Bernstein, Swanhild; Legatiuk, Dmitrii Brownian motion, martingales and Itô formula in Clifford analysis. (English) Zbl 1493.30096 Adv. Appl. Clifford Algebr. 32, No. 3, Paper No. 27, 18 p. (2022). In this proceedings paper, the authors introduce further tools of stochastic calculus in the scope of Clifford algebras. In concrete, the notions of martingales, Brownian motion and Itô calculus are introduced.Applications of this novel framework involve the solution of a Dirichlet problem (see Theorem 4.11) and a stochastic counterpart of Liouville’s theorem (see Theorem 4.13). In the remaining part of the paper a rough deduction of Itô’s formula from the classical Itô formula (see Lemma 5.4) is also obtained, without imposing any additional probabilistic orthogonality assumption.Summing up, the question of finding a faithful Itô formula, encoding the boundary values of conjugate harmonic functions in the upper half-space, remains open. A first attempt to rid the authors’ gap should start with the reformulation of the so-called Riesz-Hilbert transform as a conditional expectation of a martingale transform, representing a boundary value \(f\) with membership in \(L^p\). Reviewer: Nelson Faustino (Alfeizerão) Cited in 1 Document MSC: 30G35 Functions of hypercomplex variables and generalized variables 60H40 White noise theory 46G99 Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces) 60H15 Stochastic partial differential equations (aspects of stochastic analysis) Keywords:martingales; Brownian motion; Itô formula; stochastic Clifford analysis; Dirichlet problem PDFBibTeX XMLCite \textit{S. Bernstein} and \textit{D. Legatiuk}, Adv. Appl. Clifford Algebr. 32, No. 3, Paper No. 27, 18 p. (2022; Zbl 1493.30096) Full Text: DOI arXiv References: [1] Adams, R.; Fournier, J., Sobolev Spaces (2003), London: Academic Press, London · Zbl 1098.46001 [2] Alpay, D.; Colombo, F.; Sabadini, I., On a class of quaternionic positive definite functions and their derivatives, J. Math. Phys., 58, 033501 (2017) · Zbl 1361.60026 · doi:10.1063/1.4977082 [3] Alpay, D.; Paiva, I.; Struppa, DC, Distribution spaces and a new construction of stochastic processes associated with the Grassmann algebra, J. Math. Phys., 60, 013508 (2019) · Zbl 1418.46028 · doi:10.1063/1.5052010 [4] Alpay, D.; Cerejeiras, P.; Kähler, U., Krein reproducing kernel modules in Clifford analysis, J. Anal. Math., 143, 253-288 (2021) · Zbl 1486.30132 · doi:10.1007/s11854-021-0155-6 [5] Alpay, D.; Cerejeiras, P.; Kähler, U., Generalized Grasssmann algebras and applications to stochastic processes, Math. Methods Appl. Sci., 45, 383-401 (2022) · Zbl 1527.30031 · doi:10.1002/mma.7781 [6] Bernstein, S.: Integralgleichungen und Funktionenräume für Randwerte monogener Funktionen, Habilitation thesis, Faculty of mathematics and computer science, TU Bergakademie Freiberg (2001) · Zbl 1142.30338 [7] Bernstein, S.: Towards infinite-dimensional Clifford analysis, Mathematical Methods in the Applied Sciences (2021) (to appear) [8] Brackx, F.; Delanghe, R.; Sommen, F., Clifford analysis, Research Notes in Mathematics (1982), Boston: Pitman Advanced Publishing Program, Boston · Zbl 0529.30001 [9] Da Prato, G., An Introduction to Infinite-dimensional Analysis (2006), New York: Springer, New York · Zbl 1109.46001 · doi:10.1007/3-540-29021-4 [10] Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions (2014), Cambridge: Cambridge University Press, Cambridge · Zbl 1317.60077 · doi:10.1017/CBO9781107295513 [11] Emery, M.: Stochastic Calculus in Manifolds. Springer, Berlin. With an appendix by P.A. Meyer (1989) · Zbl 0697.60060 [12] Eriksson, S-L; Kaarakka, T., Hyperbolic harmonic functions and hyperbolic Brownian motion, Adv. Appl. Clifford Algebras, 30, 72 (2020) · Zbl 1456.60194 · doi:10.1007/s00006-020-01099-z [13] Freeman, N.: Itô Calculus and Complex Brownian Motion, Lecture Notes. University of Sheffield (2015) [14] Getoor, RK; Sharpe, MJ, Conformal martingales, Invent. Math., 16, 271-308 (1972) · Zbl 0268.60048 · doi:10.1007/BF01425714 [15] Gilbert, JE; Murray, MAM, Clifford Algebras and Dirac Operators in Harmonic Analysis (1991), Cambridge: Cambridge University Press, Cambridge · Zbl 0733.43001 · doi:10.1017/CBO9780511611582 [16] Gürlebeck, K.; Sprößig, W., Quaternionic and Clifford Calculus for Engineers and Physicists (1997), Chichester: Wiley, Chichester · Zbl 0897.30023 [17] Gürlebeck, K.; Malonek, HR, A hypercomplex derivative of monogenic functions in \({\mathbb{R}}^{n+1}\) and its applications, Complex Var., 39, 199-228 (1999) · Zbl 1019.30047 [18] Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach. Springer Science+Business Media, New York (2010) · Zbl 1198.60005 [19] Liu, W.; Röckner, M., Stochastic Partial Differential Equations: An Introduction (2015), Cham: Springer International Publishing, Cham · Zbl 1361.60002 · doi:10.1007/978-3-319-22354-4 [20] Malonek, HR, A new hypercomplex structure of the Euclidean space \({\mathbb{R}}^{m+1}\) and the concept of hypercomplex differentiability, Complex Var. Theory Appl., 14, 25-33 (1990) · Zbl 0707.30039 [21] Mörters, P.; Peres, Y., Brownian Motion (2012), Cambridge: Cambridge University Press, Cambridge · Zbl 1243.60002 [22] Øksendal, B., Stochastic Differential Equations. An Introduction with Applications (1998), New York: Springer, New York · Zbl 0897.60056 · doi:10.1007/978-3-662-03620-4 [23] Rogers, LCG; Williams, D., Diffusions. Markov Processes and Martingales (2000), Cambridge: Cambridge University Press, Cambridge · Zbl 0949.60003 · doi:10.1017/CBO9781107590120 [24] Ubøe, J., Conformal martingales and analytic functions, Math. Scand., 60, 292-309 (1987) · Zbl 0634.60042 · doi:10.7146/math.scand.a-12186 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.