Book contents
- Frontmatter
- Contents
- Preface
- List of participants
- An evolution equation for the intersection local times of superprocesses
- The Continuum random tree II: an overview
- Harmonic morphisms and the resurrection of Markov processes
- Statistics of local time and excursions for the Ornstein–Uhlenbeck process
- LP-Chen forms on loop spaces
- Convex geometry and nonconfluent Γ-martingales I: tightness and strict convexity
- Some caricatures of multiple contact diffusion-limited aggregation and the η-model
- Limits on random measures and stochastic difference equations related to mixing array of random variables
- Characterizing the weak convergence of stochastic integrals
- Stochastic differential equations involving positive noise
- Feeling the shape of a manifold with Brownian motion — the last word in 1990
- Decomposition of Dirichlet processes on Hilbert space
- A supersymmetric Feynman-Kac formula
- On long excursions of Brownian motion among Poissonian obstacles
An evolution equation for the intersection local times of superprocesses
Published online by Cambridge University Press: 31 March 2010
- Frontmatter
- Contents
- Preface
- List of participants
- An evolution equation for the intersection local times of superprocesses
- The Continuum random tree II: an overview
- Harmonic morphisms and the resurrection of Markov processes
- Statistics of local time and excursions for the Ornstein–Uhlenbeck process
- LP-Chen forms on loop spaces
- Convex geometry and nonconfluent Γ-martingales I: tightness and strict convexity
- Some caricatures of multiple contact diffusion-limited aggregation and the η-model
- Limits on random measures and stochastic difference equations related to mixing array of random variables
- Characterizing the weak convergence of stochastic integrals
- Stochastic differential equations involving positive noise
- Feeling the shape of a manifold with Brownian motion — the last word in 1990
- Decomposition of Dirichlet processes on Hilbert space
- A supersymmetric Feynman-Kac formula
- On long excursions of Brownian motion among Poissonian obstacles
Summary
Introduction
The primary aim of this paper is to establish evolution equations for the intersection local time (ILT) of the super Brownian motion and certain super stable processes. We shall proceed by carefully defining the requisite concepts and giving all of our main results in the Introduction, while leaving the proofs for later sections. The Introduction itself is divided into four sections, which treat, in turn, the definition of the superprocesses that will interest us, the definition of ILT and some previous results, our main result – a Tanaka-like evolution equation for ILT – and an Itô formula for measure-valued processes along with a description of how to use it to derive the evolution equation. Some technical lemmas make up Section 2 of the paper, while Section 3 is devoted to proofs.
In order to conserve space, we shall motivate neither the study of superprocesses per se – other than to note that they arise as infinite density limits of infinitely rapidly branching stochastic processes – nor the study of ILT – other than to note that this seems to be important for the introduction of an intrinsic dependence structure for the spatial part of a superprocess. Good motivational and background material on superprocesses can be found in Dawson (1978, 1986), Dawson, Iscoe and Perkins (1989), Ethier and Kurtz (1986), Roelly-Coppoletta (1986), Walsh (1986) and Watanabe (1968), as well as other papers in this volume. Material on ILT can be found in Adler, Feldman and Lewin (1991), Adler and Lewin (1991), Adler and Rosen (1991), Dynkin (1988) and Perkins (1988).
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- Stochastic AnalysisProceedings of the Durham Symposium on Stochastic Analysis, 1990, pp. 1 - 22Publisher: Cambridge University PressPrint publication year: 1991
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