Abstract
In this paper, we study the stochastic wave equations in the three spatial dimensions driven by a Gaussian noise which is white in time and correlated in space. Our main concern is the sample path Hölder continuity of the solution both in time variable and in space variables. The conditions are given either in terms of the mean Hölder continuity of the covariance function or in terms of its spectral measure. Some examples of the covariance functions are proved to satisfy our conditions, which include the case of the work Dalang and Sanz-Solé (Hölder–Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three. Memoirs of the American Mathematical Society Number 931 2009). In particular, we obtain the Hölder continuity results for the solution of the stochastic wave equations driven by (space inhomogeneous) fractional Brownian noises. For this particular noise, the optimality of the obtained Hölder exponents is also discussed.
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1 Introduction
We shall study the following stochastic wave equation in spatial dimension \(d=3\):
where \(t\in (0,T]\) for some fixed \(T>0\), \(x \in \mathbb {R}^3\) and \(\Delta =\frac{\partial ^2}{\partial x_1^2} +\frac{\partial ^2}{\partial x_2^2} +\frac{\partial ^2}{\partial x_3^2} \) denotes the Laplacian on \(\mathbb {R}^3\). The coefficients \(\sigma \) and \(b\) satisfy some regularity conditions which will be specified later. The Gaussian noise process \(\dot{W}\) is assumed to be white in time and with a homogeneous correlation in space. This can be informally written as
for a non-negative, non-negative definite and locally integrable function \(f\), where \(\delta \) is the Dirac delta function. We will explain in Sect. 2 how this expression can be made formal.
It is known (see, for instance, [4, Theorem 4.3]) that if \(\sigma \) and \(b\) are Lipschitz functions with linear growth and \(f\) satisfies \(\int _{|x| \le 1} f(x)/|x| dx<\infty \), then there is a unique mild solution to Eq. (1.1). Our purpose is to establish the sample path Hölder continuity both in time variable and in space variables of the solution to this equation. When \(f\) is given by a Riesz kernel \(|x|^{-\beta }\), \(\beta \in (0,2)\), the Hölder continuity of the solution has been obtained by Dalang and Sanz-Solé in their monograph [5]. Their approach is based on the fractional Sobolev imbedding theorem and the Fourier transformation technique.
In this paper, we shall consider more general Gaussian noises, and we introduce a new approach that avoids the Fourier transform. The main idea is to impose conditions on the covariance \(f\) itself. To be more precise, let \(D_w f=f(\cdot +w)\) be the shift operator. We shall show that if \(\Vert D_w f- f\Vert _{L^1(\rho )}\le C |w|^{\gamma }\) and \(\Vert D_w f +D_{-w}f -2f\Vert _{L^1 (\rho )}\le C |w|^{\gamma ^{\prime }}\) for some \(\gamma \in (0,1]\) and \(\gamma ^{\prime } \in (0,2]\), where \(\rho \) is the measure on \({\mathbb {R}}^3\) defined to be \(\rho (dz)=\mathbf{1}_{\{|z|\le 2T\} }\frac{1}{|z|}dz\), then the solution to (1.1) is locally Hölder continuous of order \(\kappa < \min (\gamma , \frac{\gamma '}{2})\) in the space variable (assuming zero initial conditions) (see Theorem 3.1).
The Hölder continuity in the time variable is more involved. Following the methodology used by Dalang and Sanz-Solé in [5], we transform the time increments into space increments, and we impose suitable assumptions on the modulus of continuity of a shift operator which are formulated integrals over \([0,T]\times (S^2)^2\), equipped with the measure \(ds \sigma (d\xi )\sigma (d\eta ) \), where \(\sigma \) is the uniform measure on the unit sphere \(S^2\) (see Theorem 4.1).
We also obtain a theorem on the Hölder continuity in the space variable using the Fourier transform technique. More precisely, we establish the Hölder continuity of order \(\kappa <\gamma \), provided the spectral measure \(\mu \) satisfies the integrability condition \(\int _{\mathbb {R}^3} \frac{ \mu (d\zeta )}{ 1+|\zeta |^{2-2\gamma }} <\infty \) and the Fourier transform of \(|\zeta |^{2\gamma } \mu (d\zeta )\) is non-negative. The non-negativity condition on this measure leads to a simple proof of the Hölder continuity in the space variable which avoids the control of the norms of the increments \(D_w f- f\) and \(D_w f +D_{-w}f -2f\) (or their respective Fourier transforms). As an application, this method provides a direct proof of the Hölder continuity in the space variable, in the case of the Riesz kernel. However, at this moment we are not able to use this approach to handle the Hölder continuity in the time variable.
To illustrate the scope of our results we provide some examples of covariance functions \(f\) which satisfy our conditions. We consider first the Riesz and Bessel kernels. Then we focus our attention to fractional noises with covariance function of the form
where \(H_1, H_2, H_3\in (1/2, 1)\) and \(\bar{\kappa }:=\sum _{i=1}^3 H_i-2\). We show (see Theorem 6.1) that, under suitable assumptions on the initial conditions, if \(\kappa _i\in (0, \min (H_i-1/2, \bar{\kappa }))\) and \(\kappa _0=\min (\kappa _1\,, \kappa _2\,, \kappa _3)\), then for any bounded rectangle \(I\subset \mathbb {R}^3\), there is a finite random variable \(K\), depending on the \(\kappa _i\)’s, such that for all \(s,t \in [0,T]\) and for all \(x,y\in I\)
To see if the Hölder exponents \(\kappa _i\)’s are optimal or not, we investigate a simple linear stochastic wave equation with additive noise. That means, we consider the Eq. (1.1) with \(v_0=\bar{v}_0=0\), \(b=0\) and \(\sigma =1\). In this situation, we prove (see Theorem 6.2 and a Kolomogorov lemma) that for any bounded rectangle \(I\subset \mathbb {R}^3\) and for any \(\kappa \in (0, \bar{\kappa })\), there is a random variable \(K_{\kappa , I}\) such that for alll \(t, s \in [0,T]\) and for all \(x,y\in I\)
On the other hand, we obtain in Theorem 6.2 a lower bound on the variance of the increments of the process \(u\) which shows that the exponent \(\bar{\kappa }\) is optimal. Notice that in the nonlinear case (see Theorem 6.1), we need the extra conditions \(\kappa _i<H_i-1/2\) for \(i=1, 2, 3\). Also, this extra condition is not necessary if \(H_i+H_j \le 3/2\) for any \(i \not =j\) (for instance, if \( H_1=H_2=H_3=H\le 3/4\)), and in this case \(\kappa _i\) coincides with the optimal constant \(\bar{\kappa }\). It would be interesting to know if the additional conditions \(\kappa _i<H_i-1/2\) are due to the nonlinearity or due to the limitation of our technique.
This paper is organized as follows. Section 2 contains some preliminary material about the noise process in Eq. (1.1). We state our basic assumptions on the covariance function \(f\) and prove a general Burkholder inequality. We also give the definition of the mild solution and state the existence and uniqueness theorem of the solution to Eq. (1.1). Section 3 contains two main results on the Hölder continuity in the space variables. One is based on the structure of the covariance function \(f\) itself and the other one uses the Fourier transform of \(f\). In Sect. 4 we prove a criterion for the Hölder continuity in the time variable. Section 5 presents some examples of covariance functions \(f\) which satisfy the conditions given in our main theorems. In the first example, \(f\) is the convolution of a Schwartz function with a Riesz kernel. In the second example, \(f\) is the Riesz kernel, which is the case studied in [5]. In the third example, \(f\) is the Bessel kernel. Section 6 deals with the case when the noise process is the formal derivative of a fractional Brownian field. The optimality of the Hölder exponents is discussed in this section. Section 6 contains some lemmas which are used in the paper.
