1 Introduction

Increased attention has been paid in recent years to the functional statistical analysis of spatiotemporal data related to environmental, ecological and health sciences applications. SPDEs offer a suitable mathematical-probabilistic context to incorporate sample information from spatiotemporal referenced data (see Christakos 2000). In particular, the prediction and extrapolation problems are addressed in an accurate and flexible way, even in the presence of spatial heterogeneities and, in general, of non-stationary or non-linear behavior in space and/or time (see, for example, Ruiz-Medina and Angulo 2007; Ruiz-Medina and Fernández-Pascual 2010).

The spatiotemporal dependence structure is often interpreted as the temporal evolution of a spatial process described in the form of a dynamic model in discrete time (see, for example, Bosq 2000; Bosq and Blanke 2007; Ruiz-Medina and Salmerón 2010 in the autoregressive Hilbertian time series context; Gelfand et al. (2005) in the functional linear context, among others). When continuous time is considered in the description of the temporal evolution of the spatial process of interest, SPDEs and, in general, stochastic pseudodifferential evolution equations are considered (see Angulo et al. 2005; Christakos 1992; Christakos and Raghu 1996; Kelbert et al. 2005; Leonenko and Ruiz-Medina 2006, among others). Spatiotemporal covariance modeling in continuous time and space also constitutes an active area of research in the last few decades, since the work of Mardia and Goodall (1993) on separable space–time covariance functions to the more recent works on nonseparable space–time covariance modeling by Gneiting (2002), Gneiting et al. (2010), Porcu et al. (2006), Porcu and Zastavnyi (2011), Stein (2005) and references therein.

This paper provides a new approach for the introduction of non-separable compactly supported spatiotemporal covariance models displaying local self-similarity and fractal behavior, based on the theory of fractional in time and in space pseudodifferential boundary-value problems. The pseudodifferential operator defining these equations allows the introduction of a compactly supported spatiotemporal covariance function family, in terms of the self-convolution of its inverse. This operator also defines the geometry of the Reproducing Kernel Hilbert Space (RKHS), that univocally characterizes the Gaussian distribution of the mean-square continuous random field solution. The continuous decay to zero at the boundary of the elements of the family of covariance functions introduced is ensured on regular bounded domains, where a continuous Green kernel exists satisfying the Dirichlet boundary conditions imposed. In particular, differentiable and non-differentiable, but continuous compactly supported covariance functions with different fractality orders in time and in space can be introduced in this framework. Parameter estimation of the elements of the spatiotemporal covariance function class introduced is achieved by computing multiple linear regression estimators in the log-wavelet domain, in terms of the empirical wavelet periodogram.

The outline of the paper is as follows. The main preliminary functional analytic elements involved are introduced in Sect. 2. In Sect. 3, the family of fractional in time and in space pseudodifferential models, with Dirichlet boundary conditions and Gaussian initial conditions, are formulated. Their random solutions, in the mean-square sense, are then derived. The main properties of the spatiotemporal covariance family introduced are analyzed in Sect. 4. The covariance function parameter estimation methodology proposed in the log-wavelet domain is described in Sect. 5. Illustration of the class of spatiotemporal compactly supported covariance functions introduced is performed by considering several numerical examples in Sect. 6. Main conclusions and interpretations are provided in Sect. 6.1.

2 Preliminaries

In the subsequent development we analyze the properties of the class of spatiotemporal covariance functions of the mean-square solution to equation

$$\begin{aligned} \frac{\partial }{\partial t}v(t,\mathbf {x})=\mathcal {L}v(t,\mathbf {x});\quad v(0,\mathbf {x})=f(\mathbf {x}), \end{aligned}$$
(1)

when \(f(\mathbf {x})=X_{0}(\mathbf {x}),\) for all \(\mathbf {x}\in D\subseteq \mathbb {R}^{d},\) with \(X_{0}(\mathbf {x})\) being a spatial zero-mean Gaussian random field, with continuous covariance function on a regular Dirichlet bounded open domain D. Operator \(\mathcal {L}\) is assumed to be a pseudodifferential operator, generating a symmetric strongly continuous transition semigroup \(\{ P_{t},\ t\ge 0\}\) (see Appendix in the Supplementary Material at http://www.editorialmanager.com/smap). This operator is defined through its symbol l in terms of the following integral equation:

$$\begin{aligned} \mathcal {L}u(\mathbf {x})=(2\pi )^{-d}\int _{\mathbb {R}^{\mathrm{d}}}\exp \left( i\left\langle \mathbf {x}, \varvec{\xi }\right\rangle \right) l(\varvec{\xi })\mathcal {F}(u)(\varvec{\xi })d\varvec{\xi },\quad \forall u\in L^{2}(\mathbb {R}^{d};d\mathbf {x}), \end{aligned}$$
(2)

where \(\mathcal {F}(u)\) denotes the Fourier transform of u. Moreover, l belongs to the space of \(C^{\infty }\) functions with bounded derivatives of all orders, and satisfies

$$\begin{aligned} C^{\prime }_{\varvec{\nu }}(1+\Vert \varvec{\xi }\Vert ^{2})^{\Vert \varvec{\nu } \Vert /2}\le |l(\varvec{\xi })|\le C_{\varvec{\nu } }(1+\Vert \varvec{\xi }\Vert ^{2})^{\Vert \varvec{\nu } \Vert /2}, \end{aligned}$$
(3)

for any multi-index \(\varvec{\nu }\in \mathbb {R}^{d}_{+},\) and positive constants \(C^{\prime }_{\varvec{\nu }}\le C_{\varvec{\nu } }.\)

