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Brownian motion, martingales and Itô formula in Clifford analysis. (English) Zbl 1493.30096

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 \(L^p\).

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)
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[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
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