[{"@context":"https:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/blog.terabox.com\/insights\/stochastic-heat-inclusion-fractional-time-bernt-oksendal#BlogPosting","mainEntityOfPage":"https:\/\/blog.terabox.com\/insights\/stochastic-heat-inclusion-fractional-time-bernt-oksendal","headline":"Solving Stochastic Heat Inclusions with Fractional Time","name":"Solving Stochastic Heat Inclusions with Fractional Time","description":"\ud83d\udcfa Today’s recommended deep-dive video: https:\/\/www.youtube.com\/watch?v=NAx8kKh5iTM Modeling Uncertainty: Solving Stochastic Heat Inclusions with Fractional TimeThe Mathematics of UncertaintyBrownian and L\u00e9vy White Noise FrameworksMild Solutions and Real-World ApplicationsKey TakeawaysQ&A Modeling Uncertainty: Solving Stochastic Heat Inclusions with Fractional Time How do we... ","datePublished":"2026-08-20","dateModified":"2026-08-20","author":{"@type":"Person","@id":"https:\/\/blog.terabox.com\/author\/flextech-admin\/#Person","name":"flextech-admin","url":"https:\/\/blog.terabox.com\/author\/flextech-admin\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/ad516503a11cd5ca435acc9bb6523536?s=150&d=mm&r=gforcedefault=1","url":"https:\/\/secure.gravatar.com\/avatar\/ad516503a11cd5ca435acc9bb6523536?s=150&d=mm&r=gforcedefault=1","height":96,"width":96}},"publisher":{"@type":"Organization","name":"terabox","logo":{"@type":"ImageObject","@id":"http:\/\/blog.terabox.com\/wp-content\/uploads\/2021\/11\/logo\u4ea7\u54c1\u540d-\u7ad6\u7248.png","url":"http:\/\/blog.terabox.com\/wp-content\/uploads\/2021\/11\/logo\u4ea7\u54c1\u540d-\u7ad6\u7248.png","width":900,"height":900}},"image":{"@type":"ImageObject","@id":"https:\/\/blog.terabox.com\/wp-content\/uploads\/2026\/08\/mqdefault-13.jpg","url":"https:\/\/blog.terabox.com\/wp-content\/uploads\/2026\/08\/mqdefault-13.jpg","height":180,"width":320},"url":"https:\/\/blog.terabox.com\/insights\/stochastic-heat-inclusion-fractional-time-bernt-oksendal","video":{"@context":"http:\/\/schema.org\/","@type":"VideoObject","@id":"https:\/\/www.youtube.com\/watch?v=NAx8kKh5iTM#VideoObject","contentUrl":"https:\/\/www.youtube.com\/watch?v=NAx8kKh5iTM","name":"B.\u00d8ksendal:Stochastic heat incl with fractional time driven by time-space Brownian & Levy white nois","description":"Date: Friday, 13 June, 2025 - 16:00 to 17:00 CEST\nTitle : The stochastic heat inclusion with fractional time driven by time-space Brownian and Levy white noise\nSpeaker: Bernt \u00d8ksendal. Department of Mathematics, University of Oslo, Norway\n\nHosted at: SISSA, International School of Advanced Studies, Trieste, Italy\nOrganizers : Pavan Pranjivan Mehta*  and Arran Fernandez** \n* SISSA, International School of Advanced Studies, Italy\n** Eastern Mediterranean University, Northern Cyprus\n\nKeywords:  fractional stochastic heat inclusion, Caputo derivative, Mittag-Leffler function, time-space Brownian white noise, time-space Levy white noise, additive noise, tempered distributions, mild solution\n\nAbstract\n\nWe study a time-fractional stochastic heat inclusion driven by additive time-space Brownian and Levy white noise. The fractional time derivative is interpreted as the Caputo derivative of order \u03b1 \u2208 (0, 2). We show the following:\n\na) If a solution exists, then it is a fixed point of a specific set-valued map.\n\nb) Conversely, any fixed point of this map is a solution of the heat inclusion.\n\nc) Finally, we show that there is at least one fixed point of this map, thereby proving that there is at least one solution of the time-fractional stochastic heat inclusion.\n\nA solution Y (t, x) is called mild if E[Y 2(t, x)] less than \u221e for all t, x. We show that the solution is mild if \u03b1 = 1 & d = 1, or \u03b1 \u2265 1 & d \u2208 {1, 2}.\n\nOn the other hand, if \u03b1 less than 1 we show that the solution is not mild for any space dimension d.\n\nBiography\n\nBernt Karsten \u00d8ksendal is a Norwegian mathematician with several years of teaching and research in stochastic analysis. He completed his undergraduate studies at the University of Oslo in 1970, and obtained his PhD from the University of California, Los Angeles (UCLA) in 1971. In 1991, he was appointed a Professor at the University of Oslo. In 1992, he was appointed an Adjunct Professor at the Norwegian School of Economics and Business Administration, Bergen, Norway. In 2017, he was appointed an Honorary Doctor at the Norwegian School of Economics.