The class of covariance functions introduced here is constructed from the composition \(\mathcal {L} ^{-1}[\mathcal {L} ^{-1}]^{*}R_{X_{0}},\) where \([\mathcal {L} ^{-1}]^{*}\) denotes the adjoint of the inverse operator \(\mathcal {L}^{-1},\) and \(R_{X_{0}}\) the covariance operator with kernel de covariance function of the spatial process \(X_{0},\) defining the random initial condition. The concept of dual random field \(\widetilde{X}\) arises, in this context, as the spatiotemporal random field whose integral covariance operator, with kernel its covariance function, is given by operator \(R_{\widetilde{X}}=R_{X_{0}}^{-1}\mathcal {L} ^{*}\mathcal {L}.\) Such an operator allows the following explicit definition of the inner product (characterizing its geometry) in the RKHS of the mean-square zero-mean Gaussian spatiotemporal solution X of the class of fractional in time and in space stochastic pseudodifferential equations studied here:

$$\begin{aligned} \left\langle f,g\right\rangle =R_{X_{0}}^{-1}\mathcal {L} ^{*}\mathcal {L}(f)(g), \end{aligned}$$
(4)

for every f, g in the RKHS of X (see also Definition 5 in the Appendix in the Supplementary Material at http://www.editorialmanager.com/smap). Note that the kernel J of \(\mathcal {L}^{*}\mathcal {L}\) satisfies

$$\begin{aligned} K_{1}\Vert \mathbf {x}-\mathbf {y}\Vert ^{-d-\gamma }\le J(\mathbf {x},\mathbf {y})\le K_{2}\Vert \mathbf {x}-\mathbf {y}\Vert ^{-d-\gamma }, \end{aligned}$$
(5)

for \(\Vert \mathbf {x}-\mathbf {y}\Vert <1,\) and \(\gamma =\Vert \varvec{\nu }\Vert \) in (3).

3 Fractional-order pseudodifferential evolution equations

We consider the following definition of Dirichlet-regular bounded open domain.

Definition 1

(see, for example, Brelot (1960), p. 137 and Theorem 32, and Fuglede (2005), p. 253). For a connected bounded open domain D with boundary \(\partial D\) we say that \(\mathbf {x}_{0}\in \partial D\) is regular iff it has a Green kernel \(G^{D}\) such that, for each \(\mathbf {x}\in D,\)

$$\begin{aligned} \lim _{\mathbf {x}\rightarrow \mathbf {x}_{0}}G^{D}(\mathbf {x},\mathbf {y})=0,\quad \forall \mathbf {y}\in D. \end{aligned}$$
(6)

The set D is regular if every point of \(\partial D\) is regular.

Remark 1

In the following we restrict our attention to simple Dirichlet-regular bounded open domains, whose symmetries allow the application of the method of separation of variables.

Let us consider the following definition of regularized fractional derivative of order \(\beta ,\) for \(\beta \in (0,1].\) This fractional derivative is given by:

$$\begin{aligned} \frac{\partial ^{{}\beta }u}{\partial t^{{}\beta }}=\left\{ \begin{array}{ll} \frac{\partial u}{\partial t}\left( t,\mathbf {x}\right) , &{} \quad \text {if }\,\beta =1 \\ \frac{1}{\varGamma \left( 1-\beta \right) }\frac{\partial }{\partial t} \int _{0}^{t}\left( t-\tau \right) ^{-\beta }u\left( \tau ,\mathbf {x}\right) d\tau -\frac{u(0,\mathbf {x})}{t^{\beta }}, &{}\quad \text {if}\, \ \beta {\in } (0,1),\ t\in (0,T] \end{array} \right. \end{aligned}$$
(7)

(see Caputo 1967; Podlubny 1999).

In the next result the class of spatiotemporal Gaussian random fields introduced is constructed from the Mittag-Leffler function, which is given by

$$\begin{aligned} E_{\beta }(z)=\sum _{j=0}^{\infty }\frac{(z)^{j}}{\varGamma (j\beta +1)},\quad z\in \mathbb {C},\quad 0<\beta \le 1. \end{aligned}$$
(8)

The properties of this function can be seen in the Appendix appearing in the Supplementary Material at http://www.editorialmanager.com/smap.

The class of Gaussian RFs with compactly supported fractional space–time covariance functions is introduced in the following theorem.

