Jorge M. Ramirez

A Generalized Taylor–Aris Formula and Skew Diffusion

This paper concerns the Taylor–Aris dispersion of a dilute solute concentration immersed in a highly heterogeneous fluid flow having possibly sharp interfaces (discontinuities) in the diffusion coefficient and flow velocity. The focus is twofold: (i) calculation of the longitudinal effective dispersion coefficient and (ii) sample path analysis of the underlying stochastic process governing the motion of solute particles. Essentially complete solutions are obtained for both problems.

Multi-skewed Brownian motion and diffusion in layered media

Multi-skewed Brownian motion B = {B t : t > 0} with skewness sequence α = {αk : k ∈ Z} and interface set S = {xk : k ∈ Z} is the solution to Xt = X0 + Bt + ∫ R L(t, x) dμ(x) with μ = ∑ k∈Z (2αk − 1)δxk . We assume that αk ∈ (0, 1) \ { 1 2 } and that S has no accumulation points. The process B generalizes skew Brownian motion to the case of an infinite set of interfaces. Namely, the paths of B behave like Brownian motion in R r S, and on B 0 = xk, the probability of reaching xk + δ before xk − δ is αk , for any δ small enough, and k ∈ Z. In this paper, a thorough analysis of the structure of B is undertaken, including the characterization of its infinitesimal generator and conditions for recurrence and positive recurrence. The associated Dirichlet form is used to relate B to a diffusion process with piecewise constant diffusion coefficient. As an application, we compute the asymptotic behavior of a diffusion process corresponding to a parabolic partial differential equation in a two-dimensional periodic layered geometry.

Advection–Dispersion Across Interfaces

This article concerns a systemic manifestation of small scale in- terfacial heterogeneities in large scale quantities of interest to a variety of diverse applications spanning the earth, biological and ecological sciences. Beginning with formulations in terms of partial differential equations gov- erning the conservative, advective-dispersive transport of mass concentra- tions in divergence form, the specific interfacial heterogeneities are intro- duced in terms of (spatial) discontinuities in the diffusion coefficient across a lower-dimensional hypersurface. A pathway to an equivalent stochastic for- mulation is then developed with special attention to the interfacial effects in various functionals such as first passage times, occupation times and local times. That an appreciable theory is achievable within a framework of ap- plications involving one-dimensional models having piecewise constant co- efficients greatly facilitates our goal of a gentle introduction to some rather dramatic mathematical consequences of interfacial effects that can be used to predict structure and to inform modeling.

Population persistence under advection–diffusion in river networks

An integro-differential equation on a tree graph is used to model the time evolution and spatial distribution of a population of organisms in a river network. Individual organisms become mobile at a constant rate, and disperse according to an advection–diffusion process with coefficients that are constant on the edges of the graph. Appropriate boundary conditions are imposed at the outlet and upstream nodes of the river network. The local rates of population growth/decay and that by which the organisms become mobile, are assumed constant in time and space. Imminent extinction of the population is understood as the situation whereby the zero solution to the integro-differential equation is stable. Lower and upper bounds for the eigenvalues of the dispersion operator, and related Sturm–Liouville problems are found. The analysis yields sufficient conditions for imminent extinction and/or persistence in terms of the values of water velocity, channel length, cross-sectional area and diffusivity throughout the river network.

Multiplicative cascades applied to PDEs (two numerical examples)

Numerical approximations to the Fourier transformed solution of partial differential equations are obtained via Monte Carlo simulation of certain random multiplicative cascades. Two particular equations are considered: linear diffusion equation and viscous Burgers equation. The algorithms proposed exploit the structure of the branching random walks in which the multiplicative cascades are defined. The results show initial numerical approximations with errors less than 5% in the leading Fourier coefficients of the solution. This approximation is then improved substantially using a Picard iteration scheme on the integral equation associated with the representation of the respective PDE in Fourier space.

Continuity of Local Time: An Applied Perspective

Continuity of local time for Brownian motion ranks among the most notable mathematical results in the theory of stochastic processes. This article addresses its implications from the point of view of applications. In particular an extension of previous results on an explicit role of continuity of (natural) local time is obtained for applications to recent classes of problems in physics, biology and finance involving discontinuities in a dispersion coefficient. The main theorem and its corollary provide physical principles that relate macro scale continuity of deterministic quantities to micro scale continuity of the (stochastic) local time.

On the path properties of a lacunary power series

A power series f(z) which converges in D = {|z| < 1} maps the radii [0, ζ) onto paths Γ(ζ), ζ ∈ T = ∂D. These are studied under several aspects in the case of the special lacunary series f(z) = z+ z2 + z4 + z8 + . . .. First the Γ(ζ) are considered as random functions on the probability space (T,B,mes /2π), where B is the σ-algebra of Borel sets and mes the Lebesgue measure. Then analytical properties of the Γ(ζ) are discussed which hold on subsets A of T with Hausdorff dimension 1 in spite of mesA = 0. Furthermore, estimates of the derivative of f and of the arc length of sections of the Γ(ζ) are given. Finally, these results are used to derive connections between the distribution of critical points of f and the overall behaviour of the paths.