Brownian Motion, Martingales and Itô Formula in Clifford Analysis
Tóm tắt
Clifford analysis has been the field of active research for several decades resulting in various methods to solve problems in pure and applied mathematics. However, the area of stochastic analysis has not been addressed in its full generality in the Clifford setting, since only a few contributions have been presented so far. Considering that the tools of stochastic analysis play an important role in the study of objects, such as positive definite functions, reproducing kernels and partial differential equations, it is important to develop tools for the study of these objects in the context of Clifford analysis. Therefore, in this work-in-progress paper, we present further steps towards stochastic Clifford analysis by studying random variables, martingales, Brownian motion, and Itô formula in the Clifford setting, as well as their applications in Clifford analysis.
Tài liệu tham khảo
Adams, R., Fournier, J.: Sobolev Spaces. Academic Press, London (2003)
Alpay, D., Colombo, F., Sabadini, I.: On a class of quaternionic positive definite functions and their derivatives. J. Math. Phys. 58, 033501 (2017)
Alpay, D., Paiva, I., Struppa, D.C.: Distribution spaces and a new construction of stochastic processes associated with the Grassmann algebra. J. Math. Phys. 60, 013508 (2019)
Alpay, D., Cerejeiras, P., Kähler, U.: Krein reproducing kernel modules in Clifford analysis. J. Anal. Math. 143, 253–288 (2021)
Alpay, D., Cerejeiras, P., Kähler, U.: Generalized Grasssmann algebras and applications to stochastic processes. Math. Methods Appl. Sci. 45, 383–401 (2022)
Bernstein, S.: Integralgleichungen und Funktionenräume für Randwerte monogener Funktionen, Habilitation thesis, Faculty of mathematics and computer science, TU Bergakademie Freiberg (2001)
Bernstein, S.: Towards infinite-dimensional Clifford analysis, Mathematical Methods in the Applied Sciences (2021) (to appear)
Brackx, F., Delanghe, R., Sommen, F.: Clifford analysis, Research Notes in Mathematics, vol. 76. Pitman Advanced Publishing Program, Boston (1982)
Da Prato, G.: An Introduction to Infinite-dimensional Analysis. Springer, New York (2006)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (2014)
Emery, M.: Stochastic Calculus in Manifolds. Springer, Berlin. With an appendix by P.A. Meyer (1989)
Eriksson, S.-L., Kaarakka, T.: Hyperbolic harmonic functions and hyperbolic Brownian motion. Adv. Appl. Clifford Algebras 30, 72 (2020)
Freeman, N.: Itô Calculus and Complex Brownian Motion, Lecture Notes. University of Sheffield (2015)
Getoor, R.K., Sharpe, M.J.: Conformal martingales. Invent. Math. 16, 271–308 (1972)
Gilbert, J.E., Murray, M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)
Gürlebeck, K., Sprößig, W.: Quaternionic and Clifford Calculus for Engineers and Physicists. Wiley, Chichester (1997)
Gürlebeck, K., Malonek, H.R.: A hypercomplex derivative of monogenic functions in \({\mathbb{R}}^{n+1}\) and its applications. Complex Var. 39, 199–228 (1999)
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)
Liu, W., Röckner, M.: Stochastic Partial Differential Equations: An Introduction. Springer International Publishing, Cham (2015)
Malonek, H.R.: 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)
Mörters, P., Peres, Y.: Brownian Motion. Cambridge University Press, Cambridge (2012)
Øksendal, B.: Stochastic Differential Equations. An Introduction with Applications. Springer, New York (1998)
Rogers, L.C.G., Williams, D.: Diffusions. Markov Processes and Martingales. Cambridge University Press, Cambridge (2000)
Ubøe, J.: Conformal martingales and analytic functions. Math. Scand. 60, 292–309 (1987)