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- Studies in non-Gaussian analysisPublication . Silva, José Luís daThis thesis presents general methods in non-Gaussian analysis in infinite dimensional spaces. As main applications we study Poisson and compound Poisson spaces. Given a probability measure μ on a co-nuclear space, we develop an abstract theory based on the generalized Appell systems which are bi-orthogonal. We study its properties as well as the generated Gelfand triples. As an example we consider the important case of Poisson measures. The product and Wick calculus are developed on this context. We provide formulas for the change of the generalized Appell system under a transformation of the measure. The L² structure for the Poisson measure, compound Poisson and Gamma measures are elaborated. We exhibit the chaos decomposition using the Fock isomorphism. We obtain the representation of the creation, annihilation operators. We construct two types of differential geometry on the configuration space over a differentiable manifold. These two geometries are related through the Dirichlet forms for Poisson measures as well as for its perturbations. Finally, we construct the internal geometry on the compound configurations space. In particular, the intrinsic gradient, the divergence and the Laplace-Beltrami operator. As a result, we may define the Dirichlet forms which are associated to a diffusion process. Consequently, we obtain the representation of the Lie algebra of vector fields with compact support. All these results extends directly for the marked Poisson spaces.
- Funções teste e funções generalizadas em dimensão 1. Descrição e caracterizaçãoPublication . Ferreira, Jorge; Gouveia, Délia; Reis, Maurício; Silva, José LuísNeste trabalho introduzimos uma família de espaços de funções teste definidas em R associadas à medida Gaussiana µ. Por dualidade obtemos a correspondente família de espaços de distribuições (ou funções generaliza das). A caracterização destas famílias à custa de funções inteiras com um certo tipo de crescimento é feita usando a transformada S. Como exemplo de aplicação apresentamos o produto de Wick entre funções generalizadas.
- The Feynman integrand for the perturbed harmonic oscillator as a Hida distributionPublication . Cunha, Mário; Drumond, Custódia; Leukert, Peter; Silva, José Luís; Westerkamp, WernerWe rwiew some basic notions and results of White Noise Analysis that are used in the con struction of the Feynman integrand as a generalized White Noise functional. We show that the Feyn man integrand for the harmonic oscillator in an external potential is a Hida distri
- Intersection local times of fractional Brownian motions with with H ∈ (0,1) as generalized white noise functionalsPublication . Drumond, Custódia; Oliveira, Maria João; Silva, José Luís daIn Rd, for any dimension d ≥ 1, expansions of self-intersection local times of fractional Brownian motions with arbitrary Hurst coefficients in (0,1) are presented. The expansions are in terms of Wick powers of white noises (corresponding to multiple Wiener integrals), being well defined in the sense of generalized white noise functionals.
- Local times for grey Brownian motionPublication . Silva, J. L. daIn this paper we study the grey Brownian motion, namely its representation and local time. First it is shown that grey Brownian motion may be represented in terms of a standard Brownian motion and then using a criterium of S. Berman, Trans. Amer. Math. Soc., 137, 277–299 (1969), we show that grey Brownian motion admits a λ-square integrable local time almost surely (λ denotes the Lebesgue measure). As a consequence we obtain the occupation formula and state possible generalizations of these results.
- Compound Poisson processes: potentials, Green measures and random timesPublication . Yuri Kondratiev; José L. da Silva; Luís da Silva, JoséIn this paper we study the existence of Green measures for Markov processes with a nonlocal jump generator. The non-singular jump kernel has no second moment and satisfies a suitable condition on its Fourier transform. We also study the same problem for certain classes of random time changes Markov processes with jump generator.