On minimal parabolic functions and time-homogeneous parabolic h-transforms

Does a minimal harmonic function $h$ remain minimal when it is viewed as a parabolic function? The question is answered for a class of long thin semi-infinite tubes $D\subset \R^d$ of variable width and minimal harmonic functions $h$ corresponding to the boundary point of $D$ ``at infinity.'' Suppose $f(u)$ is the width of the tube $u$ units away from its endpoint and $f$ is a Lipschitz function. The answer to the question is affirmative if and only if $\int^\infty f^3(u)du = \infty$. If the test fails, there exist parabolic $h$-transforms of space-time Brownian motion in $D$ with infinite lifetime which are not time-homogenous

Non-degenerate conditionings of the exit measures of super-Brownian motion

We introduce several martingale changes of measure of the law of the exit measure of super Brownian motion. These changes of measure include and generalize one arising by conditioning the exit measures to charge a point on the boun dary of a 2-dimensional domain. In the case we discuss this is a non-degenerate conditioning. We give characterizations of the new processes in terms of "immortal particle" branching processes with immigration of mass, and give application s to the study of solutions to Lu = cu^2 in D. The representations are related to those in an earlier paper, which treated the case of degenerate conditionings

A combinatorial result with applications to self-interacting random walks

We give a series of combinatorial results that can be obtained from any two collections (both indexed by $\Z\times \N$) 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