{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:50:13Z","timestamp":1760233813286,"version":"build-2065373602"},"reference-count":47,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,2,26]],"date-time":"2021-02-26T00:00:00Z","timestamp":1614297600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study spatial moduli of non-differentiability for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by space-time white noise. We use the underlying explicit kernels and spectral\/harmonic analysis, yielding spatial moduli of non-differentiability for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. On the other hand, it builds on and complements Allouba and Xiao\u2019s earlier works on spatial uniform and local moduli of continuity of time fractional SPIDEs and their gradient.<\/jats:p>","DOI":"10.3390\/sym13030380","type":"journal-article","created":{"date-parts":[[2021,2,26]],"date-time":"2021-02-26T06:47:20Z","timestamp":1614322040000},"page":"380","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Spatial Moduli of Non-Differentiability for Time-Fractional SPIDEs and Their Gradient"],"prefix":"10.3390","volume":"13","author":[{"given":"Wensheng","family":"Wang","sequence":"first","affiliation":[{"name":"School of Economics, Hangzhou Dianzi University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1250012","DOI":"10.1142\/S0219493712500128","article-title":"Interacting time-fractional and \u0394\u03bd PDEs systems via Brownian-time and Inverse-stable-L\u00e9vy-time Brownian sheets","volume":"13","author":"Allouba","year":"2013","journal-title":"Stoch. Dyn."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3915","DOI":"10.1090\/S0002-9947-09-04678-9","article-title":"Brownian subordinators and fractional Cauchy problems","volume":"361","author":"Baeumer","year":"2009","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.spa.2005.07.003","article-title":"The exit distribution of iterated Brownian motion in cones","volume":"116","author":"DeBlassie","year":"2006","journal-title":"Stoch. Process. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1975","DOI":"10.1016\/j.spa.2008.10.001","article-title":"Iterated elastic Brownian motions and fractional diffusion equations","volume":"119","author":"Beghin","year":"2009","journal-title":"Stoch. Process. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1111\/j.1365-246X.1967.tb02303.x","article-title":"Linear models of dissipation whose Q is almost frequency independent. Part II","volume":"13","author":"Caputo","year":"1967","journal-title":"Geophys. J. R. Astr. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1080\/14697688.2010.531042","article-title":"A PDE approach to jump-diffusions","volume":"11","author":"Carr","year":"2011","journal-title":"Quant. Financ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1529","DOI":"10.1214\/105051604000000404","article-title":"Iterated Brownian motion in an open set","volume":"14","author":"DeBlassie","year":"2004","journal-title":"Ann. Appl. Probab."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"021122","DOI":"10.1103\/PhysRevE.77.021122","article-title":"Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation","volume":"77","author":"Fulger","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"176","DOI":"10.3792\/pjaa.55.176","article-title":"Probabilistic construction of the solution of some higher order parabolic differential equation","volume":"55","author":"Funaki","year":"1979","journal-title":"Proc. Jpn. Acad. Ser. A Math. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"066102","DOI":"10.1103\/PhysRevE.79.066102","article-title":"Stochastic calculus for uncoupled continuous-time random walks","volume":"79","author":"Germano","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1007\/3-540-44832-2_8","article-title":"Fractional diffusion processes: Probability distribution and continuous time random walk","volume":"621","author":"Gorenflo","year":"2003","journal-title":"Lect. Notes Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1115\/1.1478062","article-title":"Application of fractional calculus to fluid mechanics","volume":"124","author":"Kulish","year":"2002","journal-title":"J. Fluids Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1317","DOI":"10.1016\/0362-546X(88)90080-6","article-title":"An inverse problem in the theory of materials with memory","volume":"12","author":"Lunardi","year":"1988","journal-title":"Nonlinear Anal."},{"key":"ref_14","first-page":"153","article-title":"The fundamental solution of the space-time fractional diffusion equation","volume":"4","author":"Mainardi","year":"2001","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"041103","DOI":"10.1103\/PhysRevE.65.041103","article-title":"Stochastic solution of space-time fractional diffusion equations","volume":"65","author":"Meerschaert","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3301","DOI":"10.1016\/j.spa.2015.04.008","article-title":"Space-time fractional stochastic partial differential equations","volume":"125","author":"Mijena","year":"2015","journal-title":"Stoch. Process. Appl."},{"key":"ref_17","first-page":"10","article-title":"Hitting probabilities of a random string, Electron","volume":"7","author":"Mueller","year":"2002","journal-title":"J. Probab."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1214\/ECP.v17-1774","article-title":"Erratum: A connection between the stochastic heat equation and fractional Brownian motion and a simple proof of a result of Talagrand","volume":"17","author":"Mueller","year":"2012","journal-title":"Electron. Commun. Probab."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1214\/08-AOP401","article-title":"Fractional diffusion equations and processes with randomly varying time","volume":"37","author":"Orsingher","year":"2009","journal-title":"Ann. Probab."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4627","DOI":"10.1090\/S0002-9947-02-03074-X","article-title":"Brownian-time processes: The PDE connection II and the corresponding Feynman-Kac formula","volume":"354","author":"Allouba","year":"2002","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1780","DOI":"10.1214\/aop\/1015345772","article-title":"Brownian-time processes: The PDE connection and the half-derivative generator","volume":"29","author":"Allouba","year":"2001","journal-title":"Ann. Probab."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1009","DOI":"10.1080\/07362994.2014.962046","article-title":"Time-changed processes governed by space-time fractional telegraph equations","volume":"32","author":"Orsingher","year":"2014","journal-title":"Stoch. Anal. