| Lévy processes are the most important processes in stochastic analysis, which have abundant mathematical structure. They have broad applications in physics, finance and other fields. Fractional Lévy processes are the natural extension of the Lévy processes, they play a key role in the problem with long range dependence property. Quantum Lévy processes are the operator-valued Lévy processes, they have profound background in quantum physics.Let {X(t), t∈IRd} be a Lévy process (or field), denote f as its characteristic exponent.By Gelfand-Vilenkin theorem, there always exists a probability measureμf on distribution space D'(IRd) withΦ(ξ) := exp{∫IRd f(ξ(t))dt},ξ∈V(IRd) as its characteristicfunctional. Furthermore, if the corresponding Lévy measure satisfies the conditionwe can prove thatμf has its measurable support in tempered distribution space S'(IRd). We call (S'(IRd),B(S'(IRd)),μf) a tempered white noise space. Through the bilinear form <, > on S'(IRd)×S(IRd), we can naturally define the tempered generalized process (or field) asWhen d = 1, we denote I<sup>βas right Riemann-Liouville fractional integral operator.In fact, we can prove that the map I<sup>β: S(IR)(?)L1(IR)∩L2(IR) is continuous. So we construct the tempered generalized fractional Lévy process asIn the white noise framework, we can view Xβ(ξ) as tempered Lévy white noise functional,which is very effective in investigating the properties of this generalized process. We prove that Xβ(ξ) is a stationary generalized process. Moreover, we give a necessary and sufficient condition on self-similarity of Xβ(ξ). At last, we discuss the conditions under which the usual fractional Lévy process is well defined, and has a integral representationasWhen d > 1, we denote Iαas Riesz potential operator. After a similar procedure, we define the tempered generalized fractional Lévy random field asWe find that the generalized random field defined above is stationary and isotropic, therefore, it is a Euclidean invariant field. In addition, all square integrable fractional random fields have long range dependence property. Finally, we propose methods to construct the anisotropic fractional Lévy fields and the multi-dimensional fractional Lévy processes.For those Lévy processes having finite moments of all orders, we also give a new method to construct quantum Lévy processes. Apart from the complex Hilbert space H = L2(IR,dt), we construct an interacting Fock spaceΓint(H). Then we redefine the creator a+, annihilator a- and conservation operator a0 on this interacting Fock space, and define the usual field operator aswhereγis a constant. Especially, whenξ= I[0,t], t > 0, we prove that X(t) := X(I[0,t]) is a standard quantum Lévy process. At last of this paper, we give some relative results on the q-deformed Lévy-Meixner polynomials with -1 |
