Regularity theory for a new class of fractional parabolic stochastic evolution equations. (English) Zbl 07914050
Summary: A new class of fractional-order parabolic stochastic evolution equations of the form \((\partial_t + A)^\gamma X(t) = {\dot{W}}^Q(t)\), \(t\in [0,T]\), \(\gamma \in (0,\infty )\), is introduced, where \(-A\) generates a \(C_0\)-semigroup on a separable Hilbert space \(H\) and the spatiotemporal driving noise \({\dot{W}}^Q\) is the formal time derivative of an \(H\)-valued cylindrical \(Q\)-Wiener process. Mild and weak solutions are defined; these concepts are shown to be equivalent and to lead to well-posed problems. Temporal and spatial regularity of the solution process \(X\) are investigated, the former being measured by mean-square or pathwise smoothness and the latter by using domains of fractional powers of \(A\). In addition, the covariance of \(X\) and its long-time behavior are analyzed. These abstract results are applied to the cases when \(A:= L^\beta\) and \(Q:={\widetilde{L}}^{-\alpha}\) are fractional powers of symmetric, strongly elliptic second-order differential operators defined on (i) bounded Euclidean domains or (ii) smooth, compact surfaces. In these cases, the Gaussian solution processes can be seen as generalizations of merely spatial (Whittle-)Matérn fields to space-time.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 35R11 | Fractional partial differential equations |
Keywords:
spatiotemporal Gaussian processes; Matérn covariance; nonlocal space-time differential operators; mild solution; mean-square differentiability; strongly continuous semigroupsSoftware:
DLMFReferences:
| [1] | Abatangelo, N.; Dupaigne, L., Nonhomogeneous boundary conditions for the spectral fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34, 2, 439-467, 2017 · Zbl 1372.35327 |
| [2] | Adams, RA; Fournier, JJF, Sobolev Spaces, Pure and Applied Mathematics (Amsterdam), 2003, Amsterdam: Elsevier/Academic Press, Amsterdam |
| [3] | Alexeeff, SE; Nychka, D.; Sain, SR; Tebaldi, C., Emulating mean patterns and variability of temperature across and within scenarios in anthropogenic climate change experiments, Clim. Change, 146, 3, 319-333, 2018 |
| [4] | Angulo, JM; Kelbert, MY; Leonenko, NN; Ruiz-Medina, MD, Spatiotemporal random fields associated with stochastic fractional Helmholtz and heat equations, Stoch. Environ. Res. Risk Assess., 22, suppl. 1, 3-13, 2008 · Zbl 1177.60052 |
| [5] | Antil, H.; Pfefferer, J.; Rogovs, S., Fractional operators with inhomogeneous boundary conditions: analysis, control, and discretization, Commun. Math. Sci., 16, 5, 1395-1426, 2018 · Zbl 06996271 |
| [6] | Bakka, H.; Vanhatalo, J.; Illian, JB; Simpson, D.; Rue, H., Non-stationary Gaussian models with physical barriers, Spat. Stat., 29, 268-288, 2019 |
| [7] | Beguin, J.; Fuglstad, GA; Mansuy, N.; Paré, D., Predicting soil properties in the Canadian boreal forest with limited data: comparison of spatial and non-spatial statistical approaches, Geoderma, 306, 195-205, 2017 |
| [8] | Bertino, L.; Evensen, G.; Wackernagel, H., Sequential data assimilation techniques in oceanography, Int. Stat. Rev., 71, 2, 223-241, 2003 · Zbl 1114.62364 |
| [9] | Biswas, A.; De León-Contreras, M.; Stinga, PR, Harnack inequalities and Hölder estimates for master equations, SIAM J. Math. Anal., 53, 2, 2319-2348, 2021 · Zbl 1462.35103 |
| [10] | Biswas, A.; Stinga, PR, Regularity estimates for nonlocal space-time master equations in bounded domains, J. Evol. Equ., 21, 1, 503-565, 2021 · Zbl 1464.35392 |
| [11] | Bolin, D.; Kirchner, K., The rational SPDE approach for Gaussian random fields with general smoothness, J. Comput. Graph. Stat., 29, 2, 274-285, 2020 · Zbl 07499255 |
