Generalized self-intersection local time for a superprocess over a stochastic flow
type: Item
instance of: scholarly article
title: Generalized self-intersection local time for a superprocess over a stochastic flow
zbMATH Open document ID: 1278.60126
DOI: 10.1214/11-AOP653
author: Aaron Heuser
published in: The Annals of Probability
publication date: 2012-08-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1102.2849
full work available at URL: https://projecteuclid.org/euclid.aop/1341401142
Mathematics Subject Classification ID: 60J68
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 6067449
zbMATH Keywords: stochastic flow
zbMATH Keywords: rapidly decreasing function
zbMATH Keywords: martingale problem
zbMATH Keywords: branching
zbMATH Keywords: resolvent
zbMATH Keywords: tempered distribution
zbMATH Keywords: Brownian motion
Wikidata QID: Q36765391
reviewed by: Ion Cuculescu
MaRDI profile type: Publication
arXiv ID: 1102.2849
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OpenAlex ID: W3098941692
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review text: Let \(b:{\mathbb R}^{d } \rightarrow {\mathbb R}^{d }\), \(c :{\mathbb R}^{d } \rightarrow \mathbb{R}^{d \times m}\) (the real \(d\times m\)-matrices) be maps satisfying a Lipschitz condition bounded together with their first and second derivatives, NEWLINE\[NEWLINE\sigma _{i j } ( x , y ) = \sum _{k = 1 } ^{m } c _{i k } ( x ) c _{j k } ( y ),NEWLINE\]NEWLINE NEWLINE\[NEWLINEa _{i j } ( x , y ) = \delta _{i j } b _{i } ( x ) b _{j }( y ) + \sigma _{i j } ( x , y ),NEWLINE\]NEWLINE NEWLINE\[NEWLINE(L \varphi ) ( x ) = ( 1 / 2 ) \sum _{i , j = 1 }^{d } a _{i j }( x , x ) ( \partial _{i j } ^{2 } \varphi ) ( x ),NEWLINE\]NEWLINE supposed uniformly elliptic, NEWLINE\[NEWLINE( \Lambda \varphi ) ( x , y ) = \sum _{i , j = 1 } ^{d }\sigma _{i j } ( x , y )( \partial _{i } \varphi ) ( x ) ( \partial _{j } \varphi ) ( x ),NEWLINE\]NEWLINE and let \(\mu _{t } \in M ({\mathbb R}^{d } )\) (finite measures for \(t \in [ 0, \infty )\); a superprocess over a stochastic flow defined by \(d Y _{t } = c ( Y _{t } ) d W _{t }\), \(W\) a Brownian motion) be the unique continuous solution of a martingale problem: for \(\varphi \in C _{K } ^{2 } ({\mathbb R}^{d })\) (K stands for compactly supported), NEWLINE\[NEWLINE\mu _{t}(\varphi)-\mu _{0 } ( \varphi )-\int _{0} ^{t } \mu _{s } ( L \varphi ) d sNEWLINE\]NEWLINE is a continuous martingale, having quadratic variation \(\int _{0} ^{t } ( \mu _{s } ( \varphi ^{2 } ) + \mu _{s } ^{2 } ( \Lambda \varphi ) ) d s\) (\(\mu _{s } ^{2 }\) means \(\mu _{s } \otimes \mu _{s }\), \(\mu _{s } ( f ) = \int f d \mu _{s }\)). This \(\mu\) was constructed by \textit{G. Skoulakis} and \textit{R. J. Adler} [Ann. Appl. Probab. 11, No. 2, 488--543 (2001; Zbl 1018.60052)] as the limit of \(\mu ^{( n ) }\), a fact essential in the proofs in this paper. NEWLINENEWLINEConsider \(\alpha = ( \alpha _{0 } ,\dotsc , \alpha _{| \alpha | } )\), \(\alpha _{i } \in [ 0 , 1 ]\), and let \(\alpha -1 = ( \alpha _{0 } ,\dotsc , \alpha _{| \alpha | - 1 } )\), \(\alpha | _…
description: scientific article; zbMATH DE number 6067449