2 Preliminaries
Consider a non-negative and non-negative definite function \(f\) which is a tempered distribution on \({\mathbb {R}}^3\) (so \(f\) is locally integrable). We know that in this case \(f\) is the Fourier transform of a non-negative tempered measure \(\mu \) on \({\mathbb {R}}^3\) (called the spectral measure of \(f\)). That is, for all \(\varphi \) belonging to the space \(\mathcal {S}({\mathbb {R}}^3)\) of rapidly decreasing \(C^{\infty }\) functions
and there is an integer \(m\ge 1\) such that
where we have denoted by \(\mathcal {F}\varphi \) the Fourier transform of \(\varphi \in \mathcal {S}({\mathbb {R}}^3)\), given by
Let \(G(t)\) be the fundamental solution of the 3-dimensional wave equation \(\displaystyle \frac{\partial ^2u}{\partial t^2}=\Delta u\). That is
for any \(t>0\), where \(\sigma _t\) denotes the uniform surface measure (with total mass \(4\pi t^2\)) on the sphere of radius \(t>0\). Sometimes it is more convenient for us to use the Fourier transform of \(G\) given by
Our basic assumption on \(f\) is
It turns out (see Lemma 6.4 and Eq. (6.21) below) that this is equivalent to
Notice that since we are in \({\mathbb {R}}^3\), condition (2.5) is satisfied if there is a \(\kappa <2\) such that in a neighborhood of \( 0\), \(f(x)\le C |x|^{-\kappa }\).
The following identities will play an important role,
for \(0<t\le s\). We refer to Lemmas 6.4 and 6.5 for proofs of these two identities.
Fix a time interval \([0,T]\). Let \(C_0^{\infty }([0,T] \times {\mathbb {R}}^{3})\) be the space of infinitely differentiable functions with compact support on \([0,T] \times {\mathbb {R}}^3\). Consider a zero mean Gaussian family of random variables \(W=\{W(\varphi ), \varphi \in C_0^{\infty }([0,T] \times {\mathbb {R}}^{3})\}\), defined in a complete probability space \((\Omega , \mathcal {F}, P)\), with covariance
Walsh’s classical theory of stochastic integration developed in [8] cannot be applied directly to the mild formulation of Eq. (1.1) since \(G\) is not a function, but a measure. We shall use the stochastic integral defined in Sect. 2.3 of [4]. We briefly summarize the construction and properties of this integral.
Let \(U\) be the completion of \(C_0^{\infty }(\mathbb {R}^3)\) endowed with the inner product
\(\varphi ,\psi \in C_0^{\infty }({\mathbb {R}}^3)\). Set \(U_T=L^2([0,T];U)\).
The Gaussian family \(W\) can be extended to the space \(U_T\). We will also denote by \(W(g)\) the Gaussian random variable associated with an element \(g \in U_T\). Set \(W_t(h)=W(\mathbf{1}_{[0,t]}h)\) for any \(t\in [0,T]\) and \(h \in U\). Then \(W=\{ W_t, t\in [0,T]\}\) is a cylindrical Wiener process in the Hilbert space \(U\). That is, for any \(h \in U\), \(\{W_t(h), t\in [0,T]\}\) is a Brownian motion with variance \(t\Vert h\Vert ^2_U\), and
Let \(\mathcal {F}_t\) be the \(\sigma \)-field generated by the random variables \(\{W_s(h), h\in U, 0\le s \le t\}\) and the \(P\)-null sets. We define the predictable \(\sigma \)-field as the \(\sigma \)-field in \(\Omega \times [0,T]\) generated by the sets \(\{A \times (s,t], 0\le s< t\le T, A\in \mathcal {F}_s\}\). Then we can define the stochastic integral of a \(U\)-valued square-integrable predictable process \(g \in L^2(\Omega \times [0,T];U)\) with respect to the cylindrical Wiener process \(W\), denoted by
and we have the isometry property
The following lemma provides a sufficient condition for a measure of the form \(\varphi (x) G(t,dx)\) to be in the space \(U\).
Lemma 2.1
Consider a Borel measurable function \(\varphi :{\mathbb {R}}^3 \rightarrow {\mathbb {R}}\), such that for some \(t>0\),
Then, \(\varphi G(t)\) belongs to \(U\) and
Furthermore, when \(\varphi \) is bounded,
Proof
Suppose first that \(\varphi \) is bounded. Then by Lemma 6.5, the equality (2.13) holds and \(\int _{{\mathbb {R}}^3}|\mathcal {F}(\varphi G(t))(\xi )|^2\mu (d\xi ) < \infty \). Let \(\psi \) be a nonnegative \(C^{\infty }\) function on \({\mathbb {R}}^3\) supported in the unit ball such that \(\int _{\mathbb {R}^3} \psi (x)dx=1\). Define \(\psi _n(x)=n^3\psi (nx)\), so
is in \(C_0^{\infty }({\mathbb {R}}^3)\), and we have
as \(n \rightarrow \infty \), by the dominated convergence theorem. This implies that \(\varphi G(t)\) is in \(U\), and (2.12) holds.
In the general case, we consider the sequence of functions \(\varphi _k(x)= \varphi (x) \mathbf {1} _{\{ |\varphi | \le k\}}\). Then \(\varphi _k(x)G(t,dx)\) belongs to \(U\), and
which clearly goes to \(0\) as \(k\) goes to infinity, by the dominated convergence theorem. \(\square \)
For any \(x\in \mathbb {R}^3\) we denote by \(G(t,x-dy)\) the shifted measure \(A\mapsto G(t,x-A)\). Clearly Lemma 2.1 holds if we replace the kernel \(G(t,dy)\) by the shifted kernel \(G(t,x-dy)\). Applying Lemma 2.1, we immediately get the following Burkholder inequality.
Lemma 2.2
Let \(Z=\{Z(t,x), (t,x)\in [0,T]\times {\mathbb {R}}^3\}\) be a predictable process such that for some \(p \ge 2\) and \(x\in {\mathbb {R}}^3\),
Then the measure-valued predictable process \(Z(s,y)G(s,x-dy)\) belongs \(L^2(\Omega \times [0,T];U)\) and there exists a positive constant \(C_p\), depending only on \(p\), such that
If we have
then an application of Hölder inequality yields
By Lemma 6.4, the above inequality can also be written as
Using the above notion of stochastic integral one can introduce the following definition:
Definition 2.3
A real-valued predictable stochastic process \(u=\{u(t, x),0 \le t \le T\,, x\in {\mathbb {R}}^3\}\) is a mild random-field solution of (1.1) if for all \(t\in (0, T]\), \(x\in {\mathbb {R}}^3\),
Consider the following condition.
(H) The coefficients \(\sigma \) and \(b\) satisfy
and
for any \(x,y \in {\mathbb {R}}^3\), \(s,t \in [0,T]\) and \(u,v \in {\mathbb {R}}\).
Then one can prove the existence and uniqueness of the solution to (1.1) in exactly the same way as in [4, Theorem 4.3].
Theorem 2.4
Suppose the condition (2.5) holds, and \(\sigma \), \(b\) satisfy the condition (H). Let \(v_0 \in C^1({\mathbb {R}}^3)\) such that \(v_0\) and \(\nabla v_0\) are bounded and \(\bar{v}_0\) is bounded and continuous. Then there exists a unique mild random-field solution \(u\) to (1.1) such that for all \(p\ge 1\),
Along the paper, \(C\) will denote a generic constant which may change from line to line.
