Studies in non-Gaussian analysis

This 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.

Susceptibility of mosquito vectors to Dirofilaria immitis on Madeira Island, Portugal

Dirofilaria immitis (Leidy, 1856), an agent of heartworm disease, is an important parasite from both the veterinary standpoint and as a model to study human filariasis. It is a mosquito-borne filarial nematode which inhabits the right ventricle and pulmonary arteries of dogs. D. immitis is an important disease agent on Madeira Island with about 30% of dogs testing positive for this worm. Nevertheless, the vectors of this parasite in Madeira have never been studied, nor has the interaction between pathogen and vector, or the environmental variables that might influence heartworm transmission. Innate susceptibility to infection is only one component of vector competence, and field isolation of naturally infected mosquitoes has shown the capability of D. immitis to exploit a great diversity of vector species under natural conditions. The purpose of this work was to determine which mosquitoes are vectors of heartworm disease, the relation between population density and environment, and the association between immune response of the vector to the filarial parasite. Seasonal abundance of Culex theileri and Culex pipiens molestus was studied. Correlation and canonical correspondence analysis were performed using abundance data of these two species with selected weather variables, including mean temperature, relative humidity and accumulated precipitation. The most important factor determining Cx. theileri abundance was accumulated precipitation, while Cx. pipiens molestus abundance did not have any relationship with weather variables. Field studies were performed to verify whether Cx. theileri Theobald functions as a natural vector of D. immitis on Madeira Island, Portugal. Cx. theileri tested positive for D. immitis for the first time. The same study was made regarding Cx. p. molestus. Two abnormal L2 stage filarial worms were found in Malpighian tubules in field caught Cx. p. molestus. In the laboratory, two strains of Cx. p. molestus were studied for their susceptibility to D. immitis. None presented infective-stage larvae. Finally, because Cx. p. molestus is an autogenous mosquito, we evaluated the reproductive costs when this mosquito mounts an immune response against D. immitis in the absence of a blood meal. This mosquito showed an active immune response when inoculated intrathoracically with microfilariae (mf) of the heartworm. The ovaries from mosquitoes undergoing melanotic encapsulation developed more eggs than those which could not melanize the mf. This fact is contradictory with some previous studies of reproductive costs in Armigeres subalbatus and Ochlerotatus trivittatus, and it was the first time that an autogenous mosquito was used to study this subject.

Affine and curvature collineations in space-time

Graham Hall