Collision Local Times, Historical Stochastic Calculus, and Competing Species
Steven N. Evans, University of California at Berkeley
Edwin A. Perkins, The University of British Columbia
Abstract
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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