3 Hölder continuity in the space variable
In this section we will prove Theorems 3.1 and 3.2 which are the main results on the Hölder continuity of the solution of Eq. (1.1) in the space variable.
Theorem 3.1
Let \(u\) be the solution to Eq. (1.1). Assume the following conditions.
-
(a)
The coefficients \(\sigma \) and \(b\) satisfy condition (H).
-
(b)
\(v_0 \in C^2({\mathbb {R}}^3)\), \(v_0\), \(\nabla v_0\) and \(\bar{v}_0\) are bounded and \(\Delta v_0\) and \(\bar{v}_0\) are Hölder continuous of orders \(\gamma _{1}\) and \(\gamma _{2}\) respectively, \(\gamma _{1}, \gamma _{2} \in (0,1]\).
-
(c)
The function \(f\) satisfies condition (2.5) and for some \(\gamma \in (0,1]\) and \(\gamma ^{\prime } \in (0,2]\) we have for all \(w\in {\mathbb {R}}^3\) such that \(|w|\le 1\)
$$\begin{aligned} \int \limits _{|z| \le 2T} \frac{|f(z+w)-f(z)|}{|z|} dz \le C |w|^{\gamma } \end{aligned}$$(3.1)and
$$\begin{aligned} \int \limits _{|z| \le 2T} \frac{|f(z+w)+f(z-w)-2f(z)|}{|z|}dz \le C |w|^{\gamma ^{\prime }}. \end{aligned}$$(3.2)
Set \(\kappa _1= \min (\gamma _1,\gamma _2,\gamma ,\frac{\gamma ^{\prime }}{2})\). Then for any \(q\ge 2\), there exists a constant \(C\) such that
for any \(x,y \in \mathbb {R}^3\).
Proof
It suffices to assume that \(|x-y|\le 1\). Set \(x-y=w\). Fix \(q \ge 2\). Then we have
For \(I_4\), since \(\bar{v}_0\) is Hölder continuous with exponent \(\gamma _{2}\) we get
For \(I_3\), we use the identity (see, for instance, [7])
Then, since \(\Delta v_0\) is Hölder continuous with exponent \(\gamma _{1}\), we get
For \(I_2\), we use the Lipschitz condition on \(b\) and Hölder’s inequality to get
For \(I_1\), we apply the Burkholder’s inequality of Lemma 2.2 to get
The main idea to estimate the above quantity is to transfer the increments of \(G\) to increments of \(f\) and \(\sigma \). We introduce the following notation
Define
and
Then by direct calculation, we can verify that \(Q=\sum _{i=1}^4 Q_{i}\). To estimate \({\mathbb {E}}|Q|^{\frac{q}{2}}\), we need to estimate \({\mathbb {E}}|Q_{i} |^{\frac{q}{2}} \) for \(i=1, \dots , 4\). For \({\mathbb {E}}|Q_{1}|^{\frac{q}{2}}\), by the assumptions on \(\sigma \), using Hölder’s inequality and identities (2.7) we have
By the condition (2.5), we get
For \({\mathbb {E}}|Q_{2}|^{\frac{q}{2}}\), we write \(f(\eta -\xi +w)-f(\eta -\xi )=\Phi _1(\eta -\xi ,w)\) and using the inequality \(ab\le \frac{a^2+b^2}{2}\) we obtain
Applying condition (3.1), identities (2.7) and Hölder’s inequality yields
For the second term we obtain
So we conclude that
The term \({\mathbb {E}}|Q_3|^{\frac{q}{2}}\) can be treated in the same way and we have
For \({\mathbb {E}}|Q_4|^{\frac{q}{2}}\), we set \(\Phi _2(\eta -\xi ,w)=f(\eta -\xi +w) +f(\eta -\xi -w) -2f(\eta -\xi )\), and using the assumption on \({\sigma }\), condition (3.2), Hölder’s inequality and the moments estimate (2.15), we have
Combining the above expression with (3.14), (3.15) and (3.16), we can write
The estimates for \(I_i\), \(i=1,2,3,4\), lead to
An application of Gronwall’s lemma yields
for any \(x\) and \(y\) in \({\mathbb {R}}^3\) such that \(|x-y|\le 1\), which completes the proof of the theorem. Notice that, as it can be checked throughout the proof, the generic constant \(C\) does not depend on \(t \in [0,T]\).
Next we give a theorem which establishes the Hölder continuity in the space variable using the Fourier transform.
Theorem 3.2
Let \(u\) be the solution to Eq. (1.1). Assume conditions (a) and (b) in Theorem 3.1. Suppose the following condition:
- (\(c^{\prime }\)):
-
For some \(\gamma \in (0,1]\), the Fourier transform of the tempered measure \(|\zeta |^{2\gamma } \mu (d\zeta )\) is a nonnegative locally integrable function and
$$\begin{aligned} \int \limits _{{\mathbb {R}}^3} \frac{\mu (d\zeta )}{1+|\zeta |^{2-2\gamma }} <\infty . \end{aligned}$$(3.19)
Set \(\kappa _1^{\prime }= \min (\gamma _1,\gamma _2,\gamma )\). Then for any \(q\ge 2\), there exists a constant \(C\) such that
for any \(x,y \in \mathbb {R}^3\).
Proof
It suffices to assume that \(|x-y|\le 1\). Set \(x-y=w\). Fix \(q \ge 2\), as in the proof of Theorem 3.1, we still express \({\mathbb {E}}|u(t,x)-u(t,y)|^q\) as \(C(I_1+I_2+I_3+I_4)\), and the estimates for \(I_2\), \(I_3\), \(I_4\) are the same as in the proof of Theorem 3.1. For \(I_1\), use the notation (3.7)–(3.13) and we need to estimate \({\mathbb {E}}|Q_i|^{\frac{q}{2}}\) for \(i=1, \dots , 4\).
The estimate for \({\mathbb {E}}|Q_1|^{\frac{q}{2}}\) is the same as in the proof of Theorem 3.1.
For \(Q_2\) we would like to apply Eq. (6.17) to \(\varphi =\Sigma _{x,y}(s,\eta )\) and \(\psi =\Sigma _{x}(s,\xi )\). Because these functions are not necessarily bounded we we introduce the truncations
for any \(k>0\). Clearly, as \(k\) tends to infinity, \(\Sigma _{x}^k(s,\xi )\) and \(\Sigma _{x,y}^k(s,\eta )\) converge pointwise to \(\Sigma _{x}(s,\xi )\) and \(\Sigma _{x,y}(s,\eta )\), respectively. Set
Then Eq. (6.17) yields
Using the estimate \(| e^{-iw\cdot \zeta }-1|\le C |w|^{\gamma }|\zeta |^{\gamma }\) for every \(0 < \gamma \le 1\), Cauchy-Schwartz’s inequality and the inequality \(\sqrt{ab} \le \frac{1}{2} (a +b)\) for any \(a,b>0\), we can write
where \(g\) is the Fourier transform of the measure \(|\cdot |^{2\gamma } \mu \), which by our hypothesis is a nonnegative locally integrable function. In the above formula, for any measure \(\nu \), \(\widetilde{\nu }\) denotes the measure \(\widetilde{\nu }(A)=\nu (-A)\). Treating \(g(\eta ) G(t-s)*G(t-s)(\eta ) d\eta \) as a new measure, and using Minkowski’s inequality, we get
where we have used the moments estimate (2.15), Eq. (2.4), the fact that \(\left( \frac{\sin (s|\xi |)}{|\xi |}\right) ^2\le \frac{C}{1+|\xi |^2}\), when \(s\in [0,T]\) and the inequality \(|\Sigma ^k_{x}(s,\xi )|\le |\Sigma _{x}(s,\xi )|\). Therefore,
Applying the dominated convergence theorem we can show that in the above inequality, as \(k\) goes to infinity, the left-hand side converges to \({\mathbb {E}}\left| Q_2\right| ^{\frac{q}{2}}\) and the expectation on the right-hand side converges to
From the expression for \(\Sigma _{x,y}(s,\xi )\) and using Minkowski’s inequality, we have
The same estimate holds for \({\mathbb {E}}|Q_3|^{\frac{q}{2}}\).
