Joseph Glover

Existence and stability of large scale nonlinear oscillations in suspension bridges

A nonlinear model of a suspension bridge is considered in which large-scale, stable oscillatory motions can be produced by constant loading and a small-scale, external oscillatory force. Louds implicit-function theoretic method for determining existence and stability of periodic solutions or nonlinear differential equations is extended to a case of a non-differentiable nonlinearity.

Riesz decompositions in Markov process theory

Riesz decompositions of excessive measures and excessive functions are obtained by probabilistic methods without regularity assumptions. The decomposition of excessive measures is given for Borel right processes. The results for excessive functions are formulated within the framework of weak duality. These results extend and generalize the pioneering work of Hunt in this area.

Constructing Markov processes with random times of birth and death

Kuznetsov [11] (see also [12]) introduced a Kolmogorov-type construction in which he constructs a stationary measure Qm from a transition semigroup Pt(x,dy) and an excessive measure m. In fact, his theorem has other interesting consequences outside of the Markovian framework, but we do not discuss these here. While Kuznetsov’s proof is “elementary”, it is rather involved. The purpose of this paper is to give an alternate construction of Qm in the case of right processes. We consider both the time homogeneous and time inhomogeneous cases. Our construction does not extend to cover the other interesting cases of Kuznetsov’s theorem, but our approach may yield some insight into the measures Qm and may aid the reader interested in recent articles [5,10] in which the measure Qm has played an important role. Mitro [13] has obtained a result similar to ours under duality hypotheses on the underlying processes, but her construction is quite different from ours.

Seminar on stochastic processes, 1987

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Topics in Energy and Potential Theory

Energy is a frustratingly delicate item in modern Markov process theory. It is a subject which commands attention, since it is linked so closely with maximum principles and Hunt’s hypothesis (H). It entered potential theory in the work of Cartan and Deny, where it enabled them to prove various delicate principles about symmetric potential kernels. It has flourished in the modern theory of Dirichlet spaces and has added to the body of knowledge concerning symmetric Markov processes. But while energy is a natural and cooperative partner in the study of symmetric potential kernels, it becomes increasingly intractable as one attempts to study more asymmetric kernels and processes. Concrete results in this domain are few. We present some topics in energy and potential theory for Markov processes with nonsymmetric potential kernels which complement several results and articles by various authors.

Node Activation Based on Link Distance (NA-BOLD) for Geographic Transmissions in Fading Channels

We consider a collection of battery-operated, low-power sensors randomly deployed within a geographic region for the purpose of sensing/monitoring the environment. In such a scenario, it is desirable to allow nodes to sleep (power off) whenever possible to conserve energy and increase the lifetime of the network. However, when the channel is subject to random variations, such as fading, there is an inherent trade-off between node activation and diversity. We consider a scenario in which a fixed number of nodes are allowed to turn on around a transmitter and the goal is to maximize the expected value of the distance that the message travels from the transmitter to the farthest receiver. We investigate the design of an algorithm to probabilistically determine whether a node should turn on based on its distance from the transmitter. This turns out to be a difficult probabilistic problem. We transform this problem into an analytical problem that provides a sub-optimal solution to the probabilistic problem. We compare the performance of this approach to a simpler protocol that turns on all the nodes within some radius of the transmitter. The results show that the link-distance based activation protocol provides significant gains in maximizing the expected value of the transmission distance to the farthest receiver, compared to the simpler protocol.

Solving semilinear partial differential equations with probabilistic potential theory

Techniques of probabilistic potential theory are applied to solve -Lu + f ( u ) = ,u, where ,u is a signed measure, f a (possibly discontinuous) function and L a second order elliptic or parabolic operator on RJ or, more generally, the infinitesi- mal generator of a Markov process. A1SO formulated are sufficient conditions guaranteeing existence of a solution to a countably infinite system of such equations.

Distance-based node activation for geographic transmissions in fading channels

In wireless multi-hop packet radio networks (MPRNs) that employ geographic transmissions, sleep schedules or node activation techniques may be used to power off some nodes to conserve energy. We consider the problem of selecting which nodes should power on to listen to a scheduled transmission when the channel suffers from random fading. We choose the objective of maximizing the expected value of the distance covered in a single transmission between a transmitter and the farthest receiver that successfully receives the packet, under a constraint on the expected number of receivers that turn on. Since there is a tradeoff between the distance of a node from the transmitter and the probability that the node receives the message correctly, we propose to use node-activation based on link-distance (NA-BOLD). We investigate optimal and sub-optimal NA-BOLD schemes and compare their performance with that of schemes that use a constant sleep schedule for every node within some radius of the transmitter. Our results show that the proposed NA-BOLD schemes achieve significantly larger transmission distances than conventional schemes.

IDENTIFYING MARKOV PROCESSES UP TO TIME CHANGE

The intertwining of Markov processes and potential theory has been apparent at least since Hunt’s fundamental trilogy on these subjects and was certainly evident even before then. The relationships between these two subjects have been investigated vigorously and profitably since then, and we intend to add to this study here. The central object of interest in potential theory is the cone of excessive functions, a positive cone of functions satisfying various axioms or principles of potential theory (see for example [7] and the references). The best known is the cone of superharmonic functions in R3 consisting of positive constants together with functions of the form ∫ |x-y|-1 μ(dy), where µ is a positive measure: this arises in Newtonian potential theory, and today’s axiomatic approach owes a great deal to abstraction and generalization of properties of this particular cone of functions. Each reasonable Markov process (X(t), Px) on E has an associated cone of excessive functions S(X) which can be obtained in an analytic manner from the semigroup P(t): throw a positive function f(x) into the cone if P(t)f(x) ≤ f(x) for all positive t and if P(t)f(x) increases to f(x) as t decreases to zero. Such a cone may contain only constant functions. We restrict our detailed discussion to transient processes (see (1.1) for a definition) so that the excessive functions separate points in the state space E. Later in this section, we discuss the non-transient case briefly.

Seminar on Stochastic Processes, 1988

The Riesz Transform Associated with Second Order Differential Operators.- The Optional Stochastic Integral.- On Brownian Excursions in Lipschitz Domains II: Local Asymptotic Distributions.- Gauge Theorem for Unbounded Domains.- Reminiscences of Some of Paul Levys Ideas in Brownian Motion and in Markov Chains.- Conditional Brownian Motion, Whitney Squares, and the Conditional Gauge Theorem.- Local Field Gaussian Measures.- Some Formulas for the Energy Functional of a Markov Process.- Note on the 3G Theorem (d=2).- The Independence of Hitting Times and Hitting Positions to Spheres for Drifted Brownian Motion.- The Exact Hausdorff Measure of Brownian Multiple Points II.- On a Stability Property of Harmonic Measures.- Behavior of Excessive Functions of Certain Diffusions Under the Action of the Transition Semigroup.- A Maximal Inequality.- Some Results for Functions of Kato Class in Domains of Infinite Measure.- Some Properties of Invariant Functions of Markov Processes.- Right Brownian Motion and Representation of Initial Problem.