Theorem 1

Let us consider the fractional in time and in space evolution equation

$$\begin{aligned}&\frac{\partial ^{\beta }}{\partial t^{\beta }}X\left( t,\mathbf {x}\right) = -\mathcal {L}X\left( t,\mathbf {x}\right) ,\quad X(0,\mathbf {x})=X_{0}(\mathbf {x}),\nonumber \\&\quad X(\mathbf {x})\underset{\text{ m.s. }}{=} 0,\ \mathbf {x}\in \partial D, \end{aligned}$$
(9)

for \(\beta \in (0,1],\) \(\varrho +\gamma \in (d/2, d),\) with \(\varrho \) being the local exponent defining the order of weak-sense differentiation of the covariance kernel of \(X_{0}\) with respect to each one of its components (see Appendix in the Supplementary Material at http://www.editorialmanager.com/smap). The mean-square solution to (9) is

$$\begin{aligned} X(t, \mathbf {x})=\sum _{k=1}^{\infty }\int _{D}E_{\beta }\left( -t^{\beta }\lambda _{k}(\mathcal {L})\right) \phi _{k}(\mathbf {x})\phi _{k}(\mathbf {y})X_{0}(\mathbf {y})d\mathbf {y},\quad \mathbf {x}\in D, \ t>0. \end{aligned}$$
(10)

Its covariance function can be computed from the following integral series:

$$\begin{aligned} r_{X}(t,s,\mathbf {x},\mathbf {y})= & {} E\left[ X(t,\mathbf {x})X(s,\mathbf {y})\right] \nonumber \\= & {} \sum _{k=1}^{\infty }\sum _{p=1}^{\infty }\int _{D\times D}E_{\beta }\left( -t^{\beta }\lambda _{k}(\mathcal {L})\right) E_{\beta }\left( -s^{\beta }\lambda _{p}(\mathcal {L})\right) \nonumber \\&\times \, \phi _{k}(\mathbf {x})\phi _{k}(\mathbf {z})B_{X_{0}}(\mathbf {z},\mathbf {v})\phi _{p}(\mathbf {v})\phi _{p}(\mathbf {y})d\mathbf {z}d\mathbf {v}, \end{aligned}$$
(11)

where for each \(k\ge 1,\) \(\phi _{k}\) satisfies \(\mathcal {L}\phi _{k}(\mathbf {x})=\lambda _{k}(\mathcal {L})\phi _{k}(\mathbf {x}),\) for \(\mathbf {x}\in D,\) and \(\phi _{k}(\mathbf {x})=0,\) for \(\mathbf {x}\in \partial D,\quad k\ge 1.\)

The proof of Theorem 1 can be found in the Supplementary Material at http://www.editorialmanager.com/smap.

4 Spatiotemporal covariance function families

In this section we apply Theorem 1 for the derivation of new spatiotemporal covariance families, with compact support, displaying local self-similarity in time and/or space. Specifically, the kernels in the parametric family introduced in Eq. (11) are constructed from convolution of the twofold convoluted Green kernel with the covariance function of the random initial condition. Applying Spectral Theorem for self-adjoint operators [(see, for example, Edwards (1965) and Dautray and Lions (1985)], such kernels are symmetric and positive semidefinite. Their properties are now derived.

Proposition 1

For \(\beta \in (0,1)\) and \(\gamma +\varrho \in (\max \{\beta d,d/2\},d),\) the following assertions hold:

  1. (i)

    For each \(t>0,\) the elements of the covariance function family (11) belong to the fractional Sobolev space \(\overline{H}^{2(\gamma +\varrho )}(D\times D),\) with \(\gamma =\Vert \varvec{\nu }\Vert \) defined as before, and \(2\varrho \) denoting the order of the fractional Sobolev space \(\overline{H}^{2\varrho }(D\times D)\) where the covariance function of the random initial condition \(X_{0}\) lies.

  2. (ii)

    For each \(\mathbf {z}\in D,\) and \(\beta >1/2,\) and for a given time interval [0, T],  \(T>0,\) the elements of the family of covariance functions (11) belong to the space of continuous functions over \([0,T]\times [0,T].\)

  3. (iii)

    For a given time interval [0, T],  \(T>0,\) the elements of the family of covariance functions (11) belong to the intersection of spaces

    $$\begin{aligned} L^{2}\left( [0,T]\times [0,T]; \overline{H}^{2}(D\times D)\cap \overline{H}^{2(\gamma +\varrho )}(D\times D)\right) \end{aligned}$$

    and

    $$\begin{aligned} C^{0}\left( [0,T]\times [0,T]; L^{2}(D\times D)\right) . \end{aligned}$$

