of the cylindrical Wiener process given by

$$(t,x)\mapsto W(t,x):=\sum_{k\in\mathbb{Z}^{d}}\zeta(t,k)e^{2\pi\mathrm{i}k\cdot x}.$$

(4)

To be precise, for every $\varphi\in L^{2}\left(\mathbb{R}\times\mathbb{T}^{d},\mathbb{C}\right)$, we define

$$\xi(\varphi):=\sum_{k\in\mathbb{Z}^{d}}\int_{-\infty}^{\infty}\hat{\varphi}(t,k)\mathrm{d}\zeta(t,k),$$

(5)

where the integral is interpreted in the sense of Itô, and $\hat{\varphi}(t,k)$ denotes $\widehat{\varphi(t,\cdot)}(k)$. According to Itô’s and Fourier’s isometries, $\xi(\varphi)$, $\varphi\in L^{2}\left(\mathbb{R}\times\mathbb{T}^{d},\mathbb{C}\right)$, is well defined and (3) is satisfied. Since $\xi(\varphi)$ is also a centered complex Gaussian random variable, this provides us with a construction of complex-valued space-time white noise. For a given $\varphi\in L^{2}\left(\mathbb{R}\times\mathbb{T}^{d},\mathbb{C}\right)$, we also use somewhat informal notation

$$\xi(\varphi):=\int_{\mathbb{R}\times\mathbb{T}^{d}}\varphi(t,x)\xi(\mathrm{d}t,\mathrm{d}x),$$

although $\xi$ is almost surely not a measure and $\xi(\varphi)$ is only defined on a set of measure one which depends on the choice of $\varphi$.

In this section, we concentrate on the stochastic heat equation with dispersion defined on the two-dimensional torus $\mathbb{T}^{2}$ as follows

$$\partial_{t}Z=\left[(\mathrm{i}+\mu)\Delta-1\right]Z+\xi,\;t>0,\;x\in\mathbb{T}^{2}.$$

(6)

We derive the regularity of the stationary solution of (6) and its Wick powers.

We set $A:=(\mathrm{i}+\mu)\Delta-1$ and let $\left\{P_{t}=e^{tA}\right\}_{t\geq 0}$ be the semigroup associated to the generator $A$. For $k=(k_{1},k_{2})\in\mathbb{Z}^{2}$, we define $|k|=\sqrt{k_{1}^{2}+k_{2}^{2}}$ and set $\rho_{k}=1+4\pi^{2}\mu|k|^{2}$, $\theta_{k}=4\pi^{2}|k|^{2}$. The distribution-valued stochastic process $\left(Z_{-\infty,t}\right)_{t>-\infty}$ is defined as a stationary solution to (6) by using Duhamel’s principle and has a formal expression as follows. For $\phi\in L^{2}(\mathbb{T}^{2},\mathbb{C})$,

	$\displaystyle Z_{-\infty,t}(\phi)$	$\displaystyle=\int_{-\infty}^{t}\int_{\mathbb{T}^{2}}\left[e^{(t-s)A}\xi(s,\cdot)\right](x)\phi(x)\mathrm{d}x\mathrm{d}s$		(7)
		$\displaystyle=\int_{-\infty}^{t}\int_{\mathbb{T}^{2}}\left\langle H(t-s,\cdot-y),\phi\right\rangle\xi(\mathrm{d}s,\mathrm{d}y),$		(8)

where $H(t,z)$ is the heat kernel defined as

$$H(t,z)=\sum_{k\in\mathbb{Z}^{2}}e^{-\left(\rho_{k}+\mathrm{i}\theta_{k}\right)t}e^{2\pi\mathrm{i}k\cdot z}=\frac{e^{-t}}{4\pi(\mathrm{i}+\mu)t}e^{-\frac{|z|^{2}}{4(\mathrm{i}+\mu)t}},\quad t\in\mathbb{R}\setminus\{0\},\quad z\in\mathbb{T}^{2}.$$

