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qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expec ..."
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Cited by 65 (2 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
Chaotic Kabanov formula for the Azéma martingales
, 2000
"... We derive the chaotic expansion of the product of nth and first order multiple stochastic integrals with respect to certain normal martingales. This is done by application of the classical and quantum product formulas for multiple stochastic integrals. Our approach extends existing results on chaot ..."
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Cited by 5 (2 self)
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We derive the chaotic expansion of the product of nth and first order multiple stochastic integrals with respect to certain normal martingales. This is done by application of the classical and quantum product formulas for multiple stochastic integrals. Our approach extends existing results on chaotic calculus for normal martingales and exhibits properties relative to multiple stochastic integrals, polynomials and Wick products, that characterize the Wiener and Poisson processes. Key words: Az'ema martingales, multiple stochastic integrals, product formulas. Mathematics Subject Classification (1991): 60G44, 60H05, 81S25. 1 Introduction The WienerIto and PoissonIto chaotic decompositions give an isometric isomorphism between the Fock space \Gamma(L 2 (IR + )) and the space of squareintegrable functionals of the process. This somorphism is constructed by association of a symmetric function f n 2 L 2 (IR + ) ffin to its multiple stochastic integral. The isometry property comes ...
DAMTP/9220 QUANTUM RANDOM WALKS AND TIMEREVERSAL
, 1992
"... ABSTRACT Classical random walks and Markov processes are easily described by Hopf algebras. It is also known that groups and Hopf algebras (quantum groups) lead to classical and quantum diffusions. We study here the more primitive notion of a quantum random walk associated to a general Hopf algebra ..."
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ABSTRACT Classical random walks and Markov processes are easily described by Hopf algebras. It is also known that groups and Hopf algebras (quantum groups) lead to classical and quantum diffusions. We study here the more primitive notion of a quantum random walk associated to a general Hopf algebra and show that it has a simple physical interpretation in quantum mechanics. This is by means of a representation theorem motivated from the theory of Kac algebras: If H is any Hopf algebra, it may be realised in Lin(H) in such a way that ∆h = W(h ⊗ 1)W −1 for an operator W. This W is interpreted as the time evolution operator for the system at time t coupled quantummechanically to the system at time t + δ. Finally, for every Hopf algebra there is a dual one, leading us to a duality operation for quantum random walks and quantum diffusions and a notion of the coentropy of an observable. The dual system has its time reversed with respect to the original system, leading us to a CTPtype theorem. 1
MARKOV PROPERTY OF MONOTONE LÉVY PROCESSES
, 2004
"... Monotone Lévy processes with additive increments are defined and studied. It is shown that these processes have natural Markov structure and their Markov transition semigroups are characterized using the monotone LévyKhintchine formula. 17 Monotone Lévy processes turn out to be related to classical ..."
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Monotone Lévy processes with additive increments are defined and studied. It is shown that these processes have natural Markov structure and their Markov transition semigroups are characterized using the monotone LévyKhintchine formula. 17 Monotone Lévy processes turn out to be related to classical Lévy processes via Attal’s “remarkable transformation.” A monotone analogue of the family of exponential martingales associated to a classical Lévy process is also defined.
Applications of Quantum Probability to Classical Stochastics
"... this paper an operatorvalued generalization of the Kolmogorov  Feller equation is discussed. The most remarkable feature of this generalization is that it also covers diffusion and Heisenberg equations as particular cases [L]. It is known that if intensity of jumps (or drift velocity, or diffusio ..."
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this paper an operatorvalued generalization of the Kolmogorov  Feller equation is discussed. The most remarkable feature of this generalization is that it also covers diffusion and Heisenberg equations as particular cases [L]. It is known that if intensity of jumps (or drift velocity, or diffusion coefficient)as a function on the phase space X of the corresponding Markov process is unbounded, then the probability measure of trajectories hitting singularities of the coefficients may be positive. If there exist a number of different ways to define the behaviour of the process after hitting a singular point then the formal evolution equation for the transition probability of the Markov jump process has a linear manifold of solutions. In the class of solutions P (x; tj\Gamma) satisfying the initial condition
Classical Markov Processes from Quantum Lévy Processes
, 1998
"... We show how classical Markov processes can be obtained from quantum L'evy processes. It is shown that quantum L'evy processes are quantum Markov processes, and sufficient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (wh ..."
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We show how classical Markov processes can be obtained from quantum L'evy processes. It is shown that quantum L'evy processes are quantum Markov processes, and sufficient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (which has the same timeordered moments as the quantum process in the vacuum state) exists whenever we can restrict to a commutative subalgebra without loosing the quantum Markov property[Kum88]. Several examples, including the Az'ema martingale, with explicit calculations are presented. In particular, the action of the generator of the classical Markov processes on polynomials or their moments are calculated using Hopf algebra duality. 1 Introduction It is an interesting question which quantum stochastic processes admit classical versions, i.e. for what families of operators (X t ) t2I there exists a classical stochastic process ( ~ X t ) t2I on some probability space(\Omega ; F ; P ) such that al...
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"... Application of a nonconservativeness criterion to the Lindblad generator of the Azéma martingale semigroup ..."
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Application of a nonconservativeness criterion to the Lindblad generator of the Azéma martingale semigroup