| [12] | Bolin, D.; Kirchner, K., Equivalence of measures and asymptotically optimal linear prediction for Gaussian random fields with fractional-order covariance operators, Bernoulli, 29, 2, 1476-1504, 2023 · Zbl 07666827 |
| [13] | Bolin, D.; Kirchner, K.; Kovács, M., Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise, BIT, 58, 4, 881-906, 2018 · Zbl 1405.65147 |
| [14] | Bolin, D.; Kirchner, K.; Kovács, M., Numerical solution of fractional elliptic stochastic PDEs with spatial white noise, IMA J. Numer. Anal., 40, 2, 1051-1073, 2020 · Zbl 1466.65181 |
| [15] | Bolin, D.; Lindgren, F., Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping, Ann. Appl. Stat., 5, 1, 523-550, 2011 · Zbl 1235.60075 |
| [16] | Bolin, D.; Lindgren, F., A comparison between Markov approximations and other methods for large spatial data sets, Comput. Stat. Data Anal., 61, 7-21, 2013 · Zbl 1349.62445 |
| [17] | Bonaccorsi, S., Fractional stochastic evolution equations with Lévy noise, Differ. Integr. Equ., 22, 11-12, 1141-1152, 2009 · Zbl 1240.60193 |
| [18] | Cameletti, M.; Lindgren, F.; Simpson, D.; Rue, H., Spatio-temporal modeling of particulate matter concentration through the SPDE approach, AStA Adv. Stat. Anal., 97, 2, 109-131, 2013 · Zbl 1443.62401 |
| [19] | Carrizo Vergara, R.; Allard, D.; Desassis, N., A general framework for SPDE-based stationary random fields, Bernoulli, 28, 1, 1-32, 2022 · Zbl 07467712 |
| [20] | Cox, S.G., Hutzenthaler, M., Jentzen, A.: Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations. Preprint arXiv:1309.5595v3 (2021) · Zbl 07853548 |
| [21] | Cox, SG; Kirchner, K., Regularity and convergence analysis in Sobolev and Hölder spaces for generalized Whittle-Matérn fields, Numer. Math., 146, 4, 819-873, 2020 · Zbl 1455.65210 |
| [22] | Cressie, N.; Huang, HC, Classes of nonseparable, spatio-temporal stationary covariance functions, J. Am. Stat. Assoc., 94, 448, 1330-1340, 1999 · Zbl 0999.62073 |
| [23] | Cressie, N.; Wikle, CK, Statistics for Spatio-temporal Data. Wiley Series in Probability and Statistics, 2011, Hoboken: Wiley, Hoboken · Zbl 1273.62017 |
| [24] | Da Prato, G.; Kwapień, S.; Zabczyk, J., Regularity of solutions of linear stochastic equations in Hilbert spaces, Stochastics, 23, 1, 1-23, 1987 · Zbl 0634.60053 |
| [25] | Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, 2014, Cambridge: Cambridge University Press, Cambridge · Zbl 1317.60077 |
| [26] | Dautray, R.; Lions, JL, Mathematical Analysis and Numerical Methods for Science and Technology, 1992, Berlin: Springer, Berlin · Zbl 0755.35001 |
| [27] | Davies, EB, Spectral Theory and Differential Operators. Cambridge Studies in Advanced Mathematics, 1995, Cambridge: Cambridge University Press, Cambridge · Zbl 0893.47004 |
| [28] | De Iaco, S.; Palma, M.; Posa, D., Spatio-temporal geostatistical modeling for French fertility predictions, Spat. Stat., 14, part B, 546-562, 2015 |
| [29] | Denk, R.; Hieber, M.; Prüss, J., \({\mathscr{R}} \)-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Am. Math. Soc., 166, 788, viii+114, 2003 · Zbl 1274.35002 |
| [30] | Desch, W., Londen, S.O.: Semilinear stochastic integral equations in \(L_p\). In: Parabolic Problems. Progress in Nonlinear Differential Equations and Their Application, vol. 80, pp. 131-166. Birkhäuser/Springer Basel AG, Basel (2011) · Zbl 1272.60047 |
| [31] | Engel, KJ; Nagel, R., One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, 2000, New York: Springer, New York · Zbl 0952.47036 |
| [32] | Evans, LC, Partial Differential Equations. Graduate Studies in Mathematics, 2010, Providence: American Mathematical Society, Providence · Zbl 1194.35001 |
| [33] | Fernández-Avilés, G.; Montero, JM, Spatio-temporal modeling of financial maps from a joint multidimensional scaling-geostatistical perspective, Expert Syst. Appl., 60, 280-293, 2016 |