For $\phi\in L^{2}(\mathbb{T}^{2},\mathbb{C})$,

		$\displaystyle\int_{-\infty}^{t}\int_{\mathbb{T}^{2}}\left|\left\langle H(t-s,\cdot-y),\phi\right\rangle\right|^{2}\mathrm{d}y\mathrm{d}s$		(9)
	$\displaystyle=$	$\displaystyle\,\int_{\mathbb{T}^{2}}\int_{\mathbb{T}^{2}}\int_{-\infty}^{t}\int_{\mathbb{T}^{2}}H(t-s,x-y)\overline{H(t-s,z-y)}\mathrm{d}y\mathrm{d}s\phi(x)\overline{\phi(z)}\mathrm{d}x\mathrm{d}z$		(10)
	$\displaystyle=$	$\displaystyle\,\sum_{k\in\mathbb{Z}^{2}}\int_{\mathbb{T}^{2}}\int_{\mathbb{T}^{2}}\int_{-\infty}^{t}e^{-2\rho_{k}\left(t-s\right)}e^{2\pi\mathrm{i}k\cdot(x-z)}\mathrm{d}s\phi(x)\overline{\phi(z)}\mathrm{d}x\mathrm{d}z$		(11)
	$\displaystyle=$	$\displaystyle\,\sum_{k\in\mathbb{Z}^{2}}\frac{1}{2\rho_{k}}\int_{\mathbb{T}^{2}}\int_{\mathbb{T}^{2}}e^{2\pi\mathrm{i}k\cdot(x-z)}\phi(x)\overline{\phi(z)}\mathrm{d}x\mathrm{d}z$		(12)
	$\displaystyle=$	$\displaystyle\,\sum_{k\in\mathbb{Z}^{2}}\frac{1}{2\rho_{k}}\left|\hat{\phi}(k)\right|^{2}\leq\left\|\phi\right\|_{L^{2}}^{2}<\infty,$		(13)

Therefore, $Z_{-\infty,t}(\phi)$ for $\phi\in L^{2}(\mathbb{T}^{2},\mathbb{C})$ is a well-defined Gaussian variable with mean zero and variance $\sum_{k\in\mathbb{Z}^{2}}\frac{1}{2\rho_{k}}\left|\hat{\phi}(k)\right|^{2}$.

We set $Z^{:0,0:}_{-\infty,t}(x)\equiv 1$ for $x\in\mathbb{T}^{2}$. For $k,l\geq 0$, $k+l>0$, we define $(k,l)$-th Wick power of $Z_{-\infty,t}(\phi)$, where $\phi\in L^{2}(\mathbb{T}^{2},\mathbb{C})$, as the complex $(k,l)$-th Wiener-Itô integral with respect to $\xi$

	$\displaystyle Z^{:k,l:}_{-\infty,t}(\phi)=$	$\displaystyle\int_{\left\{(-\infty,t])\times\mathbb{T}^{2}\right\}^{k+l}}\left\langle\prod_{j=1}^{k}H(t-s_{j},\cdot-y_{j})\prod_{j=k+1}^{k+l}\overline{H}(t-s_{j},\cdot-y_{j}),\phi\right\rangle$		(14)
		$\displaystyle\xi(\mathrm{d}s_{1},\mathrm{d}y_{1})\cdots\xi(\mathrm{d}s_{k},\mathrm{d}y_{k})\overline{\xi}(\mathrm{d}s_{k+1},\mathrm{d}y_{k+1})\cdots\overline{\xi}(\mathrm{d}s_{k+l},\mathrm{d}y_{k+l}).$		(15)

By the similar argument as (9), we know that for $\phi\in L^{2}(\mathbb{T}^{2},\mathbb{C})$,

