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Probability and Mathematical Statistics

SOME PROPERTIES OF QUANTUM LÉVY AREA IN FOCK AND NON-FOCK QUANTUM STOCHASTIC CALCULUS

Shang Chen, R. L. Hudson

Published Jan 1, 2013• 8 citations
OpenAlex

Abstract

Weconsider the analogueof Levy area, definedas an iterated stochastic integral, obtained by replacing two independent component one- dimensional Brownian motions by the mutually non-commuting momen- tum and position Brownian motions P and Q of either Fock or non-Fock quantum stochastic calculus, which are also stochastically independent in a certain sense. We show that the resulting quantum Levy area is trivially distributed in the Fock case, but has a non-trivial distribution in non-Fock quantum stochastic calculus which, after rescaling, interpolates between the trivial distribution and that of classical Levy area in the infinite tempera- ture limit. We also show that it behaves differently from the classical Levy area under a kind of time reversal, in both the Fock and non-Fock cases. 2000 AMS Mathematics Subject Classification: Primary: 81S25.

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Weconsider the analogueof Levy area, definedas an iterated stochastic integral, obtained by replacing two independent component one- dimensional Brownian motions by the mutually non-commuting momen- tum and position Brownian motions P and Q of either Fock or non-Fock quantum stochastic calculus, which are also stochastically independent in a certain sense. We show that the resulting quantum Levy area is trivially distributed in the Fock case, but has a non-trivial distribution in non-Fock quantum stochastic calculus which, after rescaling, interpolates between the trivial distribution and that of classical Levy area in the infinite tempera- ture limit. We also show that it behaves differently from the classical Levy area under a kind of time reversal, in both the Fock and non-Fock cases. 2000 AMS Mathematics Subject Classification: Primary: 81S25.
We replace the independent one-dimensional Brownian motions in the de…nition of classical Levy area by the mutually noncommuting momentum and position Brownian motions of non-Fock quantum stochastic calculus, which are independent in the sense of factorisation of joint characteristic functions. The corresponding quantum Levy area can then be constructed in a way similar to Levy’s original construction as a quantum martingale limit. Acknowledgement 1 The author thanks the referee for drawing his attention to a number of mistypes. 1 Introduction. Levy’s stochastic area [10], [11], as well as being of great intrinsic interest and possessing numerous connections with classical mathematical analysis and mathematical physics [7], has recently come to prominence in the theory of rough paths [12], [13], [5]. Here we present a quantum or noncommutative version of Levy area in which the two independent one-dimensional Brownian motions used to de…ne it are replaced by the pair of quantum Brownian motions (P;Q) ; forming the quantum Wiener process of [2]. Each of P and Q consists of a process of mutually commuting self-adjoint operators which can be simultaneously diagonalised as multiplication by a classical Brownian motion of appropriate variance in the Hilbert space of complex-valued functions square-integrable with respect to Wiener measure, But P and Q together satisfy the commutation relation P (s)Q(t) Q(t)P (s) = imin fs; tg I; (1)
We review the basic concepts of quantum probability and revise the classical and quantum stochastic (QS) calculus using the universal Itô B*-algebra approach. A non-commutative generalization of Lévy process is defined in Fock–Guichardet space in terms of the underlying B*-Itô modular algebra. The main notions and results of classical and QS analysis are reformulated in terms of the non-commutative but associative stochastic covariation of quantum integrals defined as sesquilinear forms adapted to a quantum Lévy process in Fock space. The general semi-Markov QS dynamics is defined in terms of a hemigroup of semi-morphic transformations as adjoints to a semiunitary hemigroup resolving a QS Schrödinger equation in Fock space. The corresponding semi-Markov hemigroup of completely positive contractions resolving a quantum master equation with generalized Lindblad generator as a non-commutative analogue of Feller–Kolmogorov equation is derived by taking vacuum conditional expectation in Fock space.
We study the family of causal double product integrals ∏_a<x<y<b(1+iλ2(dP_xdQ_y−dQ_xdP_y)+iμ2(dP_xdP_y+dQ_xdQ_y)), \prod\_{a < x < y < b}\Big(1 + i{\lambda \over 2}(dP\_x dQ\_y - dQ\_x dP\_y) + i {\mu \over 2}(dP\_x dP\_y + dQ\_x dQ\_y)\Big), ∏_a<x<y<b(1+i2λ​(dP_xdQ_y−dQ_xdP_y)+i2μ​(dP_xdP_y+dQ_xdQ_y)), where PPP and QQQ are the mutually noncommuting momentum and position Brownian motions of quantum stochastic calculus. The evaluation is motivated heuristically by approximating the continuous double product by a discrete product in which infinitesimals are replaced by finite increments. The latter is in turn approximated by the second quantisation of a discrete double product of rotation-like operators in different planes due to a result in \[15]. The main problem solved in this paper is the explicit evaluation of the continuum limit WWW of the latter, and showing that WWW is a unitary operator. The kernel of W−IW - IW−I is written in terms of Bessel functions, and the evaluation is achieved by working on a lattice path model and enumerating linear extensions of related partial orderings, where the enumeration turns out to be heavily related to Dyck paths and generalisations of Catalan numbers.
We study the family of causal double product integrals \begin{equation*} \prod_{a < x < y < b}\left(1 + i{λ\over 2}(dP_x dQ_y - dQ_x dP_y) + i {μ\over 2}(dP_x dP_y + dQ_x dQ_y)\right) \end{equation*} where PPP and QQQ are the mutually noncommuting momentum and position Brownian motions of quantum stochastic calculus. The evaluation is motivated heuristically by approximating the continuous double product by a discrete product in which infinitesimals are replaced by finite increments. The latter is in turn approximated by the second quantisation of a discrete double product of rotation-like operators in different planes due to a result in [Hudson-Pei2015]. The main problem solved in this paper is the explicit evaluation of the continuum limit WWW of the latter, and showing that WWW is a unitary operator. The kernel of WWW is written in terms of Bessel functions, and the evaluation is achieved by working on a lattice path model and enumerating linear extensions of related partial orderings, where the enumeration turns out to be heavily related to Dyck paths and generalisations of Catalan numbers.
Martingales are fundamental stochastic process used to model the concept of fair game. They have a multitude of applications in the real world that include, random walks, Brownian motion, gamblers fortunes and survival analysis, Just as commutative integration theory may be realised as a special case of the more general non-commutative theory for integrals, so too, we find classical probability may be realised as a limiting, special case of quantum probability theory. In this thesis we are concerned with the development of multiparameter quantum stochastic integrals extending non-commutative constructions to the general n parameter case, these being multiparameter quantum stochastic integrals over the positive n - dimensional plane, employing martingales as integrator. The thesis extends previous analogues of type one, and type two stochastic integrals, for both Clifford and quasi free representations. As with one and two dimensional parameter sets, the stochastic integrals constructed form orthogonal, centred L2 - martingales, obeying isometry properties. We further explore analogues for weakly adapted processes, properties relating to the resulting quantum stochastic integrals, develop analogues to Fubini’s theorem, and explore applications for quantum stochastic integrals in a security setting.
Quantum diffusions are non-commutative generalisations of classical diffusions on manifolds, in which the manifold i s ‘quantised’ by deforming the algebra of smooth functions thereon to a non-commutative algebra, and in which quantum stochastic calculus replaces the Itô calculus of Brownian motion. Quantum diffusions fall into two categories; inner diffusions, which are in a certain sense cohomologically trivial, and which act by conjugating by a unitary process, and outer diffusions, which do not. A natural perturbation procedure is described for linking two cohomologically equivalent quantum diffusions; it involves the construction of a new type of intertwining unitary process.