| [34] | Fuentes, M.; Chen, L.; Davis, JM, A class of nonseparable and nonstationary spatial temporal covariance functions, Environmetrics, 19, 5, 487-507, 2008 |
| [35] | Fuglstad, GA; Lindgren, F.; Simpson, D.; Rue, HV, Exploring a new class of non-stationary spatial Gaussian random fields with varying local anisotropy, Stat. Sin., 25, 1, 115-133, 2015 · Zbl 1480.62194 |
| [36] | Gneiting, T., Nonseparable, stationary covariance functions for space-time data, J. Am. Stat. Assoc., 97, 458, 590-600, 2002 · Zbl 1073.62593 |
| [37] | Haase, M., The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications, 2006, Basel: Birkhäuser Verlag, Basel · Zbl 1101.47010 |
| [38] | Handcock, MS; Wallis, JR, An approach to statistical spatial-temporal modeling of meteorological fields, J. Am. Stat. Assoc., 89, 426, 368-390, 1994 · Zbl 0798.62109 |
| [39] | Harbrecht, H., Herrmann, L., Kirchner, K., Schwab, Ch.: Multilevel approximation of Gaussian random fields: covariance compression, estimation and spatial prediction. Preprint arXiv:2103.04424v1 (2021) |
| [40] | Herrmann, L.; Kirchner, K.; Schwab, Ch, Multilevel approximation of Gaussian random fields: fast simulation, Math. Models Methods Appl. Sci., 30, 1, 181-223, 2020 · Zbl 1448.60110 |
| [41] | Hytönen, T.; van Neerven, J.; Veraar, M.; Weis, L., Analysis in Banach Spaces. Vol. I. Martingales and Littlewood-Paley Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 2016, Cham: Springer, Cham · Zbl 1366.46001 |
| [42] | Hytönen, T.; van Neerven, J.; Veraar, M.; Weis, L., Analysis in Banach Spaces. Vol. II. Probabilistic Methods and Operator Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 2017, Cham: Springer, Cham · Zbl 1402.46002 |
| [43] | Jost, G.; Heuvelink, GBM; Papritz, A., Analysing the space-time distribution of soil water storage of a forest ecosystem using spatio-temporal kriging, Geoderma, 128, 3, 258-273, 2005 |
| [44] | Kelbert, MY; Leonenko, NN; Ruiz-Medina, MD, Fractional random fields associated with stochastic fractional heat equations, Adv. Appl. Probab., 37, 1, 108-133, 2005 · Zbl 1102.60049 |
| [45] | Kirchner, K.; Lang, A.; Larsson, S., Covariance structure of parabolic stochastic partial differential equations with multiplicative Lévy noise, J. Differ. Equ., 262, 12, 5896-5927, 2017 · Zbl 1375.60107 |
| [46] | Kovács, M., Larsson, S., Urban, K.: On wavelet-Galerkin methods for semilinear parabolic equations with additive noise. In: Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer Proceedings in Mathematics and Statistics, vol. 65, pp. 481-499. Springer, Heidelberg (2013) · Zbl 1302.65028 |
| [47] | Kunstmann, P.C., Weis, L.: Maximal \(L_p\)-regularity for parabolic equations, Fourier multiplier theorems and \(H^\infty \)-functional calculus. In: Functional Analytic Methods for Evolution Equations. Lecture Notes in Mathematics, vol. 1855, pp. 65-311. Springer, Berlin (2004) · Zbl 1097.47041 |
| [48] | Lawson, AB, Hierarchical modeling in spatial epidemiology, Wiley Interdiscip. Rev. Comput. Stat., 6, 6, 405-417, 2014 · Zbl 07912744 |
| [49] | Lindgren, F., Bakka, H., Bolin, D., Krainski, E., Rue, H.: A diffusion-based spatio-temporal extension of Gaussian Matérn fields. Preprint arXiv:2006.04917v3 (2023) · Zbl 1542.62121 |
| [50] | Lindgren, F., Rue, H., Lindström, J.: An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. J. R. Stat. Soc. Ser. B Stat. Methodol. 73(4), 423-498 (2011). With discussion and a reply by the authors · Zbl 1274.62360 |
| [51] | Litsgård, M., Nyström, K.: On local regularity estimates for fractional powers of parabolic operators with time-dependent measurable coefficients. J. Evol. Equ. 23(1), Paper No. 3, 33 (2023) · Zbl 1504.35094 |