		$\displaystyle k!l!\int_{\left\{(-\infty,t])\times\mathbb{T}^{2}\right\}^{k+l}}\left|\left\langle\prod_{j=1}^{k}H(t-s_{j},\cdot-y_{j})\prod_{j=k+1}^{k+l}\overline{H}(t-s_{j},\cdot-y_{j}),\phi\right\rangle\right|^{2}$		(16)
		$\displaystyle\qquad\qquad\qquad\qquad\quad\mathrm{d}s_{1}\mathrm{d}y_{1}\cdots\mathrm{d}s_{k+l}\mathrm{d}y_{k+l}$		(17)
	$\displaystyle=$	$\displaystyle\,k!l!\sum_{j_{1},\ldots,j_{k+l}\in\mathbb{Z}^{2}}\left(\prod_{i=1}^{k+l}\frac{1}{2\rho_{j_{i}}}\right)\left|\hat{\phi}\left(\sum_{i=1}^{k+l}j_{i}\right)\right|^{2}<\infty.$		(18)

Therefore, $Z^{:k,l:}_{-\infty,t}(\phi)\in\mathscr{H}_{k,l}(\xi)$ with variance (16), where $\phi\in L^{2}(\mathbb{T}^{2},\mathbb{C})$ and $\mathscr{H}_{k,l}(\xi)$ is the complex $(k,l)$-th Wiener-Itô chaos of $\xi$.

From the definition (14), we get that

$$Z^{:l,k:}_{-\infty,t}(x)=\overline{Z^{:k,l:}_{-\infty,t}(x)},\quad x\in\mathbb{T}^{2},\quad k,l\in\mathbb{N}.$$

(19)

{proof}

Combining [27, Proposition 5] with the fact that $Z^{:k,l:}_{-\infty,\cdot}$ is spatially and temporally stationary, to prove (20), it suffices to show that there exists $0<C<\infty$ such that for every $\omega\in\mathbb{Z}^{2}$,

$$\mathrm{E}\left[\left|\hat{Z}^{:k,l:}_{-\infty,t}(\omega)\right|^{2}\right]\leq C(1+|\omega|)^{-2},$$

(22)

To prove (21), it is sufficient to prove that for any $\lambda\in(0,1)$,

$$\mathrm{E}\left[\left|\hat{Z}^{:k,l:}_{-\infty,t}(\omega)-\hat{Z}^{:k,l:}_{-\infty,s}(\omega)\right|^{2}\right]\leq C|t-s|^{\lambda}(1+|\omega|)^{-2+2\lambda},$$

(23)

uniformly in $0<|t-s|<1$ and $\omega\in\mathbb{Z}^{2}$. Then by [27, Proposition 5], we have that for every $\beta<-\lambda$, $Z^{:k,l:}_{-\infty,t}(x)\in C(\mathbb{R},\mathcal{C}^{\beta})$ and

$$\sup_{s<t}\frac{\mathrm{E}\left[\left\|Z^{:k,l:}_{-\infty,t}-Z^{:k,l:}_{-\infty,s}\right\|^{p}_{\mathcal{C}^{\beta}}\right]}{|t-s|^{\lambda p/2}}<\infty.$$

(24)

Then for any $\alpha\in(0,1)$, we take $\lambda=\alpha/2\in(0,1)$ and then $-\alpha<-\lambda$. Replacing $\beta$ with $-\alpha$ in (24), we can get that $Z^{:k,l:}_{-\infty,t}(x)\in C(\mathbb{R};\mathcal{C}^{-\alpha})$ and (21) with $\theta_{0}=\alpha/4$.

We first prove (22). For a fixed $\omega\in\mathbb{Z}^{2}$, let $\phi=e^{-2\pi\mathrm{i}\omega\cdot x}\in L^{2}(\mathbb{T}^{2},\mathbb{C})$, where $x\in\mathbb{T}^{2}$.