Consider now the term \(Q_4\). We use the truncation argument as in the estimation for \({\mathbb {E}}|Q_2|^{\frac{q}{2}}\) and we set
Then, Eq. (6.17) implies
Then we can use the same argument as before, to conclude that
Combining the moment estimates for \(E\left| Q_{i}\right| ^{\frac{q}{2}}\), \(i=1,2,3,4\), since \(|w|\le 1\) and \(0<\gamma \le 1\), we have
Finally, the estimates for \(I_i\), \(i=1, 2, 3,4\), allow us to write
An application of Gronwall’s lemma yields
for any \(x\) and \(y\) in \({\mathbb {R}}^3\) such that \(|x-y|\le 1\), which completes the proof of the theorem. Notice that, as it can be checked throughout the proof, the generic constant \(C\) does not depend on \(t \in [0,T]\). \(\square \)
Under the assumptions of Theorem 3.1 or Theorem 3.2, applying Kolmogorov’s continuity criterion, for any fixed \(t\in [0,T]\), we deduce the existence of a locally Hölder continuous version for the process \(\{u(t,x), x\in {\mathbb {R}}^3\}\) with exponent \(\kappa >0\) where \(\kappa <\kappa _1\). Namely, for any \(t\in [0,T]\) and any compact rectangle \(I\subset \mathbb {R}^3\), there exists a random variable \(K_{\kappa ,t,I}\) such that
for and \(x,y \in I \).
4 Hölder continuity in space and time variables
In this section we obtain a result on the Hölder continuity of the solution of Eq. (1.1) in both the space and time variables. Let \(S^2\) denote the unit sphere in \({\mathbb {R}}^3\) and \(\sigma (d\xi )\) the uniform measure on it. We have the following result.
Theorem 4.1
Let \(u \) be the solution to Eq. (1.1). Assume conditions (a) and (b) in Theorem 3.1. Suppose the following conditions hold.
-
(1)
For some \(0 < \nu \le 1\), \(\int _{|z|\le h}\frac{f(z)}{|z|}dz\le C h^{\nu }\) for any \(0 <h\le 2T\).
-
(2)
For some \(0< \kappa _1 \le 1\) and for any \(q\ge 2\) and \(t \in (0,T]\), we have
$$\begin{aligned} {\mathbb {E}}| u(t,x) -u(t,y) | ^q \le C |x-y|^{q\kappa _1}. \end{aligned}$$ -
(3)
Let \(\xi \) and \(\eta \) be unit vectors in \({\mathbb {R}}^3\) and \(0<h\le 1\). We have
$$\begin{aligned} \int \limits _0^T \int \limits _{S^2}\int \limits _{S^2} \Big |f\left( s(\xi +\eta ) +h(\xi +\eta ) \right) -f\left( s(\xi +\eta ) +h\eta \right) \Big | s \sigma (d\xi ) \sigma (d\eta ) ds \le C h^{\rho _1},\nonumber \\ \end{aligned}$$(4.1)for some \(\rho _1 \in (0,1]\), and
$$\begin{aligned}&\int \limits _0^T \int \limits _{S^2}\int \limits _{S^2} \nonumber \Big |f\left( s(\xi +\eta ) +h(\xi +\eta ) \right) -f\left( s(\xi +\eta ) +h\xi \right) \\&\qquad -\, f\left( s(\xi +\eta ) +h\eta \right) +f\left( s(\xi +\eta )\right) \Big | s^2 \sigma (d\xi ) \sigma (d\eta ) ds \le C h^{\rho _2}, \quad \end{aligned}$$(4.2)for some \(\rho _2\in (0,2]\).
Set \(\kappa _2= \min (\gamma _1,\gamma _2,\kappa _1,\frac{\nu +1}{2},\frac{\rho _1+\kappa _1}{2},\frac{\rho _2}{2})\). Then for any \(q\ge 2\), there exists a constant \(C\) such that
for any \(t, \bar{t} \in [0,T]\).
Proof
Fix \(x \in {\mathbb {R}}^3\) and \(q \in [2,\infty )\). For all \(0\le t\le \bar{t}\le T\) we can write, by Definition 2.3,
where
Let \(\gamma ^{\prime }=\min (\gamma _1,\gamma _2)\). By our assumptions on \(\Delta v_0\) and \(\bar{v}_0\) and by Lemma 4.9 in [5], we have
Notice that Lemma 4.9 in [5] assumes that \(x\) belongs to a bounded set in \({\mathbb {R}}^3\), but from the proof it is easy to see that the constant \(C\) does not depend on \(x\).
The term \(T_3 \) is bounded by
where
Hölder’s inequality, the linear growth of \(b\) and the moments estimate (2.15) imply
For \(T_{3,2} \), we split the integral into a difference of two integrals and then we apply the change of variables \(\frac{y}{t-s}\rightarrow y\) and \(\frac{y}{\bar{t}-s}\rightarrow y\), respectively. In this way, taking into account that \(G(t,dy)= t^{-2} G(1, t^{-1} dy)\), we get
Hence, \(T_{3,2} \le C\left( T_{3,2,1} +T_{3,2,2} \right) \), where
and
By the moments estimate (2.15) and the linear growth of \(b\), it follows that
Moreover, by the Lipschitz property of \(b\) and Hölder continuity assumption on the space variable (condition (2) in the theorem), we get
Combining the estimates for \(T_{3,1} \), \(T_{3,2,1} \) and \(T_{3,2,2} \) we conclude that
Next we estimate the term \(T_4 \) which involves a stochastic integral. Consider the decomposition
where
and
By the linear growth of \(\sigma \) and Burkholder’s inequality (Lemma 2.2), we obtain
Using Hölder’s inequality, the moments estimate (2.15) and condition (1), we can write
For \(T_{4, 2}\), for notational convenience we denote \(\bar{t}-t\) by \(h\). Applying Burkholder’s inequality (see Lemma 2.2) yields
where \(\Theta _{t,x}(s,y)=\sigma (t-s,x-y,u(t-s,x-y)) \). By making a change of variable, we can transform the integral in the space variable into an integral on the unit sphere \(S^2\). In fact, denote \(\xi =\frac{y}{|y|}\) and \(\eta =\frac{z}{|z|}\) and we recall that \(\sigma (d\xi )\) and \(\sigma (d\eta )\) denote the uniform measure on \(S^2\), so
After some rearrangements similar to those made for \(Q\) in the proof of Theorem 3.2 (see also [5] for a similar strategy), we can write
where
We estimate each \({\mathbb {E}}|R_i|^{\frac{q}{2}}\) separately.
