Motivated by its relevance for the study of perturbations of one-dimensional voter models, including stochastic Potts models at low temperature, we consider diffusively rescaled coalescing random walks with branching and killing. Our main result is convergence to a new continuum process, in which the random space–time paths of the Sun–Swart Brownian net are terminated at a Poisson cloud of killing points. We also prove existence of a percolation transition as the killing rate varies. Key issues for convergence are the relations of the discrete model killing points and their intensity measure to the continuum counterparts: these convergence issues make the addition of killing considerably more difficult for the Brownian net than for the Brownian web.
@article{newman_brownian_2015,
Author = {Newman, C. M. and Ravishankar, K. and Schertzer, E.},
Title = {Brownian net with killing},
Journal = {Stochastic Processes and their Applications},
Volume = {125},
Number = {3},
Pages = {1148--1194},
Keywords = {Brownian motion, Brownian web, Voter model
perturbations},
abstract = {Motivated by its relevance for the study of
perturbations of one-dimensional voter models,
including stochastic Potts models at low temperature,
we consider diffusively rescaled coalescing random
walks with branching and killing. Our main result is
convergence to a new continuum process, in which the
random space–time paths of the Sun–Swart Brownian
net are terminated at a Poisson cloud of killing
points. We also prove existence of a percolation
transition as the killing rate varies. Key issues for
convergence are the relations of the discrete model
killing points and their intensity measure to the
continuum counterparts: these convergence issues make
the addition of killing considerably more difficult for
the Brownian net than for the Brownian web.},
doi = {10.1016/j.spa.2014.09.018},
issn = {0304-4149},
month = mar,
url = {http://www.sciencedirect.com/science/article/pii/S0304414914002324},
urldate = {2015-08-27},
year = 2015
}