		$\displaystyle\mathrm{E}\left[\hat{Z}^{:k,l:}_{-\infty,t}(\omega)\overline{\hat{Z}^{:k,l:}_{-\infty,s}(\omega)}\right]=\mathrm{E}\left[\left\langle Z^{:k,l:}_{-\infty,t},\phi\right\rangle\overline{\left\langle Z^{:k,l:}_{-\infty,s},\phi\right\rangle}\right]$
	$\displaystyle=$	$\displaystyle\,k!l!\int_{\left\{(-\infty,t\wedge s])\times\mathbb{T}^{2}\right\}^{k+l}}\left\langle\prod_{j=1}^{k}H(t-s_{j},x_{1}-y_{j})\prod_{j=k+1}^{k+l}\overline{H}(t-s_{j},x_{1}-y_{j}),\phi\right\rangle$
		$\displaystyle\left\langle\prod_{j=1}^{k}\overline{H}(s-s_{j},x_{2}-y_{j})\prod_{j=k+1}^{k+l}H(s-s_{j},x_{2}-y_{j}),\overline{\phi}\right\rangle\mathrm{d}s_{1}\mathrm{d}y_{1}\cdots\mathrm{d}s_{k+l}\mathrm{d}y_{k+l}$
	$\displaystyle=$	$\displaystyle\,k!l!\int_{\mathbb{T}^{2}}\int_{\mathbb{T}^{2}}\left(\int_{-\infty}^{t\wedge s}\int_{\mathbb{T}^{2}}H(t-s_{j},x_{1}-y_{j})\overline{H}(s-s_{j},x_{2}-y_{j})\mathrm{d}y_{j}\mathrm{d}s_{j}\right)^{k}$
		$\displaystyle\left(\int_{-\infty}^{t\wedge s}\int_{\mathbb{T}^{2}}\overline{H}(t-s_{j},x_{1}-y_{j})H(s-s_{j},x_{2}-y_{j})y_{j}\mathrm{d}s_{j}\right)^{l}\phi(x_{1})\overline{\phi(x_{2})}\mathrm{d}x_{1}\mathrm{d}x_{2}$
	$\displaystyle=$	$\displaystyle\,k!l!\int_{\mathbb{T}^{2}}\int_{\mathbb{T}^{2}}\left(\sum_{j\in\mathbb{Z}^{2}}\frac{e^{-\rho_{j}|t-s|-\mathrm{i}\theta_{j}(t-s)}}{2\rho_{j}}e^{2\pi\mathrm{i}j\cdot(x_{1}-x_{2})}\right)^{k}$
		$\displaystyle\left(\sum_{j\in\mathbb{Z}^{2}}\frac{e^{-\rho_{j}|t-s|+\mathrm{i}\theta_{j}(t-s)}}{2\rho_{j}}e^{-2\pi\mathrm{i}j\cdot(x_{1}-x_{2})}\right)^{l}\phi(x_{1})\overline{\phi(x_{2})}\mathrm{d}x_{1}\mathrm{d}x_{2}$
		$\displaystyle=\,k!l!\sum_{j_{1}+\cdots+j_{k}-j_{k+1}-\cdots-j_{k+l}=\omega}\prod_{i=1}^{k+l}\frac{e^{-\rho_{j_{i}}|t-s|-\mathrm{i}a_{i}\theta_{j_{i}}(t-s)}}{2\rho_{j_{i}}},$

where $a_{i}=1$ for $1\leq i\leq k$ and $a_{i}=-1$ for $k+1\leq i\leq k+l$. Therefore, by [16, Lemma 10.14] or [35, Corollary C.3], we get that for every $\epsilon\in(0,1)$,

$$\mathrm{E}\left[\left|\hat{Z}^{:k,l:}_{-\infty,t}(\omega)\right|^{2}\right]=k!l!\sum_{j_{1}+\cdots+j_{k}-j_{k+1}-\cdots-j_{k+l}=\omega}\prod_{i=1}^{k+l}\frac{1}{2\rho_{j_{i}}}\leq C\frac{1}{(1+|\omega|^{2})^{1-\epsilon}}.$$

Since $\epsilon\in(0,1)$ is arbitrary, we finish the proof of (22).