    Here, \(L^{2}\left( [0,T]\times [0,T]; \overline{H}^{2}(D\times D)\cap \overline{H}^{2(\gamma +\varrho )}(D\times D)\right) \) represents the space of measurable functions on \([0,T]\times [0,T]\) with values in the intersection of fractional Sobolev spaces \(\overline{H}^{2(\gamma +\varrho )}(D\times D)\cap \overline{H}^{2}(D\times D),\) with the norm

    $$\begin{aligned}&\Vert k\Vert _{L^{2}\left( [0,T]\times [0,T]; \overline{H}^{2}(D\times D)\cap \overline{H}^{2(\gamma +\varrho )}(D\times D)\right) }\nonumber \\&\quad =\left[ \int _{0}^{T} \int _{0}^{T}\Vert k(t,s,\cdot , \cdot )\Vert _{\overline{H}^{2}(D\times D)\cap \overline{H}^{2(\gamma +\varrho )}(D\times D)}dtds\right] ^{1/2}, \end{aligned}$$
    (12)

    and \(C^{0}\left( [0,T]\times [0,T]; L^{2}(D\times D)\right) \) denotes the space of continuous functions over \([0,T]\times [0,T]\) with values in \(L^{2}(D\times D),\) with the norm

    $$\begin{aligned} \Vert k\Vert _{C^{0}\left( [0,T]\times [0,T]; L^{2}(D\times D)\right) }=\sup _{(t,s)\in [0,T]\times [0,T]}\Vert k(t,s,\cdot ,\cdot )\Vert _{L^{2}(D\times D)}. \end{aligned}$$

The proof of Proposition 1 can be found in the Supplementary Material at http://www.editorialmanager.com/smap.

5 Covariance parameter estimation based on the empirical log-wavelet periodogram

It is well known that the wavelet transform provides an optimal processing of functions displaying fractal characteristics in space and/or time. By this reason, we choose the empirical log-wavelet periodogram in space and time, as the more convenient tool for the approximation of the local self-similar exponents (local Hölder exponents) of the family of spatiotemporal covariance functions introduced. These exponents also characterize the mean-quadratic local variation properties of the temporal and spatial increments of the family of zero-mean Gaussian random fields introduced in Theorem 1.

The main elements involved in the definition of the discrete wavelet transform in space and time are summarized in the Appendix given in the Supplementary Material at http://www.editorialmanager.com/smap. We now describe the main steps required in the implementation of the wavelet-based parameter estimator of \((\beta , \gamma , \varrho ),\) characterizing the covariance function family introduced, and hence, the Gaussian distribution of the spatiotemporal random field solution (10) (see also Proposition 1).

For a given number of resolution levels \(m\,{=}\,0,1,\dots , M_{s},\) in space, and \(k\,{=}\,0,1,\dots ,M_{t},\) in time, let us denote \(\varvec{\varPsi }_{s}^{*}\,{=}\,\left( \left[ \varvec{\varPsi }_{s}^{*}\right] _{0},\dots ,\left[ \varvec{\varPsi }_{s}^{*}\right] _{M_{s}}\right) ,\) and \(\varvec{\varPsi }_{t}^{*}\,{=}\,\left( \left[ \varvec{\varPsi }_{t}^{*}\right] _{0},\dots , \left[ \varvec{\varPsi }_{t}^{*}\right] _{M_{t}}\right) .\) Hence, the corresponding wavelet projection operator in time and space is given by \(\varvec{\varPsi }^{*},{=}\,\varvec{\varPsi }_{t}^{*}\varvec{\varPsi }_{s}^{*},\) defining the diagonal details. Note that vertical details obtained from projection operators \(\varvec{\varPsi }_{s}^{*}\varvec{\varPhi }_{t}^{*}\) and \(\varvec{\varPsi }_{t}^{*}\varvec{\varPhi }_{s}^{*},\) as well as horizontal details obtained from projection operators \(\varvec{\varPhi }^{*}_{s}\varvec{\varPsi }^{*}_{t}\) and \(\varvec{\varPhi }^{*}_{t}\varvec{\varPsi }^{*}_{s},\) are not considered in the subsequent analysis, since information provided for parameters \(\beta , \gamma \) and \(\varrho \) on a regular bounded open domain D is mainly related to small scale (local) variation properties in space and time, for a fixed time interval [0, T].

Applying now \(\varvec{\varPsi }^{*}\) to the left-hand side of Eq. (9), we have

$$\begin{aligned} \varvec{\varPsi }_{t}^{*}\left[ \frac{\partial ^{\beta }}{\partial t^{\beta }}\right] \varvec{\varPsi }_{t}\varvec{\varPsi }_{t}^{*} \varvec{\varPsi }_{s}^{*}X= & {} \varvec{\varPsi }_{t}^{*}\varvec{\varPsi }_{s}^{*} (-\mathcal {L})\varvec{\varPsi }_{s}\varvec{\varPsi }_{t}\varvec{\varPsi }_{t}^{*}\varvec{\varPsi }_{s}^{*}X, \end{aligned}$$
(13)

which reflects the small-scale variation in our model. Log-wavelet multiple regression is applied for the estimation of parameter vector \((\beta ,\gamma ,\varrho )\) from \(\varvec{\varPsi }_{s}^{*}X_{0}\) and

$$\begin{aligned} \varvec{\varPsi }_{t}^{*} \varvec{\varPsi }_{s}^{*}X=\left[ \varvec{\varPsi }_{t}^{*}\left[ \frac{\partial ^{\beta }}{\partial t^{\beta }}\right] \varvec{\varPsi }_{t}-\varvec{\varPsi }_{t}^{*}\varvec{\varPsi }_{s}^{*} (-\mathcal {L})\varvec{\varPsi }_{s}\varvec{\varPsi }_{t}\right] ^{-1}\varvec{\varPsi }_{s}^{*}X_{0}, \end{aligned}$$
(14)