For \({\mathbb {E}}|R_1|^{\frac{q}{2}}\), using Hölder’s inequality, the Lipschitz condition on \(\sigma \), the assumption on the Hölder continuity on the space variable of \(u\) (condition (2)), Lemma 6.4 and condition (2.5), we have
In order to estimate \({\mathbb {E}}|R_2|^{\frac{q}{2}}\), we make the decomposition
For \(R_2^1\), using the Hölder inequality, the Lipschitz and linear growth conditions on \(\sigma \), the moments estimate (2.15), the assumption on the Hölder continuity in the space variable of \(u\) (condition 2) and condition (4.1) with the change of variable \(\eta \rightarrow -\eta \), we have
For \(R_2^2\), using Hölder’s inequality, the Lipschitz condition and linear growth conditions on \(\sigma \), the moments estimate (2.15), the assumption on the Hölder continuity in the space variable (condition (2)) and condition (1), we have
Combining the estimates for \(R_2^1\) and \(R_2^2\), we have
Similarly,
For \(R_4\), using the linear growth of \(\sigma \), the moments estimate (2.15) and the change of variable \(\eta \rightarrow -\eta \), we have
For \(R_4^1\), condition (4.2) yields
For \(R_4^2\) and \(R_4^3\), applying condition (4.1) we obtain
For \(R_4^4\), condition (1) allows us to write
When \(\nu < 1\), \(R_4^4 \le C h^{\frac{q(\nu +1)}{2}}\), when \(\nu =1\), \(R_4^4\le C h^q (\log (t+h)-\log h)^{\frac{q}{2}}\le C h^q (\log (T+h)-\log h)^{\frac{q}{2}}\le C h^{q(1-\varepsilon )}\) for any \(\varepsilon > 0\).
Combining the estimates for \(R_4^1\), \(R_4^2\), \(R_4^3\), \(R_4^4\), we have
for any \(\varepsilon > 0\). By (4.5), (4.6), (4.7), (4.8) and (4.9), we conclude that
where \(0<\rho <\min (\frac{\nu +1}{2},\frac{\rho _1+\kappa }{2},\frac{\rho _2}{2},\kappa )\). From the proof it is easy to see that the constant \(C\) in the above expression does not depend on \(x\). Then we combine the estimates of (4.3), (4.4) and (4.10) to obtain
where \(\kappa ^{\prime } \in \left( 0, \min (\gamma _1, \gamma _2, \frac{\nu +1}{2}, \frac{\rho _1+\kappa }{2}, \frac{\rho _2}{2}, \kappa )\right) \). \(\square \)
An application of Kolmogorov’s continuity criteria leads to the following Hölder continuity result in the space an time variables.
Corollary 4.2
Let \(u\) be the solution to Eq. (1.1). Assume conditions (a) and (b) in Theorem 3.1. Suppose that condition (c) of Theorem 3.1 or condition (a) of Theorem 3.2 hold. Set \(\kappa _1= \min (\gamma _1,\gamma _2,\gamma ,\frac{\gamma ^{\prime }}{2})\) in the first case and \(\kappa _1= \min (\gamma _1,\gamma _2,\gamma )\) in the second case. Suppose also that conditions (1), (2) and (3) of Theorem 3.2 hold. Set \(\kappa _2= \min (\gamma _1,\gamma _2,\kappa _1,\frac{\nu +1}{2},\frac{\rho _1+\kappa _1}{2},\frac{\rho _2}{2})\). Then, for any \(\kappa < \kappa _1\) and \(\kappa '<\kappa _2\) there exists a version of the process \(u\) which is locally Hölder continuous of order \(\kappa \) in the space variable and of order \(\kappa '\) in the time variable. That is, for any bounded rectangle \(I\subset {\mathbb {R}}^3\) we can find a random variable \(K_{\kappa , \kappa ',I}\) such that
for all \(s,t\in [0,T]\) and \(x,y\in I\).
5 Examples
In this section, we give some examples of covariance functions \(f\) satisfying the conditions in the previous theorems.
5.1 Example 1
Proposition 5.1
Let \(f\) be a non-negative and non-negative definite \(C^2\) function. Then condition (c) of Theorem 3.1 holds with \(\gamma = 1\) and \(\gamma ^{\prime }=2\).
Proof
Using some basic estimate from calculus, we have
and
The claim follows. \(\square \)
Remark 5.2
Consider the example \(f(x)=(\rho *\frac{1}{|\cdot |^{\beta }})(x)\), where \(\rho (x)\) is a nonnegative Schwartz function defined in \({\mathbb {R}}^3\) such that \((\mathcal {F}^{-1}\rho )(\xi )\ge 0\) (for example, \(\rho (x)=e^{-|x|^2}\)) and \(0 < \beta <3\). Then it is easy to see that condition (\(c^{\prime }\)) of Theorem 3.2 holds for \(0 < \gamma < \min (\frac{3-\beta }{2}, 1)\). The restriction \(\gamma < \frac{3-\beta }{2}\) comes from the fact that under the condition \(0 < \beta < 3\), the Fourier transform of \(\frac{1}{|x|^{\beta }}\) is \(\frac{C_{\beta }}{|\xi |^{3-\beta }} \) for some constant \(C_{\beta }\) which only depends on \(\beta \). We omit the details of the proof. Notice in this example, Theorem 3.2 gives a weaker result than what we would obtain using Theorem 3.1 as we have done in Proposition 5.1.
5.2 The Riesz kernel
Before giving next example, we recall some results from Dalang and Sanz-Solé [5].
Let \(\xi \), \(\eta \) be two unit vectors in \({\mathbb {R}}^3\) and let \(u\) be any point in \({\mathbb {R}}^3\). Suppose \(a\), \(b\) are positive numbers with \(a+b \in (0,3)\). Then we have for any \(h \in {\mathbb {R}}\)
and
Proposition 5.3
Let \(f(x)=|x|^{-\beta }\), \(0<\beta <2\). Then \(f\) satisfies condition (\(c^{\prime }\)) in Theorem 3.2 for any \(\gamma \in (0,\frac{2-\beta }{2})\) and \(f\) also satisfies conditions (1), (4.1) and (4.2) in Theorem 4.1 for \(\nu = 2-\beta \), any \(0 < \rho _1 < \min (2-\beta , 1)\) and \(0 < \rho _2 < 2-\beta \).
Proof
Let us first check condition (\(c^{\prime }\)) in Theorem 3.2. Since \(f(x)=|x|^{-\beta }\), we have \(\mu (d\xi )=C|\xi |^{-3+\beta }d\xi \). Then it is easy to see that
since \(0<\gamma < \frac{2-\beta }{2}\), and we have
for some positive constant \(C\), so the above expression is nonnegative. So, condition (\(c^{\prime }\)) in Theorem 3.2 holds.
To verify condition (1) in Theorem 4.1, we notice
So condition (1) in Theorem 4.1 is satisfied with \(\nu = 2-\beta \).
We turn to condition (4.1). We apply (5.1) with \(b=\rho _1<\min ((2-\beta ),1)\), \(d=3\), \(a=3-\rho _1-\beta \), \(u=s(\xi +\eta )+h\eta \) to get
For \(I_1\), making the change of variable \(w+h \rightarrow w\), using the Fourier transform (see Lemma 6.5) and noting that \(I_1\) is real positive, we can write:
Then using the change of variable \((s+h)\xi =\eta \) and the bound \(|e^{i \xi \cdot hw}|\le 1\), by direct calculation we see that \(I_1<\infty \).
