Speaker: Bernt Øksendal. Department of Mathematics, University of Oslo, Norway

Time : 16:00 - 17.00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

YouTube :  https://youtu.be/NAx8kKh5iTM

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords:  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

Abstract: We 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 α ∈ (0, 2). We show the following:

a) If a solution exists, then it is a fixed point of a specific set-valued map.

b) Conversely, any fixed point of this map is a solution of the heat inclusion.

c) 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.

A solution Y (t, x) is called mild if E[Y 2(t, x)] < ∞ for all t, x. We show that the solution is mild if α = 1 & d = 1, or α ≥ 1 & d ∈ {1, 2}.

On the other hand, if α < 1 we show that the solution is not mild for any space dimension d.

Biography: Bernt Karsten Øksendal 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.

Between 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.

Bibliography

[1] Abel, N. H. (1823): Oppløsning av et par oppgaver ved hjelp av bestemte integraler (in Norwegian). Magazin for naturvidenskaberne 55–68.
[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.
[3] Aumann, R. J. (1965): Integrals of set-valued functions. J. Math.Anal. 12, 1-12.
[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.
[5] Dalang, R. C. & Walsh, J. B. (1992): The sharp Markov property of L´evy sheets. The Annals of Probability 20 (2), 591-626.
[6] Di Nunno, G., Øksendal, B. & Proske, F. (2009): Malliavin Calculus for L´evy Processes with Applications to Finance. Springer.
[7] Hu, Y. (2019): Some recent progress on stochastic heat equations. Acta Mathematica Scientia 39B(3); 874-914.
[8] Holm, S. (2019): Waves with Power-Law Attenuation. Springer.
[9] H. Holden, B. Øksendal, J. Ubøe, and T. Zhang. Stochastic Partial Differential Equations. Birkh¨auser Boston Inc., Boston, MA, 1996.
[10] Ibe, O. C. (2013): Markov Processes for Stochastic Modelling. 2nd edition. Elsevier.
[11] Iafrate, F. & Ricciutu, C.(2024): Some families of random fileds related to multiparameter L´evy processes. J. Theoretical Probability 37: 3055-3088. https://doi.org/10.1007/s10959-024-01351-3.
[12] I.M. Gel’fand and N.Ya. Vilenkin. Generalized Functions. Vol. 4. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1964 [1977].
[13] Kochubel, A. N., Kondratiev, Y. & da Silva, J. L. (2021): On fractional heat equation. Fractional Calculus & Applied Analysis 24 (1), 73-87.
[14] Meerschaert, Mark M., Sikorskii, Alla (2019): Stochastic Models for Fractional Calculus, 2nd edition. De Greuter.
[15] Mittag-Leffler, M.G. (1903): Sur la nouvelle fonction E(x). C. R. Acad. Sci. Paris 137 554–558.
[16] Moulay Hachemi, R.Y. & Øksendal, B. (2023): The fractional stochastic heat equation driven by time-space white noise. Fractional Calculus & Applied Analysis 26, 513-532. Springer. https://doi.org/10.1007/s13540-023-00134-7.
[17] Nualart, D. & Schoutens, W. (2000): Chaotic and predictable representations for Levy processes. Stochastic Process. Appl., 90(1):109-122, 2000.

[18] Patel, A. & Kosko, B. (2007): ”Levy Noise Benefits in Neural Signal Detection,” 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP ’07, Honolulu, HI, USA, 2007, pp. III-1413-III-1416, doi: 10.1109/ICASSP.2007.367111.
[19] Pollard, H. (1948): The completely monotone character of the Mittag-Leffler function Eα(−x). Bull. Amer. Math-Soc.54, 1115-1116.
[20] B. Øksendal and F. Proske. White noise of Poisson random measures. Potential Anal., 21(4):375-403, 2004.
[21] Schneider, W.B. (1996): Completely monotone generalized Mittag-Leffler functions. Expositiones Mathematicae 14, 3-16