considering the wavelet periodogram WP(X) given by

$$\begin{aligned} WP(X)= & {} \left[ \left[ \varvec{\varPsi }_{t}^{*}\left[ \frac{\partial ^{\beta }}{\partial t^{\beta }}\right] \varvec{\varPsi }_{t}-\varvec{\varPsi }_{t}^{*}\varvec{\varPsi }_{s}^{*} (-\mathcal {L})\varvec{\varPsi }_{s}\varvec{\varPsi }_{t}\right] ^{-1}\varvec{\varPsi }_{s}^{*}X_{0}\right] ^{\mathrm {T}}\nonumber \\&\times \left[ \left[ \varvec{\varPsi }_{t}^{*}\left[ \frac{\partial ^{\beta }}{\partial t^{\beta }}\right] \varvec{\varPsi }_{t}-\varvec{\varPsi }_{t}^{*}\varvec{\varPsi }_{s}^{*} (-\mathcal {L})\varvec{\varPsi }_{s}\varvec{\varPsi }_{t}\right] ^{-1}\varvec{\varPsi }_{s}^{*}X_{0}\right] . \end{aligned}$$
(15)

Let X be the mean-square solution (10) to Eq. (9), in the case where \(\mathcal {L}=(-\varDelta )^{\gamma /2},\) and \(X_{0}=(I-\varDelta )^{-\varrho /2}\varepsilon ,\) with \(\varepsilon \) representing spatial zero-mean Gaussian white noise. The empirical local temporal (\(\beta \)), spatial (\(\gamma \)), and random initial condition spatial \((\varrho )\) fractional exponents are computed from the empirical log-wavelet periodogram, evaluated with respect to each component, applying multiple linear regression. Tables 1, 2 and 3 provide the empirical local temporal, spatial and spatiotemporal increment exponents computed from the wavelet estimate of \((\beta ,\gamma ,\varrho ),\) considering different theoretical parameter values, and a fixed number of temporal and spatial resolution levels, i.e., \(M_{t}=4,\) and \(M_{s}=6,\) of the temporal and spatial Haar wavelet transform. Note that T.E.E denotes the local temporal empirical exponents and T.T.E represents the local temporal theoretical exponents in Table 1, reflecting mean-quadratic local variation of temporal increments of X. Similarly, S.E.E denotes the local spatial empirical exponents and S.T.E represents the local spatial theoretical exponents in Table 2, reflecting mean-quadratic local variation of spatial increments of X;  S-T.E.E denotes the local spatiotemporal empirical exponents and S-T.T.E represents the local spatiotemporal theoretical exponents in Table 3, reflecting mean-quadratic local variation of spatiotemporal increments of X.

Table 1 Empirical and theoretical local temporal fractional exponents from the empirical log-wavelet periodogram
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Table 2 Empirical and theoretical local spatial fractional exponents from the empirical log-wavelet periodogram
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Table 3 Empirical and theoretical local spatiotemporal fractional exponents from the empirical log-wavelet periodogram
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Fig. 1
figure 1

Theoretical (red color) and empirical (blue color) local fractional exponents in time (top-left), in space (top-right) and in time and space (bottom) (color figure online)

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6 Numerical examples

In this section, some numerical examples of covariance functions in the family introduced are shown. In all the cases studied, a fixed truncation order \(M=25,\) and a fixed square domain \(D=[0,L]\times [0,L]\) are considered with the purpose of studying the asymptotic approximation of these covariance functions at initial times close to zero, as well as at large times (\(t,s\rightarrow \infty \)). In the first case, the influence of the values of parameter \(\varrho \) is investigated. For large t,  the influence of \(\beta \) and \(\gamma \) on the local regularity properties of the elements of the covariance function family introduced, as well as on their velocity of decay to zero at the boundary \(\partial D,\) is also analyzed (Fig. 1).

Let us first consider the zero-mean Gaussian mean-square solution to equation

$$\begin{aligned} \frac{\partial ^{\beta }}{\partial t^{\beta }}X(t,\mathbf {z})=-(I-\varDelta _{\mathcal {D}} )^{\alpha /2}(-\varDelta _{\mathcal {D}} )^{\tau /2}X(t,\mathbf {z}),\quad \gamma =\alpha +\tau {\in } (1,2),\quad \beta \in (1/2,1),\nonumber \\ \end{aligned}$$
(16)

where \(-(I-\varDelta _{\mathcal {D}} )^{\alpha /2}(-\varDelta _{\mathcal {D}} )^{\tau /2}\) denotes a fractional polynomial function of the Dirichlet Laplacian operator on \(D=[0,L]\times [0,L]\subset R^{2},\) with \(L=20,\) and with

$$\begin{aligned} X(0,\mathbf {x})=\int _{D}K(\mathbf {x},\mathbf {y})\varepsilon (\mathbf {y})d\mathbf {y},\quad \mathbf {x} \in D\subset \mathbb {R}^{2}. \end{aligned}$$
(17)