For \(I_2\), we do the same calculation, but we do not need the change of variable for \(w\). Let \(2 \varepsilon < 2-\beta -\rho _1\), then
which is finite by direct calculation.
For \(I_3\) we can write
where the inequality holds because \(|w|> 3\), \(0\le \lambda \le 1\) and \(|\xi |=1\). We can show that \(I_3<\infty \) similarly to the proof for \(I_2\) using the fact that \(\int _{|w|> 3}|w|^{\rho _1-4}dw < \infty \) since \(\rho _1<1\).
It is easy to see that \(I_1\), \(I_2\) and \(I_3\) are finite uniformly for \(0<h \le 1\). Therefore, condition (4.1) is satisfied with \(0 < \rho _1 < \min (2-\beta ,1)\).
For condition (4.2), applying (5.2), with \(d=3\), \(b=\rho _2 < 2-\beta \), \(a=3-\rho _2-\beta \), \(u=s(\xi +\eta )\), yields
For \(L_i,\, i=1,2,3,4\), we can proceed exactly in the same way as for the integrals \(I_1,\, I_2\) above. For \(L_5\), we can express
and since \(|w|>3,\, |\eta |=1\), it is easy to see that
So \(\int _{|w|>3}|w|^{\rho _2-5}dw\) is finite, and \(L_5\) is finite, by the same argument as for \(I_3\).
So condition (4.2) is satisfied with \(0 < \rho _2 < 2-\beta \). This completes the proof. \(\square \)
Notice that, with the notation of Corollary 4.2, for the Riesz kernel we can take \(\kappa _1=\kappa _2 <\frac{2-\beta }{2}\), and we deduce the local Hölder continuity of the solution \(u\) in space and time variables of order \(\kappa < \min (\gamma _1, \gamma _2, \frac{2-\beta }{2} )\). In this way we recover the result by Dalang and Sanz-Solé [5].
5.3 The Bessel kernel
In this subsection we consider the Bessel kernel defined by
for some \(\alpha >1\). The colored noise with this covariance has received some attention in the literature (see for example, [1, 2]).
Proposition 5.4
Let \(f\) be given by (5.3). Then \(f\) satisfies (2.5), (3.1), (3.2), (4.1), (4.2) and condition (1) in Theorem 4.1 for any \(0 < \gamma , \rho _1, \nu < \min (\alpha -1, 1)\) and \(0 < \gamma ^{\prime }, \rho _2 < \min (\alpha -1,2)\).
Proof
First let us check condition (2.5). We have
The change of variable \(x=2\sqrt{w}y\) gives
because \(\alpha >1\). To check condition (3.1), we note that for \(a,b \ge 0\), we have \(|e^{-a}-e^{-b}|\le |a-b|^{\gamma }(e^{-a}\vee e^{-b})\), for any \(0 \le \gamma \le 1\). So
As a consequence
For the integral \(I(y)\), with the change of variable \(z=\sqrt{w}x-y\), we have
where the last inequality follows from the fact that \(|x|^{\gamma }e^{-\frac{|x|^2}{4}}\le C e^{-\frac{|x|^2}{8}}\) and Lemma 17 in [6]. The term \(J(y)\) can be estimated in the same way using the change of variable \(z=\sqrt{w}y\), and we have
Hence,
for any \(0<\gamma < \alpha -1\). So condition (3.1) is satisfied with \(0< \gamma < \min (\alpha -1,1)\).
To check condition (3.2), note that
So we have
Here we have used the fact that \(x^2e^{-x^2}\le C e^{-\frac{x^2}{2}}\). By considering the cases \(\frac{|y|}{\sqrt{w}}\le 1\) and \(\frac{|y|}{\sqrt{w}}>1\), we obtain
for any \(0 \le \gamma ^{\prime }\le 2\). So we have
By Lemma 17 in [6], we can write
where the constant \(C\) does not depend on \(y, \lambda , \mu \). Therefore,
for any \(0<\gamma ^{\prime }<\min (\alpha -1, 2)\). As a consequence, condition (3.2) is satisfied with \(0< \gamma ^{\prime } < \min (\alpha -1,2)\).
To check condition (1) in Theorem 4.1 holds we compute
for any \(0\le \nu \le 1\). This implies that
for any \(0< \nu <\min (\alpha -1,1)\). So condition (1) in Theorem 4.1 is satisfied with \(0< \nu < \min (\alpha -1,1)\).
To check the condition (4.1), first we note that
where we have used the fact that \(|x|e^{-x^2}\le C e^{-\frac{x^2}{2}}\). By considering the cases \(\frac{h}{\sqrt{w}} \le 1\) and \(\frac{h}{\sqrt{w}} > 1\), we can write
for any \(\rho _1 \in [0,1]\). So we have
Therefore, for any \(\rho _1 \in [0,1]\).
We claim that this quantity is bounded by \(Ch^{\rho _1}\) if \(0<\rho _1 < \min (\alpha -1,1)\). To show this claim, we first estimate the quantity
Using the Fourier transform (see Lemma 6.5), the change of variables \(\xi \sqrt{w}=\eta \), and taking \(0< \varepsilon < 1\) we obtain
Similarly, we have
and
Therefore,
and (4.1) is satisfied with \(0<\rho _1 < \min (\alpha -1,1))\).
To check condition (4.2), we note that
By considering the cases \(\frac{h^2}{w}\le 1\) and \(\frac{h^2}{w}> 1\), we have for any \(0\le \rho _2 \le 2\),
Therefore, we obtain
and
We claim that when \(0<\rho _2< \min (\alpha -1,2)\), the above expression is bounded by \(h^{\frac{\rho _2}{2}}\). To show this claim, we first estimate the integral
Using the Fourier transform (see Lemma 6.5) and the change of variable \(\sqrt{w}\xi =\eta \), we obtain
where the constant \(C\) does not depend on \(\lambda \) and \(\mu \). The same estimation can be done for each of the other integrals and we obtain
Thus,
and the (4.2) is satisfied for \(0<\rho _2 < \min (\alpha -1,2)\). This completes the proof. \(\square \)
Notice that, with the notation of Corollary 4.2, for the Bessel kernel we can take \(\kappa _1=\kappa _2 <(\alpha -1)\wedge 1\), and we deduce the local Hölder continuity of the solution \(u\) in space and time variables of order \(\kappa < \min (\gamma _1, \gamma _2, \frac{\alpha -1}{2}\wedge 1 )\).
6 The fractional noise
In this section we consider the case where \(\dot{W}(t,x)\) is fractional Brownian noise in the space variable with Hurst parameters \(H_1,H_2,H_3\) in each direction. That is, suppose that \(\left\{ W(t, x), t\ge 0 , x\in \mathbb {R}^3\right\} \) is a centered Gaussian field with the covariance
where \(x=(x_1,x_2,x_3)\), \(y=(y_1,y_2,y_3)\) and
Then \(\dot{W}(t,x)\) is the formal partial derivative \(\frac{\partial ^4 W}{\partial t \partial x_1 \partial x_2 \partial x_3}(t,x)\). We will require \(\frac{1}{2}<H_i<1\), \(i=1,2,3\). This choice of noise corresponds to the covariance function
where \(H=(H_1,H_2,H_3)\) and \(c_H=\prod _{i=1}^3 H_i (2H_i-1) \). Here and in what follows for simplicity, we omit the coefficient \(c_H\) in the expression of \(f(x)\). The corresponding spectral measure is
for some constant \(C_H\) which depends only on \(H\). We will apply Theorems 3.2 and 4.1 to get the Hölder continuity of the solution to Eq. (1.1) in the space and time variables.
