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.

Physiological and structural aspects of fruit and vegetable mild processing

Over the past years fruit and vegetable industry has become interested in the application of both osmotic dehydration and vacuum impregnation as mild technologies because of their low temperature and energy requirements. Osmotic dehydration is a partial dewatering process by immersion of cellular tissue in hypertonic solution. The diffusion of water from the vegetable tissue to the solution is usually accompanied by the simultaneous solutes counter-diffusion into the tissue. Vacuum impregnation is a unit operation in which porous products are immersed in a solution and subjected to a two-steps pressure change. The first step (vacuum increase) consists of the reduction of the pressure in a solid-liquid system and the gas in the product pores is expanded, partially flowing out. When the atmospheric pressure is restored (second step), the residual gas in the pores compresses and the external liquid flows into the pores. This unit operation allows introducing specific solutes in the tissue, e.g. antioxidants, pH regulators, preservatives, cryoprotectancts. Fruit and vegetable interact dynamically with the environment and the present study attempts to enhance our understanding on the structural, physico-chemical and metabolic changes of plant tissues upon the application of technological processes (osmotic dehydration and vacuum impregnation), by following a multianalytical approach. Macro (low-frequency nuclear magnetic resonance), micro (light microscopy) and ultrastructural (transmission electron microscopy) measurements combined with textural and differential scanning calorimetry analysis allowed evaluating the effects of individual osmotic dehydration or vacuum impregnation processes on (i) the interaction between air and liquid in real plant tissues, (ii) the plant tissue water state and (iii) the cell compartments. Isothermal calorimetry, respiration and photosynthesis determinations led to investigate the metabolic changes upon the application of osmotic dehydration or vacuum impregnation. The proposed multianalytical approach should enable both better designs of processing technologies and estimations of their effects on tissue.