Lévy White Noises And Fractional Lévy Noises On Gel'fand Triple

By virtue of its self-similarity and long-range dependence,fractional Brownian mo-tion suits to model driving noises in different applications such as mathematical finance,economics,and network traffic analysis,and has evoked wide interests.Fractional Brown-ian motion can be represented as stochastic integral of a kernel function of Volterra typedetermined by the Riemann-Liouville fractional integral and derivative operator with re-spect to Brownian motion.By stochastic integration of this kernel function with respectto a general L(?)vy process,one can define fractional L(?)vy process which greatly widenthe fields of applications of fractional processes.On the other hand,for the study on stochastic differential equations driven byfractional noises in infinitely dimensional vector spaces,one need to constuct fractionalprocesses in these spaces.So far as we know,the current study only limited to stochasticevolution equations driven by fractional Brownian motions in Hilbert spaces.However,the fractional Brownian motion B^H in a Hilbert space V exists if and only if its covarianceoperator Q is nuclear.Otherwise,it will take values in a larger Hilbert space V₁ suchthat V continuously imbedded into V₁ and the imbedding is a Hilbert-Schmidt operator.So,it is more natural to consider the fractional processes taking values in the dual spaceof a countably Hilbertian nuclear space.The most suitable framework should be theGel'fand triple.Let H be a real separable Hilbert space with norm |·|₀ and inner product〈·,·〉.LetA be a positive self-adjoint operator in H such that A^-α is nuclear for someα＞0.Foreach r∈R,define |·|_r:=|A^r·|₀ and let E_r be the completion of the domain of A^rwith respect to |·|_r.Then E_r is a real Hilbert space,E_r and E_-r can be viewed as eachother's duals.Let E be the projective limit of {E_r}_r≥0 and E^* be the inductive limit of{E_-r}_r≥0.Then E is a Fr(?)chet nuclear space with E^* as its dual.We denote by〈·,·〉the canonical bilinear form on E^*×E,(?)⁺≡(?)⁺(E,E^*) the space of positive continuouslinear maps from E to E^*,then E(?)H(?)E^* is referred to as a Gel'fand triple generatedby (H,A).All the work of this dissertation is under this framework since1°It includs many usual infinitely dimensional spaces such as Schwartz space of tempered distributions,Hida space of generalized white noise functionals,and is themost natural extension of finite dimensional Euclidean spaces;2°Under this framework,we can freely use topological tensor product,Bochner-Minlos theorem,Schwartz kernel theorem,It(?) regularization theorem etc.to overcomemany difficulty in the construction.The main conclusions of this dissertation are as following:Conclusion 1 Let E(?)H(?)E^* be a real Gel'fand triple generated by (H,A) asabove.Given a∈E^*,Q∈(?)⁺ and a Borel measure v on E^* supported in E_-p for somep＞0 satisfyingThen there exists q＞0 and an infinitely divisible distributionμon E^* such thatWe call (a,Q,v) the generating triple ofμand v the L(?)vy measure.So,we obtain a general form of the charateristic functional of infinitely divisibledistributions on E^*,as a special case,we construct stable distributions on E^*.Thenwe define the E^*-valued L(?)vy process and give its L(?)vy-It(?) decomposition.Using areproducing kernel Hilbert space technique,we define stochastic integrals for a class ofoperator-valued processes with respect to L(?)vy processes.Conclusion 2 By virtue of the charateristic functional of the infinitely divisibledistribution on E^*,we introduce E^*-valued L(?)vy white noises.Let X={X_t,t∈R} bean E^*-valued L(?)vy process generated by triple (a,Q,v) with a∈E^*,Q∈(?)⁺ and L(?)vymeasure v satisfying f_{|x|-p＞1}|x|_-pdv(x)＜∞whose characteristic exponent is given byConclusion 1.Then,for any f∈L¹(R)∩L²(R),is a well-defined E^*-valued R.V.such that Therefore,{(?)(f),f∈S(R)} is a tempered E^*-valued white noise.We call it L(?)vy whitenoise on Gel'fand triple E(?)H(?)E^*.Conclusion 3 By Conclusion 2 and the continuity of the Riemann-Liouville frac-tional integration operator I_-^β,we construct a new E^*-valued (tempered) generalizedprocess as functional of E^*-valued L(?)vy white noise which is referred to as E^*-valuedfractional L(?)vy noise.Under the conditions of Conclusion 2,defineThen {(?)^β(f),f∈S(R)} is an E^*-valued (tempered) generalized process (referred to asE^*-valuedβ-fractional L(?)vy noise).Moreover,For f=1_[0,t],we obtain the E^*-valuedβ-fractional L(?)vy process:whenever the right hand side exists.Conclusion 4 When the underlying L(?)vy process is square integrable,we constructthe E^*-valuedβ-fractional L(?)vy process {X_t^β:= (?)^β(1_[0,t]),t≥0} and investigate itsdistribution and sample properties by considering it as functional of the E^*-valued L(?)vywhite noise.We prove that fractional L(?)vy process has stationary increments,long-range dependence,and give its innovational representation.In addition,we extend theconstruction to mixed fractional L(?)vy process and multi-fractional L(?)vy process.Conclusion 5 The above ideas and conclusions could be easily generalized to thecase of higher dimensions of the time parameter.Under the condition that the firstmoment of the infinitely divisible distribution on E^* exists,replacing the operator I_-^βby Riesz potential,Riesz poly-potential and multi-variate fractional integration opera-tor respectively,we constuct the E^*-valued L(?)vy random field,the isotropic fractionalL(?)vy random fields and anisotropic fractional L(?)vy random fields,and investigate theirdistribution and sample properties such as Euclidean invariance,long-range dependence,self-similarity.