Theorem 6.1
Assume conditions \((a)\) and \((b)\) in Theorem 3.1 and let \(f\) be given by (6.3) (without the constant \(c_H\)) with \(H_1+H_2+H_3>2\). Set
and choose constants \(\kappa _i>0\), \(i=0,1,2,3\) such that \(\kappa _i < \min (H_i-\frac{1}{2}, \bar{\kappa },\gamma _1,\gamma _2)\) for \( i=1,2,3\), and \(\kappa _0 \le \min (\kappa _1, \kappa _2, \kappa _3)\). Then the solution to (1.1) is locally Hölder continuous with exponent \(\kappa _0\) in the time variable and with exponent \(\kappa _i\) in the \(i\)th direction. Namely, for any bounded rectangle \(I\subset {\mathbb {R}}^3\), there exists a random variable \(K\) (depending on \(I\) and the constants \(\kappa _i\)’s), such that
for all \(t,\bar{t} \in [0,T]\), \(x,y \in I\).
Proof
First we consider the space variable. Proceeding as in the proof of Theorem 3.2, it is easy to see that if for some number \(0<\gamma \le 1\), \(\mathcal {F}\left( |\xi _1|^{2\gamma }\mu (d\xi )\right) (w)\) is a nonnegative locally integrable function and
then if \(\kappa _1 =\min (\gamma ,\gamma _1,\gamma _2)\), for any bounded rectangle \(I\subset {\mathbb {R}}^3\), and for any \(q\ge 2\), there exists a constant \(C\) such that
for any \(t\in [0,T]\) and \(x,y \in I\).
We claim that for \(0<\gamma < \min (H_1-\frac{1}{2}, \bar{\kappa })\), \(\mathcal {F}\left( |\xi _1|^{2\gamma }\mu (d\xi )\right) (w)\) is a nonnegative locally integrable function and (6.6) holds. Indeed, since \(\mu (d\xi )=|\xi _1|^{1-2H_1}|\xi _2|^{1-2H_2}|\xi _3|^{1-2H_3}d\xi \), we have
which is well defined because \(\gamma <H_1-\frac{1}{2}\). To show (6.6), we have
where the \(\alpha _i\)’s are positive with \(\alpha _1+\alpha _2+\alpha _3=1\). When \(1-2(H_1-\gamma )-2\alpha _1 < -1\), \(1-2H_2-2\alpha _2 < -1\) and \(1-2H_3-2\alpha _3 < -1\), the above three integrals are finite. It is elementary to see such \(\alpha _i\)’s exist under the condition \(\gamma < H_1+H_2+H_3-2\). The same argument holds for the other coordinates.
For the time variable, we will check conditions (1) and (3) in Theorem 4.1. To see that condition (1) in Theorem 4.1 is satisfied for some \(0 < \nu \le 1\), take positive numbers \(\varepsilon _i\), \(i=1,2,3\) such that \(\varepsilon _1+\varepsilon _2+\varepsilon _3=1\) and \(2H_i-1-\varepsilon _i>0\) for \(i=1,2,3\). Then we have
So condition (1) in Theorem 4.1 is satisfied with \(\nu =\min (2(H_1+H_2+H_3-2),1)\).
To check (4.1), let \(x=a( \xi +\eta )+ h\eta \). Then we decompose the difference \(f(x+h\xi )-f(x)\) into the sum of three terms, each of them containing an increment in one direction, and we obtain
We claim that for some \(\rho _1 \in (0,1]\), the integral on \([0,T]\times S^2 \times S^2\) of each of these three terms with respect to the measure \(s\sigma (d\xi )\sigma (d\eta )ds\) is bounded by \(C h^{\rho _1}\). To show this claim, we apply (5.1) with \(d=1\), \(b=\rho _{1}<\min (2H_1-1, 2H_2-1, 2H_3-1, 2(H_1+H_2+H_3-2))\), \(a=2H_i-\rho _1-1\) and \(u=x_i\) to the \(i\)th summand (\(i=1,2,3\)) and we get
We want to show that for \(i=1,2,3\)
We will consider only the case \(i=1\), the other two terms being similar. By splitting the integral with respect to \(w\) into two parts, one over \(|w|\le 3\), and another one over \(|w|>3\), just as we did for the Riesz kernel, we have
For integral \(I_1\), using the change of variable \(w+\xi _1\rightarrow w\) and the Fourier transform, we can write
By the change of variable \((s+h)z=x\), the bound \(|e^{iz_1hw}|\le 1\) and direct calculation, we see the integral is finite uniformly in \(0< h\le 1\).
For the integral \(I_2\) we can write
where \(\psi (y)=\left( \frac{s}{s+h}y_1,y_2,y_3\right) \), and \(G^\psi (s+h)\) denotes the image of the measure \(G(s+h)\) by the mapping \(\psi \). Then using the Fourier transform we obtain
where in the first inequality above we used the fact that \(|e^{-i\xi _1w}|\le 1\), and in the last inequality we used that fact that \(|\sin x|\le |x|^{\varepsilon }\) for any \(\varepsilon > 0\). The above integral is finite uniformly in \(w\) and \(0< h\le 1\).
For the third integral \(I_3\), we can bound \(\left| |w+\xi _1|^{\rho _{1}-1}-|w|^{\rho _{1}-1}\right| \) by \(C|w|^{\rho _{1}-2}\) as in the example of the Riesz kernel, and proceed as in the second integral \(I_2\). Applying the same argument for the other two terms, we get (6.8) with \(\rho _{1}\in (0, \min (2H_1-1,2H_2-1,2H_3-1, 2 \bar{\kappa })\). Therefore, condition (4.1) is satisfied with \(0< \rho _1 < \min (2H_1-1,2H_2-1,2H_3-1, 2 \bar{\kappa })\).
For condition (4.2), we use the inequality
Then we can apply the previous procedure to both terms on the right-hand side and the argument is the same as in the case of condition (4.1). We conclude that (4.2) is satisfied with \(0< \rho _2< \min (2H_1-1,2H_2-1,2H_3-1, 2\bar{\kappa })\).
In summary, we can take \(\nu = \min (2 \bar{\kappa },1)\) and \(\rho _1=\rho _2 \in (0, \min (2H_1-1,2H_2-1,2H_3-1, 2 \bar{\kappa }))\), and Theorem 4.2 together with the moment estimate (6.7) leads to the desired Hölder continuity in the space and time variables via an application of Kolmogorov’s continuity theorem. \(\square \)
Consider Eq. (1.1) with vanishing initial conditions \(v_0\), \(\bar{v}_0\) and coefficients \(\sigma \equiv 1\) and \(b\equiv 0\). That means, we consider the stochastic wave equation with additive fractional noise
The covariance function of the noise is given by (6.3) with \(H_i> \frac{1}{2}\) for \(i=1,2,3\) and recall that \(\bar{\kappa } =H_1+H_2+H_3-2>0\).
For this equation the solution can be written as
In this case we are going to show that \(\bar{\kappa }\) is the optimal exponent for the Hölder continuity of the solution \(u \) in the space and time variables.
Theorem 6.2
Let \(u\) be the solution to the stochastic partial differential equation (6.9). Then
-
(a)
There are two positive constants \(c_1\) and \(c_2 \) such that
$$\begin{aligned} c_1|x-y|^{2\bar{\kappa } } \le {\mathbb {E}}\left( |u(t,x)-u(t,y)|^2\right) \le c_2|x-y|^{2\bar{\kappa }} \end{aligned}$$(6.10)for all \(x,y \in \mathbb {R}^3 \) and \(t\in [0,T]\).
