Integral Transformations of Volterra Gaussian Processes

We study integral representations of Gaussian processes with a pre-specified law in terms of other Gaussian processes. The dissertation consists of an introduction and of four research articles. In the introduction, we provide an overview about Volterra Gaussian processes in general, and fractional Brownian motion in particular. In the first article, we derive a finite interval integral transformation, which changes fractional Brownian motion with a given Hurst index into fractional Brownian motion with an other Hurst index. Based on this transformation, we construct a prelimit which formally converges to an analogous, infinite interval integral transformation. In the second article, we prove this convergence rigorously and show that the infinite interval transformation is a direct consequence of the finite interval transformation. In the third article, we consider general Volterra Gaussian processes. We derive measure-preserving transformations of these processes and their inherently related bridges. Also, as a related result, we obtain a Fourier-Laguerre series expansion for the first Wiener chaos of a Gaussian martingale. In the fourth article, we derive a class of ergodic transformations of self-similar Volterra Gaussian processes.

The Emerging Phenotype: Ecological Genetics of Color, Form and Sex in the Common Frog

One of the main aims of evolutionary biology is to explain why organisms vary phenotypically as they do. Proximately, this variation arises from genetic differences and from environmental influences, the latter of which is referred to as phenotypic plasticity. Phenotypic plasticity is thus a central concept in evolutionary biology, and understanding its relative importance in causing the phenotypic variation and differentiation is important, for instance in anticipating the consequences of human induced environmental changes. The aim of this thesis was to study geographic variation and local adaptation, as well as sex ratios and environmental sex reversal, in the common frog (Rana temporaria). These themes cover three different aspects of phenotypic plasticity, which emerges as the central concept for the thesis. The first two chapters address geographic variation and local adaptation in two potentially thermally adaptive traits, namely the degree of melanism and the relative leg length. The results show that although there is an increasing latitudinal trend in the degree of melanism in wild populations across Scandinavian Peninsula, this cline has no direct genetic basis and is thus environmentally induced. The second chapter demonstrates that although there is no linear, latitudinally ordered phenotypic trend in relative leg length that would be expected under Allen s rule an ecogeographical rule linking extremity length to climatic conditions there seems to be such a trend at the genetic level, hidden under environmental effects. The first two chapters thus view phenotypic plasticity through its ecological role and evolution, and demonstrate that it can both give rise to phenotypic variation and hide evolutionary patterns in studies that focus solely on phenotypes. The last three chapters relate to phenotypic plasticity through its ecological and evolutionary role in sex determination, and consequent effects on population sex ratio, genetic recombination and the evolution of sex chromosomes. The results show that while sex ratios are strongly female biased and there is evidence of environmental sex reversals, these reversals are unlikely to have caused the sex ratio skew, at least directly. The results demonstrate that environmental sex reversal can have an effect on the evolution of sex chromosomes, as the recombination patterns between them seem to be controlled by phenotypic, rather than genetic, sex. This potentially allows Y chromosomes to recombine, lending support for the recent hypothesis suggesting that sex-reversal may play an important role on the rejuvenation of Y chromosomes.