Uniqueness for a class of one-dimensional stochastic PDEs using moment duality

We establish a duality relation for the moments of bounded solutions to a class of one-dimensional parabolic stochastic partial differential equations. The equations are driven by multiplicative space-time white noise, with a non-Lipschitz multiplicative functional. The dual process is a system of branching Brownian particles. The same method can be applied to show uniqueness in law for a class of non-Lipschitz finite dimensional stochastic differential equations

Interests and Expectations of the Confederated Tribes of the Umatilla Indian Reservation Regarding Hanford and Hanford-Affected Lands

This document updates the previous document “Scoping Report: Nuclear Risks in Tribal Communities” prepared by Confederated Tribes of the Umatilla Indian Reservation (CTUIR) in 1995. At the time, “no comprehensive or sitewide evaluation of risks and costs has been performed at Hanford.” A decade later, this is still true. It is also still true that “a full risk picture must include addressing the impacts over time.” This report provides a more docused perspective on how to establish both technically and politically defensible environmental management approach in an era of continued fiscal constraints. This was true in 1995 and is even more constraining in 2006. A major stakeholder-driven document was written in 1996 (Columbia River Comprehensive Impact Assessment, Part II). We believe that an investment by the Department of Energy (DOE) in a more effective and efficient risk assessment approach as well as increased emphasis on integration of Natural Resource Damage Assessment (NRDA) and Stewardship into the Comprehensive Environmental Response, Compensation, and Liability Act (CERCLA) process will ultimately save the DOE money by reducing future maintainence and other costs. This research was completed money allocated during Round 6 of the Citizens’ Monitoring and Technical Assessment Fund (MTA Fund). Clark University was named conservator of these works. If you have any questions or concerns please contact us at [email protected]://commons.clarku.edu/umatilla/1000/thumbnail.jp

Hitting probabilities for non-linear systems of stochastic waves

We consider a $d$-dimensional random field $u = \{u(t,x)\}$ that solves a non-linear system of stochastic wave equations in spatial dimensions $k \in \{1,2,3\}$, driven by a spatially homogeneous Gaussian noise that is white in time. We mainly consider the case where the spatial covariance is given by a Riesz kernel with exponent $\beta$. Using Malliavin calculus, we establish upper and lower bounds on the probabilities that the random field visits a deterministic subset of \IR^d, in terms, respectively, of Hausdorff measure and Newtonian capacity of this set. The dimension that appears in the Hausdorff measure is close to optimal, and shows that when $d(2-\beta) > 2(k+1)$, points are polar for $u$. Conversely, in low dimensions $d$, points are not polar. There is however an interval in which the question of polarity of points remains open.Comment: 85 page

2D growth processes: SLE and Loewner chains

This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. It starts with an informal discussion, using numerical simulations, of various examples of 2D growth processes and their connections with statistical mechanics. SLE is then introduced and Schramm's argument mapping conformally invariant interfaces to SLE is explained. A substantial part of the review is devoted to reveal the deep connections between statistical mechanics and processes, and more specifically to the present context, between 2D critical systems and SLE. Some of the SLE remarkable properties are explained, as well as the tools for computing with SLE. This review has been written with the aim of filling the gap between the mathematical and the physical literatures on the subject.Comment: A review on Stochastic Loewner evolutions for Physics Reports, 172 pages, low quality figures, better quality figures upon request to the authors, comments welcom

Signal analysis for multiple target materials through wavelet transforms

Signal identification based on different sensing systems like microwaves, infra-red, x-rays and terahertz waves is one of the classic problems in signal processing. Earlier methods had relied mainly on the amplitude spectrum obtained by these sensing techniques mainly due to non-availability of the phase information for the signals. Most of them are based on techniques like absorbance spectrum that requires a reference material\u27s signal for the test material\u27s identification. They are also sensitive to noise and highly dependent on the peak detection algorithms. Modern equipments with both amplitude and phase information provide an opportunity for time-domain signal based methods that had not been used earlier. In this thesis, the information available through time-domain signals is utilized by the use of different wavelet transform based methods. The methods have been tested for data obtained through the terahertz time-domain spectroscopy (THz-TDS), particularly because of their ability to capture the distinguishing features of the material. The methods presented here are based on the Continuous and the Discrete Wavelet Transforms. The wavelet transforms have been used to calculate time-frequency energy density in the scale-shift domain. These energy densities have then been used to identify the features described as maxima lines and ridges that are used as features for the purpose of material identification. The methods are found to be useful in the presence of noise require no pre-filtering of the signals as required in most conventional material identification techniques. They also provide a scalable method for increasing accuracy based on the computational power available. All the simulations have been done on MATLAB --Abstract, page iii