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 [Probab. Theory Related Fields 109 (1997) 343-366] and extended by Bhattacharya et al. [Trans. Amer. Math. Soc. 355 (2003) 5003-5040; IMA Vol. Math. Appl., vol. 140, 2004, in press]. 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.Comment: Published at http://dx.doi.org/10.1214/154957805100000078 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

On Normalized Multiplicative Cascades under Strong Disorder

Multiplicative cascades, under weak or strong disorder, refer to sequences of positive random measures $\mu_{n,\beta}, n = 1,2,\dots$, parameterized by a positive disorder parameter $\beta$, and defined on the Borel $\sigma$-field ${\mathcal B}$ of $\partial T = \{0,1,\dots b-1\}^\infty$ for the product topology. The normalized cascade is defined by the corresponding sequence of random probability measures $prob_{n,\beta}:= Z_{n,\beta}^{-1}\mu_{n,\beta}, n = 1,2\dots,$ normalized to a probability by the partition function $Z_{n,\beta}$. In this note, a recent result of Madaule (2011) is used to explicitly construct a family of tree indexed probability measures $prob_{\infty,\beta}$ for strong disorder parameters $\beta > \beta_c$, almost surely defined on a common probability space. Moreover, viewing $\{prob_{n,\beta}: \beta > \beta_c\}_{n=1}^\infty$ as a sequence of probability measure valued stochastic process leads to finite dimensional weak convergence in distribution to a probability measure valued process $\{prob_{\infty,\beta}: \beta > \beta_c\}$. The limit process is constructed from the tree-indexed random field of derivative martingales, and the Brunet-Derrida-Madaule decorated Poisson process. A number of corollaries are provided to illustrate the utility of this construction.Comment: 11 pages, 1 figure, submitte

Indiana Biobank and INresearch.org

poster abstractIndiana Biobank was established in July 2010 as a resource for all investigators throughout the state of Indiana. The biobank seeks to develop a statewide collection of bio specimens as a research tool to enhance translational research, to improve Hoosier health, and to promote personalized therapy. This large collection of samples is stored, catalogued, characterized and delivered to scientists to carry out translational research to improve the health of Hoosiers. INresearch.org is a secure, password-protected registry of Indiana resident volunteers who want to participate in health-related research. The goal of this registry is to match interested volunteers with researchers. The goal of INresearch.org is to improve health care by discovering new treatments and cures

Scaling and Multiscaling Exponents in Networks and Flows

The main focus of this paper is on mathematical theory and methods which have a direct bearing on problems involving multiscale phenomena. Modern technology is refining measurement and data collection to spatio-temporal scales on which observed geophysical phenomena are displayed as intrinsically highly variable and intermittant heirarchical structures,e.g. rainfall, turbulence, etc. The heirarchical structure is reflected in the occurence of a natural separation of scales which collectively manifest at some basic unit scale. Thus proper data analysis and inference require a mathematical framework which couples the variability over multiple decades of scale in which basic theoretical benchmarks can be identified and calculated. This continues the main theme of the research in this area of applied probability over the past twenty years

A Large Deviation Rate and Central Limit Theorem for Horton Ratios

Although originating in hydrology, the classical Horton analysis is based on a geometric progression that is widely used in the empirical analysis of branching patterns found in biology, atmospheric science, plant pathology, etc., and more recently in tree register allocation in computer science. The main results of this paper are a large deviation rate and a central limit theorem for Horton bifurcation ratios in a standard network model. The methods are largely self-contained. In particular, derivations of some previously known results of the theory are indicated along the way

Advection-Dispersion Across Interfaces

This article concerns a systemic manifestation of small scale interfacial 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 governing the conservative, advective-dispersive transport of mass concentrations in divergence form, the specific interfacial heterogeneities are introduced in terms of (spatial) discontinuities in the diffusion coefficient across a lower-dimensional hypersurface. A pathway to an equivalent stochastic formulation 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 applications involving one-dimensional models having piecewise constant coefficients 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.Comment: Published in at http://dx.doi.org/10.1214/13-STS442 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

We are making corrections and acknowledging colleagues that pointed out mistakes in our recent paper titled "Occupation and local times for skew Brownian motion with applications to dispersion across an interface" which was published in Annals of Applied Probability (2011) 21(1) 183-214. Specifically the corrections are: 1. The restriction of $\gamma$ to non negative values in Theorem 1.3 is not needed. But one has probabilistic interpretation only when $\gamma$ is non negative. 2. State the correct formulas in Corollary 3.3 as their were computational errors in the original formulas. We thank Pierre Etoir\'e and Miguel Martinez for pointing out these errors.Comment: Published in at http://dx.doi.org/10.1214/11-AAP775 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org