Electronic Journal of Probability

Collision Local Times, Historical Stochastic Calculus, and Competing Species

Steven Evans and Edwin Perkins

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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.

Article information

Electron. J. Probab., Volume 3 (1998), paper no. 5, 120 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J55: Local time and additive functionals 60H99: None of the above, but in this section 60G57: Random measures

super-process super-Brownian motion interaction local time historical process measure-valued Markov branching process stochastic calculus martingale measure random measure

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Evans, Steven; Perkins, Edwin. Collision Local Times, Historical Stochastic Calculus, and Competing Species. Electron. J. Probab. 3 (1998), paper no. 5, 120 pp. doi:10.1214/EJP.v3-27. https://projecteuclid.org/euclid.ejp/1454101765

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