Interfacial Phenomena and Natural Local Time

This article addresses a modification of local time for stochastic processes, to be referred to as `natural local time'. It is prompted by theoretical developments arising in mathematical treatments of recent experiments and observations of phenomena in the geophysical and biological sciences pertaining to dispersion in the presence of an interface of discontinuity in dispersion coefficients. The results illustrate new ways in which to use the theory of stochastic processes to infer macro scale parameters and behavior from micro scale observations in particular heterogeneous environments

Skew Brownian motion and branching processes applied to diffusion-advection in heterogenous media and fluid flow

This thesis contains three manuscripts addressing the application of stochastic processes to the analysis and solution of partial differential equations (PDEs) in mathematical physics. In the first manuscript, one dimensional diffusion and Burgers equation are considered. The Fourier transform of the solution to each PDE is represented as the expected value of a multiplicative functional on a branching stochastic process. Monte Carlo simulation schemes are then developed to perform accurate numerical calculations of the solution. The second manuscript considers an advection-diffusion PDE in a cylinder where the diffusion coefficient and flow velocity are constant in the direction of fluid flow, but are arbitrarily non-smooth in the transversal direction. The stochastic process associated with the PDE is constructed as a diffusion process with the appropriate infinitesimal generator. The properties of ergodic Markov processes are then used to obtain a homogenization result for the solution of the PDE. The third manuscript studies the stochastic process associated with the one-dimensional diffusion equation in the case where the diffusion coefficient is piecewise constant with a countable set of discontinuities. The resulting process generalizes skew Brownian motion to the case of countable many interfaces. Finally, some applications to advection-diffusion phenomena in two-dimensional layered media are outlined

On the constructions of the skew Brownian motion

This article summarizes the various ways one may use to construct the Skew Brownian motion, and shows their connections. Recent applications of this process in modelling and numerical simulation motivates this survey. This article ends with a brief account of related results, extensions and applications of the Skew Brownian motion.Comment: Published at http://dx.doi.org/10.1214/154957807000000013 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Monte Carlo methods for discontinuous media

International audienceThis note aims to give a brief account on some recent progress of the simulation techniques of stochastic processes associated to divergence-form opertors with discontinuous coefficients, such as the one used in the Darcy law

Local conservation laws of continuous Galerkin method for the incompressible Navier--Stokes equations in EMAC form

We consider {\it local} balances of momentum and angular momentum for the incompressible Navier-Stokes equations. First, we formulate new weak forms of the physical balances (conservation laws) of these quantities, and prove they are equivalent to the usual conservation law formulations. We then show that continuous Galerkin discretizations of the Navier-Stokes equations using the EMAC form of the nonlinearity preserve discrete analogues of the weak form conservation laws, both in the Eulerian formulation and the Lagrangian formulation (which are not equivalent after discretizations). Numerical tests illustrate the new theory

Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?

The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important problems in these fields are discussed