Boundary-crossing identities for diffusions having the time-inversion property

We review and study a one-parameter family of functional transformations, denoted by (S (β)) β∈ℝ, which, in the case β<0, provides a path realization of bridges associated to the family of diffusion processes enjoying the time-inversion property. This family includes Brownian motions, Bessel processes with a positive dimension and their conservative h-transforms. By means of these transformations, we derive an explicit and simple expression which relates the law of the boundary-crossing times for these diffusions over a given function f to those over the image of f by the mapping S (β), for some fixed β∈ℝ. We give some new examples of boundary-crossing problems for the Brownian motion and the family of Bessel processes. We also provide, in the Brownian case, an interpretation of the results obtained by the standard method of images and establish connections between the exact asymptotics for large time of the densities corresponding to various curves of each family

Nontangential limits and Fatou-type theorems on post-critically finite self-similar sets

In this paper we study the boundary limit properties of harmonic functions on $\mathbb R_+\times K$, the solutions $u(t,x)$ to the Poisson equation $$\frac{\partial^2 u}{\partial t^2} + \Delta u = 0,$$ where $K$ is a p.c.f. set and $\Delta$ its Laplacian given by a regular harmonic structure. In particular, we prove the existence of nontangential limits of the corresponding Poisson integrals, and the analogous results of the classical Fatou theorems for bounded and nontangentially bounded harmonic functions.Comment: 22 page

On h h -transforms of one-dimensional diffusions stopped upon hitting zero

For a one-dimensional diffusion on an interval for which 0 is the regular-reflecting left boundary, three kinds of conditionings to avoid zero are studied. The limit processes are $h$-transforms of the process stopped upon hitting zero, where $h$'s are the ground state, the scale function, and the renormalized zero-resolvent. Several properties of the $h$-transforms are investigated

An intermediate-depth source of hydrothermal 3He and dissolved iron in the North Pacific

We observed large water column anomalies in helium isotopes and trace metal concentrations above the Loihi Seamount. The 3He/4He of the added helium was 27.3 times the atmospheric ratio, clearly marking its origin to a primitive mantle plume. The dissolved iron to 3He ratio (dFe:3He) exported to surrounding waters was 9.3 ± 0.3 × 106. We observed the Loihi 3He and dFe “signal” at a depth of 1100 m at several stations within ∼100 – 1000 km of Loihi, which exhibited a distal dFe:3He ratio of ∼4 × 106, about half the proximal ratio. These ratios were remarkably similar to those observed over and near the Southern East Pacific Rise (SEPR) despite greatly contrasting geochemical and volcanictectonic origins. In contrast, the proximal and distal dMn:3He ratios were both ∼ 1 × 106, less than half of that observed at the SEPR. Dissolved methane was minimally enriched in waters above Loihi Seamount and was distally absent. Using an idealized regional-scale model we replicated the historically observed regional 3He distribution, requiring a hydrothermal 3He source from Loihi of 10.4 ± 4.2 mola−1, ∼2% of the global abyssal hydrothermal 3He flux. From this we compute a corresponding dFe flux of ∼40 Mmola−1. Global circulation model simulations suggest that the Loihi-influenced waters eventually upwell along the west coast of North America, also extending into the shallow northwest Pacific, making it a possibly important determinant of marine primary production in the subpolar North Pacific

The backbone decomposition for spatially dependent supercritical superprocesses

Consider any supercritical Galton-Watson process which may become extinct with positive probability. It is a well-understood and intuitively obvious phenomenon that, on the survival set, the process may be pathwise decomposed into a stochastically `thinner' Galton-Watson process, which almost surely survives and which is decorated with immigrants, at every time step, initiating independent copies of the original Galton-Watson process conditioned to become extinct. The thinner process is known as the backbone and characterizes the genealogical lines of descent of prolific individuals in the original process. Here, prolific means individuals who have at least one descendant in every subsequent generation to their own. Starting with Evans and O'Connell, there exists a cluster of literature describing the analogue of this decomposition (the so-called backbone decomposition) for a variety of different classes of superprocesses and continuous-state branching processes. Note that the latter family of stochastic processes may be seen as the total mass process of superprocesses with non-spatially dependent branching mechanism. In this article we consolidate the aforementioned collection of results concerning backbone decompositions and describe a result for a general class of supercritical superprocesses with spatially dependent branching mechanisms. Our approach exposes the commonality and robustness of many of the existing arguments in the literature

Superprocesses as models for information dissemination in the Future Internet

Future Internet will be composed by a tremendous number of potentially interconnected people and devices, offering a variety of services, applications and communication opportunities. In particular, short-range wireless communications, which are available on almost all portable devices, will enable the formation of the largest cloud of interconnected, smart computing devices mankind has ever dreamed about: the Proximate Internet. In this paper, we consider superprocesses, more specifically super Brownian motion, as a suitable mathematical model to analyse a basic problem of information dissemination arising in the context of Proximate Internet. The proposed model provides a promising analytical framework to both study theoretical properties related to the information dissemination process and to devise efficient and reliable simulation schemes for very large systems

The <i>Castalia</i> mission to Main Belt Comet 133P/Elst-Pizarro

We describe Castalia, a proposed mission to rendezvous with a Main Belt Comet (MBC), 133P/Elst-Pizarro. MBCs are a recently discovered population of apparently icy bodies within the main asteroid belt between Mars and Jupiter, which may represent the remnants of the population which supplied the early Earth with water. Castalia will perform the first exploration of this population by characterising 133P in detail, solving the puzzle of the MBC’s activity, and making the first in situ measurements of water in the asteroid belt. In many ways a successor to ESA’s highly successful Rosetta mission, Castalia will allow direct comparison between very different classes of comet, including measuring critical isotope ratios, plasma and dust properties. It will also feature the first radar system to visit a minor body, mapping the ice in the interior. Castalia was proposed, in slightly different versions, to the ESA M4 and M5 calls within the Cosmic Vision programme. We describe the science motivation for the mission, the measurements required to achieve the scientific goals, and the proposed instrument payload and spacecraft to achieve these

Patterns in random walks and Brownian motion

We ask if it is possible to find some particular continuous paths of unit length in linear Brownian motion. Beginning with a discrete version of the problem, we derive the asymptotics of the expected waiting time for several interesting patterns. These suggest corresponding results on the existence/non-existence of continuous paths embedded in Brownian motion. With further effort we are able to prove some of these existence and non-existence results by various stochastic analysis arguments. A list of open problems is presented.Comment: 31 pages, 4 figures. This paper is published at http://link.springer.com/chapter/10.1007/978-3-319-18585-9_