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In this paper, we consider families of time Markov fields (or reciprocal classes) which have the same bridges as a Brownian diffusion. We characterize each class as the set of solutions of an integration by parts formula on the space of continuous paths C[0; 1]; R-d) Our techniques provide a characterization of gradient diffusions by a duality formula and, in case of reversibility, a generalization of a result of Kolmogorov.
We develop a cluster expansion in space-time for an infinite-dimensional system of interacting diffusions where the drift term of each diffusion depends on the whole past of the trajectory; these interacting diffusions arise when considering the Langevin dynamics of a ferromagnetic system submitted to a disordered external magnetic field.
The authors analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the d-dimensional lattice. In the first part of the paper, these processes are characterized as Gibbs states on path spaces. In the second part of the paper, they study the Gibbsian character on R^{Z^d} of the law at time t of the infinite-dimensional diffusion X(t), when the initial law is Gibbsian. AMS Classifications: 60G15 , 60G60 , 60H10 , 60J60
We prove in this paper an existence result for infinite-dimensional stationary interactive Brownian diffusions. The interaction is supposed to be small in the norm ||.||∞ but otherwise is very general, being possibly non-regular and non-Markovian. Our method consists in using the characterization of such diffusions as space-time Gibbs fields so that we construct them by space-time cluster expansions in the small coupling parameter.
Adsorption layers of soluble surfactants enable and govern a variety of phenomena in surface and colloidal sciences, such as foams. The ability of a surfactant solution to form wet foam lamellae is governed by the surface dilatational rheology. Only systems having a non-vanishing imaginary part in their surface dilatational modulus, E, are able to form wet foams. The aim of this thesis is to illuminate the dissipative processes that give rise to the imaginary part of the modulus. There are two controversial models discussed in the literature. The reorientation model assumes that the surfactants adsorb in two distinct states, differing in their orientation. This model is able to describe the frequency dependence of the modulus E. However, it assumes reorientation dynamics in the millisecond time regime. In order to assess this model, we designed a SHG pump-probe experiment that addresses the orientation dynamics. Results obtained reveal that the orientation dynamics occur in the picosecond time regime, being in strong contradiction with the two states model. The second model regards the interface as an interphase. The adsorption layer consists of a topmost monolayer and an adjacent sublayer. The dissipative process is due to the molecular exchange between both layers. The assessment of this model required the design of an experiment that discriminates between the surface compositional term and the sublayer contribution. Such an experiment has been successfully designed and results on elastic and viscoelastic surfactant provided evidence for the correctness of the model. Because of its inherent surface specificity, surface SHG is a powerful analytical tool that can be used to gain information on molecular dynamics and reorganization of soluble surfactants. They are central elements of both experiments. However, they impose several structural elements of the model system. During the course of this thesis, a proper model system has been identified and characterized. The combination of several linear and nonlinear optical techniques, allowed for a detailed picture of the interfacial architecture of these surfactants.