39 research outputs found
Marked Gibbs point processes with unbounded interaction: an existence result
We construct marked Gibbs point processes in under quite
general assumptions. Firstly, we allow for interaction functionals that may be
unbounded and whose range is not assumed to be uniformly bounded. Indeed, our
typical interaction admits an a.s. finite but random range. Secondly, the
random marks -- attached to the locations in -- belong to a
general normed space . They are not bounded, but their law should
admit a super-exponential moment. The approach used here relies on the
so-called entropy method and large-deviation tools in order to prove tightness
of a family of finite-volume Gibbs point processes. An application to
infinite-dimensional interacting diffusions is also presented
Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions
A multitype Dawson-Watanabe process is conditioned, in subcritical and
critical cases, on non-extinction in the remote future. On every finite time
interval, its distribution is absolutely continuous with respect to the law of
the unconditioned process. A martingale problem characterization is also given.
Several results on the long time behavior of the conditioned mass process|the
conditioned multitype Feller branching diffusion are then proved. The general
case is first considered, where the mutation matrix which models the
interaction between the types, is irreducible. Several two-type models with
decomposable mutation matrices are analyzed too
A host-parasite multilevel interacting process and continuous approximations
We are interested in modeling some two-level population dynamics, resulting
from the interplay of ecological interactions and phenotypic variation of
individuals (or hosts) and the evolution of cells (or parasites) of two types
living in these individuals. The ecological parameters of the individual
dynamics depend on the number of cells of each type contained by the individual
and the cell dynamics depends on the trait of the invaded individual. Our
models are rooted in the microscopic description of a random (discrete)
population of individuals characterized by one or several adaptive traits and
cells characterized by their type. The population is modeled as a stochastic
point process whose generator captures the probabilistic dynamics over
continuous time of birth, mutation and death for individuals and birth and
death for cells. The interaction between individuals (resp. between cells) is
described by a competition between individual traits (resp. between cell
types). We look for tractable large population approximations. By combining
various scalings on population size, birth and death rates and mutation step,
the single microscopic model is shown to lead to contrasting nonlinear
macroscopic limits of different nature: deterministic approximations, in the
form of ordinary, integro- or partial differential equations, or probabilistic
ones, like stochastic partial differential equations or superprocesses. The
study of the long time behavior of these processes seems very hard and we only
develop some simple cases enlightening the difficulties involved
Bridges of Markov counting processes. Reciprocal classes and duality formulas
Processes having the same bridges are said to belong to the same reciprocal
class. In this article we analyze reciprocal classes of Markov counting
processes by identifying their reciprocal invariants and we characterize them
as the set of counting processes satisfying some duality formula