On the Wick product and Wong-Zakai approximations.

This thesis is a compilation of 6 papers that the author has written together with Alberto Lanconelli (chapters 3, 5 and 8) and Hyun-Jung Kim (ch 7). The logic thread that link all these chapters together is the interest to analyze and approximate the solutions of certain stochastic differential equations using the so called Wick product as the basic tool. In the first chapter we present arguably the most important achievement of this thesis; namely the generalization to multiple dimensions of a Wick-Wong-Zakai approximation theorem proposed by Hu and Oksendal. By exploiting the relationship between the Wick product and the Malliavin derivative we propose an original reduction method which allows us to approximate semi-linear systems of stochastic differential equations of the Itô type. Furthermore in chapter 4 we present a non-trivial extension of the aforementioned results to the case in which the system of stochastic differential equations are driven by a multi-dimensional fraction Brownian motion with Hurst parameter bigger than 1/2. In chapter 5 we employ our approach and present a “short time” approximation for the solution of the Zakai equation from non-linear filtering theory and provide an estimation of the speed of convergence. In chapters 6 and 7 we study some properties of the unique mild solution for the Stochastic Heat Equation driven by spatial white noise of the Wick-Skorohod type. In particular by means of our reduction method we obtain an alternative derivation of the Feynman-Kac representation for the solution, we find its optimal Hölder regularity in time and space and present a Feynman-Kac-type closed form for its spatial derivative. Chapter 8 treats a somewhat different topic; in particular we investigate some probabilistic aspects of the unique global strong solution of a two dimensional system of semi-linear stochastic differential equations describing a predator-prey model perturbed by Gaussian noise.

Stochastic physics and representation of El Niño Southern Oscillation in climate models

El Niño-Southern Oscillation (ENSO) è il maggiore fenomeno climatico che avviene a livello dell’Oceano Pacifico tropicale e che ha influenze ambientali, climatiche e socioeconomiche a larga scala. In questa tesi si ripercorrono i passi principali che sono stati fatti per tentare di comprendere un fenomeno così complesso. Per prima cosa, si sono studiati i meccanismi che ne governano la dinamica, fino alla formulazione del modello matematico chiamato Delayed Oscillator (DO) model, proposto da Suarez e Schopf nel 1988. In seguito, per tenere conto della natura caotica del sistema studiato, si è introdotto nel modello lo schema chiamato Stochastically Perturbed Parameterisation Tendencies (SPPT). Infine, si sono portati due esempi di soluzione numerica del DO, sia con che senza l’introduzione della correzione apportata dallo schema SPPT, e si è visto in che misura SPPT porta reali miglioramenti al modello studiato.

The Role Of Edaphic Environment And Climate In Structuring Phylogenetic Pattern In Seasonally Dry Tropical Plant Communities.

Seasonally dry tropical plant formations (SDTF) are likely to exhibit phylogenetic clustering owing to niche conservatism driven by a strong environmental filter (water stress), but heterogeneous edaphic environments and life histories may result in heterogeneity in degree of phylogenetic clustering. We investigated phylogenetic patterns across ecological gradients related to water availability (edaphic environment and climate) in the Caatinga, a SDTF in Brazil. Caatinga is characterized by semiarid climate and three distinct edaphic environments - sedimentary, crystalline, and inselberg -representing a decreasing gradient in soil water availability. We used two measures of phylogenetic diversity: Net Relatedness Index based on the entire phylogeny among species present in a site, reflecting long-term diversification; and Nearest Taxon Index based on the tips of the phylogeny, reflecting more recent diversification. We also evaluated woody species in contrast to herbaceous species. The main climatic variable influencing phylogenetic pattern was precipitation in the driest quarter, particularly for herbaceous species, suggesting that environmental filtering related to minimal periods of precipitation is an important driver of Caatinga biodiversity, as one might expect for a SDTF. Woody species tended to show phylogenetic clustering whereas herbaceous species tended towards phylogenetic overdispersion. We also found phylogenetic clustering in two edaphic environments (sedimentary and crystalline) in contrast to phylogenetic overdispersion in the third (inselberg). We conclude that while niche conservatism is evident in phylogenetic clustering in the Caatinga, this is not a universal pattern likely due to heterogeneity in the degree of realized environmental filtering across edaphic environments. Thus, SDTF, in spite of a strong shared environmental filter, are potentially heterogeneous in phylogenetic structuring. Our results support the need for scientifically informed conservation strategies in the Caatinga and other SDTF regions that have not previously been prioritized for conservation in order to take into account this heterogeneity.

Metodologia para a suavização de dados biomecanicos por função não parametrica ponderada local robusta

Universidade Estadual de Campinas. Faculdade de Educação Física