\n\nBetween 1992 and 1996, he held a research position as VISTA Professor, appointed by the Norwegian Academy of Science and Letters in cooperation with Den Norske Stats Oljeselskap a.s. (Statoil). In 1996, he was elected as a member of the Norwegian Academy of Science and Letters, and in the same year he won the Nansen Prize for his research in stochastic analysis and its applications. In 2002, he was elected as a member of the Royal Norwegian Science Society. In 2014, he was awarded the University of Oslo Research Prize for excellent research. He has also been chair or coordinator of many research grants and international research programmes (in both Europe and Africa) over the years.\n\nBibliography\n\n[1] Abel, N. H. (1823): Oppl\u00f8sning av et par oppgaver ved hjelp av bestemte integraler (in Norwegian). Magazin for naturvidenskaberne 55\u201368.\n[2] Adler, R.J., Monrad, D., Scissors, R.H. & Wilson, R. (1983): Representations, de-compositions and sample function continuity of random fields with independent increments. Stoch. Process. Appl. 15(1),3-30.\n[3] Aumann, R. J. (1965): Integrals of set-valued functions. J. Math.Anal. 12, 1-12.\n[4] Capelas de Oliveira, E.; Mainardi & F.Vaz, J. (2011): Models based on Mittag Leffler functions for anomalous relaxation in dielectrics. ariXir: 1106.1761 v2[cond-mat.Stat-mech] 13 Feb. 2014.\n[5] Dalang, R. C. & Walsh, J. B. (1992): The sharp Markov property of L\u00b4evy sheets. The Annals of Probability 20 (2), 591-626.\n[6] Di Nunno, G., \u00d8ksendal, B. & Proske, F. (2009): Malliavin Calculus for L\u00b4evy Processes with Applications to Finance. Springer.\n[7] Hu, Y. (2019): Some recent progress on stochastic heat equations. Acta Mathematica Scientia 39B(3); 874-914.\n[8] Holm, S. (2019): Waves with Power-Law Attenuation. Springer.\n[9] H. Holden, B. Oksendal, J. Ub\u00f8e, and T. Zhang. Stochastic Partial Differential Equations. Birkh\u00a8auser Boston Inc., Boston, MA, 1996.\n[10] Ibe, O. C. (2013): Markov Processes for Stochastic Modelling. 2nd edition. Elsevier.\n[11] Iafrate, F. & Ricciutu, C.(2024): Some families of random fileds related to multiparameter L\u00b4evy processes. J. Theoretical Probability 37: 3055-3088. https:\/\/doi.org\/10.1007\/s10959-024-01351-3.\n[12] I.M. Gel\u2019fand and N.Ya. Vilenkin. Generalized Functions. Vol. 4. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1964 [1977].\n[13] Kochubel, A. N., Kondratiev, Y. & da Silva, J. L. (2021): On fractional heat equation. Fractional Calculus & Applied Analysis 24 (1), 73-87.\n[14] Meerschaert, Mark M., Sikorskii, Alla (2019): Stochastic Models for Fractional Calculus, 2nd edition. De Greuter.\n\nFull Bibliography : https:\/\/mathlab.sissa.it\/stochastic-heat-inclusion-fractional-time-driven-time-space-brownian-and-levy-white-noise","thumbnailUrl":["https:\/\/i.ytimg.com\/vi\/NAx8kKh5iTM\/default.jpg","https:\/\/i.ytimg.com\/vi\/NAx8kKh5iTM\/mqdefault.jpg","https:\/\/i.ytimg.com\/vi\/NAx8kKh5iTM\/hqdefault.jpg","https:\/\/i.ytimg.com\/vi\/NAx8kKh5iTM\/sddefault.jpg"],"uploadDate":"2025-06-13T16:18:33+00:00","duration":"PT58M41S","embedUrl":"https:\/\/www.youtube.com\/embed\/NAx8kKh5iTM","publisher":{"@type":"Organization","@id":"https:\/\/www.youtube.com\/channel\/UC5x8tj09BHqJ5A2xlTGAOsQ#Organization","url":"https:\/\/www.youtube.com\/channel\/UC5x8tj09BHqJ5A2xlTGAOsQ","name":"Fractional Calculus Seminars @ SISSA","description":"Fractional calculus is a generalised form of the integer-order calculus. While an integer-order derivative is a local operator, a fractional derivative is a non-local operator. The notion of Brownian motion is extended to admit Levy stable processes in the case of fractional diffusion. Many different operators have been described as fractional derivatives and integrals, with different properties, and there are also further generalizations within non-local calculus. Real-world applications of non-local models can be found in turbulence, economics, electrical circuits, etc. Until recently, the theory and applications of fractional operators did not receive attention, so that many questions remain unanswered.\n\nIt is intended to touch all aspects of fractional calculus, from mathematics to simulations to applications. \n\nOrganizer: P. Pranjivan Mehta* and A. Fernandez**\nSISSA Liaison: G. Rozza*\n* International School of Advanced Studies, Italy\n** Eastern Mediterranean University, N. 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