Here, \(\varepsilon \) represents spatial Gaussian white noise with variance \(\sigma ^{2}=3/2,\) and

$$\begin{aligned} K(\mathbf {x},\mathbf {y})=\frac{1}{(\widetilde{\beta } +\Vert \mathbf {x}-\mathbf {y}\Vert ^{\widetilde{\gamma } })^{\widetilde{\alpha } }}. \end{aligned}$$
(18)

Note that the eigenvalues and eigenfunctions of the Dirichlet Laplacian operator on the square \([0,L]\times [0,L]\) are given by (see, for example, Grebenkov and Nguyen 2013)

$$\begin{aligned} \phi _{k_{1},k_{2}}(x_{1},x_{2})= & {} \sin \left( \frac{\pi (k_{1}+1)x_{1}}{L}\right) \sin \left( \frac{\pi (k_{2}+1)x_{2}}{L}\right) , \quad (k_{1},k_{2})\in \mathbb {N}_{*}^{2},\nonumber \\ \lambda _{k_{1},k_{2}}= & {} \lambda _{k_{1}}+\lambda _{k_{2}}, \quad \lambda _{k_{i}}=\frac{\pi ^{2}(k_{i}+1)^{2}}{ L^{2}},\quad i=1,2. \end{aligned}$$
(19)

The following parameter values are considered in the numerical example analyzed: for the random initial condition, \(\widetilde{\gamma }=2/4,\) \(\widetilde{\alpha }=4,\) \(\widetilde{\beta }=1,\) with \(\varrho =\widetilde{\gamma }+\widetilde{\alpha },\) and for the spatiotemporal pseudodifferential operator \(\frac{\partial ^{\beta }}{\partial t^{\beta }}+\mathcal {L}\) in (16), \(\beta =2/3,\) \(\alpha = 3/4,\) \(\tau =1/2.\) The generation of the realizations of the mean-square solution to Eq. (16) are shown in Fig. 2. The covariance function between the random variables of X located at 100 spatial nodes of the square \([0,10]\times [0,10],\) for small and large times, computed using respective negative exponential and rational approximations

$$\begin{aligned} E_{\beta }(-t^{\beta }) \left\{ \begin{array}{l} \sim \exp \left( -\frac{t^{\beta }}{\varGamma (1+\beta )}\right) ,\quad t\rightarrow 0, \\ \sim \frac{1}{t^{\beta }\varGamma (1-\beta )},\quad t\rightarrow \infty , \end{array} \right. \end{aligned}$$
(20)

when \(t\rightarrow 0,\) and when \(t\rightarrow \infty ,\) of the Mittag-Leffler function, are displayed in Fig. 3. Different cross-covariances between spatial locations in \([0,L]\times [0,L],\) \(L=20,\) and a specific spatial location in the interior of that square are also displayed, for different times, in Fig. 4.

Fig. 2
figure 2

Realizations of the solution to Eq. (16) for parameter values \(\widetilde{\gamma }=2/4,\) \(\widetilde{\alpha }=4,\) \(\widetilde{\beta }=1,\) and \(\beta =2/3,\) \(\alpha = 3/4,\) \(\tau =1/2,\) at times \(t=1/6\) (left) and \(t=1000\) (right)

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Fig. 3
figure 3

Spatiotemporal covariance function of the solution to Eq. (16), displayed in Fig. 2, at times \(t=1/6\) and \(s=1/6\) (top-left), \(t=3/6,\) and \(s=4/6\) (top-center), and \(t=1\) and \(s=1\) (top-right), and at times \(t=500\) and \(s=500\) (bottom-left), \(t=700\) and \(s=800\) (bottom-center), and \(t=1000\) and \(s=1000\) (bottom-right), between 100 spatial locations in the square \(D=[0,20]\times [0,20].\) The parameter values considered are the same as in Fig. 2

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Fig. 4
figure 4

Cross-covariances for the solution to Eq. (16) between the spatial locations in \([0,20]\times [0,20],\) and the node (1, 5) (top-left) for \((t,s)=(1/6,1/6),\) the node (20, 5) (top-center) for \((t,s)=(2/6,1/6),\) the node (15, 15) (top-right) for \((t,s)=(4/6,2/6),\) the node (1, 5) (bottom-left) for \((t,s)=(500,500),\) the node (20, 5) (top-center) for \((t,s)=(600,500),\) and the node (10, 20) (top-right) for \((t,s)=(800,600)\)

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Let us now consider as second example the following fractional in time and in space evolution equation given by

$$\begin{aligned} \frac{\partial ^{\beta }}{\partial t^{\beta }}X(t,\mathbf {z})=-(I-\varDelta _{\mathcal {D}} )^{\alpha /2}(-\varDelta _{\mathcal {D}} )^{-\tau /2}X(t,\mathbf {z}),\ \gamma =\alpha -\tau \in (1,2),\ \beta \in (1/2,1),\nonumber \\ \end{aligned}$$
(21)

with random initial condition \(X_{0}\) generated from Eqs. (17), and (18).