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"093301","DOI":"10.1063\/1.4931477","article-title":"Fractional diffusions with time-varying coefficients","volume":"56","author":"Garra","year":"2015","journal-title":"J. Math. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1239\/jap\/1091543414","article-title":"Limit theorems for continuous time random walks with infinite mean waiting times","volume":"41","author":"Meerschaert","year":"2004","journal-title":"J. Appl. Probab."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1470","DOI":"10.1016\/j.spa.2014.11.005","article-title":"Fractional time stochastic partial differential equations","volume":"125","author":"Chen","year":"2015","journal-title":"Stoch. Process. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Meerschaert, M.M., and Sikorskii, A. (2012). Stochastic Models for Fractional Calculus, de Gruyter Stud. Math. 43, De Gruyter.","DOI":"10.1515\/9783110258165"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Temam, R. (1997). Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4612-0645-3"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Tudor, C.A. (2013). Analysis of Variations for Self-Similar Processes-A Stochastic Calculus Approach, Springer.","DOI":"10.1007\/978-3-319-00936-0"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1750004","DOI":"10.1142\/S0219493717500046","article-title":"Sample path properties of the solution to the fractional-colored stochastic heat equation","volume":"17","author":"Tudor","year":"2017","journal-title":"Stoch. Dyn."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"919","DOI":"10.1215\/ijm\/1417442557","article-title":"Time-fractional and memoryful \u03942k SIEs on R+ \u00d7 Rd: How far can we push white noise?","volume":"57","author":"Allouba","year":"2013","journal-title":"Illinois J. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"413","DOI":"10.3934\/dcds.2013.33.413","article-title":"Brownian-time Brownian motion SIEs on R+ \u00d7 Rd: Ultra regular direct and lattice-limits solutions and fourth order SPDEs links","volume":"33","author":"Allouba","year":"2013","journal-title":"Discrete Contin. Dyn. Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1142\/S0219493706001864","article-title":"A Brownian-time excursion into fourth-order PDEs, linearized Kuramoto-Sivashinsky, and BTPSPDEs on R+ \u00d7 Rd","volume":"6","author":"Allouba","year":"2006","journal-title":"Stoch. Dyn."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1007\/BFb0094215","article-title":"The level sets of iterated Brownian motion","volume":"1613","author":"Burdzy","year":"1995","journal-title":"S\u00c9Minaire Probab. XXIX Lect. Notes Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1007\/BF01192468","article-title":"The Brownian snake and solutions of \u0394u = u2 in a domain","volume":"102","year":"1995","journal-title":"Probab. Theory Relat. Fields"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/S0764-4442(00)01625-6","article-title":"SPDEs law equivalence and the compact support property: Applications to the Allen-Cahn SPDE","volume":"331","author":"Allouba","year":"2000","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. I Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"6851","DOI":"10.1016\/j.jde.2015.08.033","article-title":"L-Kuramoto-Sivashinsky SPDEs in one-to-three dimensions: L-KS kernel, sharp H\u00f6lder regularity, and Swift-Hohenberg law equivalence","volume":"259","author":"Allouba","year":"2015","journal-title":"J. Differ. Equ."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/S0764-4442(00)00190-7","article-title":"Uniqueness in law for the Allen-Cahn SPDE via change of measure","volume":"330","author":"Allouba","year":"2000","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. I Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1080\/07362999808809562","article-title":"Different types of SPDEs in the eyes of Girsanov\u2019s theorem","volume":"16","author":"Allouba","year":"1998","journal-title":"Stoch. Anal. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"15521610","DOI":"10.1016\/j.jde.2017.03.027","article-title":"L-Kuramoto-Sivashinsky SPDEs v.s. time-fractional SPIDEs: Exact continuity and gradient moduli, 1\/2-derivative criticality, and laws","volume":"263","author":"Allouba","year":"2017","journal-title":"J. Differ. Equ."},{"key":"ref_40","first-page":"1","article-title":"Limsup random fractals","volume":"5","author":"Khoshnevisan","year":"2000","journal-title":"Electr. J. Probab."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/S0304-4149(03)00084-X","article-title":"A Gaussian correlation inequality and its applications to the existence of small ball constant","volume":"107","author":"Shao","year":"2003","journal-title":"Stoch. Process. Appl."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press.","DOI":"10.1142\/9781848163300"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"979","DOI":"10.1214\/08-AOP426","article-title":"Fractional Cauchy problems on bounded domains","volume":"37","author":"Meerschaert","year":"2009","journal-title":"Ann. Probab."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1090\/S0002-9947-2012-05678-9","article-title":"Fernique type inequality and moduli of continuity for anisotropic Gaussian random fields","volume":"365","author":"Meerschaert","year":"2013","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_45","unstructured":"Lai, T.L., Shao, Q.M., and Qian, L. (2007). Strong local nondeterminism and the sample path properties of Gaussian random fields. Asymptotic Theory in Probability and Statistics with Applications, Higher Education Press."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1410","DOI":"10.3150\/19-BEJ1162","article-title":"The moduli of non-differentiability of Gaussian random fields with stationary increments","volume":"26","author":"Wang","year":"2020","journal-title":"Bernoulli"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.spl.2019.02.016","article-title":"The Cs\u00f6rgo-R\u00e9v\u00e9sz moduli of non-differentiability of fractional Brownian motion","volume":"150","author":"Wang","year":"2019","journal-title":"Stat. Probab. Lett."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/3\/380\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:29:12Z","timestamp":1760160552000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/3\/380"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,26]]},"references-count":47,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["sym13030380"],"URL":"https:\/\/doi.org\/10.3390\/sym13030380","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,2,26]]}}}