Boundary Harnack Principle for Subordinate Brownian Motions

We establish a boundary Harnack principle for a large class of subordinate Brownian motion, including mixtures of symmetric stable processes, in bounded $\kappa$-fat open set (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the minimal Martin boundary of bounded $\kappa$-fat open sets with respect to these processes with their Euclidean boundary.Comment: 34 page

Rogers functions and fluctuation theory

Extending earlier work by Rogers, Wiener-Hopf factorisation is studied for a class of functions closely related to Nevanlinna-Pick functions and complete Bernstein functions. The name 'Rogers functions' is proposed for this class. Under mild additional condition, for a Rogers function f, the Wiener--Hopf factors of f(z)+q, as well as their ratios, are proved to be complete Bernstein functions in both z and q. This result has a natural interpretation in fluctuation theory of L\'evy processes: for a L\'evy process X_t with completely monotone jumps, under mild additional condition, the Laplace exponents kappa(q;z), kappa*(q;z) of ladder processes are complete Bernstein functions of both z and q. Integral representation for these Wiener--Hopf factors is studied, and a semi-explicit expression for the space-only Laplace transform of the supremum and the infimum of X_t follows.Comment: 70 pages, 2 figure

Dual quadratic differentials and entire minimal graphs in Heisenberg space

We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces $\mathbb{L}(\kappa,\tau)$ with isometry group of dimension 4, which are dual to the Abresch-Rosenberg differentials in the Riemannian counterparts $\mathbb{E}(\kappa,\tau)$, and obtain some consequences. On the one hand, we give a very short proof of the Bernstein problem in Heisenberg space, and provide a geometric description of the family of entire graphs sharing the same differential in terms of a 2-parameter conformal deformation. On the other hand, we prove that entire minimal graphs in Heisenberg space have negative Gauss curvature.Comment: 19 page

Evolution of abilities to regenerate neurons in central nervous systems

This work was supported by a Kappa Kappa Gamma Graduate Fellowship Award to C. E. H. and NIH grant NS-11861 and RCDA NS-00070 to G. D. B.Neuroscienc