The following parameter values are considered: for the random initial condition, \(\widetilde{\gamma }=3/2,\) \(\widetilde{\alpha }=2/3,\) \(\widetilde{\beta }=1,\) with \(\varrho =\widetilde{\gamma }+\widetilde{\alpha },\) and for the spatiotemporal pseudodifferential operator \(\frac{\partial ^{\beta }}{\partial t^{\beta }}+\mathcal {L}\) in (21), \(\beta =6/10,\) \(\alpha = 2.55\) and \(\tau =3/2.\) In particular, two realizations of the mean-square solution to Eq. (21) are displayed in Fig. 5 for \(t\rightarrow 0,\) and for \(t\rightarrow \infty .\) The covariance function approximations are displayed in Fig. 6, for small and large times, considering 100 spatial locations in the square \([0,10]\times [0,10].\) Different cross-covariances between the spatial locations in \([0,L]\times [0,L],\) \(L=20,\) and a specific spatial location in the interior of that square are displayed, for different times, in Fig. 7.

Fig. 5
figure 5

Realizations of the solution to Eq. (21) for parameter values \(\widetilde{\gamma }=3/2,\) \(\widetilde{\alpha }=2/3,\) \(\widetilde{\beta }=1,\) and \(\beta =6/10,\) \(\alpha = 2.55,\) \(\tau =3/2,\) for \(t=5/6\) (left) and \(t=1000\) (right)

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Fig. 6
figure 6

Spatiotemporal covariance function of the solution to Eq. (21), displayed in Fig. 5, at times \(t=1/6\) and \(s=1/6\) (top-left), \(t=3/6\) and \(s=4/6\) (top-center), and \(t=1\) and \(s=1\) (top-right), and at times \(t=500\) and \(s=500\) (bottom-left), \(t=700\) and \(s=800\) (bottom-center), and \(t=1000\) and \(s=1000\) (bottom-right), between 100 spatial locations of \(D=[0,20]\times [0,20],\) for the parameter values considered in Fig. 5

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Fig. 7
figure 7

Cross-covariances for the solution to Eq. (21) between the spatial locations in \([0,20]\times [0,20],\) and the node (15, 15) (top-left) for \((t,s)=(4/6,2/6),\) the node (20, 5) (top-center) for \((t,s)=(2/6,1/6),\) the node (10, 20) (top-right) for \((t,s)=(4/6,2/6),\) the node (15, 15) (bottom-left) for \((t,s)=(800,600),\) the node (20, 5) (top-center) for \((t,s)=(600,500),\) and the node (10, 20) (top-right) for \((t,s)=(800,600)\)

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6.1 Main conclusions and comments

It can be observed in Fig. 6 that model (21) displays the slowest decay to zero, when we approximate to the boundary of \(D= [0,20]\times [0,20],\) of the spatiotemporal covariance function, for small and large times. Note that this model corresponds to the smallest value of parameter \(\beta \) and \(\gamma =\alpha -\tau .\) The influence of the fractal parameters \(\widetilde{\alpha }\) and \(\widetilde{\gamma }\) can be appreciated at small times in the models analyzed (see Figs. 2, 5). On the other hand, at large times, when parameters \(\beta ,\) \(\alpha \) and \(\tau \) increase, the fractality of the model also increases as it can be observed in Fig. 4, where the largest \(\alpha +\tau \) value is given in comparison with model (21), as reflected in Fig. 7.

The family of non-separable compactly supported covariance models introduced from Eqs. (9) and (11) displays a fractal behavior in space which can be affected by the fractality order \(\varrho \) of the random initial condition at times close to zero, as shown in the above examples. Moreover, for large times, the fractality order increases when parameters \(\beta \) and \(\gamma \) increase. They also present a continuous slow decay to zero at the boundary of the regular domain considered. As the parameters related to the fractional order of differentiation in time and in space decrease, the velocity of decay of the covariance function to zero at the boundary becomes slower. Summarizing, the asymptotic, in time, self-similar behavior of the covariance family analyzed leads to an increasing degree of fractality for large \(\beta \) and \(\gamma ,\) and a slower decay to zero at the boundary when \(\beta \) and \(\gamma \) approaches zero. Note also that the family of covariance functions analyzed constitutes a parametric family, depending on parameter vector \((\beta ;\gamma ;\varrho )\) (in the examples defined, from parameter vector \((\beta ; \alpha , \tau ;\widetilde{\alpha },\widetilde{\gamma },\widetilde{\beta })\)). The functional form of the elements of the family of spatiotemporal compactly supported non-separable covariance functions analyzed is subject to the general restrictions of the fractional in time and in space pseudodifferential equations considered, whose elliptic and continuous spatial pseudodifferential operator generates the corresponding non-local Dirichlet form asscoiated with the dual RF. However, the family is quite flexible, since it allows to fit different degrees of spatial and temporal local regularity/singularity, as well as of decay velocity to zero at the boundary, as it can be observed in the numerical examples shown.

