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A microscopic probabilistic description of a locally regulated population and macroscopic approximations
TLDR
This work generalizes a discrete model describing a locally regulated spatial population with mortality selection by adding spatial dependence and gives a path-wise description in terms of Poisson point measures.
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Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models
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Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models.
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Stochastic particle approximations for generalized Boltzmann models and convergence estimates
fied Boltzmann equation with general motion between collisions and nonlinear bounded jump (collision) operator, and give the nonlinear martingale problem it solves. We consider various linear…
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Convergence of the fluctuations for interacting diffusions with jumps associated with boltzmann equations
We consider a nonlinear partial differential equation of the Boltzmann type, in which the nonlinearity appears in a bounded jump operator.We associate with this equation a probability measure P on…
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Propagation of chaos and fluctuations for a moderate model with smooth initial data
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Competitive or weak cooperative stochastic Lotka–Volterra systems conditioned on non-extinction
TLDR
This work studies the existence and uniqueness of a quasi-stationary distribution, that is convergence to equilibrium conditioned on non-extinction, and generalizes in two-dimensions spectral tools developed for one-dimensional generalized Feller diffusion processes.
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Martingale measures and stochastic calculus
SummaryIn this paper, martingale measures, introduced by J.B. Walsh, are investigated. We prove, with techniques of stochastic calculus, that each continuous orthogonal martingale measure is the…
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A Hilbertian approach for fluctuations on the McKean-Vlasov model
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Polymorphic evolution sequence and evolutionary branching
TLDR
It is proved that under a good combination of these two scales, the population process is approximated in the long time scale of mutations by a Markov pure jump process describing the successive trait equilibria of the population.
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