-
(b)
For any fixed \(t_0 \in (0,T]\) there are two positive constants \(c_1\) and \(c_2\) such that
$$\begin{aligned} c_1|\bar{t}-t|^{2\bar{\kappa }}\le {\mathbb {E}}\left( |u(t,x)-u(\bar{t},x)|^2\right) \le c_2|\bar{t}-t|^{2\bar{\kappa }}. \end{aligned}$$(6.11)for all \(t,\bar{t} \in [t_0,T]\) and \(x \in \mathbb {R}^3\).
Proof
For any \(x \in \mathbb {R}^3\), set \(R(x)={\mathbb {E}}\left( u(t,x)u(t,0)\right) \). It is easy to see that
Without loss of generality, we may assume that \(t=1\) and \(y=0\). We have by Lemma 6.5
The integrand is non-negative. For clarity we may assume that \(|x_1|\le |x_2|\le |x_3|\). If \(|\xi _3|\ge 1\), then \(1-\frac{\sin (2|\xi |)}{2|\xi |}\ge \frac{1}{2}\). Thus using the change of variable \(\xi x_3=\eta \), we have
If \(x\) is in a bounded interval \(I\), then there is \(L>0\) such that \(|x_3|\le L\). Thus
It is easy to see that
is a continuous function of \(u_1, u_2\) and for any \(u_1\) and \(u_2\), \(g(u_1, u_2)\) is positive. Thus
since the infimum is taken on a compact set. This proves the left-hand side inequality in (6.10).
To show the second inequality in (6.10), we can use the triangular inequality, and it suffices to show the inequality for \(x=(x_1, 0, 0)\). In this case
which is the second inequality of (6.10). Hence, (a) is proved.
Now we turn to consider (b). Let \(0\le t < \bar{t}\le T\). Then we have
where
and
Integrating with respect to the variable \(s\) yields
With the change of variable \((\bar{t}-t)\xi \rightarrow \eta \) the last integral becomes
Therefore, we have
The term \(Z_2\) is slightly more complicated. A direct integration in the variable \(s\) yields
where
and
The change of variable \((\bar{t}-t)\xi =\eta \) yields
Similarly,
Therefore,
when \(|\bar{t}-t|\) is sufficiently small and \(t \ge t_0\). So we conclude that
On the other hand, we have
Applying the substitution \(\xi (\bar{t}-t)=\eta \) to both the above integrals, we see that
Thus (b) is proved. \(\square \)
Combining the upper bound in (6.10) and (6.11), taking into account that the process \(u\) is Gaussian and applying Kolmogorov continuity criterion, for any \(\delta >0\) and any bounded rectangle \(I\subset \mathbb {R}^3\), there is a random variable \(c_{\delta , I} \) such that almost surely
The first inequalities of (6.10) and (6.11) tell us that the exponent \(\bar{\kappa }\) is the optimal.
Remark 6.3
Theorem 5.1 in [5] shows that the result obtained in Sect. 5.2 is optimal. The result in Theorem 6.2 suggests that the result in Theorem 6.1 may not be optimal. To prove the result is optimal or to find the optimal result needs further research.
7 Appendix
In this section we prove some lemmas used in this paper.
Lemma 6.4
For any \(s\ge t\)
Proof
To calculate \(\left( G(t)*G(s)\right) (dx)\), let us consider two independent random variables \(X\) and \(Y\) uniformly distributed on the spheres with radii \(s\) and \(t\) respectively with \(s\ge t\). Note that the distribution of \(X+Y\) is rotationally invariant. Consider a bounded continuous function \(\varphi \) on \({\mathbb {R}}\). We have
It is easy to see that in the above expression, the integral with respect to \(\sigma (dy)\) does not depend on \(x\), so we can take \(x=x_0=(0,0,1)\), and using spherical coordinates \(y=(\sin \phi \cos \theta ,\sin \phi \sin \theta ,\cos \phi )\), we have
Making the change of variable \(u=\sqrt{s^2+t^2+2ts\cos \phi }\), we obtain
where \(B(0,r)\) is the ball in \({\mathbb {R}}^3\) with center \(0\) and radius \(r\). So we conclude that the random variable \(X+Y\) has a density given by
Taking into account that the distributions of \(X\) and \(Y\) are given by \(\frac{1}{s}G(s,dx)\) and \(\frac{1}{t}G(t,dx)\) respectively, we easily get the desired result (6.15). \(\square \)
Our next result gives an integral identity which is used a lot in this paper. See also Theorem 5.2 in [5] for a similar result.
The following result is related to Theorem 5.2 in [5].
Lemma 6.5
Let \(\varphi \) and \(\psi \) be two bounded Borel measurable functions and assume that (2.5) holds and \(s\ge t >0\). Then for any \(w\in {\mathbb {R}}^3\) we have
Proof
Let \(\phi (x)=C \exp (\frac{1}{|x|^2-1})\mathbf{{1}}_{[0,1)}(|x|)\), where \(C\) is a normalization coefficient such that \(\int _{{\mathbb {R}}^3}\phi (x)dx=1\). Set \(\phi _{\varepsilon }(x)=\frac{1}{\varepsilon ^3}\phi (\frac{x}{\varepsilon })\), with \(\varepsilon \le t\). Using the Fourier transform we have
where \(\widetilde{\varphi G(t)}(x) =\varphi (-x) G(t,-dx)\). We are going to show that
Indeed, since \(\varphi \) and \(\psi \) are bounded, we have
Note first that the function
is supported within a ball centered at the origin with radius \(3s\) for every \(\varepsilon \le t\) and it converges to \(\frac{1}{|z|}\mathbf{{1}}_{[s-t,s+t]}\) almost everywhere. Next, for \(|x|\le 3s\) we have
where in the second integral we have used the change of variable \(z-x \rightarrow z\). Since in the second integral \(|z|< \frac{|x|}{2}\), we have \(|z+x|\ge |x|-|z|\ge \frac{|x|}{2}\), and
where in the last inequality we used the fact that \(\sup _{x \in {\mathbb {R}}^3} |x|^3 \phi (x) < \infty \). So (6.19) is proved. Then by an application of the dominated convergence theorem we have
On the other hand, the estimate (6.19) implies that the quantity in (6.18) is uniformly bounded in \(\varepsilon \). Hence, by Fatou’s lemma \(\int _{{\mathbb {R}}^3}\left| \mathcal {F}\left( \varphi G(t)\right) (\xi )\right| \left| \mathcal {F}\left( \psi G(s)\right) (\xi )\right| \mu (d\xi ) <\infty \), and by the dominated convergence, the right-hand side of (6.18) converges to
This completes the proof of the lemma. \(\square \)
In particular, if in the above lemma, take \(\varphi =\psi \), \(t=s\) and \(w=0\), then for any \(t>0\) we have
More specifically, if in addition, we take \(\varphi \equiv 1\), then we obtain
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Acknowledgments
We would like to thank the anonymous referees for their many constructive and detailed comments which helped to improve this work. D. Nualart is supported by the NSF Grant DMS1208625. Y. Hu is partially supported by a Grant from the Simons Foundation #209206.
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Hu, Y., Huang, J. & Nualart, D. On Hölder continuity of the solution of stochastic wave equations in dimension three. Stoch PDE: Anal Comp 2, 353–407 (2014). https://doi.org/10.1007/s40072-014-0035-5
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DOI: https://doi.org/10.1007/s40072-014-0035-5