Edward C. Waymire

On the formulation of an analytical approach to hydrologic response and similarity at the basin scale

Abstract The hydrologic response at the basin scale represents the temporally and spatially averaged behavior of water transport in a channel network. Some theoretical studies have recently appeared in the literature which attempt to formulate and solve the problem of how the averaging of the network geometry and the dynamics determines the hydrologic response. A review of these recent papers shows that the incorporation of the network geometry in terms of the Strahler ordered channels is not appropriate. An alternate analytical approach is formulated which requires the use of the path number classification for this purpose. This approach also makes it apparent that the analytical geomorphology (via Shreves postulates) provides a natural avenue for undertaking analytical investigations on the structure of the hydrologic response. These connections are explored in the vein of formulating and solving the problem of hydrologic similarity.

A cascade decomposition theory with applications to Markov and exchangeable cascades

A multiplicative random cascade refers to a positive T -martingale in the sense of Kahane on the ultrametric space T = {0, 1, . . . , b− 1}. A new approach to the study of multiplicative cascades is introduced. The methods apply broadly to the problems of: (i) non-degeneracy criterion, (ii) dimension spectra of carrying sets, and (iii) divergence of moments criterion. Specific applications are given to cascades generated by Markov and exchangeable processes, as well as to homogeneous independent cascades. 1. Positive T -martingales Positive T-martingales were introduced by Jean-Pierre Kahane as the general framework for independent multiplicative cascades and random coverings. Although originating in statistical theories of turbulence, the general framework also includes certain spin-glass and random polymer models as well as various other spatial distributions of interest in both probability theory and the physical sciences. For basic definitions, let T be a compact metric space with Borel sigmafield B, and let (Ω,F , P ) be a probability space together with an increasing sequence Fn, n = 1, 2, . . . , of sub-sigmafields of F . A positive T-martingale is a sequence {Qn} of B × F−measurable non-negative functions on T × Ω such that (i) For each t ∈ T, {Qn(t, ·) : n = 0, 1, . . . } is a martingale adapted to Fn, n = 0, 1, . . . ; (ii) For P -a.s. ω ∈ Ω, {Qn(·, ω) : n = 0, 1, . . . } is a sequence of Borel measurable non-negative real-valued functions on T . Let M(T ) denote the space of positive Borel measures on T and suppose that {Qn(t)} is a positive T -martingale. For σ ∈ M(T ) such that q(t) := EQn(t) ∈ L(σ), let σn ≡ Qnσ denote the random measure defined by Qnσ << σ and dQnσ dσ (t) := Qn(t), t ∈ T. Then, essentially by the martingale convergence theorem, one obtains a random Borel measure σ∞ ≡ Q∞σ such that for f ∈ C(T ), (1.1) lim n→∞ ∫ T f(t)Qn(t, ω)σ(dt) = ∫ T f(t)Q∞σ(dt, ω) a.s. Received by the editors August 18, 1994. 1991 Mathematics Subject Classification. Primary 60G57, 60G30, 60G42; Secondary 60K35, 60D05, 60J10, 60G09.

Occupation and local times for skew Brownian motion with applications to dispersion across an interface

This research was partially supported by the grant GrantCMG EAR-0724865 from the National Science Foundation.

Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations

A general method is developed to obtain conditions on initial data and forcing terms for the global existence of unique regular solutions to incompressible 3d Navier-Stokes equations. The basic idea generalizes a probabilistic approach introduced by LeJan and Sznitman (1997) to obtain weak solutions whose Fourier transform may be represented by an expected value of a stochastic cascade. A functional analytic framework is also developed which partially connects stochastic iterations and certain Picard iterates. Some local existence and uniqueness results are also obtained by contractive mapping conditions on the Picard iteration.

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.

A general decomposition theory for random cascades

This announcement describes a probabilistic approach to cascades which, in addition to providing an entirely probabilistic proof of the Kahane-Peyriere theorem for independent cascades, readily applies to general dependent cascades. Moreover, this unifies various seemingly disparate cascade decompositions, including Kahane’s T-martingale decomposition and dimension disintegration. 1. Brief history of the problem A theory of positive T-martingales was introduced in [K3] as the general framework for independent multiplicative cascades and random coverings. This includes spatial distributions of interest in both probability theory and in the physical sciences, e.g. [CCD, CK, DE, DG, DF, DM, F, GW, MS, MW, TLS, PW, S]. For basic definitions, let T be a locally compact metric space with Borel sigmafield B, and let (Ω,F , P ) be a probability space together with an increasing sequence Fn, n = 1, 2, . . . , of sub-sigmafields of F . A positive T-martingale is a sequence {Qn} of non-negative functions on T × Ω such that: (i) for each t ∈ T, {Qn(t, ·) : n = 1, 2, . . . } is a martingale adapted to Fn, n = 1, 2, . . . ; (ii) for P -a.s. ω ∈ Ω, {Qn(·, ω) : n = 1, 2, . . .} is a sequence of Borel measurable nonnegative real-valued functions on T . Let M(T ) denote the space of positive Borel measures on T , and suppose that {Qn(t)} is a positive T -martingale. For σ ∈ M(T ) let Qnσ denote the random measure defined by Qnσ << σ and dQnσ dσ (t) := Qn(t), t ∈ T. Denote the space of bounded continuous functions on T by CB(T ). Then, essentially by the martingale convergence theorem, one obtains a random Borel measure Q∞σ such that with probability one [K3]

On Weighted Heights of Random Trees

AbstractConsider the family treeT of a branching process starting from a single progenitor and conditioned to havev=v(T) edges (total progeny). To each edge we associate a weightW(e). The weights are i.i.d. random variables and independent ofT. The weighted height of a self-avoiding path inT starting at the root is the sum of the weights associated with the path. We are interested in the asymptotic distribution of the maximum weighted path height in the limit asv=n→∞. Depending on the tail of the weight distribution, we obtain the limit in three cases. In particular ify2P(W(e)> y)→0, then the limit distribution depends strongly on the tree and, in fact, is the distribution of the maximum of a Brownian excursion. If the tail of the weight distribution is regularly varying with exponent 0≤α<2, then the weight swamps the tree and the answer is the asymptotic distribution of the maximum edge weight in the tree. There is a borderline case, namely,P(W(e)> y)∼cy−2 asy→∞, in which the limit distribution exists but involves both the tree and the weights in a more complicated way.

Probability & incompressible Navier-Stokes equations: An overview of some recent developments

This is largely an attempt to provide probabilists some orientation to an important class of non-linear partial differential equations in applied mathematics, the incompressible Navier-Stokes equations. Particular focus is given to the probabilistic framework introduced by LeJan and Sznitman (1997) and extended by Bhattacharya et al (2003, 2004). In particular this is an effort to provide some foundational facts about these equations and an overview of some recent results with an indication of some new directions for probabilistic consideration.

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.

A note on the distribution of integrals of geometric Brownian motion

The purpose of this note is to identify an interesting and surprising duality between the equations governing the probability distribution and expected value functional of the stochastic process defined by where {Zs: s[greater-or-equal, slanted]0} is a one-dimensional Brownian motion with drift coefficient [mu] and diffusion coefficient [sigma]2. In particular, both expected values of the form v(t,x):=Ef(x+At), f homogeneous, as well as the probability density a(t,y) dy:=P(At[set membership, variant]dy) are shown to be governed by a pair of linear parabolic partial differential equations. Although the equations are not the backward/forward adjoint pairs one would naturally have in the general theory of Markov processes, unifying and remarkably simple derivations of these equations are provided.