Superprocesses of stochastic flows
type: Item
instance of: scholarly article
title: Superprocesses of stochastic flows
zbMATH Open document ID: 1015.60063
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: Measure-valued processes of a new type, arising as extensions of empirical processes of consistent \(k\)-point motions, are studied. More precisely, let \(b_{p}:\mathbb R^{d} \to\mathbb R^{d}\), \(1\leq p\leq d\), be bounded Borel functions, and let \(a\) be a bounded Borel mapping from \(\mathbb R^{d}\times\mathbb R^{d}\) to the set of all nonnegative definite \(d\times d\) symmetric matrices, satisfying \(a_{pq}(z_{i},z_{j}) = a_{pq}(z_{j}, z_{i})\) for all \(z_{i},z_{j}\in\mathbb R^{d}\), \(1\leq p,q\leq d\). For every \(k\geq 1\) define a differential operator \(A_{k}\) by \[ A_{k}f(z_1,\ldots,z_{k})= \frac 12 \sum^{k}_{i,j=1} \sum^{d}_{p,q=1} a_{pq}(z_{i},z_{j}){\partial^2 f\over \partial z_{ip}\partial z_{jq}}(z_1,\ldots,z_{k})+ \sum^{k}_{i=1}\sum^{d}_{p=1} b_{p}(z_{i}){\partial f \over \partial z_{ip}}(z_1,\ldots,z_{k}), \] where \(f\in C^2_{b}((\mathbb R^{d})^{k})\), \(z_{i} = (z_{i1}, \ldots,z_{id})\in\mathbb R^{d}\), \(1\leq i\leq k\). Suppose that the martingale problem for \((A_{k},C^2_{b}((\mathbb R^{d})^{k})\) is well posed and generates a unique continuous strong Markov Feller process on \((\mathbb R^{d})^{k}\). Let us denote this process, starting from \((y_1,\ldots,y_{k})\in (\mathbb R^{d})^{k}\) at time 0, by \(Y^{k}_{t} = (Y_{t}(y_1),\ldots,Y_{t}(y_{k}))\), and set \[ X^{k}_{t} = \frac 1{k}\sum^{k}_{i=1} \delta_{Y_{t} (y_{i})}. \tag{1} \] Let \(M(\mathbb R^{d})\) be the space of all finite Borel measures on \(\mathbb R^{d}\) equipped with the weak topology. It is proved that a unique continuous strong Markov Feller process \(X\) on \(M(\mathbb R^{d})\) with the following property exists: for all \(r>0\) and \(k\geq 1\), if \(X(0) = rk^{-1}\sum^{k}_{j=1}\delta _{y_{i}}\), then the law of \(X\) equals to the law of \(rX^{k}\), \(X^{k}\) being given by (1). Moreover, the generator of the process \(X\) is found and relations to stochastic coalescence and isotropic flows are discussed.
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 1905981
zbMATH Keywords: measure-valued processes
zbMATH Keywords: stochastic flows
zbMATH Keywords: coalescence
author: Zhi-Ming Ma
author: Kai-Nan Xiang
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1008956332
OpenAlex ID: W2067779620
cites work: Stochastic coalescence
cites work: Q3338999
cites work: Q3721530
cites work: Q4010320
cites work: Isotropic stochastic flows
cites work: Asymptotic expansion of stochastic flows
cites work: Constructing nonhomeomorphic stochastic flows
cites work: Rate of growth of the coalescent set in a coalescing stochastic flow
cites work: Q4010322
cites work: Q3792005
cites work: Q3142396
cites work: Three classes of infinite-dimensional diffusions
cites work: An Introduction to Branching Measure-Valued Processes
cites work: Q3660553
cites work: Q4010321
cites work: Homotopy and homology vanishing theorems and the stability of stochastic flows
cites work: Q5711199
cites work: Stochastic flows for nonlinear second-order parabolic SPDE
cites work: Coalescing and noncoalescing stochastic flows in \(R_ 1\)
cites work: Q3959169
cites work: Q3774629
cites work: Q3995203
cites work: Branching processes, random trees and superprocesses
cites work: On isotropic brownian motions
cites work: Q3828844
cites work: Q4010323
cites work: Stochastic flows on the boundaries of \(\text{SL}(n,\mathbb{R})\)
cites work: The Brownian Motion and the Canonical Stochastic Flow on a Symmetric Space
cites work: Coalescing stochastic flows on the real line
cites work: Q4866051
cites work: Q4040434
cites work: Q3862204
cites work: Correlation theory of stationary and related random functions. Volume II: Supplementary notes and references
cites work: A quasi-sure flow property and the equivalence of capacities for differential equations on the Wiener Space
DOI: 10.1214/AOP/1008956332
description: scientific article; zbMATH DE number 1905981