Stochastics

We investigate ponctual as well as L 2 distances of some stochastic processes with values in the group of homeomorphisms of a compact manifold including processes modelling time evolution of fluids. These processes are associated with operators of the form Laplace–Beltrami plus a first-order term. Several constructions are presented, in particular via coupling methods, the corresponding behaviour of the distance depending on the construction and on the drift properties.

We investigate the invariant probabilities of a possible degenerate diffusion process on a manifold. Using the support theorems of stroock, Varadhan and Kunita, the possible candidates for supports of invariant probabilities can be characterized as the invariant ...

One of the main challenges in solid mechanics lies in the passage from a heterogeneous microstructure to an approximating continuum model. In many cases (e.g. stochastic finite elements, statistical fracture mechanics), the interest lies in resolution of stress and other dependent fields over scales not infinitely larger than the typical microscale. This may be accomplished with the help of a meso-scale window which becomes the classical representative volume element (RVE) in the infinite limit. It turns out that the material properties at such a mesoscale cannot be uniquely approximated by a random field of stiffness/compliance with locally isotropic realizations, but rather two random continuum fields with locally anisotropic realizations, corresponding, respectively, to essential and natural boundary conditions on the meso-scale, need to be introduced to bound the material response from above and from below. We study the first-and second-order characteristics of these two meso-scale random fields for anti-plane elastic response of random matrix-inclusion composites over a wide range of contrasts and aspect ratios. Special attention is given to the convergence of effective responses obtained from the essential and natural boundary conditions, which sheds light on the minimum size of an RVE. Additionally, the spatial correlation structure of the crack density tensor with the meso-scale moduli is studied.

The incompressible Navier-Stokes equations (the momentum, continuity and scalar transport equations) are the fundamental equations of fluid mechanics. Of great importance to weather and climate studies is the thermohaline circulation, which is affected by gravity currents, hence we chose it as our application. First, we studied the Navier-Stokes equations (without scalar transport) using non-autonomous dynamical systems techniques, and showed the existence of recurrent or Poisson stable motions under recurrent or Poisson stable forcing, respectively. This was motivated by observed periodic and recurrent motions in nature. Next, we investigated the coupled Navier-Stokes and scalar transport equations (we may take the scalar to be salinity, say), with spatially correlated white noise on the boundary. We employed random dynamical system ideas, and showed that this system is ergodic under suitable conditions for mean salinity input flux on the boundary, Prandtl number and covariance of the noise. This addition of a random term to the boundary conditions was motivated by observed seasonal variations in the salinity flux in gravity currents. The final part of this thesis are numerical simulations studying the effects of different boundary conditions on the entrainment behavior, salinity distribution and salinity transport properties of gravity currents. The finding is that gravity currents developing under Neumann and Dirichlet boundary conditions differ most in the way they transport salinity from the middle salinity parts (roughly the middle of the current) towards the fresher part (roughly the top of the current). This study contributes to understanding the behavior of the Navier-Stokes Equations under time-periodic forcings, uncertain boundary conditions, and how gravity currents are affected by different boundary conditions.