We derive an explicit formula for the Jacobi field that is acting in an extended Fock space and corresponds to an (R-valued) Lévy process on a Riemannian manifold. The support of the measure of jumps in the Lévy-Khintchine representation for the Lévy process is supposed to have an infinite number of points. We characterize the gamma, Pascal, and Meixner processes as the only Lévy processes whose Jacobi field leaves the set of finite continuous elements of the extended Fock space invariant.
2000
This paper is devoted to the proof of chaotic representation for Gamma field and the corresponding stochastic process. It is known that a direct generalization of the chaotic representation connected with a Gamma process is impossible. Our construction is different from the classical one: for the components of our decomposition we have the representation by a many-dimensional stochastic integral (some of these components have the classical form of the multiple stochastic integral).
Abstract. We introduce notions of quasi-spherical and multi-quasispherical polynomial quaternionic equations defined in terms of the shape of the set of the solutions of the equation. We establish that everyis quasi-spherical. We get some sufficient conditions under which a quadratic quaternionic polynomial can be represented as a product of linear quaternionic polynomials (and thus the corresponding equation is multiquasi-spherical).Mathematics Subject Classification (2010). 11R52, 30G35, 12K99.
Abstract. We give complete description of possible shapes of the set of the solutions of any quaternionic equation of the form ax + xb = c. Moreover we study the set of the solutions of a quaternionic equation of the form ax 2 + x 2 b = c by the method of sections by hyperplanes perpendicular to the real axis; for every case where such section is an unbounded linear manifold a necessary and sufficient condition is found.
Mathematics Subject Classification (2010). 11R52.
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