Probabilistic foundation of nonlocal diffusion and formulation and analysis for elliptic problems on uncertain domains

In the first part of this dissertation, we study the nonlocal diffusion equation with so-called Levy measure nu as the master equation for a pure-jump Levy process. In the case nu is not in L 1(R), a relationship to fractional diffusion is established in a limit of vanishing nonlocality, which implies the convergence of a compound Poisson process to a stable process. In the case nu is not in L 1(R), the smoothing of the nonlocal operator is shown to correspond precisely to the activity of the underlying Levy process and the variation of its sample paths. We introduce volume-constrained nonlocal diffusion equations and demonstrate that they are the master equations for Levy processes restricted to a bounded domain. The ensuing variational formulation and conforming finite element method provide a powerful tool for studying both Levy processes and fractional diffusion on bounded, nonsimple geometries with volume constraints.;In the second part of this dissertation, we consider the problem of estimating the distribution of a quantity of interest computed from the solution of an elliptic partial differential equation posed on a domain o(theta) with a randomly perturbed boundary, where o(theta) is a random vector with given probability structure. We construct a piecewise smooth transformation from a partition of o(theta) to a reference domain in order to avoid the complications associated with solving the problems on o(theta). The domain decomposition formulation is exploited by localizing the effect of the randomness to boundary elements in order to achieve a computationally efficient Monte Carlo sampling procedure. An a posteriori error analysis for the approximate distribution, which includes a deterministic error for each sample and a stochastic error from the effect of sampling, is also presented. We thus provide an efficient means to estimate the distribution of a quantity of interest via a Monte Carlo sampling procedure while also providing a posteriori error estimates for each sample.