Stochastic Processes and Operator Calculus on Quantum Groups

Abstract

From the introduction: ``In this book we present some recentdevelopments in the theory of stochastic processes and operatorcalculus on quantum groups. We begin in Chapter 2 with somepreliminaries on Lie groups and related topics: representationtheory, construction of stochastic processes, Appell systems.Chapter 3 provides the necessary background on the algebraicstructures that we use in the following chapters.\par ``InChapter 4 we come to our main topic, the study of stochasticprocesses on quantum groups. First, we recall the basicterminology of non commutative or quantum probability. In Section4.2 we introduce the notion of independence for quantum randomvariables. The next section contains the definition of quantumstochastic processes with independent and stationary increments,i.e. Levy processes. As in the classical situation, theseprocesses are completely characterised by their convolutionsemigroup or by their generators. In Sections 4.4 and 4.5 wedescribe two constructions of realisations of Levy processes. Thefirst starts from the generator and gives a realisation on a BoseFock space. In order to do so, one constructs the so calledSchurmann triple of the generator, which appears as coefficientsin the quantum stochastic differential equation of the process.The second requires a convolution semigroup of normalisedfunctionals as input and uses an inductive limit procedure. Theresulting realisation is poorer from the analytical point ofview, but it has the advantage that it can be applied in a moregeneral situation, even if the semigroup is not positive. InSection 4.6, we construct convolution semigroups on quantumgroups from classical Levy processes and in Section 4.7 we provean analogue of the Feynman Kac formula for these semigroups. Thelast section (Section 4.8) deals with duality and timereversal.\par ``In Chapter 5 we show that Levy processes have anatural quantum Markov structure and use this to deal with thequestion of which quantum stochastic processes admit classicalversions, i.e. for what families of operators(X\sb t)\sb {t\in I} (of the form X \sb t=j\sb t(x), where(j\sb t) is a Levy process) there exists a classical stochasticprocess (\tilde X\sb t)\sb {t\in I} on some probability space(Ω, \scr F,P) such that all time ordered moments agree,i.e. Φ(X\sp {k\sb 1}\sb {t\sb 1}\cdots X\sp {k\sb n}\sb {t\sb n})=\Bbb E(\tilde X{}\sp {k\sb 1}\sb {t\sb 1}\cdots \tilde X{}\sp {k\sb n}\sb {t\sb n}),\tag1for all n,k\sb 1,\dots,k\sb n\in\Bbb N,t\sb 1\leq \dots\leq t\sb n\in I, for the (vacuum) stateΦ.\par ``A famous example of a classical version of aquantum Levy process is the Azema martingale. It is well knownthat on a commutative algebra the quantum Markov property issufficient for the existence of classical versions. We show thatwe can obtain quantum Markov processes on commutative subalgebrasfrom quantum Levy processes that are not commutative, and thatthere also exist commutative processes that are not Markovian. Wealso show that this approach leads to a powerful tool for theexplicit calculation of the classical generators ormeasures.\par ``The next two chapters investigate the relationbetween Levy processes and evolution equations. In Chapter 6, wedefine and construct diffusions on braided spaces, and we showthat their shifted moments sequences are polynomials (the socalled Appell polynomials). These polynomials satisfy equationsof the form (\partial\sb t L)u=0, where L is an operatorconsisting of linear and quadratic terms in the braided partials\partial\sp i.\par ``In Chapter 7 we consider stochasticprocesses on quantum groups that are related to evolutionequations of the form \partial \sb tu=Lu, with some differencedifferential operator L. For the equations considered inSection 7.1, u is an element of a quantum or braided group\scr A. We recall that solutions of these equations can begiven as Appell systems or shifted moments of the associatedprocess, and show how these can be calculated explicitly on theq affine group and on a braided analogue of the Heisenberg Weylgroup. In Section 7.2, we define a Wigner map from functionals ona quantum group or braided group to a `Wigner' density on theundeformed space. We prove that the densities associated in thisway to (pseudo ) Levy processes satisfy a Fokker Planck typeequation. In the one dimensional case these coincide with theevolution equations of Section 7.1, but in the general case weget new equations.\par ``In Chapter 8, we turn to thecharacterisation of certain probability laws and convolutionsemigroups on nilpotent quantum groups and on nilpotent braidedgroups. In Section 8.1 we determine the functionals which satisfyan analogue of the Bernstein property, i.e. that the sum anddifference of independent random variables are also independent,on several braided groups. This extends results obtained on Liegroups by D. Neuenschwander et al. As for Lie groups this classturns out to be too small to constitute a satisfactory definitionof `Gaussianity'. Therefore we turn to convolutionsemigroups.\par ``Chapter 9 presents the results of Franz,Neuenschwander, and Schott for the problem of phase retrieval onnilpotent quantum groups and nilpotent braided groups: Given thesymmetrisation μ*\overlineμ and the first moments of aunital functional μ on \scr A, when is it possible toretrieve the original functional μ from these data? Thesomewhat surprising answer is that in this framework theretrieval is always possible (provided that the quantum orbraided group is `sufficiently' noncommutative, e.g. if q isnot a root of unity).\par ``In Chapter 10 we present severallimit theorems. In Section 10.1, the general results for limittheorems on bialgebras due to M. Schurmann [White noise onbialgebras, Lecture Notes in Math., 1544, Springer, Berlin, 1993;MR 95i:81128] are presented. Then a randomised q central (orq commutative) limit theorem on a family of bialgebras with onecomplex parameter is shown [U. Franz, in Quantum probability(Gdansk, 1997), 183 189, Polish Acad. Sci., Warsaw, 1998; MR99k:81142]. Section 10.3 is devoted to Woronowicz' results onconvergence of convolution products of probability measures tothe Haar functional on compact quantum groups. A q centrallimit theorem for U\sb q({\rm su}(2)) has been proved by R.Lenczewski. This result is stated in Section 10.4 as well as aweak law of large numbers which derives easily from the centrallimit theorem. Then we recall Neuenschwander and Schott's resultson domains of attraction for q transformed random variables inSection 10.5.''\par Contents: Preface; 1. Introduction; 2.Preliminaries on Lie groups; 3. Hopf algebras, quantum groups andbraided spaces; 4. Stochastic processes on quantum groups; 5.Markov structure of quantum Levy processes; 6. Diffusions onbraided spaces; 7. Evolution equations and Levy processes onquantum groups; 8. Gauss laws in the sense of Bernstein onquantum groups; 9. Phase retrieval for probability distributionson quantum groups and braided groups; 10. Limit theorems onquantum groups; Bibliography; Index