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Characterization Theorems for Differential Operators on White Noise Spaces
Full text linkWe characterize through their action on stochastic exponentials
the class of white noise operators which are derivations with respect to both
the point-wise and Wick products. We define the class of second order differential
operators and second order Wick differential operators and we characterize
the white noise operators belonging to both classes. We find that the
intersection of these two classes, in the first and second order cases, is identified
by a skewness condition on the coefficients of the differential operator.
Our technique relies on simple algebraic properties of commutators and on
the Gaussian structure of our white noise space. Our approach is suitable to
study differential operators of any orde
Identification of the theory of orthogonal polynomials in d -indeterminates with the theory of 3 -diagonal symmetric interacting Fock spaces on Cd
Full text linkThe identification mentioned in the title allows a formulation of the multidimensional Favard lemma different from the ones currently used in the literature and which parallels the original 1-dimensional formulation in the sense that the positive Jacobi sequence is replaced by a sequence of positive Hermitean (square) matrices and the real Jacobi sequence by a sequence of positive definite kernels. The above result opens the way to the program of a purely algebraic classification of probability measures on Rd with moments of any order and more generally of states on the polynomial algebra on Rd. The quantum decomposition of classical real-valued random variables with all moments is one of the main ingredients in the proof
QWN-convolution operators with application to differential equations
No full textIn this paper we introduce a quantum white noise (QWN) convolution calculus
over a nuclear algebra of operators. We use this calculus to discuss new solutions
of some linear and non-linear differential equation
New insight on the structural architecture of northern Tunisia with a multidisciplinary approach. Association of Miocene piggy-back basin and active Plio-quaternary out-of-sequence thrusts
No full textIn order to better understand the intricate structural architecture of northern Tunisia, we use a multidisciplinary approach that includes field observations, geomorphic, sedimentological, and gravity data across this area. We offer a model of progressive thrusting in which four deformation phases occurred in the context of plate convergence (i) Uplift of the Jebel Ichkeul Triassic succession via a NE-trending thrust fault during the Eocene; (ii) Advance of Numidian thrust sheets during the late Oligocene-Middle Miocene, causing the formation of the Lake Ichkeul/Jalta as being piggy-back basin; (iii) Initial out-of-sequence thrusting during the Late Miocene, resulting in the formation of the Sejnane and El Mejel basins with syntectonic Pliocene infill; and (iv) Subsequent out-of-sequence thrusting during the Early Pleistocene, leading to the formation of the Oued (River) Zyatine basin with a syntectonic Quaternary deposits. We used the gravity data analysis to mark the thicknesses of syntectonic deposits and their bounding thrust faults. The kinematic analysis indicates the occurrence of progressive multiphase out-of-sequence thrusting events with thrusts verging to the southeast. The combination of climate and mechanical fractures induced gradually long-term damage and changed the original landscape of the evidenced out-of-sequence thrusts. Furthermore, the existence of local normal faults in the backlimbs and reverse faults in the forelimbs, together with the drainage network type, earthquakes, the slope erosion, and the aggraditional terrace system, implies a gradual landscape evolution of thrust topography. This physical evolution occurred following an active tectonics across the exhibited out-of-sequence thrusting since the Pliocene
A quantum approach to Laplace operators
Full text linkIn this paper, a theory of stochastic processes generated by quantum extensions of Laplacians is developed. Representations of the associated heat semigroups are discussed by means of suitable time shifts. In particular the quantum Brownian motion associated to the Levy-Laplacian is obtained as the usual Volterra-Gross Laplacian using the Cesaro Hilbert space as initial space of our process as well as multiplicity space of the associated white noise
Characterization Theorems for Differential Operators on White Noise Spaces
Full text linkWe characterize through their action on stochastic exponentials
the class of white noise operators which are derivations with respect to both
the point-wise and Wick products. We define the class of second order differential
operators and second order Wick differential operators and we characterize
the white noise operators belonging to both classes. We find that the
intersection of these two classes, in the first and second order cases, is identified
by a skewness condition on the coefficients of the differential operator.
Our technique relies on simple algebraic properties of commutators and on
the Gaussian structure of our white noise space. Our approach is suitable to
study differential operators of any orde
Cesaro Hilbert space and the Levy laplacian
Full text linkIn this paper we introduce a new scalar product on distribution spaces based on the Cesaro mean of a sequence. We then use this scalar product to construct a family of separable Hilbert spaces , called Cesaro Hilbert spaces and naturally associated to the Levy Laplacian. Finally we use the
essentially infinite dimensional character of the Levy Laplacian
to construct a class of solutions of the Levy heat equation which has no finite dimensional (or ``regular'' infinite dimensional) analogue
White noise Lévy-Meixner processes through a transfer principal from one-mode to one-mode type interacting fock spaces
No full textConsider the Lévy–Meixner one-mode interacting Fock space {ΓLM, 〈 ⋅, ⋅ 〉LM}. Inspired by a derivative formula appearing in 〈 ⋅, ⋅ 〉LM, we define scalar products 〈 ⋅, ⋅ 〉LM , nin symmetric n-particle spaces. Then, we introduce a class of one-mode type interacting Fock spaces [Formula: see text] naturally associated to the one-dimensional infinitely divisible distributions with Lévy–Meixner type {μr; r > 0}. The Fourier transform in generalized joint eigenvectors of a family [Formula: see text] of Lévy–Meixner Jacobi fields provides a way to explicit a unitary isomorphism 𝔘LMbetween [Formula: see text] and the so-called Lévy–Meixner white noise space [Formula: see text]. We derive a chaotic decomposition property of the quadratic integrable functionals of the Lévy–Meixner white noise processes in terms of an appropriate Wick tensor product. For their stochastic regularity, we give explicit form and sharp estimate of the associated Donsker's delta function.</jats:p
Identification of the theory of orthogonal polynomials in d -indeterminates with the theory of 3 -diagonal symmetric interacting Fock spaces on ℂd
Full text link The identification mentioned in the title allows a formulation of the multidimensional Favard lemma different from the ones currently used in the literature and which parallels the original 1-dimensional formulation in the sense that the positive Jacobi sequence is replaced by a sequence of positive Hermitean (square) matrices and the real Jacobi sequence by a sequence of positive definite kernels. The above result opens the way to the program of a purely algebraic classification of probability measures on [Formula: see text] with moments of any order and more generally of states on the polynomial algebra on [Formula: see text]. The quantum decomposition of classical real-valued random variables with all moments is one of the main ingredients in the proof. </jats:p
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