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1998 Collision Local Times, Historical Stochastic Calculus, and Competing Species
Steven Evans, Edwin Perkins
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Electron. J. Probab. 3: 1-120 (1998). DOI: 10.1214/EJP.v3-27


Branching measure-valued diffusion models are investigated that can be regarded as pairs of historical Brownian motions modified by a competitive interaction mechanism under which individuals from each population have their longevity or fertility adversely affected by collisions with individuals from the other population. For 3 or fewer spatial dimensions, such processes are constructed using a new fixed-point technique as the unique solution of a strong equation driven by another pair of more explicitly constructible measure-valued diffusions. This existence and uniqueness is used to establish well-posedness of the related martingale problem and hence the strong Markov property for solutions. Previous work of the authors has shown that in 4 or more dimensions models with the analogous definition do not exist.


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Steven Evans. Edwin Perkins. "Collision Local Times, Historical Stochastic Calculus, and Competing Species." Electron. J. Probab. 3 1 - 120, 1998.


Accepted: 8 April 1998; Published: 1998
First available in Project Euclid: 29 January 2016

zbMATH: 0899.60081
MathSciNet: MR1615329
Digital Object Identifier: 10.1214/EJP.v3-27

Primary: 60K35
Secondary: 60G57 , 60H99 , 60J55

Keywords: Historical process , interaction , Local time , martingale measure , measure-valued Markov branching process , random measure , stochastic calculus , Super-Brownian motion , super-process

Vol.3 • 1998
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