Open AccessPosted Content
On the conditioned exit measures of super-Brownian motion
Thomas S. Salisbury,John Verzani +1 more
TLDR
In this article, the authors present a martingale related to the exit measures of super-Brownian motion, which is shown to be identical to a measure generated by a non-homogeneous branching particle system with immigration of mass.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.read more
Citations
More filters
Journal Article
Random Trees, Levy Processes and Spatial Branching Processes
TL;DR: In this article, the genealogical structure of general critical or subcritical continuous-state branching processes is investigated, and it is shown that whenever a sequence of rescaled Galton-Watson processes converges in distribution, their genealogies also converge to the continuous branching structure coded by the appropriate height process.
Book
Diffusions, Superdiffusions and Partial Differential Equations
TL;DR: In this article, the authors discuss the relationship between Markov processes and elliptic differential equations, including linear elliptic equations and diffusions, as well as branching exit Markov systems and superprocesses.
Journal ArticleDOI
Local extinction versus local exponential growth for spatial branching processes
TL;DR: In this article, it was shown that either λc≤0 and X exhibits local extinction or λ c>0 and there is an exponential growth of mass on compacts of D with rate λC. This completes the local extinction criterion obtained by Pinsky [Ann. Probab. 24 (1996) 237--267] and a recent result on the local growth of Mass under a spectral assumption given by Englander and Turaev [ANN. 30 (2002) 683--722].
Journal ArticleDOI
A decomposition of the (1 + β)-superprocess conditioned on survival
TL;DR: A study of the (infinite-variance) (1 + β)-superprocess, conditioned on survival until some fixed time T, which sees a Poisson number of immortal trees and the rate of immigration along the branches is no longer deterministic.
Journal ArticleDOI
Basic properties of critical lognormal multiplicative chaos
TL;DR: In this article, the authors studied one-dimensional exact scaling lognormal multiplicative chaos measures at criticality and determined the exact asymptotics of the right tail of the distribution of the total mass of the measure.
References
More filters
Journal ArticleDOI
A probabilistic approach to one class of nonlinear differential equations
TL;DR: In this article, the authors established connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes, and improved results on superprocesses by using the recent progress in the theory of removable singularities for differential equations.
Journal ArticleDOI
Conditional gauge and potential theory for the Schrödinger operator
TL;DR: In this paper, theoreme de jauge conditionnelle a des operateurs plus generaux et a des domaines moins reguliers is defined, i.e.
Journal ArticleDOI
Two representations of a conditioned superprocess
TL;DR: In this paper, the authors consider a class of measure-valued Markov processes constructed by taking a superprocess over some underlying Markov process and conditioning it to stay alive forever.
Book ChapterDOI
A Path-Valued Markov Process and its Connections with Partial Differential Equations
TL;DR: In this article, a path-valued Markov process (ξ t ) is related to the nonlinear equation Δu = u 2 in the same way as Brownian motion is to the Laplace equation, and a key tool is the exit measure, which describes the way all these paths exit a given domain.
Journal ArticleDOI
Path processes and historical superprocesses
TL;DR: In this paper, the authors developed a theory of superprocesses over path processes whose core is the integration with respect to measure functionals, and applied this theory to historical super-processes.