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Brownian motion, martingales and Itô formula in Clifford analysis. (English) Zbl 07511198

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 it is also obtained a rough deduction of Itô’s formula from the classical Itô formula (see Lemma 5.4.), 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 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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