In addition, the functional form of the covariance family introduced changes with the domain. Specifically, in the above numerical examples, we have used the eigenfunctions of the Dirichlet Laplacian operator on the rectangle, which are defined in terms of sine functions. However, the eigenfunctions of the Dirichlet Laplacian operator strongly depend on the geometry of the domain. For example, let us consider D to be a disk. That is, \(D=\{(x_{1},x_{2})\in \mathbb {R}^{2};\quad \Vert \mathbf {x}\Vert <R\}.\) The following fractional in time an in space evolution equation with Dirichlet boundary conditions on the disk can then be considered:

$$\begin{aligned} \frac{\partial ^{\beta }}{\partial t^{\beta }}X(t,\mathbf {x})= & {} -\mathcal {L}X(t, \mathbf {x}), \quad X(\mathbf {x})\underset{\text{ m.s }}{=}0,\quad \text{ for }\ \Vert \mathbf {x}\Vert =R, \nonumber \\ \mathbf {x}= & {} (x_{1},x_{2})\in D=\{(x_{1},x_{2})\in \mathbb {R}^{2};\quad |\mathbf {x}|<R\},\quad t\in \mathbb {R}_{+}. \end{aligned}$$
(22)

In polar coordinates, \(x_{1}=r\cos \varphi ,\) \(x_{2}=r\sin \varphi ,\) the Laplace operator \(\varDelta \) admits the expression \( \varDelta =\frac{\partial ^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial }{ \partial r}+\frac{1}{r^{2}}\frac{\partial ^{2}}{\partial \varphi ^{2}}.\) The fundamental solution (in polar coordinates) to Eq. (22) is then given by

$$\begin{aligned} G_{t-s}(r,\varphi ,r^{\prime },\varphi ^{\prime })=\sum _{n=0}^{\infty }\sum _{k=1}^{\infty }\sum _{l=1}^{2}E_{\beta }\left( -(t-s)^{\beta }\left( \alpha _{n,k}^{2}/R^{2}\right) \right) u_{n,k,l}(r,\varphi )u_{n,k,l}(r^{\prime },\varphi ^{\prime }), \end{aligned}$$

where

$$\begin{aligned} u_{n,k,l}(r,\varphi )=J_{n}(r\alpha _{n,k}/R)\times \left\{ \begin{array}{l} \cos (n\varphi ),\quad l=1 \\ \sin (n\varphi ),\quad l=2 \ (n\ne 0), \end{array} \right. \end{aligned}$$

with \(J_{n}\) and \(J_{n}^{\prime }\) being the Bessel function of the first kind and order n and its derivative, respectively, and the coefficients \(\{\alpha _{n,k}\}\) are set by the boundary conditions at \(r=0,\) and \(r=R.\) In particular, as particular example, consider \(\mathcal {L}=(-\varDelta )^{\gamma /2},\) with parameter values \(\beta =8/10,\) \(\gamma =11/10,\) \(\widetilde{\alpha } =0.8,\) \(\widetilde{\beta }=1,\) and \(\widetilde{\gamma }=3/2.\) Two realizations of the mean-square solution at time \(t=1/6\), its covariance function between a subset of 10 points in the interior of the disk of radius \(R=10,\) at times \(t=1/6\) and \(s=5/6,\) and the cross-covariance function between the points of \(D=\{\mathbf {x}: \Vert \mathbf {x}\Vert \le 20\},\) and the node (15, 15) in the interior of D,  for times \(t=4/6\) and \(s=2/6,\) are displayed in Fig. 8. We can see the influence of the geometry of the domain in the results displayed in Fig. 8. Hence, different parametric functional families with compact support can be introduced depending of the geometrical characteristics of the regular domain considered.

Fig. 8
figure 8

Two realizations of the solution to Eq. (22) for parameter values \(\widetilde{\gamma }=3/2,\) \(\widetilde{\alpha }=0.8,\) \(\widetilde{\beta }=1,\) and \(\beta =8/10,\) \(\gamma =11/10,\) for \(t=1/6\) (top-left and -right). Covariance function at \(t=1/6\) and \(s=5/6\) between 10 spatial locations in the interior of the disk of radius \(R=10,\) at times \(t=1/6\) and \(s=5/6\) (bottom-left). Cross-covariance function between 20 spatial locations of \(D=\{\mathbf {x}: \Vert \mathbf {x}\Vert \le 20\}\) and the node numbered as (15, 15) in the interior of D,  for times \(t=4/6\) and \(s=2/6\) (bottom-right)

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Summarizing, this paper opens a new research line in the introduction of new families of compactly supported covariance functions from the theory of fractional in time and in space stochastic partial differential equations. We understand here the concept of compactly supported covariance functions as the class of kernels that vanish in a continuous way outside the boundary of a bounded Dirichlet regular domain. The special case where such equations involve an elliptic bounded fractional spatial pseudodifferential operator is addressed. This case allows the connection with the theory of non-local Dirichlet forms through the duality condition introduced, and the concept of strong Feller semigroup. However, the results presented can be extended to a more general setting in terms of a wider class of fractional in time and in space evolution equations, beyond the duality condition and the ellipticity assumption on the spatial fractional pseudodifferential operator \(\mathcal {L}.\)