| [52] | Liu, W.; Röckner, M., Stochastic Partial Differential Equations: An Introduction. Universitext, 2015, Cham: Springer, Cham · Zbl 1361.60002 |
| [53] | Martínez Carracedo, C.; Sanz Alix, M., The Theory of Fractional Powers of Operators. North-Holland Mathematics Studies, 2001, Amsterdam: North-Holland, Amsterdam · Zbl 0971.47011 |
| [54] | Matérn, B.: Spatial variation: stochastic models and their application to some problems in forest surveys and other sampling investigations. Meddelanden Från Statens Skogsforskningsinstitut, Band 49, Nr. 5, Stockholm (1960) |
| [55] | Mateu, J.; Porcu, E.; Gregori, P., Recent advances to model anisotropic space-time data, Stat. Methods Appl., 17, 2, 209-223, 2008 · Zbl 1184.62197 |
| [56] | Mejia, AF; Yue, Y.; Bolin, D.; Lindgren, F.; Lindquist, MA, A Bayesian general linear modeling approach to cortical surface fMRI data analysis, J. Am. Stat. Assoc., 115, 530, 501-520, 2020 · Zbl 1445.62283 |
| [57] | Nyström, K.; Sande, O., Extension properties and boundary estimates for a fractional heat operator, Nonlinear Anal., 140, 29-37, 2016 · Zbl 1381.35230 |
| [58] | Oberhettinger, F.; Badii, L., Tables of Laplace Transforms, 1973, New York: Springer, New York · Zbl 0285.65079 |
| [59] | Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST handbook of mathematical functions. U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge (2010) · Zbl 1198.00002 |
| [60] | Ouhabaz, EM, Analysis of Heat Equations on Domains. London Mathematical Society Monographs Series, 2005, Princeton: Princeton University Press, Princeton · Zbl 1082.35003 |
| [61] | Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 1983, New York: Springer, New York · Zbl 0516.47023 |
| [62] | Pereira, S., Turkman, K.F., Correia, L., Rue, H.: Unemployment estimation: spatial point referenced methods and models. Spat. Stat. 41, Paper No. 100345 (2021) |
| [63] | Peszat, S.; Zabczyk, J., Stochastic Partial Differential Equations with Lévy Noise. Encyclopedia of Mathematics and its Applications, 2007, Cambridge: Cambridge University Press, Cambridge · Zbl 1205.60122 |
| [64] | Porcu, E., Furrer, R., Nychka, D.: 30 years of space-time covariance functions. Wiley Interdiscip. Rev. Comput. Stat. 13(2), e1512, 24 (2021) · Zbl 07910735 |
| [65] | Porcu, E.; Mateu, J.; Bevilacqua, M., Covariance functions that are stationary or nonstationary in space and stationary in time, Stat. Neerl., 61, 3, 358-382, 2007 · Zbl 1121.62082 |
| [66] | Sang, H.; Jun, M.; Huang, JZ, Covariance approximation for large multivariate spatial data sets with an application to multiple climate model errors, Ann. Appl. Stat., 5, 4, 2519-2548, 2011 · Zbl 1234.62071 |
| [67] | Sigrist, F.; Künsch, HR; Stahel, WA, Stochastic partial differential equation based modelling of large space-time data sets, J. R. Stat. Soc. Ser. B Stat. Methodol., 77, 1, 3-33, 2015 · Zbl 1414.62405 |
| [68] | Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton (1970) · Zbl 0207.13501 |
| [69] | Stein, ML, Interpolation of Spatial Data. Springer Series in Statistics, 1999, New York: Springer, New York · Zbl 0924.62100 |
| [70] | Stein, ML, Space-time covariance functions, J. Am. Stat. Assoc., 100, 469, 310-321, 2005 · Zbl 1117.62431 |
| [71] | Stinga, PR; Torrea, JL, Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation, SIAM J. Math. Anal., 49, 5, 3893-3924, 2017 · Zbl 1386.35419 |
| [72] | Taylor, M.E.: Pseudodifferential Operators. Princeton Mathematical Series, No. 34. Princeton University Press, Princeton (1981) · Zbl 0453.47026 |
| [73] | Whittle, P., Stochastic processes in several dimensions, Bull. Inst. Int. Stat., 40, 974-994, 1963 · Zbl 0129.10603 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
