Thomas S. Salisbury

Construction of right processes from excursions

SummaryIn a previous work, the author obtained the strong Markov property of a process from conditions on its excursion process. The Ray and right properties are obtained here under similar conditions, using the Ray-Knight compactification.

On the conditioned exit measures of super Brownian motion

Abstract. In this paper we present a martingale related to the exit measures of super Brownian motion. By changing measure with this martingale in the canonical way we have a new process associated with the conditioned exit measure. This measure is shown to be identical to a measure generated by a non-homogeneous branching particle system with immigration of mass. An application is given to the problem of conditioning the exit measure to hit a number of specified points on the boundary of a domain. The results are similar in flavor to the “immortal particle” picture of conditioned super Brownian motion but more general, as the change of measure is given by a martingale which need not arise from a single harmonic function.

On the Ito excursion process

SummaryNecessary and sufficient conditions are given, for a process to be the excursion process of some strong Markov process. These are modifications of necessary conditions of Itô, which are here shown by example not to be sufficient.

Random Walks in Degenerate Random Environments

We study the asymptotic behaviour of random walks in i.i.d. random environments on

Martin boundaries of sectorial domains

\Z^d

A combinatorial result with applications to self-interacting random walks

. The environments need not be elliptic, so some steps may not be available to the random walker. We prove a monotonicity result for the velocity (when it exists) for any 2-valued environment, and show that this does not hold for 3-valued environments without additional assumptions. We give a proof of directional transience and the existence of positive speeds under strong, but non-trivial conditions on the distribution of the environment. Our results include generalisations (to the non-elliptic setting) of 0-1 laws for directional transience, and in 2-dimensions the existence of a deterministic limiting velocity.

ENERGY, AND INTERSECTIONS OF MARKOV CHAINS*

LetD be a domain inR2 whose complement is contained in a pair of rays leaving the origin. That is,D contains two sectors whose base angles sum to 2π. We use balayage to give an integral test that determines if the origin splits into exactly two minimal Martin boundary points, one approached through each sector. This test is related to other integral tests due to Benedicks and Chevallier, the former in the special case of a Denjoy domain. We then generalise our test, replacing the pair of rays by an arbitrary number.

Degenerate random environments

We give a series of combinatorial results that can be obtained from any two collections (both indexed by ZxN) of left and right pointing arrows that satisfy some natural relationship. When applied to certain self-interacting random walk couplings, these allow us to reprove some known transience and recurrence results for some simple models. We also obtain new results for one-dimensional multi-excited random walks and for random walks in random environments in all dimensions.

Conditioning super-Brownian motion on its boundary statistics, and fragmentation

In an earlier paper, the author and Fitzsimmons resolved several problems about the intersections of Brownian motions and Levy processes, using projection techniques to construct certain measures of finite energy. The present paper aims to make these techniques more accessible, by treating an analogous but simpler problem. Upper bounds will be obtained on intersection probabilities for discrete time Markov chains. This will be applied to obtain a Wiener type test for intersections of random walks within given sets.

A Martin boundary in the plane

We consider connectivity properties of certain i.i.d. random environments on i¾?d, where at each location some steps may not be available. Site percolation and oriented percolation are examples of such environments. In these models, one of the quantities most often studied is the random set of vertices that can be reached from the origin by following a connected path. More generally, for the models we consider, multiple different types of connectivity are of interest, including: the set of vertices that can be reached from the origin; the set of vertices from which the origin can be reached; the intersection of the two. As with percolation models, many of the models we consider admit, or are expected to admit phase transitions. Among the main results of the paper is a proof of the existence of phase transitions for some two-dimensional models that are non-monotone in their underlying parameter, and an improved bound on the critical value for oriented site percolation on the triangular lattice. The connectivity of the random directed graphs provides a foundation for understanding the asymptotic properties of random walks in these random environments, which we study in a second paper.